A dynamic inventory/maintenance model

A dynamic inventory/maintenance model

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A dynamic inventory/maintenance model
Bates, Jonathan J
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[Tampa, Fla.]
University of South Florida
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Subjects / Keywords:
Predicted demand forecasting
Reliability eest
Weibull estimation
Stocking policy
Dissertations, Academic -- Industrial Engineering -- Doctoral -- USF ( lcsh )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


ABSTRACT: A model is proposed to provide inventory and maintenance guidance for a system of operating parts. This model is capable of handling a system with multiple operating components, unknown part lifetime failure distribution, and separately maintained parts. In this model, part reliability characteristics are used along with system costs to predict the required stocking levels and part replacement times. Two maintenance strategies are presented that have the unique characteristic of allowing flexible scheduling of replacements. A case study is completed comparing developed stocking policies to an existing policy. An estimation selection method is introduced and fit into the model for computing Weibull distribution parameters when part reliability is not well known. An algorithm is displayed that describes the implementation of the system model and data from practical case scenarios are conducted using this algorithm.
Dissertation (Ph.D.)--University of South Florida, 2007.
Includes bibliographical references.
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by Jonathan J. Bates.

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A dynamic inventory/maintenance model
h [electronic resource] /
by Jonathan J. Bates.
[Tampa, Fla.] :
b University of South Florida,
3 520
ABSTRACT: A model is proposed to provide inventory and maintenance guidance for a system of operating parts. This model is capable of handling a system with multiple operating components, unknown part lifetime failure distribution, and separately maintained parts. In this model, part reliability characteristics are used along with system costs to predict the required stocking levels and part replacement times. Two maintenance strategies are presented that have the unique characteristic of allowing flexible scheduling of replacements. A case study is completed comparing developed stocking policies to an existing policy. An estimation selection method is introduced and fit into the model for computing Weibull distribution parameters when part reliability is not well known. An algorithm is displayed that describes the implementation of the system model and data from practical case scenarios are conducted using this algorithm.
Dissertation (Ph.D.)--University of South Florida, 2007.
Includes bibliographical references.
Text (Electronic dissertation) in PDF format.
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
Title from PDF of title page.
Document formatted into pages; contains 125 pages.
Includes vita.
Adviser: Jose L. Zayas-Castro, Ph.D.
Predicted demand forecasting.
Reliability eest.
Weibull estimation.
Stocking policy.
0 690
Dissertations, Academic
x Industrial Engineering
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2214


A Dynamic Inventory/Maintenance Model by Jonathan J. Bates A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Management Systems Eng ineering College of Engineering University of South Florida Major Professor: Jose L. Zayas-Castro, Ph.D. Tapas K. Das, Ph.D. Thom Hodgson, Ph.D. Louis Martin-Vega, Ph.D. Alex Savachkin, Ph.D. Carlos Smith, Ph.D. Date of Approval: October 24, 2007 Keywords: Predicted Demand Forecasting, Censoring, Reliability Test, Weibull Estimation, Stocking Policy Copyright 2007, Jonathan J. Bates


DEDICATION To my beautiful wife Melissa and our daughter Made lynn who brings color to my life. Praise be to God for making all things possible.


ACKNOWLEDGEMENTS I would like to thank Dr. Zayas-Castro for his dire ction and support throughout the course of this research. I also appreciate and support his leadership and vision for the IMSE Department at University of South Florida. I would also like to thank my committee members for their guidance. Outside of the academic environment, I thank the U. S. Coast Guard for the opportunity to pursue this effort and the continued support throughout the research process. My family, friends, and supervisors were also incredibly supportive while encouraging and prodding me to keep going.


i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv ABSTRACT v CHAPTER 1 INTRODUCTION 1 1.1 Motivation 1 1.2 System Description 3 1.3 Research Objectives and Contributions 4 CHAPTER 2 INVENTORY THEORY 6 2.1 ( s, S ) Inventory Policy 6 2.2 Optimal Stationary Solutions 9 2.3 Approximate Stationary Solutions 13 2.4 Non-stationary Solution 15 CHAPTER 3 MAINTENANCE POLICIES 18 3.1 Maintenance Policy 18 3.2 Joint Inventory/Maintenance Policies 19 3.3 Optimal Replacement Policy 23 3.4 System Maintenance Policy 25 CHAPTER 4 REPLACEMENT AND DEMAND DISTRIBUTION 27 4.1 Calculating the Replacement Probabilities 27 4.2 Constructing the Demand Distribution: GrayÂ’s Code 29 4.3 Replacement Probability and Demand Distributi on Example 34 4.4 Stationarity Analysis 36 CHAPTER 5 PARAMETRIC ESTIMATION 37 5.1 Censoring 37 5.2 Methods of Estimation 38 5.2.1 Maximum Likelihood Estimation (MLE) 39 5.2.2 Kaplan-Meier Estimator (KME) 42 5.2.3 Piecewise Exponential Estimator (PEXE) 43 5.2.4 Fldes, Rejt, and Winter Estimator (FRWE) 44 5.2.5 Klein, Lee, and Moeschberger Partially Para metric Estimator (KLM) 44


ii 5.2.6 White Estimator (WH) 45 5.2.7 Bain and Engelhardt Estimator (BE) 46 5.2.8 Modified Profile Maximum Likelihood Estimat or (MPMLE) 47 5.2.9 Ross Estimator (ROSS) 47 5.3 Estimation Analysis 48 CHAPTER 6 SYSTEM ANALYSIS 54 6.1 System Introduction 54 6.2.1 Stocking Policy Analysis 54 6.2.2 Stocking Policy with Failure Estimation 61 System Algorithm 61 Policy Analysis 66 CHAPTER 7 CONCLUSION AND FUTURE RESEARCH 69 7.1 Conclusion 69 7.2 Research Extensions 70 REFERENCES 73 APPENDICES 80 Appendix A: Initial Estimation Selection Analysis 81 Appendix B: Weibull Parameter Database 84 Appendix C: Final Estimation Selection Analysis 85 Appendix D: Kabir and Al-Olayan Comparison for b=1.5 87 Appendix E: Kabir and Al-Olayan Comparison for b=2.0 90 Appendix F: Kabir and Al-Olayan Comparison for b=3.0 93 Appendix G: Cost Comparison with Additional CR/UR Values 96 Appendix H: Matlab Code 97 Appendix I: Run 10 Distribution Parameters 122 Appendix J: Run 11 Distribution Parameters 124 ABOUT THE AUTHOR End Page


iii LIST OF TABLES Table 1 Wagner, O'Hagan, and Lundh Solutions 12 Table 2 Modified Feng and Xiao Solutions 12 Table 3 Probability of Replacement 35 Table 4 Demand Distribution 35 Table 5 Stationarity Test 36 Table 6 Initial Estimation Method Selection Results 50 Table 7 Kabir and Al-Olayan Policy Values 58 Table 8 System Results 66 Table 9 Poorly Estimated Parameters 67


iv LIST OF FIGURES Figure 1 ( s, S ) Inventory Example 7 Figure 2 ( s, S ) Inventory Solution Method 9 Figure 3 ( s, S ) Solution Modification 11 Figure 4 System Model 63


v A DYNAMIC INVENTORY/MAINTENANCE MODEL Jonathan J. Bates ABSTRACT A model is proposed to provide inventory and maint enance guidance for a system of operating parts. This model is capable of handl ing a system with multiple operating components, unknown part lifetime failure distribut ion, and separately maintained parts. In this model, part reliability characteristics are used along with system costs to predict the required stocking levels and part replacement t imes. Two maintenance strategies are presented that have the unique characteristic of al lowing flexible scheduling of replacements. A case study is completed comparing developed stocking policies to an existing policy. An estimation selection method is introduced and fit into the model for computing Weibull distribution parameters when part reliability is not well known. An algorithm is displayed that describes the implement ation of the system model and data from practical case scenarios are conducted using t his algorithm.


1 CHAPTER 1 INTRODUCTION 1.1 Motivation Fleet oriented organizations, that is, organization s that utilize mobile, complex assets to perform organizational goals, rely on pro per maintenance and suffer greatly if the operational time of those assets is diminished. Examples of “fleet oriented organizations” include airlines, shipping companies (utilizing trucks, airplanes, ships etc.), and military organizations among others. Ad ditionally, as these assets increase in complexity and cost, the intricacy and cost of spar e parts will also increase. Costs associated with maintenance and reliability, includ ing inventory costs, are on the rise. The commercial airline industry spent over $36.1 bi llion in 2004 on maintenance and reliability and has a large amount of money tied up in unnecessary inventory [1]. The market analysis firm AeroStrategy is forecasting a 5.6% growth rate in maintenance and reliability spending for the commercial airline ind ustry over the next decade and by 2014, this type of spending will exceed $62 billion [2]. These problems are not limited to the commercial airline industry. Every year, the U.S. General Accounting Office (GAO) issues reports to the U.S. Congress citing the need for improvement and high costs of maintenance and reliability within the Department o f Defense. These reports recommend that Congress and listed government agencies take s pecific actions to decrease costs while providing the public the same level of servic e [3-6].


2 Inventory theory has been studied for years, with a significant amount of work completed since the 1950’s. Despite all this effor t, successful practical application has not been achieved in most industries. Optimal inve ntory policies generally require “complete knowledge” of the part’s demand distribut ion. This notion of complete knowledge is a phrase that suggests a distribution’ s type and parameters are confidently known. However, it is important to be able to expl ain the reasons for the demand. In the retail goods world, demand represents the active de sire of a consumer to own a product. With many products, the consumer’s desire may vary over time or with seasons. When dealing with a dynamic market, understanding the ev ents that produce product desirability is just as important as having complet e historical knowledge of the demand distribution. These causes for demand can be used to predict changes in demand, rather than updating the distribution with recent historic al information. It can be argued that complete knowledge is not obtained until the underl ying causes for demand are known. What are the causes for a demand on inventory? Thi s study will specifically look at the area of spare parts demand. For spare or re parable parts, the underlying causes for demand can be found in the maintenance policies and reliability information of the individual components. A demand is created when ei ther a part is required due to maintenance or the part failed and therefore a repl acement is needed. This research is guided by a unique concept, building a methodology to model a spare parts inventory/maintenance system using the causes for d emand, the maintenance and failure events. Using this idea, an approach is outlined f or a jointly modeled inventory and maintenance system that is subject to existing or e xpected operational and budgetary constraints.


3 The building blocks for this methodology include co ntributions from inventory and reliability theory. Chapter 2 contains an intr oduction of some inventory theory concepts, a review of existing research, and soluti on methods to select an optimal stationary inventory policy, approximate solutions, and a non-stationary inventory policy. The selection of a maintenance policy is discussed in Chapter 3 along with a review of relative research in the area of joint inventory/ma intenance policies. The construction of the predicted demand forecast is discussed in Chapt er 4. Chapter 5 outlines an estimation selection method that will be used to develop the p art failure distribution. The system model is assembled, tested, and analyzed in Chapter 6, followed by concluding remarks and a discussion of possible research extensions in Chapter 7. 1.2 System Description The following is a description of the system being modeled in very general terms; this general description is employed to widen the u tilization of the study. The “system” refers to a collection of n operating parts either in various geographical loc ations or grouped together. The term “parts” can refer to a functioning collection of parts such as an engine or a component level part such as a pisto n. The parts have generally the same, but not identical operational schedules. Additiona lly, the “birth-dates” of the parts can vary. The failure rates of the parts are identical and are either of the constant or increasing type, but do not have to be well known. This allowance makes it possible for this model to be employed by organizations that do not have well established reliability programs. The system also includes an inventory of parts that are available for installation if an operating part replacement is ne cessary. As a part fails or is replaced


4 due to maintenance, a demand is placed on the inven tory stock. The costs associated with the maintenance and inventory, along with syst em variables, are introduced in later chapters. 1.3 Research Objectives and Contributions Only a limited number of studies have been complete d examining the interaction of maintenance, reliability, and inventory concepts The interaction between them is clearly evident and a holistic study is necessary t o ensure an efficient system approach is available to organizations and managers. The appli cation of this study is targeted for use in enterprises such as commercial and public transp ortation, militaries, and organizations that may operate fleets of vehicles. However, the completed research and resulting decision tool would be useful to any organization t hat carries part inventories for use in an equipment maintenance program including manufact uring and utility plants. The objective is to provide a strategy to obtain a combined inventory/maintenance model, where the primary input parameters of part reliability and inventory /maintenance costs determine the stocking levels and maintenance policy. This strategy can serve as a decision tool for organizations to make justifiable and cost saving policy changes. The combined model should be dynamic in that the invent ory and maintenance policies may vary with changing operational conditions and organ izational objectives. The difficult task in this research is to provide a linking mecha nism between inventory and maintenance in a way that provides decision makers with the ability to adopt new policies under anticipated or expected conditions. Often, o rganizations that maintain their own equipment do not have sufficient information to pre dict when parts will fail. Existing


5 literature in maintenance planning often assumes th at failure times are known perfectly. This study addresses this issue and provides a meth od to implement this model given little or zero part reliability information The area of combined inventory/maintenance modeling is not a complete field and most studies that have been conducted deal only with simplified and static parameters, such as known failure distributions and limited allowable spares. Existing studies often assume that machines are maintained s imultaneously. An integrated approach to determine a maintenance policy and inve ntory stocking rule for a dynamic system is not available in the existing literature. The approach in this study is to present a solution with a wide range of applicability. The c ontributions of this research include advancements and additions to the available academi c literature in inventory theory and reliability analysis, as well as a practical soluti on to a significant problem that exists throughout many industries. Although the resulting methodology will utilize several existing ideas, the manner in which these ideas are weaved together along with some unique concepts has not been accomplished to date.


6 CHAPTER 2 INVENTORY THEORY 2.1 ( s, S ) Inventory Policy A common aspect of most spare parts inventories is the demand of these parts is relatively low or intermittent. The ( s, S ) inventory system has been shown in several studies to be the best performing for items with su ch demand [7]. The ( s, S ) model is periodic-single item inventory system and is an opt imal policy for systems that meet the following assumptions: independent and identically distributed demand, ordering costs are linear plus a fixed setup cost, and all other p eriodic costs are linear (shortage and holding) [8]. The contributions of the existing ( s, S ) inventory model literature can be grouped into three general categories: model formulation an d characteristics, optimal solutions, and approximate solutions. Arrow et al. [9] first formulated this model by examining a dynamic system with demand as a random variable wit h known distribution. The orderup-to and reorder levels are determined as function s of ordering cost, penalty cost, and demand distribution. [10] showed that the ( s, S ) policy is always the optimal policy if holding and shortage costs are linear [8, 11, 12] c ontribute to the area of model characteristics and derive bounds on the modelÂ’s pa rameters, s and S In this model, s is the safety stock level and S is the maximum replenishment level. If inventories are reviewed on a periodic b asis, the on-hand level plus the amount


7 on order, hereby called sr, could have the range sr greater than s or sr less than or equal to s where s again is the safety stock level. If sr falls into the second range, an order should be placed of quantity S – sr, where S is the maximum replenishment level. In contrast to a periodic review model, if the inventory is continuo usly reviewed, the order quantity is always equal to S – s and sr will never be less then s Figure 1 shows an example of an ( s, S ) inventory model where S = 15 and s = 5. At time unit 13, the value sr is equal to s so an order of S – sr (10 units) is placed. At time unit 14, sr is equal to 15, 5 units on-hand and 10 units on order. This order is delivered and is added to the on-hand amount at time unit 17. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 123456789101112131415161718192021 Time Inventory Units On Order On Hand Figure 1 ( s, S ) Inventory Example The solution to the single-item, stochastic invento ry problem was first conceived through a small inventory control conference at RAN D during the summer of 1950 [13]. This conference was organized by the Office of Nava l Research and it brought together


8 Jacob Marschak and Kenneth J. Arrow. From the disc ussion during this conference, Arrow and Marschak thought it was important to inve stigate a realistic model that was a combination of the two main kinds of existing inven tory models. The first existing inventory model was one where inventory could be ca rried over from one period to the next to fulfill a constant demand. The other model was a one-period model with stochastic demand. The combination of these two mo dels represented a more realistic scenario where inventory is carried over from one p eriod to the next and demand is random. At this time, only solutions of the (s, S) form were considered and the underlying problem was identified as a Markov Decis ion Process. The third author of [9], Theodore Harris, was brought on to analyze thi s Markov process and subsequently provided the method to determine the discounted exp ected cost under a fixed (s, S) policy. “Optimal Inventory Policy” inspired a lar ge amount of research in dynamic inventory policy including several papers showing t he theoretical optimality of the (s, S) policy as well as several variations of optimal and approximate solutions to the problem. Additionally the topic of dynamic programming, as c oined by Richard Bellman, has an origin credited to this paper [13]. It is interest ing to note that Kenneth J. Arrow first referred to this two-bin policy as the (S, s) policy and this remained the models title until Donald Iglehart’s paper in 1963 re-titled it as the (s, S) policy [8]. This rearranged title seems to currently dominate although Arrow seems to prefer the latter (S, s) designation [13]. Many papers were written related to the topi c of optimal inventory theory over the next few years, including [11, 14-23], some of whic h are part of the now fabled “Stanford Studies”, but it was not until [10], that a proof w as offered for the optimality of the (s, S) model.


9 2.2 Optimal Stationary Solutions In 1959, Herbert Scarf offered a proof of optimalit y by showing that if holding and shortage costs can be shown or assumed to be li near, there exists an optimal pair (s, S) that minimizes the expected costs over an infinite horizon, thus making the (s, S) policy optimal. He then proposed to solve for this optimal pair, the minimum of the one period expected cost G(y) must be determined, where ( ) ()() GyczLy =+is equal to the ordering cost of z units and the holding and penalty costs of incurre d for y units. The value y* that minimizes G(y) is the optimal order-up-to parameter S To solve for the reorder level s simply solve the following equation) S ( G K )s( G + = where K is the fixed setup/ordering cost. This solution method is shown in Figure 2. Throughout this chapter and the rest of this study, the following i nventory cost definitions will be used: h – inventory holding cost p – inventory shortage cost K – inventory ordering cost l – part lead-time Figure 2 ( s, S ) Inventory Solution Method


10 In a recent paper, Feng and Xiao [24] developed a n ew algorithm to search for the optimal pair of (s, S) following a tradition of papers including [25-29]. This new algorithm is compared to the most previous approach provided by [29] with numerical tests that show an average savings in evaluation ef fort of 30%. The proposed algorithm can be successfully implemented for a system with z ero lead-time. A clarification to this algorithm was presented in [30] providing a simple modification to implement Feng and Xiao’s algorithm when a fixed lead-time exists. Th is modification is outlined in the remainder of this section. Following [24], let () Gy equal the average one-period holding and shortage c ost when the inventory position at the beginning of the period is y ; and 0 y equal the largest y minimizing ).y ( G The modification to this algorithm is in the sele ction of y0. As stated in [24], y0 can be obtained by solving a one-period newsboy pr oblem or the solution to the inequality: () ()00 1, p yy ph F<£F+ + (2.1) where ( ) F is an arbitrary CDF of demand and unit ordering co sts are ignored. When a fixed lead-time exists, [24] states that the cost f unction )y ( G can be redefined as in [25]: 11 01 1 0 ()()()()1 () ()()0 y kky n k hykkpkyky Gy pkykyll ljj j ++ ==+ + = -+- = -£ (2.2) where ) k(nj is the n-fold convolution of ) k( j or the demand distribution. When no lead-time is present or 0, l = 1 n = and it is common to drop the superscripts of this cost


11 function. However, [24] fails to state that using the solution to a one-period newsboy will not find the largest y minimizing )y ( Gn. To obtain y0 when a lead-time exists, the lower bound of S or S must first be determined. This can be done by sol ving for S using the inequality given in [25]: 11 (1)(). p SS phll++F-<£F + (2.3) [8] showed that )y ( Gn is a convex function with a minimum value at y0. Therefore, with S known, y0 can be described as 0 y yS =+D (2.4) whereyD is the largest positive integer such that ()(1). nn yyGSGS +D>+D+ (2.5) Thus, y0 can be found as shown in Figure 3 and Feng and Xia o’s algorithm can be run to completion to solve for s* and S* the optimal pair. This modification provides a m ore general case for a system with a fixed lead-time or with zero lead-time. Figure 3 ( s, S ) Solution Modification


12 Using the modified Feng and Xiao algorithm, the cas e study shown in Table 3 of [31] is duplicated here as Table 1 where s* and S* are the optimal pair minimizing the long-run average cost function c ( s, S ). Table 2 duplicates the numerical results of fou r cases from Feng and Xiao with the additional result s for fixed lead-times of 1 and 2. Table 1 Wagner, O'Hagan. and Lundh Solutions l S y0 s* S* c(s*,S*) 0 14 14 9 14 13.7929 1 17 26 20 27 15.9607 Table 2 Modified Feng and Xiao Solutions l =0 m S y0 s* S* c(s*,S*) 10 14 14 6 40 35.0215 15 20 20 10 49 42.6978 20 26 26 14 62 49.173 25 32 32 19 56 54.2621 l =1 m S y0 s* S* c(s*,S*) 10 17 26 16 51 36.0974 15 17 37 25 65 44.0712 20 35 48 35 83 50.7336 25 44 59 44 83 56.4817 l =2 m S y0 s* S* c(s*,S*) 10 23 37 26 62 37.0426 15 36 54 41 82 45.2529 20 48 70 55 104 52.0842 25 61 86 70 109 58.3301


13 2.3 Approximate Stationary Solutions Approximate solutions to the ( s, S ) inventory policy general require only limited demand information to determine the inventory param eters. Many approximate methods have been developed and three of the more commonly studied approximate solutions are presented here [31-37]. These methods make use of the demand mean m and variance s along with the inventory costs in determining the i nventory values. The first approximate ( s, S ) solution, known as the Normal Approximation, was presented by Donald Roberts in [38]. This approximation utilizes an iterative pro cedure to solve for the inventory parameters s and S This iterative procedure is described in followi ng five steps. 1) Set 2 K Q h m = 2) Solve 1, 2LhQ u h p m m = + n where ( ) 1. L mlm =+ 3) Solve LL su ms =+ where 1. Lssl =+ 4) Redefine Q as () 22 LLNKp QIu hhmm ms =++ nwhere ()() 2.51 2t N u Ituedt pm -=- or the standardized Normal loss integral. 5) Repeat steps 2, 3, and 4 until the inventory values s and SsQ =+ converge. Additional remarks concerning the implementation of the Normal approximation are given in [39].


14 The second approximation provides heuristic decisio n rules for the operating parameters s and S [40]. Five examples from [41] are solved and resu lts from this heuristic are compared to the optimal values. In t his method, defining 21 h K p q hm + n = (2.6) the inventory values are 22222 21 22122LLqpqq s ph sms ms m -- =++-++ + (2.7) and 22222 2 22122LLqpqq S ph msms ms m -- =++-++ + (2.8) [42] published a revision of an earlier method [43] known as the Revised Power Approximation. In the previous approximation, a fi tted regression model was developed by adjusting the method of Roberts [38] to 288 know n inventory policies. [44] shows how the original Power Approximation can be adjuste d to prove useful under the following differing conditions; non-stationary dema nd, correlated demand, or stochastic lead-times. Solving for this method, the initial i nventory values are set as 0.183 .9731.0632.192p LL L p LD s p D h p hms s s r =++- n n (2.9) and


15 0 LL p S ph ms =+ +n (2.10) where .116 .506 2 .494 2 1.31. L p LK D hs m m =+ n n (2.11) The final approximate inventory values are then det ermined as () 0 001.5 min,1.5 p pD sif s D sSifm m > = £ (2.12) and () () 0 001.5 min,1.5 p p p pD sDif S D sDSifm m +> = +£ (2.13) 2.4 Non-stationary Solution The optimality of the ( s, S ) inventory policy under linear purchasing costs wa s extended to non-stationary demand distributions by Samuel Karlin [23]. Karlin presented a policy that utilizes critical numbers to determin e if an order is placed or not. This policy allows the critical number to vary from peri od to period as demand changes. Only a limited number of studies have been complete d that present unique solutions to this non-stationary inventory policy a nd a dynamic programming solution is the only known optimal solution. Sethi and Cheng [ 45] present a dynamic programming solution for both finite and infinite horizon probl ems under Markovian demand. A


16 number of nearly optimal heuristics have been prese nted [46, 47]. Bollapragada and Morton [46] compared their heuristic with Askin [47 ] and an optimal solution obtained through dynamic programming. This study found the Bollapragada and Morton method differed from the optimal solution only 1.7% over t he cases studied while the Askin method had an error of 2%. Additionally, the study showed that the Bollapragada and Morton heuristic is computationally more efficient. To solve for the optimal inventory policy under non -stationary demand, a recursive dynamic programming method can be utilize d [46]. Let ( ) x JN k ,be the optimal cost from period k to period N if during each period an optimal policy is followe d. Therefore, ()()() { } ,,1,min,kNkkNk JxLxEJxkN lx+r =+- (2.14) and ( ) ( ) ,, NNN JxLx l= (2.15) where ( ) ()()() ()() ()10 101, 1,2,1111 0 ,1111 k k NkNk kkkkkkk kk NNNNNEJx LxLS LSll x x l xx xxjx jx xjx+= -=+ ++++++ = ----r -= r -+-+ +- (2.16) Also, ( ) x Lkl,is the expected one-period costs where orders place d in period k will be received at the beginning of l+k While 1 + k l this expected one-period cost is


17 ( ) ()()()()()() ()()(), 1, 1, 0 1 1, 0 0 0k k kk x kkkkkkkk x kkkkLx hxLxxpxLxxx pxLxxxl l l x x l xxzjxzj xzj + + = =+ + == rr -+-+-+-> r -+-£ (2.17) and when 1 + < k l () ()()()() ()()0 1 0 0 0k k kx kkkk x kkhxxpxxx Lx pxxxxx ll xxjxj xj ==+ = -+-> = -£ (2.18) To determine the optimal cost, a recursive dynamic programming approach is used. First, ( ) x JN N is evaluated for all values of x between the lower and upper bounds of SN and the lowest value obtained yields the order-up-to parame ter SN. Then ( ) ( ) ( ) 1,1,, NNNNNNkJxLxEJSl x -r =+(2.19) is evaluated to determine SN-1 and this is repeated for ( ) x JN N ,2 -, ( ) x JN N ,3 -, …, ( ) x JN k to determine SN-2, SN-3, …, Sk. The reorder level parameter is determined by sol ving for the largest x less than Sk such that ( ) ( ) ( ) ,,1, kNkkkNkkJxKLSEJSlx+ r £++(2.20) Due to the property of K -convexity originally shown by Scarf [10], this val ue is the reorder level for period k


18 CHAPTER 3 MAINTENANCE POLICIES 3.1 Maintenance Policy Spare parts inventories exist to allow maintenance personnel access to the parts required to properly sustain equipment operations. The demand rate of spare parts is largely dictated by the usage of the equipment in t erms of both time and operational environment. The maintenance policy that is applie d to the system also plays a key role in the demand rate of a spare part. Obviously, as the operational time increases on a piece of equipment, the possible chance of failure and subsequent repair also increase. “Run to failure” or Failure Based Maintenance (FBM) is a common maintenance philosophy. Other maintenance policies include pre ventative maintenance and condition based maintenance. Preventative maintenance is a p olicy that assigns replacement or repair of system parts at assigned units of time an d/or cycles regardless of the condition. In condition-based maintenance, inspections or meas urements are conducted on the equipment and replacement occurs when a certain con dition is found. As stated in Chapter 2, the optimal inventory compu tations require exact knowledge of the demand distribution. The ideas pr esented in this chapter and Chapter 4, are an effort to define the relationship shared wit h a component’s maintenance, failure rate, and inventory policy. To do this, let us exa mine a simple component that is governed by an increasing failure rate and due to c osts associated with a failure event, it


19 is less expensive to replace the component prior to failure. The maintenance and failure events are mutually exclusive either the part is maintained prior to failure or the part fails and requires replacement. The probability of this part being replaced is determined by the addition of these two mutually exclusive eve nts. Using the probability axiom for the addition of mutually exclusive events, { } { } { } F P M P F M P + = , a combined distribution can be obtained. This comb ined distribution describes the probability of a part replacement or the probabilit y of a part demand. This idea will be expanded upon in Chapter 4. The following section identifies some of the studie s that have been completed in the area of joint inventory/maintenance policies. If a componentÂ’s time to failure is known confidently, the optimal timing for the maint enance event can be determined to minimize maintenance costs. Section three of this chapter describes how the optimal replacement time is determined. Once the optimal r eplacement time is determined, a maintenance schedule can be developed for the syste m of n parts. The maintenance schedule for each part is described in section four of this chapter. 3.2 Joint Inventory/Maintenance Policies Attempts have been made to link inventory and maint enance policies but additional research is required. Most existing stu dies are limited and are not suitable for a dynamic environment. Some limitations found in e xisting studies include assumptions that the failure distribution is known and constant spares or machines are limited to one, or all operating machines are maintained at the sam e time.


20 One of the first papers in the combined study of in ventory and maintenance policies was presented by Falkner in 1968 [48]. In this model, a single component has an increasing failure rate and it is considered more e conomical to replace a component prior to failure than to allow the component to fail. A dynamic programming problem is derived and solved to minimize the expected machine operating cost over a finite horizon. Given a known lifetime distribution funct ion, an optimal inventory level and reorder level is calculated to minimize costs. Thomas and Osaki [49] evaluated a system comprised of a maintained component with one allowable spare unit. The expected cost p er unit time is derived using ordering (normal and emergency), shortage, and holding costs An increasing failure rate is used to determine an expected cycle time for preventativ e replacement. Results show that the optimal policy is either to replace the unit as soo n as the ordered spare is delivered or not to replace until after failure. The results depend on given conditions on shortage costs, ordering costs, and lifetime distribution. Another study by Thomas and Osaki [50], sought an optimal ordering policy, or time to order 0 t for a one-machine system with one available spare. Cases are presented where, given varying failure rates, it is optimal to order just prior, just after, or some time after a part is placed in service. This research is limited to systems with one allowable spare and kno wn lifetime distributions. Osaki, Dohi, and Kaio have completed a long line of papers dealing with the stocking policies for a one unit system [51-58]. [59] is the first of two articles by Armstrong and Atkins discussing the joint optimization of maintenance and inventory policies. Their initial model consists of a single component subject to random failure with an allowance of one spare unit. Using a


21 constant lead-time, maintenance costs for replaceme nt and breakage occur along with inventory costs for holding and shortage. The obje ctive function, or Joint Cost Function (JCF), is derived giving the expected operating cos t per unit time. The JCF is composed of the expected cost per cycle in the numerator and the expected cycle length in the denominator. The expected cost per cycle is develo ped through the addition of the expected replacement and breakage costs per cycle, along with the expected shortage and holding costs per cycle. The JCF is minimized to p roduce the optimal ordering and maintenance policies. Characteristics of the JCF ar e given under certain conditions. Results show that the joint optimization gives an a verage improvement of 3% over a sequentially optimized system. Armstrong and Atkin s strongly recommend joint optimization when the inventory costs dominate the maintenance costs and when the leadtime is large. Additionally, they found sequ ential optimization in some cases can yield good results and recommend that inventory man agers be aware of maintenance policies. Armstrong and Atkins [60] offered an ext ension to include replacing the fixed replacement cost with a cost function and set up th e problem with a service level constraint. This work also incorporated separate l ead-times for scheduled versus unscheduled orders as well as random lead-times. Sarker and Haque [61] used simulation to show joint ly optimized polices produce better results than separately or sequentially opti mized policies. Cost savings of 2.81 – 8.77% are shown for case data. A jointly optimal a pproach lowered ordering cost, holding cost, and failure replacement costs. Sarke r and Haque commented on the research conducted in optimal inventory/maintenance stating, “relatively little effort has been exerted to their (maintenance and inventory) j oint optimization…” Their simulation


22 model uses the following assumptions: operating uni ts are statistically identical (failure is revealed instantly), spares do not deteriorate, rep lacement time is stochastic, increasing failure rate (non-exponential), unit cost is consta nt and is ignored, emergency orders are placed when stocking level is at or below zero, mai ntenance policy is of the block replacement type, and inventory is of the continuou s review type. A significant amount of papers have examined mainte nance and provisioning policies under a block replacement policy [62, 63]. Chelbi and Dound proposed a computational procedure to determine the optimal re placement period T and optimal inventory threshold s for one unit or a set of identical units under a b lock replacement policy. The optimal policies were determined by mi nimizing the total average cost per unit time over an infinite horizon. Also presented are expressions for inventory costs including: average total holding cost, average tota l shortage cost, and total inventory management cost per unit time [64]. In a block rep lacement policy, a failed component is replaced at the time of failure and all n operating components are replaced simultaneously at some predetermined time interval. This type of policy is useful for applications where all operating mechanisms can be offline during the same period to perform maintenance. However, it may not be feasib le to have all mechanisms offline at the same time. This policy also assumes that the m achinery began operating at the same time and continue to follow the same operational sc hedule. Kabir and Al-Olayan [65] states joint ordering and maintenance policies commonly are based on a single machine system and a maximum spare allowance of 1. A simulation study of ordering and maintenance poli cies for multiple machines under a (s, S) ordering policy is conducted and results indicate that the expected cost under


23 separate policies are higher than if policies are j ointly derived. Cases show a percent savings between 0 and 21 percent, where the amount of savings depended on the values of inventory and replacement costs and the failure distribution parameters. Using the results, regression analysis determined that holdin g and shortage inventory costs have the greatest influence on the optimal policies. Additi onally, the preventive and failure replacement costs, as well as the failure distribut ion shape have considerable influence on the stocking policy. [66] further explored the Barlow and Proschan age-b ased preventative replacement policy. Optimal values of the decision variables (t1, s, S) are sought through minimization of the expected total cost per period, where t1 is the time of preventative replacement. This research compared results betwee n a jointly optimal (t1, s, S) policy and a Barlow-Proschan age replacement policy suppor ted by an optimal (s, S) inventory policy. The system studied allows for multiple spa res and assumes a known lifetime distribution. This is the most relevant work to th e system described in this study. The policies presented in this study are unique in that they are developed for a multiple component system where the parts are not maintained by block replacement; rather, each part is maintained separately. The policies presen ted in this paper will be used in Chapter 6 for a comparison case study. 3.3 Optimal Replacement Policy Let Cf equal the cost incurred when a part is replaced du e to failure, and Cp be the cost when a part is replaced due to preventive main tenance. If pf CC < the general maintenance policy is to replace the part at failur e or at an optimal replacement age t


24 whichever occurs first. The optimal replacement ag e is solved by minimizing the expected maintenance cost per unit of time as defin ed by Barlow and Hunter [67]. The expected maintenance cost per unit of time is found by dividing the expected maintenance cost per cycle by the expected length o f the cycle. Let the expected maintenance cost per cycle equal []* *0 ()()()() t fpfp t ECCftdtCftdtCFtCRt =+=+ (3.1) and the expected cycle length equal []* ** 0 0 ()()(). t t t ETtftdttftdtRtdt =+= (3.2) The expected maintenance cost per unit of time is t hen [ ] [] 0 ()() (). (')' fp t CFtCRt EC Zt ET Rtdt + == (3.3) The optimal replacement age may also be solved by s etting the ratio f fp C CC -equal to the equation: *0 ()()()(). t LthtRtdtRt =+ (3.4) For a Weibull distribution, this expression is then *1 0 ()expexp. tttt Ltdtb bbb hhhhrr =-+nn n (3.5) A closed-form solution to the integral of R(t) is not available and it is suggested that the SimpsonÂ’s Rule be used to evaluate this expression [68].


25 3.4 System Maintenance Policy This study will implement a practical maintenance p olicy using the guidance of the optimal replacement policy described in the pre vious section. In most applications, it is impractical to guarantee that a part will be rep laced at the exact optimal time. The following reasons are offered to justify this state ment: 1) Maintenance personnel may not be available at the exact replacement time, 2) The component may be in an operational state and not available for replacement, and 3) Rep lacement or repair may need to be delayed due to a spare not being available. Two maintenance policies were introduced in [69] an d will be used to allow for flexible maintenance scheduling. In first policy, called the Normal maintenance strategy, the replacement time is a random variable following a Normal Distribution with a mean equal to the optimal replacement time. The varianc e of the distribution can be adjusted to ensure the probability of replacements occur in a d esired range of time. Additionally, the variance of this distribution should be adjusted so that the distribution is an adequate representation of actual replacement times. The se cond policy will be called the Uniform maintenance strategy and it utilizes a Uniform dist ribution to model the maintenance performance. In this strategy, the optimal replace ment time will be again be used to establish the time range for maintenance completion For the Normal maintenance strategy, a maintenance planner is given two decision variables to develop the maintenance sched ule. These variables are PC or probability of completion, and CR completion range. CR describes the time range that a replacement should be completed, given as a percent age of the mean or optimal completion time. For example, if the mean is 3, a .10 CR = yields a completion range of


26 CRtCR mmmm-££+ (3.6) or 2.73.3 t ££ PC describes the desired or actual probability of rep lacement during the completion range and is defined using the expre ssion { } PCPCRtCR mmmm=-££+ (3.7) The Completion Range ( CR ) can be used to set a up a confidence limit for th e Normal maintenance strategy and using this limit the strat egy variance can be determined. For a Normal Distribution, the two-sided 95% confidence i nterval can be represented as: ( ) 1 .975, ms F where 1 F is the inverse standardized normal function. If w e let .95 PC = the value CR m is then equal to ( ) 1.975 s F and () 1 .975 CR m s = F The optimal replacement time is at three years. A maintenance policy is established to ensure there is a 95% probability th at the replacement will occur within 10% of the optimal replacement time. This maintenance policy can be described by a Normal Distribution with 3 m = and () 13.1 .153. .9751.96 CR m s === F For the Uniform maintenance strategy, a variable UR or Uniform Range will be employed to define the Uniform distribution paramet ers as ** ttUR - and ** ttUR + respectively. Using the same optimal replacement t ime from the previous example and .10 UR = this strategy would be modeled with a random vari able uniformally distributed with parameters 2.9 and 3.1


27 CHAPTER 4 REPLACEMENT AND DEMAND DISTRIBUTION 4.1 Calculating the Replacement Probabilities As first described in Chapter 3, operating parts in the system can either fail or be removed through maintenance. These two stochastic events govern the demand placed on inventory. Only one of these events can occur o n each part, for if one of the events transpires, the part is removed from service. In p robability theory these events are said to be mutually exclusive. Let the removal due to a ma intenance event be designated as M and a removal due to failure be F then ( ) 0 PMF = and the solution to MF is the third axiom of probability: ( ) ( ) () PMFPMPF =+ Therefore, the replacement probability of each part is determined by summing t he probability of being maintained and the probability of failing. If part i P has been in operation for a length of time i t we can determine the probability of maintenance or failure during the ne xt period D through conditioning. The conditional probability of event A occurring given event B has occurred, as long as ( ) 0 PB > is () ( ) () PAB PAB PB =, where ( ) PAB is the probability of A and B occurring simultaneously. Let A indicate the replacement of part i due to maintenance or failure during period t D and B denote that part i has not been maintained or failed up to time i t the probabilities of maintenance and failure of par t i during the next period t D are,


28 () ( ) () () () ()() (|) ()1i i ii iitt M iiii iiiiiii t iiiM tttt M M tt M ftdt PtMtt PMPtMttMt PMtFt ftdt tdt Rtl+D +D+D<£+D =<£+D>=== >= (4.1) and () ( ) () () () ()() (|) ()1 ,i i ii iitt F iiii iiiiiii t iiiF tttt F F tt F ftdt PtFtt PFPtFttFt PFtFt ftdt tdt Rtl+D +D+D<£+D =<£+D>=== >= (4.2) where ( ) Rt and ( ) t l are the reliability and failure rate of maintenance and failure. Recalling the third axiom of probability, the repla cement probability of part i is, ()()()() ii iitttt iiMF tt PMPFtdttdt ll+D+D+=+ (4.3) In order to solve for the k -Period non-stationary inventory policy as discusse d in Chapter 2.4, the demand distributions for k future periods are required. The k -period demand distributions will be constructed using the replacement probabilities of each individual part as described above. The replacemen t probabilities can be solved for k periods using the expression, ( ) ( ) () ( ) () ( ) 11 ,, i i i itjttjt ikjikjMF tjttjt PMPFtdttdt ll++D++D ++ +D+D+=+ (4.4) where k is the current period and 0,1,. jn = When applying this expression the following rules are used: 1) If ( ) ( ) ,, 1 ikjikjPMPF++ += for any j set ,1 0, ikjt++ = and 2) Let ikj R + represent any replacement or ( ) ( ) ( ) ,,, ikjikjikjPRPMPF+++=+ if ( ) 1 ikjPR+ > set ( ) 1. ikjPR+ =


29 Rule 1 indicates a predicted replacement during per iod k+j and subsequent part renewal at the beginning of period k+j+1 Rule 2 is needed to ensure that the second axiom of probability is maintained for ( ) ikj PR+ 4.2 Constructing the Demand Distribution: GrayÂ’s C ode Once ikj P + has been determined for all i and j the period demand can be determined. The period demand will be represented as a discrete probability distribution, since the inventory can only be depleted by discret e units. To construct a discrete demand distribution, the probability of the random variable X where X represents a discrete part demand, is calculated for each possib le X Determining the probability of demand in this case is not a trivial matter, rememb er, ,1, ikjikj PP +++ for 11 in =. For each X all the combinations of part replacements that yi eld a demand of X must be considered, a collection of !()! n XnX combinations; making it necessary to construct a total of 2n unique combinations. For each possible x, where 0,1,2,,, xn = the products of each combination are summed to determine the val ue for the probability of demand ( ) kj x j+. The general expression to determine ( ) kj x j+ is ()()()()1,,1,,11 11 1 ()11, mnmmnmq nn mPXxPPPPdddd-= r ==-(4.5) where !()! n q xnx = -,1 0im ifpartifailsincombinationm otherwised = and 1 n im i xm d= =" The following example is provided to further explai n.


30 For 4 n = the combinations of all demand possibilities are shown below where ik P is the probability of replacement and ( ) 1 ik P is the probability of survival of part i during period k : ()()()()1,14,11,14,11 11 1,4,1,4, 1(0)11kkkkk mPXPPPPdddd-= r ==--= ( ) ( ) ( ) ( ) ()()()()001010 1111 1,4,1,4,1234 111111kkkk PPPPPPPP ----=---()()()()1,4,1,4,4 11 1,4,1,4, 1(1)11mmmmkkkkk mPXPPPPdddd-= r ==--= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,12,13,14,11,12,13,14,11111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPPdddddddd---=----+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,42,43,44,41,42,43,44,4 1111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP dddddddd -------( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 100011101010 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP---=----+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 010010111010 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 001010101110 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) () ( ) 0001101011 10 1,2,3,4,1,2,34,1111kkkkkkkkPPPPPPPP ------()()()()1,4,1,4,6 11 1,4,1,4, 1(2)11mmmmkkkkk mPXPPPPdddd-= r ==--= ( ) ( ) ( ) ( ) ( ) ( ) () ( ) 1,12,13,14,11,12,1 4,1 3,111 1 1 1,2,3,4,1,2,34,1111kkkkkkkkPPPPPPPPdddddd d d-=----+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,62,63,64,61,62,63,64,6 1111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP dddddddd -------( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 110011111010 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP---=----+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 011010111110 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 001110101111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 100111101011 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 101011101110 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 010110111011 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+


31 ()()()()1,4,1,4,4 11 1,4,1,4, 1(3)11mmmmkkkkk mPXPPPPdddd-= r ==--= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,12,13,14,11,12,13,14,11111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPPdddddddd---=----+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,42,43,44,41,42,43,44,4 1111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP dddddddd -------( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 111011111110 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP---=----+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 011110111111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 101111101111 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP-------+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 110111111011 1,2,3,4,1,2,3,4,1111kkkkkkkkPPPPPPPP ------()()()()1,14,11,14,11 11 1,4,1,4, 1(4)11kkkkk mPXPPPPdddd-= r ==--= ( ) ( ) ( ) ( ) 1111 1,2,3,4, kkkkPPPP = Note: A total of 4 216 = unique combinations are listed above. With an exponential increase in possibilities, a co mbinatorial technique known as Gray code of order n, or GrayÂ’s code, is necessary to ensure all combinations are identified as n becomes large (the possible combinations exceed 1 million for 20 n = ). This code makes use of a n-tuple of 0Â’s and 1Â’s along with a systematic algorithm t o produce all unique combinations. For this case, th e digit 1 will represent the probability of a part replacement and 0 will represent the prob ability of a part survival during the next period. For each combination, the sum of the n-tuple will identify the demand amount produced by that combination. GrayÂ’s code was first demonstrated in 1878 by a Fre nch engineer Emile Baudot and was patented by the Bell Labs researcher Frank Gray in 1953 [70]. The algorithm to generate GrayÂ’s code is:


32 Begin with n-tuple 12 000 naaa = While12 100 naaa follow the following steps: 1) Add the series of digits. If the sum is even, chan ge n a 2) Else, find j where 1 ja = and 0 ia = for all i where ,1,,1 innj =-+ and change 1 j a Note: if jn = change 1 n a The following example will illustrate GrayÂ’s code a nd the use of the above algorithm for 4 n = 1 a 2 a 3 a 4 a 0 0 0 0 (sum is even change n a ) 0 0 0 1 (see note) 0 0 1 1 (sum is even change n a ) 0 0 1 0 (3 1 a = and4 0 a = change 2 a ) 0 1 1 0 (sum is even change n a ) 0 1 1 1 (see note) 0 1 0 1 (sum is even change 4 a ) 0 1 0 0 (2 1 a = ,3 0 a = and4 0 a = change 1 a ) 1 1 0 0 (sum is even change n a ) 1 1 0 1 (see note) 1 1 1 1 (sum is even change n a ) 1 1 1 0 (see note) 1 0 1 0 (sum is even change n a ) 1 0 1 1 (see note) 1 0 0 1 (sum is even change n a ) 1 0 0 0 (end). Assigning the values1, .1 kP = 2, .05 kP = ,3, .15 kP = ,4, .75 kP = and using the series produced by GrayÂ’s code, the values for the discret e demand distribution are determined as:


33 1, k P 2, k P 3, k P 4, k P 0 0 0 0 ( ) ( ) ( ) ( ) . == 0 0 0 1 ( ) ( ) ( ) . == 0 0 1 1 ( ) ( ) . == 0 0 1 0 ( ) ( ) ( ) . == 0 1 1 0 ( ) ( ) . == 0 1 1 1 ( ) . == 0 1 0 1 ( ) ( ) . == 0 1 0 0 ( ) ( ) ( ) . == 1 1 0 0 ( ) ( ) . == 1 1 0 1 ( ) . == 1 1 1 1 . == 1 1 1 0 ( ) . == 1 0 1 0 ( ) ( ) . == 1 0 1 1 ( ) .1.95.1575.0107 == 1 0 0 1 ( ) ( ) . == 1 0 0 0 ( ) ( ) ( ) . == (0).1817 kj= (1).5451.0321.0096.0202.6070 kj=+++= (2).0962.0017.0287.0011.0036.0606.1919 kj=+++++= (3).0051.0032.0002.0107.0192 kj=+++= (4).0006 kj= The increase in computational effort to assemble th e collection of 2n combinations requires a short discussion on the upper bound of n in the application of this model. The ( s, S ) inventory policy is best suited for intermittent or low demand items and this type of demand supports a system where n is relatively small. Throughout the development o f this model, a general system size of 30 £ n was envisioned. This “upper bound” is not a restricting number but is provided as a practical m easure.


34 4.3 Replacement Probability and Demand Distribution Example As shown in the previous section, the demand distr ibution from period to period is influenced by the replacement probabilities and age s of each part. If the replacement probabilities change from period to period, the dem and distribution will also change. To further examine how the replacement probabilities a nd demand distribution change, let us examine a series of consecutive periods for a syste m of four parts. These four parts have an identical failure rate following a Weibull distr ibution with shape parameter b of 3 and scale parameter h of 6. The expected time of the maintenance event i s found by solving for the optimal replacement time as shown in Chapte r 3.3. For this example, let 2 fC = and 1 pC = Using the equation *1 0 expexp, t f fpC ttt dt CCb bbb hhhhrr =-+nn n (4.6) the optimal replacement time t* is found to be 4.69. Letting the decision variabl e PC and CR equal .95 and .10, the variation of the maintenanc e policy is .24. If the age of the parts are 1 .5, t = 2 1.7, t =3 3.1, t = and 3 4.6, t = the probability of replacement for each part during the next period of length .25 is: () 1 21 .75 .75 1, 2 .5 .51 exp1.0014, 2 2kt tt P dtdtbm mb sshh ps-rr=--F+= n n () 1 21 1.95 1.95 2, 2 1.7 1.71 exp1.0116, 2 2kt tt P dtdtbm mb sshh ps-rr=--F+= n n () 1 21 3.35 3.35 3, 2 3.1 3.11 exp1.0361, 2 2kt tt P dtdtbm mb sshh ps-rr=--F+= n n


35 and () 1 21 4.85 4.85 4, 2 4.6 4.61 exp11.0. 2 2kt tt P dtdtbm mb sshh ps-rr=--F+= n n Using the probabilities, the demand distribution fo r the next period is determined to be (0)0, kj = (1).9514, kj= (2).0481, kj= (3).0005, kj= and (4)0. kj = Using this same process, the probability of replacement of each par t and demand distribution were determined for next four periods and the results ar e shown in Tables 4 and 5 respectively. Note: The age of Part 4 was initially close to the optimal replacement time and 4, 1.0. kP = Using the rules given in Chapter 4.1, the part age at the beginning of the next period is 0. Table 3 Probability of Replacement j=0j=1j=2j=3j=4 P1,k+j0.00140.00270.00440.00660.0092 P2,k+j0.01160.0150.01880.0230.0277 P3,k+j0.03610.0420.04840.06170.1365 P4,k+j10.00010.00050.00140.0027 Table 4 Demand Distribution j=0j=1j=2j=3j=4jk+j(0) 00.94110.92910.90940.8296jk+j(1) 0.95140.05810.06970.08850.1647jk+j(2) 0.04810.00080.00120.00210.0056jk+j(3) 0.00050000jk+j(4) 00000


36 4.4 Stationarity Analysis Once the demand distribution has been estimated, t he inventory policy can be determined using either a stationary or non-station ary solution. A stationary demand is one that follows the same demand distribution from period to period. A non-stationary demand has a varying demand distribution from perio d to period. To distinguish between a stationary and non-stationary demand, the Kolmogo rov-Smirnov or KS test can be used to test the distribution from one period to the nex t. The hypothesis of the KS test is that the distributions are the same and the test statist ic is 1 max, kjkjKSDD+++=-where kj D + is the demand distribution cdf for period k+j The rejection criteria for a two-sided test is 2 kp KS c < A KS test was performed using the distributions f rom the previous section and the KS statistic, p-value, and result are displ ayed in Table 5. Table 5 Stationarity Test KS Statp-valueResult Period k to k+ 1 0.94110.02Hypothesis rejected Period k+ 1 to k+ 2 0.0121Could not reject Period k+ 2 to k+ 3 0.01971Could not reject In this scenario, the demand distribution is not st ationary and the non-stationary solution method from Chapter 2.4 should be used to determine the inventory policy.


37 CHAPTER 5 PARAMETRIC ESTIMATION 5.1 Censoring Part lifetime data is typically in the form of a co mplete or censored data set. A complete data set is obtained if all items are allo wed to operate until failure. However, it is often impractical in reliability testing to allo w all items to operate until failure. Censoring occurs when an item is removed from servi ce for any reason prior to failure. The most common forms of censorship are right, left and interval censoring, with right being the most common. Right censored data describ es a data set where the initial time placed in operation is known for all items but the exact failure times of one or more items are unknown due to removal from service. Left cens oring is less common then right and occurs when failure times are known for all items b ut the exact times that one or more items were placed into service are unknown. Lastly interval censoring occurs if failure times are not exactly known but grouped into interv als. Right censoring is further divided into three sub-c ategories: Type I or time censoring, Type II or order statistic censoring, an d random censoring. In Type I censoring, one or more items are removed from servi ce at a specified time. Type II censoring occurs when one or more items are removed from service after a specified number of failures have occurred. Random censoring occurs when one or more items are removed from service before failure at independentl y random times. With random


38 censoring, there are two or more probabilistic dist ributions in play and this type of censoring is also referred to as a competing risks model. These competing risks are the two or more stochastic events that could end the se rvice life of each item. In this study, the competing risks are failure and random removal from service prior to failure. In the system being modeled, there are two stochastic even ts that will cause the part to be removed from operation. Replacement due to a failu re may occur if a part stops adequately performing its desired function. The se cond competing risk is replacement at some stochastic interval due to a maintenance polic y. The purpose of reliability testing is to gain knowl edge of the governing failure distribution for a particular item. The data colle cted is used to determine the distribution type and parameters that best describes the expecte d life of that item. Once lifetime data is collected and the type of censorship is determin ed, the distribution type and parameters can be estimated. In this study, the Weibull distr ibution is assumed to best describe the lifetime of the item. The Weibull distribution is often used in reliability analysis due to its flexibility in modeling failure rates. It has been used to model the life of electronic components, semi-conductors, pumps, motors, ball be arings, fatigued materials, as well as various biological organisms. Using this assump tion, several methods can be used to determine the estimated Weibull parameters. 5.2 Methods of Estimation Reineke completed a dissertation in 1998 that exam ined estimation methods for lifetime data under random right censoring levels o f 25%, 50%, and 75% [71]. This study tested seven parametric, nonparametric, and s emi-parametric methods and


39 concluded that the parametric Maximum Likelihood Es timator (MLE) is the best method given that the distribution type is correctly speci fied. Additionally, the Piecewise Exponential Estimator (PEXE) [72] and Fldes, Rejt, and Winter Estimator (FRWE) [73] methods are the best nonparametric estimators for all tested censoring levels. With the exception of the Klein, Lee, and Moeschberger ( KLM) Partial Parametric Estimator [74], the semi-parametric methods performed poorly and were not recommended for use. The MLE, PEXE, FRWE, and Klein, Lee, and Moeschberg er Partially Parametric (KLM) methods will be further tested, along with the comm only used Kaplan-Meier Estimator (KME) [75], and are described in the following sect ions. Due to the nature of this study, it is important t o know the best performing estimator given a high level of censoring, lower va lues of n and a shape parameter that best models mechanical parts. Reineke states that the censoring level does have an effect on estimator performance; however, the study did no t comment on how the part sample size n and the shape parameter affect the performance. T he analysis described later in this chapter will add to the work completed by Rein eke, specifically for highly-censored cases. The following sections outline several esti mators that are used in this study. 5.2.1 Maximum Likelihood Estimation (MLE) Given a censored set of data where ti represents a failure of part i and ci represents a censored time of part i let 1=id if ii tc £ and 0=idif ii tc > The likelihood function is then ()()1 1 (,),,, ii n ii iLufuRu dd qqq==' (5.1)


40 where q is the vector of parameters to be estimated and u includes all failure and censored times. The log likelihood function is the n ()()1 11 ln(,)ln,ln,. i i nn ii iiLufuRu dd qqq===+ (5.2) Since the density function is the product of the ha zard and reliability functions or ()()() fthtRt = the log likelihood function can be rewritten in t he form ()()()1 111ln(,)ln,ln,ln, ii i nnn iii iiiLuhuRuRu ddd qqqq ====++ (5.3) and simplifies to ()()11 ln(,)ln,ln,. inn ii iiLuhuRud qqq ===+ (5.4) If set U contains all values of u where 1 id = and using the property ()log(), HtRt =the expression can again be rewritten as ()()1 ln(,)ln,,. n ii iUiLuhuHu qqq ==(5.5) In the case of the Weibull distribution, the log li kelihood function is ()()()1 1 ln,ln, n ii iUiLuuu bb qblll=r =(5.6) where b and l are the shape and scale parameters respectively. T his expression can then be simplified to ()()[]1 1ln,lnln1ln(1)ln lnln(1)ln.n ii iU i n ii iU i Lu uu uu bb bbqblblbl bblbl = ==++-+-r =++-(5.7) If there is r observed failures or values in set U the log likelihood function becomes


41 ()1 ln,lnln(1)ln. n ii iUi Lurruu bb qbblbl==++-(5.8) To solve for the estimates of the shape and scale p arameters, the partial derivatives are determined for each parameter and solved for zero u sing the following expressions: ( ) 1 1ln, 0 n i iLu r ubbq b bl ll= =-= (5.9) and ( ) ()()1ln, lnlnln0. n ii i iUiLu r ruuubq lll bb= =++-= (5.10) Unfortunately, there is no closed form solution to the simultaneous solution of these two equations. However, l can be solved in terms of b using the expression 1 1 n i ir u b bl= = n (5.11) Applying this expression for l to the second partial shown above and simplifying y ields () 1 1ln ln0. n ii i i n iU i iruu r gu ub bb b= = =+-= (5.12) To solve for b the iterative Newton-Raphson procedure could be u sed. In this case, each subsequent 1 i b + is ( ) () 1 ii i ig g b bb b +=¢ (5.13) where


42 () ()2 2 2 2 111 1 lnln. nnn iiiii n iii i irr guuuuu ubbb bb b=== = r ¢ =--nnn n (5.14) The procedure is terminated when 1 ii bb + is less than some small value e To determine the initial value for0b, the following process recommended in Leemis [76] can be used: while the observed number of failures or 2 r set ( ) ( ) () 0 2 iii iU iU XXYY XXb -= (5.15) where ln, ii Xu = 1 lnln, () i in Y nu r + = n X and Y are the sample means, and ( ) i nu is the number of items operating prior to time i u The Maximum Likelihood Estimator of the lifetime distribution is ()() 1exp. MLEFxxbl r =-(5.16) 5.2.2 Kaplan-Meier Estimator (KME) The KME, also known as the product-limit estimator, was introduced in 1958 and is one of the most significant contributions in rel iability theory. This paper is ranked as one of the most cited papers [77] of all time, havi ng been cited over 29,000 times [78]. When no censoring is present, the KME simply reduce s to the empirical distribution function. The KME is defined as:


43 () 1 1 100 1, 1m jj KME m j j mu nr Fxuxu n xu= < =-££ > (5.17) where m is the total number of failures, 11 00 ii ijj jj nnsr -===-and sj and rj are given the value of 1 if the jth time represents a censored and failed item respect ively and zero otherwise. 5.2.3 Piecewise Exponential Estimator (PEXE) Let ui represent the ordered failure times, cij represent the ordered censoring times, and ki be the total number of censored observations betwee n failures. 1 2 11,11,12,12,21,1, 1,11,0 .r r kkrrrkr rrk cctccttcct cc+++ <<<<<<<<<<<<<< <<< For the interval between successive failures, a con stant failure rate is estimated and this rate is used to fit an exponential estimator of the lifetime distribution on each interval. These fitted functions are then pieced together to form a piecewise function from 0 to the time of the rth failure. Beyond the rth failure, an exponential fit with a hazard rate equ al to the previous interval is often used. The constan t failure rate for each interval is: ()() ,1 1 1 11 1i k i ijijii j jz cunikuu = == -+-+-n (5.18) and the PEXE of the lifetime function is


44 () ( ) (){}()(){}()(){}1 1121 12 1122111112211exp 0 exp exp ,3,, expi PEXE iiii rrrzxxu zuzxuuxu Fx zuzuuzxtuxuir zuzuuzxtxu---££ -+-££ r = -+-++-££= r -+-++-> r (5.19) 5.2.4 Fldes, Rejt, and Winter Estimator (FRWE) The FRWE kernel density estimator is defined as () 11 n i FRWEi i nn xt fxk hh= =D n (5.20) where 1 5 n hn s = s is the standard deviation of the failure set, Di is the vertical jump of the KME from ti-1 to ti, and k(x) is the standard normal kernel estimator or 211 ()exp. 2 2 kxx p =- n (5.21) The FRWE estimator of the lifetime distribution fun ction is then ()() x FRWEFRWE Fxftdt -= (5.22) 5.2.5 Klein, Lee, and Moeschberger Partially Parame tric Estimator (KLM) Let ( ) R q represent a parametric survivor function where q is a vector of estimated parameters from some previous estimation method. In this case, the MLE survivor estimate will be used, where for q consists of the estimates for b and l


45 Recalling that 1 =id if ii tc £ and 0 =idif ii tc > the KLM estimator of the lifetime distribution function is () ()() 1 n i i KLMxR Fx n fq= = (5.23) where ()() () () 1 01. 0 i iii ii iifXx xRifXxand Rx ifXxand RXqqd q d q > =£= £= (5.24) 5.2.6 White Estimator (WH) The equations to estimate the Weibull shape and sca le parameters for the White Estimator (WH) are () () () 1 2 11m iii i WH m WH ii i XXYYw b XXwb= =-== (5.25) and ln, WHWHWH uYbX a==(5.26) where 1 1 m ii i m i i Xw X w = == (5.27)


46 1 1 m ii i m i i Yw Y w = == (5.28) (){} 1 lnln, expexpi iX EX r = -r n (5.29) ln, ii Yx = (5.30) and () 1 vari iw X = (5.31) ( ) i EX and ( ) var i X can be found in Table 1 and Table 2 respectively in [79]. The WH estimator of the lifetime distribution function is then: () 1exp. WHWH WHx Fxba r =-n (5.32) 5.2.7 Bain and Engelhardt Estimator (BE) Using the same notation as above, the equations to estimate the Weibull shape and scale parameters for the Bain and Engelhardt Estima tor (BE) [80] are 1 1 1m ir BE i BErn YY b nk b= ==(5.33) and [ ] ln, BEBEmmBE uYEXb a==- (5.34) where


47 [][]()1 11 m mnim ikEXEX n= =-n (5.35) The BE estimator of the lifetime distribution funct ion is then () 1exp. BEBE BEx Fxba r =-n (5.36) Note: The WH and BE methods are undefined at a cens oring level of n -1. 5.2.8 Modified Profile Maximum Likelihood Estimator (MPMLE) The modified profile score function for b in the case of censored data is () 1 ** 1112 loglog, rrr miiii iiir Sryyyy bbb b====-+ nn (5.37) where r is the total number of censored observations and t he notation 11 () rr iir ii wwnrw ===+(5.38) is used. The modified b is then found by solving ( ) 0. mSb = The scale parameter can be solved in the same manner as in the MLE method [81] 5.2.9 Ross Estimator (ROSS) The Ross Maximum Likelihood [82] unbiasing method c onsists of the same scale parameter found using the MLE method and shape para meter 1.37 1 1.92MLE ROSS n rr b b= + (5.39) The ROSS estimator of the lifetime distribution func tion is then


48 () 1exp. ROSSROSS MLEx Fxba r =-n (5.40) 5.3 Estimation Analysis Expanding Reineke’s [71] analysis, the MLE, KME, FR WE, PEXE, and KLM methods were evaluated under the following censoring levels, sample sizes, and Weibull shape parameters: Censoring Levels – (60%-70%], (70%-80%], (80%-90%], (90%-100%] Sample Size – 5, 6, 8, 10, 15, 20, 25, 30, 35, 40, 50, 60, 70, 80, 90, 100 Weibull Shape Parameter – 3, 4, 5 Data was simulated for all 765 combinations and the I ntegrated Square Error (ISE) was determined for each case. ISE is defined here as t he integrated squared difference between the estimated and true distribution function s or ()()()2 ISEFFxFxdx +-r=(5.41) This process was then repeated up to 10,000 times to collect a suitable sample size to determine which method performed best for each case. The ISE values for each case were then evaluated using a pair-wise Kruskal-Wallis ( KW) test comparison. The KW test is a nonparametric version of a one-way analysi s of variance and is used to determine if the values of one sample are different then the values of another sample. To perform a KW test, the samples are combined into a single samp le, the values are sorted from smallest to largest, and a rank is assigned to each value. The average rank of each sample is then compared using the test statistic


49 () 2 1 121 12K ii iN KWnR NN=+ =+ n (5.42) where K is the number of samples under comparison, n is the size of the sample, and i R is the average rank of the sample. The null hypothesi s for this test is that the distributions are the same and it is rejected if 2 1 KKWc > A total of 10 pair-wise comparisons were made for each test combination and the methods were ranked from best performing to worst. If the hypothesis for two or more methods cou ld not be rejected, these methods were given an equal rank. If the hypothesis was reje cted, the method with the smaller mean was given a higher rank. The analysis was conducted initially replicating ea ch test case 1,000 times. This initial analysis showed that for values of n greater than 30, the MLE method is the best performing regardless of censoring level or shape p arameter. For these same test cases, the FRWE method had the second best performance. T he results including mean ISE, ISE standard deviation, and rank for all 765 combin ations are shown in Appendix A. However, for values of n equal to or less than 30, such a simple conclusion was not apparent. To increase the ability of the KW test to distinguish between paired methods, a second analysis incorporating 10,000 replications wa s conducted for all combinations of 30 n £ and the best performing method for each combination is shown in Table 6 [83].


50 Table 6 Initial Estimation Method Selection Results n=5 n=6 Censoring Level =2 =3 =4 Censoring Level =2 =3 =4 (60%-70%] FRWE FRWE FRWE (60%-70%] PEXE PEXE PEXE (80%-90%] FRWE FRWE FRWE (80%-90%] PEXE PEXE PEXE n=10 n=10 Censoring Level =2 =3 =4 Censoring Level =2 =3 =4 (60%-70%] PEXE MLE MLE (60%-70%] PEXE MLE MLE (70%-80%] PEXE MLE FRWE (70%-80%] PEXE MLE FRWE (80%-90%] KLM KLM FRWE (80%-90%] KLM KLM FRWE (90%-100%] KLM KLM FRWE n=15 n=20 Censoring Level =2 =3 =4 Censoring Level =2 =3 =4 (60%-70%] MLE MLE MLE (60%-70%] MLE FRWE FRWE (70%-80%] PEXE MLE MLE (70%-80%] MLE MLE FRWE (80%-90%] PEXE MLE MLE (80%-90%] PEXE MLE MLE (90%-100%] KLM KME KME (90%-100%] PEXE PEXE KLM n=25 n=30 Censoring Level =2 =3 =4 Censoring Level =2 =3 =4 (60%-70%] MLE FRWE FRWE (60%-70%] MLE MLE FRWE (70%-80%] MLE MLE MLE (70%-80%] MLE MLE MLE (80%-90%] MLE MLE MLE (80%-90%] MLE MLE MLE (90%-100%] PEXE PEXE KLM (90%-100%] PEXE KLM KLM An examination of this table fails to provide simple rules to follow in choosing an estimation method except that as the MLE begins to dominate as n increases. The well known bias increase with the MLE explains the decreas ed performance with smaller values of n In practice, this table may not be useful since one of the entering arguments, b is an unknown value in a reliability test. However, t he range of the shape parameter could be estimated using expert selection or throug h existing component values from published data as found in [84] and displayed in App endix B. An improved application would be to provide a tool for selecting an estimati on method given an assumed shape


51 parameter range, n and known level of censoring. This tool could be used at the completion of a reliability test, when the censored amount is known. Using this strategy, a second analysis was performed to determine the best performing method given a value of n censoring amount, and shape parameter range. Four additional methods were incorporated into this analysis to allow for increased competition. These methods were outlined earlier in 5.2.6-5.2.9. Montanari et al. [85] tested a total of six methods including the WH, BE, and ROSS methods for sample sizes of 6, 10, and 20; shape parameters of .5, 1, and 10 ; and for censoring levels of 30% and 50%. Their results show that the WH and BE estimato r performed well at higher levels of censoring. The ROSS method was shown to perform s atisfactorily but was not recommended for higher levels of censoring. Yang a nd Xie’s [86] MPMLE method was shown to have less bias than the MLE and ROSS methods and had increased efficiency with lower sample sizes and heavier censoring. A total of nine estimation methods were tested for al l combinations of the following values: Censoring Levels – n-1, n-2 ,…, 1 Sample Size ( n ) – 5, 10, 15, 20, 30 Weibull Shape Parameter ( ) b ranges – 1-1.45, 1.5-1.95, 2-2.45, 2.5-2.95, 3-3.45 3.5-3.95, 4-4.45 Once again, the ISE for each value combination was ca lculated using 10,000 simulated estimations and a KW test was completed for all 36 pa ir-wise comparisons. Throughout the simulation, the shape parameter values were uni formly distributed between the specified range values over the 10,000 replications The best performing method was


52 again determined by evaluating the rank value in th e KW test. The p-value was calculated for each comparison and used to determin e the significance of the results. The results of this second analysis are displayed in Appendix C. In these tables, an asterisk indicates that a method is not signific antly different than one or more other methods at a 95% confidence level. Generally for th ese cases, the best performing method displayed slightly outperforms a method that is the best performing method at a neighboring censoring level. For example, in the c ase of 20, n = 1.51.95, b =and censored number of 8, the rank values indicated the MPMLE method has the best performing, followed by ROSS, and then MLE. The p-va lue for the comparison of MPMLE to ROSS was .78 and .54 for the comparison of M PMLE to MLE. The p-values indicate that the MPMLE is not significantly differ ent than the ROSS or MLE; however, it was selected as the best performing due to a lower rank value. The ROSS and MLE method are selected as the best performing at the n eighboring censoring value. To obtain more significant results, the replication number co uld be increased, but 10,000 was used as a reasonable choice for all tests. The results shown in Appendix C show that for heavie r censoring levels, the KME and PEXE methods tend to dominate as best perform ing. The FRWE also performs well at high censoring levels at lower val ues of n On the opposite end of the censoring level, the MPMLE and ROSS methods perform well and the MLE is overwhelmingly favored in the middle ranges. At smal ler values of n and lower censoring levels, the WH, BE, and to a lesser degre e FRWE perform well. For n greater than 10 and censoring less than 45%, the pair-wise c omparison of the MPMLE and


53 ROSS methods yielded high p-values indicating that t he ISE ranks of these two methods are not significantly distinguishable. This analysis shows that for varying value combinat ions, several estimation methods should be employed and carefully selected g iven the exact scenario. This is clearly evident as the censoring level changes withi n a fixed b value range. This occurrence will happen often in a practical scenario as the number of censored items will fluctuate from test to test. For a reliability test of randomly right censored Weibull distributed data, the results obtained by this analysis should be used to select the best performing estimation method. In the cases where the shape parameter rang e does not match that tested in this analysis, an similar comparison of several estimati on methods should be conducted. A further examination of selection methods for a part icularly important when high or low censoring is found.


54 CHAPTER 6 SYSTEM ANALYSIS 6.1 System Introduction Combining the information from Chapters 2-4, a gene ralized approach using component failure distribution to determine the sto cking policy for any multiplecomponent system subject to individual replacement times is introduced in this Chapter. As earlier stated in Chapter 3.2, Kabir and Al-OlayanÂ’ s study [66] presented one of the few policies using multiple components that do not f ollow a block replacement strategy. Using this study as a baseline, a total of 49 develo ped stocking policies are tested against the policy values from [66] and the policy costs ar e compared. In this analysis, the lifetime failure distribution of the components is assumed to be well known. Section 6.2.2 will expand upon these policies when the compon ent failure information is unknown. 6.2.1 Stocking Policy Analysis To develop our stocking policies, three methods of compiling inventory demand information is used. Two of the three methods empl oy the predictive forecasting method introduced in Chapter 4. The first method will be c alled the Failure Forecast and uses only the lifetime failure parameters (equation 4.2) while the second, called the Combined


55 Forecast, uses both the failure information and the maintenance distribution information (equation 4.3). The third method simply uses histo rical demand information. Using the information from these three methods, the Power, Normal, and Naddor ( s, S ) approximations introduced in Chapter 2.3 were used to produce the following inventory policies: 1) Historical Power Approximation 2) Historical Normal Approximation 3) Historical Naddor Approximation 4) Forecasted Failure Power Approximation 5) Forecasted Failure Normal Approximation 6) Forecasted Failure Naddor Approximation 7) Combined Forecast Power Approximation 8) Combined Forecast Normal Approximation 9) Combined Forecast Naddor Approximation. As shown in Chapter 2.3, these approximations require the holding cost, penalty cost, order cost, lead-time, demand mean m and demand variation s to determine the ( s, S ) inventory parameters. For the Forecasted Failure a nd Combined Forecast policies, the expected demand and variation was calculated for the future 30 k = periods. A second group of inventory policies were developed using these approximations by determining the expected m and s during only the future 1 l + periods. These six policies are called the: 10) Forecasted Failure Lead Power Approximation 11) Forecasted Failure Lead Normal Approximation


56 12) Forecasted Failure Lead Naddor Approximation 13) Combined Forecast Lead Power Approximation 14) Combined Forecast Lead Normal Approximation 15) Combined Forecast Lead Naddor Approximation. The recursive dynamic programming approach introdu ced in Chapter 2.4 was combined with the Forecasted Failure and Combined Fo recast for 30 k = periods providing two additional inventory policies: 16) Forecasted Failure Non-Stationary Solution 17) Combined Forecast Non-Stationary Solution. Lastly, the optimal ( s, S ) solution presented in Chapter 2.2 was solved to ob tain the following inventory policies: 18) Forecasted Failure Optimal Solution 19) Combined Forecast Optimal Solution. For these policies, the discrete probability densit y functions for the future k periods were determined and used to find a single average discre te probability density function. For the inventory policies tested in this Chapter, the stationarity characteristic of the demand distribution was ignored. For the majority of the s cenarios studied, the demand from period to period was shown to be non-stationary usin g the techniques presented in Chapter 4.4. These 19 inventory policies were then coupled with t wo or three maintenance strategies. The first maintenance strategy, called the Fixed strategy, is to replace the part at the optimal replacement time determined by equat ion 3.5 or upon failure. The second and third maintenance strategies were introduced in Chapter 3.4 and allow for practical


57 maintenance scheduling. Using the Fixed, Normal, and Uniform maintenance strategies, a total of 49 stocking policies were developed. Inv entory policies 1 – 6, 10 – 12, 16, and 18 were coupled with all three maintenance strategies Policies 7 – 9, 13 – 15, 17, and 19 were tested under the Normal and Uniform strategies o nly. These policies all use the Combined Forecast approach which incorporates these two strategies into the demand prediction. The 49 stocking policies will be referr ed to by the numerical listing shown previously for the inventory policy following by the maintenance strategy. For example, Policy 1 Normal refers to a stocking policy whose inv entory parameters are calculated by the Revised Power Approximation using historical dema nd that is subject to a Normal maintenance strategy. The 49 stocking policies were tested against the st ocking policies presented in [66] under varying inventory and maintenance costs. These costs, along with the optimal replacement time t* and inventory policy values, are displayed in Tabl e 7. These cases were repeated for Weibull parameter ( ) bh values of (1.5,100), (2,100), and (3,100). The comparison results located in Appendix D, E, and F display the cost difference between the stocking policies presented in this stud y and the stocking policy of Kabir and Al-Olayan. The optimal replacement time determined b y equation 3.4 and the inventory policy values are also displayed in these Appendices The inventory policy values displayed are the minimum and maximum values calcul ated during the comparison, as these values vary from period to period as latest i nformation is used to determine the inventory policies. The inventory cost accounting for stocking policie s developed in this study differ slightly from that used by Kabir and Al-Olayan. The p olicies developed in this study are


58 all periodically reviewed where the policies in Kabir and Al-Olayan are continuous. In a periodic policy, holding and shortage costs are tot aled for full periods. Holding costs are not accumulated for parts that are removed from inv entory during the period. Similarly, shortage costs will be added for each period a part is short and for the period that the part is received. In a continuous policy, the costs are computed for the exact time it is held or short. The cost accounting differences between the se two policies does not give either policy a significant advantage. Table 7: Kabir and Al-Olayan Policy Values Run Ordering ( K )Penalty ( p )Holding ( h ) Maintenance ( C m )Failure ( C f ) t*sS 18.7513.50.6875255521002216.2528.50.6875255510003316.2513.51.5625255521002416.2513.50.6875255521002516.2513.50.687535851000368.7528.51.562525552100278.7528.50.687535552201288.7528.50.687525851000398.7513.51.5625355521002 108.7513.51.5625358521002118.7513.50.68753585210021216.2528.51.56253555210021316.2528.51.56252585100031416.2528.50.68753585100031516.2513.81.5625358521002168.7528.51.56253585210021720211.125307021002185211.1253070210021912.5361.1253070100032012.561.1253070210022112.52123070210022212.5210.253070220132312.5211.1254070210022412.5211.1252070100032512.5211.12530100100032612.5211.1253040210022712.5211.125307021002 Inventory valuesMaintenance costsInventory values


59 The lead-time for part orders in the Kabir and Al-Ola yan policies is stochastic and defined by the Weibull parameters (3.2,10). The pe riod length for the periodically reviewed policies is the mean value of this stochast ic lead-time. This value is the solution to 1 1. b h G+ n For this period length, the lead-time for the per iodic policies was typically one or two periods. The comparison results were obtained by compiling s tocking policy costs for a simulated 1,000 periods which is equivalent to a tim e length of 1 10001. b h G+ n For the studied Weibull parameters, the simulation length wa s: 1.5 – 1491 time units, 2.0 – 1989 time units, and 3 – 2983 time units. A beginning sp an of 15 periods was used to initialize part ages. At the conclusion of this beginning span all costs and inventory orders were reset to zero. The historical demand information b egan with information from the previous ten periods and accumulated information up until the 1000th period. As stated before, the conditional probabilities of failure us ed in the predicted demand forecasting for policies 4 – 9 and 16 – 17 were calculated for 30 k = future periods. In this comparison, the values used for the Normal and Uniform maintenance strategies as discussed in Chapter 3.4 were .99, PC = .025, CR = and .025. UR = Using these parameters for the first run of 1.5, b = the Normal maintenance strategy allows for maintenance scheduling where 99% of the replacement times are normally distributed between 177.1 – 186.2 time units. Likewise, the Unifo rm maintenance strategy allows replacement times to occur uniformly distributed be tween 177.1 and 186.2 time units. If


60 this maintenance window is unrealistic or needs to be widened or reduced, the CR value could be increased or decreased. The results in Appendix D – F show that stocking Poli cy 10 Fixed and 12 Fixed have the best performance against the Kabir and Al-Ola yan policies over the entirety of the runs. Policy 10 Fixed has a maximum average di fference of 8% for 1.5 b = and returns an average cost savings of 1% for 3.0. b = Policy 12 Fixed has a maximum average difference of 6% for 2.0 b = and equals the Kabir and Al-Olayan cost for 3.0. b = In general, Policies 4 – 6, 13 – 16, 18, and 19 p erformed poorly for all maintenance strategies and cost groups. Although no t as cost effective as when coupled with a Fixed maintenance strategy, Policy 10 and 12 show strong performances when a flexible maintenance policy is applied as seen in t he results for Policy 10 Normal/Uniform and Policy 12 Normal/Uniform. Appendix G shows additional cost analysis results for CR/UR values of .05, .075, and .1 with 3.0. b = These additional parameter values increase the sc heduling window up to 10% of the optimal replacement time. A similar comparis on could be used to make an informed decision when altering maintenance policy to allow for a wider scheduling window. In these results, the cost diffe rence between Policy 12 Normal with CR value of .05 is not substantially different than t he same policy with a CR value of .025 although it expands the scheduling window by 100%. Several of the policies developed in this study pe rformed poorly against the Kabir and Al-Olayan policy. Policy 10 and 12 performed well with all maintenance strategies and will be used along with Policy 11 in the following section. A total of five operating parts were used in this stocking policy comparison, which is a relatively small system. A


61 larger system would provide an increased sample size of parts that would better reflect the predicted demand forecasts used to build the pr esented stocking policies. Note: The analysis conducted in this section was firs t presented in [69]. 6.2.2 Stocking Policy with Failure Estimation The system outlined in the following sections bring together the ideas presented throughout this study. The comparison in the previ ous section was completed with the assumption that the part lifetime failure distribut ion is known. Using this same periodic review of the inventory/maintenance system, the esti mation selection method presented in Chapter 5 can be applied to a system where the fa ilure distribution is unknown. For any system matching the description given in Chapte r 1.2, a dynamic inventory/maintenance system can be employed to sel ect the inventory and maintenance policies from one period to the next. System Algorithm The proposed system model is outlined in Figure 4. The system values shown on the right side of the figure should be known prior t o implementing the algorithm. The system cost values such as Cf, Cp, h, p, and K can be updated as they may change over the life of the model. This study does not discuss how these cost values are assigned, but all are commonly used and their values should be av ailable in any existing system. To track the ages of each part, an identifier such as a serial number should be assigned to each part. When Part i is replaced, the replacement part is then assigned the same identifier i


62 The variables on the left side of the figure are ca lled decision variables. These values can be adjusted by a maintenance manager to ensure adequate time is available to perform the maintenance. The maintenance manager s hould then ensure these values accurately reflect the maintenance performance. The following is a further description of each step in the algorithm. Step 0 is carried out only when the model is first applied to a system. Upon the completion of Step 5, the algorithm returns to Step 1 and Step 0 is ne ver visited again. Step 0 : This model was developed under the assumption that t he reliability information for the part is not well known. However, values for the failu re distribution parameters need to be entered to initiate the algorithm. These values can be determined as suggested in Chapter 5.3. Step 1 : If the failure distribution parameters or maintena nce cost values have changed since the previous iteration, the optimal replaceme nt time will be re-calculated. If no changes to the distribution parameters or cost valu es have occurred, the optimal replacement time will not change and this step can b e bypassed. Step 2 : If the optimal replacement time or decision variab les have changed since the previous iteration, the system maintenance policy wi ll be re-calculated. The new optimal replacement time t* becomes the mean or expected time for part mainten ance and the maintenance strategy values are determined as shown in Chapter 3.4. This new maintenance policy should be applied to the current period.


63 Figure 4 System Model Step 0 : Using existing failure information or expert analysis, an initial failure distribution is used to develop the system maintenance policy and predicted demand. Step 1 : Given failure distribution parameters and system cost values Cf and Cp, determine the optimal replacement time t as shown in Chapter 3.4. Cf – failure replacement cost Cp – maintenance replacement cost System Values Decision Variables Step 2 : Given t* and decision variables CR UR, and PC determine the system maintenance policy as shown in Chapter 3.4. Step 3 : Given system values, failure distribution, and maintenance strategy, determine the predicted demand for future periods as described in Chapter 4 n – number of operating parts ti – age of part i h – inventory holding cost per period k p – inventory shortage cost per period k K – inventory ordering cost l – part lead time Step 5 : If a failure or replacement occurred during the previous period, re-estimate the failure distribution parameters using the estimation selection method outline in Chapter 5. n’ – cumulative total number of failures, replacements, and censored data Step 4 : Given predicted demand for future periods, determine the stocking policy as shown in Chapter 6. 2.1. CR – Completion range UR – Uniform Range PC – Probability of Completion


64 Step 3 : If any values from Step 1 – 2 or any system values have changed, the stocking policy values will need to be re-calculated. If all these values have not changed, the conditional failure probabilities and predicted dem and remain the same as calculated during the previous iteration. The age of each par t should be updated and recorded after every iteration. Step 4: If the predicted demand or any system values have c hanged, the inventory policy will need to be re-calculated using the appropriate inventory solution method. The new reorder level and order-up-to amount should be appl ied for the current period. Step 5 : At the conclusion of any period containing a part r eplacement, Step 5 or the reestimation of the failure distribution parameters s hould be completed. The completion of Step 5 then triggers Steps 1 – 4 to be completed at the beginning of the following period. The need to complete Step 5 could also triggered by a significant number of failures. This immediate re-estimation would provide increased model performance prior to the completion of the current period. If a part replac ement occurred during the previous period, either due to failure or maintenance, a new estimate for the distribution parameters will be calculated using the appropriate estimation method for the value of n and the cumulative censored number. Through each r eplacement, the reliability data increases and the distribution parameter estimation improves. The cumulative reliability data, n’ contains the age of a part at each maintenance re placement, the age of a part at each failure, and the age of the censored parts at the time of a part failure. The censored


65 data dilutes the parameter estimation. Therefore, whenever a part is replaced due to maintenance, the censored times that were recorded earlier during that partÂ’s life are removed from nÂ’ Each record in nÂ’ consists of the time of failure or censor and the mode indicator, where 1 indicates a failure and 0 indicat es a censor. Additionally, each record in n Â’ has a subscript i,k where i indicates the part identifier and k indicates the period when the data was collected. The following example is provided to further explain how nÂ’ is updated. At the beginning of period 1, four parts have ages ,1.61= t ,6.22= t,5.53= t and .9.54= t The period length is .25 years. During period 1, Part 3 fails and reliability data is gathered with Parts 1, 2, and 4 being censored. ( ) ( ) ( ) ( ) 1,4 1,3 1,2 1,10, 15.6 1, 75.5 0, 85.2 0, 35.6 = n During period 2, Part 4 fails and reliability data i s gathered with Parts 1, 2, and 3 being censored. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2,4 2,3 2,2 1,4 1,3 1,2 1,11, 40.6 0, 25., 0, 10.3 0, 60.6 0, 15.6 1, 75.5 0, 85.2 0, 35.6 '2,1= n During period 3, Part 1 is replaced through maintena nce and this censored time is included in the reliability data while the censored values for Part 1 at the failure of Part 3 and 4 are discarded. The discarded data was previo usly listed with the subscripts 1,1 and 1,2. ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3,1 2,4 2,3 2,2 1,4 1,3 1,20, 85.6 1, 40.6 0, 25., 0, 10.3 0, 15.6 1, 75.5 0, 85.2 = n


66 Policy Analysis Using the system algorithm described in the previou s section, Policies 10 – 12 coupled with the Fixed, Normal, and Uniform maintenanc e strategies were tested using the inventory and maintenance costs from Run 1 list ed in Table 7. The Matlab code for the analysis completed in this section can be found in Appendix H. For the initial distribution parameters required in Step 0, 11 shap e and scale parameter pairs were used. The accumulated costs using these parameter pairs f or a simulated length of 100 periods are displayed in Table 8 and Table 9. Run 1 of Table 8 is the base run using the system a lgorithm with known failure distribution parameters. In this run, the distribu tion estimation procedure was bypassed Table 8: System Results ShapeScaleP10P11P12P10P11P12P10P11P12 Run 13.01005553.16787.05288.69640.17524.59174.74948 .95121.84837.1 Run 22.51004926.05833.25654.814578.28052.312087.313 101.26212.811201.0 Run 33.51004518.54911.34498.25161.25397.25058.55607 .95543.15546.1 Run 43.0908262.77336.75709.35101.75263.35372.46119. 96619.25590.3 Run 52.5904818.25865.14510.64988.15536.45404.45609. 85490.65726.6 Run 63.5905780.06551.15578.16689.34954.46513.39481. 95393.68327.3 Run 73.01106043.97326.75757.06187.65334.35826.74904 .74846.64646.9 Run 82.51104864.96781.95349.011043.87225.57926.1511 2.75048.94634.0 Run 93.511013429.87613.19408.25848.05490.55582.6549 2.85255.65212.1 6580.56527.45808.27449.75906.76721.46928.85551.3636 0.5 3016.8931.21542.73490.41105.92351.42883.8590.82273. 9 19%-4%10%-23%-21%-27%40%8%31% D (Run 1 Mean)= Fixed Maintenance StrategyNormal Maintenance Strate gy Uniform Maintenance Strategy Step 0 Values Mean Cost (Run 2-9)= Std Dev (Run 2-9)= and the true distribution parameters were used to de velop the predicted demand forecast. In Runs 2 – 9 of Table 8, the Step 0 values were var ied to allow Step 5 of the algorithm to be tested. The Step 0 values represent the initial parameter estimates that could be used when applying this system and were varied 17% for the shape parameter and 10% for


67 the scale parameter in Runs 2 – 9. These values wer e used to represent a well estimated initial distribution. The mean values for all poli cies are shown along with the standard deviation and percent cost difference between the ba se run cost and mean value of Runs 2 – 9. A negative value indicates a cost savings over the base run while a positive value represents a cost increase. From the information displayed in Table 8, Policy 11 Uniform shows excellent performance when an initial estimate for the distrib ution parameters is used. Policy 11 Uniform returns the lowest mean value and standard d eviation for Runs 2 – 9. The low value for standard deviations indicates that the po licy has the most stable performance across varying initial estimates. This policy also returned a low average cost increase as compared to an identical system with known parameters Table 9: Poorly Estimated Parameters ShapeScaleP10P11P12P10P11P12P10P11P12 Run 103.05011192.68556.58810.314756.110215.310364.7 14852.19977.510970.9 Run 113.01505256.16286.45084.04398.64675.24536.1467 4.04718.84674.7 Fixed Maintenance StrategyNormal Maintenance Strate gy Uniform Maintenance Strategy Step 0 Values The run scenarios in Table 9 began with a poorly es timated initial scale parameter. These cost figures show that a conservat ive estimate of the scale paramter returns inflated costs. In this run, many componen ts were replaced due to maintenance which led to a highly-censored distribution estimati on. The operational lifetime of the parts were shortened and frequent preventive replace ments increased the inventory and maintenance costs which led to a high total cost. W hereas in Run 11, the high scale value allowed more early period failures and subsequently an improved distribution estimation.


68 Appendix I shows the distribution parameters and opti mal replacement times as estimated for each period of Run 10. Likewise, Appendix J shows these values for Run 11. An examination of these Appendices reveals that the ove r-estimated scale value in Run 11 allows the estimation procedure to quickly close in on the actual parameter values. This comparison lends evidence to suggest that an over-e stimated scale parameter may be an effective strategy when the distribution parameters are unknown.


69 CHAPTER 7 CONCLUSION AND FUTURE RESEARCH 7.1 Conclusion This research effort first presented ideas and then proposed a system model to further expand the body of work in combined or joint ly modeled inventory/maintenance systems. Several unique concepts were introduced an d applied to build this system model, including using the lifetime failure charact eristics of parts to develop a predicted demand forecast, maintenance strategies that allow f lexible scheduling, and an estimation selection method for highly-censored Weibull data. Several stocking policies were tested against existi ng policies presented in a study by Kabir and Al-Olayan. This testing showed that the p redicted demand over a lead-time coupled with a flexible maintenance strategy provide s adequate results as highlighted in Chapter 6.2.1. This comparison adds to the limited amount of existing studies dealing with multiple components subject to differing replac ement times. As indicated earlier, most stocking policy studies examining multiple com ponents limit the replacement strategy to a block replacement type. Applying these stocking policies to a part system wit h unknown failure distribution parameters, the estimation selection m ethod described in Chapter 5 allowed this dynamic system to produce inventory and mainte nance guidance. Section presented an algorithm that can be applied to any s ystem matching the description


70 outlined in Section 1.2. The proposed system algor ithm performed well in several scenarios that began with limited or non-existent pa rt reliability data. The best results were obtained from Policy 11 Uniform as shown in Tabl e 8. This policy returned the most stable performance and lowest costs for a stock ing policy that allows for flexible maintenance scheduling. Flexible maintenance strategies are used to provide a scheduling window for part replacements. This idea is not often used in liter ature but is almost always used in application. Small expansions of this scheduling wi ndow are shown in Appendix G to have limited effect on costs. 7.2 Research Extensions Additional complexity could be included to further t he applicability of this study. Such complexity could include adding to and refinin g the system cost values. For example, this study utilized a single variable to a ccount for the cost of a part failure. This single variable could be broken down into sub-costs including, but not limited to a cost to account for the operational time lost, cost to dete rmine the cause of failure, and emergency travel costs of maintenance personnel. Ot her system costs, such as the inventory holding costs, could also be expanded. F urther analysis with multiple replications of the proposed algorithm could be com pleted using these additional costs to build a sensitivity analysis. A supplementary extension to this research would inc lude adding increased complexity to the stocking policy. A more complex p olicy would include spares that are new and/or comprised of previously used parts that h ave been overhauled. The inventory


71 stock would then be replenished by orders placed and by parts returned from overhaul. The failure characteristic of these overhauled part s may or may not then vary from that of a new one. This system utilized a rather simple policy where pa rts either fail or are replaced prior to failure. A major extension to this researc h would include the introduction of Condition Based Monitoring (CBM), Predictive Mainte nance, and Reliability Centered Maintenance techniques into the system model. Thes e techniques are widely used in industry and their inclusion would provide even furt her applicability. The idea of imperfect maintenance would also increase the comple xity of this model. Imperfect maintenance is any maintenance action that is perfo rmed on the part, whether it is the action of replacing the part or minor preventive ma intenance in between replacements, which causes damage or a condition that accelerates failure. Table 9 suggests that the performance of the system algorithm is excellent when an over-estimated scale parameter is used for the i nitial parameter value. This overestimation would normally allow a larger amount of e arly failures. These failures however provide early reliability data to determine better parameter estimates. Additional testing of this or other initial estimati on strategies would also be beneficial to this study. If the over-estimation strategy is sho wn to be an effective option, failure costs and a increased demand on inventory could be antici pated and planned for. A few ideas discussed in this study could have some a pplication in medical studies. Organ transplant surgeries can be viewed as part replacements, where the transplanted organ takes on the role of a “spare pa rt”. The system algorithm could be applied to a medical system of n patients that have failing parts or organs that pl ace a


72 demand on the inventory stock. Such a system is si milar to the one in this study in that there are multiple operating components that have v arying replacement times. Finally, a case study using existing data or real-t ime data should be completed to validate this system model. Analysis of real-time d ata would allow the model to showcase the dynamic characteristics, specifically t he ability to allow flexible maintenance scheduling. The data obtained from the actual replacement times could then be used to determine if the modeled maintenance str ategy accurately reflects the observed replacement times.


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81 Appendix A: Initial Estimation Selection Analysis MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.07500.10230.07160.07890.09680.06130.07300.04840.0 6460.06800.03780.04650.03060.04510.0428 St Dev 0.12650.13430.09980.11820.12810.08720.08440.06200.0 7660.08090.05350.04950.03690.04920.0488 Rank 351241412314132 .8-.9Mean 0.19840.08610.07090.09300.08810.13990.06390.04490.1 0290.06600.19600.08460.07040.09010.0865 St Dev 0.18620.10960.07560.12980.11500.16540.09230.05890.1 1630.09610.18510.10930.07470.12400.1144 Rank 321214213231121 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.06270.07960.05990.05860.07440.05050.05890.03940.0 5030.05400.03260.03970.02580.03700.0354 St Dev 0.09220.09940.07140.07580.09400.07540.07430.05050.0 6410.07000.04530.04580.03190.04310.0424 Rank 352141412314132 .8-.9Mean 0.16200.07290.07010.06790.07460.10680.05090.03960.0 7380.05290.07310.03480.02540.08050.0367 St Dev 0.16960.09520.06290.08760.10000.12980.07730.04970.0 8160.08130.08270.04830.03070.08290.0518 Rank 322123112132142 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.20550.25640.34530.17620.23040.06650.08450.10350.0 6780.07550.03220.04960.03660.04850.0429 St Dev 0.29900.34380.29200.25720.32630.07980.09310.09170.0 8360.08630.04310.04860.04200.05100.0441 Rank 245131452315243 .7-.8Mean 0.14230.17210.44460.10250.14690.04510.05890.12080.0 5100.05030.03870.04960.03750.04640.0422 St Dev 0.22770.26800.28810.17100.24700.06370.07250.08780.0 7160.06460.04640.04930.04020.05040.0433 Rank 245131342125143 .8-.9Mean 0.08410.09500.56560.08350.08070.04190.03070.13130.1 1400.02880.10280.05030.04410.10180.0536 St Dev 0.21720.18830.24370.15340.20030.05860.05230.07460.1 5130.05560.10660.06290.04280.10070.0688 Rank 345213254142132 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.03840.05390.04270.04310.05050.02990.04600.03540.0 4160.04100.02920.03980.02560.04130.0341 St Dev 0.05270.05850.04750.05090.05710.03960.04450.03990.0 4570.04230.03820.04090.02980.04540.0371 Rank 152341524312132 .7-.8Mean 0.04240.05210.04810.03980.04760.03350.04520.03700.0 3970.03940.02900.03760.02560.03720.0318 St Dev 0.05360.05750.04780.04740.05420.04030.04430.03980.0 4390.04050.04100.04340.02920.04520.0389 Rank 243131524334152 .8-.9Mean 0.04500.04850.06190.03520.04430.03970.04580.04390.0 3710.03990.03610.04380.03630.04020.0366 St Dev 0.06690.06720.05390.05130.06320.05160.05270.04170.0 4440.04770.04540.04780.03660.04580.0419 Rank 134122541335142 .9-1Mean 0.07200.03760.07930.02970.03820.07660.03980.05640.0 4850.04120.07620.03920.04290.06910.0417 St Dev 0.10140.06510.05460.04290.06720.09380.05920.04670.0 5030.06250.08600.05370.03830.06590.0582 Rank 532144143242132 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.11000.15060.31490.10570.12550.03880.05160.08700.0 4030.04360.02040.03280.02180.04130.0263 St Dev 0.14700.16550.23210.16860.16380.04070.04530.06820.0 5610.04350.02310.02740.02450.04090.0242 Rank 135421452314253 .7-.8Mean 0.08720.12160.38340.08060.09790.02990.04130.09810.0 3250.03440.02380.03390.02710.03850.0269 St Dev 0.13270.14790.24430.14630.14420.03520.04010.06730.0 4860.03750.02840.03310.02800.04220.0281 Rank 245131452314352 .8-.9Mean 0.04550.06220.54120.03140.04580.01500.02150.11820.0 1730.01760.02800.03630.03040.03560.0288 St Dev 0.07410.08470.21130.07060.07840.02200.02510.05630.0 2440.02190.03790.04200.02990.04380.0358 Rank 254131452315342 .9-1Mean 0.02290.03210.59640.03010.02090.02520.01250.11830.0 3320.01250.04350.02480.04360.03300.0260 St Dev 0.05930.05300.15450.04980.05350.02230.01640.05590.0 3780.01850.05610.03850.03360.03030.0413 Rank 245313154241532 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.02320.03200.02940.03190.02830.01650.02260.01560.0 2650.01870.01100.01620.00990.02280.0126 St Dev 0.02760.02980.03080.03310.02890.02290.02500.01760.0 3090.02220.01430.01650.01140.02390.0139 Rank 153422415313142 .7-.8Mean 0.02460.03090.03880.02880.02670.01490.01980.01690.0 2160.01610.01060.01520.01060.01980.0116 St Dev 0.03440.03640.03590.03650.03330.02150.02350.01740.0 2690.02050.01510.01740.01190.02310.0144 Rank 145321434224153 .8-.9Mean 0.02100.02480.05470.02090.02150.01300.01660.02100.0 1590.01340.00970.01320.01260.01460.0101 St Dev 0.03570.03670.04230.03340.03400.02010.02200.01900.0 2220.01910.01480.01700.01210.01960.0142 Rank 245131453214332 .9-1Mean 0.01530.01560.07980.01100.01390.01040.01100.03030.0 0840.00930.00940.01000.01650.00840.0083 St Dev 0.02940.02860.04430.02250.02700.01750.01700.02100.0 1370.01570.01540.01470.01320.01280.0135 Rank 435122341112311 n=20 Beta=2 Beta=3 Beta=4 n=15 Beta=2 Beta=3 Beta=4 n=10 Beta=2 Beta=3 Beta=4 n=8 Beta=2 Beta=3 Beta=4 n=6 Beta=2 Beta=3 Beta=4 n=5 Beta=2 Beta=3 Beta=4


82 Appendix A (continued): Initial Estimation Selectio n Analysis MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0206470.0283070.026755040.03024480.02442020.01357 30.01912030.013257380.02478110.01538080.0091820.014 02450.00847830.0223890.010482 St Dev 0.0244420.0263880.026785050.03093160.02467250.01829 10.02013260.014413550.02683330.01769220.0118330.014 13330.00971150.02288920.0115547 Rank 143422415313142 .7-.8Mean 0.0199580.0251380.039018610.02438560.02126930.01157 80.01594490.015709910.01851290.01251950.0084770.012 47040.00932680.01768440.0091983 St Dev 0.0297290.0315790.033387190.03258970.02828260.01699 10.01907970.015552290.02272240.01637180.0116970.013 89820.00970740.01985660.0112829 Rank 145321334214352 .8-.9Mean 0.0164540.0200110.052383970.01783050.01694480.00996 90.01319020.019998540.01356950.01037480.0078930.011 18230.01142130.01339280.0082419 St Dev 0.0277270.0290330.038559940.02798820.02649540.01525 40.01682020.017102520.01820210.01449770.0121620.014 26650.01082320.01733160.011704 Rank 134221354213442 .9-1Mean 0.0097120.009970.084915850.00687910.0086420.0072520 .00772270.032156220.00574870.00642690.0064190.00700 950.01751090.00580130.0056904 St Dev 0.0192660.0188840.038523550.01524850.01763550.01282 50.0124160.019261520.01028750.01144130.0106160.0106 4860.01224910.00943590.0094779 Rank 452131231112311 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0180020.0249480.024704020.0281730.0211110.010910. 01602730.011412310.02252530.01244620.0077950.012157 90.00735060.02179360.0089015 St Dev 0.0217580.0237820.024166760.02936330.02211040.01465 90.01669120.012371870.02354690.01432040.0097640.011 78650.00807380.02093290.0095907 Rank 143521425324153 .7-.8Mean 0.0173560.0226280.034107050.02353340.01874670.01025 80.01455460.013613850.0186420.01112770.0075150.0115 6340.00801380.01816620.0081658 St Dev 0.0258380.0276850.02939260.03042670.02472570.014738 0.0167040.013475010.02163370.01427520.0100130.01237 370.0082530.01867670.0096457 Rank 125421435213242 .8-.9Mean 0.0129370.0162130.051069910.01504060.01338970.00830 60.01139180.018996580.01246790.00868060.0065320.009 58040.01050490.01262380.0068649 St Dev 0.021670.0229590.035464450.02289610.02070660.013217 0.01482970.015402210.01701390.01266670.0098670.0118 6650.00951830.01535630.0095428 Rank 145321343213442 .9-1Mean 0.0081420.0095960.077610050.00717450.00798290.00576 80.00727940.02891690.00618240.00569960.0052030.0068 4420.01538150.00656770.0051157 St Dev 0.0155140.0161550.037042890.01391170.01477020.01008 0.01086910.017667350.01051150.00966290.0086110.0094 6580.01089310.00968910.0081654 Rank 345122453124531 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0228370.0321040.024787150.03136240.03089160.00948 20.01472360.010771640.01702440.01392070.0063440.009 69080.00713720.01287050.0091548 St Dev 0.0263890.02770.026012860.02861750.02783340.0101330 .01105050.010336940.01263230.01111740.0072770.00777 920.00731240.01064150.0077901 Rank 132331425314253 .7-.8Mean 0.0222430.0314760.023988460.03112340.03013420.00979 50.0150870.011146740.01915580.01402210.0058390.0090 4220.00664110.01290290.0083432 St Dev 0.0241590.0260540.024322630.02806820.02617440.01121 80.01233070.01141620.01600680.01213540.0070070.0072 1830.00682090.00983810.0072092 Rank 132331425314253 .8-.9Mean 0.0232770.0319270.024839570.03209810.03034930.00977 50.01490560.010757720.01950170.01372550.0060270.009 30440.00675890.01496450.0085018 St Dev 0.0298640.0299710.027900520.0324790.03010020.011194 0.0119840.011316780.01513980.01210020.0072870.00772 460.00719920.01172750.0076638 Rank 132331425314253 .9-1Mean 0.0218490.0306410.024227340.03100860.02892050.00968 10.014890.010522190.02036690.01360870.0062730.00971 360.00709780.01639270.008719 St Dev 0.0251950.0269590.025391050.02969650.02717260.01207 20.01300980.011735610.01754720.01296960.0068070.007 45010.00702960.01256880.0072808 Rank 132331425314253 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0192580.0280940.021402670.02747130.02710710.00855 10.01317440.009868240.0156870.01250970.0051260.0079 8260.00597950.0107810.007563 St Dev 0.0212290.0236150.021964110.02447460.02356430.01012 60.01094270.010284210.01262350.01097440.00580.00632 50.00598590.00784570.0063342 Rank 132331425314253 .7-.8Mean 0.0201610.0287870.02198450.02848390.02762730.008650 .01317660.009764080.0161020.01234690.0050980.007947 30.00582240.01164330.0074149 St Dev 0.0213880.0224850.021213110.02444480.02272060.00980 80.01043260.009945850.01262070.01033940.0057150.006 01340.0057630.00869480.0060184 Rank 132331425314253 .8-.9Mean 0.0193790.0278750.021401260.02840520.02652310.00877 70.01350630.009969020.01738660.01250970.0050890.007 99470.0058570.01272240.0073839 St Dev 0.0204290.021540.020358060.02398940.02172420.010448 0.01125480.010528330.01359570.0113240.0055710.00604 830.00565380.00937920.0060072 Rank 132331425314253 .9-1Mean 0.0195810.0273550.021377640.02794950.02587590.00857 90.01311590.009572330.01791180.01199290.0052590.008 21480.00602380.01423340.0074578 St Dev 0.0228180.0237540.022357480.02693760.02391250.01017 90.01099290.010265180.01449260.01088810.0059630.006 42710.00616260.01077120.0063658 Rank 132331425314253 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0159430.0238550.01849060.02409080.02317220.006611 0.01023130.007850670.01230450.00982980.004290.00662 530.00512980.00898770.0063263 St Dev 0.0182090.0201260.019224530.02109330.02022910.00740 30.00794870.007619080.00935030.00801830.0047670.005 12670.00496470.00640980.0051475 Rank 132331425314253 .7-.8Mean 0.0171430.0247590.0193370.02521610.0238650.0072660. 0110510.008402770.013820.01048110.0042410.00664470. 00506740.00994030.0062914 St Dev 0.0189130.0198610.018825680.02073720.01986130.00812 20.00850980.008158150.0101120.00856190.0047570.0052 2240.00494550.00712440.0052205 Rank 132331425314253 .8-.9Mean 0.0165660.0238860.018443790.02443930.02294270.00735 50.01120190.008433340.01454530.01053580.0042980.006 61050.00501830.01075530.0061819 n=50 Beta=2 Beta=3 Beta=4 n=40 Beta=2 Beta=3 Beta=4 n=35 Beta=2 Beta=3 Beta=4 n=30 Beta=2 Beta=3 Beta=4 n=25 Beta=2 Beta=3 Beta=4


83 Appendix A (continued): Initial Estimation Selectio n Analysis MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0143120.0205010.016271390.02063350.01999690.00575 80.00886040.006888830.01047180.00858690.0035240.005 4440.00428660.00730290.0052298 St Dev 0.0170850.0180810.017419730.01812740.01814910.00691 40.00730660.007017810.00797020.00734970.0038160.004 10230.00403810.00495670.0041096 Rank 132331324313243 .7-.8Mean 0.0139930.0205730.016120160.02124130.0198980.005820 .00904030.007062860.01151080.00865530.0038530.00572 650.00451390.00829150.0054656 St Dev 0.0157970.0167640.01621530.01777130.0167950.0064180 .00711730.006885820.00862580.00713830.0047190.00484 80.00470310.00605960.0048544 Rank 132331425314253 .8-.9Mean 0.0144420.021160.016961920.02259810.02052320.005785 0.00891670.00685960.01191860.00843840.0037460.00574 260.00445270.00900560.0054614 St Dev 0.0167790.018480.017769150.02003190.01853860.006278 0.00687670.006529230.00851380.00687150.0044230.0047 2490.004570.0066990.0047533 Rank 132331425314253 .9-1Mean 0.0143490.0206360.016298280.02223020.01980070.01408 20.02033790.016220960.02211480.01953110.0033890.005 42260.00411160.00975150.0050452 St Dev 0.0163160.0168640.01625390.01862930.01688960.015685 0.01688890.016200090.01896690.01692140.0038470.0042 1120.0039710.00659680.0042137 Rank 132331424314253 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0100320.0150580.012228350.01564270.0147370.004403 0.00686990.005513070.00812970.00668910.0025670.0039 6050.00317660.0052710.0038463 St Dev 0.0114390.0123190.012088070.01258730.01235870.00489 40.0052630.005139640.00566520.00526940.0028870.0030 9470.00299870.00369510.0031046 Rank 132331324313243 .7-.8Mean 0.0101630.0152830.01215910.01602440.01485860.004122 0.00654810.005142510.00817670.00632960.0026680.0041 1720.00329270.00586820.0039789 St Dev 0.011290.0121860.011670960.01272590.01220040.004997 0.00534360.005187640.00594830.00534820.0028620.0031 0710.00298680.00386190.0031053 Rank 132331324312345 .8-.9Mean 0.0107840.01570.012569250.01698870.01519980.0044510 .0067820.005344350.00904210.00651520.0106930.015644 40.01269360.01710610.0152082 St Dev 0.0119910.0128250.012318570.01360170.01282250.00537 40.00597770.005736380.00691680.00599490.0118020.012 60760.0123140.01371340.012658 Rank 132431425313243 .9-1Mean 0.0102630.0152210.012066880.01680570.01468110.01022 80.01533420.012288740.01705490.01483320.0106140.015 75880.01260840.01759550.0151984 St Dev 0.0115850.012270.011718260.01351680.01228730.011192 0.01181380.01147150.01257820.01188790.0128650.01359 530.01304410.01463470.0135931 Rank 132431324313243 MLEiseKMEiseFRWEisePEXEiseKLMiseMLEiseKMEiseFRWEise PEXEiseKLMiseMLEiseKMEiseFRWEisePEXEiseKLMise .6-.7Mean 0.0080920.0123790.010289290.01288440.0121560.003416 0.00540070.004372070.00631670.00528060.0021270.0032 9580.00272970.004310.0032151 St Dev 0.0091110.010260.009984850.01047410.01027080.004345 0.00455430.004452560.00469680.0045550.0026450.00273 250.00273730.00319310.0027325 Rank 132331324313243 .7-.8Mean 0.0088170.013080.01071340.01389920.01281550.0087740 .01293050.010719880.01381710.01264710.002080.003273 90.00272210.00470820.0031781 St Dev 0.0102990.0110370.010899480.01159580.01106580.01083 70.01115060.011112680.01157160.01116530.002280.0024 8440.00240160.00295160.0024936 Rank 132431324313243 .8-.9Mean 0.0085170.0128880.010638330.01403480.0125740.008814 0.01292520.010662860.01408590.01257870.0084210.0125 6610.01017260.01354740.0122496 St Dev 0.0094330.0103530.009915190.01094110.0103740.010935 0.01120660.011139560.01197570.01121730.0096640.0106 10.0104290.01092840.0106093 Rank 132431324313243 n=100 Beta=2 Beta=3 Beta=4 n=80 Beta=2 Beta=3 Beta=4 n=60 Beta=2 Beta=3 Beta=4


84 Appendix B: Weibull Parameter Database ComponentLowTypicalHighLow TypicalHighBall bearing0.71.33.514,00040,000250,000Roller bearings0.71.33.59,00050,000125,000Sleeve bearing0.71310,00050,000143,000Belts, drive0.51.22.89,00030,00091,000Bellows, hydraulic0.51.3314,00050,000100,000Bolts0.5310125,000300,000100,000,000Clutches, friction0.51.4367,000100,000500,000Clutches, magnetic0.811.6100,000150,000333,000Couplings0.82625,00075,000333,000Couplings, gear0.82.5425,00075,0001,250,000Cylinders, hydraulic123.89,000,000900,000200,000,00 0 Diaphragm, metal0.53650,00065,000500,000Diaphragm, rubber0.51.11.450,00060,000300,000Gaskets, hydraulics0.51.11.4700,00075,0003,300,000Filter, oil0.51.11.420,00025,000125,000Gears0.52633,00075,000500,000Impellers, pumps0.52.56125,000150,0001,400,000Joints, mechanical0.51.261,400,000150,00010,000,000Knife edges, fulcrum0.5161,700,0002,000,00016,700,0 00 Liner, recip. comp. cyl.0.51.8320,00050,000300,000Nuts0.51.11.414,00050,000500,000"O"-rings, elastomeric0.51.11.45,00020,00033,000Packings, recip. comp. rod0.51.11.45,00020,00033,00 0 Pins0.51.4517,00050,000170,000Pivots0.51.45300,000400,0001,400,000Pistons, engines0.51.4320,00075,000170,000Pumps, lubricators0.51.11.413,00050,000125,000Seals, mechanical0.81.443,00025,00050,000Shafts, cent. pumps0.81.2350,00050,000300,000Springs0.51.1314,00025,0005,000,000Vibration mounts0.51.12.217,00050,000200,000Wear rings, cent. pumps0.51.1410,00050,00090,000Valves, recip comp.0.51.443,00040,00080,000Machinery EquipmentCircuit breakers0.51.5367,000100,0001,400,000Compressors, centrifugal0.51.9320,00060,000120,000Compressor blades0.52.53400,000800,0001,500,000Compressor vanes0.534500,0001,000,0002,000,000Diaphgram couplings0.524125,000300,000600,000Gas turb. comp. blades/vanes,000250,000300,000 Gas turb. blades/vanes0.91.62.710,000125,000160,000Motors, AC0.51.231,000100,000200,000Motors, DC0.51.2310050,000100,000 Weibull ShapeWeibull Scale Eta (hours)






87 Appendix D: Kabir and Al-Olayan Comparison for b bb b=1.5 P1 FixedP1 NormalP1 UniformP2 FixedP2 NormalP2 Unif ormP3 FixedP3 NormalP3 Uniform Run t* DDD sS DDD sS DDD sS 1181.713%13%16%1.3-2.44.8-6.75%6%9%0.7-1.24.3-5.810 %7%10%1.5-2.64.3-6.1 2181.7-5%-5%-9%0.7-1.84.2-6.8-7%-4%-9%0.4-0.94.0-6. 2-11%-10%-13%0.7-1.83.6-6.0 3181.735%37%35%1.1-1.93.6-5.124%34%30%0.5-1.33.0-4. 834%38%34%1.5-2.13.3-4.8 4181.728%25%25%1.8-2.36.4-7.521%18%15%1.1-1.65.8-7. 222%20%20%1.7-2.25.7-6.8 5154.67%6%6%0.4-1.63.9-6.91%1%0%0.3-0.93.8-6.54%4%5 %0.6-2.13.5-6.7 6181.737%39%38%1.8-2.83.9-5.222%27%26%1.1-1.53.3-4. 317%20%21%1.6-2.23.0-3.9 7412.416%17%16%1.2-3.04.4-7.111%12%11%0.6-1.24.1-5. 610%9%9%1.0-2.73.6-5.9 899.81%4%3%1.7-2.85.1-7.0-7%-5%-4%0.8-1.24.5-6.0-3% 0%-1%1.5-2.54.3-5.9 9412.445%44%43%1.5-2.53.6-5.031%28%25%1.0-1.53.3-4. 236%33%32%1.6-2.43.2-4.2 10154.633%30%30%0.8-1.62.8-4.322%24%20%0.5-1.02.7-4 .032%28%28%1.2-2.02.6-4.0 11154.618%22%16%2.2-3.55.5-7.510%17%8%1.2-1.84.7-6. 113%18%10%1.9-2.94.6-6.1 12412.422%25%23%1.2-2.63.6-5.98%9%12%0.8-1.53.3-5.0 11%12%14%1.1-2.23.0-4.8 1399.829%31%28%3.1-7.16.6-11.311%12%11%1.7-2.65.4-7 .212%16%10%2.6-5.35.4-8.3 14154.61%2%1%1.6-2.46.4-8.3-4%-1%-2%0.8-1.25.8-7.50 %1%0%1.6-2.35.7-7.4 15154.618%21%19%0.6-1.33.5-4.816%21%17%0.5-0.83.6-4 .523%26%23%1.0-2.03.4-4.9 16154.621%18%20%1.2-1.93.5-4.613%9%11%0.7-1.03.2-4. 012%10%13%1.2-1.92.9-3.8 17164.627%23%21%0.9-2.34.6-7.417%16%13%0.6-1.24.4-6 .525%24%20%1.2-3.04.3-7.2 18164.618%21%23%1.3-3.02.8-5.17%13%15%0.7-1.52.4-4. 16%11%15%1.2-2.22.1-3.7 19164.69%7%9%2.3-4.15.3-7.6-2%-4%-2%1.2-1.84.3-5.74%-7%-4%1.8-2.94.0-5.6 20164.620%21%18%0.6-1.73.1-5.317%18%14%0.2-1.02.8-4 .933%31%31%1.2-2.63.2-5.6 21164.618%18%20%0.6-1.32.3-3.88%12%15%0.3-0.82.0-3. 517%17%20%1.0-1.72.1-3.5 22164.69%11%10%2.4-4.28.8-12.14%5%7%1.2-1.97.9-10.1 3%5%7%1.6-3.37.3-10.2 23296.821%22%25%1.2-2.93.4-6.515%12%17%0.6-1.62.8-5 .415%13%18%1.2-2.52.7-5.3 2499.723%27%26%2.1-3.15.2-6.812%11%13%1.2-1.84.5-5. 912%16%13%1.8-2.64.3-5.7 25102.116%21%20%1.2-2.04.5-5.712%15%13%0.7-1.04.1-5 .010%14%15%1.3-2.24.0-5.1 26872.611%12%11%1.1-1.74.2-5.23%10%4%0.7-0.94.0-4.6 10%11%10%1.2-1.93.7-4.7 27164.626%29%25%1.7-2.84.8-6.418%20%17%1.1-1.64.4-5 .519%21%16%1.5-2.44.1-5.3 19%20%19%11%12%11%14%14%14% P4 FixedP4 NormalP4 UniformP5 FixedP5 NormalP5 Unif ormP6 FixedP6 UniformP6 NormalP7 Normal Run DDD sS DDD sS DDD sS D sS 117%16%41%1.3-1.95.6-6.712%11%31%0.8-1.15.4-6.311%3 9%13%1.0-1.45.4-6.312%1.3-2.15.3-6.6 20%0%27%1.4-2.07.2-8.7-3%0%21%0.8-1.17.0-8.3-5%16%7%0.8-1.16.1-7.3-1%1.4-2.16.9-8.4 359%61%80%2.0-2.95.8-7.348%54%62%1.7-2.25.8-7.046%7 1%54%1.9-2.65.3-6.648%1.7-3.25.4-7.4 437%34%49%2.0-2.97.7-9.632%31%44%1.6-2.27.8-9.328%3 7%24%1.5-2.16.8-8.429%1.9-3.07.5-9.2 511%12%23%1.3-1.87.0-8.59%9%17%0.8-1.17.0-8.24%13%5 %0.7-1.06.0-7.210%1.2-2.16.7-8.5 638%42%96%2.2-3.05.0-6.228%41%78%1.7-2.24.7-5.728%1 00%36%2.2-2.94.4-5.445%2.1-4.24.8-7.2 721%22%66%1.5-2.15.7-7.117%21%56%0.8-1.25.4-6.613%7 1%18%1.1-1.64.8-6.024%1.4-2.15.6-7.2 8-6%0%38%1.4-1.85.6-6.50%-2%27%0.8-1.05.4-6.0-8%41% -4%1.1-1.34.7-5.43%1.7-3.85.9-8.6 955%55%79%2.0-3.04.8-6.345%44%65%1.6-2.34.7-5.847%7 9%47%2.1-2.94.4-5.655%2.0-3.24.8-6.6 1039%39%61%1.3-1.84.1-5.032%31%46%0.8-1.13.9-4.532% 60%30%1.3-1.73.6-4.337%1.1-2.53.8-5.6 1120%25%34%2.1-2.86.4-7.616%21%27%1.7-2.16.2-7.315% 33%20%1.9-2.45.6-6.723%2.0-5.16.1-9.9 1234%37%71%2.2-3.26.0-7.826%28%59%1.7-2.35.8-7.325% 69%27%2.0-2.85.3-6.835%2.1-3.25.9-7.7 1319%21%52%2.1-2.75.9-7.012%17%35%1.6-2.05.7-6.69%5 0%12%1.9-2.45.2-6.129%2.4-7.26.2-11.8 142%6%29%1.4-1.97.2-8.6-1%3%22%0.8-1.17.0-8.2-2%18% 1%0.7-1.16.1-7.24%1.4-2.57.0-8.8 1532%34%50%1.2-1.75.0-6.126%27%37%0.8-1.14.9-5.723% 46%28%1.0-1.44.4-5.330%1.0-1.74.6-5.9 1621%20%65%1.4-2.04.2-5.214%19%48%0.8-1.13.9-4.613% 67%12%1.4-1.83.5-4.321%1.4-2.44.1-5.4 1730%31%47%1.3-1.86.3-7.625%22%38%0.8-1.16.2-7.219% 39%19%0.9-1.25.4-6.527%1.2-2.06.0-7.4 1821%27%73%2.2-2.94.7-5.813%20%57%1.7-2.24.4-5.315% 76%19%2.3-2.94.1-5.025%2.1-4.74.5-7.5 193%2%44%2.2-3.06.2-7.53%0%33%1.7-2.26.0-7.1-1%47%4%2.0-2.65.4-6.55%2.2-6.46.0-11.0 2040%47%52%1.8-2.75.8-7.336%37%44%1.6-2.25.9-7.132% 43%39%1.7-2.45.5-6.729%1.4-2.15.2-6.4 2129%33%67%1.3-1.94.3-5.321%28%50%0.8-1.14.0-4.825% 68%31%1.3-1.73.7-4.632%1.1-1.93.9-5.1 229%11%31%2.2-3.110.6-13.08%11%28%1.6-2.210.6-12.84 %26%7%1.0-1.59.0-11.110%2.2-4.510.3-13.8 2330%32%59%2.2-3.16.2-7.825%25%51%1.7-2.36.0-7.423% 58%25%2.0-2.75.4-6.829%2.1-3.46.1-7.8 2419%22%56%2.1-2.76.1-7.017%21%41%1.7-2.05.9-6.711% 54%16%1.9-2.35.4-6.234%2.5-7.26.5-11.9 2517%21%49%1.4-1.75.3-6.113%19%38%0.8-1.05.1-5.79%4 8%14%1.1-1.44.5-5.223%1.4-3.25.3-7.7 2622%25%60%1.4-2.25.3-7.116%21%42%0.8-1.25.4-6.512% 56%16%1.1-1.74.6-6.023%1.3-2.05.3-6.7 2732%33%57%2.2-2.96.1-7.424%29%46%1.7-2.26.0-7.023% 57%29%2.0-2.55.4-6.533%2.0-3.75.8-8.1 24%26%54% 19%22%42%17%51%19% 25% Policy (P) Avg Difference=


88 Appendix D (continued): Kabir and Al-Olayan Compari son for b bb b=1.5 P7 UniformP8 NormalP8 UniformP9 NormalP9 Uniform Run t* D sS D sS D sS D sS D sS 1181.715%0.8-2.64.1-7.411%0.8-1.25.3-6.612%0.6-1.44 .1-6.88%1.0-2.04.6-6.012%0.7-2.73.6-6.8 2181.7-3%1.0-3.65.8-10.5-1%0.8-1.26.8-8.0-4%0.6-1.7 5.8-9.3-6%0.8-1.65.9-7.3-7%0.6-3.64.9-9.6 3181.752%1.2-3.74.4-8.245%1.6-2.25.5-6.945%1.2-2.54 .6-7.547%1.8-3.35.1-6.948%1.3-3.84.1-7.6 4181.730%1.2-4.05.8-10.727%1.6-2.17.6-8.927%1.0-2.7 5.9-10.220%1.4-2.76.6-8.321%0.9-3.85.1-9.7 5154.69%0.7-2.05.2-8.48%0.8-1.36.7-8.28%0.5- 6181.741%1.6-4.03.9-7.134%1.6-2.44.6-6.133%1.2-2.53 .7-6.233%2.1-3.34.2-5.631%1.6-3.33.3-5.7 7412.423%1.5-2.25.8-7.319%0.8-1.25.4-6.718%0.9-1.35 .5-6.915%1.1-1.64.7-6.014%1.1-1.74.8-6.1 899.8-2%0.4-3.42.7-8.2-3%1.0-1.65.7-7.2-3%0.2-1.62. 7-7.0-3%1.3-3.45.0-7.2-5%0.4-3.02.3-6.9 9412.454%2.1-3.24.9-6.644%1.6-2.64.6-6.446%1.7-2.64 .7-6.346%2.1-3.34.3-6.046%2.2-3.24.4-6.0 10154.635%0.6-2.02.8-5.030%0.8-1.43.7-4.927%0.5-1.2 2.8-4.729%1.3-2.93.4-5.329%0.8-2.32.5-4.7 11154.614%0.9-4.03.7-8.620%1.6-2.76.0-8.010%0.8-2.4 3.8-7.518%1.8-4.25.4-8.28%0.9-3.43.3-7.2 12412.436%2.1-3.55.9-8.127%1.7-2.45.8-7.430%1.6-2.6 5.7-7.824%1.9-2.95.2-6.826%1.9-3.15.2-7.1 1399.817%0.9-7.33.4-11.821%1.8-3.26.1-8.512%0.7-3.0 3.4-8.218%2.1-5.65.4-9.111%1.0-5.53.0-9.0 14154.61%0.8-2.55.1-8.74%0.9-1.36.8-8.22%0.5-1.35.1 -8.11%0.8-2.35.9-7.90%0.5-2.44.4-7.8 15154.625%0.5-1.73.3-5.928%0.8-1.24.7-5.824%0.4-1.2 3.4-5.826%1.0-2.34.2-5.924%0.7-2.23.1-5.9 16154.620%0.9-2.63.1-5.713%0.9-1.33.8-4.715%0.6-1.4 3.0-5.012%1.4-2.23.5-4.514%1.0-2.42.6-4.8 17164.625%0.8-3.64.9-9.822%0.8-1.26.0-7.124%0.6-1.8 4.9-8.417%0.9-2.15.2-6.920%0.7-4.64.3-9.8 18164.631%1.6-5.93.5-8.817%1.6-2.64.2-6.023%1.1-2.9 3.3-6.618%2.1-3.63.9-5.522%1.6-4.12.9-6.1 19164.69%1.4-8.14.5-13.0-1%1.6-3.05.8-8.54%1.1-3.54 .4-9.3-5%1.9-4.55.1-8.2-1%1.2-5.63.8-9.4 20164.629%0.7-2.63.7-7.233%1.5-2.05.6-6.833%0.8-2.4 4.0-7.531%1.6-2.55.2-6.534%1.0-3.33.6-7.4 21164.634%0.5-3.12.5-6.524%0.8-1.23.8-4.828%0.4-1.6 2.5-5.420%1.2-2.13.5-4.628%0.7-3.42.2-6.0 22164.613%1.3-6.67.2-16.510%1.6-2.510.2-12.610% 23296.837%2.0-3.75.9-8.325%1.7-2.45.9-7.432%1.6-2.5 5.8-7.719%1.9-2.95.4-6.730%1.8-3.25.2-7.1 2499.722%0.7-7.22.9-12.024%1.9-3.16.3-8.416%0.6-3.2 3.0-8.521%2.2-5.65.7-9.213%0.8-5.62.6-9.3 25102.116%0.3-1.82.3-6.020%0.9-1.65.2-6.514%0.2-1.1 2.4-5.818%1.3-3.44.6-7.011%0.4-1.72.0-5.3 26872.624%1.4-2.05.3-6.717%0.8-1.25.1-6.216%0.8-1.2 5.1-6.215%1.1-1.64.5-5.714%1.1-1.64.6-5.7 27164.631%1.0-4.73.8-9.329%1.6-2.45.7-7.224%0.8-2.8 3.8-8.027%1.8-3.25.1-6.821%1.0-3.93.3-7.7 24% 20% 20%18% 18% P10 FixedP10 NormalP10 UniformP11 FixedP11 NormalP1 1 UniformP12 FixedP12 NormalP12 UniformP13 Normal Run DDD sS DDD sS DDD sS D sS 15%5%7%0.6-1.53.4-5.621%3%6%0.3-0.83.3-5.2-1%-1%0%0 .5-1.02.9-4.711%0.6-6.73.3-13.0 2-11%-8%-6%0.7-1.54.5-7.315%-8%-8%0.4-0.84.5-7.0-13 %-9%-5%0.3-0.73.8-6.1-6%0.6-7.54.4-16.0 326%36%31%1.0-2.43.7-6.350%34%26%0.8-1.73.7-5.819%2 8%24%0.9-1.93.3-5.446%0.9-14.53.3-21.1 420%16%16%0.9-2.34.8-8.051%16%12%0.7-1.64.7-7.722%1 6%8%0.6-1.54.1-6.920%1.0-6.54.8-13.9 50%4%11%0.6-1.54.3-7.222%1%9%0.3-0.84.3-6.8-4%-3%5% 0.2-0.63.6-6.014%0.6-13.84.5-24.4 614%27%20%1.0-2.32.9-5.135%28%16%0.7-1.62.8-4.614%2 3%14%1.1-2.22.5-4.434%1.1-7.93.0-11.5 78%13%22%0.7-1.73.5-6.128%15%24%0.4-0.93.3-5.68%9%2 0%0.5-1.22.9-5.09%0.6-1.73.3-6.1 8-7%-5%-6%0.6-1.23.3-5.17%-1%-5%0.3-0.73.2-4.7-5%-5 %-5%0.5-0.92.8-4.247%0.6-20.03.4-27.8 928%28%28%1.0-2.42.9-5.340%24%17%0.7-1.72.8-4.822%2 2%18%1.1-2.22.5-4.626%1.0-2.62.8-5.6 1025%24%29%0.6-1.42.5-4.129%19%30%0.4-0.72.4-3.618% 17%25%0.7-1.22.1-3.434%0.6-5.62.5-9.7 119%13%11%1.0-2.14.0-6.219%15%10%0.8-1.53.9-5.95%11 %8%0.9-1.73.4-5.324%1.0-7.23.9-12.7 1212%16%12%1.1-2.63.8-6.730%16%3%0.8-1.83.7-6.25%10 %2%1.0-2.23.3-5.714%1.0-3.13.6-7.6 132%10%10%1.0-2.03.5-5.621%15%19%0.7-1.43.4-5.21%7% 4%0.9-1.73.0-4.898%1.0-31.73.6-38.6 14-6%-2%-3%0.6-1.54.4-7.217%-3%-6%0.3-0.84.4-6.8-9% -2%-4%0.3-0.73.0-6.010%0.7-7.74.5-16.4 1515%23%20%0.6-1.43.1-5.130%15%19%0.3-0.73.0-4.76%1 1%15%0.5-1.02.6-4.226%0.6-4.93.1-10.5 168%10%9%0.6-1.52.5-4.317%5%5%0.3-0.82.3-3.82%5%2%0 .7-1.42.1-3.533%0.6-17.52.4-22.6 1713%15%18%0.6-1.53.9-6.343%13%12%0.3-0.83.9-6.011% 12%9%0.4-0.83.3-5.334%0.7-14.34.1-23.4 187%16%12%1.1-2.22.8-4.715%14%9%0.8-1.62.7-4.22%13% 7%1.2-2.22.4-4.023%1.0-8.52.6-11.7 19-8%-9%-1%1.1-2.33.8-6.27%-3%-2%0.8-1.63.7-5.8-10% -13%-4%1.0-1.93.3-5.315%1.0-19.63.6-25.7 2020%26%23%1.0-2.33.9-6.238%23%20%0.8-1.63.8-5.914% 20%18%0.9-1.73.4-5.426%1.0-4.53.7-10.1 2112%19%22%0.7-1.52.7-4.525%12%22%0.4-0.82.5-4.06%1 0%16%0.7-1.32.3-3.732%0.7-13.92.7-19.3 224%6%3%1.0-2.46.6-11.131%7%4%0.7-1.76.6-10.817%19% 10%0.3-1.05.5-9.314%1.1-21.07.0-34.0 2312%14%18%1.1-2.53.9-6.732%10%14%0.8-1.83.8-6.29%1 1%9%1.0-2.13.4-5.79%1.1-3.33.8-8.1 241%10%14%1.1-1.93.8-5.623%14%21%0.8-1.43.7-5.20%10 %13%1.0-1.63.3-4.883%0.9-31.53.6-38.7 254%11%12%0.6-1.33.3-4.919%12%18%0.4-0.73.2-4.52%8% 14%0.6-0.92.8-4.168%0.6-15.63.1-22.9 266%13%10%0.7-1.93.3-6.226%5%5%0.4-1.03.2-5.6-3%1%0 %0.6-1.32.8-5.16%0.6-1.73.2-5.8 2712%22%13%1.1-2.23.9-6.130%19%8%0.8-1.63.8-5.710%1 5%4%1.0-1.93.4-5.226%1.0-7.83.8-13.5 8%13%13% 27%12%11% 5%9%8% 29% Policy (P) Avg Difference=


89 Appendix D (continued): Kabir and Al-Olayan Compari son for b bb b=1.5 P13 UniformP14 NormalP14 UniformP15 NormalP15 Unifo rmP16 FixedP16 Normal Run t* D sS D sS D sS D sS D sS DD 1181.711%0.6-6.63.4-12.84%0.3-2.93.2-9.97%0.3-2.93. 3-9.85%0.5-7.32.8-12.35%0.5-7.22.8-12.129%24% 2181.7-3%0.6-7.54.3-15.8-6%0.3-2.94.3-12.2-4%0.3-2. 94.2-12.5-9%0.3-8.63.7-15.5-7%0.3-8.53.6-15.335%37% 3181.737%1.0-7.03.5-12.740%0.7-6.03.3-14.033%0.7-4. 33.5-10.838%0.8-12.82.9-17.830%0.9-6.83.1-11.649%53 % 4181.729%1.0-6.64.7-14.019%0.7-3.44.8-11.820%0.7-3. 74.7-12.517%0.6-6.44.2-12.722%0.6-6.54.0-12.835%32% 5154.68%0.5-5.83.9-14.46%0.4-4.64.4-15.97%0.3-2.73. 9-12.018%0.3-20.13.8-28.09%0.2-8.43.3-15.614%13% 6181.728%0.8-7.72.5-11.130%0.8-4.32.8-9.422%0.6-3.7 2.4-8.421%1.2-5.72.6-8.821%1.0-5.12.2-7.967%70% 7412.49%0.7-1.73.7-6.015%0.3-0.93.2-5.66%0.4-0.93.5 -5.59%0.5-1.22.8-5.07%0.6-1.23.1-5.07%9% 899.821%0.4-19.92.6-27.74%0.3-5.63.3-15.93%0.2-5.62 .6-15.826%0.5-16.42.9-21.89%0.4-16.42.2-21.85%-2% 9412.425%1.1-2.53.1-5.423%.7-1.82.7-5.022%0.8-1.72. 9-4.919%1.1-2.52.5-4.820%1.2-2.32.7-4.674%73% 10154.631%0.6-5.62.4-9.722%0.4-2.62.3-7.422%0.3-2.6 2.2-7.429%0.745.92.1-8.927%0.7-5.92.0-8.939%34% 11154.617%1.0-7.33.8-12.716%0.7-3.93.8-10.99%0.7-3. 93.7-10.817%0.9-5.93.4-10.59%0.9-5.93.3-10.424%28% 12412.414%1.0-2.43.5-6.38%0.8-2.43.5-7.315%0.7-1.73 .4-5.98%1.0-2.83.1-6.69%0.9-2.03.1-5.446%46% 1399.860%0.7-31.72.8-38.729%0.7-7.93.5-17.421%0.5-8 .02.7-17.549%0.9-19.33.1-23.928%0.7-19.42.4-24.027% 29% 14154.64%0.6-7.54.4-16.02%0.4-3.14.5-12.7-1%0.3-2.9 4.3-12.36%0.8-8.73.8-15.90%0.3-8.63.7-15.536%42% 15154.628%0.5-8.92.9-15.820%0.3-2.43.0-8.715%0.3-3. 72.8-11.528%0.5-6.62.6-11.123%0.5-11.42.5-16.620%23 % 16154.634%0.6-17.42.3-22.610%0.3-5.12.3-12.414% 4% 17164.622%0.6-6.13.9-13.520%0.4-4.74.0-14.817%0.3-2 .83.8-10.732%0.4-18.13.4-24.820%0.4-7.83.3-13.941%3 4% 18164.623%0.9-8.32.4-11.516%0.7-4.02.5-8.519%0.6-3. 92.3-8.411%1.1-5.52.3-7.916%1.0-5.52.1-7.846%49% 19164.65%0.8-8.53.3-13.5-5%0.7-6.13.5-14.8-1%0.6-4. 03.2-10.7-5%0.9-11.73.1-16.0-4%0.8-5.82.8-10.316%14 % 20164.624%0.8-4.33.4-9.824%0.7-3.63.6-9.923%0.6-3.4 3.3-9.631%0.8-5.43.3-10.523%0.7-5.23.0-10.136%38% 21164.626%0.5-5.92.3-10.216%0.4-4.52.5-11.218%0.3-2 .62.1-7.726%0.7-12.62.3-16.119%0.6-6.01.9-9.126%28% 22164.616%0.9-16.76.4-30.18%0.8-6.17.0-20.810%0.7-6 .06.4-21.618%0.3-18.65.9-28.917%0.2-14.75.4-26.24%7 % 23296.815%1.0-3.63.8-8.510%0.8-2.53.7-7.815%0.8-2.7 3.7-8.17%1.0-2.93.3-7.111%1.0-3.23.3-7.440%38% 2499.754%0.6-31.52.8-38.727%0.7-7.93.5-17.624%0.5-7 .92.7-17.641%0.9-19.73.1-24.427%0.7-19.72.4-24.429% 31% 25102.147%0.5-15.52.6-22.726%0.3-4.93.0-13.720% %45% 26872.68%0.7-1.63.3-5.73%0.3-0.93.1-5.32%0.4-0.93.2 -5.20%0.5-1.22.7-4.8-3%0.6-1.22.8-4.728%32% 27164.630%0.9-16.23.4-22.221%0.8-4.33.7-11.116% 43% 23% 15% 14%20% 15% 32%32% P16 UniformP17 NormalP17 UniformP18 FixedP18 Normal P18 UniformP19 NormalP19 Uniform Run D sS D sS D sS DDD sS DD sS 130%0.01.0-2.017%0.01.0-3.026%0.0-1.01.0-3.026%26%3 0%1.0-2.07.0-8.020%23%1.0-2.06.0-8.0 235%0.02.0-3.031%0.0-1.02.0-4.026%0.0-1.02.0-4.09%1 1%9%2.09.0-10.011%8%2.09.0-10.0 350%3.0-4.05.0-6.055%3.0-7.05.0-9.054%3.0-5.05.0-7. 060%66%66%2.0-3.07.0-8.062%63%2.0-3.06.0-8.0 431%3.0-4.06.033%3.0-5.06.0-8.033%3.0-5.05.0-8.049% 47%47%2.0-3.09.0-11.034%35%2.0-3.09.0-10.0 515%0.01.0-2.05%0.0-1.00.011%0.0-1.01.0-4.017%17%19 %1.0-2.08.0-10.012%15%0.0-2.00.0-9.0 669%4.0-5.06.070%4.0-6.05.0-7.070%4.0-6.05.0-7.064% 68%68%3.0-4.06.0-8.057%57%2.0-4.06.0-8.0 78%0.02.0-3.08%0.02.0-3.04%0.02.0-3.034%38%35%2.0-3 .07.0-9.037%36%2.0-3.07.0-8.0 8-1%0.02.0-11%0.0-3.02.0-5.0-4%0.0-2.01.0-4.02%8%7% 2.07.0-8.06%6%0.0-3.00.0-9.0 972%4.05.0-6.072%4.05.0-6.072%4.05.0-6.072%75%74%2. 0-3.06.0-7.071%73%2.0-3.06.0-8.0 1035%0.01.0-2.022%0.0-1.01.0-3.027%0.0-1.01.0-3.039 %40%40%1.0-2.05.0-6.039%39%1.0-2.04.0-6.0 1121%4.06.030%4.0-6.05.0-8.024%4.0-6.05.0-8.029%34% 26%3.08.0-9.029%24%2.0-4.07.0-9.0 1247%4.0-5.06.046%4.0-5.06.0-7.047%4.05.0-6.054%58% 55%0.0-4.00.0-9.055%57%0.0-4.00.0-9.0 1327%4.05.0-6.037%4.0-7.05.0-8.033%4.0-7.05.0-8.033 %37%34%3.07.0-8.033%31%3.07.0-9.0 1439%0.02.0-3.030%0.0-1.02.0-4.031%0.0-1.02.0-4.011 %13%13%2.09.0-10.013%12%2.09.0-10.0 1518%0.01.0-2.016%0.0-1.01.0-3.014%0.0-1.01.0-3.033 %36%35%1.06.0-7.034%35%1.05.0-7.0 164%0.01.0-2.0-1%0.0-2.01.0-4.02%0.0-2.01.0-4.035%3 8%39%2.05.0-6.027%31%1.0-2.05.0-6.0 1730%0.01.0-2.024%0.0-2.01.0-4.022%0.0-1.01.0-3.041 %39%40%1.0-2.08.0-9.030%30%1.0-2.07.0-8.0 1852%4.0-5.06.049%4.0-6.05.0-7.052%4.0-6.05.0-8.039 %46%47%3.0-4.06.0-7.036%38%0.0-4.00.0-7.0 1917%4.0-5.06.0-7.014%4.0-7.06.0-9.017%4.0-6.06.0-8 .021%21%23%3.0-4.08.0-9.015%16%0.0-5.00.0-10.0 2037%2.0-3.05.038%2.0-4.05.0-7.038%2.0-4.05.0-7.042 %46%45%1.0-2.07.0-8.032%32%1.0-2.06.0-7.0 2132%0.01.0-2.019%0.0-2.01.0-4.028%0.0-1.01.0-3.039 %45%47%1.0-2.05.0-6.032%35%0.0-2.00.0-6.0 227%4.0-5.06.0-7.07%4.0-6.06.0-8.07%4.0-6.06.0-8.01 8%20%21%3.0-4.013.0-15.016%16%3.0-4.012.0-15.0 2341%4.06.038%4.0-5.06.0-7.041%4.0-5.06.0-7.047%47% 49%3.0-4.08.0-9.046%47%0.0-4.00.0-9.0 2430%4.06.039%4.0-7.05.0-8.037%4.0-8.05.0-9.035%37% 37%3.08.037%35%2.0-4.07.0-9.0 2547%0.01.0-2.019%0.0-2.01.0-4.028%0.0-2.01.0-4.030 %34%33%1.0-2.07.025%27%1.0-2.06.0-8.0 2627%0.01.0-2.034%0.01.0-2.026%0.01.0-2.039%42%40%1 .0-2.07.0-8.045%40%1.0-2.07.0-8.0 2739%4.06.046%4.0-6.05.0-7.043%4.0-7.05.0-8.048%51% 45%3.08.043%41%2.0-3.07.0-9.0 32% 29%30% 36%39%38% 33%33% Avg Difference= Policy (P)


90 Appendix E: Kabir and Al-Olayan Comparison for b bb b=2.0 P1 FixedP1 NormalP1 UniformP2 FixedP2 NormalP2 Unif ormP3 FixedP3 NormalP3 Uniform Run t* DDD sS DDD sS DDD sS 198.117%19%16%1.0-1.74.3-5.615%19%13%1.0-1.74.3-5.6 15%17%13%1.0-1.93.8-5.1 298.118%18%18%3.1-4.98.1-10.57%9%2%3.1-4.98.1-10.57 %7%7%2.4-4.06.6-8.5 398.133%34%35%0.9-1.44.0-5.027%26%27%0.9-1.44.0-5.0 33%34%35%1.6-2.34.0-5.0 498.118%18%22%1.2-1.85.7-7.216%16%18%1.2-1.85.7-7.2 20%19%23%1.8-2.85.5-7.2 588.921%16%13%1.6-5.67.2-12.815%10%6%1.6-5.67.2-12. 826%22%19%2.1-9.17.0-14.3 698.119%22%20%1.9-4.74.5-7.82%5%3%1.9-4.74.5-7.85%9 %7%1.9-4.03.7-6.1 7154.439%36%41%2.7-4.76.2-8.723%20%26%2.7-4.76.2-8. 722%20%28%1.9-3.24.7-6.3 866.95%5%7%2.5-4.26.6-8.9-3%-6%0%2.5-4.26.6-8.9-1%0 %2%2.1-3.85.5-7.4 9154.445%42%43%1.5-2.43.8-4.932%34%32%1.5-2.43.8-4. 944%40%41%1.7-2.33.4-4.2 1098.125%25%26%1.3-1.93.6-4.712%13%11%1.3-1.93.6-4. 723%24%23%1.8-2.43.4-4.4 1198.18%10%10%1.2-2.24.6-6.34%7%8%1.2-2.24.6-6.39%1 1%12%1.5-2.64.1-5.8 12154.419%23%20%1.1-1.64.0-4.917%15%17%1.1-1.64.0-4 .918%22%18%1.3-1.73.5-4.4 1366.942%37%40%1.7-3.35.0-7.524%19%23%1.7-3.35.0-7. 530%25%29%2.0-3.64.5-6.8 1498.120%18%23%3.4-8.28.9-14.412%13%11%3.4-8.28.9-1 4.48%8%11%2.6-6.67.3-11.1 1598.138%38%39%2.4-3.76.0-8.223%22%22%2.4-3.76.0-8. 229%26%28%2.6-3.85.6-7.6 1698.140%43%41%3.4-5.75.9-8.518%18%23%3.4-5.75.9-8. 513%17%16%2.6-3.74.3-5.6 1792.421%16%16%1.6-3.66.3-8.617%12%12%1.6-3.66.3-8. 628%25%27%2.0-5.36.1-8.9 1892.418%16%18%1.5-2.23.5-4.57%1%5%1.5-2.23.5-4.55% 1%5%1.5-2.02.8-3.6 1992.419%18%20%2.1-2.95.7-7.313%10%8%2.1-2.95.7-7.3 15%16%15%1.9-2.64.8-6.2 2092.415%13%16%0.4-0.93.6-4.815%12%16%0.4-0.93.6-4. 824%21%26%1.3-2.24.1-5.5 2192.439%41%40%2.3-3.55.0-6.522%23%21%2.3-3.55.0-6. 527%28%26%2.2-3.14.3-5.4 2292.416%20%17%3.0-3.910.5-12.011%19%18%3.0-3.910.5 -12.010%13%11%1.9-2.88.8-10.1 23129.831%31%31%0.7-3.62.6-6.921%20%23%0.7-3.62.6-6 .921%22%23%0.8-3.02.2-5.3 2465.447%49%52%2.8-5.76.2-10.124%23%29%2.8-5.76.2-1 0.132%30%34%2.4-4.55.0-7.9 2567.927%29%28%1.9-3.05.9-7.315%15%15%1.9-3.05.9-7. 320%19%21%2.0-3.25.3-6.7 26225.740%39%39%2.3-3.25.5-6.928%29%29%2.3-3.25.5-6 .927%27%29%2.0-2.74.5-5.7 2792.419%20%17%2.7-4.26.0-8.14%6%5%2.4-4.26.0-8.17% 9%7%2.3-3.44.8-6.5 26%26%26%15%15%16%19%19%20% P4 FixedP4 NormalP4 UniformP5 FixedP5 NormalP5 Unif ormP6 FixedP6 NormalP6 UniformP7 Normal Run DDD sS DDD sS DDD sS D sS 159%62%82%2.9-3.98.5-10.643%47%65%1.8-2.48.0-9.848% 50%74%2.7-3.57.6-9.227%1.5-4.25.7-9.2 260%66%92%5.1-6.912.7-16.041%48%71%3.3-4.611.7-14.9 41%43%73%3.8-5.310.7-13.622%2.6-8.48.4-15.3 395%95%110%2.3-3.37.4-9.379%85%101%1.7-2.37.2-8.988 %93%113%2.6-3.47.1-8.748%1.1-3.05.0-7.5 462%63%80%2.6-3.710.2-12.852%52%72%1.7-2.410.0-12.3 49%49%67%2.4-3.19.4-11.526%1.3-3.87.1-10.7 544%44%53%2.6-3.710.1-12.838%37%44%1.7-2.49.9-12.33 8%35%45%2.4-3.19.4-11.617%1.4-3.67.3-10.4 666%69%115%3.2-4.26.8-8.640%43%88%1.8-2.46.1-7.552% 52%108%2.9-3.85.7-7.123%1.6-4.74.4-7.9 7100%103%149%5.5-8.311.1-15.672%77%116%3.4-5.510.014.272%73%127%4.2-6.79.0-12.941%2.7-6.87.1-11.9 818%19%60%3.4-4.19.0-10.46%8%47%1.8-2.28.2-9.58%10% 59%2.7-3.37.5-8.712%1.9-11.76.3-17.5 9148%150%173%4.4-6.58.1-11.1116%121%146%3.3-4.97.610.3130%135%161%4.3-6.37.2-9.960%2.3-4.05.2-7.3 1066%73%92%2.5-3.66.2-8.050%53%70%1.7-2.45.8-7.363% 67%91%2.7-3.75.7-7.230%1.3-3.54.2-6.8 1136%38%56%2.9-4.08.4-10.624%28%45%1.7-2.47.9-9.827 %30%52%2.6-3.57.5-9.316%1.5-4.25.8-9.2 1295%99%125%2.9-4.57.8-10.971%77%104%1.7-2.77.3-9.9 80%85%118%2.7-4.06.9-9.537%1.5-2.85.4-7.3 1365%61%103%2.9-3.77.8-9.441%45%77%1.7-2.27.3-8.649 %47%95%2.7-3.36.9-8.240%1.6-9.05.6-14.3 1452%53%83%5.2-6.712.8-15.836%37%64%3.3-4.511.9-14. 734%35%67%3.9-5.210.8-13.421%2.7-8.68.6-15.5 1590%92%109%4.0-5.49.0-11.377%77%90%3.1-4.38.7-10.9 81%83%100%4.0-5.38.4-10.540%2.1-5.66.0-10.1 1693%100%142%5.1-6.68.8-10.963%68%103%3.3-4.57.7-9. 768%74%122%4.3-5.77.1-9.049%2.7-8.35.5-11.6 1762%62%83%2.7-3.79.3-11.548%48%66%1.7-2.39.0-10.94 9%49%67%2.5-3.28.5-10.321%1.4-3.86.4-9.7 1851%52%93%3.2-4.26.5-8.133%33%75%1.8-2.45.6-7.038% 38%97%2.9-3.85.3-6.616%1.6-7.74.1-10.9 1954%56%101%3.2-4.48.3-10.635%37%80%1.8-20.47.6-9.5 38%42%95%2.7-3.67.0-8.919%1.6-5.15.6-9.8 2057%57%72%1.8-2.67.0-8.851%52%64%1.5-2.17.3-8.962% 60%73%2.4-3.17.3-8.818%0.8-3.24.7-8.2 21121%127%153%4.4-6.08.2-10.690%100%121%3.2- 2240%45%61%5.5-7.116.6-20.431%36%51%3.3-4.615.7-19. 427%31%50%3.6-4.714.1-17.424%2.7-10.111.2-20.3 23101%105%129%4.8-7.010.0-13.577%84%103%3.3- 2485%88%125%4.7-5.89.9-11.760%62%96%3.3-4.19.1-10.8 65%65%110%4.1-5.08.5-10.155%2.6-9.66.8-14.6 2547%48%79%2.8-3.68.0-9.631%33%59%1.7-2.27.4-8.834% 36%69%2.6-3.37.0-8.429%1.6-5.85.7-11.2 26148%161%186%4.9-7.810.1-14.1113%125%151%3.3-5.69. 3-13.6118%126%162%4.2-6.98.7-12.882%3.0-4.77.6-9.3 2756%60%77%4.7-6.39.9-12.439%46%57%3.2-4.59.1-11.54 3%48%66%4.0-5.58.5-10.723%2.4-7.36.4-12.0 73%76%103%54%58%82%59%61%93% 33% Policy (P) Avg Difference=


91 Appendix E (continued): Kabir and Al-Olayan Compari son for b bb b=2.0 P7 UniformP8 NormalP8 UniformP9 NormalP9 Uniform Run t* DDD sS DDD sS DDD sS 198.122%0.4-4.42.8-9.718%1.0-1.85.6-7.414%0.3-2.02. 9-7.921%1.3-4.85.0-8.614%0.5-5.02.5-9.2 298.117%1.2-6.25.1-14.69%1.8-3.38.2-11.310%0.8-4.05 .0-13.48%1.8-6.77.2-12.23%0.8-4.94.3-12.3 398.142%0.4-2.93.0-7.542%0.9-1.65.0-7.136%0.4-1.73. 2-7.444%1.2-4.44.6-8.040%0.6-4.32.8-7.9 498.126%0.5-2.94.4-11.122%0.9-1.77.2-9.222%0.4-10.9 4.5-10.824%1.0-5.76.4-11.419%0.4-3.93.9-9.8 588.911%0.3-2.83.4-10.816%1.0-1.67.3-9.29%0.2-1.83. 5-10.617%1.0-5.66.5-11.18%0.3-3.33.0-9.7 698.115%0.4-4.81.1-8.113%1.0-1.94.1-5.99%0.1-2.11.2 -6.612%1.5-4.03.7-6.27%0.5-4.11.0-6.4 7154.443%1.9-5.45.3-10.932%1.9-3.26.8-9.338%1.3-3.5 5.1-10.227%2.2-4.76.0-8.831%1.5-4.04.4-9.2 866.94%0.4-11.72.3-17.5-3%1.1-3.36.0-9.9-1%0.2-3.32 .3-9.92%1.4-10.05.3-13.9-3%0.5-10.01.9-13.9 9154.457%1.4-4.53.7-8.555%1.9-2.75.1-6.448%1.2-3.63 .7-8.151%2.4-3.74.7-6.348%1.6-4.53.4-7.7 1098.120%0.0-3.00.8-7.226%1.0-1.64.1-5.516%0.0-2.00 .8-6.628%1.5-4.03.8-6.319%0.4-3.10.7-6.4 1198.112%0.3-2.72.6-8.111%1.0-1.85.7-7.88%0.3-1.62. 7-7.713%1.3-4.85.1-8.68%0.5-2.92.3-7.1 12154.434%1.0-2.74.3-7.833%1.0-1.65.3-6.632%0.7-1.7 4.3-7.230%1.3-2.84.7-6.628%0.9-2.53.8-6.7 1366.928%0.3-6.02.3-11.222%1.0-2.75.4-8.421%0.2-2.5 2.3-8.231%1.4-9.24.9-12.721%0.5-6.32.0-10.2 1498.121%0.6-6.23.6-13.611%1.9-3.38.4-11.614%0.5-3. 73.6-12.812%1.9-6.87.3-12.413%0.5-5.03.0-11.6 1598.134%0.5-4.22.6-9.432%1.8-3.06.1-8.627%0.5-3.42 .7-9.337%2.1-5.45.6-8.930%0.7-4.22.4-8.7 1698.132%0.0-6.10.3-9.325%1.9-3.25.1-7.421%0.0-3.60 .3-8.327%2.4-5.24.7-7.518%0.5-4.60.3-7.7 1792.419%0.1-4.11.5-10.517%0.9-1.76.4-8.614%0.1-2.0 1.6-9.618%1.1-5.25.7-9.914%0.3-5.21.3-10.6 1892.415%0.3-5.01.6-7.95%1.0-2.43.8-6.59%0.2-2.11.5 -6.26%1.6-5.63.4-7.58%0.6-4.01.3-6.0 1992.417%0.4-3.72.6-8.810%1.0-1.95.3-7.512%0.3-1.92 .6-8.110%1.4-4.74.7-8.28%0.5-3.32.2-7.4 2092.422%0.1-1.92.2-7.220%0.7-1.35.4-6.625%0.2-1.62 .4-7.429%1.1-6.54.8-10.327%0.4-3.12.2-7.1 2192.438%0.5-5.42.0-9.839%1.8-3.15.1-7.431%0.4-4.12 .1-9.139%2.3-5.14.7-7.631%0.8-5.11.8-8.7 2292.418%0.8-7.05.3-17.318%1.9-3.511.0-14.317%0.5-3 .65.3-16.716%1.4-9.09.4-17.412%0.3-5.94.4-14.7 23129.838%1.2-6.34.1-12.632%1.9-2.66.4-7.830%1.0-4. 64.1-11.832%2.2-4.25.7-7.829%1.2-5.53.6-11.0 2465.432%0.2-4.31.3-9.632%2.0-3.56.6-9.439%0.1-3.21 .3-9.135%2.3-7.25.9-10.823%0.4-3.71.1-8.3 2567.920%0.3-5.52.3-10.518%1.0-2.45.6-8.313%0.2-2.2 2.3-8.121%1.4-6.35.0-10.315%0.5-6.02.0-9.7 26225.777%2.7-4.56.8-9.961%2.3-2.97.4-8.361%2.0-3.3 6.4-9.358%2.7-3.96.7-7.854%2.3-3.95.8-8.6 2792.414%0.7-5.42.8-10.213%1.8-3.16.2-8.79%0.5-3.22 .9-9.013%2.1-5.75.5-9.28%0.7-4.42.5-8.5 27% 23% 22%24% 20% P10 FixedP10 NormalP10 UniformP11 FixedP11 NormalP1 1 UniformP12 FixedP12 NormalP12 UniformP13 Normal Run DDD sS DDD sS DDD sS D sS 15%14%9%0.4-1.62.9-5.924%11%6%0.3-0.82.8-5.54%11%5% 0.4-1.12.4-4.984%0.4-31.22.9-40.5 2-1%1%1%0.7-2.63.9-8.727%6%2%0.5-1.84.0-8.4-1%4%0%0 .4-1.83.3-7.573%0.8-38.04.4-48.6 318%25%26%0.4-1.62.6-5.542%21%19%0.3-0.92.6-5.110%1 6%20%0.4-1.12.2-4.780%0.4-12.02.6-19.1 48%13%16%0.4-1.63.5-7.641%10%16%0.2-0.93.6-7.24%8%1 1%0.2-0.73.0-6.367%0.4-14.23.5-25.0 54%8%1%0.3-1.63.3-7.631%6%2%0.2-0.93.4-7.34%3%1%0.2 -0.72.8-6.463%0.4-27.83.7-40.5 6-11%-2%6%0.4-1.62.0-4.46%-5%1%0.3-0.81.9-3.9-11%-3 %-4%0.6-1.41.7-3.7129%0.4-35.12.0-41.2 720%24%27%0.8-3.83.4-9.240%23%27%0.6-2.73.3-8.517%1 4%29%0.8-3.22.9-7.933%0.9-14.13.6-20.9 8-9%0%2%0.5-1.22.8-5.011%4%5%0.3-0.72.8-4.7-4%5%6%0 .4-0.92.4-4.2147%0.4-69.32.6-80.1 929%36%37%0.8-3.12.5-6.339%31%30%0.6-2.22.5-5.823%3 1%31%1.0-2.92.3-5.644%0.6-6.42.2-9.9 105%13%10%0.4-1.61.8-4.516%9%7%0.2-0.91.8-4.01%7%6% 0.5-1.41.6-3.879%0.3-13.61.8-18.7 11-2%4%4%0.4-1.62.6-6.012%4%3%0.2-0.92.6-5.6-4%2%2% 0.4-1.12.2-5.078%0.4-31.22.7-40.5 1214%24%21%0.4-2.22.4-6.834%17%16%0.2-1.22.3-6.16%1 5%14%0.4-1.62.0-5.634%0.4-10.02.6-17.0 136%11%17%0.4-1.22.3-4.725%8%17%0.2-0.72.3-4.412%3% 14%0.4-1.02.0-4.0192%0.3-53.72.3-63.4 143%5%9%0.7-2.44.1-8.430%14%12%0.6-1.74.2-8.18%18%1 2%0.5-1.73.5-7.265%07-19.54.2-28.5 1511%15%15%0.7-2.22.9-6.129%15%15%0.5-1.72.9-5.88%9 %11%0.7-1.92.6-5.4101%0.7-21.63.0-28.6 167%17%14%0.8-2.42.4-5.216%22%19%0.6-1.72.3-4.85%14 %15%0.9-2.32.1-4.5125%0.7-36.82.4-41.8 174%11%8%0.4-1.53.2-6.633%5%4%0.2-0.83.2-6.33%1%2%0 .3-0.92.7-5.589%0.4-28.83.0-39.8 18-5%-2%3%0.4-1.51.8-4.010%-6%5%0.2-0.81.7-3.5-9%-6 %1%0.6-1.51.5-3.3111%0.4-36.91.8-42.3 19-4%5%6%0.4-1.62.3-5.720%3%9%0.2-0.92.3-5.2-2%4%6% 0.4-1.22.0-4.7113%0.4-37.92.5-46.6 209%13%15%0.4-1.42.7-5.428%7%11%0.3-0.82.7-5.05%5%9 %0.3-0.92.3-4.670%0.3-32.42.2-42.4 219%20%19%0.6-2.42.2-5.427%20%17%0.5-1.72.2-5.05%16 %16%0.8-2.22.0-4.7104%0.8-16.02.7-20.6 227%14%12%0.7-2.55.7-11.439%21%18%0.6-1.85.8-11.231 %37%32%0.2-1.14.8-9.778%0.8-48.46.0-63.1 2315%24%21%0.7-3.13.0-7.832%20%21%0.6-2.33.0-7.211% 19%17%0.7-2.72.7-6.746%0.8-16.83.2-22.9 2415%18%27%0.7-1.93.0-5.544%38%41%0.5-1.43.0-5.213% 24%27%0.7-1.62.6-4.7180%0.8-32.93.2-40.1 253%4%11%0.3-1.22.3-4.824%5%11%0.2-0.72.3-4.57%7%8% 0.4-0.92.0-4.1107%0.4-31.42.7-40.1 2629%41%42%0.8-4.03.2-9.350%34%35%0.6-2.93.2-8.519% 31%29%0.8-3.42.8-8.030%0.7-3.73.0-8.8 27-2%5%1%0.7-2.42.9-6.413%6%2%0.5-1.72.9-6.1-2%2%0% 0.7-2.02.5-5.665%0.7-28.63.1-35.9 7%13%14% 28%13%14% 6%11%12% 88% Policy (P) Avg Difference=

PAGE 100

92 Appendix E (continued): Kabir and Al-Olayan Compari son for b bb b=2.0 P13 UniformP14 NormalP14 UniformP15 NormalP15 Unifo rmP16 FixedP16 Normal Run t* D sS D sS D sS D sS D sS DD 198.145%0.3-15.82.1-23.530%0.3-7.92.8-20.019%0.2-4. 82.1-13.871%0.4-29.72.4-35.736%0.3-16.51.8-21.751%5 1% 298.129%0.5-19.33.3-28.115%0.6-8.74.4-22.212%0.4-5. 93.3-16.840%0.6-26.63.8-33.815%0.4-14.62.8-21.29%12 % 398.164%0.3-11.92.3-18.945%0.3-4.12.5-12.141%0.2-4. 02.3-12.095%0.4-15.02.2-20.071%0.4-14.92.0-19.947%4 1% 498.142%0.4-14.23.3-24.531%0.2-4.83.6-16.224%0.2-4. 53.3-15.289%0.2-20.73.0-28.743%0.2-20.72.8-28.245%4 1% 588.932%0.3-14.22.9-24.924%0.3-7.53.7-21.614%0.2-4. 72.9-16.187%0.2-37.93.1-46.645%0.2-20.72.4-28.641%3 3% 698.148%0.3-17.91.7-23.027%0.2-0.81.9-18.011%0.5-5. 01.7-12.361%0.6-22.31.7-25.722%0.5-12.81.5-15.9-2%1 0% 7154.432%0.7-8.93.2-14.322%0.7-5.83.6-14.829%0.5-4. 03.2-11.219%0.8-9.43.1-14.824%0.7-5.92.8-10.546%47% 866.975%0.3-69.32.0-80.122%0.2-12.82.6-31.711%0.1-1 2.82.0-31.779%0.4-47.32.2-53.739%0.4-47.31.7-53.720 %36% 9154.442%0.6-6.42.2-10.432%0.5-3.72.2-8.233%0.5-4.1 2.2-8.935%0.8-5.32.0-8.235%0.8-5.72.0-8.974%76% 1098.148%0.2-13.71.4-18.934%0.2-4.31.7-10.725%0.1-4 .41.4-11.066%0.5-12.41.5-15.742%0.5-12.61.2-15.934% 34% 1198.124%0.4-6.72.6-12.921%0.2-7.92.7-19.912%0.2-2. 92.6-9.968%0.4-29.72.3-35.722%0.4-7.32.2-12.333%34% 12154.424%0.4-6.72.3-12.318%0.3-4.42.6-12.815%0.2-2 .82.3-9.128%0.5-10.12.2-15.515%0.4-6.92.0-11.233%33 % 1366.992%0.2-31.51.9-39.850%0.2-11.32.3-26.034% 06%89% 1498.154%0.4-19.63.1-28.316%0.6-6.64.2-18.618%0.3-6 .13.1-16.936%0.4-14.83.6-21.731%0.4-14.82.6-21.510% 10% 1598.153%0.5-12.82.3-18.442%0.6-6.83.0-15.530%0.4-5 .22.3-12.479%0.7-17.32.7-22.244%0.5-10.92.0-14.929% 28% 1698.168%0.2-19.01.2-23.333%0.6-8.12.3-16.527%0.2-5 .81.3-12.248%0.9-16.82.1-19.729%0.6-10.41.1-13.243% 45% 1792.452%0.1-28.81.7-40.030%0.2-7.53.0-20.422%0.1-7 .71.7-20.8105%0.3-33.72.5-41.057%0.3-33.81.4-41.353 %42% 1892.461%0.3-18.91.4-23.519%0.2-8.11.7-18.015%0.2-5 .31.3-12.543%0.6-20.91.5-23.826%0.5-12.31.2-15.00%1 % 1992.444%0.3-10.12.3-16.328%0.2-8.52.5-21.716%0.2-3 .52.2-11.071%0.4-28.62.1-34.027%0.4-9.01.9-13.522%2 5% 2092.438%0.3-9.02.2-16.227%0.5-5.32.3-18.124%0.2-2. 72.2-10.6104%0.3-41.22.0-47.649%0.3-14.61.9-19.947% 41% 2192.488%0.5-17.32.1-21.946%0.6-5.52.6-11.735%0.4-5 .22.0-11.261%1.0-10.62.4-13.748%0.7-11.01.8-14.055% 61% 2292.430%0.5-21.64.8-34.723%0.6-10.96.0-31.718% %14% 23129.854%0.7-22.12.9-29.325%0.6-5.33.1-12.829% %43% 2465.487%0.4-16.82.1-22.955%0.6-8.03.2-17.839%0.3-5 .72.1-13.597%0.8-20.22.8-24.853%0.5-11.71.8-16.033% 33% 2567.973%0.1-20.81.4-28.930%0.3-7.82.7-19.325%0.1-6 .31.4-17.087%0.5-28.22.3-33.758%0.3-20.01.1-25.571% 64% 26225.728%0.8-4.03.2-9.333%0.6-2.83.0-8.425%0.6-3.1 3.2-8.925%0.7-3.32.6-7.723%0.8-3.62.8-8.164%67% 2792.433%0.4-16.92.2-23.019%0.6-8.03.0-17.812%0.3-5 .62.2-13.534%0.8-18.52.7-23.517%0.5-11.81.9-16.118% 21% 50% 30% 23%65% 37% 38%38% P16 UniformP17 NormalP17 UniformP18 FixedP18 Normal P18 UniformP19 NormalP19 Uniform Run D sS D sS D sS DDD sS DD sS 144%01.0-2.015%0.0-3.01.0-5.022%0.0-2.01.0-4.065%68 %67%0.0-4.00.0-11.032%29%1.0-3.06.0-9.0 211%4.0-5.06.0-7.016%4.0-7.05.0-9.014%4.0-7.05.0-8. 067%71%71%0.0-7.00.0-17.024%23%0.0-5.00.0-13.0 342%01.0-2.017%0.0-1.01.0-4.027%0.0-1.01.0-4.0106%1 09%108%0.0-3.00.0-10.048%47%0.0-2.00.0-8.0 447%01.0-2.015%0.0-2.01.0-4.033%0.0-1.01.0-4.066%68 %71%3.0-4.011.0-14.029%32%0.0-3.00.0-11.0 528%01.0-2.011%0.0-2.01.0-5.015%0.0-1.01.0-4.050%48 %44%3.0-4.011.0-14.018%16%0.0-3.00.0-10.0 610%01.0-2.0-11%0.0-3.01.0-4.01%0.0-2.01.0-4.073%76 %78%0.0-4.00.0-9.024%25%1.0-3.05.0-7.0 751%4.0-6.06.0-7.046%4.0-6.06.0-8.051%4.0-6.06.0-8. 0108%110%114%0.0-9.00.0-16.054%60%3.0-5.09.0-11.0 833%01.0-2.0-12%0.0-5.01.0-6.017%0.0-5.01.0-6.024%2 5%27%0.0-4.00.0-11.04%7%0.0-5.00.0-11.0 977%45.0-6.076%3.0-6.05.0-7.078%3.0-6.05.0-7.0156%1 65%163%0.0-7.00.0-12.072%77%2.0-3.06.0-7.0 1032%01.0-2.04%0.0-2.01.0-4.010%0.0-2.01.0-4.068%74 %75%0.0-4.00.0-9.028%27%1.0-2.04.0-6.0 1134%01.0-2.05%0.0-3.01.0-5.018%0.0-1.01.0-3.039%41 %44%0.0-4.00.0-11.017%18%1.0-3.06.0-9.0 1238%01.0-3.036%0.0-1.01.0-4.029%0.0-1.01.0-3.0105% 110%109%3.0-5.09.0-11.054%54%1.0-2.06.0-8.0 1395%01.0-2.01%0.0-5.01.0-6.040%0.0-3.01.0-5.068%68 %72%3.0-4.09.0-10.030%38%0.0-4.00.0-10.0 1413%4.0-5.06.0-7.015%4.0-7.05.0-9.018%4.0-7.05.0-9 .059%59%62%0.0-7.00.0-17.021%25%0.0-5.00.0-13.0 1528%3.0-4.05.0-6.040%3.0-7.05.0-9.040%3.0-6.05.0-8 .098%101%102%0.0-5.00.0-12.037%41%2.0-4.06.0-9.0 1645%4.0-5.05.0-6.051%4.0-7.05.0-8.048%4.0-7.05.0-8 .0100%108%108%5.0-7.09.0-12.037%36%2.0-4.06.0-9.0 1742%01.0-2.04%0.0-3.01.0-5.023%0.0-3.01.0-5.068%67 %67%0.0-4.00.0-12.022%27%1.0-3.07.0-10.0 188%01.0-2.0-13%0.0-3.01.0-5.0-1%0.0-2.01.0-4.061%6 0%63%3.0-4.07.0-9.016%19%1.0-4.05.0-7.0 1925%01.0-3.0-3%0.0-3.01.0-5.022%0.0-2.01.0-4.062%6 3%63%0.0-5.00.0-11.024%26%1.0-3.06.0-9.0 2046%01.0-2.015%0.0-5.00.0-6.028%00.0-3.061%59%65%2 .0-3.08.0-9.022%28%0.0-3.05.0-7.0 2158%45.0-6.072%4.0-7.05.0-8.066%3.0-7.05.0-8.0130% 139%139%4.0-6.09.0-11.049%48%2.0-4.06.0-9.0 2212%4.0-5.06.0-7.016%4.0-7.06.0-9.013%4.0-7.05.0-9 .045%49%46%6.0-8.018.0-22.025%21%3.0-6.012.0-16.0 2343%4.0-5.05.0-7.047%4.0-7.05.0-8.047%4.0-7.05.0-8 .0108%114%113%0.0-7.00.0-14.050%50%3.0-4.08.0-9.0 2436%45.0-6.045%4.0-7.05.0-8.043%4.0-7.05.0-8.092%9 5%97%0.0-6.00.0-12.045%49%0.0-5.00.0-11.0 2568%01.0-2.0-1%0.0-3.01.0-5.028%0.0-3.01.0-5.051%5 5%55%0.0-4.00.0-10.024%26%1.0-3.06.0-9.0 2666%4.0-5.05.0-7.065%4.0-5.05.0-7.065%4.0-5.05.0-7 .0158%167%166%0.0-8.00.0-15.096%96%0.0-4.00.0-10.0 2718%45.0-6.026%4.0-7.05.0-8.023%4.0-7.05.0-8.062%6 7%65%0.0-7.00.0-13.024%22%2.0-4.07.0-10.0 39% 22%30% 80%83%83% 34%36% Policy (P) Avg Difference=

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93 Appendix F: Kabir and Al-Olayan Comparison for b bb b=3.0 P1 FixedP1 NormalP1 UniformP2 FixedP2 NormalP2 Unif ormP3 FixedP3 NormalP3 Uniform Run t* DDD sS DDD sS DDD sS 176.011%11%12%1.7-2.55.3-6.86%8%9%0.8-1.24.5-5.710% 11%11%4.9-6.31.8-2.9 276.019%20%17%2.0-3.07.0-8.812%13%13%0.9-1.35.9-7.4 17%19%18%6.3-8.21.8-3.3 376.055%51%52%1.8-3.05.1-6.833%28%31%1.3-1.74.9-5.9 40%35%37%4.8-6.32.0-3.2 476.025%25%25%2.8-5.58.1-11.815%15%16%1.4-2.46.4-8. 919%19%18%7.2-10.72.6-5.6 571.517%17%21%1.8-7.67.4-15.112%9%15%1.0-2.16.2-9.1 23%22%27%7.3-17.22.3-12.1 676.028%24%25%2.0-3.84.4-6.816%10%9%0.9-1.53.5-4.82 2%19%20%3.5-5.31.9-3.4 799.218%18%18%1.6-2.25.1-6.012%18%14%0.8-1.04.5-5.1 7%8%7%4.2-5.01.4-2.0 859.813%11%13%2.8-5.97.2-10.9-1%-5%0%1.3-2.15.5-7.7 3%2%5%6.1-9.02.4-5.3 999.223%21%24%0.8-1.43.0-3.916%12%16%0.6-0.92.6-3.7 23%21%24%2.8-3.71.1-1.8 1071.517%19%13%1.4-2.93.8-6.014%16%10%0.7-1.33.2-4. 625%27%23%3.5-5.51.9-3.4 1171.515%13%10%1.6-2.75.7-7.15%3%1%0.9-1.34.9-6.011 %10%6%5.1-6.61.6-3.1 1299.223%23%25%1.2-1.74.3-5.117%16%15%0.7-0.94.0-4. 616%15%19%3.8-4.61.3-1.9 1359.893%92%96%4.6-19.18.7-24.431%32%34%1.9-4.55.810.958%58%60%7.0-15.43.7-12.1 1471.56%6%7%1.6-2.46.9-8.11%0%4%0.9-1.26.0-7.35%4%6 %6.0-7.31.3-2.4 1571.523%26%24%1.3-3.54.9-7.99%11%9%0.6-1.34.1-5.82 4%28%26%4.9-8.41.8-5.2 1671.532%33%34%3.8-8.86.1-12.06%9%9%1.5-2.94.1-6.97 %8%8%4.1-7.32.6-5.3 1773.337%37%41%2.8-5.47.1-10.122%20%20%1.4-2.35.8-7 .833%30%31%6.1-9.22.7-5.0 1873.338%38%38%4.0-8.76.4-11.43%6%2%1.7-2.84.2-6.34 %6%3%4.3-6.52.9-4.8 1973.353%50%51%3.7-8.97.1-13.321%17%18%1.6-2.95.2-8 .025%22%21%5.1-8.82.6-5.7 2073.315%16%14%0.8-2.94.3-7.611%12%10%0.0-0.73.5-5. 036%37%34%5.2-9.51.9-6.1 2173.365%65%62%3.1-7.15.9-10.334%33%31%1.4-2.44.1-6 .139%39%36%4.7-7.32.8-5.1 2273.311%11%6%3.8-6.610.8-15.32%1%-4%1.5-2.58.5-11. 63%4%-1%9.0-13.02.9-5.9 2389.819%20%17%1.7-3.95.2-7.78%11%6%0.9-1.44.3-5.41 4%10%8%4.7-6.91.8-4.5 2459.040%39%39%2.2-6.16.3-11.120%17%19%1.2-2.15.4-7 .429%27%34%5.7-10.22.3-6.7 2560.430%31%27%2.3-5.16.4-9.914%16%10%1.2-2.05.1-7. 231%32%27%5.8-9.12.4-5.6 26121.732%34%33%1.5-2.94.4-6.523%25%26%1.0-1.74.1-5 .621%23%24%3.7-5.41.4-2.5 2773.332%32%33%3.1-4.66.6-8.614%13%14%1.6-2.15.3-6. 519%17%19%5.3-6.82.6-3.7 29%29%29%14%14%13%21%20%20% P4 FixedP4 NormalP4 UniformP5 FixedP5 NormalP5 Unif ormP6 FixedP6 NormalP6 UniformP7 Normal Run DDD sS DDD sS DDD sS D sS 172%74%95%3.7-7.99.8-14.438%41%61%2.2-3.28.9-10.968 %71%98%3.5-8.68.8-13.717%1.4-5.75.4-11.5 291%89%121%4.1-9.112.4-18.147%49%81%2.2-3.411.4-13. 887%85%122%3.4-10.511.0-17.726%1.5-6.77.1-14.5 3201%192%208%5.2-10.110.6-16.1126%126%143%2.8-5.010 .0-12.1179%176%198%5.1-9.49.9-14.053%1.8-12.05.5-17 .2 4124%123%142%5.6-11.613.8-20.472%76%89%3.9-5.313.015.6114%112%129%5.0-11.312.5-18.641%2.1-14.37.7-22. 2 570%65%86%3.7-7.011.9-16.043%42%59%2.2-3.111.2-13.0 85%79%96%4.1-10.311.6-17.641%1.9-8.98.8-17.6 6131%130%179%4.1-8.48.0-12.761%58%99%2.2-3.36.9-8.8 100%93%156%3.7-6.96.7-9.830%1.5-6.84.2-10.5 798%97%137%3.8-10.210.0-16.852%55%88%2.1-3.59.1-11. 876%75%123%3.0-8.98.3-13.931%1.8-6.46.2-11.7 869%63%108%5.4-10.211.5-16.924%23%61%2.5-3.59.1-11. 850%49%100%4.6-8.99.7-14.014%1.7-15.46.0-21.6 9126%125%147%3.0-6.27.1-10.582%85%105%2.0-3.06.6-8. 4120%119%146%3.2-6.56.4-9.538%1.4-3.94.2-7.3 10106%104%119%3.5-6.87.5-11.251%57%72%2.2-3.06.7-8. 398%96%119%3.8-7.06.9-10.162%1.9-8.35.2-12.6 1157%55%70%4.0-8.110.1-14.728%29%41%2.3-3.39.2-10.9 56%55%72%4.0-8.99.2-13.933%2.1-9.77.1-16.2 12120%120%151%3.4-7.58.9-13.474%75%105%2.1-3.28.2-1 0.2110%107%147%3.1-7.77.8-12.237%1.5-3.85.5-8.5 13196%195%233%7.5-15.513.1-21.693%96%129%4.3-5.710. 8-13.4136%137%181%6.2-11.010.7-15.583%2.4-12.46.3-1 7.5 1463%58%86%4.5-5.512.8-17.428%28%53%2.3-3.311.6-13. 459%57%89%4.0-9.611.6-16.935%2.3-10.29.1-18.9 1591%93%107%3.1-6.18.6-12.161%64%77%2.1-2.98.1-9.61 10%109%128%3.7-8.18.4-12.860%1.7-7.16.2-13.1 16169%168%209%7.0-15.311.1-19.669%70%105%4.2-5.69.3 -11.991%93%140%5.6-9.18.7-12.1135%3.6-29.76.9-34.4 17148%144%166%5.9-12.413.2-20.183%80%103%4.0-5.512. 0-14.6123%119%146%5.2-11.011.7-17.142%2.2-14.67.0-2 1.6 18166%162%198%7.0-14.710.6-18.664%66%96%4.1-5.58.611.483%80%136%5.5-8.58.0-11.136%2.5-18.94.9-22.3 19176%168%215%7.2-18.012.8-24.281%79%119%4.1-6.110. 9-14.7103%103%159%5.4-11.110.2-15.747%2.6-12.26.4-1 7.2 2070%69%78%2.3-4.27.9-10.457%58%64%1.8-2.38.0-9.310 8%106%112%3.4-7.58.6-12.619%0.7-4.74.5-10.1 21232%227%256%5.7-12.410.0-17.0121%123%147%4.0-5.38 .9-11.3168%163%199%5.2-9.08.6-12.264%2.1-9.65.0-13. 3 2264%63%76%7.2-17.619.4-30.628%31%41%4.1-6.017.8-21 .952%50%66%5.1-15.616.7-26.513%2.7-13.310.8-24.4 2395%91%113%3.5-8.19.2-14.355%56%76%2.1-3.38.5-10.6 90%86%111%3.2-8.58.2-13.237%1.7-5.46.0-10.5 24116%111%153%4.4-8.010.0-14.160%57%96%2.3-3.38.7-1 0.3110%106%155%4.5-8.49.2-13.037%1.5-8.85.5-14.7 2594%92%120%4.3-8.010.0-14.148%49%71%2.3-3.38.7-10. 589%88%118%4.4-8.49.1-13.037%1.5-9.75.5-15.6 26191%194%224%5.2-12.610.9-18.6118%120%150%3.8-5.81 0.2-13.7149%149%175%4.5-9.49.5-14.363%2.6-5.26.7-9. 8 27153%148%175%6.3-13.612.0-19.777%75%100%4.0-5.710. 6-13.5109%105%138%5.3-10.110.1-14.739%2.3-10.16.1-1 5.0 122%119%147%64%65%90%101%99%132% 43% Policy (P) Avg Difference=

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94 Appendix F (continued): Kabir and Al-Olayan Compari son for b bb b=3.0 P7 UniformP8 NormalP8 UniformP9 NormalP9 Uniform Run t* D DD s SD D Ds S DD D sS 176.0-3%0.0-5.91.0-11.68%0.9-2.55.3-9.1-3%0.0-2.41. 0-8.615%1.2-6.54.8-10.8-7%0.3-6.60.9-11.0 276.02%0.0-6.81.3-14.714%0.9-2.66.9-11.62%0.0-2.61. 4-11.021%1.0-8.06.1-14.00%0.3-7.91.1-14.2 376.026%0.1-8.61.2-13.934%1.6-3.15.6-9.026%0.1-3.61 .3-9.745%1.9-10.25.2-13.819%0.4-8.01.1-12.1 476.015%0.1-5.41.5-12.219%1.7-3.97.6-12.010%0.1-2.8 1.6-10.031%1.7-3.76.8-19.35%0.3-5.41.3-11.1 571.519%0.0-6.51.1-15.126%1.3-3.38.7-12.315%0.0-2.8 1.2-11.849%1.4-13.67.8-20.122%0.2-9.61.0-16.6 676.0-4%0.0-4.10.7-7.410%0.9-2.43.9-6.9-6%0.0-1.80. 7-5.617%1.5-5.63.6-8.0-14%0.4-3.60.6-6.0 799.225%0.5-4.92.7-10.320%1.0-2.35.8-8.316%0.3-2.12 .7-8.321%1.3-5.75.1-9.715%0.5-4.32.3-8.7 859.87%0.0-10.60.5-16.7-3%1.0-3.75.7-11.7-7%0.0-3.3 0.5-10.93%1.3-12.75.0-16.7-3%0.4-9.20.4-13.5 999.230%0.3-3.71.7-7.133%1.0-1.84.1-5.722%0.2-1.91. 8-6.040%1.5-4.33.9-6.728%0.6-4.21.6-6.5 1071.517%0.0-5.20.6-9.234%1.3-2.94.9-8.15%0.0-2.40. 6-7.158%2.0-8.24.7-11.016%0.4-5.50.5-8.4 1171.55%0.0-8.50.7-15.014%1.3-3.36.8-10.6-3%0.0-3.3 0.8-10.432%1.8-10.56.2-15.31%0.3-9.30.6-14.1 1299.229%0.3-3.82.4-8.528%1.0-1.85.3-7.020%0.3-1.82 .4-7.129%1.4-4.14.8-7.721%0.5-4.12.1-7.7 1359.865%0.0-15.40.6-21.330%1.8-4.66.1-11.326%0.0-5 .50.6-12.845%2.2-8.75.5-12.335%0.4-10.80.5-15.2 1471.510%0.0-7.71.1-16.112%1.3-3.58.8-13.63%0.0-3.0 1.1-12.333%1.5-12.17.8-18.67%0.3-8.80.9-15.6 1571.519%0.0-5.40.3-11.137%1.3-2.86.2-9.312%0.0-2.5 0.3-8.874%1.8-9.55.8-13.923%0.4-7.30.2-11.8 1671.543%0.1-11.50.6-15.639%2.6-7.06.3-14.612%0.1-4 .60.6-10.155%3.2-14.25.8-17.011%0.5-7.30.5-10.1 1773.340%0.5-8.93.1-14.718%1.7-4.17.0-11.724%0.4-4. 03.1-11.826%1.8-12.46.2-17.229%0.5-0.82.7-12.6 1873.337%0.1-12.90.6-16.57%1.7-4.54.5-8.68%0.1-4.80 .6-10.28%2.3-8.74.0-10.75%0.5-7.50.5-9.9 1973.343%0.6-13.82.4-19.820%1.8-3.96.0-10.116%0.4-5 .42.3-13.319%2.1-7.65.3-11.218%0.6-9.11.9-13.8 2073.315%0.0-3.41.8-9.217%0.3-1.54.8-7.116%0.1-1.82 .0-8.136%1.1-8.94.6-12.725%0.4-6.21.8-11.1 2173.354%0.4-9.81.8-13.834%1.7-3.44.9-8.132%0.3-4.0 1.8-9.041%2.2-6.44.5-9.337%0.7-7.01.6-9.8 2273.312%0.1-16.61.7-29.44%1.8-4.210.5-16.21%0.1-5. 61.7-20.66%1.4-11.99.1-20.96%0.2-14.71.4-25.4 2389.817%0.6-4.13.1-9.124%1.1-2.25.7-7.78%0.4-2.03. 1-7.531%1.4-5.95.2-9.612%0.7-4.32.7-8.3 2459.03%0.0-5.10.6-10.318%1.0-3.05.3-9.711%0.0-2.30 .6-8.133%1.3-9.24.8-13.4-1%0.4-5.50.5-9.5 2560.4-2%0.0-5.70.5-11.016%1.0-3.25.3-10.21%0.0-2.4 0.5-8.134%1.3-10.14.8-14.1-5%0.4-6.20.4-10.1 26121.755%1.1-6.53.6-11.647%2.0-2.86.6-8.142%0.8-3. 43.5-9.247%2.3-4.35.9-8.041%1.0-5.33.1-9.4 2773.339%0.5-9.22.2-14.313%1.7-3.75.8-9.816%0.4-4.2 2.2-10.817%2.0-7.45.3-10.920%0.6-7.01.9-11.3 23% 21% 12%32% 14% P10 FixedP10 NormalP10 UniformP11 FixedP11 NormalP1 1 UniformP12 FixedP12 NormalP12 UniformP13 Normal Run DDD sS DDD sS DDD sS D sS 1-12%-2%-2%0.2-1.61.7-6.214%-4%-1%0.1-0.91.7-5.8-11 %-2%-3%0.3-1.21.4-5.2146%0.1-31.61.4-40.9 2-4%8%6%0.2-1.72.2-7.937%9%11%0.1-0.92.3-7.610%5%5% 0.2-0.91.8-6.6197%0.1-36.91.9-49.5 39%19%23%0.3-2.42.0-6.634%19%24%0.3-2.02.0-6.55%14% 19%0.5-2.21.8-6.0165%0.3-39.42.0-47.4 44%10%11%0.4-2.62.9-9.138%16%11%0.3-2.03.0-8.93%16% 10%0.3-1.92.5-8.0175%0.3-51.62.5-63.7 53%-1%8%0.2-1.52.2-7.431%-3%6%0.1-0.92.3-7.21%-1%4% 0.2-0.71.9-6.389%0.2-28.12.4-40.8 6-13%-2%1%0.2-1.71.3-4.66%2%1%0.1-0.91.3-4.2-14%-4% -2%0.5-1.61.1-3.9253%0.2-35.91.1-42.0 73%18%13%0.2-2.51.9-7.923%18%17%0.1-1.41.9-7.20%16% 11%0.4-1.91.6-6.6111%0.2-40.61.7-50.0 8-5%0%4%0.1-1.21.5-5.017%2%5%0.1-0.71.5-4.72%3%5%0. 3-0.91.3-4.2155%0.1-70.51.4-81.3 99%20%22%0.2-2.41.3-6.020%11%17%0.1-1.41.3-5.37%15% 17%0.5-2.21.2-5.1116%0.2-27.11.4-33.2 10-13%0%-5%0.1-1.41.0-4.20%0%-6%0.1-0.81.0-3.9-13%3%-8%0.4-1.30.9-3.6155%0.2-27.31.2-33.4 11-11%-4%-4%0.1-1.51.5-5.88%-4%-5%0.1-0.91.5-5.5-8% -3%-9%0.3-1.11.2-4.9107%0.2-31.81.8-41.0 124%17%19%0.2-2.51.6-7.424%11%14%0.1-1.41.7-6.7-3%1 2%13%0.4-1.91.4-6.286%0.1-16.11.2-23.0 137%19%26%0.2-2.31.5-6.244%36%50%0.2-1.71.6-6.08%34 %42%0.5-2.01.3-5.4252%0.4-34.12.0-40.9 14-9%-6%-3%0.2-1.52.5-7.420%-2%-7%0.1-0.82.6-7.1-2% -2%3%0.2-0.82.1-6.2100%0.2-36.92.6-49.5 15-7%5%1%0.2-1.51.6-5.514%4%0%0.1-0.91.6-5.2-8%2%-4 %0.3-1.21.4-4.8151%0.2-24.31.6-32.6 16-13%12%2%0.3-2.61.5-5.63%20%13%0.3-1.91.5-5.2-16% 6%-2%0.6-2.61.3-4.9222%0.3-44.51.3-50.4 177%6%12%0.3-2.72.4-8.336%10%21%0.3-2.12.5-8.15%12% 15%0.4-2.12.1-7.3150%0.3-45.12.3-55.7 18-12%1%0%0.4-2.71.6-5.45%9%13%0.3-2.01.6-4.9-12%-3 %-4%0.7-2.71.4-4.7184%0.3-68.51.3-73.7 193%12%15%0.3-3.21.8-8.029%24%23%0.2-2.41.8-7.5-1%1 7%23%0.5-2.81.6-6.9196%0.4-40.82.0-47.9 20-2%8%3%0.1-1.51.5-5.718%6%2%0.1-0.91.6-5.4-3%3%-1 %0.3-1.11.4-5.0117%0.1-32.91.3-43.0 211%22%15%0.3-2.61.6-5.919%25%21%0.3-2.01.6-5.62%17 %7%0.6-2.61.4-5.3232%0.3-30.81.5-36.0 22-5%1%-2%0.3-2.93.5-12.431%12%10%0.2-2.13.6-12.232 %23%17%0.1-1.42.9-10.571%0.4-43.84.0-59.0 23-1%11%5%0.2-2.01.7-6.721%6%2%0.1-1.11.7-6.1-3%6%1%0.4-1.61.4-5.6108%0.2-31.91.8-40.5 24-2%9%11%0.2-1.01.6-4.517%9%10%0.1-0.61.6-4.34%7%1 1%0.4-0.81.3-3.8195%0.1-32.01.4-40.6 25-3%11%4%0.1-1.21.5-4.921%13%3%0.1-0.71.5-4.73%14% 1%0.4-1.01.2-4.2196%0.1-32.01.5-40.6 2632%43%47%0.4-4.92.2-10.854%37%37%0.3-3.72.3-10.02 2%33%34%0.6-4.42.0-9.464%0.3-13.92.0-20.0 270%3%7%0.3-2.82.0-7.223%7%11%0.3-2.12.0-6.9-3%6%1% 0.5-2.51.7-6.3157%0.4-34.02.2-41.0 -1%9%9% 22%11%11% 0%9%8% 154% Policy (P) Avg Difference=

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95 Appendix F (continued): Kabir and Al-Olayan Compari son for b bb b=3.0 P13 UniformP14 NormalP14 UniformP15 NormalP15 Unifo rmP16 FixedP16 Normal Run t* D sS D sS D sS D sS D sS DD 176.065%0.1-16.11.3-23.833%0.1-7.91.4-19.916%0.1-5. 01.3-14.2129%0.3-30.01.2-35.956%0.3-16.71.1-22.152% 49% 276.083%0.1-18.81.7-29.243%0.1-8.61.9-24.729%0.1-5. 51.8-17.8186%0.2-38.91.6-47.381%0.2-21.31.4-28.8106 %111% 376.081%0.3-13.01.7-19.363%0.3-8.72.1-19.447%0.2-5. 61.7-13.2125%0.5-28.01.8-33.173%0.4-11.51.5-16.541% 40% 476.062%0.3-23.42.5-34.045%0.2-10.42.6-25.126%0.2-7 .62.6-19.8134%0.3-42.32.2-50.152%0.3-21.72.1-29.818 %19% 571.554%0.1-14.31.9-24.831%0.1-7.42.4-21.524%0.1-4. 71.9-15.9127%0.2-38.22.0-46.871%0.2-20.91.6-28.576% 58% 676.074%0.1-18.21.0-23.252%0.1-8.21.1-18.618%0.1-5. 01.0-12.3131%0.4-22.60.9-26.137%0.4-12.90.8-16.031% 39% 799.271%0.2-20.61.7-28.329%0.1-6.81.7-23.524%0.1-5. 71.7-16.172%0.3-30.31.4-36.345%0.3-16.81.4-22.212%3 4% 859.888%0.1-56.811-67.124%0.1-12.91.4-31.97%0.1-11. 21.2-28.596%0.3-48.01.1-54.345%0.3-40.21.0-46.594%8 9% 999.255%0.1-13.80.7-18.839%0.1-6.61.4-15.325%0.0-4. 10.7-10.389%0.5-21.81.2-25.447%0.4-12.50.6-15.624%2 6% 1071.562%0.1-14.00.7-19.041%0.1-6.71.2-15.317%0.0-4 .40.7-10.9118%0.4-21.91.0-25.450%0.4-12.60.6-15.832 %32% 1171.523%0.1-16.11.4-23.718%0.1-7.81.8-19.70%0.1-5. 01.4-14.692%0.3-30.11.5-35.919%0.3-16.71.1-22.045%4 2% 1299.284%0.2-16.31.7-23.230%0.1-4.91.3-13.733%0.1-4 .91.7-13.571%0.4-15.41.1-20.168%0.4-15.61.4-20.230% 39% 1359.8162%0.2-39.81.2-47.167%0.3-8.42.1-18.440% 7%31% 1471.559%0.1-37.21.6-49.824%0.1-8.62.6-24.611%0.1-8 .51.7-24.4101%0.2-38.92.2-47.358%0.2-39.21.4-47.684 %78% 1571.567%0.0-14.61.1-21.744%0.1-6.11.6-16.223%0.0-4 .11.1-12.2166%0.3-27.01.3-32.381%0.3-17.51.0-22.349 %48% 1671.5103%0.2-37.11.1-42.338%0.2-10.91.3-21.721%0.2 -8.51.1-17.268%0.6-21.01.2-24.627%0.5-17.20.9-20.22 3%25% 1773.392%0.3-28.52.3-37.545%0.3-10.62.4-24.331% 2%23% 1873.387%0.2-39.81.1-44.229%0.3-11.61.3-22.814% %28% 1973.381%0.3-21.41.7-27.346%0.3-8.62.0-19.524%0.2-6 .11.7-14.674%0.5-20.81.7-25.334%0.5-12.21.5-16.229% 29% 2073.335%0.1-9.11.3-16.235%0.1-5.91.3-18.119%0.1-3. 11.3-11.3186%0.3-41.81.1-48.266%0.3-14.81.1-19.955% 47% 2173.3102%0.3-18.61.4-23.471%0.3-7.51.5-15.439% 4%57% 2273.342%0.1-44.02.0-59.114%0.3-9.74.1-28.56%0.1-9. 22.0-27.651%0.1-36.53.4-47.427%0.1-36.71.6-47.2-1%1 % 2389.856%0.2-16.41.6-23.432%0.1-7.91.8-19.520%0.1-4 .91.6-13.493%0.4-28.51.5-33.948%0.4-16.01.3-20.742% 46% 2459.093%0.1-21.00.9-28.952%0.1-7.81.5-19.233%0.0-6 .10.9-16.1161%0.4-28.61.2-33.980%0.4-20.10.8-25.411 6%113% 2560.481%0.1-28.11.3-36.753%0.1-7.91.5-19.619%0.1-7 .41.4-18.7159%0.4-28.61.3-33.966%0.3-25.81.1-31.395 %91% 26121.755%0.4-8.62.1-15.240%0.3-5.32.0-12.837%0.3-5 .52.1-13.343%0.5-10.21.7-14.838%0.5-7.31.8-12.962%6 4% 2773.361%0.3-17.01.8-22.940%0.3-8.12.3-17.917%0.2-5 .61.9-13.577%0.5-20.61.9-25.132%0.5-11.71.6-16.224% 24% 73% 40% 23%109% 52% 47%47% P16 UniformP17 NormalP17 UniformP18 FixedP18 Normal P18 UniformP19 NormalP19 Uniform Run D sS D sS D sS DDD sS DD sS 152%01.0-2.0-12%0.0-3.00.0-5.011%0.0-2.00.0-4.053%5 6%56%0.0-4.00.0-11.015%17%1.0-3.06.0-10.0 2109%01.0-3.03%0.0-3.01.0-5.047%0.0-2.01.0-4.064%67 %66%0.0-5.00.0-14.024%27%1.0-4.08.0-13.0 340%3.0-4.05.0-6.061%3.0-8.05.0-10.056%3.0-6.05.0-8 .0140%135%138%0.0-6.00.0-12.039%50%0.0-4.00.0-10.0 420%3.0-4.05.0-6.023%3.0-7.05.0-9.020%3.0-7.05.0-9. 086%86%88%0.0-6.00.0-16.028%32%2.0-5.08.0-15.0 566%01.0-2.019%0.0-2.01.0-5.048%0.0-1.01.0-4.056%54 %57%3.0-4.012.0-14.029%35%1.0-4.09.0-13.0 636%01.0-2.0-16%0.0-3.01.0-5.013%0.0-2.01.0-4.084%8 5%86%0.0-5.00.0-9.018%22%1.0-3.04.0-8.0 729%0.0-1.01.0-3.010%0.0-3.01.0-5.06%0.0-2.01.0-4.0 72%74%74%0.0-5.00.0-12.031%36%0.0-4.00.0-10.0 8101%01.0-2.0-18%0.0-5.01.0-6.036%0.0-4.01.0-5.039% 38%38%4.0-6.011.0-12.03%7%0.0-8.00.0-15.0 926%00.0-2.0-7%0.0-3.00.0-4.010%0.0-2.00.0-4.0104%1 03%103%0.0-4.00.0-9.038%42%1.0-2.05.0-6.0 1028%00.0-2.0-14%0.0-3.00.0-4.0-3%0.0-3.00.0-4.069% 74%69%0.0-4.00.0-8.038%37%0.0-3.00.0-9.0 1138%01.0-2.0-12%0.0-3.01.0-5.06%0.0-2.00.0-4.040%4 0%36%0.0-4.00.0-12.020%17%0.0-4.00.0-12.0 1245%01.0-3.07%0.0-2.00.0-4.018%0.0-2.01.0-4.098%10 2%103%3.0-4.09.0-11.042%48%1.0-3.06.0-8.0 1330%45.0-6.048%4.0-8.05.0-9.042%4.0-8.05.0-10.0113 %113%114%6.0-7.012.0-13.039%51%0.0-6.00.0-11.0 1486%01.0-2.0-7%0.0-3.01.0-5.025%0.0-4.01.0-5.042%4 2%44%0.0-5.00.0-14.021%25%2.0-5.09.0-14.0 1544%00.0-2.013%0.0-2.00.0-4.028%0.0-1.00.0-4.072%7 5%72%0.0-3.00.0-10.046%45%0.0-3.00.0-10.0 1626%4.0-5.05.0-6.035%4.0-8.05.0-10.032%4.0- 1726%3.0-4.05.0-6.028%3.0-8.05.0-10.030%3.0- 1826%4.0-5.05.0-6.034%4.0-7.05.0-8.030%4.0-8.05.0-9 .086%87%87%0.0-7.00.0-11.017%19%0.0-8.00.0-10.0 1928%4.0-5.05.0-7.037%4.0-7.05.0-9.032%4.0-7.05.0-9 .0104%104%102%0.0-8.00.0-15.029%34%0.0-6.00.0-11.0 2049%00.0-2.028%0.0-5.00.0-6.038%0.0-1.00.0-4.061%5 9%60%2.0-3.08.0-10.019%21%0.0-3.05.0-10.0 2154%3.0-4.05.0-6.0.74%3.0-7.05.0-9.066%3.0- 22-5%4.0-5.06.0-7.04%4.0-7.05.0-9.0-2%4.0-8.05.0-10 .036%37%31%6.0-8.019.0-22.08%5%2.0-7.011.0-17.0 2337%01.0-3.00%0.0-3.01.0-5.010%0.0-2.01.0-4.074%76 %72%0.0-5.00.0-11.037%34%1.0-3.06.0-9.0 24120%01.0-2.0-16%0.0-3.01.0-5.037%0.0-3.00.0-4.079 %77%82%3.0-4.010.0-11.024%35%1.0-4.05.0-11.0 2584%01.0-2.0-8%0.0-3.01.0-5.023%0.0-2.01.0-4.063%6 5%61%3.0-5.010.0-11.025%25%1.0-4.05.0-11.0 2665%4.0-5.05.0-7.064%4.0-7.05.0-8.064%4.0-6.05.0-8 .0147%148%151%5.0-7.011.0-14.069%68%0.0-4.00.0-9.0 2726%4.0-5.05.0-6.035%4.0-7.05.0-8.033%4.0-7.05.0-8 .092%90%94%5.0-7.012.0-14.023%27%0.0-5.00.0-10.0 48% 15%28% 82%83%82% 30%33% Policy (P) Avg Difference=

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96 Appendix G: Cost Comparison with Additional CR/UR V alues Policy (P)P1 NormalP1 UniformP2 NormalP2 UniformP3 NormalP3 Uniform CR/UR Value D D D D D D 0.02529%29%14%13%20%20% 0.0529%29%13%13%21%21% 0.07530%31%14%15%21%21% 0.128%28%13%13%19%19% P4 NormalP4 UniformP5 NormalP5 UniformP6 NormalP6 U niform CR/UR Value D D D D D D 0.025119%147%65%90%99%132% 0.05123%150%66%91%99%133% 0.075125%153%68%94%102%136% 0.1124%152%67%92%100%133% P7 NormalP7 UniformP8 NormalP8 UniformP9 NormalP9 U niform CR/UR Value D D D D D D 0.02543%23%21%12%32%14% 0.0545%30%22%16%32%21% 0.07548%55%24%28%34%40% 0.151%61%24%31%35%44% P10 NormalP10 UniformP11 NormalP11 UniformP12 Norma lP12 Uniform CR/UR Value D D D D D D 0.0259%9%11%11%9%8% 0.0510%10%12%13%9%9% 0.07510%12%12%13%11%11% 0.111%11%14%14%10%11% P13 NormalP13 UniformP14 NormalP14 UniformP15 Norma lP15 Uniform CR/UR Value D D D D D D 0.025154%73%40%23%109%52% 0.05185%115%43%34%124%78% 0.075184%129%47%44%123%90% 0.1196%128%47%45%124%86% P16 NormalP16 UniformP17 NormalP17 UniformP18 Norma lP18 Uniform CR/UR Value D D D D D D 0.02547%48%15%28%83%82% 0.0543%43%16%24%84%83% 0.07546%47%18%24%86%86% 0.139%40%18%22%84%84% P19 NormalP19 Uniform CR/UR Value D D 0.02530%33% 0.0531%34% 0.07533%37% 0.133%37%

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97 Appendix H: Matlab Code Text in italics is added for code explanation Parameter definition and values Penalty_Cost=13.5; Holding_Cost=.6875; Ordering_Cost=8.75; Cf=55; Cp=25; Demand_History_Fixed=xlsread('demand data.xls','Deman d'); h=Holding_Cost; p=Penalty_Cost; K=Ordering_Cost; Shape=3; Shape_Fixed=3; Shape_Normal=3; Shape_Uniform=3; Scale=100; Scale_Fixed=90; Scale_Normal=90; Scale_Uniform=90; PC=.99; CR=.025; UR=.025; k=30; Scale_Lead=10; Shape_Lead=3.2; Period_Maintenance_Fixed=0; Period_Failure_Fixed=0; Period_Maintenance_Normal=0; Period_Failure_Normal=0; Period_Maintenance_Uniform=0; Period_Failure_Uniform=0; Period_Holding_FO_Lead_Fixed_PowerAppx=0; Period_Holding_FO_Lead_Fixed_NormalAppx=0; Period_Holding_FO_Lead_Fixed_NaddorAppx=0; Period_Holding_FO_Lead_Normal_PowerAppx=0; Period_Holding_FO_Lead_Normal_NormalAppx=0; Period_Holding_FO_Lead_Normal_NaddorAppx=0; Period_Holding_FO_Lead_Uniform_PowerAppx=0; Period_Holding_FO_Lead_Uniform_NormalAppx=0; Period_Holding_FO_Lead_Uniform_NaddorAppx=0; Period_Shortage_FO_Lead_Fixed_PowerAppx=0; Period_Shortage_FO_Lead_Fixed_NormalAppx=0;

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98 Appendix H (continued): Matlab Code Period_Shortage_FO_Lead_Fixed_NaddorAppx=0; Period_Shortage_FO_Lead_Normal_PowerAppx=0; Period_Shortage_FO_Lead_Normal_NormalAppx=0; Period_Shortage_FO_Lead_Normal_NaddorAppx=0; Period_Shortage_FO_Lead_Uniform_PowerAppx=0; Period_Shortage_FO_Lead_Uniform_NormalAppx=0; Period_Shortage_FO_Lead_Uniform_NaddorAppx=0; Period_Order_FO_Lead_Fixed_PowerAppx=0; Period_Order_FO_Lead_Fixed_NormalAppx=0; Period_Order_FO_Lead_Fixed_NaddorAppx=0; Period_Order_FO_Lead_Normal_PowerAppx=0; Period_Order_FO_Lead_Normal_NormalAppx=0; Period_Order_FO_Lead_Normal_NaddorAppx=0; Period_Order_FO_Lead_Uniform_PowerAppx=0; Period_Order_FO_Lead_Uniform_NormalAppx=0; Period_Order_FO_Lead_Uniform_NaddorAppx=0; On_Hand_FO_Lead_Fixed_PowerAppx=ceil(S_FO_Lead_Fixed_Power Appx); On_Hand_FO_Lead_Fixed_NormalAppx=ceil(S_FO_Lead_Fixed_Norm alAppx); On_Hand_FO_Lead_Fixed_NaddorAppx=ceil(S_FO_Lead_Fixed_Nadd orAppx); On_Hand_FO_Lead_Normal_PowerAppx=ceil(S_FO_Lead_Normal_Power Appx); On_Hand_FO_Lead_Normal_NormalAppx=ceil(S_FO_Lead_Normal_Norm alAppx); On_Hand_FO_Lead_Normal_NaddorAppx=ceil(S_FO_Lead_Normal_Nad dorAppx); On_Hand_FO_Lead_Uniform_PowerAppx=ceil(S_FO_Lead_Uniform_Po werAppx); On_Hand_FO_Lead_Uniform_NormalAppx=ceil(S_FO_Lead_Uniform_N ormalAppx); On_Hand_FO_Lead_Uniform_NaddorAppx=ceil(S_FO_Lead_Uniform_N addorAppx); On_Order_FO_Lead_Fixed_PowerAppx=0; On_Order_FO_Lead_Fixed_NormalAppx=0; On_Order_FO_Lead_Fixed_NaddorAppx=0; On_Order_FO_Lead_Normal_PowerAppx=0; On_Order_FO_Lead_Normal_NormalAppx=0; On_Order_FO_Lead_Normal_NaddorAppx=0; On_Order_FO_Lead_Uniform_PowerAppx=0; On_Order_FO_Lead_Uniform_NormalAppx=0; On_Order_FO_Lead_Uniform_NaddorAppx=0; Z_FO_Lead_Fixed_PowerAppx=On_Hand_FO_Lead_Fixed_PowerAppx+On _Order_F O_Lead_Fixed_PowerAppx; Z_FO_Lead_Fixed_NormalAppx=On_Hand_FO_Lead_Fixed_NormalApp x+On_Order_ FO_Lead_Fixed_NormalAppx; Z_FO_Lead_Fixed_NaddorAppx=On_Hand_FO_Lead_Fixed_NaddorApp x+On_Order_ FO_Lead_Fixed_NaddorAppx; Z_FO_Lead_Normal_PowerAppx=On_Hand_FO_Lead_Normal_PowerAppx +On_Order _FO_Lead_Normal_PowerAppx; Z_FO_Lead_Uniform_PowerAppx=On_Hand_FO_Lead_Uniform_PowerAppx +On_Ord

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99 Appendix H (continued): Matlab Code er_FO_Lead_Uniform_PowerAppx; Z_FO_Lead_Normal_NormalAppx=On_Hand_FO_Lead_Normal_NormalAppx +On_Or der_FO_Lead_Normal_NormalAppx; Z_FO_Lead_Uniform_NormalAppx=On_Hand_FO_Lead_Uniform_Normal Appx+On_ Order_FO_Lead_Uniform_NormalAppx; Z_FO_Lead_Normal_NaddorAppx=On_Hand_FO_Lead_Normal_NaddorAppx +On_Or der_FO_Lead_Normal_NaddorAppx; Z_FO_Lead_Uniform_NaddorAppx=On_Hand_FO_Lead_Uniform_Naddor Appx+On_O rder_FO_Lead_Uniform_NaddorAppx; Simulation_Length=100; reset=15; history=10; Lead_Orders_FO_Lead_Fixed_PowerAppx=zeros(Simulation_Le ngth+lamda+1,1); Lead_Orders_FO_Lead_Normal_PowerAppx=zeros(Simulation_Le ngth+lamda+1,1); Lead_Orders_FO_Lead_Uniform_PowerAppx=zeros(Simulation_L ength+lamda+1,1); Lead_Orders_FO_Lead_Fixed_NormalAppx=zeros(Simulation_L ength+lamda+1,1); Lead_Orders_FO_Lead_Normal_NormalAppx=zeros(Simulation_L ength+lamda+1,1); Lead_Orders_FO_Lead_Uniform_NormalAppx=zeros(Simulation_ Length+lamda+1,1); Lead_Orders_FO_Lead_Fixed_NaddorAppx=zeros(Simulation_L ength+lamda+1,1); Lead_Orders_FO_Lead_Normal_NaddorAppx=zeros(Simulation_L ength+lamda+1,1); Lead_Orders_FO_Lead_Uniform_NaddorAppx=zeros(Simulation_ Length+lamda+1,1); Determine initial maintenance strategy tstar=Optimal_Replacement_Time(Shape_Fixed,Scale_Fix ed,Cp,Cf); tstar_Fixed=tstar; tstar_Normal=tstar; tstar_Uniform=tstar; sigma=Maintenance_Policy(PC,CR,tstar); Uniform_a=tstar-UR*tstar; Uniform_b=tstar+UR*tstar; t=xlsread('part data.xls','Part Ages'); n=length(t); Determine period length [M,V] = weibstat(1/(Scale_Lead^Shape_Lead),Shape_Lead ); deltat=M; lamda=ceil((weibrnd((1/(Scale_Lead^Shape_Lead)),Shap e_Lead))/deltat); Determine initial replacement modes (failed or maintaine d) for all parts t_Fixed=Replacement_mode_Fixed(t,Shape,Scale,tstar) ; Data_Combined_Fixed(:,1)=t_Fixed(:,1); Data_Combined_Fixed(:,2)=0; t_Normal=Replacement_mode_Normal(t,Shape,Scale,tstar, sigma);

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100 Appendix H (continued): Matlab Code Data_Combined_Normal(:,1)=t_Normal(:,1); Data_Combined_Normal(:,2)=0; t_Uniform=Replacement_mode_Uniform(t,Shape,Scale,tsta r,Uniform_a,Uniform_b); Data_Combined_Uniform(:,1)=t_Uniform(:,1); Data_Combined_Uniform(:,2)=0; Determine the conditional probabilities and demand dis tribution for future periods Pconditional_FO_Fixed=Conditional_Failure_Only(n,t_Fi xed,k,deltat,Shape_Fixed,Scale _Fixed); Pconditional_FO_Normal=Conditional_Failure_Only(n,t_No rmal,k,deltat,Shape_Normal ,Scale_Normal); Pconditional_FO_Uniform=Conditional_Failure_Only(n,t_Un iform,k,deltat,Shape_Unifo rm,Scale_Uniform); Demand_Failure_Only_Fixed=Demand_Distribution(n,k,Pcond itional_FO_Fixed); Demand_Failure_Only_Normal=Demand_Distribution(n,k,Pcon ditional_FO_Normal); Demand_Failure_Only_Uniform=Demand_Distribution(n,k,Pcon ditional_FO_Uniform); FO_Demand_pdf_Fixed=kExpected_Demand(n,k,Demand_Failure _Only_Fixed); FO_Demand_pdf_Normal=kExpected_Demand(n,k,Demand_Failure _Only_Normal); FO_Demand_pdf_Uniform=kExpected_Demand(n,k,Demand_Failur e_Only_Uniform); Determine the initial inventory values Demand_FO_Lead_Fixed=Expected_Demand(n,lamda+1,Demand_ Failure_Only_Fixed) ; [s_FO_Lead_Fixed_PowerAppx S_FO_Lead_Fixed_PowerAppx] =Power_Approximation(lamda,p,h,K,Demand_FO_Lead_Fixed); [s_FO_Lead_Fixed_NormalAppx S_FO_Lead_Fixed_NormalAppx] =Normal_Approximation(lamda,p,h,K,Demand_FO_Lead_Fixed) ; [s_FO_Lead_Fixed_NaddorAppx S_FO_Lead_Fixed_NaddorAppx] =Naddor_Approximation(lamda,p,h,K,Demand_FO_Lead_Fixed) ; FO_Lead_Fixed_Exp_Demand=Demand_FO_Lead_Fixed(1,1); Demand_FO_Lead_Normal=Expected_Demand(n,lamda+1,Demand_F ailure_Only_Nor mal); [s_FO_Lead_Normal_PowerAppx S_FO_Lead_Normal_PowerAppx] =Power_Approximation(lamda,p,h,K,Demand_FO_Lead_Normal); [s_FO_Lead_Normal_NormalAppx S_FO_Lead_Normal_NormalAppx] =Normal_Approximation(lamda,p,h,K,Demand_FO_Lead_Normal) ; [s_FO_Lead_Normal_NaddorAppx S_FO_Lead_Normal_NaddorAppx] =Naddor_Approximation(lamda,p,h,K,Demand_FO_Lead_Normal) ; FO_Lead_Normal_Exp_Demand=Demand_FO_Lead_Normal(1,1); Demand_FO_Lead_Uniform=Expected_Demand(n,lamda+1,Demand_ Failure_Only_Uni form); [s_FO_Lead_Uniform_PowerAppx S_FO_Lead_Uniform_PowerAppx] =Power_Approximation(lamda,p,h,K,Demand_FO_Lead_Uniform);

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101 Appendix H (continued): Matlab Code [s_FO_Lead_Uniform_NormalAppx S_FO_Lead_Uniform_NormalAppx ] =Normal_Approximation(lamda,p,h,K,Demand_FO_Lead_Uniform ); [s_FO_Lead_Uniform_NaddorAppx S_FO_Lead_Uniform_NaddorAppx ] =Naddor_Approximation(lamda,p,h,K,Demand_FO_Lead_Unifor m); FO_Lead_Uniform_Exp_Demand=Demand_FO_Lead_Uniform(1,1); Begin simulated periods Iteration=1; while Iteration
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102 Appendix H (continued): Matlab Code Period_Order_FO_Lead_Normal_NaddorAppx=0; Period_Holding_FO_Lead_Uniform_PowerAppx=0; Period_Shortage_FO_Lead_Uniform_PowerAppx=0; Period_Order_FO_Lead_Uniform_PowerAppx=0; Period_Holding_FO_Lead_Uniform_NormalAppx=0; Period_Shortage_FO_Lead_Uniform_NormalAppx=0; Period_Order_FO_Lead_Uniform_NormalAppx=0; Period_Holding_FO_Lead_Uniform_NaddorAppx=0; Period_Shortage_FO_Lead_Uniform_NaddorAppx=0; Period_Order_FO_Lead_Uniform_NaddorAppx=0; Check to see if orders are received if Lead_Orders_FO_Lead_Fixed_PowerAppx(Iteration)>0 On_Hand_FO_Lead_Fixed_PowerAppx=On_Hand_FO_Lead_Fixed_PowerAp p x+Lead_Orders_FO_Lead_Fixed_PowerAppx(Iteration); On_Order_FO_Lead_Fixed_PowerAppx=On_Order_FO_Lead_Fixed_Powe rApp x-Lead_Orders_FO_Lead_Fixed_PowerAppx(Iteration); end if Lead_Orders_FO_Lead_Normal_PowerAppx(Iteration)>0 On_Hand_FO_Lead_Normal_PowerAppx=On_Hand_FO_Lead_Normal_PowerAppx+Lead_Orders_FO_Lead_Normal_PowerAppx(Iteration); On_Order_FO_Lead_Normal_PowerAppx=On_Order_FO_Lead_Normal_Po wer Appx-Lead_Orders_FO_Lead_Normal_PowerAppx(Iteration); end if Lead_Orders_FO_Lead_Uniform_PowerAppx(Iteration)>0 On_Hand_FO_Lead_Uniform_PowerAppx=On_Hand_FO_Lead_Uniform_Po we rAppx+Lead_Orders_FO_Lead_Uniform_PowerAppx(Iteration) ; On_Order_FO_Lead_Uniform_PowerAppx=On_Order_FO_Lead_Uniform_P ow erAppx-Lead_Orders_FO_Lead_Uniform_PowerAppx(Iteration); end if Lead_Orders_FO_Lead_Fixed_NormalAppx(Iteration)>0 On_Hand_FO_Lead_Fixed_NormalAppx=On_Hand_FO_Lead_Fixed_Norma lA ppx+Lead_Orders_FO_Lead_Fixed_NormalAppx(Iteration); On_Order_FO_Lead_Fixed_NormalAppx=On_Order_FO_Lead_Fixed_Nor malA ppx-Lead_Orders_FO_Lead_Fixed_NormalAppx(Iteration); end if Lead_Orders_FO_Lead_Normal_NormalAppx(Iteration)>0 On_Hand_FO_Lead_Normal_NormalAppx=On_Hand_FO_Lead_Normal_NormalAppx+Lead_Orders_FO_Lead_Normal_NormalAppx(Iteration); On_Order_FO_Lead_Normal_NormalAppx=On_Order_FO_Lead_Normal_Nor m alAppx-Lead_Orders_FO_Lead_Normal_NormalAppx(Iteration); end

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103 Appendix H (continued): Matlab Code if Lead_Orders_FO_Lead_Uniform_NormalAppx(Iteration)>0 On_Hand_FO_Lead_Uniform_NormalAppx=On_Hand_FO_Lead_Uniform_NormalAppx+Lead_Orders_FO_Lead_Uniform_NormalAppx(Iteration) ; On_Order_FO_Lead_Uniform_NormalAppx=On_Order_FO_Lead_Uniform_ Nor malAppx-Lead_Orders_FO_Lead_Uniform_NormalAppx(Iteration) ; end if Lead_Orders_FO_Lead_Fixed_NaddorAppx(Iteration)>0 On_Hand_FO_Lead_Fixed_NaddorAppx=On_Hand_FO_Lead_Fixed_Naddo rAp px+Lead_Orders_FO_Lead_Fixed_NaddorAppx(Iteration); On_Order_FO_Lead_Fixed_NaddorAppx=On_Order_FO_Lead_Fixed_Nad dorA ppx-Lead_Orders_FO_Lead_Fixed_NaddorAppx(Iteration); end if Lead_Orders_FO_Lead_Normal_NaddorAppx(Iteration)>0 On_Hand_FO_Lead_Normal_NaddorAppx=On_Hand_FO_Lead_Normal_Nadd o rAppx+Lead_Orders_FO_Lead_Normal_NaddorAppx(Iteration); On_Order_FO_Lead_Normal_NaddorAppx=On_Order_FO_Lead_Normal_Nad d orAppx-Lead_Orders_FO_Lead_Normal_NaddorAppx(Iteration); end if Lead_Orders_FO_Lead_Uniform_NaddorAppx(Iteration)>0 On_Hand_FO_Lead_Uniform_NaddorAppx=On_Hand_FO_Lead_Uniform_NaddorAppx+Lead_Orders_FO_Lead_Uniform_NaddorAppx(Iteration ); On_Order_FO_Lead_Uniform_NaddorAppx=On_Order_FO_Lead_Uniform_ Na ddorAppx-Lead_Orders_FO_Lead_Uniform_NaddorAppx(Iteratio n); end Check to see if parts fail or maintained during per iod [output t_Fixed]= Check_t(t_Fixed,Shape,Scale,tstar_Fixed,deltat,On_Han d_FO_Lead_Fixed_PowerAppx, On_Hand_FO_Lead_Fixed_NormalAppx,On_Hand_FO_Lead_Fixed_Naddo rAppx); Period_Failure_Fixed=output(1); Period_Maintenance_Fixed=output(2); On_Hand_FO_Lead_Fixed_PowerAppx=output(3); On_Hand_FO_Lead_Fixed_NormalAppx=output(4); On_Hand_FO_Lead_Fixed_NaddorAppx=output(5); Z_FO_Lead_Fixed_PowerAppx=On_Hand_FO_Lead_Fixed_PowerAppx+On _Order_F O_Lead_Fixed_PowerAppx; Z_FO_Lead_Fixed_NormalAppx=On_Hand_FO_Lead_Fixed_NormalApp x+On_Order_ FO_Lead_Fixed_NormalAppx; Z_FO_Lead_Fixed_NaddorAppx=On_Hand_FO_Lead_Fixed_NaddorApp x+On_Order_ FO_Lead_Fixed_NaddorAppx; [output t_Normal] =Check_t_Normal(t_Normal,Shape,Scale,tstar_Normal,sig ma,deltat,On_Hand_FO_Lead

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104 Appendix H (continued): Matlab Code _Normal_PowerAppx,On_Hand_FO_Lead_Normal_NormalAppx,On_Hand_FO_ Lead_ Normal_NaddorAppx); Period_Failure_Normal=output(1); Period_Maintenance_Normal=output(2); On_Hand_FO_Lead_Normal_PowerAppx=output(3); On_Hand_FO_Lead_Normal_NormalAppx=output(4); On_Hand_FO_Lead_Normal_NaddorAppx=output(5); Z_FO_Lead_Normal_PowerAppx=On_Hand_FO_Lead_Normal_PowerAppx +On_Order _FO_Lead_Normal_PowerAppx; Z_FO_Lead_Normal_NormalAppx=On_Hand_FO_Lead_Normal_NormalAppx +On_Or der_FO_Lead_Normal_NormalAppx; Z_FO_Lead_Normal_NaddorAppx=On_Hand_FO_Lead_Normal_NaddorAppx +On_Or der_FO_Lead_Normal_NaddorAppx; [output t_Uniform]= Check_t_Uniform(t_Uniform,Shape,Scale,Uniform_a,Uniform _b,deltat,On_Hand_FO_L ead_Uniform_PowerAppx,On_Hand_FO_Lead_Uniform_NormalAppx,On_ Hand_FO_L ead_Uniform_NaddorAppx); Period_Failure_Uniform=output(1); Period_Maintenance_Uniform=output(2); On_Hand_FO_Lead_Uniform_PowerAppx=output(3); On_Hand_FO_Lead_Uniform_NormalAppx=output(4); On_Hand_FO_Lead_Uniform_NaddorAppx=output(5); Z_FO_Lead_Uniform_PowerAppx=On_Hand_FO_Lead_Uniform_PowerAppx +On_Ord er_FO_Lead_Uniform_PowerAppx; Z_FO_Lead_Uniform_NormalAppx=On_Hand_FO_Lead_Uniform_Normal Appx+On_ Order_FO_Lead_Uniform_NormalAppx; Z_FO_Lead_Uniform_NaddorAppx=On_Hand_FO_Lead_Uniform_Naddor Appx+On_O rder_FO_Lead_Uniform_NaddorAppx; If a replacement occurs, re-estimate distribution va lues and optimal replacement times if Period_Maintenance_Fixed+Period_Failure_Fixed>0 [t_Fixed Failure_Data_Fixed Replacement_Data_Fixed Period_Censored_Fixed]=Add_Data_Fixed(t_Fixed,Shape,S cale,tstar_Fixed); Data_Combined_Fixed=Combine_Data2(n,Data_Combined_Fixe d,Failure_Data_ Fixed,Replacement_Data_Fixed,Period_Censored_Fixed); failures=0; censored=0; [max b]=size(Data_Combined_Fixed); Combined_Fixed=Data_Combined_Fixed; scrub=0; for i=1:max if Data_Combined_Fixed(i,2)==1

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105 Appendix H (continued): Matlab Code failures=failures+1; else if Data_Combined_Fixed(i,2)>=0 censored=censored+1; else Combined_Fixed(i-scrub,:)=[]; scrub=scrub+1; end end end Combined_Fixed=Combined_Fixed censored=censored failures=failures Shape_Fixed=Shape_Fixed best_method=input('Enter preferred estimation metho d (1-MLE, 2-KME, 3PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE, 10-S imulate, 11NA): '); if best_method==1 Shape_Old=Shape_Fixed; Scale_Old=Scale_Fixed; [Shape_Fixed Scale_Fixed]=MLE_Estimato r(Combined_Fixed); ok=input('check: 1-Yes, 2-No: ') if ok==1 tstar_Fixed=Optimal_Replacement_Time(Shape_Fixed,Sca le_Fixe d,Cp,Cf); else Shape_Fixed=Shape_Old Scale_Fixed=Scale_Old best_method=input('Enter preferred estimation metho d (1-MLE, 2KME, 3-PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE, 10-Simulate): '); end end if best_method==10 [max b]=size(Combined_Fixed); flag=0; while flag==0 Total_Iterations=input('Enter sim ulation length: '); Best_Estimator(max,Shape_Fixed,ce nsored,Total_Iterations) flag=input('Simulation sufficient ? (0-No, 1-Yes): '); end best_method=input('Enter preferred estimation metho d (1-MLE, 2-KME, 3-PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE): );

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106 Appendix H (continued): Matlab Code end if best_method==2 tstar_Fixed=Optimal_Replacement_Time_KME (Combined_Fixed,Cf,Cp); end if best_method==3 tstar_Fixed=Optimal_Replacement_Time_PEXE(Combined_F ixed,Cf,Cp); end if best_method==4 tstar_Fixed=Optimal_Replacement_Time_FRWE(Combined_F ixed,Cf,Cp ); end if best_method==5 tstar_Fixed=Optimal_Replacement_Time_KLM(Combined_Fi xed,Cf,Cp); end if best_method==6 [Shape_Fixed Scale_Fixed]=ROSS_Estimato r(Combined_Fixed); tstar_Fixed=Optimal_Replacement_Time(Shape_Fixed,Sca le_Fixed,Cp,C f); end if best_method==7 [Shape_Fixed Scale_Fixed]=WH_Estimator( Combined_Fixed); tstar_Fixed=Optimal_Replacement_Time(Shape_Fixed,Sca le_Fixed,Cp,C f); end if best_method==8 [Shape_Fixed Scale_Fixed]=BE_Estimator (Combined_Fixed); tstar_Fixed=Optimal_Replacement_Time(Shape_Fixed,Sca le_Fixed,Cp,C f); end if best_method==9 [Shape_Fixed Scale_Fixed]=MPMLE_Estima tor(Combined_Fixed); tstar_Fixed=Optimal_Replacement_Time(Shape_Fixed,Sca le_Fixed,Cp,C f); end end if Period_Maintenance_Uniform+Period_Failure_Uniform> 0 [t_Uniform Failure_Data_Uniform Replacement_Data_Uniform Period_Censored_Uniform]=Add_Data_Uniform(t_Uniform,Shap e,Scale,Unifor m_a,Uniform_b);

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107 Appendix H (continued): Matlab Code Data_Combined_Uniform=Combine_Data2(n,Data_Combined_Uni form,Failure_ Data_Uniform,Replacement_Data_Uniform,Period_Censored_Un iform); failures=0; censored=0; [max b]=size(Data_Combined_Uniform); Combined_Uniform=Data_Combined_Uniform; scrub=0; for i=1:max if Data_Combined_Uniform(i,2)==1 failures=failures+1; else if Data_Combined_Uniform(i,2)>=0 censored=censored+1; else Combined_Uniform(i-scrub,:)=[] ; scrub=scrub+1; end end end Combined_Uniform=Combined_Uniform censored=censored failures=failures Shape_Uniform=Shape_Uniform best_method=input('Enter preferred estimation metho d (1-MLE, 2-KME, 3PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE, 10-S imulate, 11NA): '); if best_method==1 Shape_Old=Shape_Uniform; Scale_Old=Scale_Uniform; [Shape_Uniform Scale_Uniform]=MLE_Estima tor(Combined_Uniform); ok=input('check: 1-Yes, 2-No: ') if ok==1 tstar_Uniform=Optimal_Replacement_Time(Shape_Uniform,S cale _Uniform,Cp,Cf); else Shape_Uniform=Shape_Old Scale_Uniform=Scale_Old best_method=input('Enter preferred estimation metho d (1-MLE, 2KME, 3-PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE, 10-Simulate): '); end end if best_method==10

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108 Appendix H (continued): Matlab Code [max b]=size(Combined_Uniform); flag=0; while flag==0 Total_Iterations=input('Enter sim ulation length: '); Best_Estimator(max,Shape_Uniform,c ensored,Total_Iterations) flag=input('Simulation sufficient ? (0-No, 1-Yes): '); end best_method=input('Enter preferred estimation metho d (1-MLE, 2-KME, 3-PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE): ); end if best_method==2 tstar_Uniform=Optimal_Replacement_Time_KME(Combined_Un iform,C f,Cp); end if best_method==3 tstar_Uniform=Optimal_Replacement_Time_PEXE(Combined_ Uniform, Cf,Cp); end if best_method==4 tstar_Uniform=Optimal_Replacement_Time_FRWE(Combined_ Uniform, Cf,Cp); end if best_method==5 tstar_Uniform=Optimal_Replacement_Time_KLM(Combined_Un iform,C f,Cp); end if best_method==6 [Shape_Uniform Scale_Uniform]=ROSS_Estima tor(Combined_Uniform); tstar_Uniform=Optimal_Replacement_Time(Shape_Uniform,S cale_Unifo rm,Cp,Cf); end if best_method==7 [Shape_Uniform Scale_Uniform]=WH_Estimato r(Combined_Uniform); tstar_Uniform=Optimal_Replacement_Time(Shape_Uniform,S cale_Unifo rm,Cp,Cf); end if best_method==8 [Shape_Uniform Scale_Uniform]=BE_Estimat or(Combined_Uniform); tstar_Uniform=Optimal_Replacement_Time(Shape_Uniform,S cale_Unifo rm,Cp,Cf); end if best_method==9

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109 Appendix H (continued): Matlab Code [Shape_Uniform Scale_Uniform]= MPMLE_Estimator(Combined_Uniform); tstar_Uniform=Optimal_Replacement_Time(Shape_Uniform,S cale_Unifo rm,Cp,Cf); end Uniform_a=tstar_Uniform-UR*tstar_Uniform; end if Period_Maintenance_Normal+Period_Failure_Normal>0 [t_Normal Failure_Data_Normal Replacement_Data_Normal Period_Censored_Normal]=Add_Data_Normal(t_Normal,Shape,S cale,tstar_Nor mal,sigma); Data_Combined_Normal=Combine_Data2(n,Data_Combined_Norm al,Failure_D ata_Normal,Replacement_Data_Normal,Period_Censored_Norm al); failures=0; censored=0; [max b]=size(Data_Combined_Normal); Combined_Normal=Data_Combined_Normal; scrub=0; for i=1:max if Data_Combined_Normal(i,2)==1 failures=failures+1; else if Data_Combined_Normal(i,2)>=0 censored=censored+1; else Combined_Normal(i-scrub,:)=[]; scrub=scrub+1; end end end Combined_Normal=Combined_Normal censored=censored failures=failures Shape_Normal=Shape_Normal best_method=input('Enter preferred estimation metho d (1-MLE, 2-KME, 3PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE, 10-S imulate, 11NA): '); if best_method==10 [max b]=size(Combined_Normal); flag=0; while flag==0

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110 Appendix H (continued): Matlab Code Total_Iterations=input('Enter sim ulation length: '); Best_Estimator(max,Shape_Normal,ce nsored,Total_Iterations) flag=input('Simulation sufficient ? (0-No, 1-Yes): '); end best_method=input('Enter preferred estimation metho d (1-MLE, 2-KME, 3-PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE): ); end if best_method==1 Shape_Old=Shape_Normal; Scale_Old=Scale_Normal; [Shape_Normal Scale_Normal]=MLE_Estimat or(Combined_Normal); ok=input('check: 1-Yes, 2-No: ') if ok==1 tstar_Normal=Optimal_Replacement_Time(Shape_Normal,Sca le_Normal, Cp,Cf); else Shape_Normal=Shape_Old Scale_Normal=Scale_Old best_method=input('Enter preferred estimation metho d (1-MLE, 2KME, 3-PEXE, 4-FRWE, 5-KLM, 6-ROSS, 7-WH, 8-BE, 9-MPMLE, 10-Simulate): '); end end if best_method==2 tstar_Normal=Optimal_Replacement_Time_KME(Combined_Norm al,Cf, Cp); end if best_method==3 tstar_Normal=Optimal_Replacement_Time_PEXE(Combined_No rmal,Cf ,Cp); end if best_method==4 tstar_Normal=Optimal_Replacement_Time_FRWE(Combined_No rmal,C f,Cp); end if best_method==5 tstar_Normal=Optimal_Replacement_Time_KLM(Combined_Nor mal,Cf, Cp); end if best_method==6 [Shape_Normal Scale_Normal]=ROSS_Estimat or(Combined_Normal); tstar_Normal=Optimal_Replacement_Time(Shape_Normal,Sca le_Normal, Cp,Cf);

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111 Appendix H (continued): Matlab Code end if best_method==7 [Shape_Normal Scale_Normal]=WH_Estimato r(Combined_Normal); tstar_Normal=Optimal_Replacement_Time(Shape_Normal,Sca le_Normal, Cp,Cf); end if best_method==8 [Shape_Normal Scale_Normal]=BE_Estimato r(Combined_Normal); tstar_Normal=Optimal_Replacement_Time(Shape_Normal,Sca le_Normal, Cp,Cf); end if best_method==9 [Shape_Normal Scale_Normal]=MPMLE_Estim ator(Combined_Normal); tstar_Normal=Optimal_Replacement_Time(Shape_Normal,Sca le_Normal, Cp,Cf); end sigma=Maintenance_Policy(PC,CR,tstar_Normal ); end Determine the conditional probabilities and demand dis tribution for future periods Pconditional_FO_Fixed=Conditional_Failure_Only(n,t_Fi xed,k,deltat,Shape_Fixed,Scale _Fixed); Pconditional_FO_Normal=Conditional_Failure_Only(n,t_No rmal,k,deltat,Shape_Normal ,Scale_Normal); Pconditional_FO_Uniform=Conditional_Failure_Only(n,t_Un iform,k,deltat,Shape_Unifo rm,Scale_Uniform); Demand_Failure_Only_Fixed=Demand_Distribution(n,k,Pcond itional_FO_Fixed); Demand_Failure_Only_Normal=Demand_Distribution(n,k,Pcon ditional_FO_Normal); Demand_Failure_Only_Uniform=Demand_Distribution(n,k,Pcon ditional_FO_Uniform); FO_Demand_pdf_Fixed=kExpected_Demand(n,k,Demand_Failure _Only_Fixed); FO_Demand_pdf_Normal=kExpected_Demand(n,k,Demand_Failure _Only_Normal); FO_Demand_pdf_Uniform=kExpected_Demand(n,k,Demand_Failur e_Only_Uniform); Determine the initial inventory values Demand_FO_Lead_Fixed=Expected_Demand(n,lamda+1,Demand_ Failure_Only_Fixed) ; [s_FO_Lead_Fixed_PowerAppx S_FO_Lead_Fixed_PowerAppx]= Power_Approximation(lamda,p,h,K,Demand_FO_Lead_Fixed); [s_FO_Lead_Fixed_NormalAppx S_FO_Lead_Fixed_NormalAppx]= Normal_Approximation(lamda,p,h,K,Demand_FO_Lead_Fixed);

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112 Appendix H (continued): Matlab Code [s_FO_Lead_Fixed_NaddorAppx S_FO_Lead_Fixed_NaddorAppx]= Naddor_Approximation(lamda,p,h,K,Demand_FO_Lead_Fixed); FO_Lead_Fixed_Exp_Demand=Demand_FO_Lead_Fixed(1,1); Demand_FO_Lead_Normal=Expected_Demand(n,lamda+1,Demand_F ailure_Only_Nor mal); [s_FO_Lead_Normal_PowerAppx S_FO_Lead_Normal_PowerAppx]= Power_Approximation(lamda,p,h,K,Demand_FO_Lead_Normal); [s_FO_Lead_Normal_NormalAppx S_FO_Lead_Normal_NormalAppx]= Normal_Approximation(lamda,p,h,K,Demand_FO_Lead_Normal); [s_FO_Lead_Normal_NaddorAppx S_FO_Lead_Normal_NaddorAppx]= Naddor_Approximation(lamda,p,h,K,Demand_FO_Lead_Normal); FO_Lead_Normal_Exp_Demand=Demand_FO_Lead_Normal(1,1); Demand_FO_Lead_Uniform=Expected_Demand(n,lamda+1,Demand_ Failure_Only_Uni form); [s_FO_Lead_Uniform_PowerAppx S_FO_Lead_Uniform_PowerAppx]= Power_Approximation(lamda,p,h,K,Demand_FO_Lead_Uniform); [s_FO_Lead_Uniform_NormalAppx S_FO_Lead_Uniform_NormalAppx ]= Normal_Approximation(lamda,p,h,K,Demand_FO_Lead_Uniform); [s_FO_Lead_Uniform_NaddorAppx S_FO_Lead_Uniform_NaddorAppx ]= Naddor_Approximation(lamda,p,h,K,Demand_FO_Lead_Uniform) ; FO_Lead_Uniform_Exp_Demand=Demand_FO_Lead_Uniform(1,1); Check if order needs to be placed if Z_FO_Lead_Fixed_PowerAppx<=s_FO_Lead_Fixed_PowerAppx Lead_Orders_FO_Lead_Fixed_PowerAppx(Iteration+lamda)=ce il(S_FO_Lead_F ixed_PowerAppx-Z_FO_Lead_Fixed_PowerAppx); Period_Order_FO_Lead_Fixed_PowerAppx=1; end if Z_FO_Lead_Normal_PowerAppx<=s_FO_Lead_Normal_PowerAppx Lead_Orders_FO_Lead_Normal_PowerAppx(Iteration+lamda)=ce il(S_FO_Lead _Normal_PowerAppx-Z_FO_Lead_Normal_PowerAppx); Period_Order_FO_Lead_Normal_PowerAppx=1; end if Z_FO_Lead_Uniform_PowerAppx<=s_FO_Lead_Uniform_PowerAppx Lead_Orders_FO_Lead_Uniform_PowerAppx(Iteration+lamda)=c eil(S_FO_Lead _Uniform_PowerAppx-Z_FO_Lead_Uniform_PowerAppx); Period_Order_FO_Lead_Uniform_PowerAppx=1; end if Z_FO_Lead_Fixed_NormalAppx<=s_FO_Lead_Fixed_NormalAppx Lead_Orders_FO_Lead_Fixed_NormalAppx(Iteration+lamda)=c eil(S_FO_Lead_ Fixed_NormalAppx-Z_FO_Lead_Fixed_NormalAppx); Period_Order_FO_Lead_Fixed_NormalAppx=1; end

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113 Appendix H (continued): Matlab Code if Z_FO_Lead_Normal_NormalAppx<=s_FO_Lead_Normal_NormalAppx Lead_Orders_FO_Lead_Normal_NormalAppx(Iteration+lamda)=c eil(S_FO_Lea d_Normal_NormalAppx-Z_FO_Lead_Normal_NormalAppx); Period_Order_FO_Lead_Normal_NormalAppx=1; end if Z_FO_Lead_Uniform_NormalAppx<=s_FO_Lead_Uniform_NormalAp px Lead_Orders_FO_Lead_Uniform_NormalAppx(Iteration+lamda)= ceil(S_FO_Lea d_Uniform_NormalAppx-Z_FO_Lead_Uniform_NormalAppx); Period_Order_FO_Lead_Uniform_NormalAppx=1; end if Z_FO_Lead_Fixed_NaddorAppx<=s_FO_Lead_Fixed_NaddorAppx Lead_Orders_FO_Lead_Fixed_NaddorAppx(Iteration+lamda)=c eil(S_FO_Lead_ Fixed_NaddorAppx-Z_FO_Lead_Fixed_NaddorAppx); Period_Order_FO_Lead_Fixed_NaddorAppx=1; end if Z_FO_Lead_Normal_NaddorAppx<=s_FO_Lead_Normal_NaddorApp x Lead_Orders_FO_Lead_Normal_NaddorAppx(Iteration+lamda)=c eil(S_FO_Lea d_Normal_NaddorAppx-Z_FO_Lead_Normal_NaddorAppx); Period_Order_FO_Lead_Normal_NaddorAppx=1; end if Z_FO_Lead_Uniform_NaddorAppx<=s_FO_Lead_Uniform_NaddorAp px Lead_Orders_FO_Lead_Uniform_NaddorAppx(Iteration+lamda)= ceil(S_FO_Lea d_Uniform_NaddorAppx-Z_FO_Lead_Uniform_NaddorAppx); Period_Order_FO_Lead_Uniform_NaddorAppx=1; end Total up holding costs if On_Hand_FO_Lead_Fixed_PowerAppx>0 Period_Holding_FO_Lead_Fixed_PowerAppx=On_Hand_FO_Lead_Fix ed_Powe rAppx; end if On_Hand_FO_Lead_Normal_PowerAppx>0 Period_Holding_FO_Lead_Normal_PowerAppx=On_Hand_FO_Lead_Norm al_P owerAppx; end if On_Hand_FO_Lead_Uniform_PowerAppx>0 Period_Holding_FO_Lead_Uniform_PowerAppx=On_Hand_FO_Lead_Un iform _PowerAppx; end if On_Hand_FO_Lead_Fixed_NormalAppx>0 Period_Holding_FO_Lead_Fixed_NormalAppx=On_Hand_FO_Lead_Fi xed_Nor malAppx; end

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114 Appendix H (continued): Matlab Code if On_Hand_FO_Lead_Normal_NormalAppx>0 Period_Holding_FO_Lead_Normal_NormalAppx=On_Hand_FO_Lead_N ormal_ NormalAppx; end if On_Hand_FO_Lead_Uniform_NormalAppx>0 Period_Holding_FO_Lead_Uniform_NormalAppx=On_Hand_FO_Lead_U nifor m_NormalAppx; end if On_Hand_FO_Lead_Fixed_NaddorAppx>0 Period_Holding_FO_Lead_Fixed_NaddorAppx=On_Hand_FO_Lead_F ixed_Nad dorAppx; end if On_Hand_FO_Lead_Normal_NaddorAppx>0 Period_Holding_FO_Lead_Normal_NaddorAppx=On_Hand_FO_Lead_No rmal_ NaddorAppx; end if On_Hand_FO_Lead_Uniform_NaddorAppx>0 Period_Holding_FO_Lead_Uniform_NaddorAppx=On_Hand_FO_Lead_U nifor m_NaddorAppx; end Total up shortage costs if On_Hand_FO_Lead_Fixed_PowerAppx<0 Period_Shortage_FO_Lead_Fixed_PowerAppx=abs(On_Hand_FO_Le ad_Fixed_ PowerAppx); end On_Order_FO_Lead_Fixed_PowerAppx=On_Order_FO_Lead_Fixed_Powe rAppx+sum( Lead_Orders_FO_Lead_Fixed_PowerAppx(Iteration+lamda:Sim ulation_Length+lamda)) ; Z_FO_Lead_Fixed_PowerAppx=On_Hand_FO_Lead_Fixed_PowerAppx+On _Order_F O_Lead_Fixed_PowerAppx; if On_Hand_FO_Lead_Normal_PowerAppx<0 Period_Shortage_FO_Lead_Normal_PowerAppx=abs(On_Hand_FO_L ead_Nor mal_PowerAppx); end On_Order_FO_Lead_Normal_PowerAppx=On_Order_FO_Lead_Normal_Po werAppx+s um(Lead_Orders_FO_Lead_Normal_PowerAppx(Iteration+lamda: Simulation_Length+la mda)); Z_FO_Lead_Normal_PowerAppx=On_Hand_FO_Lead_Normal_PowerAppx +On_Order _FO_Lead_Normal_PowerAppx; if On_Hand_FO_Lead_Uniform_PowerAppx<0 Period_Shortage_FO_Lead_Uniform_PowerAppx=abs(On_Hand_FO_ Lead_Unif orm_PowerAppx);

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115 Appendix H (continued): Matlab Code end On_Order_FO_Lead_Uniform_PowerAppx=On_Order_FO_Lead_Uniform_P owerAppx +sum(Lead_Orders_FO_Lead_Uniform_PowerAppx(Iteration+la mda:Simulation_Lengt h+lamda)); Z_FO_Lead_Uniform_PowerAppx=On_Hand_FO_Lead_Uniform_PowerAppx +On_Ord er_FO_Lead_Uniform_PowerAppx; if On_Hand_FO_Lead_Fixed_NormalAppx<0 Period_Shortage_FO_Lead_Fixed_NormalAppx=abs(On_Hand_FO_L ead_Fixed _NormalAppx); end On_Order_FO_Lead_Fixed_NormalAppx=On_Order_FO_Lead_Fixed_Nor malAppx+su m(Lead_Orders_FO_Lead_Fixed_NormalAppx(Iteration+lamda: Simulation_Length+lam da)); Z_FO_Lead_Fixed_NormalAppx=On_Hand_FO_Lead_Fixed_NormalApp x+On_Order_ FO_Lead_Fixed_NormalAppx; if On_Hand_FO_Lead_Normal_NormalAppx<0 Period_Shortage_FO_Lead_Normal_NormalAppx=abs(On_Hand_FO_L ead_Nor mal_NormalAppx); end On_Order_FO_Lead_Normal_NormalAppx=On_Order_FO_Lead_Normal_Nor malAppx +sum(Lead_Orders_FO_Lead_Normal_NormalAppx(Iteration+la mda:Simulation_Lengt h+lamda)); Z_FO_Lead_Normal_NormalAppx=On_Hand_FO_Lead_Normal_NormalAppx +On_Or der_FO_Lead_Normal_NormalAppx; if On_Hand_FO_Lead_Uniform_NormalAppx<0 Period_Shortage_FO_Lead_Uniform_NormalAppx=abs(On_Hand_FO_ Lead_Un iform_NormalAppx); end On_Order_FO_Lead_Uniform_NormalAppx=On_Order_FO_Lead_Uniform_ NormalAp px+sum(Lead_Orders_FO_Lead_Uniform_NormalAppx(Iteration+ lamda:Simulation_Le ngth+lamda)); Z_FO_Lead_Uniform_NormalAppx=On_Hand_FO_Lead_Uniform_Normal Appx+On_ Order_FO_Lead_Uniform_NormalAppx; if On_Hand_FO_Lead_Fixed_NaddorAppx<0 Period_Shortage_FO_Lead_Fixed_NaddorAppx=abs(On_Hand_FO_L ead_Fixed _NaddorAppx); end On_Order_FO_Lead_Fixed_NaddorAppx=On_Order_FO_Lead_Fixed_Nad dorAppx+su m(Lead_Orders_FO_Lead_Fixed_NaddorAppx(Iteration+lamda: Simulation_Length+lam da)); Z_FO_Lead_Fixed_NaddorAppx=On_Hand_FO_Lead_Fixed_NaddorApp x+On_Order_ FO_Lead_Fixed_NaddorAppx; if On_Hand_FO_Lead_Normal_NaddorAppx<0

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116 Appendix H (continued): Matlab Code Period_Shortage_FO_Lead_Normal_NaddorAppx=abs(On_Hand_F O_Lead_Nor mal_NaddorAppx); end On_Order_FO_Lead_Normal_NaddorAppx=On_Order_FO_Lead_Normal_Nad dorAppx +sum(Lead_Orders_FO_Lead_Normal_NaddorAppx(Iteration+la mda:Simulation_Lengt h+lamda)); Z_FO_Lead_Normal_NaddorAppx=On_Hand_FO_Lead_Normal_NaddorAppx +On_Or der_FO_Lead_Normal_NaddorAppx; if On_Hand_FO_Lead_Uniform_NaddorAppx<0 Period_Shortage_FO_Lead_Uniform_NaddorAppx=abs(On_Hand_FO_ Lead_Un iform_NaddorAppx); end On_Order_FO_Lead_Uniform_NaddorAppx=On_Order_FO_Lead_Uniform_ NaddorAp px+sum(Lead_Orders_FO_Lead_Uniform_NaddorAppx(Iteration +lamda:Simulation_Le ngth+lamda)); Z_FO_Lead_Uniform_NaddorAppx=On_Hand_FO_Lead_Uniform_Naddor Appx+On_O rder_FO_Lead_Uniform_NaddorAppx; Record inventory values and costs Iteration=Iteration+1 Inventory_Parameters(Iteration,1)=On_Hand_FO_Lead_Fix ed_PowerAppx; Inventory_Parameters(Iteration,2)=s_FO_Lead_Fixed_Po werAppx; Inventory_Parameters(Iteration,3)=S_FO_Lead_Fixed_Po werAppx; Inventory_Parameters(Iteration,4)=On_Order_FO_Lead_F ixed_PowerAppx; Inventory_Parameters(Iteration,5)=Period_Holding_FO_L ead_Fixed_PowerAppx; Inventory_Parameters(Iteration,6)=Period_Shortage_F O_Lead_Fixed_PowerAppx; Inventory_Parameters(Iteration-1,7)=Period_Order_FO_L ead_Fixed_PowerAppx; Inventory_Parameters(Iteration,8)=On_Hand_FO_Lead_Fix ed_NormalAppx; Inventory_Parameters(Iteration,9)=s_FO_Lead_Fixed_Nor malAppx; Inventory_Parameters(Iteration,10)=S_FO_Lead_Fixed_No rmalAppx; Inventory_Parameters(Iteration,11)=On_Order_FO_Lead_F ixed_NormalAppx; Inventory_Parameters(Iteration,12)=Period_Holding_FO_ Lead_Fixed_NormalAppx; Inventory_Parameters(Iteration,13)=Period_Shortage_ FO_Lead_Fixed_NormalAppx; Inventory_Parameters(Iteration-1,14)=Period_Order_FO_ Lead_Fixed_NormalAppx; Inventory_Parameters(Iteration,15)=On_Hand_FO_Lead_F ixed_NaddorAppx; Inventory_Parameters(Iteration,16)=s_FO_Lead_Fixed_Na ddorAppx; Inventory_Parameters(Iteration,17)=S_FO_Lead_Fixed_Na ddorAppx; Inventory_Parameters(Iteration,18)=On_Order_FO_Lead_F ixed_NaddorAppx; Inventory_Parameters(Iteration,19)=Period_Holding_FO_ Lead_Fixed_NaddorAppx; Inventory_Parameters(Iteration,20)=Period_Shortage_ FO_Lead_Fixed_NaddorAppx; Inventory_Parameters(Iteration-1,21)=Period_Order_FO_ Lead_Fixed_NaddorAppx; Inventory_Parameters(Iteration,22)=On_Hand_FO_Lead_No rmal_PowerAppx; Inventory_Parameters(Iteration,23)=s_FO_Lead_Normal_P owerAppx;

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117 Appendix H (continued): Matlab Code Inventory_Parameters(Iteration,24)=S_FO_Lead_Normal_P owerAppx; Inventory_Parameters(Iteration,25)=On_Order_FO_Lead_No rmal_PowerAppx; Inventory_Parameters(Iteration,26)=Period_Holding_FO_ Lead_Normal_PowerAppx; Inventory_Parameters(Iteration,27)=Period_Shortage_ FO_Lead_Normal_PowerAppx; Inventory_Parameters(Iteration-1,28)=Period_Order_FO_ Lead_Normal_PowerAppx; Inventory_Parameters(Iteration,29)=On_Hand_FO_Lead_No rmal_NormalAppx; Inventory_Parameters(Iteration,30)=s_FO_Lead_Normal_No rmalAppx; Inventory_Parameters(Iteration,31)=S_FO_Lead_Normal_No rmalAppx; Inventory_Parameters(Iteration,32)=On_Order_FO_Lead_No rmal_NormalAppx; Inventory_Parameters(Iteration,33)=Period_Holding_FO_ Lead_Normal_NormalAppx; Inventory_Parameters(Iteration,34)=Period_Shortage_ FO_Lead_Normal_NormalAppx; Inventory_Parameters(Iteration-1,35)=Period_Order_FO_ Lead_Normal_NormalAppx; Inventory_Parameters(Iteration,36)=On_Hand_FO_Lead_No rmal_NaddorAppx; Inventory_Parameters(Iteration,37)=s_FO_Lead_Normal_Na ddorAppx; Inventory_Parameters(Iteration,38)=S_FO_Lead_Normal_Na ddorAppx; Inventory_Parameters(Iteration,39)=On_Order_FO_Lead_No rmal_NaddorAppx; Inventory_Parameters(Iteration,40)=Period_Holding_FO_ Lead_Normal_NaddorAppx; Inventory_Parameters(Iteration,41)=Period_Shortage_ FO_Lead_Normal_NaddorAppx; Inventory_Parameters(Iteration-1,42)=Period_Order_FO_ Lead_Normal_NaddorAppx; Inventory_Parameters(Iteration,43)=On_Hand_FO_Lead_Un iform_PowerAppx; Inventory_Parameters(Iteration,44)=s_FO_Lead_Uniform_ PowerAppx; Inventory_Parameters(Iteration,45)=S_FO_Lead_Uniform_ PowerAppx; Inventory_Parameters(Iteration,46)=On_Order_FO_Lead_Un iform_PowerAppx; Inventory_Parameters(Iteration,47)=Period_Holding_FO_ Lead_Uniform_PowerAppx; Inventory_Parameters(Iteration,48)=Period_Shortage_ FO_Lead_Uniform_PowerAppx; Inventory_Parameters(Iteration-1,49)=Period_Order_FO_ Lead_Uniform_PowerAppx; Inventory_Parameters(Iteration,50)=On_Hand_FO_Lead_Un iform_NormalAppx; Inventory_Parameters(Iteration,51)=s_FO_Lead_Uniform_ NormalAppx; Inventory_Parameters(Iteration,52)=S_FO_Lead_Uniform_ NormalAppx; Inventory_Parameters(Iteration,53)=On_Order_FO_Lead_Un iform_NormalAppx; Inventory_Parameters(Iteration,54)=Period_Holding_FO_ Lead_Uniform_NormalAppx; Inventory_Parameters(Iteration,55)=Period_Shortage_ FO_Lead_Uniform_NormalAppx; Inventory_Parameters(Iteration-1,56)=Period_Order_FO_ Lead_Uniform_NormalAppx; Inventory_Parameters(Iteration,57)=On_Hand_FO_Lead_Un iform_NaddorAppx; Inventory_Parameters(Iteration,58)=s_FO_Lead_Uniform _NaddorAppx; Inventory_Parameters(Iteration,59)=S_FO_Lead_Uniform_ NaddorAppx; Inventory_Parameters(Iteration,60)=On_Order_FO_Lead_Un iform_NaddorAppx; Inventory_Parameters(Iteration,61)=Period_Holding_FO_ Lead_Uniform_NaddorAppx; Inventory_Parameters(Iteration,62)=Period_Shortage_ FO_Lead_Uniform_NaddorAppx; Inventory_Parameters(Iteration-1,63)=Period_Order_FO_ Lead_Uniform_NaddorAppx; Record maintenance values and costs Policy_Parameters(Iteration-1,1)=Period_Maintenance _Fixed;

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118 Appendix H (continued): Matlab Code Policy_Parameters(Iteration-1,2)=Period_Failure_Fix ed; Policy_Parameters(Iteration,3)=FO_Lead_Fixed_Exp_Dema nd; Policy_Parameters(Iteration,4)=abs(Period_Maintenan ce_Fixed+Period_Failure_FixedFO_Lead_Fixed_Exp_Demand); Policy_Parameters(Iteration-1,5)=Period_Maintenance _Fixed; Policy_Parameters(Iteration-1,6)=Period_Failure_Fix ed; Policy_Parameters(Iteration,7)=FO_Lead_Fixed_Exp_Dema nd; Policy_Parameters(Iteration,8)=abs(Period_Maintenan ce_Fixed+Period_Failure_FixedFO_Lead_Fixed_Exp_Demand); Policy_Parameters(Iteration-1,9)=Period_Maintenance _Fixed; Policy_Parameters(Iteration-1,10)=Period_Failure_Fi xed; Policy_Parameters(Iteration,11)=FO_Lead_Fixed_Exp_Dem and; Policy_Parameters(Iteration,12)=abs(Period_Maintena nce_Fixed+Period_Failure_FixedFO_Lead_Fixed_Exp_Demand); Policy_Parameters(Iteration-1,13)=Period_Maintenanc e_Normal; Policy_Parameters(Iteration-1,14)=Period_Failure_Nor mal; Policy_Parameters(Iteration,15)=FO_Lead_Normal_Exp_Dem and; Policy_Parameters(Iteration,16)=abs(Period_Maintena nce_Normal+Period_Failure_Nor mal-FO_Lead_Normal_Exp_Demand); Policy_Parameters(Iteration-1,17)=Period_Maintenanc e_Normal; Policy_Parameters(Iteration-1,18)=Period_Failure_Nor mal; Policy_Parameters(Iteration,19)=FO_Lead_Normal_Exp_Dem and; Policy_Parameters(Iteration,20)=abs(Period_Maintena nce_Normal+Period_Failure_Nor mal-FO_Lead_Normal_Exp_Demand); Policy_Parameters(Iteration-1,21)=Period_Maintenanc e_Normal; Policy_Parameters(Iteration-1,22)=Period_Failure_Nor mal; Policy_Parameters(Iteration,23)=FO_Lead_Normal_Exp_Dem and; Policy_Parameters(Iteration,24)=abs(Period_Maintena nce_Normal+Period_Failure_Nor mal-FO_Lead_Normal_Exp_Demand); Policy_Parameters(Iteration-1,25)=Period_Maintenanc e_Uniform; Policy_Parameters(Iteration-1,26)=Period_Failure_Uni form; Policy_Parameters(Iteration,27)=FO_Lead_Uniform_Exp_De mand; Policy_Parameters(Iteration,28)=abs(Period_Maintena nce_Uniform+Period_Failure_Uni form-FO_Lead_Uniform_Exp_Demand); Policy_Parameters(Iteration-1,29)=Period_Maintenanc e_Uniform; Policy_Parameters(Iteration-1,30)=Period_Failure_Uni form; Policy_Parameters(Iteration,31)=FO_Lead_Uniform_Exp_De mand; Policy_Parameters(Iteration,32)=abs(Period_Maintena nce_Uniform+Period_Failure_Uni form-FO_Lead_Uniform_Exp_Demand); Policy_Parameters(Iteration-1,33)=Period_Maintenanc e_Uniform; Policy_Parameters(Iteration-1,34)=Period_Failure_Uni form; Policy_Parameters(Iteration,35)=FO_Lead_Uniform_Exp_De mand;

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119 Appendix H (continued): Matlab Code Policy_Parameters(Iteration,36)=abs(Period_Maintena nce_Uniform+Period_Failure_Uni form-FO_Lead_Uniform_Exp_Demand); Record distribution values Distribution_Parameters(Iteration,1)=Shape_Fixed; Distribution_Parameters(Iteration,2)=Scale_Fixed; Distribution_Parameters(Iteration,3)=tstar_Fixed; Distribution_Parameters(Iteration,4)=Shape_Normal; Distribution_Parameters(Iteration,5)=Scale_Normal; Distribution_Parameters(Iteration,6)=tstar_Normal; Distribution_Parameters(Iteration,7)=Shape_Uniform; Distribution_Parameters(Iteration,8)=Scale_Uniform; Distribution_Parameters(Iteration,9)=tstar_Uniform; Simulation initialization procedure if Iteration==reset Inventory_Parameters(1,:)=Inventory_Parame ters(Iteration,:); Policy_Parameters(1,:)=Policy_Parameters(I teration,:); Iteration=1; reset=0; On_Hand_FO_Lead_Fixed_PowerAppx=ceil(S_FO_Lead_Fixed_Power Appx); On_Hand_FO_Lead_Fixed_NormalAppx=ceil(S_FO_Lead_Fixed_Nor malAppx) On_Hand_FO_Lead_Fixed_NaddorAppx=ceil(S_FO_Lead_Fixed_Nad dorAppx); On_Hand_FO_Lead_Normal_PowerAppx=ceil(S_FO_Lead_Normal_Power App x); On_Hand_FO_Lead_Normal_NormalAppx=ceil(S_FO_Lead_Normal_Norm alA ppx); On_Hand_FO_Lead_Normal_NaddorAppx=ceil(S_FO_Lead_Normal_Nad dorAp px); On_Hand_FO_Lead_Uniform_PowerAppx=ceil(S_FO_Lead_Uniform_Po werAp px); On_Hand_FO_Lead_Uniform_NormalAppx=ceil(S_FO_Lead_Uniform_N ormal Appx); On_Hand_FO_Lead_Uniform_NaddorAppx=ceil(S_FO_Lead_Uniform_N addor Appx); On_Order_FO_Lead_Fixed_PowerAppx=0; On_Order_FO_Lead_Fixed_NormalAppx=0; On_Order_FO_Lead_Fixed_NaddorAppx=0; On_Order_FO_Lead_Normal_PowerAppx=0; On_Order_FO_Lead_Normal_NormalAppx=0; On_Order_FO_Lead_Normal_NaddorAppx=0; On_Order_FO_Lead_Uniform_PowerAppx=0; On_Order_FO_Lead_Uniform_NormalAppx=0; On_Order_FO_Lead_Uniform_NaddorAppx=0;

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120 Appendix H (continued): Matlab Code Z_FO_Lead_Fixed_PowerAppx=On_Hand_FO_Lead_Fixed_PowerAppx+On _O rder_FO_Lead_Fixed_PowerAppx; Z_FO_Lead_Fixed_NormalAppx=On_Hand_FO_Lead_Fixed_NormalApp x+On _Order_FO_Lead_Fixed_NormalAppx; Z_FO_Lead_Fixed_NaddorAppx=On_Hand_FO_Lead_Fixed_NaddorApp x+On_ Order_FO_Lead_Fixed_NaddorAppx; Z_FO_Lead_Normal_PowerAppx=On_Hand_FO_Lead_Normal_PowerAppx +O n_Order_FO_Lead_Normal_PowerAppx; Z_FO_Lead_Uniform_PowerAppx=On_Hand_FO_Lead_Uniform_PowerAppx + On_Order_FO_Lead_Uniform_PowerAppx; Z_FO_Lead_Normal_NormalAppx=On_Hand_FO_Lead_Normal_NormalAppx + On_Order_FO_Lead_Normal_NormalAppx; Z_FO_Lead_Uniform_NormalAppx=On_Hand_FO_Lead_Uniform_Normal App x+On_Order_FO_Lead_Uniform_NormalAppx; Z_FO_Lead_Normal_NaddorAppx=On_Hand_FO_Lead_Normal_NaddorAppx + On_Order_FO_Lead_Normal_NaddorAppx; Z_FO_Lead_Uniform_NaddorAppx=On_Hand_FO_Lead_Uniform_Naddor Appx +On_Order_FO_Lead_Uniform_NaddorAppx; Lead_Orders_FO_Lead_Fixed_PowerAppx=zeros(Simulation_Le ngth+lamda+1, 1); Lead_Orders_FO_Lead_Normal_PowerAppx=zeros(Simulation_Le ngth+lamda+ 1,1); Lead_Orders_FO_Lead_Uniform_PowerAppx=zeros(Simulation_L ength+lamda +1,1); Lead_Orders_FO_Lead_Fixed_NormalAppx=zeros(Simulation_L ength+lamda+1 ,1); Lead_Orders_FO_Lead_Normal_NormalAppx=zeros(Simulation_L ength+lamda +1,1); Lead_Orders_FO_Lead_Uniform_NormalAppx=zeros(Simulation_ Length+lamda +1,1); Lead_Orders_FO_Lead_Fixed_NaddorAppx=zeros(Simulation_L ength+lamda+1 ,1); Lead_Orders_FO_Lead_Normal_NaddorAppx=zeros(Simulation_L ength+lamda +1,1); Lead_Orders_FO_Lead_Uniform_NaddorAppx=zeros(Simulation_ Length+lamda +1,1); t=t_Fixed(:,1); end End of simulation End Sum up all costs and record j=1;

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121 Appendix H (continued): Matlab Code for i=0:4:32 Output(5,j)=sum(Policy_Parameters(:,i+1)); Output(6,j)=sum(Policy_Parameters(:,i+2)); Output(7,j)=sum(Policy_Parameters(:,i+4))/Itera tion; j=j+1; end j=1; for i=0:7:56 Output(1,j)=sum(Inventory_Parameters(:,i+5))*h*delta t+sum(Inventory_Paramet ers(:,i+6))*p*deltat+sum(Inventory_Parameters(:,i+7 ))*K+Output(5,j)*Cp+Outp ut(6,j)*Cf; Output(2,j)=sum(Inventory_Parameters(:,i+5)); Output(3,j)=sum(Inventory_Parameters(:,i+6)); Output(4,j)=sum(Inventory_Parameters(:,i+7)); Output(8,j)=tstar; j=j+1; end

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122 Appendix I: Run 10 Distribution Parameters PeriodShapeScalet*ShapeScalet*ShapeScalet* Policy 10Policy 11Policy 12

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123 Appendix I (continued): Run 10 Distribution Paramet ers PeriodShapeScalet*ShapeScalet*ShapeScalet* 522.3109.895.03.3115.486.03.1128.297.0532.5103.785.53.4110.482.03.3124.693.0542.5103.785.53.4110.482.03.3124.693.0552.5102.984.03.4110.482.03.3127.095.0562.5102.984.03.4115.786.03.3127.095.0572.5102.984.03.4115.786.03.2133.8100.5582.5106.587.53.5111.082.03.2133.8100.5592.5106.587.53.5111.082.03.8110.581.0602.5107.887.53.6105.678.03.8110.581.0612.5107.887.53.6105.678.03.8110.581.0622.5110.991.03.5103.276.53.8110.581.0632.5110.991.03.5103.276.53.7110.981.5642.4113.994.53.6104.177.03.7110.981.5652.4113.994.53.6104.177.03.7110.981.5662.5110.589.53.6104.177.03.7110.981.5672.5110.589.53.6104.177.03.7110.981.5682.5110.589.53.7105.878.03.8107.779.0692.5111.791.53.7105.878.03.8107.779.0702.5111.791.53.6107.079.03.8108.179.5712.4111.793.03.6107.079.03.8108.179.5722.4111.793.03.6106.778.53.9103.876.0732.4111.793.03.6106.778.53.9103.876.0742.4114.795.53.4106.379.03.9106.378.0752.4114.795.53.4106.379.03.9106.378.0762.5112.893.03.4107.780.03.9106.378.0772.5112.893.03.4107.780.04.0104.576.5782.5113.693.53.4107.680.04.0104.576.5792.5113.693.53.4107.680.04.1105.277.0802.6106.785.03.4106.279.04.1105.277.0812.6106.785.03.4106.279.04.1105.477.0822.6104.883.53.4106.279.04.1105.477.0832.6104.883.53.4107.880.04.1106.478.0842.7105.083.53.4107.880.04.1106.478.0852.7105.083.53.5105.178.04.1106.478.0862.7106.884.53.5105.178.03.8106.178.0872.7106.884.53.5102.275.53.8106.178.0882.7108.186.03.5102.275.53.9106.478.0892.7108.186.03.5102.275.53.9106.478.0902.6109.387.53.6101.275.03.9106.478.0912.6109.387.53.6101.275.03.7105.578.0922.6108.887.03.6101.975.53.7105.578.0932.6108.887.03.6101.975.53.6104.477.0942.6108.887.03.6101.975.53.6104.477.0952.7105.883.53.6102.475.53.4104.077.0962.7105.883.53.6102.475.53.4104.077.0972.7107.585.03.6101.375.03.4104.077.0982.7107.585.03.6101.375.03.5105.678.5992.7107.585.03.5101.475.03.5105.678.5 Mean=2.5108.788.33.5105.978.53.7109.981.1 Policy 10Policy 11Policy 12

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124 Appendix J: Run 11 Distribution Parameters PeriodShapeScale t* ShapeScale t* ShapeScale t* 13.0150.079.06.791.168.54.895.670.022.0169.9169.07.392.870.54.894.469.532.0169.9169.07.392.870.54.894.469.542.1172.3162.57.392.870.54.894.469.552.1172.3162.57.392.870.54.3105.577.062.2148.9131.55.5113.884.04.3105.577.072.2148.9131.55.5113.884.04.1103.776.082.2151.2133.05.2112.182.54.1103.776.092.2151.2133.05.2112.182.53.9107.078.5 102.2122.4107.55.2111.982.53.9107.078.5112.2122.4107.55.2111.982.53.0106.681.5122.2114.4101.05.2111.982.53.0106.681.5132.2114.4101.05.8112.383.53.1108.682.5142.4109.192.05.8112.383.53.1108.682.5152.4109.192.05.9113.384.53.1108.682.5162.4109.192.05.9113.384.53.3102.376.5172.4109.192.05.7105.478.03.3102.376.5182.6107.986.05.7105.478.03.3102.376.5192.6107.986.05.8105.478.53.5101.875.5202.6107.986.05.8105.478.53.5101.875.5212.6107.986.05.8105.478.53.5103.376.5222.4117.599.55.8105.478.53.5103.376.5232.4117.599.55.8106.379.03.5103.376.5242.3122.2107.05.8106.379.03.5106.478.5252.3122.2107.05.7106.679.03.5106.478.5262.3118.8101.55.7106.679.03.5106.478.5272.3118.8101.55.7106.679.03.4106.078.5282.3118.8101.55.6106.379.03.4106.078.5292.3118.8101.55.6106.379.03.4106.078.5302.4123.2104.05.9106.779.53.4106.078.5312.4123.2104.05.9106.779.53.4106.078.5322.5118.598.04.9108.680.03.6107.279.0332.5118.598.04.9108.680.03.6107.279.0342.5118.598.04.9108.479.53.5108.880.5352.5121.699.54.9108.479.53.5108.880.5362.5121.699.54.9108.479.53.5106.979.5372.5119.898.05.1105.477.53.5106.979.5382.5119.898.05.1105.477.53.5107.880.0392.5119.898.05.4103.376.53.5107.880.0402.6119.996.55.4103.376.53.4106.979.5412.6119.996.55.6102.776.03.4106.979.5422.6119.396.05.6102.776.03.4106.979.5432.6119.396.05.6102.576.03.5105.478.0442.6119.396.05.6102.576.03.5105.478.0452.6118.095.05.6102.576.03.7102.775.5462.6118.095.05.7103.076.53.7102.775.5472.6118.095.05.7103.076.53.7102.775.5482.6115.092.55.8103.477.03.7105.577.5492.6115.092.55.8103.477.03.7105.577.5502.6115.092.55.9104.477.53.7105.878.0512.6116.492.55.9104.477.53.7105.878.0 Policy 10Policy 11Policy 12

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125 Appendix J (continued): Run 11 Distribution Paramet ers PeriodShapeScale t* ShapeScale t* ShapeScale t* 522.6116.492.55.9104.477.53.7105.878.0532.6116.492.56.0104.478.03.7107.079.0542.6116.893.56.0104.478.03.7107.079.0552.6116.893.56.0104.478.03.6105.778.0562.7115.491.55.5104.677.53.6105.778.0572.7115.491.55.5104.677.53.6105.778.0582.6115.892.55.4103.576.53.7106.878.5592.6115.892.55.4103.576.53.7106.878.5602.7115.891.55.4103.576.53.7107.479.0612.7115.891.55.4103.576.53.7107.479.0622.7115.891.55.5103.476.53.7108.379.5632.7115.891.55.5103.476.53.7108.379.5642.7115.891.55.5103.476.53.7105.077.5652.8113.388.05.5103.877.03.7105.077.5662.8113.388.05.5103.877.03.8101.775.0672.8113.388.05.5102.676.03.8101.775.0682.8114.389.05.5102.676.03.8101.775.0692.8114.389.05.6102.776.03.8101.775.0702.8113.988.55.6102.776.03.9102.975.5712.8113.988.55.6102.776.03.9102.975.5722.8113.688.05.7103.276.53.5102.175.5732.8113.688.05.7103.276.53.5102.175.5742.8113.688.05.7103.276.53.5102.476.0752.8113.688.05.2102.375.53.5102.476.0762.8113.688.05.2102.375.53.5101.875.5772.8113.788.55.2102.175.03.5101.875.5782.8113.788.55.2102.175.03.6102.676.0792.8114.088.05.2102.275.53.6102.676.0802.8114.088.05.2102.275.53.6101.274.5812.8114.088.05.2101.575.03.6101.274.5822.9113.587.55.2101.575.03.6102.375.5832.9113.587.55.2101.575.03.6102.375.5842.9114.588.05.3100.974.53.6101.675.0852.9114.588.05.3100.974.53.6101.675.0863.0110.184.55.3100.974.53.6101.675.0873.0110.184.55.3100.974.53.6101.675.0883.0110.184.55.4101.475.03.7101.574.5893.0110.384.05.4101.475.03.7101.574.5903.0110.384.05.4101.475.03.7101.675.0913.0110.384.05.3100.974.53.7101.675.0923.0112.585.55.3100.974.53.7102.675.5933.0112.585.55.3100.974.53.7102.675.5943.0112.585.55.1100.774.03.7102.675.5953.0113.586.55.1100.774.03.7102.675.5963.0113.586.55.1100.774.03.7102.675.5973.0110.984.05.1100.774.03.7102.675.5983.0110.984.05.299.973.53.6102.775.5993.0110.984.05.299.973.53.6102.775.5 Mean=2.6119.296.85.6104.077.13.6104.177.0 Policy 10Policy 11Policy 12

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ABOUT THE AUTHOR Jonathan J. Bates received a B.S. in Naval Architectu re and Marine Engineering from the United States Coast Guard Academy in 1998. As an officer in the United States Coast Guard, he has served two years as Assistant Engi neer Officer on USCGC VIGILANT (WMEC 617), three years as Port Engineer at Naval Engineering Support Unit Portsmouth, Virginia, and is currently serving a s Engineer Officer onboard USCGC CONFIDENCE (WMEC 619). In 2003, he earned a M.S. in Oce an Engineering from Virginia Tech and a Professional Engineer license fr om the State of Florida in 2004. Upon completion of this dissertation, he received a Ph.D. in Industrial Engineering from the University of South Florida.


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