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Continuous time disaggregation in hierarchical production planning

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Continuous time disaggregation in hierarchical production planning
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Al-Tamimi, Rami Salhab
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Production schedule
Family level
Minimize setup cost
Production cycle
Stochastic demand
Out-of-stock percentage
Dissertations, Academic -- Industrial Engineering -- Masters -- USF   ( lcsh )
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Abstract:
ABSTRACT: One of the objectives of disaggregation in hierarchical production planning is to minimize the setup costs incurred when changing production from one family to another. In this research, the setup costs are reduced by determining a production schedule that minimizes the number of setups during the planning horizon. Previous solutions to the disaggregation problem have considered discrete-time, and more recently continuous-time formulations. This research extends the continuous time disaggregation approach by incorporating production schedules allowing backorder. A mathematical formulation and a solution algorithm are presented and the computational complexity and convergence properties of the algorithm are discussed. Experimental results, using both deterministic and stochastic demand patterns, which demonstrate the efficacies of the solution approach are provided.
Thesis:
Thesis (M.S.I.E.)--University of South Florida, 2006.
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Includes bibliographical references.
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by Rami Salhab Al-Tamimi.
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Continuous Time Disaggregation in Hierarchical Production Planning by Rami Salhab Al-Tamimi A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Management Systems Engineering College of Engineering University of South Florida Major Professor: Ali Yalcin, Ph.D. Susana Lai-Yuen, Ph.D. Michael Weng, Ph.D. Date of Approval: November 1, 2006 Keywords: production schedule, family level, minimize setup cost, production cycle, stochastic demand, out-of-stock Percentage Copyright 2006, Rami Salhab Al-Tamimi

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Dedication To my family

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Acknowledgements I would like to thank, my advisor, Dr. Ali Yalcin for his continual guidance and support throughout this thesis. I also would like to thank my committee members Dr. Susana Lai-Yuen and Dr. Michael Weng for their kind reviews and support. Finally, I would like to thank my family, especially my fiance, for supporting me in my educational pursuits and my friends for their encouragement.

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i Table of Contents List of Tables................................................................................................................. iii List of Figures................................................................................................................ iv Abstract....................................................................................................................... ....v Chapter 1. Introduction....................................................................................................1 1.1 Hierarchical Production Planning.............................................................1 1.2 Aggregate Planning and Disaggregation...................................................2 1.3 Continuous Time Disaggregation..............................................................3 1.4 Motivation................................................................................................4 1.5 Research Objective...................................................................................4 1.6 Thesis Organization..................................................................................5 Chapter 2. Literature Review...........................................................................................6 2.1 Discrete Time Disaggregation with Deterministic Demands.....................7 2.2 Discrete Time Disaggregation with Stochastic Demands........................11 2.3 Continuous Time Disaggregation............................................................14 Chapter 3. Continuous Time Disaggregation and Backorders.........................................16 3.1 The Deterministic Continuous Time Disaggregation...............................16 3.2 Deterministic Continuous Time Disaggregation Allowing Backorder.....19 3.3 Experimental Results..............................................................................30 3.3.1 Comparison of Algorithms A1B and RKM..................................32 3.3.2 Comparison of Algorithms A1 and A1B (Compression Effect)....36 Chapter 4. Convergence Conditions and Computational Issues......................................39 4.1 Introduction............................................................................................39 4.2 General Convergence Expression...........................................................42 4.3 Convergence Example and Discussion....................................................46 4.4 Comparison of Algorithms A1 and A1B Convergence Expressions........49 4.5 A Modification to Algorithm A1B and Comparison to RKM..................50 4.6 Computational Experience with Algorithm A1B.....................................52 Chapter 5. Continuous Time Disaggregation and Demand Uncertainty..........................55 5.1 Algorithm A1B in the Presence of Demand Uncertainty.........................55 5.2 Incorporating a Service Level Parameter in the Problem Formulation.....58

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ii Chapter 6. Conclusions and Future Work.......................................................................63 6.1 Conclusions............................................................................................63 6.2 Future Work...........................................................................................65 References..................................................................................................................... 66

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iii List of Tables Table 3.1 Example Data.................................................................................................25 Table 3.2 Paired Comparisons of Number of Setups: n =3..............................................33 Table 3.3 Paired Comparisons of Number of Setups: n =6..............................................33 Table 3.4 Paired Comparisons of Number of Setups: n =9..............................................34 Table 3.5 Analysis of Differences in Number of Setups: v = 0.2....................................35 Table 3.6 Analysis of Differences in Number of Setups v = 0.5.....................................36 Table 3.7 Comparisons of Number of Setups: Compression Effect Data Sets.................37 Table 3.8 Analysis of Differences in Number of Setups v = 0.5.....................................37 Table 4.1 Paired Comparisons of Number of Setups: n = 3............................................51 Table 4.2 Analysis of Differences in Number of Setups v = 0.5.....................................52 Table 4.3 Computational Experiment for Algorithm A1B, v = 0.5, Iavg = 2000 ............... 53 Table 4.4 Computational Experiment for Algorithm A1, v = 0.5, Iavg = 2000 ................. 53 Table 5.1 Paired Comparison of Out-of-stock Percentage..............................................56 Table 5.2 Analysis of Differences in Out-of-stock Percentage........................................57 Table 5.3 Analysis of Differences Based on Demand Variability...................................58 Table 5.4 Comparison of Different Service Levels n = 6................................................59

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iv List of Figures Figure 1.1 Hierarchical Production Planning Levels.........................................................1 Figure 3.1 Algorithm A1B Flowchart............................................................................24 Figure 3.2 The Design for the Experiment.....................................................................31 Figure 3.3 Inventory Versus Time Under Algorithm A1................................................38 Figure 3.4 Inventory Versus Time Under Algorithm A1B..............................................38 Figure 4.1 Steps 6 and 7 from Algorithm A1B...............................................................41 Figure 4.2 Typical Relationship Between Tand ] [ iDfor Two Successive Iterations ......44 Figure 5.1 Inventory Versus Time Z1 = 0 and Z2 = 0......................................................61 Figure 5.2 Inventory Versus Time Z1 = -0.2 and Z2 = 0.2...............................................61 Figure 5.3 Inventory Versus Time Z1 = -0.5 and Z2 = 0.5...............................................62

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v Continuous Time Disaggregation in Hierarchal Production Planning Rami Salhab Al-Tamimi ABSTRACT One of the objectives of disaggregation in hierarchical production planning is to minimize the setup costs incurred when changing production from one family to another. In this research, the setup costs are reduced by determining a production schedule that minimizes the number of setups during the planning horizon. Previous solutions to the disaggregation problem have considered discrete-time, and more recently continuoustime formulations. This research extends the continuous time disaggregation approach by incorporating production schedules allowing backorder. A mathematical formulation and a solution algorithm are presented and the computational complexity and convergence properties of the algorithm are discussed. Experimental results, using both deterministic and stochastic demand patterns, which demonstrate the efficacies of the solution approach are provided.

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1 Chapter 1. Introduction 1.1 Hierarchical Production Planning Production planning problems involve the scheduling of a large number of items. Formulating such problems using mathematical programming models including all these items is an overwhelming process. Moreover, this process includes complex decisions that cannot be made all at once. An alternative solution to this process is the hierarchical production planning approach (HPP) where the planning decisions are made in a sequence. In HPP, the overall problem is divided into sub problems, where each sub problem is related to a different hierarchal level and solved sequentially. Hax and Meal [15] recommend three levels of HPP, as shown in Figure 1.1. The first level contains the items which are the final products to be delivered to the customers. The second level is the family level which is a group of items that share a common manufacturing setup. The third level is the type level which is a group of families that have similar costs per unit of production time, and similar seasonal demand patterns. Figure 1.1 Hierarchical Production Planning Levels Item 1 Item 2 Family 1 Type 1 Family 2 Family 3 Item 3 Item 4 Type Level Family Level Item Level

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2 An example for HPP would be the planning of production in a dairy plant. In the dairy plant, there are several dairy types like milk, ice cream, butter, cheese, eggnog and yogurt. Each of these types includes several families that have similar costs per unit of production time and similar seasonal demands patterns. For instance, cheese type includes string cheese, cheddar cheese, cottage cheese, and cream cheese families. Each cheese family contains large number of items. These items differ from each other by characteristic such as color, packaging, labels, and accessories. As an example, cottage cheese contains items like creamed cottage cheese, 2% and 1% cottage cheeses, dry-curd cottage cheese, etc. These cottage cheese items differ from each others by the label, the milk fat percentage and the drying temperature. Nevertheless, all the items within the same family share a common manufacturing setup which is associated with a setup cost when changing over to another family. For example, the production of the cheddar cheese family needs specific types of mixers, cutters and dryers. Some of these machines need to be cleaned, replaced or their parts need to be changed before starting the production of another cheese family. 1.2 Aggregate Planning and Disaggregation HPP consists of two steps: aggregate planning and disaggregation. Aggregate planning is done at the types’ level with an objective of minimizing the total costs that include production cost, inventory holding cost and labor cost according to Bitran and Hax [5]. In the dairy plant example, each type has its own production cost, inventory holding cost and labor cost. As an example, the milk production cost is less than the cheese production cost while it has more inventory holding cost. All these costs are

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3 included in the objective function of the aggregate planning to minimize the total cost. The output of the aggregate planning is the number of units to be produced of each type during each period. Disaggregation is done at the family level. The purpose of disaggregation is to determine a production schedule for the families with the objective of minimizing the setup costs that are incurred during production changes from one family to another. In addition, it also aims at maintaining consistency and feasibility between production decisions that were made during aggregate planning and during the process of disaggregation itself. In the dairy plant example, changing the production from the cheddar cheese family to the cottage cheese family has a certain setup cost. The disaggregation objective is to minimize the total setup cost within the production schedule by minimizing the number of setups. 1.3 Continuous Time Disaggregation Most of the previous studies of disaggregation include discrete time formulations. These formulations determine the production schedule for the families on a single period basis by solving the disaggregation problem at the beginning of each period. Yalcin and Boucher [31] introduce a formulation that deals with the planning horizon as one continuous time interval. This continuous time disaggregation methodology looks forward to the entire planning horizon to establish common cycles of production. During each cycle, each family is produced once. Therefore, increasing the cycle’s length decreases the number of setups for each family. As a result, the total setup costs are minimized. This stems from the idea in the previous section that minimizing the

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4 setup costs in disaggregation can be achieved by minimizing the total number of setups during the planning horizon. 1.4 Motivation Due to market variation, in most practical disaggregation problems, there is a degree of uncertainty associated with the demand for each family. The uncertainty in demand may result in some families to go to backorder requiring the need for disaggregation formulations that allow for backorder. The continuous time disaggregation formulation introduced by Yalcin and Boucher [31] shows significant improvements over the discrete time disaggregation methods in minimizing the number of setups. However, this formulation does not allow backorder and does not accommodate demand uncertainty. Additionally, the case of high demand variability and low initial inventory often results in a “compression effect” [31] where the solution tends to have more frequent setups when multiple families reach their run-out of inventory times at approximately the same time. Allowing families to go to backorder can resolve the compression effect problem, further reducing the number of setups and in turn the total setup costs. 1.5 Research Objective The goal of this thesis is to provide a continuous time disaggregation solution that allows backorder and accommodates demand uncertainty. Within this goal, the proposed disaggregation solution will be compared with the previous approaches through

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5 experimental designs and computational properties of the solution algorithm will be discussed. 1.6 Thesis Organization The rest of the thesis is organized as follows: Chapter 2 reviews the previous work in literature concerning disaggregation. Chapter 3 describes the specific problem to be addressed, discusses the theoretical foundations and presents the continuous disaggregation formulation allowing backorder. Chapter 4 discusses the convergence conditions of the continuous-time disaggregation allowing backorder. Chapter 5 addresses issues associated with demand uncertainty. Finally, chapter 6 includes the conclusion of this research and the directions for future research.

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6 Chapter 2. Literature Review Hierarchical Production Planning (HPP) was first introduced by Hax and Meal [15] for a multiple plant, multiple product seasonal demand model. For this model, three different hierarchical levels are introduced: items, families, and types. Items are the final products. Families are group of items that share the same equipment and require a setup cost when changing the production from one family to another. Types are group of families, which have similar costs per unit of production time. The HPP is also comprised of important planning components known as “aggregation” and “disaggregation.” Aggregate planning is done at the type level in order to minimize the total cost, while disaggregation is done at the family level in an attempt to minimize the setup costs. Minimizing the setup costs is procured by determining the intermittent production schedules during the planning horizon. For recent discussion about HPP, see Askin and Goldberg [1] and Nahimas [23]. In this literature review, previous approaches for disaggregation are divided into three categories. The first is discrete time disaggregation with deterministic demands. The second is discrete time disaggregation with stochastic demands. The last is continuous time disaggregation.

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7 2.1 Discrete Time Disaggregation with Deterministic Demands Bitran and Hax [5] propose the following family disaggregation model, that is solved for each product type i at the beginning of each period: Problem Pi Minimize ,oJ j j j jY d S Subject to: =oJ j i jX Y*, ,j j jub Y lb ,oJ j where: Yj: number of units of family j to be produced, Yj is the only unknown variable and it is the output of the disaggregation model. Sj: setup cost for family j, dj: forecast demand for family j, lbj and ubj: upper and lower bounds for the quantity Yj, Xi *: total amount to be allocated among all the families belonging to type i. Xi has been determined by the aggregate planning at the types level. The lower bound, which defines the minimum production required for family j, is given by: [ ] j j L j j j jSS AI d d d lb + Š + + + =+) ... ( 0 max1 2 1 ,, where dj+1+…+ dj+L+1 is the total forecasted demand for family j during the production lead time L plus the review period (assumed equal one), AIj is the current available inventory for family j, and SSj is the required safety stock for family j.

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8 The upper bound is given by: j j jAI OS ub Š =, where OSj is the overstock limit of family j. Bitran and Hax [6] show that the first constraint of problem Pi can be relaxed without changing the optimum solution. The constraint can be substituted by the following: oJ j i jX Y* Following this formulation, an algorithm is introduced. This algorithm is a Regular Knapsack Method (RKM), because the optimal value of at least one variable is determined in each iteration. The algorithm determines the production quantities of the families in the following period, so only those families that run-out of inventory in the current period are considered for production. Three cases according to the upper and lower bounds are considered. The first case is when the summation of the upper bound is less than Xi *. In this case, all the families are produced to their upper bound limit, and producing families that are not considered for production at this period fills the remaining capacity. The second case is when the summation of the lower bounds is more than Xi *. In this case backorders occur. The backorders are distributed among all the families that are considered for production using the following formula: o J j j j J j j i j jJ j lb lb lb X lb Yo o Š = +, ) (* The last case is when Xi lies between the lower and the upper bounds, then Algorithm RKM is run to determine the production quantities.

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9 Bitran et al. [7, 8] present a hierarchical approach to model a planning and scheduling production problem in single stage and in two-stage manufacturing environments. They propose some modification to the original HPP system and introduce a mixed integer programming formulation to solve the problem. Furthermore, they consider the infeasibility that can be introduced in the aggregate problem by the disaggregation procedure used. Infeasibility occurs when the solution of the disaggregation is not feasible at the aggregation level. Example of infeasibility is a solution of one period that results in the need of more production than that allocated by the aggregate problem to satisfy the demands of the next periods. A Look Ahead Feasibility Rule (LAFR) is introduced. This rule looks ahead one period to prevent the infeasibility when applying the disaggregation procedure at this period. Saad [29] presents some modifications to Hax and Meal [15] approach to include multi-plant multi-marketing area systems. In his approach, product types are assigned to the plants and within each plant products are grouped in families. He includes marketing and logistics decisions in the aggregate planning formulation to find the production quantity for each plant. Then, the disaggregation is done at the plant level using a mixed integer linear programming formulation. Graves [14] presents a mixed integer programming model to solve both the aggregate planning at the product type level and the family disaggregation. He proposes a Lagrangean relaxation to solve the problem where the two planning levels are linked with an inventory consistency relationship. In case of high setup costs, Graves’s test results show a more effective solution over Bitran et al. [6]. However, this should be balanced against the computational requirements and the complexity of his algorithm.

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10 Ozdamar et al. [26] extend Bitran et al. [6] to resolve backorders that occur in the RKM family disaggregation procedure. These backorders are the result of infeasibility when the production quantities assigned in the aggregate planning are infeasible at the disaggregation level. They propose a heuristic modification that eliminates this condition and reduces the total setup costs. Axsater [4] and Erschler [12] present conditions of consistency of the disaggregation procedure. Given a feasible aggregate plan, the disaggregation procedure is consistent if it satisfies the demands in the first period, and it retains the feasibility of the aggregate plan for the rest of the planning horizon. A modified LAFR that looks ahead over the entire horizon is then discussed. Qui and Burch [27] and Qui [28] demonstrate a solution model to a real world production planning and scheduling in a fiber plant using a combination of hierarchical production planning and an expert system. At the aggregate level, the monthly lot size, machine assignment and the ending inventory level for each individual product are determined using the given forecast demand, machine capacities, and the total inventory target. At the disaggregation level, with the objective of minimizing the total setup costs, the model uses the outputs of the aggregate model to determine the sequence in which the products are produced on their assigned machine. The disaggregation model is formulated as a mixed integer linear program. Discrete disaggregation approaches with deterministic demand have been implemented widely in modern manufacturing systems. Dempster et al. [11] provide an analytical review for disaggregation approaches using detailed production planning, job shop scheduling, distribution system design, and vehicle routing scheduling applications.

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11 Tsubone and Sugawara [30] present an HPP that includes feedback between the hierarchical levels based on human judgments for motor industry applications. Nguyen and Dupont [25] consider an HPP approach in steel manufacturing firm which works on demands according to each customer’s requirements. Akturk and Wilson [2] apply an HPP approach in cellular manufacturing system by introducing several enhancements to Hax and Meal [15] HPP approach. Neureuther et al. [24] introduce a non-linear disaggregation model for a strictly make-to-order steal fabrication plant. 2.2 Discrete Time Disaggregation with Stochastic Demands Bitran et al. [9] consider a manufacturing of style goods application. Style goods have very short selling seasons and the demands are changing over time. This characteristic requires a continuous revision of the forecasted demands over the planning horizon. They formulate the problem as a mixed integer stochastic program. As this formulation is hard to solve, two-stage hierarchical approach is proposed to solve this problem. Furthermore, the products are grouped into families where the means of the demands are assumed to be known. The first stage of the approach is the aggregate problem which is formulated as a deterministic mixed integer program that provides a lower bound on the optimal solution. The solution to this problem determines the set of product families to be produced in each period. The second stage is a disaggregation stage where item lot sizes are determined for families scheduled in each period. Matsuo [20] extends this approach by a stochastic sequencing formulation that simultaneously determines product sequence. His sequencing rule based on the ratio of the inventory

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12 holding cost plus the expected value of additional information per unit time to the expected resource consumption of a family. Ari and Axsater [3] consider HPP disaggregation in the case of stochastic demands where the item demands are assumed to be independent stochastic variables with known distributions. Moreover, they are known for the current period, but are still unknown for the rest of the planning horizon. A dynamic programming algorithm is introduced with an objective of maximizing the probability of feasibility of the aggregate plan. The optimal solution is proved to be the same as minimizing the differences between the inventory levels at the end of each period which is referred to as the simple disaggregation rule. This rule is applicable for many common demand distributions. However, according to the authors, because of the set up costs it is, in general, not possible to apply the simple disaggregation rule making the inventories as equal as possible. Lasserre and Merce [19] consider a robust aggregate plan that provides a disaggregation policy that can handle the variation in the demands. The items demands are known for the current period, while for the rest of the planning horizon they vary within given bounds. According to the authors, the aggregate plan is said to be robust if and only if for any potential demand there exists a feasible disaggregation policy. Necessary and sufficient conditions for robustness are presented. These conditions are converted to constraints in aggregate planning, which take into account the uncertainty of the demands. Gfrerer and Zapfel [13] propose a generalization for the robustness concept by introducing two sets of sufficient conditions of robustness using the concept of static

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13 and dynamic demand schemes. In addition, simplifications to these conditions are introduced for implementation purposes. Another stochastic linear programming approach is introduced by Kira [18]. This approach is a modification of Bitran’s and Hax’s [5] formulation that elaborate penalties for infeasibility. Two penalties are introduced: surplus and shortage, which give an upper and a lower bound for the production for each family. Their formulation shows some improvement over Bitran and Hax [5] formulation with the largest average improvement occurring with the highest expected demand and the smallest with the lowest expected demand. Yan [32-36] explores the hierarchical stochastic production planning problem of flexible automated work shops in an agile manufacturing environment. Yan [32] and [35] present a nonlinear model of stochastic production with delay interaction. They have developed a software package named SPID with an algorithm based on their model. In Yan [33], a stochastic nonlinear programming model whose constraints are linear and a piecewise linear is formulated. This model is transformed into a deterministic nonlinear programming model then into a linear programming one to simplify the solution. Two algorithms are proposed to solve this model. The first algorithm is Karmarkar algorithm [17]. The second is an interaction/prediction algorithm. Yan proposes that this approach is capable of solving more complicated manufacturing systems with multi-period multiproduct problems. In Yan [34], the approach is implemented to a multi-period multiproduct problem. A software package named SIPA has been developed based on the stochastic interaction/prediction algorithm.

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14 Incorporating stochastic demand patterns has gained attention in modern manufacturing systems. Dempster et al. [11] present a multistage stochastic model for a job shop design problem. Ciarallo et al. [10] discuss the characteristics of the effect of random capacity and demand uncertainty on periodic review, production planning decisions. Kasilingam [16] proposes a non-linear programming formulation for solving the capacity allocation problem considering stochastic demands in air cargo yield management. Mulvey and Ruszczynski [22] consider a decomposition method for multistage stochastic scheduling and transportation problem. Yokoyama and Lewis [38] introduce two heuristic algorithms for a multi-product multi-machine production system where the demand in each period is a mutually independent random variable whose probability distribution is known. Yin et al. [37] present a stochastic approach to incorporate demand uncertainties in paper industry. 2.3 Continuous Time Disaggregation Yalcin and Boucher [31] introduce a continuous time solution method to the disaggregation problem. This continuous time solution looks forward to the entire planning horizon to establish common cycles of production. During each cycle each family is produced once. This solution is formulated as a set of linear equations that can be solved simultaneously. The formulation is solved for T,] [it, i = 1, …, n and given by: T Max, s.t. ) 1 ...( 2 1 0 ] [ ] [] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = + Š Š ++n i D T t P t t Ii i i i i i, (1-1)

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15 n i D T t P t T Ii i i i i= + Š Š + 0 ] [ ] [] [ ] [ ] [ ] [ 0 ], [, (1-2) n i r ti i... 2 1 ,] [ ] [= (1-3) where ] [iP is the average production rate over the period t[i] to t[i+1]; ] [iD is the average demand rate over the period 0 to ] [] [T ti+; ] [it and ] 1 [+it is the time to start and stop producing family i; n is the number of families considered for disaggregation; and ] [ir is the run-out of inventory time for family i that can be found by solving the following equality. 0] [0 ], [ 0 ], [= Š=dt D Iir t t i i The objective of the above formulation is to maximize the length of the cycle time T Since each family is produced once in each cycle, maximizing the cycle time decreases the number of setups during the planning horizon, as a result the setup costs are minimized. The constraints (1-1) and (1-2) of the formulation ensure that supply meets demand over the cycle. The last constraint (1-3) ensures that the production of family i begins before it runs out of inventory. Several properties for the optimal solution are considered. ] [ iP and ] [ iD depend on the value of T Therefore, an initial value for T has to be estimated before solving the formulation. The proposed estimation is the time when the last family runs out. An iterative algorithm is proposed to solve this formulation. The experimental results of Yalcin and Boucher [31] algorithm demonstrate a better performance and a significant difference in the number of the setups of the proposed algorithm over the Algorithm RKM.

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16Chapter 3. Continuous Time Disaggregation and Backorders This chapter discusses the fundamentals of the continuous time disaggregation formulation. Then, it presents the continuous disaggregation formulation allowing backorder along with experimental results. This chapter is divided into 3 sections. Section 3.1 discusses in detail the continuous time disaggregation formulation and algorithm introduced by Yalcin and Boucher [31] which forms the basis for the work in this thesis. Section 3.2 introduces a modified formulation that allows families to go to backorder, some interesting properties of the formulation, and an iterative algorithm to solve the formulated problem. In section 3.3, experimental results comparing the performance of the formulation allowing backorders are illustrated. 3.1 The Deterministic Continuous Time Disaggregation Continuous time disaggregation is formulated as a set of linear equations as follows: T Max (3-1a) s.t. ) 1 ...( 2 1 0 ] [ ] [] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = + Š Š ++n i D T t P t t Ii i i i i i, (3-1b) n i D T t P t T Ii i i i i= + Š Š + 0 ] [ ] [] [ ] [ ] [ ] [ 0 ], [, (3-1c) n i r ti i... 2 1 ,] [ ] [= (3-1d)

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17 where ] [ iP is the average production rate over the period t[i] to t[i+1] ; ] [ iD is the average demand rate over the period 0 to ] [] [T ti+; ] [ it is the time to start producing family i ; ] 1 [ + it is the time to stop producing family i ; n is the number of families considered for disaggregation; and ] [ ir is the run-out of inventory time that can be found by solving the following equality: 0] [0 ], [ 0 ], [= Š=dt D Iir t t i i (3-1e) The objective function of this formulation is maximizing the cycle time. Maximizing the cycle time decreases the number of setups during the planning horizon, and as a result the setup costs are minimized. Constraints (3-1a) and (3-1b) ensure that inventory and production meets the demand over the cycle T. For each family the summation of its current inventory and its production during the cycle should be equal to or greater than the demand in that cycle. The last constraint (3-1d) ensures that family production will start before it runs out of inventory. Although the previous constraints will guarantee that by the end of the cycle all the demands will be satisfied, they do not prevent the families from going to backorder before the start of their production during the cycle time. This constraint will ensure that families will not go to backorder by forcing the production to start before the run-out of inventory time of each family. The deterministic continuous time formulation is iterative. To solve for the cycle time, initial values for the average demands should be calculated. However; the average demand for each family depends on the value of the cycle time which requires an initial value for T As some variables have to be determined by a decision variable, an

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18 algorithmic approach is used. The following algorithm is called A1, and is used to solve the deterministic continuous time disaggregation formulation: Algorithm A1 Step 1: Initialize a counter, j =0, and a convergence tolerance level, Step 2: Order families in ascending order of their run-out times. Step 3: Place all families in the set F, which is an ordered set of families currently being considered for disaggregation. For n families in F, let k = n Step 4: Initialize T0 = r[n]. Initialize ] [ iP and ] [ iD as follows: 0[][], 0 01T iitPPdt T= 0[][], 0 01T iitDDdt T=. Step 5: Increment j. Solve the following equation. ) 1 ...( 2 1 0 ] [ ] [] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = = + Š Š ++n i D T t P t t Ii i i i i i Step 6: Using solution values from Step 5, compute, for all families in F: dt P t t Pi it t t i i i i+Š =+] 1 [ [] ], [ ] [ ] 1 [ ] [1 [][][], [] 01itT iit iDDdt tT+= + Increment j Step 7: Solve the following equation set: [],0[1][][][][] [],0[][1][1][1][1][][]0,1,2,3...(2) [][]0ikiiii kkkkkkIttPtTDik IrtPtTDŠ ŠŠŠŠ+ŠŠ+==Š +ŠŠ+=

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19 If < ŠŠ 1 j jT T, go to Step 8. Else, go to Step 6. Step 8: Evaluate the conditioni r ti i ] [ ] [. If true, stop. This is a valid solution. If false, go to Step 9. Step 9: Set k = the largest number i in F for which ] [ ] [ i ir t> is true. Delete from F all families wherek i >. Go to Step 1. 3.2 Deterministic Continuous Time Disaggregation Allowing Backorder The last constraint in the deterministic continuous time formulation prevents families from going to backorder. If this constraint is eliminated, families are allowed to start their production before or after their run-out of inventory time during a given cycle which allows the families to go to backorder. The formulation is as follows: T Max, (3-2a) s.t. ) 1 ...( 2 1 0 ] [ ] [] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = + Š Š ++n i D T t P t t Ii i i i i i, (3-2b) n i D T t P t T Ii i i i i= + Š Š +, 0 ] [ ] [] [ ] [ ] [ ] [ 0 ], [, (3-2c) Although the only difference between the two formulations is the last constraint, deleting this constraint changes the properties of the optimal solution. Several properties of the deterministic continuous time formulation allowing backorder are described below. Property 1: In any optimal solution, equations (3-2b) and (3-2c) will be equalities. Although any family can go to backorder during the production cycle, at the end of the cycle all the demands should be satisfied.

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20Proof: Let P be the average aggregate production rate over the period 0-T, i.e., } ) ( ] ) [( { 1] [ ] [ ] [ ] [ 1 1 ] 1 [ n n i i n i iP t T P t t T P Š + Š =Š = +. Since ] [ iP and ] [ iD are in the same aggregate units, equations (3-2b) and (3-2c) can be combined as follows: = = =+ +n i i i n i i n i iD T D t P T I1 ] [ ] [ 1 ] [ 1 0 ], [, (3-3) If there existed a solution to equations (3-2b) and (3-2c) in which the inequality of equation (3-3) holds, then the solution could be improved by increasing T until equation (3-3) becomes an equality. Therefore, any optimal solution is a solution in which constraints (3-2b) and (3-2c) are equalities. The significance of this property is that an optimal solution can be obtained from a solution of n simultaneous equations of the form: ) 1 ...( 2 1 0 ] [ ] [] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = = + Š Š ++n i D T t P t t Ii i i i i i (3-4a) n i D T t P t T Ii i i i i= = + Š Š + 0 ] [ ] [] [ ] [ ] [ ] [ 0 ], [ (3-4b) and t[1] = 0. Since t[1] = 0, for n products, there will be n simultaneous equations with n unknowns ( t[i], i = 2,3,… n and T ). Note that, in equation (3-3), the left hand side includes the supply of production time allocated to the cycle, T ; i.e., P T The right hand side is the demand during the cycle, T Since, in any optimal solution, equation (3-3) must be an equality, the period T will be determined by equilibrium of the supply and the demand. In effect, more demand can be serviced by increasing the production cycle time, T on the right hand side of equation (3-3).

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21 Property 2: If, in an optimal solution to equations (3-4a) and (3-4b),[][]n nr t<, then the cycle time of the first ( n -1) families will be increased by solving equation (3-4a) alone and letting [][]n nr t=. However; if, in an optimal solution to equations (3-4a) and (3-4b), [][]n nr t>, then the cycle time of the first ( n -1) families will be decreased by solving equation (3-4a) alone. Proof: In Property 1, we noted that more demand can be serviced by increasing the production cycle time, T When equations (3-4a) and (3-4b) are solved, the total production time allocated to the first ( n -1) families is: ] 1 [ ) ] 1 [ ] [ ( 2 1 ] [ ) ] [ ] 1 [ ( 1 1 0 ], [ Š Š Š + Š = Š + + Š =n P n t n t n i i P i t i t n i i I (3-5) If equation (3-4b) is dropped and [][]n nr t= is imposed, the total production time allocated to the first ( n -1) families is: ] 1 [ ) ] 1 [ ] [ ( 2 1 ] [ ) ] [ ] 1 [ ( 1 1 0 ], [ Š Š Š + Š = Š + + Š =n P n t n r n i i P i t i t n i i I (3-6) Subtracting (3-5) from (3-6) yields: ] 1 [ ] [ ] 1 [ ] [ Š Š Š n P n t n P n r (3-7) When [][]n nr t, T will be greater under (3-5) than (3-6).

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22 Property 2 suggests that maximizing the cycle time T first require solving equations (3-4a) and (3-4b) for all families n Secondly, the value of []nt is compared to[]nr. If [][]n nr t> then this is the solution that maximizes T and the algorithm is run again at time T If [][]n nr t< then letting [][]n nr t= and solving equation (3-4a) alone will result in larger cycle time T and the algorithm will be run again at time []nr. The deterministic continuous time formulation allowing backorder is also iterative since an initial value of T should be assigned to calculate the initial values of the average demands. The following algorithm (Algorithm A1B) is proposed as an algorithmic approach for the solution of this formulation. Algorithm A1B Step 1: Initialize a counter, j = 0, and a convergence tolerance level,. j is the iteration index. is the level of convergence. If the difference between two successive cycle time, T is the algorithm considers T as an optimal solution. Step 2: Order families in ascending order of their run-out times r[i], i = 1, …, n Where n is the number of families, and r[i] is the solution to: 0] [0 ], [ 0 ], [= Š=dt D Iir t t i i Step 3: Place all families in the set F, which is an ordered set of families currently being considered for disaggregation. Step 4: Initialize T0 = r[n]. Initialize ] [ iP and ] [ iD as follows:

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23 0[][], 0 01T iitPPdt T= =00 ], [ 0 ] [1T t i idt D T D Step 5: Increment j Solve equation set (3-4). Step 6: Using solution values, compute, for all families in F: dt P t t Pi it t t i i i i+Š =+] 1 [ [] ], [ ] [ ] 1 [ ] [1 ++ =j iT t t i j i idt D T t D] [0 ], [ ] [ ] [1 Step 7: Increment j Solve equation set (3-4). If ŠŠ 1 j jT T go to step 6. Step 8: If [][]n nr t> this is a valid solution (end), else go to step 9. Step 9: Set j = 0. Initialize T0 = r[n]. Initialize ] [ iP and ] [ iD i = 1, …, n as follows: 0[][], 0 01T iitPPdt T= =00 ], [ 0 ] [1T t i idt D T D Step 10: Increment j Solve equation set (3-4a). (Note (3-4b) is not included) Step 11: Using solution values, compute, for all families in F: dt P t t Pi it t t i i i i+Š =+] 1 [ [] ], [ ] [ ] 1 [ ] [1 ++ =j iT t t i j i idt D T t D] [0 ], [ ] [ ] [1 Step 12: Increment j Solve equation set 3-4a. If ŠŠ 1 j jT T go to step 11, else end this is a valid solution.

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24 If the valid solution is the result of solving equation set (3-4) then the algorithm is run again at time T when family n ends its production. However; if the valid solution is the result of only solving 3-4a then the algorithm is run again at time[]nr when family n-1 ends its production. Algorithm A1B is demonstrated using a flowchart in Figure 3.1 followed by a numerical example. Start Order families according to their run out times Compute P(i)and D(i) |Tj– Tj-1| j = j + 1 Solve equation set (3-4) End No Yes j = 0 T0= r(n)Compute P(i)and D(i) t(n) < r(n) Yes Compute P(i)and D(i) |Tj– Tj-1| j = j + 1 Solve equation (3-4a) Yes j = 0 T0= r(n)Compute P(i)and D(i) No No j = j + 1 Solve Equation Set (3-4) j = j + 1 Solve Equation (3-4a) Figure 3.1 Algorithm A1B Flowchart

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25 Example Problem: Consider the data of Table (3.1) showing the initial inventory, I0 and product demand, Dt for three products over six periods. Pt is the aggregate production rate. Table 3.1 Example Data Period Dt Family A Dt Family B Dt Family C Pt Production 1 1054.2838.3807.93000.0 2 1170.0843.21188.83000.0 3 1158.9982.11154.63000.0 4 1026.01170.51029.83000.0 5 1129.61052.91086.03000.0 6 885.1863.51108.43000.0 I0 2000.01000.00.0 Step 1: Initialize a counter, j = 0, and a convergence tolerance level,= 0.01. Step 2: The number of families n = 3. Compute each family run-out time. rA = 1.808, rB = 1.192, and rC = 0. Order them in ascending order. r[1] = rC = 0; r[2 ]= rB = 1.192; r[3] = rA = 1.808. Step 3: Place all families in the set F = {C, B, A}. Step 4: Initialize T as the last family in set F run-out time; T0 = r[3] = 1.808. Initialize ] [ iP and ] [ iD i = 1, …, n ,as follows: [] 3000 808 1 ) 3000 )( 808 0 ( 30001= + = P, which is the same for all i. 0 1106 808 1 ) 0 1170 )( 08 8 0 ( 2 1054 5 840 808 1 ) 2 843 )( 08 8 0 ( 3 838 2 978 808 1 ) 8 1188 )( 808 0 ( 9 807] 3 [ ] 2 [ ] 1 [= + = = + = = + = D D D

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26 Step 5: Increment j ; j = 1. Solve for1T ,] 1 [t, ] 2 [t, and ] 3 [t. 0 ] [ ] [] 1 [ ] 1 [ ] 1 [ ] 1 [ ] 2 [ 0 ], 1 [= + Š Š + D T t P t t I 0 ] [ ] [] 2 [ ] 2 [ ] 2 [ ] 2 [ ] 3 [ 0 ], 2 [= + Š Š + D T t P t t I 0 ] [ ] [] 3 [ ] 3 [ ] 3 [ ] 3 [ 0 ], 3 [= + Š Š + D T t P t T I 0 ] 2 978 ][ [ ] 3000 ][ [ 01 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t 0 ] 5 840 ][ [ ] 3000 ][ [ 10001 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t t 0 ] 0 1106 ][ [ ] 3000 ][ [ 20001 ] 3 [ ] 3 [ 1= + Š Š + T t t T Solution: 1T = 3.47, ] 1 [t= 0, ] 2 [t= 1.13, and ] 3 [t= 2.09. Step 6: Using the solution from step 5, compute ] [iP and ] [iD for all i [] 3000 =iP 3 1085 56 5 ) 1 885 )( 56 0 ( 6 1129 0 1026 9 1158 0 1170 2 1054 9 970 60 4 ) 8 1052 )( 60 0 ( 5 1170 1 982 2 843 3 838 6 047 1 47 3 ) 8 1029 )( 47 0 ( 6 1154 8 1188 9 807] 3 [ ] 2 [ ] 1 [= + + + + + = = + + + + = = + + + = D D D Step 7: Increment j ; j = 2. Solve for2T ,] 1 [t, ] 2 [t, and ] 3 [t. 0 ] 6 1047 ][ [ ] 3000 ][ [ 02 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t 0 ] 9 970 ][ [ ] 3000 ][ [ 10002 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t t

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270 ] 3 1085 ][ [ ] 3000 ][ [ 20002 ] 3 [ ] 3 [ 2= + Š Š + T t t T Solution: 2T = 2.59, ] 1 [t= 0, ] 2 [t= 0.91, and ] 3 [t= 1.71. Š1 2T T, go to step 6. Step 6: Using the solution from step 7, compute ] [iP and ] [iD for all i [] 3000 =iP 2 1104 20 4 ) 6 1129 )( 20 0 ( 0 1026 9 1158 0 1170 2 1054 3 928 50 3 ) 5 1170 )( 50 0 ( 1 982 2 843 3 838 2 1034 59 2 ) 6 1154 )( 59 0 ( 8 1188 9 807] 3 [ ] 2 [ ] 1 [= + + + + = = + + + = = + + = D D D Step 7: Increment j ; j = 3. Solve for3T ,] 1 [t, ] 2 [t, and ] 3 [t. 0 ] 2 1034 ][ [ ] 3000 ][ [ 03 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t 0 ] 3 928 ][ [ ] 3000 ][ [ 10003 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t t 0 ] 2 1104 ][ [ ] 3000 ][ [ 20003 ] 3 [ ] 3 [ 3= + Š Š + T t t T Solution: 3T = 2.74, ] 1 [t= 0, ] 2 [t= 0.95, and ] 3 [t= 1.76. Š2 3T T, go to step 6. Steps 6 and 7 are repeated till convergence occurs at8T 8T = 2.69, ] 1 [D = 1038.7, ] 2 [D = 936.88, ] 3 [D = 1104.92, ] 1 [t= 0, ] 2 [t= 0.93, and ] 3 [t= 1.73. Step 8: [][]3 3r t < solving 3-4a alone will result in larger T Go to step 9.

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28 Step 9: Set j = 0. Initialize T0 = r[3] = 1.808. Initialize ] [iP and ] [iD i = 1, …, n ,as follows: [] 30001= P which is the same for all i 5 840 2 978] 2 [ ] 1 [= = D D Step 10: Increment j ; j = 1. Solve for1T ,] 1 [t, and ] 2 [t. 0 ] [ ] [] 1 [ ] 1 [ ] 1 [ ] 1 [ ] 2 [ 0 ], 1 [= + Š Š + D T t P t t I 0 ] [ ] [] 2 [ ] 2 [ ] 2 [ ] 2 [ ] 3 [ 0 ], 2 [= + Š Š + D T t P t r I 0 ] 2 978 ][ [ ] 3000 ][ [ 01 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t 0 ] 5 840 ][ [ ] 3000 ][ [ 10001 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t r Solution: 1T = 3.07, ] 1 [t= 0 and] 2 [t= 1.00. Step 11: Using the solution from step 10, compute ] [iP and ] [iD for all i [] 3000 =iP 2 960 07 4 ) 8 1052 )( 60 0 ( 5 1170 1 982 2 843 3 838 0 050 1 07 3 ) 8 1029 )( 07 0 ( 6 1154 8 1188 9 807] 2 [ ] 1 [= + + + + = = + + + = D D Step 12: Increment j ; j = 2. Solve for2T ,] 1 [t, and ] 2 [t. 0 ] 0 1050 ][ [ ] 3000 ][ [ 01 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t

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290 ] 2 960 ][ [ ] 3000 ][ [ 10001 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t r Solution: 2T = 2.74, ] 1 [t= 0, and ] 2 [t= 0.96. Š1 2T T, go to step 11. Step 11: Using the solution from step 12, compute ] [iP and ] [iD for all i [] 3000 =iP 2 941 70 3 ) 5 1170 )( 70 0 ( 1 982 2 843 3 838 5 040 1 74 2 ) 6 1154 )( 74 0 ( 8 1188 9 807] 2 [ ] 1 [= + + + = = + + = D D Step 12: Increment j ; j = 3. Solve for3T ,] 1 [t, and ] 2 [t. 0 ] 5 1040 ][ [ ] 3000 ][ [ 01 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t 0 ] 2 941 ][ [ ] 3000 ][ [ 10001 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t r Solution: 2T = 2.78, ] 1 [t= 0, and ] 2 [t= 0.97. Š2 3T T, go to step 11. Step 11: Using the solution from step 12, compute ] [iP and ] [iD for all i [] 3000 =iP 4 944 75 3 ) 5 1170 )( 75 0 ( 1 982 2 843 3 838 3 042 1 78 2 ) 6 1154 )( 78 0 ( 8 1188 9 807] 2 [ ] 1 [= + + + = = + + = D D

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30 Step 12: Increment j ; j = 4. Solve for4T ,] 1 [t, and ] 2 [t. 0 ] 3 1042 ][ [ ] 3000 ][ [ 01 ] 1 [ ] 1 [ ] 2 [= + Š Š + T t t t 0 ] 4 944 ][ [ ] 3000 ][ [ 10001 ] 2 [ ] 2 [ ] 3 [= + Š Š + T t t r Solution: 2T = 2.78, ] 1 [t= 0, and ] 2 [t= 0.96. < ŠŠ1j jT T Convergence occurs. *T=4T= 2.78, ] 1 [t= 0, and ] 2 [t= 0.96. At this point, the solution is accepted up to[]3r. When the time reaches[]3r, the algorithm is run again to find the production periods for the next cycle. Recalling property 2, in this example, when solving equation set (3-4), it is found that [][]3 3r t <. As a result, solving equation 3-4a alone results in *T= 2.79 which is larger than *T= 2.69 when solving equation set (3-4). 3.3 Experimental Results Following the experimental design and the data sets in [31], Algorithms A1B and RKM are compared. Figure 3.1 illustrates the experimental design. Three control parameters are used to generate different data sets for the experiment. These parameters are: the number of families n, the amount of the average initial inventory Iavg, and the demand variability v.

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31 Figure 3.2 The Design for the Experiment In Algorithm A1B, the number of families n determines the number of setups during each production cycle. For instance, if the number of families is n, the number of setups will be either n or n-1 during each production cycle. Three levels are used in the experimental data sets for the number of families: three, six and nine. The level of the average initial inventory is a factor that constrains the number of setups during the planning horizon. The average initial inventory (Iavg) determines the run-out of inventory time for each family and affects the length of the production cycle since at the end of the cycle time all the demands during that cycle must be satisfied. If the initial inventory is high, the production time available for each family will be longer. Three total initial inventory levels are used: one, two and three periods of supply. Two levels of demand variability are used: low variability and high variability. A uniform distribution is used to generate the demands for each family for a 12 period planning horizon which is assumed to reoccur every 12 periods. The average demand which is the mean of the distribution is 1000 units. The distribution limits are as follows: n v I avg 9 6 3 0.5 0.2 1 2 3

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32 D v D D D v D D + = Š =max min For low variability v is set to 0.2, and for high variability is set to 0.5. The production rate is defined as a function of the number of families and the average demand. It is set as 3000, 6000 and 9000 for three, six and nine families. 3.3.1 Comparison of Algorithms A1B and RKM Algorithm A1B is compared with Algorithm RKM since both algorithms allow backorder. The comparison results of the number of setups are summarized in Tables 3.2, 3.3, and 3.4. The data sets are grouped in the tables according to the number of families, and then each table is divided according to the variability and the initial inventory levels. The results show that the number of setups using Algorithm A1B is less than the number of setups using Algorithm RKM for all data sets. Note that, as the number of families increases the difference between the numbers of setups for the two algorithms decreases. Table 3.2, 3.3, and 3.4 show that the average number of setups using Algorithm A1B is between 37.3-59.6% for n = 3, 18.4-33.7% for n = 6, and 16.1-31.7% for n = 9 fewer than the average number of setups using Algorithm RKM. However, the average initial inventory and the demand variability do not affect the difference between the numbers of setups between the two algorithms.

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33 Table 3.2 Paired Comparisons of Number of Setups: n=3 Iavg=1000 units/product Inv. Dist. 2000, 1000, 0 Iavg=2000 units/product Inv. Dist. 4000, 2000, 0 Iavg=3000 units/product Inv. Dist. 6000, 3000, 0 v=.2 RKM A1B RKM A1B RKM A1B 23 NA* 12 7 11 5 22 12 12 6 11 4 21 13 12 7 11 5 17 12 12 6 12 5 20 12 12 6 10 4 18 12 12 6 12 4 v=.5 RKM A1B RKM A1B RKM A1B 25 NA* 11 7 11 5 20 NA* 12 6 11 4 20 NA* 12 NA* 11 NA* 19 11 12 NA* 12 NA* 17 NA* 12 NA* 10 4 19 NA* 12 6 12 NA* *Non-convergence occurs. Will be discussed in chapter 4. Table 3.3 Paired Comparisons of Number of Setups: n=6 Iavg=1000 units/product Inv. Dist. 2000, 1000, 0 Iavg=2000 units/product Inv. Dist. 4000, 2000, 0 Iavg=3000 units/product Inv. Dist. 6000, 3000, 0 v=.2 RKM A1B RKM A1B RKM A1B 36 33 24 17 12 11 39 31 23 15 14 10 39 32 23 16 13 10 38 28 25 15 12 10 36 30 24 15 14 10 39 31 23 16 12 11 v=.5 RKM A1B RKM A1B RKM A1B 48 NA* 23 17 16 11 43 32 24 14 17 9 38 32 21 16 16 10 37 24 19 14 12 10 33 27 22 14 15 10 42 31 21 16 16 11 *Non-convergence occurs. Will be discussed in chapter 4.

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34 Table 3.4 Paired Comparisons of Number of Setups: n=9 Iavg=1000 units/product Inv. Dist. 2000, 1000, 0 Iavg=2000 units/product Inv. Dist. 4000, 2000, 0 Iavg=3000 units/product Inv. Dist. 6000, 3000, 0 v=.2 RKM A1B RKM A1B RKM A1B 59 50 38 25 29 17 54 47 35 24 25 15 56 50 35 25 22 16 58 45 35 24 23 16 60 46 31 24 19 16 58 51 43 25 24 17 v=.5 RKM A1B RKM A1B RKM A1B 57 52 35 25 25 18 58 46 31 20 27 14 62 48 33 25 20 16 51 43 30 23 22 16 57 43 32 23 23 15 66 NA* 41 27 24 17 *Non-convergence occurs. Will be discussed in chapter 4. A paired-sample t-test [21] is performed in the differences between means,d, for paired trials. The test considers the hypothesis: 0 : 0 :1 0 =d dH H The test statistics is: n S d td/0= Where: ) 1 ( ) ( 11B A x RKM x d d n dj j j n j jŠ = == ) ( RKM xj = number of setups on trial j for Algorithm RKM

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35) 1 ( B A xj = number of setups on trial j for Algorithm A1B 1 ) 1 (2 1 1 2Š Š = = =m d n d Sn j j n j j d For each trial, a sample size of m = 3, 5 or 6 paired comparisons is run. The test statistics for two-tail test with 2, 4 and 5 degree of freedom and a 95% confidence interval are 4.303, 2.776 and 2.571 respectively. The results of the statistical test are shown in Table 3.5 and Table 3.6. All comparisons are statistically significant above the 95% level of confidence. Therefore, it is concluded that using Algorithm A1B the number of setups is less than the number of setups using Algorithm RKM. Table 3.5 Analysis of Differences in Number of Setups: v = 0.2 Iavg = 1000 n = 3 n = 6 n = 9 Avg. (RKM) 20.17 37.83 57.50 Avg. (A1B) 12.20 30.83 48.17 d 7.40 7.00 9.33 Sd 1.95 2.37 3.39 t0 9.30* 7.25* 6.75* Iavg = 2000 Avg. (RKM) 12.00 23.67 36.17 Avg. (A1B) 6.33 15.67 24.50 d 5.67 8.00 11.67 Sd 0.52 1.26 3.67 t0 26.88* 15.49* 7.79* Iavg = 3000 Avg. (RKM) 11.17 12.83 23.67 Avg. (A1B) 4.50 10.33 16.17 d 6.67 2.50 7.50 Sd 0.82 1.38 3.15 T0 20.00* 4.44* 5.84* Significant above 0.95 CI.

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36 Table 3.6 Analysis of Differences in Number of Setups v = 0.5 Iavg = 1000 n = 3 n = 6 n = 9 Avg. (RKM) 20.00 40.17 58.50 Avg. (A1B) 11.00 29.20 46.40 d 8.00 9.40 10.60 Sd NA 3.21 3.97 T0 NA 7.17* 6.53* Iavg = 2000 Avg. (RKM) 11.83 21.67 33.67 Avg. (A1B) 6.33 15.17 23.83 d 5.33 6.50 9.83 Sd 1.15 2.07 2.48 T0 11.31* 7.68* 9.70* Iavg = 3000 Avg. (RKM) 11.17 15.33 23.50 Avg. (A1B) 4.33 10.17 16.00 d 6.33 5.17 7.50 Sd 0.58 1.94 3.02 T0 26.87* 6.52* 6.09* Significant above 0.95 CI. 3.3.2 Comparison of Algorithms A1 and A1B (Compression Effect) Algorithm A1 does not allow backorder. As a result, the case of having a high variability and a low initial inventory leads to a “compression effect” [31] where multiple families reach their run-out of inventory times at approximately the same time. The compression effect forces Algorithm A1 to setup families more frequently to prevent them from going to backorder. As a result, the number of setups under Algorithm A1 increases and, in some cases, Algorithm RKM results in less number of setups when the compression effect is present. On the other hand, Algorithm A1B shows a significant difference in the number of setups from Algorithm RKM. Table 3.7 shows comparison of the number of setups for the data sets where compression effect is present. A pairedsample t-test is performed for the difference in means of the number of setups under A1

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37 and A1B. Table 3.8 summarizes the statistical analysis results which show significant differences between the means above the 95% level of confidence. According to the results, the average number of setups using Algorithm A1B is between 16.5-17% less than the average number of setups using Algorithm A1. Table 3.7 Comparisons of Number of Setups: Compression Effect Data Sets Iavg=1000 units/product Inv. Dist. 2000, 1000, 0 v =.5 N = 6 n = 9 RKM A1 A1B RKM A1 A1B 39 43 32 57 66 52 36 38 32 58 53 46 29 37 24 62 61 48 30 33 27 51 49 43 42 42 31 57 49 43 Table 3.8 Analysis of Differences in Number of Setups v = 0.5 Iavg = 1000 n = 6 n = 9 Avg. (A1) 35.20 55.60 Avg. (A1B) 29.20 46.40 d 6.00 9.20 Sd 3.16 3.96 T0 4.65* 5.69* Significant above 0.95 CI. To fully illustrate the effectiveness of Algorithm A1B in reducing compression effect period, inventory versus time graphs are shown in Figures 3.2 and 3.3 for Algorithms A1 and A1B for a family with initial inventory of 1200 units and n = 6. The inventory increase in the figures represents the duration when this family is in production. At time 2.9 and 5.9 in Figure 3.4, there are two short time production periods which result in two setups. These two setups disappeared in Figure 3.5 by allowing the family to

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38 go to backorder. It is concluded that allowing backorders using Algorithm A1B resolves the compression effect problem presented in Algorithm A1. Figure 3.3 Inventory Versus Time Under Algorithm A1 Figure 3.4 Inventory Versus Time Under Algorithm A1B Inventory vs Time-500 0 500 1000 1500 2000 2500 0.00.92.13.04.14.95.96.98.410.111.6 Time(period)Inventory (unit) A1B Inventory vs Time-500 0 500 1000 1500 2000 2500 0.01.32.63.84.85.96.77.78.810.211.6 Time(period)Inventory (unit) A1

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39Chapter 4. Convergence Conditions and Computational Issues This chapter discusses the convergence conditions and the computational issues of the continuous disaggregation algorithm allowing backorder (Algorithm A1B) presented in Chapter 3. It is divided into 6 sections. Section 4.1 presents an introduction to the convergence issue for Algorithm A1B. Section 4.2 presents a derivation for the general convergence expression for Algorithm A1B. Section 4.3 states special cases for convergence with an example. Section 4.4 compares the convergence conditions for Algorithm A1 and Algorithm A1B. In section 4.5, a modification to Algorithm A1B is proposed to partially resolve the convergence problem. Section 4.6 compares the computational requirements for Algorithm A1 and Algorithm A1B. 4.1 Introduction In Chapter 3, some data sets in the experimental design do not converge to an optimal cycle time *T For these cases the number of setups is indicated as not available (NA) in Tables 3.2, 3.3 and 3.4. The non-convergence occurs when Algorithm A1B alternates between step 6 and 7 and never proceed to step 8. Recall step 6 and 7 from Algorithm A1B:

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40 Step 6: Using solution values, compute, for all families in F: dt P t t Pi it t t i i i i+Š =+] 1 [ [] ], [ ] [ ] 1 [ ] [1 ++ =j iT t t i j i idt D T t D] [0 ], [ ] [ ] [1 Step 7: Increment j Solve the following equation set: ) 1 ...( 2 1 0 ] [ ] [] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = = + Š Š ++n i D T t P t t Ii i i i i i n i D T t P t T Ii i i i i= = + Š Š + 0 ] [ ] [] [ ] [ ] [ ] [ 0 ], [ If ŠŠ 1 j jT T go to step 6. Note that in step 7, the algorithm solves for cycle time jT using the average production rate P and the average demand D values found in step 6. Then it compares the new cycle time jT and the previous cycle time 1 Š jT using the condition ŠŠ 1 j jT T, where is the convergence tolerance level. If the difference between the two cycle times is within the tolerance level, < ŠŠ 1 j jT T the algorithm proceeds to step 8. If the difference is not within the tolerance level, ŠŠ 1 j jT T, the algorithm goes back to step 6 to find new P and D values using the new cycle time. Then it proceeds to step 7 to start a new iteration. These steps are illustrated in Figure 4.1.

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41 Compute P(i)and D(i) |Tj– Tj-1| j = j + 1 Solve equation set (3-4) for Tj Yes No Step 6 Step 7 Step 8 Figure 4.1 Steps 6 and 7 from Algorithm A1B When the algorithm is converging to the optimal cycle time *T, the absolute value of the difference between two successive cycle times, 1 ŠŠ = j j jT T T becomes smaller and smaller until the difference is within the tolerance level However; for some cases Algorithm A1B alternates between two or more cycle times and never reaches the optimal cycle time *T These cases cause jT to remain greater than which results in a non-convergence condition. For Algorithm A1B to converge to an optimal cycle time *T 1 +jT for iteration j +1 must be less than jT for iteration j and the condition 11< + j jT T must be satisfied.

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42 In the following sections a general expression for convergence conditions for Algorithm A1B is obtained and the convergence conditions are further explored. 4.2 General Convergence Expression The first step to obtain a general expression for the convergence conditions is to start with the two family case. Equation set (3-4) for 2 families is written as follows: 0 ] 0 [ ] 0 [] 1 [ ] 2 [ ] 1 [= + Š Š + D T P t I (4-1a) 0 ] [ ] [] 2 [ ] 2 [ ] 2 [ ] 2 [= + Š Š + D T t P t T I (4-1b) For the jth iteration in step 7 in Algorithm A1B, equation set (4-1) can be rewritten as follows: 0 ] ][ [ ] [1 ], 1 [ ] 1 [ ], 2 [ ] 2 [ ] 1 [= + + Š + +Š j j jD D T T P t t I (4-2a) 0 ] ][ [ ] [1 ], 2 [ ] 2 [ ], 2 [ ] 2 [ ], 2 [ ] 2 [ ] 2 [= + + + + Š Š Š + +Š j j j j jD D T T t t P t t T T I (4-2b) where jT jt], 2 [, 1 ], 1 [ ŠjD and 1 ], 2 [ ŠjD are deviations from the optimal values at iteration j 1 ], 1 [ ŠjD and 1 ], 2 [ ŠjD lag jT by one iteration since the algorithm uses 1 Š jT to estimate 1 ], 1 [ ŠjD and 1 ], 2 [ ŠjD which are used to compute jT in the following iteration. From equation (4-2a): P D D T D T D T P t I tj j j j] [1 ], 1 [ ] 1 [ 1 ], 1 [ * ] 1 [ ] 1 [ ] 2 [ ] 1 [ ], 2 [ Š Š + + + + Š Š = (4-3) From equation (4-1a) it is known that

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43 0* ] 1 [ * ] 2 [ ] 1 [= Š + D T P t I (4-4) Using equation (4-4), equation (4-3) reduces to: P D D T D T tj j j j] [1 ], 1 [ ] 1 [ 1 ], 1 [ ], 2 [ Š Š + + = (4-5) Similarly using equations (4-1b) and (4-2b): 1 ], 2 [ ] 2 [ 1 ], 2 [ 2 1 ], 2 [ 1 ], 2 [ * ] 2 [ ], 2 [ Š Š Š Š Š Š Š Š Š Š = j j j j j j j j jD D P D t D T D T D T P T t (4-6) jt], 2 [ can be eliminated by equating the expressions in equations (4-5) and (4-6). After rearranging and simplifying terms, an equation for jT is obtained as follows: ] ][ [ ] [ ] [ ] [], 2 [ ] 2 [ ], 1 [ ] 1 [ ], 2 [ ] 2 [ 2 ], 2 [ ], 2 [ ] 2 [ ], 2 [ 1 j j j j j j j jD D P D D P D D P t T D P D D P D T T + + + Š Š Š + + + + = + (4-7) Recall that the value of jD], 1 [ and jD], 2 [ are estimated from jT of the previous iteration, j Also, j i i j iD D D], [ ] [ ], [ + = For any iteration, the following relationship holds: ), (* ] [ ], [T T m D Dj i i j iŠ + = (4-8) where im is the slope of the line connecting the two points ) (], [ j j iT D and ) (* ] [T Di as shown in Figure 4.1 which depicts a typical relationship between T and ] [ iDfor two successive iterations.

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44 Figure 4.2 Typical Relationship Between Tand ] [ iDfor Two Successive Iterations Equation (4.8) can be rewritten as: i T m Dj i j i = ,], [ (4-9) Substituting equation (4-9) into equation (4-7), the following expression is obtained relating the values of T for two successive iterations in step 7: ] ][ [ ] [ ] [ ] [2 ] 2 [ 1 ] 1 [ 2 ] 2 [ 2 2 2 ] 2 [ 1 1 j j j j j j jT m D P T m D P T m D P t T m P T m D P T m T T + + + Š Š Š + + + + = + (4-10) Expression (4-10) is arranged to obtain the following expression: ) )( ( 2 ] [ ) (2 ] 2 [ 1 ] 1 [ 2 2 2 2 ] 2 [ 1 1 j j j j j jT m D P T m D P P P t T m T m D P T m T T + + + + Š + + + + = + (4-11) The second step to obtain a general expression for the convergence conditions is to follow the same approach for the two family case to find the expressions for three and four family cases which are presented in equation (4-12) and (4-13). ] [ i D T T jT ] [ iD j iD], [ Average Demand vs. Cycle Time 1 + jT 1 ], [ + j iD i T D mj j i i = ,], [ j i D ], [ jT

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45n = 3 ) )( )( ( 2 ] [ ) ( ] [ ) )( (3 ] 3 [ 2 ] 2 [ 1 ] 1 [ 3 2 3 3 3 ] 3 [ 2 2 3 ] 3 [ 2 ] 2 [ 1 1 j j j j j j j j j jT m D P T m D P T m D P P P t T m T m D P P t T m T m D P T m D P T m T T + + + + + + Š + + + + + + + + + + = + (4-12) n = 4 + + + + + + + + Š + + + + + + + + + + + + + + + + + + = +) )( ( ) )( ( 2 ] [ ) ( ] [ ) )( ( ] [ ) )( )( (4 ] 4 [ 3 ] 3 [ 2 ] 2 [ 1 ] 1 [ 4 3 4 4 4 ] 4 [ 2 3 3 4 ] 4 [ 3 ] 3 [ 2 2 4 ] 4 [ 3 ] 3 [ 2 ] 2 [ 1 1 j j j j j j j j j j j j j j jT m D P T m D P T m D P T m D P P P t T m T m D P P t T m T m D P T m D P P t T m T m D P T m D P T m D P T m T T(4-13) From equations (4-11), (4-12), and (4-13), the general expression for k families is: = Š = Š + = Š + + + Š + + + + + = k i j i i k k i k j k k k i l j l l i j i i j jT m D P P P t T m T m D P P t T m T T1 ] [ 1 1 1 * 1 ] [ 1 * 1) ( 2 ] [ ) ( ] [ (4-14) As indicated in Section 4.1, the general expression for n = k families will converge to an optimal cycle time *Twhen: = Š = Š + = Š + + + Š + + + + + = k i j i i k k i k j k k k i l j l l i j i i j jT m D P P P t T m T m D P P t T m T T1 ] [ 1 1 1 * 1 ] [ 1 * 1) ( 2 ] [ ) ( ] [ <1 (4-15)

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464.3 Convergence Example and Discussion The general expression (4-15) is evaluated using the example problem in Section 2 of Chapter 3. Using the results from the example at the first iteration; i.e., j = 1 and at the optimal cycle time *T the general expression is evaluated as follows: *T = 2.69, ] 1 [D= 1038.7, ] 2 [D= 936.88, ] 3 [D= 1104.92, ] 1 [ *t = 0, ] 2 [ *t = 0.93, ] 3 [ *t = 1.73, P = 3000. 1T = 3.472.69 = 0.78 41 11 78 0 7 1038 6 10471= Š = m, 62 43 78 0 88 936 9 9702= Š = m 15 25 78 0 92 1104 3 10853Š = Š = m In this example n = 3, following the general expression: 1T m = (11.41) (2.69) = 30.69 ] [* 2 2jt T m + = (43.62) (2.69 + 0.93) = 157.90 ] [* 3 3jt T m + = (-25.15) (2.69 + 1.73) = -111.16 ) (1 ] 1 [jT m D P + + = 3000 + 1038.70 + (11.41) (0.78) = 4047.60 ) (2 ] 2 [jT m D P + + = 3000 + 936.88 + (43.62) (0.78) = 3970.90 ) (3 ] 3 [jT m D P + + = 3000 + 1104.92 + (-25.15) (0.78) = 4085.30

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47 The convergence condition: 1 ) )( )( ( 2 ] [ ) ( ] [ ) )( (3 ] 3 [ 2 ] 2 [ 1 ] 1 [ 3 2 3 3 3 ] 3 [ 2 2 3 ] 3 [ 2 ] 2 [ 1 1< + + + + + + Š + + + + + + + + + + = + j j j j j j j j j jT m D P T m D P T m D P P P t T m T m D P P t T m T m D P T m D P T m T T ) 30 4085 )( 90 3970 )( 60 4047 ( ) 3000 ( 2 ) 3000 )( 16 111 ( ) 30 4085 )( 3000 )( 90 157 ( ) 30 4085 )( 90 3970 )( 69 30 (3 2 1 2Š Š + + = T T<1 1 123 0 6 1166145340 14326295421 2< Š = Š = T T Note that the condition for the convergence is satisfied. Furthermore, 2T = 01 0 ) 123 0 ( ) 78 0 ( ) 123 0 (1Š = Š = Š T This result agrees with the result from the example problem in Chapter 3, where 2T = 2T *T = 2.59 – 2.69 = -0.01. The general expression (4-15) converges when the absolute value of the numerator is less than the absolute value of the denominator. = Š = Š + = Š + + Š < + + + + +k i j i i k k i k j k k k i l j l l i j i iT m D P P P t T m T m D P P t T m1 ] [ 1 1 1 * 1 ] [ 1 *) ( 2 ] [ ) ( ] [ In the above expression, P and ] [ iD are always positive values. j iT m equal j iD], [ for all i which is a small value compared to P and ] [ iD. This concludes that the

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48 term ) (* ] [ j i iT m D P + + is always positive. im can be positive or negative and relatively large or small depending on the demand’s variation. Recall that i T D mj j i i = ,], [. The absolute value of the numerator depends on im as it is the only term that can have a negative value in the numerator. Also, when some of the im s have negative values and the others have positive values the numerator tends to be smaller. Recall that im depends on the differences between the demands averages over the iterations which depend on the demand variability. It is concluded that the numerator decreases when the demand variability is decreasing, and increases when the demand variability is increasing. In the experiments in Chapter 3, two levels of demand variability are considered v = 0.2 and v = 0.5. In the experimental results, more data sets do not converge to an optimal cycle time when v = 0.5 due to the higher demand variability. In the denominator, the production rate, P is assumed to equal the summation of the demands’ averages = k i iD1 ] [ in each period, which without loss of generality, is equal the number of the families multiplied by the average demand for a single family ( ] [ 1 ] [ i k i iD k D P =). Notice that as k increases the negative part of the denominator ( = + +k i j i iT m D P1 ] [) () becomes larger than the positive part ( k P 2) and the whole denominator increases. In the experiments in Chapter 3, three levels for the number of families are used n = 3, n = 6, and n = 9. In the experimental results, more data sets converge to an optimal cycle time when the number of families increases.

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494.4 Comparison of Algorithms A1 and A1B Convergence Expressions Yalcin and Boucher [31] provide a general convergence expression for the A1 formulation for k families as follows: () 2 1 ] 1 [ 1 ] 1 [ 2 1 1 1 ] 1 [ 1) )( 2 ( ] ] [ ) )( 2 [(Š Š Š Š Š Š Š Š Š Š + + + + + Š + + + + Š Š = k j k k j k k k j k k j k k j jP T m D T m D P families k for nator denomi P t T m T m D P families k for numerator T T(4-16) When k = 3 families in equation (4-16), ) 2 ( ) 2 ( families k for nator denomi families k for numerator Š Š forms the seed for the expression and equals jT m D t T m + +1 ] 1 [ 1 1] [ In order to get a general expression similar to the A1B formulation general convergence expression (4-14), the seed is substituted and further rearrangement is made to obtain the following general expression: ( ) ( ) Š = Š Š Š Š + = Š Š = Š Š Š Š + = Š + + + + + + Š + + + + + = 2 1 2 1 ] 1 [ 1 1 ] [ 1 ] [ 2 1 2 1 1 1 1 ] [ 1 * 1) ( ] [ ) ( ] [k i k j k k k i l j l l i j i i k i k j k k k i l j l l i j i i j jP T m D T m D P P T m D P t T m T m D P P t T m T T (4-17) The two general convergence expressions (4-14) and (4-17) have the same numerator patterns. However; the general convergence expression in equation (4-17) for Algorithm A1 does not contain the last family term as the summation is up to k -2 families only unlike the general convergence expression in equation (4-14) where it is up to k -1 families. Moreover; P is raised to the power of k -2 in equation (4-17) while it is k -1 in equation (4-14). These two factors cause the numerator of equation (4-17) to be smaller than the numerator of equation (4-14) for the same data set. The denominator in equation (4-17) for Algorithm A1 contains a positive and a negative term. However, the denominator in equation (4-14) for Algorithm A1 contain

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50 only a negative term. This difference causes the denominator in equation (4-14) to have a greater absolute value than the numerator. Since the general convergence expression for Algorithm A1B has a larger numerator and a smaller denominator compared to the general convergence expression for Algorithm A1 for the same data set, it is concluded that Algorithm A1B is less “convergent” compared to Algorithm A1 which explains the presence of data sets that converge using Algorithm A1 but not A1B. 4.5 A Modification to Algorithm A1B and Comparison to RKM As indicated in section 4.1 the non-convergence occurs between step 6 and 7 of Algorithm A1B. A proposed modification is to evaluate an extra condition ( 11 Š j jT T ) for non-convergence in step 7 after solving equation set (3-4). If the condition does not hold then the algorithm is converging to an optimal cycle time *T However, if it holds, then the algorithm is not going to converge to an optimal cycle time *T by solving equation set (3-4). Therefore, the algorithm proceeds and solves equation set (3-4a). Note that 1 ŠŠ = j j jT T T, 2 1 1 Š Š ŠŠ = j j jT T T and T0 is an estimation so at least three iteration are needed to evaluate the condition 11 Š j jT T The modified step 7 is as follows:

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51 Step 7: Increment j Solve equation set 3-4. If j 3 then [If 11 Š j jT T go to step 9]. If Š 1 j jT T, go to step 6 The modification is applied to algorithm A1B to populate the missing data in Table 3.2 where v = .5 as shown in Table 4.1. Following that the missing data in Table 3.6 is filled as shown in Table 4.2. All the comparisons in Table 4.2 are still statistically significant above the 95% level of confidence. The results show that the number of setups using the modified Algorithm A1B when there is a non-convergence data set is less than the number of setups using Algorithm RKM. Table 4.1 Paired Comparisons of Number of Setups: n = 3 Iavg=1000 units/product Inv. Dist. 2000, 1000, 0 Iavg=2000 units/product Inv. Dist. 4000, 2000, 0 Iavg=3000 units/product Inv. Dist. 6000, 3000, 0 v =.5 RKM A1B RKM A1B RKM A1B 25 16 11 7 11 5 20 12 12 6 11 4 20 14 12 7 11 5 19 11 12 6 12 5 17 10 12 6 10 4 19 12 12 6 12 5

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52 Table 4.2 Analysis of Differences in Number of Setups v = 0.5 Iavg = 1000 n = 3 Avg. (RKM) 20.00 Avg. (A1B) 12.50 d 7.50 Sd 1.05 T0 17.52* Iavg = 2000 n = 3 Avg. (RKM) 11.83 Avg. (A1B) 6.33 d 5.5 Sd 0.84 T0 16.10* Iavg = 3000 n = 3 Avg. (RKM) 11.17 Avg. (A1B) 4.67 d 6.5 Sd 0.55 T0 29.07* Significant above 0.95 CI. 4.6 Computational Experience with Algorithm A1B Experience with Algorithm A1B indicates a very rapid convergence. It also requires minimal CPU time. For all the trials presented in this thesis as well as some larger test problems, including 18 families, the CPU time to solve a cycle was always less than a second. Computations were performed on a Pentium IV, 3.2 GHz processor with 512 MB of RAM. Since this algorithm is intended to for use in off-line scheduling, the computation time is more than satisfactory. Some computational parameters for Algorithm A1B are compared with Algorithm A1 to evaluate the computational performance of Algorithm A1B. Typical sets of results are shown in Table 4.3 and Table 4.4 which examine different numbers of families n for cases where the variability of demand v = 0.5, and the initial average inventory Iavg =

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53 2000. For each number of families six trials are examined. The number of cycles is the summation of the cycles for all the trials. Table 4.3 Computational Experiment for Algorithm A1B, v = 0.5, Iavg = 2000 Number of families n 3 6 9 18 Number of trials 6 6 6 6 Number of cycles 18 18 19 19 Number of iterations/cycle Average 13.17 9.06 7.58 6.42 Maximum 78 46 21 10 Table 4.4 Computational Experiment for Algorithm A1, v = 0.5, Iavg = 2000 Number of families n 3 6 9 18 Number of trials 6 6 6 6 Number of cycles 26 32 30 32 Number of iterations/cycle Average 4.19 3.78 3.63 3.59 Maximum 6 5 5 4 Note that in Table 4.3, as the number of families increases the average number of iterations decreases. Recall that in Section 4.3, it is concluded that as the number of families increases Algorithm A1B requires fewer iterations to converge to the optimal cycle time. In Algorithm A1B, the average number of cycles for each trial is 3 compared to 5 in Algorithm A1. It is because in each cycle in Algorithm A1B n or n -1 families are set for production as stated in property 2 for A1B formulation. However, in each cycle in Algorithm A1 n -1 families or less are set for production following step 9 in the algorithm where all the families that has ] [ ] [ i ir t > are deleted from the production set F. Step 9 is included below for convenience:

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54 Step 9: Set k = the largest number i in F for which ] [ ] [ i ir t > is true. Delete from F all families wherek i >. Go to Step 1. As indicated in Section 4.4, the general expression for convergence of Algorithm A1 has a smaller absolute value than the general expression for convergence of Algorithm A1B for the same data set. This explains why the average number of iterations per cycle in Algorithm A1B is larger than Algorithm A1 as Algorithm A1 needs fewer iterations to converge to the optimal cycle time.

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55Chapter 5. Continuous Time Disaggregation and Demand Uncertainty This chapter considers the demand uncertainty in the continuous time disaggregation allowing backorder with experimental results. It is divided into 2 sections. Section 5.1 incorporate demand uncertainty into Algorithm A1B, and compares the performance of Algorithm A1B in the presence of demand uncertainty and Algorithm A1B with deterministic demand. Section 5.2 incorporates a desired service level parameter into the problem formulation and Algorithm A1B. 5.1 Algorithm A1B in the Presence of Demand Uncertainty Allowing backorders in HPP’s disaggregation step allows incorporating demand uncertainty into the problem formulation and solution. Demand uncertainty is incorporated into the problem assuming that the actual demands are known at the beginning of each period. Demand for each family is assumed to be an independent random variable which follows a specific distribution and only the information about demand’s mean and variation is known for the remainder of the planning horizon. Since it is assumed that actual demands become available at the beginning of each period, the time when Algorithm A1B is set to run again is an important consideration. The policy proposed here is to check if there is any new actual demand information before changing over between the families. If there is new information then the algorithm is run again before a change-over is performed and the next family starts its production.

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56 This policy keeps the algorithm updated with the new demand information without interfering with its continuous behavior. Another policy is set the algorithm to run again whenever there is new demand information. Since it is assumed that the actual demands become available at the beginning of each period, this policy sets the algorithm to run at the beginning of each period converting the algorithm from a continuous time to a discrete time algorithm. The experimental design in Chapter 3 is used again to evaluate the effectiveness of using Algorithm A1B in the presence of demand uncertainty. In Chapter 3, three control parameters are used to generate different data sets for the experiment. These parameters are: the number of families n with three levels: three, six and nine, the amount of the average initial inventory Iavg with three levels: one, two and three periods of supply, and the demand variability v with two levels: high variability (0.5) and low variability (0.2). The algorithm is compared with Algorithm A1B with deterministic demand based on the percentage of the time of each family is out-of-stock during the entire planning horizon as shown in Table 5.1. Table 5.1 Paired Comparison of Out-of-stock Percentage n = 3 n = 6 n = 9 v 0.2 0.5 0.2 0.5 0.2 0.5 Iavg = 1000 A1B deterministic 1.97% 4.06% 2.55% 5.80% 2.48% 5.16% A1B uncertain demand 3.00% 5. 22% 4.39% 7.85% 3.89% 6.79% Iavg = 2000 A1B deterministic 1.38% 3.04% 1.41% 3.62% 1.34% 2.74% A1B uncertain demand 2.86% 6. 26% 2.79% 5.16% 2.25% 4.31% Iavg = 3000 A1B deterministic 1.12% 1.99% 1.44% 3.03% 1.40% 2.88% A1B uncertain demand 1.87% 4. 52% 1.92% 4.43% 1.64% 3.28%

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57 The results show that the averages of percentage of time each family is out-ofstock during the entire planning horizon in the presence of demand uncertainty are greater than those with deterministic demands patterns for all data sets. A one-tail test statistic is constructed to check the significant of the results of Table 5.1. The test statistic for one-tail test with 17 degree of freedom and a 99% confidence interval is 2.567. The test results show that the comparisons in Table 5.1 are statistically significant above the 99% confidence interval as shown in Table 5.2. These results are expected as the demand uncertainty tends to increase the time the family is out-of-stock. Furthermore, it confirms that the incomplete knowledge of the demand decrease the service level associated with the families, highlighting the importance of the decision associated with when Algorithm A1B should be re-run. Table 5.2 Analysis of Differences in Out-of-stock Percentage Avg. (A1B deterministic demand) 2.63% Avg. (A1B uncertain demand) 4.02% d 1.39% Sd 0.01 T0 7.95 Significant above 0.99 CI. As stated in Chapter 3, two levels of demand variability are used in the experimental design: low variability v = 0.2 and high variability v = 0.5. Based on the data in Table 5.1, a one-tail test statistics is constructed to check the effect of the demand variability on the percentage of the time of each family being out-of-stock during the entire planning horizon. The test statistic for one-tail test with 8 degree of freedom and a 99% confidence interval is 2.896. Table 5.3 shows the results of the one-tail test which states that, as the demand variability increases, the difference between the percentage of out-of-stock time during the entire planning horizon between Algorithm A1B in the

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58 presence of demand uncertainty and Algorithm A1B with deterministic demand increases. This result shows that families with high actual demand variability have poorer service level than families with low demand variability. Table 5.3 Analysis of Differences Based on Demand Variability Avg. ( v = 0.2) 1.06% Avg. ( v = 0.5) 1.72% d 0.66% Sd 0.01 T0 2.93* Significant above 0.99 CI. 5.2 Incorporating a Service Level Parameter in the Problem Formulation In Section 5.1, the experimental results show that demand uncertainty decreases the service levels associated with the families and show that families with higher actual demand variability has poorer service level. In this section, a service level parameter associated with the production characteristic and the importance of satisfying the demand of each family is introduced. The service level parameter is incorporated in the problem formulation as follows: T Max (5-1a) s.t. [] ) 1 ...( 2 1 0 1 ] [ ] [] [ ] [ ] [ ] [ ] [ ] 1 [ 0 ], [Š = + + Š Š ++n i D Z T t P t t Ii i i i i i i, (5-1b) [] n i D Z T t P t T Ii i i i i i= + + Š Š + 0 1 ] [ ] [] [ ] [ ] [ ] [ ] [ 0 ], [, (5-1c) where] [ iZ is a constant and equal to ] [ ] [ i iv c where: ] [ ic is the coefficient of variation of the ith product family in the production cycle,

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59 and ] [ iv is a service level parameter which can have a positive or negative value determined by the production planner. Notice that the term [] [ ] iZ + 1 is multiplied by ] [ iD of each family in the formulation constraints. Also notice that a positive ] [ iZ increases the demand part of the constraint which causes the formulation solution to allocate a higher amount of production time for the family. The term[] [ ] iZ + 1 is accommodated in Algorithm A1B when computing the average demand as follows: ++ + =j iT t t i i j i idt D Z T t D] [0 ], [ ] [ ] [ ] [] 1 [ 1 (5-3) The data sets for 6 families from the experimental design of Chapter 3 are run to evaluate the effect of incorporating the service level parameter. The results are compared bases on the percentage of the time that each family is out-of-stock, the maximum inventory, and the average inventory. In the experiment, three levels of Z are used: -0.2, 0, and 0.2. The experimental results are summarized in Table 5.4. Table 5.4 Comparison of Different Service Levels n = 6 Percentage of Out-of-stock time* Imax* Iavg* Iavg Z v -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 0.2 14.23% 7.77%2.07%1,7962,144 2,586691 912 1,228 1000 0.5 13.01% 8.28%3.78%1,8822,158 2,667699 867 1,133 0.2 11.26% 7.48%0.71%3,4274,101 5,1631,451 1,7822,670 2000 0.5 9.27% 8.95%1.81%3,4704,094 5,2591,486 1,7052,593 0.2 6.25% 6.92%1.23%4,9105,996 7,6602,224 2,4684,216 3000 0.5 7.12% 8.63%1.86%4,9865,881 7,6292,281 2,4824,054 Average of 6 trials

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60 Table 5.4 shows that increasing Z[i] associated with family i decreases the percentage of the time the family is out-of-stock as more production time is allocated to the family. However; allocating more production time also increases the maximum and the average inventory of the family. An interesting observation is that the increase in the initial inventory does not have any significant affect on the percentage of the time the family is out-of-stock although it increases the maximum and average inventory of the families. Also, notice that the demand variability does not have any effect on the experimental results. Figures 5.1, 5.2 and 5.3 show an inventory versus time graphs for two families with different service level parameters. In Figure 5.1, Z = 0 is used for both of the families. In Figure 5.2, Z = -0.2 is used for one family and Z = 0.2 is used for the other one. Then, In Figure 5.3, Z = -0.5 and Z = 0.5 are used. The inventory increase in the graphs portions with positive slope represents the duration when this family is in production. Note that as Z increases the family average and maximum inventory increases. Furthermore; the percentage of time the family is out-of-stock increases when Z decreases. This clearly shown in Figure 5.3 where the family with Z = 0.5 has a maximum inventory of 8000 compared to only 2000 for the family with Z = -0.5. Also, notice that in Figure 5.1, when both of the families have the same service level parameters, they have almost the same amount of time assigned for production during each setup and the same number of setups. In figure 5.2, the time assigned for production during each setup for the family with greater Z is increased while it is decreased for the other family. This is more obvious in Figure 5.3, where the increase in

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61 the time assigned for production during each setup for the family with greater Z reduces a complete setup and the decrease of the time assigned for production during each setup for the other family adds one more setup in order to satisfy the demand. Figure 5.1 Inventory Versus Time Z1 = 0 and Z2 = 0 Figure 5.2 Inventory Versus Time Z1 = -0.2 and Z2 = 0.2 Inventory vs Time-500 0 500 1000 1500 2000 2500 3000 3500 4000 45000.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.3 7.1 7.9 8.7 9.5 10.211.1 12.012.7 Time(period)Inventory (unit) Z1= 0 Z2= 0 Inventory vs Time-2000 -1000 0 1000 2000 3000 4000 5000 6000 0.01.01.82.93.84.75.46.17.07.88.79.3 10.2 11.111.912.9 Time (period)Inventory (unit) Z1 = -0.2 Z2= 0. 2

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62 Figure 5.3 Inventory Versus Time Z1 = -0.5 and Z2 = 0.5 The results in this section illustrate that choosing a certain service level parameter for a family affects the percentage of the time the family is out-of-stock, its maximum and average inventory, and the number of setups during the planning horizon. Furthermore, these results highlight some future research topics which are discussed in the next chapter. Inventory vs Time-400 0 -200 0 0 2000 4000 6000 8000 10000 0.01.62.33.95.06.16.87.58.28.79.210.211.112.4 Time (period)Inventory (unit) Z1 = -0.5 Z2= 0. 5

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63Chapter 6. Conclusions and Future Work This chapter summarizes the results and the conclusions of the thesis and outlines the directions for further research. Section 6.1 summarizes the results and the conclusions derived from this research. Section 6.2 presents the directions for future research in the continuous time disaggregation area. 6.1 Conclusions This thesis extends the continuous-time disaggregation approach for hierarchical production planning introduced by Yalcin and Boucher [31]. To the best of our knowledge, the algorithm presented in this thesis is the only continuous algorithm that allows backorder and considers demand uncertainty. In Chapter 3, a continuous-time disaggregation formulation that allows backorder is provided and an algorithm (Algorithm A1B) based on this formulation is presented. At the end of Chapter 3, a design of experiment is used to show the effectiveness of the Algorithm A1B over the Regular Knapsack Method Algorithm [6] in reducing the number of setups, and to demonstrate how allowing backorder eliminates the compression effect problem presented in [31]. Chapter 4 discusses the convergence conditions and computational requirements associated with Algorithm A1B. A general expression for convergence is obtained. It is

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64 concluded that increasing the number of families and decreasing the demand variability allows Algorithm A1B to converge to an optimal cycle time more readily. A comparison for convergence conditions and other computational requirements of the Algorithm A1B and Algorithm A1 presented in [31] shows that Algorithm A1B has more strict convergence conditions compared Algorithm A1. Finally, a modification to detect and resolve the non-convergence data sets is introduced for Algorithm A1B. Chapter 5 incorporates demand uncertainty into continuous-time disaggregation. The experimental results confirm that uncertainty associated with demand increases the percentage of the time the family is out of stock leading to decrease service levels. A family service level parameter which can be used to capture the production planners’ desired risk-level of being out-of-stock for each family is introduced. Experiments show that service level parameter for a family can be effectively used to decrease the percentage of the time the family is out-of-stock. In the disaggregation process, the measure of efficiency is the minimization of the number of setups during the planning horizon. The results of this thesis show that continuous-time disaggregation allowing backorder approach is effective over discretetime disaggregation approaches in minimizing the number of setups. From industry application perspective, where the setup costs are high, the continuous-time disaggregation can be very useful in minimizing the costs and incorporating demand uncertainty. The experimental results in this thesis were designed using constant production rates and uniform demand distribution for simplicity. However continuous-times

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65 disaggregation can cope with variable production rates and other demand distribution that might exist in industry applications. 6.2 Future Work The presented continuous time disaggregation is considered at the disaggregation done at the family level of HPP. A future research direction can extend this research to consider the disaggregation done at the items levels. A future research can consider a formal proof associated with the detection of non-convergence. Possibly, to check the demand patterns of the data set before starting the disaggregation process. It is concluded that the time when to set the algorithm to run again in the presence of demand uncertainty is an important consideration. A future study can reconsider the time when new demand information is available to study different policies for the time when the algorithm is set to run again. The service level parameter introduced in Chapter 5 could be implemented in a real world application. The families preferred specifications set by the planners such as aimed average inventory, maximum and minimum inventory allowed and percentage of time the family is out-of-stock can be used to derive the service level parameter for each family. Furthermore, as the service level parameter affects the number of setups associated with each family, this research can also be extended to include the situation when different setup costs are associated when changing over between the families.

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66References [1] Askin, R.G. and J.B. Goldberg (2002) Design and analysis of lean production systems Wiley, New York, NY, 130-164. [2] Akturk, M.S. and G.R Wilson (1998) A hierarchical model for the cell loading problem of cellular manufacturing systems. International Journal of Production Research, 36, 2005-2023. [3] Ari, E. A. and S. Axsater (1988) Disaggregation under uncertainty in hierarchical production planning. European Journal of Operational Research 35, 182-186. [4] Axsater, S. (1981) Aggregation of product data for hierarchical production, Operations Research 29, 744-755. [5] Bitran, G. R. and A. C. Hax (1977) On the design of hierarchical production control systems, Decision Science 8, 28-55. [6] Bitran, G. R. and A. C. Hax (1981) Disaggregation and resource allocation using convex knapsack problems with bounded variables, Management Science 27, 431-441. [7] Bitran, G. R., E. A Hass, and A. C. Hax (1981) Hierarchical production planning: a single stage system, Operations Research 29, 717-743. [8] Bitran, G. R., E. A Hass, and A. C. Hax (1982) Hierarchical production planning: a two stage system, Operations Research 30, 232-251. [9] Bitran, G. R., E. A Hass, and H. Matsuo (1986) Production planning of style goods with high setup costs and forecast revisions, Operations Management 34, 226-236. [10] Ciarallo, F.W., R. Akella, and T.E. Morton (1994) A periodic review, production planning model with uncertain capacity and uncertain demand-optimality of extended myopic policies. Management Science, 40, 320-332. [11] Dempster, M. A. H.; Fisher, M. L., Jansen, L. Jansen; Lageweg, B. J., Lenstra, J. K., and Rinnooy Kan, A. H. G. (1981), Analytical evaluation of hierarchical planning systems. Operations Research 29, 707-716.

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Continuous time disaggregation in hierarchical production planning
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ABSTRACT: One of the objectives of disaggregation in hierarchical production planning is to minimize the setup costs incurred when changing production from one family to another. In this research, the setup costs are reduced by determining a production schedule that minimizes the number of setups during the planning horizon. Previous solutions to the disaggregation problem have considered discrete-time, and more recently continuous-time formulations. This research extends the continuous time disaggregation approach by incorporating production schedules allowing backorder. A mathematical formulation and a solution algorithm are presented and the computational complexity and convergence properties of the algorithm are discussed. Experimental results, using both deterministic and stochastic demand patterns, which demonstrate the efficacies of the solution approach are provided.
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