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Optimal discreteintime inventory control of a single deteriorating product with partial backlogging
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ABSTRACT: The implicit assumption in conventional inventory models is that the stored products maintain the same utility forever, i.e., they can be stored for an infinite period of time without losing their value or characteristics. However, generally speaking, almost all products experience some sort of deterioration over time. Some products have very small deterioration rates, and henceforth the effect of such deterioration can be neglected. Some products may be subject to significant rates of deterioration. Fruits, vegetables, drugs, alcohol and radioactive materials are examples that can experience significant deterioration during storage. Therefore the effect of deterioration must be explicitly taken into account in developing inventory models for such products. In most existing deteriorating inventory models, time is treated as a continuous variable, which is not exactly the case in practice. In reallife problems time factor is always measured on a discrete scale only, i.e. in terms of complete units of days, weeks, etc. In this research, we present several discreteintime inventory models and identify optimal ordering policies for a single deteriorating product by minimizing the expected overall costs over the planning horizon. The various conditions have been considered, e.g. periodic review, timevarying deterioration rate, waitingtimedependent partial backlogging, timedependent demand, stochastic demand etc. The objective of our research is twofold: (a) To obtain optimal order quantity and useful insights for the inventory control of a single deteriorating product over a discrete time horizon with deterministic demand, variable deterioration rates and waitingtimedependent partial backlogging ratios; (b) To identify optimal ordering policy for a single deteriorating product over a finite horizon with stochastic demand and partial backlogging. The explicit ordering policy will be developed for some special cases. Through computational experiments and sensitivity analysis, a thorough and insightful understanding of deteriorating inventory management will be achieved.
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Advisor: Michael Weng, Ph.D.
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Deteriorating inventory control
Deterministic demand
Periodic review
Stochastic demand
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Service level
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Optimal DiscreteinTime Inventory Control of a Single Deteriorating Product with Partial Backlogging by Yang Tan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Management Systems Engineering College of Engineering University of South Florida Major Professor: Michael Weng, Ph.D. Alan Hevner, Ph.D. Hao Zheng, Ph.D. Grisselle Centeno, Ph.D. Bo Zeng, Ph.D. Date of Approval: October 29, 2010 Keywords: deteriorating inventory control, deterministic demand, periodic review, stochastic demand, base stock policy, ( s S ) policy, service level Copyright 2010, Yang Tan
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Dedication To my beloved parents
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Acknowledgements I would like to truly thank my advisor Prof. Michael Weng for offering me such a wonderful opportunity to conduct research and pursue my doctorate degree under his superior guidance. Prof. WengÂ’s countless encouragement and critical mentoring in my research work have prepared me with confidence both in research and my personal life. I would like to thank my committee memb ers Prof. Alan Hevn er, Prof. Grisselle Centeno, Prof. Bo Zeng and Prof. Hao Zheng for their valuable advice in my research direction and methodologies. A special thank goes to Prof. Qiong Zhang for her kind help in serving as the Chairperson. Additionally, I would like to show my sincere appreciation to Prof. Jose ZayasCastro for his enormous encouragement on my Ph.D. study. I would also like to thank all other faculty and staff members, Prof. Alex Savachkin, Prof. Paul Schnitzler, Prof. Patricia Zarate, Ms. Gloria Hanshaw, Ms. Jackie Stephens, and Mr. Rafael Urena in the IMSE department for their generous help during my Ph.D. studies. Meanwhile, I would like to express my appreciation to Ms. Catherine Burton for her valuable correctio ns on my dissertation draft. Finally, I wish to express my purehearted appreciation to my beloved parents Fenglin Tan and Shuxia Yang and my sweet wife Ying Zhang. Without their constant support, continual encouragement and endless love, I would have never accomplished my study and achieved what I have so far. I am indebted to them tremendously and love them forever.
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i Table of Contents List of Tables iii List of Figures v Abstract vii Chapter 1 Introduction 1 1.1 Background 1 1.2 Inventory Control Models 4 1.2.1 Classification Criteria 4 1.2.2 Deterministic Inventory Models 5 1.2.2.1 The Economic Order Quantity (EOQ) Model 5 1.2.2.2 The WagnerWhitin Model 7 1.2.3 Stochastic Inventory Models 9 1.2.3.1 OnePeriod Stochastic Inventory Model 9 1.2.3.2 MultiPeriod Stochastic Inventory Model 10 1.3 Inventory Control for Deteriorating Items 11 1.4 Dissertation Outline 13 Chapter 2 Literature Review 16 2.1 Introduction 16 2.2 Deteriorating Inventory Models 16 2.3 PeriodicReview Stochastic Inventory Control Problem 26 2.4 Stochastic LotSizing Problem 29 2.5 Summary 32 Chapter 3 A DiscreteinTime Inventory Model with Deterioration and Backlog 33 3.1 Assumptions and Notations 33 3.2 Model 34 3.3 A Numerical Example 39 3.4 Sensitivity Analysis 41 3.5 Conclusion 49
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ii Chapter 4 A DiscreteinTime Deteriorating Inventory Model with TimeVarying Demand, Variable Deterioration Rate and WaitingTimeDependent Partial Backlogging 50 4.1 Assumptions and Notations 50 4.2 Model 51 4.3 Numerical Examples 56 4.4 Sensitivity Analysis 59 4.5 Conclusions 65 Chapter 5 PeriodicReview Stochastic Inventory Control Problem 67 5.1 Problem Description 67 5.2 Notations 68 5.3 Model 69 5.4 Base Stock Policy 71 5.5 Optimality of Base Stock Policies 71 5.6 Explicit Optimal Base Stock Level 75 5.7 Summary 77 Chapter 6 Stochastic LotSizing Problem 78 6.1 Notations 79 6.2 Model 80 6.2.1 Formulation 80 6.2.2 ( s S ) Policy 81 6.2.3 K Convex Functions 81 6.2.4 Optimality of ( s S ) Policy 82 6.3 Summary 88 Chapter 7 The Stochastic LotSizing Problem with Deterioration and ServiceLevel Constraints 90 7.1 Assumptions and Notations 91 7.2 Model 92 7.3 A Numerical Example 97 7.4 Summary 104 Chapter 8 Conclusions and Future Work 105 8.1 Conclusions 105 8.2 Future Work 107 Cited References 108 About the Author End Page
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iii List of Tables Table 3.1 Different Optimal Values w.r.t. 42 Table 3.2 Different Optimal Values w.r.t. c 44 Table 3.3 Different Optimal Values w.r.t. h 45 Table 3.4 Different Optimal Values w.r.t. b 46 Table 3.5 Different Optimal Values w.r.t. T 47 Table 4.1 First Computational Results for M ( k ) Values 57 Table 4.2 Second Computational Results for M ( k ) Values 58 Table 4.3 The Optimal Solutions for 3 Demand Patterns 59 Table 4.4 Different Optimal Values w.r.t. 60 Table 4.5 Different Optimal Values w.r.t. 61 Table 4.6 Different Optimal Values w.r.t. 62 Table 4.7 Different Optimal Values w.r.t. Cycle Time T 63
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iv Table 7.1 Forecasted Values of Demands 98 Table 7.2 The Calculated Values of the Inverse Cumulative Distribution Function ) (1 ) ( G 99 Table 7.3 Results for c = 0 100 Table 7.4 TarimÂ’s Results for c = 0 100 Table 7.5 Results for c = 4 101 Table 7.6 TarimÂ’s Results for c = 4 101 Table 7.7 Results for c = 6 102 Table 7.8 TarimÂ’s Results for c = 6 102 Table 7.9 Service Level and Actual Fill Rate 103 Table 7.10 Worst Scenario Cost 103
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v List of Figures Figure 1.1 Inventory Levels for the EOQ Model 6 Figure 2.1 WuÂ’s (2002) Inventory System 19 Figure 3.1 Inventory Level of the System 34 Figure 3.2 Total Monthly Cost C ( t1) as a Function of t1 41 Figure 3.3 Sensitivity Analysis w.r.t. 43 Figure 3.4 Sensitivity Analysis w.r.t. c 44 Figure 3.5 Sensitivity Analysis w.r.t. h 45 Figure 3.6 Sensitivity Analysis w.r.t. b 46 Figure 3.7 Sensitivity Analysis w.r.t. T 47 Figure 4.1 Inventory Level for the Model over the Cycle 52 Figure 4.2 Sensitivity Analysis w.r.t. 60 Figure 4.3 Sensitivity Analysis w.r.t. 62
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vi Figure 4.4 Sensitivity Analysis w.r.t. 63 Figure 4.5 Sensitivity Analysis w.r.t. Cycle Time T 64 Figure 5.1 Base Stock Policy 71 Figure 7.1 Illustration of Notation Index 93
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vii Abstract The implicit assumption in conventional inventory models is that the stored products maintain the same utility forever, i.e., they can be stored for an infinite period of time without losing their value or characteris tics. However, generally speaking, almost all products experience some sort of deterioration over time. Some products have very small deterioration rates, and henceforth the effect of such deterioration can be neglected. Some products may be subject to significant rates of deterioration. Fruits, vegetables, drugs, alcohol and radioactive materials are examples that can experience significant deterioration during storage. Therefore the effect of deterioration must be explicitly taken into account in developing inventory models for such products. In most existing deteriorating inventory models, time is treated as a continuous variable, which is not exactly the case in practice. In reallife problems time factor is always measured on a discrete scale only, i.e. in terms of complete units of days, weeks, etc. In this research, we present several discreteintime inventory models and identify optimal ordering policies for a single deteriorating product by minimizing the expected overall costs over the planning horizon. The various conditions have been considered, e.g. periodic review, timevarying deterioration rate, waitingtimedependent partial backlogging, timedependent demand, stochastic demand etc. The objective of our research is twofold: (a) To obtain optimal order quantity and useful insights for the inventory control of a single deteriorating product over a discrete time horizon with
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viii deterministic demand, variable deterioration rates and waitingtimedependent partial backlogging ratios; (b) To identify optimal ordering policy for a single deteriorating product over a finite horizon with stochastic demand and partial backlogging. The explicit ordering policy will be developed for some special cases. Through computational experiments and sensitivity analysis, a thorough and insightful understanding of deteriorating inventory management will be achieved.
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1 Chapter 1 Introduction 1.1 Background Nowadays, for most successful, wellorganized businesses, inventory control systems play a critical role in ensuring that adequate inventories are on hand to satisfy their customer demand. In general, the inventories can be classified into the following four categories (Nahmias, 2001): (1) Raw materials. The raw materials are the resources required in the production or processing activity of a firm. (2) Components. The components correspond to items that have not yet reached completion in the production process. Sometimes components are referred to as subassemblies. (3) Workinprocess. This is the inventory either waiting in the system for processing or being processed. Workinprocess inventories include component inventories and may include some raw materials inventories as well. (4) Finished goods. The finished goods are also known as end items, which are the final products of the production process. As inventories are expensive and need careful control, a fundamental question arises: Why do organizations hold inventories? Generally speaking, the main reason why inventories are held is to provide a buffer between uncertain supply and demand. For
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2 example, if a customer places an order and there are no items available immediately, it is very likely that the customer will go somewhere else and may never return. So the uncertainty of external demand is the most important reason to ho ld inventories. There are also some other reasons. For instance, if the fixed setup/ordering costs are high, it would be economical to produce/order a relatively large quantity and store them for future use. Also, sometimes if the price of a product increases over a short period of time, then it will be more economical to buy a large amount of the product at current price and put into storage than to pay a higher price in future. Since it is necessary to hold some inventories on hand, then it is essential to manage inventories economically. Inventory control consists of all the activities and procedures used to ensure the right amount of products is held in stock (Waters, 1992). A frequent objective of inventory control is to provide a moderate amount of inventories at minimum cost, so inventory control often relies on a tradeoff between conflicting costs. The relevant costs that are considered in most inventory systems are as follows (Nahmias, 2001). (1) Fixed Order Cost. It is incurred independent of the size of the order as long as the order quantity is not zero. (2) Unit Purchasing Cost. It is incurred on a perunit basis. (3) Inventory Holding Cost. It is also known as the inventory carrying cost, which is the sum of all costs that are proportional to the amount of inventory physically on hand. Some of the components of the holding cost include (i) Cost of providing the physical space to store the items. (i i) Taxes and insuranc e. (iii) Opportunity cost of alternative investment. The inventory holding cost ($/unit/year) is usually
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3 measured by the product of unit purchasing cost and annual interest rate. The interest rate is an aggregated term co mprised of some components like cost of capital, taxes and insurance, and cost of storage. (4) Backlogging Cost. It includes whatever bookkeeping and/or delay costs that might be involved and Â“lossofgoodwillÂ” cost. (Orders that cannot be filled immediately are held on the books until the next shipment arrives.) (5) Penalty Cost for Lost Sales. It includes the lost profit that would have been made from the sale and Â“lossofgoodwillÂ” cost. As one can see, the "lossofgoodwill" cost is included in either backlogging or penalty cost, and is a measure of customer unsatisfaction. Estimation of the Â“lossofgoodwillÂ” cost can be very difficult in practice. By using cost minimization as an optimization criterion, the following two questions which reflect the fundamental problem of inventory control can be answered. (1) When should an order be placed? (2) How many should be ordered? Regarding the time of ordering, there are two distinct inventory systems with different timing of replenishment: a periodic review system and continuous review system. A periodic review system allows inventory levels to be checked at discrete times periodically, and the order size is subject to change according to the variation in demand in each period. This system is often used in supermarkets, where stocks are reviewed at the end of each day and any sold units are replaced. A continuous review system allows the level of inventory to be monitored continuously and an order is placed whenever the inventory decreases to a specified level. The time between two consecutive orders is
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4 subject to change according to the variati on in demand over time. Minimizing the total costs of the inventory system can yield the optimal quantity of ordering. 1.2 Inventory Control Models 1.2.1 Classification Criteria The inventory control models in the literature can be classified according to the following criteria (Ravindran, 2008). (1) Stocking location: Single stocking location (single location models) and more than one stocking location (multiechelon inventory models). (2) Supply process lead times: Deterministic lead time and stochastic lead time. (3) Demand: Deterministic demand and stochastic demand. The stochastic demand can be stationary or nonstationary. St ationary stochastic demand means all demand parameters are constant over time. If the parameters change over time, then the demand is said to be nonstationary. (4) Capacities: Uncapacitated inventory models and capacitated inventory models. (5) Number of items: Single product inventory models and multiple product inventory models. (6) Sourcing options: Single sourcing and multiple sourcing. In this proposal, we are considering sing le stocking location, zero lead time, deterministic/stochastic demand, uncapacita ted, single product, and single sourcing inventory models. One commonly used criterion is demand pattern. Based on this, the inventory control models can be classified as deterministic inventory models and stochastic inventory models.
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5 1.2.2 Deterministic Inventory Models Deterministic inventory models assume that the demand is fixed and known. Two most famous models of this kind are presented in the following section. 1.2.2.1 The Economic Order Quantity (EOQ) Model The EOQ model lays the foundation for all inventory models. It is the most important analysis of inventory control and describes the important tradeoff between fixed order cost and holding cost. The first reference to the EOQ model is by Harris (1913) but this model was popularized by Wilson (1934). In this model, the demand per unit tim e is assumed to be a known constant R A constant fixed cost is incurred whenever an order is placed. An order of quantity Q is placed whenever the onhand inventory becomes zero and the replenishment time is assumed to be zero. The unit purchasing cost c is constant and known. The onhand inventory is charged with a constant holding cost h per unit per unit time. Shortages are not allowed. The objective is to determine Q so as to minimize the total average cost. The relationship between order and onhand inventory can be depicted in Figure 1.1, where T (= Q / R ) is called the cycle time. Without loss of generality, it is assumed that the initial inventory is zero. There are two cost components: ordering cost and inventory holding cost. Since all cycle are identical, it is only necessary to derive these costs in a cycle. The ordering cost consists of a fixed order cost K and purchasing cost cQ The average inventory in a cycle is Q /2. Therefore, the total cost per unit time, C ( Q ), is 2 2 / 2 ) ( hQ Rc Q KR hQ R Q cQ K hQ T cQ K Q C
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6 Figure 1.1 Inventory Levels for the EOQ Model In C ( Q ), only ordering quantity Q is a decision variable. The first and second order derivatives of C ( Q ) are given by 2 ) (2h Q KR Q C and 0 2 ) (3 Q KR Q C for 0 Q. Since 0 ) ( Q C, C ( Q ) is a convex function of Q for Q > 0. The optimal value of Q occurs where 0 ) ( Q C. So the optimal value of Q can be derived as h KR Q 2* The Q* is known as the economic order quantity. One of the main strengths of the EOQ model is that the average total cost in crease only slightly for any order quantity Q close to EOQ Q*. As a matter of fact, C ( Q*) = )] 1 ( 2 1 [ C ( Q*). For example, = 3/2 (or 2/3), C ( Q*) = 1.08 C ( Q*). That is, when the order quantity Q is either 50% higher than the EOQ or the EOQ is 50% higher than the order quantity, the resulting average Time Inventory Level Q T
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7 total cost is only about 8% above the optimal cost. This is very important when there is uncertainty in data, or some other factors prevent the calculated EOQ from the true optimal value, a close estimation will yield good results too (Waters, 1992). 1.2.2.2 The WagnerWhitin Model Wagner and Whitin (1958) consider a finite planning horizon, discrete time dynamic lot sizing problem for a single product. Demand is known but varying for each period, i.e. timevarying demand. There are no shortages and backorders. No capacity constraints are considered. A positive fixed order cost is incurred each time an order is placed. In each period, you have to decide if an order should be placed and how many to order. The objective is to find the optimal orde r quantity in each peri od so that the total costs over the planning horizon are minimized. Notations are as follows. (1) N = number of periods in the planning horizon. (2) tK= fixed order cost in period t (3) th= inventory holding cost per unit remaining at the end of period t (4) td= demand in period t (5) M = a large number. (6) tQ= order quantity in period t (7) ty= binary variable. (8) tI= inventory remaining at the end of period t 0I denotes the initial inventory.
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8 The mixed integer programming (MIP) model can be formulated as follows. Minimize N t t t N t t tI h y K1 1 Subject to t t t tI D Q I 1 t tQ My for all t 0 t tI Q for all t ty binary for all t The follows are the main conclusions drawn from this dynamic lot size model (Ravindran, 2008; Wagner and Whitin, 1958). (1) Inventory is held over a period if and only if the ordering costs are bigger than the holding costs. (2) Replenishment occurs only when the inventory level goes to zero (zero inventory ordering property). Consequently, order quantity must cover demand over an integer number of periods. (3) 01 t tI Q for all t (4) If the ending inventory of one period is positive, then this ending inventory level is at least the next periodÂ’s demand. The maximum amount is all remaining periodÂ’s demand. (5) If the ending inventory of one period is zero, then an order must be placed in the next period. The lower bound of this order quantity is equal to the next periodÂ’s demand and the upper bound is the total demand of all remaining periods.
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9 1.2.3 Stochastic Inventory Models Stochastic inventory models assume that the demand follows some known distributions. 1.2.3.1 OnePeriod Stochastic Inventory Model This model is originated in terms of a newsboy who must decide how many units (newspapers) to buy at the beginning of a day before selling them on a street corner during the day. The problem arises because the customer demand is uncertain. If the newsboy buys too many, he will end up w ith unsold stock of newspapers which is valueless at the end of the day. If he buys too few, he will have unsatisfied demand which could have yielded a profit. The newsboyÂ’s objective is to determine a proper number of units to buy so as to maximize his total profit per day. Notations and assumptions are as follows. (1) co is the cost of unit inventory that is left at the end of the period (overage cost). (2) cu is the cost of unit unsatisfied demand (underage cost). (3) The demand D is a continuous nonnegative random variable with a density function ) ( x and a cumulative distribution function ) ( x (4) The decision variable Q is the amount of units ordered at the beginning of the period. (5) C ( Q D ) is the total cost incurred at the end of the period. If Q units are purchased and the demand is D then the leftover at the end of the period will be D Q if Q > D and the unsatisfied demand will be Q D if Q < D. Hence, the total overage and underage cost can be written as ) 0 max( ) 0 max( ) ( Q D c D Q c D Q Cu o
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10 Then the expected cost function is 00) ( ) 0 max( ) ( ) 0 max( )) ( ( ) ( x Q x c dx x x Q c D Q C E Q Cu o dx x Q x c dx x x Q cQ u o 00) ( ) ( ) ( ) ( Applying the LeibnizÂ’s rule, we can obtain the first and second order derivatives of the expected cost function C ( Q ) as )) ( 1 ( ) ( ) ( ) 1 ( ) ( 1 ) (00Q c Q c dx x c dx x c Q Cu o Q u o and 0 ) ( ) ( ) ( Q c c Q Cu o for all 0 Q. Since 0 ) ( Q C, it follows that C ( Q ) is a convex function of Q The optimal value of Q occurs where 0 ) ( Q C. So the optimal value of Q can be computed by u o uc c c Q ) (*. The righthand side of the above equation is called critical ratio, and such an optimal solution is called a cr itical ratio solution. Since uc and oc are both positive, then this critical ratio is strictly between 0 and 1. So the above equation is always solvable for a continuous demand distribution. 1.2.3.2 MultiPeriod Stochastic Inventory Model In this model, it assumes that the system wi ll be run for a finite number of periods. Demand in each period is stochastic. Any remaining inventory left at the end of one period can be used in the following period. An inventory holding cost will be incurred for any positive leftover stock. If demand exceeds the onhand in ventory, the excess amount
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11 will be backlogged, which means the customers are willing to wait for the next replenishment. A penalty cost will be incurr ed for any backlogging amount. The system is under periodic review, i.e. the inventory is checked at the beginning of each period and a decision is made on how many to order. If the model does not include a fixed order cost, it will be classified as periodicreview stochastic inventory model. If the model does include a fixed order cost that is incurred for any nonzero order quantity, it will be classified as stochastic lotsizing inventory model. A base stock policy is found to be optimal for periodicreview stochastic inventory model. The decision rule for this policy is as follows: if the inventory is below the base stock, then you should order up to this base stock level; if the inventory is above the base stock, then you order nothing. A ( s S ) policy is proved to be optimal for stochastic lotsizing inventory model. The decision rule for this policy is as following: if the inventory level is below s then you should order up to S ; otherwise do nothing. Those two types of models will be discussed in detail in Chapter 2. 1.3 Inventory Control for Deteriorating Items As previous sections indicate, one implicit assumption in most existing inventory models is that products can be stored indefinitely to meet the future demands. However, in general, almost all items deteriorate to a certain extend over time. If the rate of deterioration is small and negligible, its effect can be ignored. Nevertheless, there are many products in the real world that are subject to a significant rate of deterioration. For example, some commonly used products like fruits, vegetables, meat, foodstuffs, perfumes, drugs, alcohol, gasoline, radioactive substances, photographic films, electronic
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12 components, etc., can experience significant deterioration. Hence, the impact of product deterioration should be considered explicitly in developing inventory models for those types of products. In general, deterioration is defined as decay, damage, spoilage, evaporation, obsolesce or loss of utility of an item such that it cannot be used for its original purpose. The inventory models dealing with deterioration can be classified into two categories (Dave, 1985). The first category includes models with agedependent ongoing deterioration (i.e., the items have fixed lifetimes). Such models are referred to as perishable inventory models. The second category consists of models with ageindependent ongoing deterioration (i.e., the products have random lifetimes). Such models are called deteriorating inventory models. Milk, fish and blood are examples of category one, while perfume, alcohol and gasoline are examples of category two. It was Van Zyl (1964) who first started the research work of perishable products inventory with a lifetime of two periods. This work was extended by Fries (1975) and Nahmias (1975) independently by considering lifetime of m ( 2) periods. Then some researchers (e.g. Ishii et al., 1981; Nose et al., 1981, 1984) considered this perishable inventory model with a nonzero lead time. An excellent review of such perishable inventory models is given by Nahmias (1982). Ghare and Schrader (1963) were the first to start the research of deteriorating product inventory by developing a model with exponential deterioration of inventory. Covert and Philip (1973) and Philip (1974) studied models under the assumption that the time of deterioration of an item follows a Weibull distribution. Since then, many researchers have devoted to the deteriorating inventory issues. Dave (1981) studied an
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13 inventory system with finite planning horizon, multiple replenishments, constant deterioration rate and linearly changing demand rate. Hariga (1996) considered a finite planning horizon inventory model with constant deterioration rate and timevarying demand. Wu (2001) studied a generalized EOQ model with Weibull distribution deterioration, ramp type demand rate and variable backlogging ratio. Deng (2005) generalized WuÂ’s model by eliminating an u nnecessary condition to ensure the existence of the minimum solution. Interested readers can refer to Raafat (1991), Shah and Shah (2000), and Goyal and Giri (2001) for excellent reviews of such models. For the deteriorating inventory problems, there is little literature dealing with discrete time and stochastic demand, so we conducted a discreteintime inventory research with various demand pa tterns to mitigate this gap. 1.4 Dissertation Outline The dissertation is organized as follows. Chapter 2 reviews the existing literature th at is relevant to the dissertation research. In chapter 3, a deterministic inventory model is developed for a single deteriorating product in which we consider time as a discrete variable, constant cycle time, constant demand rate, constant deterioration rate (deterioration is a constant fraction of onhand inventory), complete backlogging and the time point at which the inventory level goes to zero is a noninteger. The sensit ivity analysis is conducted to provide some insights about deterministic inventory control for deteriorating items. In chapter 4, a deterministic inventory model is developed for a single deteriorating product in which we consider time as a discrete variable, constant cycle
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14 time, timevarying demand pattern, variable de terioration rate (deterioration is a fixed but various in each period fraction of on hand inventory), waitingtimedependent partial backlogging ratio, and the time point at which the inventory level goes to zero is an integer. The sensitivity analysis is conducted through computational experiments and some insights are provided. In chapter 5, a periodicreview stochas tic inventory model is developed for a single deteriorating product in which we consider finite planning horizon, periodic review, independent and identically distributed demand (i.i.d.) in each period, constant deterioration rate (deterioration is constant fraction of ending inventory), and constant partial backlogging ratio. The costs considered are unit purchasing cost, unit holding cost, unit backlogging cost, and unit penalty cost. We test whether the base stock level policy applies, and if it does, under what conditions. The explicit ordering policy is developed for a special case. In chapter 6, a stochastic lotsizing inventory model is developed for a single deteriorating product in which we consider finite planning horizon, periodic review, independent and identically distributed demand in each period, constant deterioration rate (deterioration is a constant fraction of ending inventory), and constant partial backlogging ratio. The costs considered are fixed order cost, unit purchasing cost, unit holding cost, unit backlogging cost, and unit penalty cost. We test whether the ( s S ) policy still holds, and if it does, under what conditions. In chapter 7, a mixed integer programming (MIP) model is built for a more general case, with fixed order costs, nonstationary stochastic demands and servicelevel constraints. Some numerical examples are presented and solved by programming in
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15 C++/CPLEX to provide explicit ordering policies and comparisons with nondeteriorating case. Chapter 8 concludes the dissertation and discusses possible future work.
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16 Chapter 2 Literature Review 2.1 Introduction There has been tremendous research work done for deteriorating inventory control problems during recent decades. A general li terature review of deteriorating inventory models is given in section 2.2. As mentioned in Chapter 1, our research includes adding deterioration and partial backlogging to trad itional periodicreview stochastic inventory model and stochastic lotsizing model, so the periodicreview stochastic inventory control problems and the stochastic lotsizing problems are reviewed in sections 2.3, 2.4 and 2.5 respectively. 2.2 Deteriorating Inventory Models As mentioned in Chapter 1, these types of models deal with products that have random life times, i.e. the amount of deterioration is a function of onhand inventory level. Ghare and Schrader (1963) were the first to start the analysis of deteriorating inventory problems by developing an EOQ model with constant rate of deterioration. They formulated that the differential equation describing the inventory level ) ( t I over a cycle is d t I dt t dI ) ( ) (,
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17 where is the constant deterioration rate and d is the constant demand rate. The optimal inventory cycle time T* is determined by the following equation 0 22 T dhc dhc T K c d where c is the unit purchasing cost, K is the fixed order cost, and h is the inventory holding cost. The optimal order quantity *Q is then 22 * *T d dT Q The first EOQ model with varying rate of deterioration was developed by Covert and Philip (1973). They assume that the time to deterioration of an item follows a twoparameter Weibull distribution. The Weibull density function they use is ) exp( ) (1 t t x f where 0 is the scale parameter, 0 is the shape parameter, and 0 t is the time to deterioration. This function can stand for a decreasing, constant or increasing rate of deterioration. Using same notations as the previous model, the differential equation describing the inventory level over a cycle is d t I t dt t dI ) ( ) (1. The optimal cycle time T* is obtained by solving the following equation 1 2 ) 1 (0 2 ) exp( ) 1 (n n nT K T hd n n T n cd The optimal order quantity is then 0 ) 1 *( *) 1 ( !n n nn n T d Q
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18 The computational procedure for obtaining the optimal cycle time is given by the authors. Since then, many researchers have devoted to the deteriorating inventory issues and numerous models that consider the effect of deterioration were developed. Dave and Patel (1981) studied an inventory system with finite planning horizon, multiple replenishments but equal order cycles, constant deterioration rate, no shortages and linearly changing demand rate. The objective is to find the optimal number of replenishments. Sachan (1984) extended Dave and PatelÂ’s (1981) model by allowing shortages and complete backlogging. He also corrected some approximation errors for their model. BahariKashani (1989) extended Dave and PatelÂ’s (1981) model by dropping off the assumption that the planning horizon is divided into multiple equal ordering cycles, i.e. the replenishment cycle length can be varying. A heuristic method was developed to find the near optimal solution. Pal etc. (1993) developed an inventory model with infinite planning horizon, constant deterioration rate, deterministic demand rate which is a known function of the instantaneous inventory level, and no shortages. The objective is to determine the optimal cycl e length. Chakrabarty etc. (1998) extended Covert and PhilipÂ’s (1973) model by consid ering threeparameter Weibull distribution deterioration, shortages and timevarying demand. They provided a procedure to find the optimal cycle time. Wu (2000) considered a deteriorating inventory model with fixed cycle length, single replenishment per cycle, timevarying demand, Weibull distribution deterioration, and complete backlogging. The objective is to find the optimal time point at which inventory falls to zero, and hence identify the optimal order quantity. There are
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19 two excellent review papers describing such models developed before 2001 in detail Raafat (1991) and Goyal and Giri (2001). More recently, some inventory models for deteriorating items take into consideration of the effect of partial backlogging. Wu (2002) extended WuÂ’s (2000) model by considering that the shortages are partially backlogged and the partial backlogging ratio is dependent on the length of the waitingtime until the next replenishment. The inventory system developed is illustrated by Figure 2.1. Figure 2.1 WuÂ’s (2002) Inventory System The differential equations describing inventory level over time are formulated as follows. ) ( ) ( ) (1t D t I t dt t dI 10 t t where 1) ( t t is Weibull distribution deterioration, ) ( t D is the timevarying demand rate, and 1t is the time point at which inventory level falls to zero. Inventory Level Time S 0 t1 T
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20 ) ( 1 ) ( ) (t T t D dt t dI T t t 1. where ) ( 1 1t T is the partial backlogging ratio which decreases as waiting time ( T t ) increases. The optimal 1t was found by minimizing the total costs per unit time ) (1t C. Unfortunately, it is very difficult to show that ) (1t C is a convex function for all 1t and the calculation of 1t is not very straightforward. Teng etc. (2005) establish a deteriorating EOQ model in which the demand rate is a function of the onhand inventory and no shortages are allowed. There are three possible cases for this inventory problem and they establish the necessary and sufficient conditions for each case. Moreover, an algorithm to determine the optimal replenishment cycle time and ordering quantity is proposed to maximize the total profit. Teng and Chang (2005) establish an EPQ model for deteriorating items when demand rate is function of both stock level and selling price per unit. They provide the necessary conditions to determine an optimal solution that maximizes profits for the EPQ model. Chang etc. (2006) establish a finite horizon EOQ model with deterioration for a retailer to determine its optimal selling price, replenishment number and replenishment schedule. They prove that the optimal replenishment schedule exists and unique. A simple algorithm is provided to find the optimal solutions. Dye etc. (2006) consider an inventory system with nonconstant purchase cost, timevarying demand, and partial backlogging rate which linearly depends on the total number of customers in the waiting line. They also provide a simple solution procedure to find the optimal replenishment schedule. Hou (2006) derives a deteriorating inventory model with stockdependent demand and
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21 complete backlogging under inflation and time discounting of money over a finite planning horizon. They show that the total cost function is convex and an algorithm is presented to determine the optimal order quantity. Jolai etc. (2006) derive the optimal production over a finite planning horizon for items that follow a Weibull distribution deterioration with a stockdependent demand, fixed partial backlogging rate and under inflation. Manna and Chaudhuri (2006) develop an orderlevel inventory system for deteriorating items with ramp type demand rate, finite producti on rate and timedependent deterioration rate. The models of no shortage case and shortage case are discussed. Mandal etc. (2006) consider a deteriorating inventory model with finite demand rate and limited storage space. It is solved by modified geometric programming method and nonlinear programming method. Ouyang etc. (2006) establish a general EOQ model for deteriorating items with waitingtime dependent partial backlogging and permissible delay in payments. They mathematically prove that the total cost function is strictly pseudoconvex so that the optimal not only exists but also is unique. Yang (2006) considers an inventory system with constant demand rate, Waitingtimedependent partial backlogging and two warehouses under inflation. The own warehouse (OW) has a fixed capacity. The rented warehouse (RW) has unlimited capacity. The inventory holding and deterioration costs in RW are higher than those in OW. They prove that the optimal solution not only exists but also is unique. Dye etc. (2007) study an inventory system with constant demand rate, waitingtimedependent partial backlogging and two warehouses. A rented warehouse is used when the ordering quantity exceeds the capacity of the owned warehouse. They obtain the condition when to rent the warehouse and provide simple solution procedures for finding the maximum total profit per unit time.
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22 Dye (2007) develops a deteriorating inventory model with sellingpricedependent demand rate, timedependent deterioration rate and waitingtimedependent partial backlogging. They proved that the optimal replenishment schedule not only exists but also is unique for any given selling price. An algorithm to find the optimal selling price and replenishment schedule for the proposed model is developed. Balkhi and Tadj (2008) establish a generalized economic order quantity model for deteriorating items with timevarying demand, timevarying deterioration rate and waitingtimedependent partial backlogging. The cost parameters are also assumed to be general functions of time. Necessary and sufficient conditions for a unique optimal solution are derived. Chern etc. (2008) consider a deteriorating inventory system with finite planning horizon, timevarying demand rate and waitingtimedepen dent partial backlogging under inflation. They provide an algorithm for determining the optimal replenishment number and schedule. Rong etc. (2008) study a deteriorating inventory system with pricedependent demand, partial/fully backlogging, imprecise leadtime and two warehouses. Holding cost at rented warehouse decreases with the increase of distance from the market place. The optimal solutions are derived by maximizing the average profit. Roy (2008) develops a deteriorating inventory model with sellingpricedependent demand rate, time proportional deterioration rate and timedependent holding cost. They considered both shortage case and no shortage case. Shah and Acharya (2008) formulate an orderlevel lotsize inventory model for a timedependent deterioration and exponentially declining demand. The optimal solutions are obtained through minimizing the total cost per unit time. Lee and Hsu (2009) develop a twowarehouse inventory model for deteriorating items with timedependent demand and a finite replenishment rate within a finite
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23 planning horizon. One approach to determine the number of production cycles and replenishment schedule is developed which permits variation in production cycle times. Shah and Shukla (2009) study a deteriorating inventory model with constant demand rate, constant deterioration rate and waitingtimedependent partial backlogging. The convexity of the total cost function is shown numerically and the optimal order quantity is obtained through minimizing the total cost Skouri etc. (2009) consider a deteriorating inventory model with general ramp type de mand rate, Weibull deterioration rate and waitingtimedependent partial backlogging. The optimal replenishment policy is derived under two different replenishment policies: (a) starting with no shortages and (b) starting with shortages. Yang etc. (2010) extended HouÂ’s (2006) model by considering an deteriorating inventory lotsize model under inflation with stockdependent demand rate, constant deterioration rate, and waitingtimedependent partial backlogging. They proved that there exists a unique replenishment schedule and a good estimate for finding the optimal replenishment number is provided. Geetha and Uthayakumar (2010) studied an EOQ based model for deteriorating products with permissible delay in payments. They considered constant demand rate, noninstantaneous deterioration, and waitingtimedependent partial backlogging. The necessary and sufficient conditions of the existence and uniqueness of the optimal solutions are provided. An uptodate review paper is published recently by Li etc. (2010). One common feature of the above deteriorating inventory models is that time is treated as a continuous variable, which may not always be the case in practice. For example, in some reallife problems, time may be better treated as a discrete variable and a result of say 6.65891 hours is difficult to measure and confusing. In this regard, some
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24 researchers have attempted to study the deteriorating inventory by considering time as a discrete variable. Dave (1978) develops a discreteintime EOQ model for deteriorating items with constant demand rate, constant deterioration rate and no shortage. The optimal solution for cycle time is derived. Dave (1979) establishes a discreteintime orderlevel inventory model for deteriorating items with constant demand rate, constant deterioration rate, complete backlogging and predetermined fixed cycle length. Since time is considered as a discrete variable, the difference equations describing the inventory levels at different time points are formulated as d t I t I t I ) ( ) ( ) 1 ( 1 , 2 1 01 t t and d t I t I ) ( ) 1 (, T t t t , 1 ,1 1 where 1t is the time at which inventory level goes to zero. The total cost per unit time ) (1t C is derived. Since 1t is an integer, the necessary and sufficient conditions for ) (1t C to have a global minimum at 1t are ) ( 0 ) 1 (* 1 1t C t C and 0 ) (1 2 t C for all T t , 2 1 01 where ) ( ) 1 ( ) (1 1 1t C t C t C and )) ( ( ) (1 1 2t C t C The sufficient condition was satisfied and the optimal 1t is derived by testing the necessary condition. Dave and Jaiswal (1980) consider a discreteintime probabilistic inventory model for deteriorating items with stationary uniform demand, constant deterioration rate, and no shortages. The optimal solution for cycle time is derived. Dave and Shah (1982) extend Dave and Jaiswal's (1980) model by allowing lead time equal to one scheduling
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25 period. The optimal solution for cycle time is derived. Rengarajan and Vartak (1983) extended Dave's (1979) model by allowing timedependent demand which occurs at the end of each period. They find that the initial stock level is not at all affected by the nature of demand. Dave (1984) generalized Dave and Shah's (1982) model by allowing lead time equal to a fixed constant. The optimal solution for cycle time is derived. Dave (1985) establishes a discreteintime deteriorating inventory model with demand rate linearly changing with time, constant deterioration rate, no shortages and finite planning horizon. The optimal replenishment number and schedule are derived. Dave (1987) considers three inventory systems, i.e. EOQ, orderlevel and orderlevel lotsize inventory systems, for deteriorating items with finite replenishment rate, constant demand rate and constant deterioration rate. The EOQ does not allow shortage. The orderlevel system allows shortage but assume the cycle time is a predetermined constant. The orderlevel lotsize system allows shortage and does not consider the scheduling period as a prescribed constant. Searching procedures for finding the optimal solutions are derived for all three systems. Dave (1988) studies a discreteintime deteriorating inventory model with constant demand rate, constant deterioration rate and no shortages under permissible delay in payments. There is no charge if th e due amount is paid within this permitted settlement period. The solution procedure for optimal cycle time is provided. Dave (1990) considers a discreteintime deteriorating inventory model with stationary uniform and instantaneous demand occurring at the beginning of the scheduling period, constant deterioration rate and no shortages. Three inventory models are developed, i.e. a model with no lead time, a model with a deterministic lead time and a model with a lead time equal to a multiple of the scheduling peri od. The optimal solution for cycle time is
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26 derived for each model. Shah and Shah (1998) consider a discreteintime deteriorating inventory model with stationary uniform demand, constant deterioration rate and no shortages under permissible delay in payments. The optimal cycle time is derived by solving a sequence of inequalities. Shah (1998) considers a discreteintime deteriorating inventory model with stationary uniform demand, constant deterioration rate and no shortages. The unit purchase cost will increase from a specified future date. The procedure to find the optimal cycle time is provided. 2.3 PeriodicReview Stochastic Inventory Control Problem In this type of problem, the planning horizon is finite. Demand in each period is stochastic. The system is under periodic review, i.e. the inventory level is checked at the beginning of each period and a decision is made on how many to order. The objective is to determine the optimal ordering policy at the beginning of each period with minimum expected overall purchasing, holding and shortage costs. Arrow (1958) started this type of research by considering that the demands that arise in successive periods are independent and identically distributed random variables with known distribution functions. Recently, Po rteus (2002) reconsidered this problem in his book with stochastic i.i.d demand. The complete proof was provided and the explicit optimal base stock level was computed for some special case. Since our proposed model will be derived based on PorteusÂ’s work, the detail review of his model is presented next. Notations in PorteusÂ’ Model are as follows. (1) c Â– unit purchasing cost ($/unit) (2) Hc Â– unit holding cost, charged against positive ending inventory ($/unit) (3) Pc Â– unit shortage cost ($/unit)
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27 (4) D Â– generic random variable representing demand, which is i.i.d over every period (5) Â– demand density distribution (6) x Â– inventory level before ordering (the state of the system) (7) y Â– inventory level after ordering (the decision variable) (8) N Â– the length of the planning horizon (9) Â– oneperiod discount factor (] 1 0 ( ) (10) ) 0 max( x x (11) ) 0 min(x x The expected oneperiod holding and shortage cost function is ) ( ) ( D y El y L where ) ( ) ( x c x c x lP H. He derived the optimality equations for N t 1 as 0 1) ( ) ( ) ( ) ( min ) ( dD D D y f y L x y c x ft x y t where ) (1x fN is the terminal value function ) ( x v. Porteus proved that if the terminal value function is convex, then the optimal policy in each period is characterized by a single critical number, which is called base stock policy. That is, order up to the base stock level if the current inventory is less than the base stock and order nothing otherwise. The explicit optimal base stock level was de rived for the case that the terminal value function has a slope of Â– c
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28 Karlin (1960) considered that the demands in su ccessive periods are not identically distributed. He prov ed that the optimal ordering policy in each period is a base stock policy, but the base stock level may vary in successive periods. Karlin (1960) studied this dynamic stochastic inventory problem with demand distribution varying over successive periods in a cyclical fashion. The optimal ordering policies are derived for both backlogging and nonbacklogging cases. Iglehart (1964) considered the dynamic inventory system with demand distribution possessing a density belonging to either exponential or range family of densities and having an unknown parameter. They applied a Bayesian estimation method to obtain the optimal ordering policies as the amount of demand information varies. Azoury (1985) considered the periodic review inventory problem in which one or more parameters of the demand distribution are unknown but with a known prior distribution chosen from the natural conjugate family. An explicit form of the optimal ordering policy is given. Gavirneni (2004) considered this type of problem with i.i.d demand and fluctuating purchasing cost. He showed that an order up to policy is optimal and proposed a method to predict the effectiveness of myopic heuristics. Bertsimas and Thiele (2006) proposed a general methodology based on robust optimization to study this type of stochastic inventory problem without assuming a specific distribution of the demand. They showed that the structure of the optimal robust policy is of the same base stock character. Levi etc. (2007) considered this type of stochastic inventory problem under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. They desc ribed how to compute the optimal policies based only on the observed samples of the demands.
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29 2.4 Stochastic LotSizing Problem This type of problem is just the periodicreview stochastic inventory control problem with fixed order cost. It was Scarf (1960) who first established the optimal ordering policy structure for the stochastic lotsizing problem with independent and identically distributed demands in successive periods. This problem was restu died by Porteus (2002) and a complete proof was provided. Since our proposed model builds on PorteusÂ’s work, the detail review of his model is given below. Notations in PorteusÂ’ Model are as follows. (1) c Â– unit purchasing cost ($/unit) (2) Hc Â– unit holding cost, charged against positive ending inventory ($/unit) (3) Pc Â– unit shortage cost ($/unit) (4) K Â– fixed order cost ($/order) (5) D Â– generic random variable representing demand, which is i.i.d over every period (6) Â– demand density distribution (7) x Â– inventory level before ordering (the state of the system) (8) y Â– inventory level after ordering (the decision variable) (9) N Â– the length of the planning horizon (10) Â– oneperiod discount factor (] 1 0 ( ) (11) ) 0 max(x x (12) ) 0 min( x x
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30 The expected oneperiod holding and shortage cost function is ) ( ) ( D y El y L where ) ( ) (x c x c x lP H. The optimality equations for N t 1 was derived as )] ( [ min ), ( min ) ( y G K x G cx x ft x y t t where 0 1) ( ) ( ) ( ) (dD D D y f y L cy y Gt t and ) (1x fN is the terminal value function ) ( x v. Porteus proved that if the terminal value function is continuous and K convex, then a ( s S ) policy is optimal in each period. That is, order up to inventory level S if the current inventory is less than the level s and order nothing otherwise. Schal (1976) generalized ScarfÂ’s result by finding some new conditions for the optimality of an ( s S ) policy and a special case without assuming particular demand distributions was obtained. Iyer etc. (1992) analyzed the deterministic ( s S ) inventory problem which is to determine parameters s and S such that implementing this ( s S ) policy results in the minimum possible total costs given a set of demands for n periods. A polynomial time algorithm for finding an optimal ( s S ) for the deterministic problem was provided. Sox (1997) considered the case in which the demand is random and the costs are nonstationary. He modeled the problem as a mixed integer nonlinear program. An optimal solution algorithm was developed. Then Gallego etc. (2000) studied the finite ordering capacity and they showed that the optimal capacitated policy has an ( s S )like structure. Sobel and Zhang (2001) considered that the demands arrive simultaneously from a deterministic source and a random source. The deterministic demand has to be
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31 satisfied immediately and demand from a stochastic source can be backlogged if necessary. They proved that a modified ( s S ) policy is optimal assuming that the stochastic demand is satisfied immediately if there is sufficient stock on hand. Dellaert and Melo (2003) considered a stochastic manufacturing system with only partial knowledge on future demand because customers tend to order in advance of their actual needs. A Markov decision model was formulated to find the optimal policy. Two approximate strategies for obtaining nearoptimal production lot sizes were proposed. More recently, Ozer and Wei (2004) consider ed a capacitated production system faced by a manufacturer who has the ability to obtain advance demand information. The capacity constraint is that the number of production periods is limited. Two cases were analyzed. When there is no fixed cost, the optimal policy is of a statedependent modified base stock policy. When there is positive fixed cost, they analyzed a class of production policies under which the manager is restricted to either producing at full capacity or not at all. Bensoussan etc. (2006) considered the effect of information delay between the current time and the time of the most recent inventory level known to the inventory manager. A constant delay and a random were both discussed. The optimal ordering policy is base stock level when there is no fixed order cost and (s, S) policy when fixed order cost exists. When there are nonhomogeneous stochastic demands and fixed order costs, the computation of the optimal ordering polici es are extremely difficult, so a lot of researchers have attempted to develop some efficient computational algorithms to approximate the optimal ordering policies. Most of them are based on heuristic methods (Porteus 1985, Bollapragada and Morton 1999, Levi etc. 2007), while Gavirneni and
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32 Tayur (2001) developed an efficient solution method Â– Direct Derivative Estimation (DDE) Â– for computing optimal orderupto levels for a discrete time nonstationary inventory control model. 2.5 Summary The following gaps are identified from literature review. (1) Literatures for discreteintime deteriorating inventory models are very limited. (2) No closedform solutions for deteriorating inventory problems. (3) When considering timevarying demand, variable deterioration rate, and waitingtimedependent partial backlogging, the co nvexity of total cost function is not proved. (4) No literatures for multiperiod stochastic inventory model with deterioration and partial backlogging. (5) No literatures for stochastic deteriorating inventory control under servicelevel constraints. Our dissertation completes the current litera ture by filling up all those gaps. The results of this research have the potential to positively influence industrial engineering and management science curricula related to production and inventory control, large scale optimization, optimization modeling, and supply chain management. Furthermore our work could enhance student learning by providing practical examples and by development of casestudies on the design of control systems for deteriorating inventory.
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33 Chapter 3 A DiscreteinTime Inventory Model with Deterioration and Backlog There is a single product with a constant rate of demand. Any physical inventory experiences a constant rate of deterioration, and any unsatisfied demand is completely backlogged. A previous model was presented to determine the optimal orderlevel for a given constant cycle time, where time is treated as a discrete variable. However, the optimal solution derived is only valid under th e restriction that the physical inventory level goes to zero at an integer time. This is totally unnecessary from the practical view point. This chapter relaxes this restriction. Furthermore, a closedform equation is derived to compute the optimal solution, while no closedform solution was presented in literature (even for the integerrestrictive ca se). This greatly reduces the computational effort to identify the optimal solutions, and makes sensitivity analysis possible. Some insights are provided thr ough sensitivity analysis. 3.1 Assumptions and Notations The following assumptions are made. (1) The cycle has T unit times, where T is a known constant. (2) The demand rate of R units per unit time is a known constant. (3) Rate of replenishment is infinite and lead time is zero. The fixed lotsize q raises the inventory at the beginning of each cycle to stock level S (4) Shortages are made up immediat ely after a fresh lot arrives.
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34 (5) There is neither repair nor replacement for the deteriorated items in the inventory during a cycle. (6) The unit purchasing cost c inventory carrying cost h per unit per unit time and the shortage cost b per unit per unit time are known and constant. (7) The deterioration rate is a constant (i.e., a constant fraction of the onhand inventory deteriorates per unit time). (8) The inventory level at any time t within the cycle is denoted as I ( t ), T t 0. 3.2 Model Figure 3.1 Inventory Level of the System As illustrated in Figure 3.1, the fixed lotsize q raises the inventory at the beginning of a cycle to stock level S This initial inventory is gradually reduced due to both demand and deterioration. At time 1t t for all T t 10, the inventory level goes to zero, i.e., 0 ) (1 t I. All demands occurring after time 1t are fully backlogged and will q S . . .0 T 1t 1t1t
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35 be fulfilled by the new order from the next cycl e. Therefore, the total backlog is equal to ) (1t T R Since the lotsize q raises the initial inventory to S during every cycle, we have S t T R q ) (1 (3.1) With the depletion of inventory determined by both demand and deterioration, the difference equations describing the inventory level ) ( t I of the system from time 0 to time 1t are R t I t I t I ) ( ) ( ) 1 ( ] 1 0 [1 t t. (3.2) Equation (3.2) can be rewritten as R t I t I ) ( ) ( ] 1 0 [1 t t (3.3) where ) ( ) 1 ( ) ( t I t I t I (3.4) Since there are no units held in inventory from time 1t to time T the deterioration will not exist during this time span. Hence, the difference equations describing the inventory level ) ( t I of the system from time 1t to time T are R t I t I ) ( ) 1 (, ] 1 [1 T t t (3.5) which can be rewritten as R t I ) (,] 1 [1 T t t (3.6) Solving (3.3) and using the boundary condition 0 ) (1 t I, we obtain ] 1 ) 1 [( ) (1 t tR t I ,] 0 [1t t (3.7)
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36 Similarly, from equation (3.6) we get ) ( ) (1t t R t I ,] [1T t t (3.8) At time 0, the inventory level S I ) 0 (. Substituting this into (3.7) yields ] 1 ) 1 [(1 tR S (3.9) Then from equation (3.1), we get ] 1 ) 1 [( ) (11 tR t T R q (3.10) Since the lotsize q is depleted by both demand and deterioration, and the total demand during one cycle is RT then the total number of units, ) (1t D, that deteriorate during a cycle T will be 1 1] 1 ) 1 [( ) (1Rt R RT q t Dt (3.11) Given unit cost c the average deterioration cost per unit time is T t cD ) (1 (3.12) When 1t is restricted to an integer, the average number of units in inventory during T can be computed as follows. )] ( ) ( 2 ) 0 ( [ 2 1 ) (1 1 1 11t I j I I T t It j j (3.13) Note that (3.13) differs from DaveÂ’s in that the mean inventory in a period is set here as the average of the beginning and ending inventories for the period. However, when t1 can be any real value, to compute ) (1t I becomes much more complicated if not impossible. This dilemma can be resolved by using the Trapezoid method. Since I ( t ) is a polynomial
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37 function of t it is a smooth function. Then (3.13) can be approximated by integration. In particular, the following is valid (Bronshtein, et al., 1997) ) (1t I = ) ) ( ( 110dt t I Tt, (3.14) where is the approximation error and can be computed by = t1I "( )/12, 0 t1. I "( t ) = ) 1 ( ln ) 1 (21 t tR, 0 t t1. Therefore, I "( ) max{ I "( t ): 0 t t1} = ) 1 ( ln ) 1 (21 tR, and the error term is bounded by ) 1 ( ln ) 1 ( 122 11 tR t. The integration is calculated as ) 1 ln( )] 1 ln( 1 ) 1 [( ) ( 11 01 1 T t R dt t I Tt t. Then the relative error, RE ( ), of approximating ) (1t I by dt t I Tt10) ( 1is bounded by RE ( ) = 10) ( 1 /tdt t I T T )] 1 ln( ) 1 ( 1 [ 12 ) 1 ( ln ) 1 (1 3 11 1 t tt t. (3.15) When this error bound is small, the approximation is good. In doing so, the average inventory is computed by ) (1t I = ) 1 ln( )] 1 ln( 1 ) 1 [(11 T t Rt. (3.16) The use of this approximation will greatly simplify the computation of ) (1t I. More importantly, this approximation will lead to a closedform solution for the optimal t1, which will be given at the end of this section.
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38 From equation (3.7), the average number of units in inventory during T is calculated as follows: ) 1 ln( )] 1 ln( 1 ) 1 [( ) ( 1 ) (1 0 11 1 T t R dt t I T t It t (3.17) The average inventory holding cost per unit time is ) (1t I h (3.18) From equation (3.8), the average shortage during T is 2 1 1) ( 2 )] ( [ 1 ) (1t T T R dt t I T t ST t (3.19) Hence, the average shortage cost per unit time is ) (1t S b (3.20) Adding up costs given by (3.12), (3.18) and (3.20) leads to the following total average cost of the system per unit time during one cycle 2 1 1 1 1 1 1 1) ( 2 ) 1 ln( )] 1 ln( 1 ) 1 [( ] 1 ) 1 [( ) ( ) ( ) ( ) (1 1t T T bR T t hR T cRt T cR t S b t I h T t cD t Ct t (3.21) The firstorder derivative of ) (1t C is given by T cR T T t bR T hR T cR t Ct t ) ( ] 1 ) 1 [( ) 1 ln( ) 1 ( ) (1 11 1 (3.22)
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39 Similarly, the secondorder derivative of ) (1t C is given by T bR T hR T cR t Ct t ) 1 ln( ) 1 ( )] 1 [ln( ) 1 ( ) (1 12 1 (3.23) Since all the parameters are positive and 1 0 it is obvious to see from equation (3.23) that 0 ) (1 "t C, for all ] 0 [ T t Therefore, ) (1t C is a strictly convex function. This means that the solution to the equation 0 ) (1 t C is optimal to ] 0 [ ) ( min1 11T t t Ct Solving equation 0 ) (1 t C, we get the optimal 1t, denoted as 1t, as follows: T b h W b c t ) 1 ln(* 1 (3.24) where W is ] ) 1 ln( ) 1 ln( [) )( 1 ln( b T b h ce b h c LambertW W (3.25) The Lambert W function is defined as x e x LambertWx LambertW)] ( [) (, for e x 1 The result obtained here show s that the optimal value of t1 is not affected by the demand rate. Substituting 1t into equations (3.9) and (3.10) yields the optimal lotsize and orderlevel. 3.3 A Numerical Example The same example from DaveÂ’s paper is used to illustrate our methodology. The values of the parameters are given as follows.
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40 (1) R = 200 units/month; (2) c = $80.00/unit; (3) h = $1.00/unit/month; (4) b = $9.00/unit/month; (5) T = 12 months; (6) 05 0 Substituting these numbers into equation (3.24) and solving by Maple 10.0, we obtain the following optimal value of 1t as 93 6* 1t months From (3.9) and (3.10), the optimal orderlevel and lotsize are 1708* S and 2722* q. Also, substituting 1t into equation (3.21), the minimum average cost per month obtained by this method is month t C / $4534.13 ) (* 1. By DaveÂ’s method, 1t is computed as 7 months. Substituting this 1t value into equation (3.21), one gets the corresponding total monthly cost at $4534.82, which is almost the same as the optimal total monthly cost given above. This should be expected, as the optimal 1t of 6.93 months is so close to 7 months. As a matter of fact, the total monthly cost as a function of t1 is pretty flat around the optimal 1t as shown in Figure 3.2. This result echoes the behavior of the wellknown EOQ model. By Equation (3.15), the relative error, RE( ), is computed as RE(0.05) < 0.17%. Since the inventory holding cost is only one of 3 cost components, the actual impact of
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41 the approximation by Equation (3.16) on the optimal total cost is far less than 0.17%. Therefore, the use of approximation (3.16) is appropriate. Figure 3.2 Total Monthly Cost C ( t1) as a Function of t1 3.4 Sensitivity Analysis The sensitivity of 1t, ) (* 1t C, *q and *s with respect to parameters c h b and T is studied numerically in this section. The results are illustrated in Figure 3.3 to Figure 3.7. We use the values given in Section 4 as the base values for the various parameters. It is worth mentioning that the time unit considered in this paper is month. Moreover, to put the values of 1t in a perspective relative to cycle time T we present the computational results using the ratio T t* 1 instead of the absolute value of 1t. The ratio 5 0* 1 T t means that the inventory goes to zero in the middle of a cycle T A ratio below or above 0.5 indicates the inventory goes to zero early or late in a cycle, respectively. Alternatively, a small ratio T t* 1 means a large backlog, while a large ratio indicates less backlogging. Total monthly cost C ( t1) t1
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42 The sensitivity study is conducted as follows. Only one parameter value is altered at a time, while holding all the other parameters constant. The range of is chosen from 0.001 to 0.5. This range represents a very little to significant deterioration. The ranges of c and b considered are, respectively, 40 ~ 360 ($/unit) and 3 ~ 24 ($/unit/month). The range of T is set to be from 3 to 24 (months). In practice, the annual inventory holding cost per unit can usually be expressed as the product of the unit purchasing cost c and an annual interest rate I (i.e., h=Ic ), where the annual interest rate is an aggregated term comprised of cost of capital, taxes and insurance, cost of storage, and breakage and spoilage, etc. In the sensitivity study, we co nsider the range of the annual interest rate from 1% to 75%. This leads to the range of h values from 0.067 to 5 ($/month). The computational results and analyses are represented next. Table 3.1 Different Optimal Values w.r.t. 0.001 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 t1* 0.891 0.5775 0.395 0.2875 0.2191 0.171 0.1366 0.11 0.089 0.07 0.058 S 2152 1708 1295.5 1002.5 798.3 694.9 529.93 437.63 363.6 303 249 q 2412 2721.4 2747.5 2712.5 2672.3 2634 2601.9 2573.6 2549 2529 2509 C ( t1*) 1168 4534.1 6504.3 7696.7 8479.3 9026 9428.6 9734.0 9972 10162 10315
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43 Figure 3.3 Sensitivity Analysis w.r.t. (a) (b) T t* 1 (1) From Table 3.1 and Fig. 3.3(a), as increases from 0, the optimal ratio T t* 1 decreases rapidly from almost 1 until reaches about 0.2, and then the pace of decrease levels off and approaches close to 0. That is, for a rapidly deteriorating product (i.e., a large ), it is better to have substa ntial backlogging to keep costs low, as expected. From Fig. 3.3(b), interestingly, as increases, the optimal lotsize *q increases at first, peaks at around = 0.08, and then decreases slowly. In general, *q stays pretty steady, for all However the optimal orderlevel *s keeps decreasing at a much faster pace as increases. Moreover, the total optimal cost ) (* 1t C increases very dramatically as increases from 0, and levels off after around = 0.3. This indicates th at at small values of a little reduction of can lead to significant improvem ent of cost reduction.
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44 Table 3.2 Different Optimal Values w.r.t. c c 40 80 120 160 200 240 280 320 360 t1* 0.6908 0.5775 0.496667 0.436667 0.388333 0.35 0.318333 0.290833 0.2675 S 2119.7 1708 1430.3 1233.45 1080.05 961.59 865.82 784.15 715.93 q 2861.7 2721.4 2638.3 2585.45 2548.05 2521.59 2501.82 2486.15 2473.93 C ( t1*) 3251.8 4534.1 5456.69 6157.28 6709.53 7157.04 7527.48 7839.42 8105.79 Figure 3.4 Sensitivity Analysis w.r.t. c (a) (b) T t* 1 (2) As shown in Table 3.2 and Figure 3.4, there is a decrease in the optimal ratio T t* 1 with an increase in the unit price c. The optimal lotsize *q and the optimal orderlevel *s decrease as the unit price c increases. It should be noted that *s is more sensitive than *q with respect to the parameter c. In other words, the change of *q is smaller than that of *s with the increase of c. In fact, *q is almost flat, as c changes. The optimal total monthly cost ) (* 1t C increases more significantly as c increases.
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45 Table 3.3 Different Optimal Values w.r.t. h h 0.067 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t1* 0.62 0.6 0.5775 0.55666 0.5375 0.52 0.5033 0.4875 0.473 0.46 0.44 S 1861 1786.9 1708 1634.64 1568.5 1508.9 1452.6 1399.7 1352 1309 1265 q 2771 2746.9 2721.4 2698.64 2678.5 2660.9 2644.6 2629.7 2616 2605 2593 C ( t1*) 4066 4292.6 4534.1 4757.1 4963.9 5156.19 5335.5 5503.1 5660 5808 5947 (a) (b) Figure 3.5 Sensitivity Analysis w.r.t. h (3) From Table 3.3 and Fig. 3.5, when h increases from 0.067 to 5, the optimal ratio T t* 1 decreases almost linearly within a rela tively small range from 0.62 to 0.45. The trends of changes in ) (* 1t C, *q and *swith respect to inventory holding cost h are similar to the trends with respect to c but in a more or less linear fashion. Again, *s is a little more sensitive to the changes in h than in *q.
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46 Table 3.4 Different Optimal Values w.r.t. b b 3 6 9 12 15 18 21 24 t1* 0.321667 0.481667 0.5775 0.641667 0.689167 0.725 0.753333 0.775833 S 875.81 1380.44 1708 1937.29 2113.44 2249.78 2359.73 2448.42 q 2503.81 2624.44 2721.4 2797.29 2859.44 2909.78 2951.73 2986.42 C ( t1*) 2484.78 3746.72 4534.13 5078.34 5478.98 5787.11 6031.86 6231.15 Figure 3.6 Sensitivity Analysis w.r.t. b (a) (b) T t* 1 (4) As shown in Table 3.4 and Fig. 3.6, the optimal ratio T t* 1 increases pretty dramatically as the backlogging cost b increases from 3 to about 12, and then the slope of the increase becomes steady as b further increases. The total monthly cost ) (* 1t C and the optimal orderlevel *s behave in a very similar manner with respect to b. However, the lotsize*qslightly increases in an almost linear fashion as b increases. This phenomenon is reasonable, since the higher the backlogging cost is, the less backlogging should be expected, which translates into larger ratio
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47 T t* 1, and larger q*, s* and ) (* 1t C. Again, s* is more sensitive to b than q*, especially when b is small. Table 3.5 Different Optimal Values w.r.t. T T 3 6 9 12 15 18 21 24 t1* 0.573333 0.586667 0.584444 0.5775 0.569333 0.560556 0.551429 0.542083 S* 368.93 791.52 1238.82 1708 2189.7 2711.64 3244.7 3796.07 q* 624.93 1287.52 1986.82 2721.4 3490.7 4293.64 5128.7 5994.07 C(t1*) 1260.61 2314.63 3405.17 4534.13 5701.29 6906 8147.47 9424.8 Figure 3.7 Sensitivity Analysis w.r.t. T (a) (b) T t* 1 (5) From Table 3.5 and Figure 3.7, as T increases, the optimal ratio T t* 1 almost remains constant around 0.56. The optimal ) (* 1t C, *q and *s all increase dramatically in more or less linear fashions, with respect to T. This is somewhat expected, as larger T leads to higher inventory and more backlogging, since only one replenishment is considered in a cycle of T time units.
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48 In conclusion, the optimal cost ) (* 1t C always increases as the value of any parameter increases. The parameter plays a very important role in total cost, which explains why it is essential to consider the effect of deterioration for products that deteriorate. The parameter T is also very important too because all the optimal values are very sensitive to it as mentioned in part (5). In real life, the inventory carrying cost and the shortage backlogging cost may be assessed as a given proportion to the unit price of goods. To examine the system behavior for such cases, this paper considers fixing the ratio c:h:b while altering all three parameter values simultaneously. In particular, consider c h1 and c b2, where 1 and 2 are constants. Substituting these into equations (3.24) and (3.25) leads to the following. T e LambertW tT 2 1 ) )( 1 ln( 2 1 2 1) 1 ln( ] ) 1 ln( ) 1 ln( [ 12 2 1. As one can see clearly from this equation, the optimal 1t is now independent of all three parameters c, h, and b, provided that 1and 2 are constants. Consequently, the optimal *s and *q are also independent of all c, h, and b, by equations (3.9) and (3.10). Similarly, the following equation can be obtained by using equation (3.21). 2 1 2 1 1 1 1) ( 2 ) 1 ln( )) 1 ln( 1 ) 1 (( ) 1 ) 1 (( ) (* 1 1t T T R T t R T Rt T R c t Ct t (3.26)
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49 The term in the brackets of equation (3.26) is independent of all c, h, and b. Therefore, the optimal cost ) (* 1t C is a linear function of c (h or b), provided that 1and 2 are constants. This insight is very significant in the sense that for companies with a fixed ratio b h c: :, the total cost depends strictly linearly on the unit price of a product and the optimal lotsize stays constant. 3.5 Conclusion In this chapter, we generalized an existing inventory model by allowing the time at which inventory level reaches zero to be noninteger. Explicit formulas for orderlevel, lotsize, and total cost are derived. As a result, the computation is much simplified. Furthermore, the derived explicit total cost equation makes sensitivity analysis possible. From the sensitivity analysis, it is found that the cost ) (* 1t C always increases as any parameter value increases. The deterioration rate contributes pretty significantly to total cost, which explains why it is imperative to address the effect of deterioration for products that do deteriorate. The cycle time T also plays a rather important role as all the optimal solutions are very sensitive to it. This should be expected, though, as a single replenishment is restricted for each cycle. Another insight is that the optimal cost ) (* 1t C is a linear function of c, provided that c h1 and c b2, where 1and 2 are constants. This finding is pretty significant in the sense that for companies with a fixed ratio b h c : :, the total cost depends strictly linearly on the unit price of a product.
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50 Chapter 4 A DiscreteinTime Deteriorating Inventory Model with TimeVarying Demand, Variable Deterioration Rate and Waitin gTimeDependent Partial Backlogging A new inventory system is considered for a single deteriorating item in which the demand is varying over time, unsatisfied demands are partially backlogged depending on the waiting time up to the next replenishment, and deterioration is assumed to be a variable fraction of the on hand inventory at the beginning of each period. Time is treated as a discrete variable because in real life we always consider time on a discrete scale, i.e. in terms of complete units of days, weeks, months, etc. (Dave, 1979). Under this consideration, we are able to derive explicit solutions based on the sufficient optimality condition. The necessary optimality condition is easily proved to be true, while it is not provable in continuous time case. The first example demonstrates that our model can reduce to DaveÂ’s (1979) by considering constant demand, constant deterioration rate and complete backlogging. The second example considers a more general case with variable deterioration rate and waitingtimedependent partial backlogging. 4.1 Assumptions and Notations The model under consideration is developed with the following assumptions. (1) The cycle time of T periods is known and constant. (2) The demand of iR units in period i is given and occurs at the end of the period.
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51 (3) iS denotes the inventory level at time point ) , 2 1 0 ( T i i The inventory level becomes zero at the end of time period k, i.e. 0 kS. (4) The replenishment rate is infinite and le ad time is zero. At the beginning of a cycle, a fixed lotsize Q raises the inventory level to 0S. (5) A constant fraction i of the beginning onhand inventory in period ) , 2 1 ( T i i deteriorates during that period. (6) The fraction of shortages backlogged for period i is denoted as B(i), for i = 1, 2, Â…, T, where i is the waiting time until the next replenishment. It is assumed B(i) is a decreasing function of i. One possible form of ) (iB could be i 1 1 where the backlogging parameter is a positive constant (Chang and Dye, 1999). This function guarantees that 1 ) ( 0 iB for all i, and when the waiting time i is zero for i = T, 1 ) ( TB, which is complete backlogging. (7) There is neither repair nor replac ement for the deteriorated items. (8) The unit purchasing cost c, inventory holding cost h per unit per unit time, backlogging cost b per unit per unit time, and the penalty cost p per unit lost sale are all known and constant. 4.2 Model As illustrated in Figure 4.8, the fixed lotsize Q raises the inventory at the beginning of the cycle to stock level 0S. This inventory of 0S is gradually reduced due to both demand and deterioration. By assumption, 0 kS for some k. Demands occurring after time k are partially backlogged.
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52 Figure 4.1 Inventory Level for the Model over the Cycle With the depletion of inventory by both demand and deterioration, the difference equation describing the inventory level for period ) 0 1 ( k k i i is i i i i iR S S S 1 1 0 1 k k i, i.e. i i i i iS R S 1 11 0 1 k k i. From this recursive relation, we can obtain k k kR S 11 since 0 kS. Similarly we can get 1 1 1 21 ) 1 )( 1 ( k k k k k kR R S ) 1 )( 1 ( ) 1 )( 1 )( 1 (2 1 1 2 1 3 k k k k k k k kR R S . T Q 0S. . .0 k
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53 1 1 1 2 1 1 1 1 01 ) 1 ( ) 1 )( 1 ( ) 1 ( ) 1 )( 1 ( R R R Sk k k k k k Therefore, the inventory levels from time 0 to time k can be rewritten as j i n n k i j j iR S1 1 1) 1 (, k i , 1 0 (4.1) The orderlevel of the inventory system is j n n k j jR S1 1 1 0) 1 (. (4.2) Since there are no units held in inventory from time k to time T the deterioration will not exist during this time span. By assumptions, the shortages are partially backlogged and the partial backlogging ratio is determined by the waiting time until the next replenishment. Hence, the difference equation describing the inventory level for period ) 1 ( T k k i i is ) (1 1 1 i i i iB R S S T k k i 1 From this recursive relation, we can obtain ) (1 1 1 k k kB R S since 0 kS. Similarly, we can get ) ( ) (2 2 1 1 2 k k k k kB R B R S ) ( ) ( ) (3 3 2 2 1 1 3 k k k k k k kB R B R B R S . ) ( ) ( ) (2 2 1 1 T T k k k k TB R B R B R S
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54 So the general expression of inventory levels from time k+1 to time T is i k j j j iB R S1) (, T k k i , 2 1 (4.3) The lotsize Q is the summation of the orderleve l and total backlogging amount, i.e. T k j j jB R S Q1 0) ( (4.4) From Equation (4.1), the average number of units in inventory per time unit during a cycle is 1 0 1 1 1 1 0 1) 1 ( 1 1 ) (k i j i n n k i j j k i iR T S T k I. (4.5) The average inventory holding cost per unit time is ) (1t I h (4.6) Since deterioration rate varies each period, according to Equation (4.1), the number of units that deteriorate during a cycle is k i j i n n k i j j i k i i iR S k D1 1 1 1) 1 ( ) ( (4.7) Given unit cost c, the average deterioration cost per unit time is T t cD ) (1 (4.8) From Equation (4.2), the average backlogging per unit time during a cycle is T k i i k j j j T k i iB R T S T k I11 1 2) ( 1 ) ( 1 ) (. (4.9)
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55 Hence, the average backlogging cost per unit time is ) (2k bI. (4.10) Since the partial backlogging ration is ) (iB the number of lost sales per cycle will be T k i i iB R k LS1) ( 1 ) (. (4.11) Therefore, the average penalty cost for lost sales per unit time is T k pLS ) (. (4.12) Adding up costs given by (4.6), (4.8), (4.10) and (4.12) leads to the following total average cost of the system per unit time during one cycle T k i i i T k i i k j j j k i j i n n k i j j k i j i n n k i j j iB R T p B R T b R T h R T c T k pLS k bI k hI T k cD k C1 11 1 0 1 1 1 1 1 2 1) ( 1 ) ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) ( (4.13) Since k is a nonnegative integer, the following two conditions should be satisfied by the optimal value of k denoted by *k, that minimizes ) ( k C (Sasieni et al., 1959): ) ( 0 ) 1 (* *k C k C forT k , 1 0 and (4.14) 0 ) (2 k C, for T k, 1 0. (4.15) Using Equation (4.13), we can obtain ) ( k C and ) (2k C as 1 1 1 1 1 1 1 1 1 1 1) ( 1 ) )( ( ) 1 ( ) 1 ( ) ( ) 1 ( ) (k i k k k i n n k i n n k i i kB p k T bB h c T R k C k C k C (4.16)
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56 and ) ( ) 1 ( ) (2k C k C k C ) )( ( ) 1 ( ) ( ) 1 )( ( 1 ) 1 ( ) 1 ( ) (1 2 1 1 1 2 2 1 1 2 1bT p bk B bT p k b B h c h ck k k i k k k i n k n i (4.17) Since all the parameters are positive, 1 0 and ) ( ) (1 2 k kB B it is obvious to see from Equation (4.17) that 0 ) (2 k C, for all T k , 1 0 Therefore, condition (4.15) is satisfied. If conditio n (4.14) is also satisfied, then ) (k Cwill be minimized at *k. Using Equations (4.13) and (4.16), the condition (4.14) simplifies to ) ( ) 1 (* *k M p k M (4.18) where 1 1 1 1 1) )( ( ) 1 ( ) ( ) (k i k k i n n ibT p bk B h c k M (4.19) The result obtained here shows that the optimal value of k is not affected by the demand rate and pattern. Substituting *k into Equations (4.2) and (4.4) yields the optimal orderlevel and lotsize. 4.3 Numerical Examples This section presents several numerica l examples to illustrate the developed method. The first example uses the same basic data from DaveÂ’s 1979 paper. However, additional data, regarding lost sales penalty cost p and partial backlogging ratio B ( i), must be added. The data are given below. (1) R = 200 units/month; (2) c = $80.00/unit; (3) h = $1.00/unit/month;
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57 (4) b = $9.00/unit/month; (5) p = $50/unit; (6) 1 ) ( iB ; (7) T = 12 months; (8) 05 0 Such a data setting represents a constant demand rate, a constant deterioration rate, and complete backlogging. In doing so, our results can be compared to DaveÂ’s. Note that complete backlogging means no lost sales. Therefore, the results for this case are independent of p values. The M ( k ) values are computed by using Equation (4.19) and presented in Table 4.1 below (for k = 1 to 12). Table 4.1 First Computational Results for M ( k ) Values k 1 2 3 4 5 6 7 8 9 10 11 12 M ( k ) 38 23.3 8.2 7.24 23.04 39.2 55.73 72.67 90.02 107 126 144.8 Since the value of p (=50) is between ) 6 ( M and ) 7 ( M, the optimal *k= 7 months, by condition (18). Substituting this op timal value into Equations (4.2) and (4.4), one can obtain the optimal orderlevel 1728* 0 S units, and the optimal lotsize 2728* Q units. These results conform to DaveÂ’s. As mentioned earlier, these results are independent of lost sales penalty cost p since 1 ) ( iB for all i In fact, various values of p were tried, and the same k (= 7 months) was obtained. The rest of this section extends to a cas e with variable deterioration rates and waitingtimedependent partial backlogging ratios. Since the optimal value of k is
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58 independent of demand rate and pattern, the constant demand rate is considered in the illustration of computation. But demand patterns do have an effect on optimal orderlevel, lotsize, lost sales, etc. Therefore, at the end of the section, two other different demand patterns are included to illustrate such effects. The new data are given below. 1 ii, 12 , 2 1 i (K. Skouri, 2009); ) ( 1 1 ) (i T Bi 12 , 2 1i (Chang and Dye, 1999). = 0.003 and = 2 are used in our computational experiments. These values lead to deterioration rates slightly smaller than 5%, on the average. In addition, = 0.5 is considered. This value results in backlogging ratios B ( i) gradually increase to 1 as i increases to T That is, the percentage of backlogging gradually decreases as waiting time increases. Such considerations are reasonable in the practical sense, as one gets less patient when waiting time becomes longer. The M ( k ) values are computed and presented in Table 4.2 below (for k = 1 to 12). Table 4.2 Second Computational Results for M ( k ) Values k 1 2 3 4 5 6 7 8 9 10 11 12 M ( k ) 7.4 4.6 1.2 2.87 7.84 13.77 20.85 29.38 39.85 53.18 71.46 100.68 By condition (4.18), p = 50 leads to 10*k months. This *k is bigger than that of the previous case, because the average deterioration rate is small (< 0.05), and there is a large penalty cost for lost sales. So it is economical to keep more inventories and to have less lost sales. The corresponding optimal orderlevel 0S, the optimal lotsize Q*, the total lost sales ) (*k LS and the total amount of deterioration ) (*k D during one cycle
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59 are presented in Table 4.3. This table also pr esents all the optimal values under two other demand patterns: linearly increasing and linearly decreasing demands, while keeping the total demands the same at 2400 units/cycle. As shown from the table, the optimal *k stays the same (10 months) for all demand patterns. But the rest are all different. The optimal lotsizes are almost the same for all three demand patterns, since the total demands are the same and the differences between deterioration and lost sales for all three cases are very close (See the last colu mn of Table 4.3). However, the optimal orderlevels differ dramatically. In particular, the decreasing demand case has less lost sales, while the increasing demand case has much more lost sales. The lost sales and deterioration quantities increase as the demand pattern changes from decreasing to uniform, or from uniform to increasing, as expected. Table 4.3 The Optimal Solutio ns for 3 Demand Patterns Demand Pattern k** 0S Q*) (*k D) (*k LS) ( ) (* *k LS k D 200 iR, for all i 10 2300.9 2634.3 301 66.7 234.3 i Ri13 400 for all i 10 2056.2 2651.1 363.9 112.8 251.1 ) 13 ( 13 400 i Ri for all i 10 2545.7 2617.5 238 20.5 217.5 4.4 Sensitivity Analysis This section first presents the sensitivity of *k, ) (*k C, *Q, 0S and LS with respect to parameters , and T through numerical experiments. The sensitivity of the quantities with respect to parameters c h b and p is analytically studied and presented at the end of this section.
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60 The parameter values of the second case in Section 4 are used as the base values. It is worth mentioning that the time unit consid ered in this paper is month. The sensitivity study is conducted as follows. Only one pa rameter value is altered at a time, while holding all the other parameters constant. The values of parameters we select are mostly the critical values at which the optimal *k changes. The computational results and analyses are presented next. Table 4.4 Different Optimal Values w.r.t. 0 0.25 0.5 0.75 1 5 20 k 10 10 10 10 11 11 11 S 2300.9 2300.9 2300.9 2300.9 2600.9 2600.9 2600.9 q 2700.9 2660.9 2634.3 2615.2 2800.9 2800.9 2800.9 LS 0 40 66.7 85.7 0 0 0 C ( k *) 3509.6 3616.3 3687.4 3738.2 4121.1 4121.1 4121.1 Figure 4.2 Sensitivity Analysis w.r.t. The range of is chosen from 0 to 20. While = 0 means a complete backlogging, > 5 results in very little ba cklogging. Consequently, as increases from 0 (i.e., the (a) (b) k C ( k ) Q* 0S LS
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61 backlogging ratio decreases from 1), the optimal *k jumps from 10 to 11 at =1, and then stays constant at 11, afterwards. This is somewhat expected because when 1, the partial backlogging ratio is small, which means a lot of lost sales. With lost sales penalty cost of $50/unit which is high compared to inventory and backlogging costs, there should be little or no lost sales. When *k = 11, there are no lost sales. From Figure 4.2(b), the optimal orderlevel *s has exactly the same trend as *k because affects *s directly through *k. The lotsize *Q decreases at the beginning as increases from 0 to 1, and then jumps to a constant after reaches 1. This is because for *k = 10, the optimal lotsize decreases as partial backlogging amount decreases, and *k=11 represents complete backlogging. The amount of lost sales LS on the other hand, increases rapidly at the beginning when *k=10, and becomes zero after reaches 1 (i.e. *k=11). The total average cost per month ) (*k C increases quickly as changes from 0 to 1, and then remains constant. Table 4.5 Different Optimal Values w.r.t. 0.0001 0.002 0.003 0.005 0.006 0.008 0.011 0.016 0.025 0.041 k 12 11 10 9 8 7 6 5 4 3 S 2414.6 2453.3 2300.9 2197.7 1946.2 1727.9 1505.8 1284.9 1063.6 824.1 q 2614.6 2653.3 2634.3 2631 2459.5 2307.9 2142.9 1972.1 1795.2 1595.7 LS 0 0 66.7 166.7 286.7 420 562.9 712.9 868.4 1028.4 C ( k *) 1555.6 3064.3 3687.4 4117.1 5045.8 5576 6144.3 6774.8 7448.4 8037.5
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62 Figure 4.3 Sensitivity Analysis w.r.t. The range of considered is from 0.0001 to 0.041. This range represents very little to significant deterioration. From Figure 4.3(a), the optimal *k decreases dramatically as increases. That is, for a rapidly deteriorating product, it is better to have higher backlogging amount and even more lost sales to keep costs low, as expected. From Fig. 4.4.2(b), when increases, the optimal orderlevel 0S and lotsize Q* decrease, while the lost sales LS increase. The total average monthly cost ) (*k C increases rapidly as increases. This indicates that a little reduct ion of deterioration ca n lead to significant improvement of cost reduction. This is especially true when the deterioration rate is small. Table 4.6 Different Optimal Values w.r.t. 1 1.8 2 2.2 2.3 2.5 2.7 k 12 11 10 9 8 7 6 S 2447.5 2444.9 2300.9 2151.3 1902.6 1702.5 1468.5 q 2647.5 2644.9 2634.3 2584.7 2415.9 2282.5 2105.6 LS 0 0 66.7 166.7 286.7 420 562.9 C ( k *) 1784.8 2997.8 3687.4 4513.2 4746.7 5409.1 5896 (a) (b) k C ( k ) Q * 0S LS
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63 Figure 4.4 Sensitivity Analysis w.r.t. Figure 4.4 depicts the sensitivity with respect to The range of considered is from 1 to 2.7. For = 1, 003 0 is a very small constant, for all i = 1, 2, Â…, 12. On the other extreme, when = 2.7, 7 1081 0 i (12 , 2 1 i) which indicates very significant deterioration. As one can see from Fig. 4, the changes in all quantities become significant only when > 1.9. This is because the deterioration rates are kind of small when 1.9. Table 4.7 Different Optimal Values w.r.t. Cycle Time T T 6 9 12 15 18 24 k 6 9 10 11 12 12 S 1271 2020.8 2300.9 2600.9 2924.2 2924.2 q 1471 2220.8 2634.3 3114.3 3561.3 3796.3 LS 0 0 66.7 286.7 562.9 1527.9 C ( k *) 994.1 2463 3687.4 5980.5 8742 14387.1 (a) (b) k C ( k ) Q * 0S LS
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64 Figure 4.5 Sensitivity Analysis w.r.t. Cycle Time T The range of T considered is set to be from 6 to 24 (months). From Fig. 4.4(a), the optimal *k increases as T increases. This should be expected, as only one replenishment is considered in one cycle. The optimal orderlevel 0S and lotsize Q* increase in the same fashion with respect to T Again, this is somewhat expected, as larger T leads to higher inventory and more backlogging and lost sales, since only one replenishment is allowed in a cycle of T time units. Similarly, the amount of lost sales LS increases as T increases and the total average monthly cost ) (*k C increases rapidly. All these point to one conclusion: it is better to keep cycle time small. The rest of this section will discuss the sensitivity with respect to parameters c h b and p In real life, the inventory carrying cost, the shortage backlogging cost and the penalty cost for lost sale may be assessed as some given proportions to the unit price of goods. To examine the system behavior for such cases, this paper considers fixing the ratio c : h : b : p while altering all four parameter va lues simultaneously. In particular, (a) (b) k C ( k ) Q* 0S LS
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65 consider c h1 c b2 and c p3 where 1 2 and 3 are constants. Substituting these into Equations (4.18) and (4.19) leads to the following. 1 1 2 3 2 1 1 1 1 3 1 2 3 2 2 1 1 1* * *) )( ( ) 1 ( ) ( ) )( ( ) 1 ( ) (k i k k i n n i k i k k i n n iT k B T k B As one can see clearly from the above, the optimal *k is now independent of all four parameters c h b and p provided that 1, 2 and 3 are constants. Consequently, the optimal 0S, *Q and LS are also independent of all c h b and p by Equations (4.2), (4.4) and (4.11), respectively. Similarly, the following equation can be derived from Equation (4.13). T k i i i T k i i k j j j k i j i n n k i j j k i j i n n k i j j iB R T B R T R T R T c k C1 3 11 2 1 0 1 1 1 1 1 1) ( 1 ) ( ) 1 ( ) 1 ( 1 ) ( The terms in the brackets of the above equation are independent of all c h b and p Therefore, the optimal cost ) (*k C is a linear function of c ( h b or p ), provided that 1, 2 and 3 are constants. This insight is very significant in the sense that for companies with a fixed ratio p b h c : : :, the total cost depends strictly linearly on the unit price of a product and the optimal lotsize is independent of the product price when the cycle time T is given. 4.5 Conclusions A discreteintime deteriorating inventory model with timevarying demands, variable deterioration rates and waitingtimedependent partial backlogging ratios is
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66 addressed in this paper. This model deals with a very general case and solutions can be computed easily. This model extends a model in the literature where there is a constant deterioration rate and complete backlogging. Through extensive numerical experiments, a sensitivity study is conducted to illustrat e the robustness of the proposed model. From the sensitivity analysis, it is found that the optimal total cost always increases as any parameter value increases. The deterioration rate (through parameters and ) has a significant impact on total cost. This explains why it is imperative to address the effect of deterioration for products that do deteriorate. The cycle time T also plays a rather important role as all the optimal solu tions are very sensitiv e to it. In addition, T should be set as small as possible. All these should be expected, though, as a single replenishment is restricted for each cycle. Another rather interesting insight is that the optimal total cost is a linear fu nction of unit product price c and the optimal k* is independent of c h, b and p provided that inventory holding cost c h1, backlog cost c b2 and lost sales penalty costc p3 where 1, 2 and 3 are constants. This finding is pretty significant in the sense that for companies with a fixed ratio p b h c: : :, the total cost depends strictly linearly on the unit price of a product and the optimal lotsize is independent of the product price when the cycle time T is fixed.
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67 Chapter 5 PeriodicReview Stochastic Inventory Control Problem A finite horizon inventory model for a single product is considered. The system is under periodic review and there is no fixed order cost associated with any placed order. The demand in successive periods is independent and identically distributed. A constant fraction of any positive leftover stock is deteriorated at the end of each period. Any unsatisfied demand is partially backlogge d and fulfilled immediately as a new order arrives. It was proved (Porteus, 2002) that a base stock policy is optimal under complete backlogging and nondeterioration. Then this chapter can be treated as a generalization of PorteusÂ’ model by considering deterioration a nd partial backlogging. It is shown that the base stock policy is still optimal as long as the terminal value function is convex and secondorder differentiable. The explicit base stock level is derived for a special case. 5.1 Problem Description The problem studied in this chapter is as follows. There is a single product. The product has a random life and will deteriorate over time. The system will be run for N periods. The demand Di in period i (= 1, 2, Â…, N ) is stochastic. At the beginning of each period, one needs to decide if it is necessary to place an order, and if so, how much to order. When an order is placed, there is no fixed cost. There is a perunit cost, though. Any onhand inventory at the end of a period can be used in the next period. Any
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68 unsatisfied demand can be partially backlogged until fulfilled, or lost. The order decisions are made such that the total expected longrun cost is minimized. It is assumed that (1) All demands are independent and identically distributed. (2) A constant fraction of the positive leftover stock will deteriorate. (3) The excess demand will be partially backlogged at the end of the period A penalty cost will be incurred for any backlogging and lost sale amount. The system is under periodic review, i.e. the inventory level is checked at the beginning of each period and a decision is made on how many to order. Porteus (2002) has shown that a base stock level policy is optimal under complete backlogging and nondeterioration. We will fo llow the same logic of PorteusÂ’ and some of his proof. Our objective is to identify under what conditions the base stock level policy still holds when deterioration and partia l backlogging are taken into account. 5.2 Notations (1) c Â– unit purchasing cost ($/unit) (2) h Â– unit holding cost, charged against positive ending inventory ($/unit) (3) b Â– unit backlogging cost, charged against shortages backlogged at the end of a period ($/unit) (4) p Â– penalty cost of a lost sale including lost profit ($/unit) (5) D Â– generic random variable representing demand, which is i.i.d over every period (6) Â– oneperiod demand distribution (7) Â– demand density distribution
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69 (8) x Â– inventory level before ordering (the state of the system) (9) y Â– inventory level after ordering (the decision variable) (10) Â– contant fraction of positive leftover stock at the end of the period that is deteriorated (11) Â– constant fraction of unsatisfied de mand during a period that is backlogged (12) N Â– the length of the planning horizon (13) ) 0 max( x x (14) ) 0 min( x x 5.3 Model To build the model, we first study the oneperiod problem. Expected oneperiod holding, backlogging, shortage and deteriorating cost function of level y of inventory after ordering is as follows ) ( ) )( ( ) )( ( ) )( 1 ( ) ( ) ( ) ( ) ( D y El y D p p b D y c h E y D p y D b D y c D y h E y L where ) )( ( ) ( ) ( x p p b x c h x l If the inventory level at the end of period N is x then the terminal cost ) ( x v is incurred. We assume the terminal cost function is convex and secondorder differentiable, one example can be cx x v ) (. This case happens if we can obtain reimbursement of the unit cost for each leftover unit and must incur the unit cost for each unit backlogged.
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70 The optimality equations for this model, for N t 1, consist of purchasing cost from inventory x to inventory y, oneperiod expected cost and minimum expected cost in future: 0 1) ( ) )( 1 ( ) ( ) ( ) ( min ) ( dD D D y D y D y f y L x y c x ft x y t where ) ( ) (1x v x fN Let 0 1) ( ) )( 1 ( ) ( ) ( ) ( dD D D y D y D y f y L cy y Gt t then cx y G x ft x y t ) ( min ) (. Thus, the optimal decision starting with inventory level x in period t can be determined by minimizing ) ( y Gt over x y y. In PorteusÂ’ book (2002), 0 1) ( ) ( ) ( ) ( dD D D y f y L cy y Gt t As one can see, there are two additional terms ) ( D y and ) )( 1 ( D y in the optimality equations if we consider the effects of deterioration and partial backlogging. These two terms complicates the proof.
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71 5.4 Base Stock Policy As illustrated by Figure 5.1, if tG is given as the above figure, then, if the inventory level is lower than tS, then we should order up to it. If the inventory level is higher than tS, we should order nothing. In other words, the decision rule is a base stock policy, which can be represented as follows (Porteus, 2002): otherwise if ) (t tS x x S x Figure 5.1 Base Stock Policy 5.5 Optimality of Base Stock Policies Lemma 1 If 1tf is convex and secondorder differentiable, then the following holds. (a) tG is convex. (b) A base stock policy is optimal in period t. Indeed, any minimize of tG is an optimal base stock level. ) ( x Gt tS x
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72 (c) tf is convex and secondorder differentiable. Proof (a) tG can be written as: ] ) )( 1 ( ) ( [ ) ( ) (1 D y D y D y Ef D y El cy y Gt t We know l is convex, then according to the following Theorem 1, ) ( D y El is convex. Theorem 1 Suppose that l is a convex function defined on R and the real valued function H is defined on R by ) ( ) ( D y l E R HD where D is a random variable with a density Then H is convex on R without assuming that l is differentiable everywhere. Proof Let 1y and 2y be arbitrary elements of y. Let 1 0 p and let p q 1. Then ) ( of convexity ) ( ) ( ) ( )] ( ) ( [ ) ( ) ( ) ( ) ( ) ( ) (2 1 0 2 1 0 2 1 0 2 0 1 2 1D qy py l E l dD D D qy py l dD D D y ql D y pl dD D D y l q dD D D y l p D y l qE D y l pED D D which implies ) ( D y l EDis convex. Proof of Thm 1 is completed. To show ] ) )( 1 ( ) ( [1 D y D y D y Eft is convex, since 1tf is convex and secondorder differentiable, we have ] ) )( 1 ( ) ( [1D y D y D y Eft y t y tdD D D y f dD D D y f ) ( )] ( [ ) ( )] )( 1 [(1 0 1
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73 )] ( [ ) ( )] ( [ )] )( 1 [( ) ( )] )( 1 [(1 1 1 0 1y y f dD D D y f y y f dD D D y ft y t t y t 1 1 0 1 1 1 1 0 1 of convexity 0 ) ( )] ( [ ) ( )] )( 1 [( )] ( [ ) ( )] ( [ )] )( 1 [( ) ( )] )( 1 [( t y t y t t y t t y tf dD D D y f dD D D y f y y f dD D D y f y y f dD D D y f So ] ) )( 1 ( ) ( [1 D y D y D y Eft is convex. Then tG is the sum of three convex functions and therefore convex itself. Proof of (a) is completed. (b) (Porteus, 2002) We know cx y G x ft x y t ) ( min ) (. Let tS denote a minimizer of ) ( y Gt over all real y If tS x then the minimizing x y is at tS y whereas, if tS x then the minimizing y is at x y That is, a base stock policy with base stock level tS is optimal for period t Proof of (b) is completed. (c) According to the following Theorem 2, tf is convex. Theorem 2 (Heyman and Sobel, 1984) If X is a convex set, Y(x) is nonempty set for every X x the set ) ( ) ( x Y y X x y x C is a convex set, ) ( y x g is a convex function on C, ) ( inf ) () (y x g x fx Y y and ) ( x f for every X x then f is a convex function on X.
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74 Proof Let x and x be arbitrary elements of X. Let 1 0 and let 1. Select arbitrary 0 By the definition of f, there must exist ) ( x Y y such that ) ( ) ( x f y x g and ) ( ) ( x f y x g. Then, C] ) y y x x [( ) x x f( C] on ) g( of [convexity ) ( ] and of s [propertie ) ( ) ( ) ( ) ( y y x x g y y y x g y x g x f x f Assume the inequality does not hold for 0 that is ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) ( ) (* *x x f z x x f z z x x f x f x f x x f x f x f Since can be arbitrarily small, a contradictio n is reached. So the inequality must hold for 0 which means f is convex. Proof of Thm 2 is completed. Since all three terms of tf are secondorder differentiable, then tf is secondorder differentiable itself. Proof of (c) is completed. Lemma 2 (Porteus, 2002) A base stock policy is optimal in each period of a finitehorizon problem. Proof By assumption, the terminal va lue function is convex and secondorder differentiable. Thus, by Lemma 1, NG is convex and a base stock policy is optimal for period N. By Lemma 1 (c), Nf is convex as well. Thus, the argument iterates backward through the periods in the sequence 1 1 N N t
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75 5.6 Explicit Optimal Base Stock Level Although we have already illustrated that a base stock policy is optimal for this periodicreview stochastic inventory control problem, it is very difficult to derive the explicit form of base stock level. However, the explicit solution can be obtained for a very special case. As we mentioned before, if we assume the terminal value function is cx x v ) (, then the explicit optimal base stock level can be derived as following. Let us examine the oneperiod problem at the end of the time horizon. The expected ordering, holding, backlogging, shortage and deterioration cost, less any expected salvage value, in that period, starting with zero inventory and ordering y units can be written as y y NdD D y D c p p b dD D D y c c c h cy D D y D y D y c y L cy y G0 0) ( ) )( ( ) ( ) )( ( ) ( ] ) )( 1 ( ) ( )[ ( ) ( ) ( Let S denote a solution to 0 ) ( S GN, then S is a minimize of NG and can be found as ) 1 ( ) ( ) 1 ( ) 1 ( ) ( p c b h p c b S (5.1) Examine Nf by plugging in the optimal decision for each state (Porteus, 2002) otherwise cx x G S x if cx S G x fN N N) ( ) ( ) ( Therefore, otherwise c x G S x if c x fN N) ( ) ( (5.2) That is, ) ( x fN has a slope of c for S x
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76 Lemma 3 If 1 tf is convex and c x ft ) (1 for S x where S is as defined in (5.1), then the following hold. (a) S minimizes ) ( y Gt over all real y. (b) The optimal base stock level in period t is also S. (c) tf is convex and c x f ) ( for S x Proof (a) As in Lemma 1, tG is convex. To see that S is a minimize of tG, 0 )] ( 1 [ ) ( ) 1 ( )] ( 1 )[ ( ) ( ) ( ) ( )] ( [ ) ( )] )( 1 [( ) ( ) (0 1 1 S c S c S p p b S c h c dD D D S f dD D D S f S L c S GS S t t t Hence, S must be a minimize of tG, and, therefore, by Lemma 1, part (b) must also hold. (c) Lemma 1 ensures that tf is convex. By calculating the consequences of using the optimal base stock level in period t as was done in (5.2) for period N we get otherwise c x G S x if c x ft t) ( ) ( Proof of Lemma 3 is completed. Theorem 3 (Porteus, 2002) If the terminal value function v has a slope of c, then a base stock policy with base stock level S defined by (2.1) is optimal for every t Proof By assumption, the terminal value function is convex and has a slope of c Thus, by Lemma 3, the optimal base stock level in period N is S. By Lemma 3 (c), Nf is convex and has a slope of c (if x
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77 5.7 Summary A finite horizon inventory model for a single product is considered. The system is under periodic review and there is no fixed order cost associated with any placed order. The demand in successive periods is independent and identically distributed. A constant fraction of any positive leftover stock is deteriorated at the end of each period. Any unsatisfied demand is partially backlogge d and fulfilled immediately as a new order arrives. It was proved (Porteus, 2002) that a base stock policy is optimal under complete backlogging and nondeterioration. Then this chapter can be treated as a generalization of PorteusÂ’ model by considering deterioration and partial backlogging. It was shown that the base stock policy is still optimal as long as the terminal value function is convex and secondorder differentiable. The explicit base stock level was derived for a special case.
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78 Chapter 6 Stochastic LotSizing Problem The problem concerned here can be described as follows. It is a single product, single location problem. The system will be run for N periods. The product has a random life and will deteriorate over time. The custom er demand in each period is stochastic. At the beginning of each period, one needs to decide if it is necessary to place an order, and if so, how much to order. A fixed order cost is incurred whenever an order is placed. There is a perunit cost associated with each order too. Any onhand inventory at the end of a period can be used in the next period. Any unsatisfied demand can be partially backlogged until fulfilled, or lost. The orde r decisions are made such that the total expected longrun cost is minimized. It is assumed that (1) All demands are independent and identically distributed. (2) A constant fraction of the positive leftover stock will deteriorate. (3) The excess demand will be partially backlogged at the end of the period A penalty cost will be incurred for any backlogging and lost sale amount. The system is under periodic review, i.e. the inventory level is checked at the beginning of each period and a decision is made on how many to order. As one can see, the only difference between Chapter 5 and Chapter 6 is th at the fixed order cost will be considered explicitly in this chapter.
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79 Porteus (2002) has shown that a ( s S ) ordering policy (see Chapter 2) is optimal under complete backlogging and nondeterior ation. We will follow the same logic of PorteusÂ’ and some of his proof. Our objective is to identify under what conditions the ( s S ) policy still holds when deterioration and partial backlogging are taken into account. 6.1 Notations (1) c Â– unit purchasing cost ($/unit) (2) h Â– unit holding cost, charged against positive ending inventory ($/unit) (3) b Â– unit backlogging cost, charged against shortages backlogged at the end of a period ($/unit) (4) p Â– penalty cost of a lost sale including lost profit ($/unit) (5) K Â– fixed ordering cost (6) Â–one period discount factor (7) D Â– generic random variable representing demand, which is i.i.d over each period (8) Â– oneperiod demand distribution (9) Â– demand density distribution (10) x Â– inventory level before ordering (the state of the system) (11) y Â– inventory level after ordering (the decision variable) (12) Â– constant fraction of positive leftover stock at the end of the period that is deteriorated
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80 (13) Â– constant fraction of unsatisfied demand during a period that is backlogged (14) N Â– the length of the planning horizon (15) ) 0 max( x x (16) ) 0 min( x x 6.2 Model 6.2.1 Formulation Let us first examine the oneperiod problem. Expected oneperiod holding, backlogging, shortage and deteriorating cost function of level y of inventory after ordering is ) ( ) )( ( ) )( ( ) )( 1 ( ) ( ) ( ) ( ) ( D y El y D p p b D y c h E y D p y D b D y c D y h E y L where ) )( ( ) ( ) ( x p p b x c h x l Let 0 1) ( ) )( 1 ( ) ( ) ( ) ( dD D D y D y D y f y L cy y Gt t then the optimality equations (OE) will be ) ( min ), ( min ) ( y G K x G cx x ft x y t t That is, there is no fixed order cost associated with zero order and a fixed order cost K will be incurred if we order something. We have to make a choice on placing an order or not. In PorteusÂ’ book, 0 1) ( ) ( ) ( ) ( dD D D y f y L cy y Gt t As one can see, if we consider the effects of deterioration and partial backlogging, there will be two additional
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81 terms ) ( D y and ) )( 1 ( D y in the optimality equations. These two terms make the proof more challenging. Letting ) ( min ), ( min ) (*y G K x G x Gt x y t t then the OE can be rewritten as ) ( ) (*x G cx x ft t 6.2.2 ( s S ) Policy Since there is a fixed ordering cost incurred for any nonzero order, then the ordering cost function is concave, which is different from PeriodicReview Stochastic LotSizing model. If there is no deterioration and backlogging is complete, a ( s S ) policy will be optimal in each period. This policy means whenever the inventory is below some amount s we will place an order to bring the inventory level up to S (where S s ). The order quantity is greater than or equal to S s If the inventory level is above s we will not place an order. This ensures that the fixed ordering cost only occurs for a certain amount (i.e. S s ). The order will not be placed if the amount is too small. 6.2.3 K Convex Functions A function f : RR (a real valued function of a single real variable) is K convex if K 0, and for each y x 1 0 and 1 (Porteus, 2002): ) ( ) ( ) (y f K x f y x f The following Lemma (Porteus, 2002) provides some important properties of K convex function. Lemma 6.1 (a) If f is K convex and is a positive scalar, then f is k convex for all K k (b) The sum of a K convex function and a k convex function is ( K + k )convex.
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82 (c) If f is K convex, y x and ) ( ) ( y f K x f then ) ( ) ( y f K z f for all ] [y x z. 6.2.4 Optimality of ( s S ) Policy In this section, the proof is presen ted about under what conditions the ( s S ) policy is still optimal in each period. Lemma 6.2 If 1tf is a continuous K convex function and 1, then the following hold. (a) tG is a continuous K convex function. (b) A ( s S ) policy is optimal in period t (c) tG is a continuous K convex function. (d) tf is a continuous K convex function. Proof (a) From section 6.2.1, we know that 0 1) ( ) )( 1 ( ) ( ) ( ) (dD D D y D y D y f y L cy y Gt t .Define 0 1) ( ) )( 1 ( ) ( ) (dD D D y D y D y f y gt then ) ( ) ( ) ( y g y L cy y Gt If we can show that for each 2 1y y 1 0 and 1, the following holds: ) ( ) ( ) (2 1 2 1y g K y g y y g
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83 Then according to Lemma 6.1 (a), ) ( y g is a K convex function. Since c y and ) ( y L are convex, then according to Lemma 6.1 (b), it can be shown that ) (y Gt is k convex. The following is to show that ) ( ) ( ) (2 1 2 1y g K y g y y g ) (2 1y y g 0 2 1 2 1 2 1 1) ( ) )( 1 ( ) (dD D D y y D y y D y y ft 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 1) ( ) )( 1 ( ) (2 1 2 1 2 1 1 y y tdD D D y y D y y D y y f 2) ( ) (2 1 1 y tdD D D y y f 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 2 1 2 1 1) ( ) ( ) ( ) )( 1 (2 1 1 2 1 1 y y y t y y y tdD D D y y f dD D D y y f 2) ( ) (2 1 1 y tdD D D y y f 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 1 1) ( ) )( 1 ( ) )( 1 (2 1 1 1 y y y t tdD D k D y f D y f
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84 2 2 1) ( ) ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2) ( ) (2 1 1 y tdD D D y y f 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 1 1) ( ) )( 1 ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2 2 1) ( ) )( 1 ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2) ( ) (2 1 1 y tdD D D y y f ) ( ) (2 1y g K y g Proof of (a) is completed. (b) Please refer to (Porteus, 2002). (c) Please refer to (Porteus, 2002). (d) Please refer to (Porteus, 2002). The proof is completed. Lemma 6.3 If 1tf is a continuous decreasing K convex function, 1 and p p b c h p c p b xi ) ( (N i, 2 1 ) where ix is the inventory level before ordering for each period, then the following hold.
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85 (a) tG is a continuous K convex function. (b) A ( s S ) policy is optimal in period t (c) tG is a continuous K convex function. (d) tf is a continuous decreasing K convex function. Proof (a) As of Lemma 6.2, the objective is to show that for each 2 1y y 1 0 and 1, ) ( ) ( ) (2 1 2 1y g K y g y y g ) (2 1y y g 0 2 1 2 1 2 1 1) ( ) )( 1 ( ) (dD D D y y D y y D y y ft 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 1) ( ) )( 1 ( ) (2 1 2 1 2 1 1 y y tdD D D y y D y y D y y f 2) ( ) (2 1 1 y tdD D D y y f 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 2 1 2 1 1) ( ) ( ) ( ) )( 1 (2 1 1 2 1 1 y y y t y y y tdD D D y y f dD D D y y f 2) ( ) (2 1 1 y tdD D D y y f 10 2 1 1) ( ) )( 1 (y tdD D D y y f
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86 2 1 1) ( ) )( 1 ( ) )( 1 (2 1 1 1 y y y t tdD D k D y f D y f 2 2 1) ( ) ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2) ( ) (2 1 1 y tdD D D y y f ) ( ) (2 1y g K y g 2 1 1 1) ( ) ( ) ( ) )( 1 (1 1 0 1 1 y y y t y tdD D D y f dD D D y f 2 2 2 1) ( ) ( ) ( ) (1 1 1 1 y t y y y tdD D D y f dD D D y f 2 1 1 1) ( ) )( 1 ( ) ( ) )( 1 (2 1 0 2 1 y y y t y tdD D D y f dD D D y f 2 2 2 1) ( ) ( ) ( ) )( 1 (2 1 2 1 y t y y y tdD D D y f dD D D y f 2 2 2 1 2 1 1 1) ( ) ( ) ( ) (0 y y y y y y y ydD D k dD D k dD D k dD D k 10 2 1 1 1) ( ) )( 1 ( ) )( 1 (y t tdD D k D y f D y f 2 1 1) ( ) )( 1 ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2 2 1) ( ) )( 1 ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2) ( ) ( ) (2 1 1 1 y t tdD D k D y f D y f 10 2 1 1) ( ) )( 1 (y tdD D D y y f 2 1 1) ( ) )( 1 ( ) (2 1 1 1 y y y t tdD D k D y f D y f
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87 2 2 1) ( ) )( 1 ( ) (2 1 1 1 y y y t tdD D k D y f D y f 2) ( ) (2 1 1 y tdD D D y y f Since 1 and 1tf is a decreasing function according to the assumption, the following must hold: 2 1 1 2 1 1) ( ) )( 1 ( ) ( ) (1 1 1 1 y y y t y y y tdD D D y f dD D D y f and 2 2 1 2 2 1) ( ) ( ) ( ) )( 1 (2 1 2 1 y y y t y y y tdD D D y f dD D D y f Then ) ( ) ( ) (2 1 2 1y g K y g y y g i.e. ) ( y g is a Kconvex function. The proof of (a) is completed. (b) Please refer to (Porteus, 2002). (c) Please refer to (Porteus, 2002). (d) According to section 6.2.1, we have: ) ( min ), ( min ) ( y G K x G cx x ft x y t t where 0 1) ( ) )( 1 ( ) ( ) ( ) (dD D D y D y D y f x L cx x Gt t and x xdD D x D p p b dD D D x c h x D p x D b D x c D x h E x L0) ( ) )( ( ) ( ) )( ( ) )( 1 ( ) ( ) ( ) ( ) ( To prove ) (x Gt is a decreasing function, we only need to prove ) ( x L cx is decreasing because ) (1x ft is given to be a decreasing function.
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88 The first derivative of ) (x L cx is given by: ) ( ) ( ) ( ) ( ) ( ) ( x p p x p p x b b x c x h c x L cx ) ( ) ( x p p b c h p p b c since p p b c h p c p b y ) (1, then 0 ) ( x L cx. So ) ( x L cx is a decreasing function, and therefore, tG is a decreasing function. Since ) ( ) ( min y G K y G Kt t x y which is a constant, we obtain that ) ( min ), ( min ) ( y G K x G cx x ft x y t t is a decreasing function. From part (3) we know that tG is kconvex, then ) ( x ft is the summation of a convex function and a k convex function, therefore, k convex itself. The proof is completed. 6.3 Summary A finite horizon inventory model for a single product is considered in this chapter. The system is under periodic review and there is a fixed order cost associated with any nonzero order. The demand in successive periods is independent and identically distributed. A constant fraction of any positive leftover stock is deteriorated at the end of each period. Any unsatisfied demand is partially backlogged and fulfilled immediately as a new order arrives. Porteus (2002) proved that a ( s S ) policy is optimal under complete backlogging and nondeterioration. This chapter generalized PorteusÂ’ model by considering deterioration and partial backlogging. It was shown that the ( s S ) policy is still optimal for the following two cases:
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89 (1) Tv is a continuous Kconvex function and 1. (2) Tv is a continuous decreasing K convex function, 1 and p p b c h p c p b xi ) ( (N i, 2 1 ) where ix is the inventory level before ordering for each period. One drawback is that the two conditions ar e both rigid to some level. The explicit form of ( s S ) is very difficult to obtain. We will di scuss how to overcome this in Chapter 8.
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90 Chapter 7 The Stochastic LotSizing Prob lem with Deterioration and ServiceLevel Constraints The problem considered in this chapter can be described as follows. It is a single product, single location problem. The product has a random life and will deteriorate over time. The system will be run for N periods. The customer demand in each period is stochastic. At the beginning of each period, one needs to decide if it is necessary to place an order, and if so, how much to order. A fixed order cost is incurred whenever an order is placed. There is a perunit cost associated with each order too. Any onhand inventory at the end of a period can be used for the next period. The net inventory at the end of each period not being negative is set to be a probability of at least (service level). It is assumed that the value is quite high, so that this service level incorporates the perception of the cost of backorders already and shortage cost can be neglected. The system is under periodic review, i.e. the inventory level is checked at the beginning of each period and a decision is made on how many to order. Bookbinder and Tan (1988) first studied this problem without considering the effect of deterioration. They developed a strategy called Â“staticdynamic uncertaintyÂ” strategy, in which they determine the number of replenishments at the beginning of the planning horizon, and then obtain the order quantity based on the realized demand. Tarim and Kingsman (2004) improved this strategy by presenting a mixedinteger programming
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91 formulation that simultaneously determines both number of replenishments and order quantity in a single step and gives the optimal solution rather than the heuristic results given by Bookbinder and Tan. In this chapter, we follow Tarim and King smanÂ’s method but take into account of the effect of deterioration. The computational results were obtained and 100 random cases were simulated for th e optimal ordering policy. 7.1 Assumptions and Notations (1) The demand td in period t is a random variable with known probability density) (t td g. The demands are independent and the distribution of demand may vary from period to period. (2) Lead time is zero, i.e. the replenishment order tX arrives immediately at the beginning of period t before the demand in that period occurs. (3) The service level is i.e. the probability that at the end of each period the net inventory will not be negative is set to be at least It is assumed that the value is quite high, so that this service level incorporates the perception of the cost of backorders already and shortage cost can be neglected. (4) A constant fraction of positive leftover stock is deteriorated over one period. (5) The length of the planning horizon is N (6) The inventory level at the end of period t is denoted as tI. The initial onhand inventory is 0I. (7) The unit purchasing cost c unit holding cost h and fixed ordering cost a are known.
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92 7.2 Model The problem can be formulated as minimizing the total expected cost ] [TC E over the planning horizon, as following: Min] [ TC E 12) ( ) ( ) ( ) ( ) ( ) ( ) ) ( (2 1 1 2 2 1 1 ddd N N t N N t t tNd d d d d d d g d g d g cX I c h a (7.1) subject to 0 0 1 otherwise X ift t N t , 1 (7.2) ) )( 1 (1 t t t td X I I N t , 1 (7.3) ) 0 Pr(tI, N t , 1 (7.4) 0 t tI X, N t , 1 (7.5) The objective function (7.1) consists of fixed order cost, inventory holding cost, deterioration cost and purchasing cost. Constraint (7.2) means that t takes the value of 1 if an order is placed in period t and 0 otherwise. Constraint (7.3) shows the balance of the inventory flow, i.e. inventory level is determined by order quantity, demand and deterioration. Constraint (7.4) guarantees that the service level is at least Constraint (7.5) makes sure all the decision variables are positively defined.
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93 From the recursive relationship of inventory levels described by Equation (7.3), we can obtain the general expression of inventory level at period t as k t k k t k t k k t t td X I I 1 1 1 1 0) 1 ( ) 1 ( ) 1 ( (7.6) Figure 7.1 Illustration of Notation Index We assume that there are m reviews over the N period planning horizon with orders arriving at mT T T , ,2 1, where 1 j jT T, N Tm. Set 11 T and 11 N Tm. Define iTR as the orderupto level to which inventory level should be reached after replenishment at the beginning of the i th review period iT. Then according to Fig.7.1 and equation (7.6), we have the following relationship: t T k k k t T T t ti i id R I1 1) 1 ( ) 1 ( 1 i iT t T, , 1 m i (7.7) If there is no replenishment in period t then tR is explained as the opening stock level in period t The relationship between tR and tI is as following: ) )( 1 (t t td R I (7.8)
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94 From Figure 7.1, it is clear that if there is no replenishment in period t tR will be equal to 1tI. This relationship can be defined by the following two inequalities: t t tM I R 1, N t , 1 (7.9) 1 t tI R, N t , 1 (7.10) Applying Equation (7.7) to the servicelevel constraint ) 0 Pr(tI, we have ) 0 ) 1 ( ) 1 Pr((1 1 t T k k k t T T ti i id R i.e. ) ) 1 ( ) 1 ( Pr(1 1i i iT T t t T k k k tR d i.e. ) ( ) 1 (1 ) 1 ( ) 1 ( ) 1 ( 11 1 t i T i T t i T i T t i id d d T T tG R, where ) () 1 ( ) 1 ( ) 1 (1 1y Gt i T i T t i T i T td d d is the cumulative distribution function of t T T t T T td d d t Di i i i) 1 ( ) 1 ( ) 1 ( ) (1 1 Since t T k k k t T T t ti i id R I1 1) 1 ( ) 1 ( then t T k k k t d d d ti t i T i T t i T i T td G I1 1 ) 1 ( ) 1 ( ) 1 () 1 ( ) (1 1 1 i iT t T, m i , 1. (7.11) In constraint (7.11), ) (1 ) 1 ( ) 1 ( ) 1 (1 1 t i T i T t i T i T td d dGcan only be determined if the replenishment timing Ti is known. But Ti is also a decision variable in our model, so there
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95 is circularity here. Tarim and Kingsman addressed this issue by defining a new binary variable to formulate this problem as a mixedinteger programming model. Since the planning horizon is finite and consists of N periods, the following binary variable Ptj is defined to calculate all relevant cases of ) (1 ) 1 ( ) 1 ( ) 1 (1 1 t i T i T t i T i T td d dG. otherwise 0 1 period in was period prior to order recent most the if 1 j t t Ptj Then tj t j d d d d d dP G Gt j t j j t j t i T i T t i T i T t 1 1 ) 1 ( ) 1 ( ) 1 ( 1 ) 1 ( ) 1 ( ) 1 () ( ) (2 1 1 1 1 N t , 1. So from Equation (7.11), we have tj t j t j t k k k t d d d tP d G It j t j j t j 11 1 1 ) 1 ( ) 1 ( ) 1 () ) 1 ( ) ( (2 1 1 N t , 1 (7.12) According to Bookbinder and TanÂ’s (1988) argument, ) (1 ) 1 ( ) 1 ( ) 1 (2 1 1 t j t j j t jd d dG can be calculated as t j t k k k t t j t k k k td E C z d E1 2 / 1 1 1 2 95 0 1]) ) 1 [( ( ] ) 1 [( The following constraint (7.13) guarantees that there can at most be only one most recent order received prior to period t and constraint (7.14) depicts the relationship between Ptj and (Tarim and Kingsman, 2004). t j tjP11 N t , 1. (7.13)
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96 t j t k k j t tjP2 1 N t , 1, t j , 1 (7.14) In order to determine the timing of the replenishments before any demands become known under the Â“staticdynamic uncertaintyÂ” strategy, the expected values of the stochastic variables tI, tR and td must be applied. Total cost in (7.1) can also be expressed as N t t t t t N t t t tI R c I c h a cX I c h a TC1 1 1)) ( ) ( ( ) ) ( ( Taking expected values of both sides yields the following N t t t t tI E R E c I E c h a TC E1 1]) [ ] [ ( ] [ ) ( ] [ Putting together constraints (7.8), (7.9), (7 .10), (7.12), (7.13), and (7.14), the final model, denoted by M1, is obtained as follows. Minimizing (7.1) subject to (7.2) (7.5) is equal to solving the model M1. In the rest of this chapter, we will use this new model M1 to do computational tests. Model M1 is expressed as follows. Min N t t t t tI E R E c I E c h a TC E1 1]) [ ] [ ( ] [ ) ( ] [ Subject to ]) [ ] [ )( 1 ( ] [t t td E R E I E N t , 1 ] [ ] [1 t tI E R E, N t , 1.
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97 t t tM I E R E ] [ ] [1, N t , 1. tj t j t j t k k k t d d d tP d E G I Et j t j j t j 11 1 1 ) 1 ( ) 1 ( ) 1 () ] [ ) 1 ( ) ( ( ] [2 1 1 N t , 1. t j tjP11, N t , 1. t j t k k j t tjP2 1 N t , 1, t j , 1 0 ] [ ], [t tR E I E, 1 0 tj tP N t , 1 t j , 1 7.3 A Numerical Example To examine if our new model works, the same numerical example from Tarim and KingsmanÂ’s paper is applied except that a deterioration rate is added based on our extension. The purpose of this numerical example is to compare our model with the model without deterioration and show how the effect of deterioration plays a role in decisionmaking. Since the objective of the final model is the total expected cost, a simulation of random cases will be conducte d to show how the optimal ordering policy works on each individual case. Then the comparison of each individual cost with the expected cost will reveal how the actual cost deviates from the optimal cost in practice. Also, the actual fill rate can be calculated to compare with the designated service level.
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98 (1) order a/ 2500 $ (2) period unit h / / 1 $ (3) 05 0 (4) 95 0 (5) 645 195 0z (6) 333 0 / t tC Table 7.1 Forecasted Values of Demands Period ( k ) 1 2 3 4 5 6 7 8 9 10 E [ dk] 800 850 700 200 800 700 650 600 500 200 Table 7.1 gives the forecasted values of expected demands. It is assumed that the initial inventory level is zero and the demand in each period is assumed to be normally distributed about the forecasted value under a constant coefficient of variation C=0.333. Since the service level =0.95, then the z value for standardized normal distribution is 1.645.
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99 Table 7.2 The Calculated Values of the Inverse Cumulative Distribution Function ) (1 ) ( G a. t j 1 2 3 4 5 1 1239 0 0 0 0 2 1316 2234.595 0 0 0 3 1084 2092.902 2935.981 0 0 4 310 1245.395 2198.945 2998.065 0 5 1239 1440.418 2189.761 3044.783 3782.77 6 1084 2025.997 2215.069 2902.639 3695.909 7 1006 1824.389 2681.901 2860.181 3501.309 8 929 1689.138 2434.231 3230.4 3399.004 9 774 1485.342 2181.765 2876.348 3623.417 10 310 957.3219 1626.003 2284.564 2942.64 b. t j 6 7 8 9 10 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 6 4386.898 0 0 0 0 7 4243.853 4893.81 0 0 0 8 4000.861 4699.103 5312.113 0 0 9 3783.181 4350.952 5010.124 5589.814 0 10 3651.025 3802.74 4341.529 4967.113 5517.409 Table 7.2 shows the calculated ) (1 ) 1 ( ) 1 ( ) 1 (2 1 1 t j t j j t jd d dG values. For example, the element for t =6 and j =2 is calculated as 2025.997. That means the opening
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100 inventory level in period t j +1=5 is 2025.997 and this inventory can satisfy the demands from period 5 to period 6 with a probability of at least =0.95. Table 7.3 Results for 0 c Period ( t ) 1 2 3 4 5 6 7 8 9 10 Order decision (Delta) 1 0 1 0 1 0 0 1 0 0 Closing inv. level ( E [ It]) 1507 624 600 380 2098 1328 644 1164 631 409 Opening inv. level ( E [ Rt]) 2386 1507 1332 600 3009 2098 1328 1825 1164 631 Orderuptolevel 2386 1332 3009 1825 Total expected cost 19390 Table 7.4 TarimÂ’s Results for c =0. (Tarim and Kingsman, 2004) Period ( t ) 1 2 3 4 5 6 7 8 9 10 Order decision (Delta) 1 0 1 0 1 0 0 1 0 0 Closing inv. Level ( E [ It]) 1490 640 599 399 2033 1333 683 1142 642 442 Opening inv. Level (E[ Rt]) 2290 1490 1299 599 2833 2033 1333 1742 1142 642 Orderuptolevel 2290 1299 2833 1742 Total expected cost 19404 (*) 2nd row is added. Table 7.3 and Table 7.4 show both our results and TarimÂ’s results when the unit purchasing cost is not considered. From the tables, it is clear that the optimal replenishment timings for both models are the same, which are in periods 1, 3, 5 and 8. The orderupto level is higher in our model. This is because the effect of deterioration plays an important role and we have to raise the inventory level much higher since it is determined by both demand and deterioration. The total expected cost is a little bit lower in our model because the unit purchasing cost is not considered and hence the deterioration cost is not considered either.
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101 Table 7.5 Results for 4 c Period ( t ) 1 2 3 4 5 6 7 8 9 10 Order decision (Delta) 1 0 1 0 1 0 1 0 1 0 Closing inv. Level ( E [ It]) 1507 624 600 380 1295 566 1096 471 497 282 Opening inv. Level ( E [ Rt]) 2386 1507 1332 600 2164 1295 1804 1096 1023 497 Orderuptolevel 2386 1332 2164 1804 1023 Total expected cost 47957 Table 7.6 TarimÂ’s Results for 4 c Period ( t ) 1 2 3 4 5 6 7 8 9 10 Order decision (Delta) 1 0 1 0 1 0 1 0 1 0 Closing inv. level ( E [ It]) 1490 640 599 399 1283 583 1085 485 495 295 Opening inv. level ( E [ Rt]) 2290 1490 1299 599 2083 1283 1735 1085 995 495 Orderuptolevel 2290 1299 2083 1735 995 Total expected cost 45036 Table 7.5 and Table 7.6 show both our results and TarimÂ’s results when the unit purchasing cost is at c =4. From both tables, we can see that the replenishments arrive in the same periods, which are periods 1, 3, 5, 7 and 9. The opening inventory level of our model is higher than that of TarimÂ’s model, which means the orderupto level is also higher in our model. This is because the i nventory level in our model is determined by both demand and deterioration, so it needs to be raised much higher since deterioration plays a very important role. The total expected cost is much higher in our model because the effect of deterioration is taken into acc ount explicitly and the unit purchasing cost is nonzero.
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102 Table 7.7 Results for 6 c Period ( t ) 1 2 3 4 5 6 7 8 9 10 Order decision (Delta) 1 0 1 0 1 0 1 0 1 0 Closing inv. Level ( E [ It]) 1508 624.6 600.4 380.4 1296 566 1097 471.6 497.2 282.3 Opening inv. Level ( E [ Rt]) 2387 1508 1332 600.4 2164 1296 1804 1097 1023 497.2 Orderuptolevel 2387 1332 2164 1804 1023 Total expected cost 62025.2 Table 7.8 TarimÂ’s Results for 6 c Period ( t ) 1 2 3 4 5 6 7 8 9 10 Order decision (Delta) 1 0 1 0 1 0 1 0 1 0 Closing inv. level ( E [It]) 1490 640 599 399 1283 583 1085 485 495 295 Opening inv. level ( E [ Rt]) 2290 1490 1299 599 2083 1283 1735 1085 995 495 Orderuptolevel 2290 1299 2083 1735 995 Total expected cost 57624 Table 7.7 and Table 7.8 show both our results and TarimÂ’s results when the unit purchasing cost is set at c =6. Similarly, we can see from both tables that the replenishments still arrive in the same periods which are periods 1, 3, 5, 7 and 9. The orderupto level is higher in our model than th at of TarimÂ’s model. The reason is similar, i.e., the inventory level has to be raised higher since it is depleted by both demand and deterioration. The total expected cost is much higher in our model than that of TarimÂ’s model and the difference of the total costs is higher than the case when c =4. This is because the effect of deterioration is taken into account explicitly and the unit purchasing cost in this case is higher than the previous one.
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103 Table 7.9 Service Level and Actual Fill Rate Service Level 95% 90% 85% Average Fill Rate 99.70% 99.10% 98.40% A simulation of 100 random cases is conducted for the case of c =4. As shown in Table 7.5, the optimal ordering policy for the case of c =4 is that replenishments take place in periods 1, 3, 5, 7 and 9, and the inventory levels will be raised to 2386.8, 1332, 2164, 1804.2, and 1023.4 respectively. This ordering policy will guarantee the minimum expected longrun cost. In this simulation, we test 100 random cas es each for service levels of 95%, 90% and 85%. Through the random experiment, it is found out that the actual fill rate is much higher than the service level, which one can see from Table 7.9. Actually, there is only one case for 95% service level that fill rate is lower than 95%. Table 7.10 Worst Scenario Cost Service Level 95% 90% 85% Expected Cost 47957.5 46222.9 45047.4 Worst Scenario Cost 51038.4 49306.4 48129.6 Percentage Increase 6.42% 6.67% 6.84% Table 7.10 shows that the total expected cost decreases as service level decreases. From our optimal ordering policy, one can see that it guarantees a minimum expected cost, but not necessarily minimum cost for each case. From the simulation, the worst scenario, i.e. case with maximum cost, is illustrated in Table 7.10 for each service level. As shown in Table 10, the worst case cost is about 6% higher than the total expected cost for each service level, and this percentage increases as service level decreases.
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104 7.4 Summary A stochastic lotsizing problem with deterioration and servicelevel constraints is considered in this chapter. The system is under periodic review. The demand in successive periods is independent but not necessarily identical. A constant fraction of any onhand inventory is deteriorated over on e period. The service level is assumed to incorporate the perception of the cost of backlogging so that the shortage cost can be ignored. It was found that the orderupto leve l is higher in our model than that of Tarim and KingsmanÂ’s because the inventory is depleted by both demand and deterioration. Through a simulation of 100 random cases, it wa s found that the actual fill rate is much higher than the service level and the total expected cost decreases as service level decreases.
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105 Chapter 8 Conclusions and Future Work 8.1 Conclusions This dissertation presents several discreteintime deteriorating inventory models and identifies optimal ordering quantities or policies for a single deteriorating product under deterministic or stochastic demand by minimizing the expected overall costs over the planning horizon. The various conditions have been considered, e.g. periodic review, timevarying deterioration rate, waitingtim edependent partial backlogging, timedependent demand, stochastic demand, servicelevel constraints etc. The computational experiments and sensitivity analysis bring a thorough and insightful understanding of the inventory control for deteriorating products. Th e major contributions of this dissertation are summarized as follows. Deterministic inventory control for a single deteriorating product. Under the conditions of constant demand and constant deterioration rate, a closedform equation is derived to compute the optimal solution, while no closedform solution was presented in lite rature. This greatly reduces the computational effort to identify the optimal solutions, and make s sensitivity analysis possible. Under the conditions of timedependent demand, timedependent deterioration rate and waitingtimedependent partial backlogging, we are able to derive explicit solutions based on the sufficient optimali ty condition. The necessary optimality
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106 condition is easily proved to be true, while it is not provable in continuous time case as stated by Wu (2002). Stochastic inventory control for a single deteriorating product. To the best of our knowledge, all the deteriorating inventory control models in the existing literature are dealing with deterministic customer de mand. Hence, this dissertation fills the vacancy by studying the deteriorating inventory models with stochastic customer demand and periodic review. All the finding s are clearly stated in Chapter 5 and Chapter 6. Stochastic inventory control for a single deteriorating product under servicelevel constraints. The deteriorating inventory control model with stochastic demand and servicelevel constraints is first studied in this dissertation. It was found that the actual fill rate is much higher than the service level and the total expected cost decreases as service level decreases. The 21st Century Engineering Grand Challenges, identified by National Academy of Engineering, requires innovative approaches to effectively use and manage finite resources. This will positively impact the su stainability, health, security and living of different species. Of particular importance is the reduction and management of various wastes generated in the world. Our resear ch addresses this challenge by optimally controlling the inventory of deteriorating products and minimizing the enormous cost due to deterioration. Our efforts could be applied and yield positive results in a variety of sectors that stock deteriorating products.
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107 8.2 Future Work This study aims to (a) obtain optimal or der quantity and useful insights for the inventory control of a single deteriorating product over a discrete time horizon with deterministic demand, variable deterioration rates and waitingtimedependent partial backlogging ratios and (b) identify optimal ordering policy for a single deteriorating product over a finite horizon with stochastic demand and partial backlogging. For part (a), the demand patterns we consider are constant and timevarying demands. In future, more demand patterns could be taken into consideration, such as stockdependent demand, pricedependent demand, ramp type demand, etc. The partial backlogging behaviors we consider are constant and waitingtimedep endent partial backlogging. For future research, more partial backlogging behaviors could be studied, such as exponential partial backlogging ratio, shortagedependent partial backlogging ratio, etc. For the stochastic lotsizing model with deterioration and backlogging in part (b), the conditions for the ( s S ) policy to hold are very restrictive and the explicit optimal ordering policy is rather difficult to develop. One possible future work will be to study this problem by discretizing the customer demand into several discrete scenarios. For example, we could have three customer demand scenarios, low customer demand, high customer demand and most likely customer demand. This customer demand uncertainty could be resolved by applying multistage mixedinteger st ochastic programming techniques, but the computation will be timeconsuming since the si ze of the problem grows exponentially as the number of scenarios increases. A superior future research study could concentrate on the development of efficient computationa l algorithms for this type of problem.
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About the Author Yang Tan received a BachelorÂ’s Degree in Industrial Engineering from Tianjin University, Tianjin, China in 2005, and a Master Degree in Industrial Engineering at University of South Florida, Tampa in 2009. He is currently a Ph.D. candidate in the department of Industrial and Management Systems Engineering at University of South Florida, Tampa. While in the Ph.D. program at University of South Florida, Yang Tan focuses on the research of optimal discreteintime inventory control for deteriorating products under partial backlogging. He has submitted two journal papers. He also made paper presentations at annual meetings of IIE and INFORMS. He is a member of INFORMS and IIE.
