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Modeling demand uncertainty and processing time variability for multiproduct chemical batch process
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by Rishi Darira.
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[Tampa, Fla.] :
University of South Florida,
2004.
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Thesis (M.S.I.E.)University of South Florida, 2004.
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ABSTRACT: Most of the literature on scheduling of multiproduct batch process does not consider the uncertainties in demand and variability in processing times. We develop a simulation based variable production schedule model for a multiproduct batch facility assuming zero wait transfer policy and single product campaign. The model incorporates the demand uncertainties and processing time variability. The impact of demand uncertainties is evaluated in terms of total annual cost, which comprises of the backorder and inventory costs per year. The effect of variability in processing time is measured by the annual production time. We also develop a constant production schedule model that has uncertain demand arrival, but the schedule is independent of demand variations. We compare the variable production schedule model with constant production schedule model in terms of the total annual cost incurred and subsequent results are presented. The conclusion drawn from this comparison is that the total annual cost can be significantly reduced when the demand uncertainties are accounted for in the production schedule.
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Coadviser: Khator, Suresh
Coadviser: Sunol, Aydin
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multiperiod.
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zero wait policy.
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x Industrial Engineering
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Modeling Demand Uncertainty and Processing Time Variability for MultiProduct Chemical Batch Process by Rishi Darira 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 CoMajor Professor: Suresh Khator, Ph.D. CoMajor Professor: Aydin Sunol, Ph.D. Ali Yalcin, Ph.D. Date of Approval: July 6, 2004 Keywords: zero wait policy, single camp aign, simulation, scheduling, multiperiod Copyright 2004, Rishi Darira
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TABLE OF CONTENTS LIST OF TABLES iv LIST OF FIGURES v ABSTRACT vi CHAPTER 1 INTRODUCTION 1 1.1 Multiple Product Batch Plants 1 1.2 Types of Campaigns 1 1.3 Transfer Policies 2 1.4 Uncertainties in Data 2 1.5 Simulation as Modeling Tool 3 1.6 Thesis Organization 3 CHAPTER 2 LITERATURE REVIEW 4 2.1 Process Design 4 2.2 Scheduling MultiProduct Plant 5 2.2.1 Unlimited Storage and Zero Wait Policies 5 2.2.2 Finite Intermediate Storage 5 2.3 Single Product Campaign 5 2.4 Mixed Product Campaign 5 2.5 Sequence Determination 6 2.6 Uncertainties in Demand 6 2.7 Variability in Processing Times 6 2.8 Markov Chain Approach to Unpaced Transfer Lines 7 2.9 Use of Simulation in Batch Process 7 2.10 Summary 7 i
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CHAPTER 3 MODELING FEATURES 9 3.1 Single Product Campaign Model with Zero Wait Policy 9 3.2 Modeling Uncertainties 11 3.2.1 Uncertain Demand Arrival 11 3.2.2 Processing Time Variation 12 3.3 Operating Policy 12 3.3.1 Extra Capacity Utilization: Use of Inventory 13 3.3.2 Backordering Due to Insufficient Capacity 14 3.3.3 Modeling Zero Wait Policy 14 3.4 Constant Production Model 14 3.5 Summary 16 CHAPTER 4 MODEL DEVELOPMENT 17 4.1 Modeling Method 17 4.2 ARENA as Simulation Software 17 4.3 Example Problem 18 4.4 Assumptions 20 4.5 Model Formulation 20 4.5.1 Demand Arrival 20 4.5.2 Evaluation 21 4.5.3 Inventory Logic 21 4.5.4 Backorder Logic 22 4.5.5 Planning 23 4.5.6 Production 23 4.6 Performance Measure 23 4.7 Model with Fixed Production Schedule 24 4.8 Summary 25 ii
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CHAPTER 5 RESULTS AND ANALYSIS 26 5.1 Uncertain Demand 26 5.1.1 Performance Measure for Uncertain Demand 28 5.1.2 Results and Analysis of Demand Distribution 28 5.2 Sensitivity of Total Cost to Different BackorderInventory Cost Ratio 30 5.3 Backorder and Inventory Variations 31 5.4 Higher Variation for Normal and Uniform Distribution 32 5.4.1 Results and Analysis of Comparable C.V. 33 5.5 Variable Processing Times 34 5.5.1 Performance Measure for Variable Processing Times 35 5.5.2 Results and Analysis of Variable Processing Times 35 5.6 Comparison with Constant Production Schedule Model 36 5.6.1 Results and Analysis of Comparison 37 5.7 Summary 38 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 6.1 Summary and Conclusions 39 6.2 Future Research 40 REFERENCES 42 APPENDICES 45 Appendix A. Variable Schedule Model 46 Appendix B. Constant Production Model 52 Appendix C. Results for Total Cost with Low Demand Variability 53 Appendix D. Results for Total Cost with High Demand Variability 55 Appendix E. Result for Variable Processing Times 57 Appendix F. Results for Two Factorial Design 59 iii
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LIST OF TABLES Table 3.1 Demand Distribution 11 Table 3.2 Coefficients of Variation in Processing Times 12 Table 4.1 Annual Demand 18 Table 4.2 Net Profit 18 Table 4.3 Clean up Times in Hours 19 Table 4.4 Processing Times in Hours 19 Table 4.5 Demand Arrival per Cycle 21 Table 4.6 Minimum Demand Ratios and Processing Times 22 Table 5.1 Mean and Standard Deviation with Normal Distribution 27 Table 5.2 Demand in Batches with Uniform Distribution 27 Table 5.3 Coefficient of Variation Associated with Poisson distribution 28 Table 5.4 Total Annual Cost in Dollars per Kg 28 Table 5.5 Comparison of Total Costs for BackorderInventory Cost Ratio 30 Table 5.6 Average Annual Backorder and Inventory in Batches 31 Table 5.7 Mean and Standard Deviation for Normal Distribution 32 Table 5.8 Mean and Range for Uniform Distribution 33 Table 5.9 Levels of Processing Time Variability 34 Table 5.10 Mean and Standard Deviation in Hours 35 Table 5.11 Total Cost in Thousands of Dollars for Production Schedules 37 Table C.1 Total Cost in Thousands of Dollars for Low Demand Variability 53 Table D.1 Total Cost in Thousands of Dollars for High Demand Variability 55 Table E.1 Annual Production Time in Hours 57 Table F.1 Total Cost in Thousands of Dollars for Comparison 59 iv
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LIST OF FIGURES Figure 3.1 Flowchart of the Variable Schedule Model 15 Figure 4.1 Process Layout 19 Figure 4.2 Sub Models 20 Figure 4.3 Model with Fixed Production Schedule 24 Figure 5.1 Analysis of Variance for Variable Demand 29 Figure 5.2 Box Plot of Treatment Levels 30 Figure 5.3 Plot of Total Cost Versus Backorder Inventory Cost Ratio 31 Figure 5.4 Analysis of Variance for Demand with Higher Variability 33 Figure 5.5 Box Plot Showing Mean and Variability for Demand Distributions 34 Figure 5.6 Analysis of Variance for Variation in Processing Times 35 Figure 5.7 Box Plot of Levels 36 Figure 5.8 Analysis of Variance for Total Cost 37 Figure A.1 Demand Arrival Sub Model 46 Figure A.2 Evaluation Sub Model 47 Figure A.3 Inventory Sub Model 48 Figure A.4 Backorder Sub Model 49 Figure A.5 Planning Sub Model 50 Figure A.6 Production Sub Model 51 Figure B.1 Inventory and Backorder Sub Model 52 v
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MODELING DEMAND UNCERTAINTY AND PROCESSING TIME VARIABILITY FOR MULTIPRODUCT CHEMICAL BATCH PROCESS Rishi Darira ABSTRACT Most of the literature on scheduling of multiproduct batch process does not consider the uncertainties in demand and variability in processing times. We develop a simulation based variable production schedule model for a multiproduct batch facility assuming zero wait transfer policy and single product campaign. The model incorporates the demand uncertainties and processing time variability. The impact of demand uncertainties is evaluated in terms of total annual cost, which comprises of the backorder and inventory costs per year. The effect of variability in processing time is measured by the annual production time. We also develop a constant production schedule model that has uncertain demand arrival, but the schedule is independent of demand variations. We compare the variable production schedule model with constant production schedule model in terms of the total annual cost incurred and subsequent results are presented. The conclusion drawn from this comparison is that the total annual cost can be significantly reduced when the demand uncertainties are accounted for in the production schedule. vi
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CHAPTER 1 INTRODUCTION Chemical processes are divided into two types continuous and batch. Chemicals are manufactured in batches if production volumes are small. Batch processes are used in the manufacture of pharmaceutical products, food, and certain types of chemicals. Batch process is divided into multiproduct plant or multi purpose plant. There are several decisions involved in batch process. We would concentrate on the scheduling aspect which can have a large economic impact. In most problems involving scheduling of batch plants it is assumed that the problem data can be predetermined. We would explore scheduling by incorporating uncertainties in data. The basic structure of the batch process and scheduling concepts are explained in this chapter. 1.1 MultipleProduct Batch Plants When a batch process is used to manufacture more than two products, two types of plants arises, flowshop plants in which all products have the same recipe, and job shop plants where the products do not require same recipe. Flowshop plants are known as multiproduct plants and job shop plants are known as multipurpose plants. 1.2 Types of Campaigns The entire scheduling horizon is divided into planning periods in which cycles of predetermined set of batches are produced. One option is to use single product campaigns in which all batches of a given product which was predetermined to be produced in a period are manufactured before switching to another product. The other option is to use mixed product campaigns in which the batches are produced according to some selected sequence. The inventory is lower when mixed product campaign is used but its efficiency depends on the cleanup times between successive products. If the clean up times are 1
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significantly large in comparison to the processing time then it is advisable to use single product campaign to maximize production. Optimal sequence determination is also an important aspect of batch process scheduling. An effective algorithm was proposed by Birewar and Grossmann [1989] to determine optimal sequence. 1.3 Transfer Policies The main types of transfer policies are explained below 1. Zero wait transfer This transfer policy assumes that the batch at any stage would be transferred immediately to the next stage. It is often used when intermediate storage vessel is not available due to economic constraints or when storage is not allowed for the intermediate product. The zero wait transfer because of its constraint of no wait is associated with higher cycle time in comparison to other transfer policies. This policy has forces idle time between successive batches and successive products [Rekalaitis et al, 1992]. 2. No Intermediate Storage This transfer policy allows holding the material inside the vessel until the next stage is idle. Storage vessels are not required in this policy thus reducing the investment cost but the production time increases due to holding. 3. Finite Intermediate Storage This policy predetermines the approximate quantity of batches that require storage. Optimal storage equipment can be purchase therefore reducing the investment cost. 4. Unlimited Intermediate Storage This policy assumes that batches can be stored without any capacity limit in the storage vessel. It is associated with lowest cycle time but requires large capital investment. 1.4 Uncertainties in Data Most of the scheduling models that have been developed assume that all the data are predetermined. Such models are said to be deterministic. In chemical plants there are many factors such as equipment availability, processing times, demands of products and costs causing uncertainty [Balasubramaniam and Grossmann, 2001]. These uncertainties 2
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are common and can have undesirable costs attached. Thus these uncertainties should be taken into account while scheduling. 1.5 Simulation as Modeling Tool Mathematical programming involves making simplified assumptions. Simulation is a process which mimics the process. It provides fast analysis of the schedule. Using simulation for testing a schedule is economical. Simulation can be used for planning the process, process development, process design and the manufacturing stage. 1.6 Thesis Organization The organization of the thesis is as follows. Chapter 2 reviews the prior work in the area of scheduling, scheduling with uncertainties in demand, processing time variation and simulation of multiproduct batch process. Chapter 3 describes the variable schedule model which incorporates uncertainties in demand and processing time variability. Various performance measures used to judge the effectiveness of the schedule are explained. A constant schedule model is also described which would be compared to the variable schedule model. Chapter 4 describes the modeling methodology used. It also describes the formation of variable schedule model and constant schedule model for an example problem. The results obtained are listed in Chapter 5. A statistical analysis of the simulation runs is made leading to appropriate interpretations. Finally, Chapter 6 summarizes the entire thesis. It also includes the conclusions and areas of future research. 3
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CHAPTER 2 LITERATURE REVIEW Batch processing constitutes a significant fraction in the chemical process industries. For example 80 percent of pharmaceutical and 65 percent of the food and beverage processes are batch processes [Reeve, 1992]. Despite the development of design models and techniques for batch processes, there is lack of comprehensive methodologies that can properly address the many aspects involved in batch process [Reklaitis, 1990]. A comprehensive overview on scheduling and planning of batch process was presented in the book by Reklaitis et al [1992] 2.1 Process Design A mixed integer nonlinear programming problem was proposed by Graham et al [1979] that minimized equipment cost. It also found optimal number of equipments and their sizes. Zero wait policy was implemented. Branch and bound method could be used to solve optimization problems, but it has limitations. Process merging was implemented by Yeh and Reklaitis [1987] for single product plants. Birewar and Grossman [1989] showed that process merging leads to lower investment cost. It was assumed that the equipment sizes were continuous but practically equipment with standard discrete sizes is available. Standard equipment sizes were considered by Voudouris and Grossmann [1991]. Karimi and Reklaitis [1985] show that storage size effects cycle time and batch size. The design of multiproduct plants with intermediate storage was shown by Modi and Karimi [1989]. Simulation can also be used to identify need of intermediate storage and campaign. Birewar and Grossmann [1990] introduced scheduling at the design stage. 4
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2.2 Scheduling Multi Product Plant Each product is produced in batches that determine the size factor of equipments thus each batch is considered as a different entity for scheduling. All products follow the same process stages. The unlimited intermediate storage, no intermediate storage, zero wait and finite storage conditions are all treated as the options for transfer policies. 2.2.1 Unlimited Storage and Zero Wait Policies Kuriyan and Reklaitis [1989] developed algorithms for the scheduling of multi product batch plants with unlimited intermediate storage and zero wait policies. Minimizing make span was their objective. A twostep method was used. The first step simplified sequencing problem and the second step evaluated the sequence. 2.2.2 Finite Intermediate Storage Many multi product batch plants operate under finite storage capacity policy. Kuriyan, Joglekar and Reklaitis [1987] studied the use of finite intermediate storage in detail. They used simulation to evaluate the production time. The study showed that simulation could be effectively used for scheduling. 2.3 Single Product Campaign Wellons and Reklaitis [1989] proposed a MINLP algorithm for determining the sequence of a single product plant. The objective was to maximize the average production rate of the plant. 2.4 Mixed Product Campaign The output of a multi product plant can be increased by changing long single product campaigns with combinations of batches of different products. The combinations are repeated periodically. Mixed product campaigns studied by Birewar and Grossmann [1989]. The set up times have an effect on the use of mixed product campaigns. The inventory levels are lower when this policy is implemented. 5
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2.5 Sequence Determination Campaign lengths are determined by the demand size. The sequence of batches in the campaign is an important factor to determine. Birewar and Grossmann [1989] proposed a simple algorithm to determine optimal sequence. The objective was determination of minimum cycle time. Inventory costs and process dictated by due dates has also been studied. Sahinidis and Grossmann [1991] evaluated the problem of a multiproduct scheduling for constant demands. They developed an algorithm with cost minimization as an objective. 2.6 Uncertainties in Demand Petkov and Maranas [1998] proposed a method for designing a multi product batch plant operating uncertain single product campaign. Demand follows Normal distribution. A deterministic mixedinteger nonlinear programming problem was developed. Ierapetritou and Pistikopoulos [1995] discussed the problem of designing multi product batch plants with uncertain demands. They developed a nonlinear optimization model with the objective function including investment cost, net sales and penalty for unsatisfied demand. 2.7 Variability in Processing Times Pistikopoulos et al [1996] proposed a twostage stochastic programming formulation. The objective function included processing costs, sales and a penalty term accounting for not meeting demand. Straub and Grossmann discussed the problem of evaluation and optimization of probabilistic features in batch plants. They proposed computational methods to determine size and number of equipment to maximize expected probabilistic uncertainty. Constraint included the investment. Honkomp, Mockus and Reklaitis [1997] compared three different models. The uniform time discretized model, non uniform discretized model and model with scheduling uncertainty. 6
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2.8 Markov Chain Approach to Unpaced Transfer Lines Many researchers have studied unpaced transfer lines using Markov chain approach. A parallel can be drawn between a transfer line and a chemical manufacturing facility. A transfer line is a serial production system with multiple identical stages with same processing time distribution. Initial studies concentrated on transfer lines with variability in processing times with no buffers (zerowait) and no breakdowns. Due to the process time variability workstations are starved until the upstream station finishes its operation. Similarly workstations are blocked until the downstream station finishes and can pass on its job. It was observed that the system throughput decreases with the increase in the processing time variability (measured by coefficient of variation) and the number of stages. The impact of increase in coefficient of variation was very significant while the impact of number of stages levels off after 5 or 6 stages [Conway et al, 1988]. It has been reported by the authors that other than few simple cases which could be solved using analytical approach, simulation was invariably used to study such systems. There were other studies which included buffers (nonzero wait) to increase the productivity. The effect of breakdowns on transfer lines with constant processing time was also studied [Askin and Standridge, 1993]. 2.9 Use of Simulation in Batch Process Rippin [1983] has reviewed the literature on optimal operation of batch process equipments. He has provided his view of methods available for designing and operating a batch plant. The benefits of simulation at various stages of the commercialization process are categorized by Petrides, Koulouris and Lagonikos [2002]. Simulation can be effectively used for planning the process, process development, process design and the manufacturing stage. 2.10 Summary The work done for the design and scheduling of a multiproduct batch plant by researchers was reviewed in this chapter. The design aspect involves allocation of tasks to the resources and the use of intermediate storage if possible. The use of intermediate 7
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storage decreases the cycle time but it increases the investment cost. Algorithms optimizing the use of intermediate storage with objective to minimize the cost were studied. The scheduling of a multiproduct batch plant deals with the sequencing of products. Birewar and Grossmann [1989] have developed effective algorithm that determines optimal sequence of products while incorporating cleanup times and slack times in the algorithm with minimizing the cycle time as the objective. A lot of work in operations planning has been done to develop deterministic models. The problem data is known in advance in deterministic models. There can be uncertainty in real problems in factors like processing times and demands. The common approach to deal with these uncertainties is with probabilistic model. Operations planning considering uncertainties in various factors are an important aspect as this can have considerable economic implications. This research incorporates uncertainties in processing times and variable demands of products in a multiproduct batch plant with multiple stages. The next chapter describes the model features and performance measures. 8
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CHAPTER 3 MODELING FEATURES The scheduling of a multiproduct batch process was discussed in the previous chapters. This chapter describes a variable schedule model based on multiproduct batch process. The model features and its performance measures are shown below. 3.1 Single Product Campaign Model with Zero Wait Policy In a multiproduct plant all products follow the same sequence of operation. The first step is to decide a transfer policy suitable for the process. The different types of transfer policies are described in chapter one. If we select policies like unlimited intermediate storage policies or finite intermediate storage transfer policy then investments would have to be made on storage vessels. The constraints on the investment cost lead us to assume zero wait policy The type of production campaign is decided next. The campaign type selected is used for manufacturing a specified number of batches for the various products in a planning period or cycle. One option is to use single product campaigns in which all batches of a given product are manufactured before changing to another product. The other option is to use mixed product campaigns in which batches are produced according to a selected sequence during a production cycle. The inventory levels of individual products in single product campaign are greater than the mixed product campaign. Thus inventorycarrying costs are higher in single product campaigns. Efficiency of the mixed product campaign depends on the cleanup times that are needed after successive products. If clean up times are very small compared to the processing time, mixedproduct campaign would be preferred over single product campaign. In this research, we assume that the clean up times are significantly large thus we select single product campaign. The optimal sequence of products during a cycle for is determined with the 9
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help of an algorithm and a graphical method proposed by Birewar and Grossmann [1989]. Their algorithm for single product campaign is categorized below. It explains that some idle time is forced in zero wait policy. In their algorithm, SL ikj denotes the minimum forced idle time between the batches of product i and k in stage j. These slack times (forced idle time) between two products could be predetermined by calculation. The processing time of process i in stage j was indicated by t ij. The total number of batches of product i produced is n i The cycle time (CT) for unlimited intermediate storage is given in Equation 3.1. CT = max j=1... M ( Npin1 i t ij ) (3.1) The slack time was accounted for by making the cyclic sequence consist of pairs of batches of products. For each pair, NPRS ik was defined as the number of times product i is followed by product k (i, k = 1Np) in the schedule. Equation 3.2 shows the succeeding constraint and 3.3 shows the preceding constraint for sequencing. Equation 3.4 gives the cycle times of each stage. The equality sign in Equation 3.5 enforces single product campaign. Min CT Subject to Npk1 NPRS ik = n i i= 1 N P (3.2) Npi1 NPRS ik = n k k=1 N P (3.3) CT n Npi1 i t ij + NPRS Npi1 Npk1 ik SL ikj j= 1 M (3.4) NPRS ii = n i 1 i = 1 N P (3.5) CT 0, NPRS ik 0 i, k = 1 N P (3.6) The solution of this algorithm provides the values of NPRS ik for combinations of products i and k. The optimal sequence could then be calculated by a simple graphical method. 10
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3.2 Modeling Uncertainties This objective of this research is to study the impact of the two of the uncertain conditions that a chemical processing facility faces: uncertain demand and variable processing time. The following subsections outline how these have been modeled in this thesis. 3.2.1 Uncertain Demand Arrival The total scheduling horizon would be divided into equal periods, which are called as the planning periods. The demand arriving at each period is assumed to be variable. The Poisson distribution can model this variability. It is a discrete distribution which gives the average number of events in the given time interval. The Poisson distribution is determined by one parameter lambda, indicating the mean of the distribution. The variability in demand can also be modeled by a continuous distribution. Normal distribution can be used to model demand variability. It is largely accepted that Normal distribution covers the important features of uncertain demands. Theoretical reasoning in use of Normal distribution can be based on central limit theorem as the demands are affected by a large number of probabilistic events. [Petkov and Maranas, 1998]. Uniform distribution can also be used to model demand arrival. The three distributions would be compared in terms of their effect on the performance measure. The three distributions used to model the demand variation in this thesis are given in Table 3.1. Table 3.1: Demand Distribution Demand Distribution Uniform Normal Poisson 11
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3.2.2 Processing Time Variation In chemical processes the processing times at each stage varies depending on factors such as the efficiency of the equipment and quality of material being processed. This variation is continuously distributed. The uncertainty in processing times would be modeled using Normal distribution. The Normal is a twoparameter probability distribution described by mean and standard deviation ( and ). The uncertainty in processing times is assumed to be at two levels high and low. The values of coefficient of variation corresponding to the level of uncertainty are indicated in Table 3.1. For example when coefficient of variation is 0.05 (low level of uncertainty) and the mean processing time is 7 hours then the standard deviation is 0.35 hours (from Equation 3.7), therefore 95 percent of the times the processing times will lie between 6.3 hours and 7.7 hours (2 limit). When the coefficient of variation is 0.1 (high level of uncertainty) with mean processing time as 7 hours the standard deviation is 0.7 hours thus 95 percent of time the processing time would lie within 5.6 hours and 8.4 hours. The effect of theses variations when they are at high level or low level would be modeled and analyzed. The variability is shown in Table 3.2. Table 3.2: Coefficients of Variation in Processing Times Level of Uncertainty Coefficient of Variation None 0.00 Low 0.05 High 0.10 3.3 Operating Policy The batch size is assumed to be constant for each product. An optimal design of the process with the objective of minimizing the cycle time is considered. The demand is assumed to arrive at the beginning of the planning period. The demands have to be met by the end of the planning period. The following two cases arise for this model. 12
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3.3.1 Extra Capacity Utilization: Use of Inventory If the demands were met before the end of planning period, the remaining time in that period can be used to utilize the capacity. The product would then be stored as inventory. Thus we have to determine which product should be produced and in what quantity. The time remaining in a particular planning period is calculated. We determine the minimum demand ratios of the products. For example consider four products have to be produced and their mean periodic demand is 14, 7, 7 and 21 respectively. Thus their minimum weekly demand ratio will be 2, 1, 1 and 3. The mean production time for this minimum demand ratio is then calculated. This production time is then compared to the time remaining in a period. If the time remaining is less than the production time of the products in their minimum demand ratio then the product with the highest demand is produced first. The remaining time is divided by the cycle time of the product with highest demand. This would give the number of batches of that product that can be made. We limit its production till its minimum demand ratio that is 3 units as in example explained. The product with the second highest demand is produced next considering its minimum demand ratio. Thus the production continues till the time remaining is not sufficient to produce any more batches. The capacity would not be utilized for the residual time remaining. If the time remaining is greater than the production time of the products in their minimum demand ratios then we check if the multiples of the minimum demand ratio can be produced. Based on the above example it would be checked if 4, 2, 2 and 6 batches of the products could be produced. If enough time is remaining for this then we produce it. We produce in higher demand ratios because the clean up times are reduced. They are produced till the time remaining is less than the production time of a particular ratio. The lower ratio is then considered till there is not enough time remaining to produce the minimum demand ratio The production then continues according to the scenario described above when time remaining is less than the production time of the minimum demand ratio cycle time. 13
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3.3.2 Backordering Due to Insufficient Capacity If the demands are not satisfied in a particular planning period then the options are to either consider lost sales or backorder the product so that it is produced in the next period. Lost sales lead to loss of profit and goodwill among customers. The customers also tend to wait for some more time since they have already waited for the time of planning period. Thus the unsatisfied demand in a planning period in this model would be considered backordered. The backordered products are added to their respective new period demands and they are produced in the next cycle according to the sequence minimizing the cycle time. Figure 3.1 shows the flowchart of the process. 3.3.3 Modeling Zero Wait Policy We assume zero wait policy which results in forced idle times between batches of products. We have to make sure that there is no waiting or queues of intermediate products. This would be done by forced delay between successive batches and between successive products. The forced delay between successive batches would be the cycle time of that particular product. The delay between successive products would include the cycle time and the clean up times. For the case with variable processing times zero wait policy would be implemented by holding the batch in a stage until the next stage is idle. 3.4 Constant Production Model Most of the literature for scheduling of a multiproduct batch plant does not incorporate uncertainties in demand. A constant production model would also be developed in this thesis. This model would have uncertainties in demand but the schedule would be fixed. If excess demand arrives in a planning period then the extra batches are backordered and if insufficient demand arrives then inventory will be allowed to build. 14
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Periodic Deman d Inventory and Backorder Update N o New Backorder Demand Met? Yes New Inventory Planning Production Production Figure 3.1: Flowchart of the Variable Schedule Model 15
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3.5 Summary This chapter outlined the model features. The modeling problem includes a multi product batch plant. Zero wait policy is assumed due to investment constraints, because for other transfer policies investment has to be made on storage vessels. Single product campaign is implemented to minimize to production time. The operating optimal sequence is obtained from the algorithm proposed by Birewar and Grossmann [1989]. Uncertainties in demand arrival and processing time variability would be modeled. Various distributions for demand arrival and their effects on the performance measures would be analyzed. A constant production model would be developed which would be compared to the variable schedule model. The next chapter explains the model development and the performance measures used. 16
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CHAPTER 4 MODEL DEVELOPMENT A multiproduct batch facility with multiple stages is modeled. Zero wait transfer policy is assumed. The entire scheduling cycle is divided into equal planning periods. Single product campaign is assumed within a cycle. The processing times and the demands of products are assumed to be variable. If demands are not met in a planning period the product is backordered. If the demand is met before the end of planning period then the excess capacity is utilized to make products for inventory to be used in the next cycle. A simulation model using ARENA simulation software is developed. The model is then used as a vehicle to determine the impact of the demand uncertainties on the backorders and inventory accumulation and the processing time variability on production cycle time. Finally, the performance of the variable schedule model will be compared with the constant schedule model. 4.1 Modeling Method Simulation was used to model the process. Simulation mimics the behavior of real systems, usually on a computer with appropriate software. Simulation has become very popular with the evolution of computers and software. The purpose of simulation is to mimic the uncertainties in demand arrival and the processing times. The effect of these uncertainties is evaluated. 4.2 ARENA as Simulation Software Arena is simulation software which has flexible model building capabilities. It has excellent features for statistics collection. The model with uncertainties in demand and processing times in a multistage batch process with zero wait policy is created by using the Basic process, advanced process, advanced transfer and blocks templates. 17
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4.3 Example Problem The example problem deals with manufacturing of four different pharmaceutical products in a fivestage batch process. Since production costs are essentially fixed due to production recipe, capital investment is the main economic factor in the design of such a batch process. Given that the plant must be capable of manufacturing four different products, a major consideration in the design. This involves determining the length of production campaigns as well as the sequencing of the products. In analyzing various alternatives for scheduling, different plant configurations are considered. This example problem was also solved by Voudouris and Grossmann [1993] by developing an MINLP algorithm. The annual demand and net profit of the four products are shown in Table 4.1 and Table 4.2 respectively. Table 4.1: Annual Demand Product A 400,000 kg/yr Product B 200,000 kg/yr Product C 200,000 kg/yr Product D 600,000 kg/yr Table 4.2: Net Profit Product A 0.60 $/kg Product B 0.65 $/kg Product C 0.70 $/kg Product D 0.55 $/kg The following steps were required to produce the four products, according to the flow diagram given in Figure 4.1 The reactor and the crystallizer have longer cycle time and are generally the bottleneck stages. In order to reduce the idle time for other stages as 18
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well as the batch cycle time, parallel equipment has been used at both these stages as can be seen from Figure 4.1. Recoverer Purifier Crystallizer A Crystallizer B Reactor A Centrifuge Reactor B Figure 4.1: Process Layout The clean up times are assumed to be the same for each piece of equipment. However the clean up times depend on the sequence of products as shown in Table 4.3. Table 4.4 indicates the processing times at each processing stage. Table 4.3: Clean up Times in Hours Product A Product B Product C Product D Product A 0 0.2 0.2 0.2 Product B 0.5 0 0.5 0.5 Product C 0.5 0.5 0 0.5 Product D 2 2 2 0 Table 4.4: Processing Times in Hours Product Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 A 4.5 2.5 1.5 3.75 0.83 B 5.5 2.5 1.5 1.5 0.83 C 3.75 2.5 1.5 5.75 0.83 D 7.25 2.5 1.5 8.5 0.83 19
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4.4 Assumptions 1. The time to transfer products between units and the time to fill and empty vessels is negligible. 2. Demand arrives in batches. The batch size for all the products is the same. The batch size is assumed to be 600 kilograms based on economic design considerations. 3. Backorder cost is two times the cost of inventory. 4. The demand is assumed to arrive at the beginning of the period and is satisfied at the end of the planning period 5. The process design is optimal. 4.5 Model Formulation There are four different products which require the same set of processing stages. The products are manufactured in five stages. The cycle time is calculated to be 171.58 hrs. The model is run for 50 cycles that constitutes one years production. The model is divided into six sub models as shown in Figure 4.2. Evaluation Production Demand Arrival Inventory Policy Backorder Policy Planning Figure 4.2: Sub Models 4.5.1 Demand Arrival Interarrival time between demands is constant 171.58 hours (cycle time). The demand occurs in batches. The demand is assumed to arrive at the beginning of the period. The average demand size of each product in batches is shown in Table 4.5. The modules and the associated flow for each of the sub models are given in Appendix A. 20
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Table 4.5: Demand Arrival per Cycle Product Mean (Batches) A 14 B 7 C 7 D 21 4.5.2 Evaluation When a new periodic demand for a product arrives, a check is made with the current periods product inventory. The current inventory is then subtracted from the demand size. If there were batches of products backordered from the previous cycle for this product then the backordered batches are added to the current demand size. This gives us the total number of batches to be produced in a particular cycle. Time taken to produce the total number of batches in that period is then calculated based on the longest processing time at any stage during the processing of a product (see Table 5). This value is compared to the expected cycle time of 171.58 hours. 4.5.3 Inventory Logic If the total time required to complete the production needs of a cycle is less than the planning period, the remaining time in that period is utilized by producing extra batches for next periods demands. The product would then be stored as inventory. The process of deciding which products to be made and how many batches to be made is given below. The time remaining in a particular planning period after meeting the current periods demand is based on the minimum demand ratios of the product. The minimum demand ratios of the four products are shown in Table 4.7. The mean production time for this minimum demand ratio is then calculated. This production time is then compared to the time remaining in a period. If the time remaining is less than the total production time for the minimum demand ratio batch size then the product with the highest demand (Product D) is produced first. In order to calculate the number of batches, the time remaining is divided by the cycle time of the product. 21
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Table 4.6: Minimum Demand Ratios and Processing Times Product Mean Demand Ratio Cycle Time Process Time A 14 2 2.5 5 B 7 1 2.75 2.75 C 7 1 2.88 2.88 D 21 3 4.25 12.75 We limit its production to its minimum demand ratio value (3 batches for Product D). The product with the second highest demand (Product A) is produced next considering its minimum demand ratio batch size. If time remaining is not sufficient to produce one batch of product with highest demand then the product with minimum cycle time is produced if possible. Thus the production continues till the remaining time is not sufficient to produce any more batches. The remaining capacity cant be utilized. If the time remaining is greater than the production time of the products in their minimum demand ratio, we check if the multiples of the minimum demand ratio can be produced. Based on the above example it would be checked if 4, 2, 2 and 6 batches of the products could be produced. If enough time is remaining for this then we produce it. This is modeled by implementing a loop in the logic. This continues till there is not enough time to produce all the products in their demand ratio. The production then continues according to the scenario described in the previous paragraph. 4.5.4 Backorder Logic If the production time of the demand size exceeds the expected time then some batches are backordered till the next period. Batches are backordered according to the same rule as the inventory build up. 22
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4.5.5 Planning In this sub model the number of batches of each product is assigned a number. The first batch of the first product is sent for production. A forced delay is implemented on subsequent batches. The time of forced delay is the cycle time of that product. This forced delay is implemented so that there is no waiting at the stages following the Zero wait policy. After the last batch of the first product is released for production, the next products first batch is signaled after a delay of the slack time. The slack time is calculated by constructing Gantt charts. This prevents waiting in stages. The slack time includes the cycle time of the product and the clean up times between successive products. Due to variability in processing times if the production does not end in a planning period the next cycle is not released. When the last batch of the last product is released for production, it signals the next cycles release. 4.5.6 Production The production submodel consists of the five process stages. Zero wait policy is modeled in the planning stage, but due to uncertainties in processing time there is bound to be some waiting. Thus, this is modeled by incorporating hold and signal modules. A batch is not released if the next stage is busy. As soon as the next stage is idle, this sends a signal to the previous stage for the batches release. Thus, there is no waiting at the resources. 4.6 Performance Measures The performance of the process would be measured in terms of 1. Total Cost: This is the total cost of carrying inventory and backorder per year. Backorder cost is assumed to be twice that of inventory cost. 2. Annual Production Time: This is the time taken to satisfy the demand for 50 cycles. 23
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4.7 Model with Fixed Production Schedule Most of the literature for scheduling of multiproduct batch plant does not include demand uncertainties. It would be interesting to see if the constant production schedule misinterprets the annual inventory and backorder generated due to demand uncertainties. Thus a simulation model was developed for the same example problem where the production schedule is fixed. The constant production schedule model has uncertain demand arrival, but the schedule is independent of demand variations. Total cost for inventory and backorder is set as the performance measure. The model layout is shown in Figure 4.3 Evaluation Production Demand Arrival Inventory and Backorder Planning Figure 4.3: Model with Fixed Production Schedule 24 The demand arrival sub model is similar to the previous model described in Section 4.5.1. In the evaluation sub model the demand size is not compared to the expected value. The schedule is constant. The production is not flexible to variations in the demand size. When a demand arrives it is compared with the fixed schedule (as used in existing models found in literature). If the demand size is more than that being produced then the excess batches are considered backordered. If the demand size is less than the set number of batches in the production schedule then the extra batches produced, are considered as inventory. Example: if the fixed schedule is to produce 14, 7 ,7 and 21 batches in every cycle for products A, B, C and D respectively. If a new periods demand arrival is 16, 8, 5 and 24 batches for products A, B, C and D respectively then 2 batches of A, 1 batch of B and 3 batches of D will be backordered and 2 batches of C will be made for inventory provided there is no existing inventory for Products A, B and D. This logic is modeled in the inventory and backorder sub model. The planning and production sub model incorporates zero wait policy and uncertain processing times identical to the earlier model.
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4.8 Summary In this chapter, an Arena based simulation model was developed to mimic the process of multiproduct batch production with zero wait policy and single product campaign. This model had a variable production schedule which incorporated demand uncertainties and variation in processing time. The model features, formulation, assumptions and performance measures were discussed. A constant schedule model with uncertainties in demand was also developed to see if it misinterprets the annual inventory and backorder generated due to demand uncertainties. The next chapter shows the results and analysis. 25
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CHAPTER 5 RESULTS AND ANALYSIS The previous chapter explained the model formulation and the performance measures of the process. The model was run and the results obtained are presented in this chapter. Variability in demand and processing times are analyzed for the variable schedule model. Sensitivity analysis is performed for the backorder to inventory cost ratio and the quantity of backorder and inventory. Demand distributions with higher uncertainty are also analyzed. Finally a two factorial design is presented to see the effect of the variable demand on the variable production schedule and the constant production schedule. Coefficient of variation (CV) will be used as a measure of uncertainty or variability associated with the demand and processing times in this thesis. The CV is a better measure of variability since it takes into account the mean value as well. The CV is defined as C.V = / (5.1) Where, is the standard deviation and is the mean. 5.1 Uncertain Demand The mean demand for the four products is given earlier in Table 4.6. For products A, B, C and D these values are 14, 7, 7, and 21 batches respectively. The effect of demand uncertainty is analyzed for different distributions shown below. 1. Normal Distribution The standard deviation assumed for the periodic demand is shown in Table 5.1 which results into a coefficient of variation of approximately 0.14. 26
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Table 5.1: Mean and Standard Deviation with Normal Distribution Product Mean Std. Deviation A 14 2 B 7 1 C 7 1 D 21 3 2. Uniform Distribution A uniformly distributed random variable, uniform with range [a, b], where a and b are real numbers, and a < b, has a variance given by Variance = (b a) 2 / 12 (5.2) Table 5.2 shows the range assumed for the demand arrival of the four products following uniform distribution. The coefficient of variation is approximately 0.16. Table 5.2: Demand in Batches with Uniform Distribution Product Range A 14 4 B 7 2 C 7 2 D 21 6 3. Poisson Distribution The Poisson distribution is used to model the number of random occurrences of the demand arrival in a specified unit of time. The variance is equal to the mean (standard deviation will be equal to ) for Poisson distribution. The coefficient of variation for this distribution then can be given as C.V. = / = / = 1 / (5.3) 27
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The coefficient of variation with each products demand arrival is shown in Table 5.3. It can be observed from the Table that the larger the mean the smaller the coefficient of variation. Table 5.3: Coefficient of Variation Associated with Poisson Distribution Product Mean C.V. A 14 0.27 B 7 0.38 C 7 0.38 D 21 0.22 5.1.1 Performance Measure for Uncertain Demand The performance measure for this experiment is the Total Annual Cost. It comprises of average annual backorder and average annual inventory of the four products. The costs are shown in Table 5.4. The Equation is given by Total Annual Cost = 2.4 Average annual inventory Batch Size + 4.8 Average annual backorder Batch Size (5.4) Table 5.4: Total Annual Cost in Dollars per Kg Type Average Yearly Cost per kg Inventory $ 2.40 Backorder $ 4.80 5.1.2 Results and Analysis of Demand Distribution The simulation was run for 50 cycles. Each cycle represents a year of production. The run was replicated 100 times. The results are shown in Appendix C. Minitab Statistical Software was used to carry out the analysis of variance. A formal test of hypothesis of no 28
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differences in treatment means is performed. The analysis of variance is summarized in Figure 5.1 Oneway ANOVA: Uniform, Normal, Poisson Analysis of Variance Source DF SS MS F P Factor 2 40838 20419 25.11 0.000 Error 297 241531 813 Total 299 282369 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +++Uniform 100 37.03 22.41 (*) Normal 100 30.89 21.30 (*) Poisson 100 58.13 38.52 (*) +++Pooled StDev = 28.52 36 48 60 Figure 5.1: Analysis of Variance for Variable Demand From Figure 5.1 we observe that the mean square (MS) of the factor is much larger than the error mean square. This indicates that it is unlikely that the treatments means are equal. The F ratio = 25.11 obtained is compared to an appropriate upper tail percentage point of the F 2, 297 distribution after selecting as 0.05. We get F 0.05, 2, 297 is approximately 3.05. Since 25.11 (F ratio) > 3.05 we reject the hypothesis that all the means are equal and conclude that the treatment means differ; thus the different distributions in the demand arrival significantly affects the total annual cost. The P value is very small, P < 0.01. The average total cost when demand distribution follows Poisson is 57 % and 88 % greater than when it follows uniform and Normal distributions respectively. The box plot in Figure 5.2 depicts the mean and standard deviation for the different distributions. It can also be observed from the box plot that the confidence interval for Poisson is much higher in this case. 29
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PoissonNormalUniform 2001000 Boxplots of Uniform Poisson(means are indicated by solid circles) Figure 5.2: Box Plot of Treatment Levels 5.2 Sensitivity of Total Cost to Different Backorder Inventory Cost Ratio The backorder cost was assumed to be twice the inventory cost in earlier results. Table 5.5 shows the values for the annual total cost for different distributions with the backorder cost same, twice and three times that of inventory cost. Figure 5.3 shows the plot of total cost versus the backorder to inventory ratio Table 5.5: Comparison of Total Costs for BackorderInventory Cost Ratio Total Cost in Thousands of Dollars Back/Inv Ratio Uniform Normal Poisson 1 25.55 21.34 39.78 2 37.03 30.89 58.13 3 48.51 40.65 76.51 30
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0102030405060708090123Backorder Inventory RatioTotal Cost ($) Uniform Normal Poisson Figure 5.3: Plot of Total Cost Versus Backorder Inventory Cost Ratio From the Figure we observe that the difference in the total cost persists with different ratios. 5.3 Backorder and Inventory Variations The variation in the average annual backorder and inventory in batches with the different demand distributions are shown in Table 5.6. Table 5.6: Average Annual Backorder and Inventory in Batches Backorder Inventory Uniform 7.97 9.77 Normal 6.70 8.12 Poisson 12.75 14.87 Normal distribution has the lowest inventory accumulation and backorders. The Poisson distribution resulted into the highest level of inventory which was 52 % and 83 % higher in comparison to uniform and Normal distributions. Uniform and Normal distributions 31
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have the same amount of backorder. The backorder is about 60 % higher in Poisson distribution when compared to other distributions. Poisson distribution in this case has the highest inventory accumulation and the largest amount of backorder thus resulting in higher total annual cost. 5.4 Higher Variation for Normal and Uniform Distribution It must be pointed that in the previous experiment to assess the impact of different distributions, Normal and uniform distributions had much lower variability (CV of 0.14 and 0.16 with Normal and uniform respectively) compared to Poisson distribution (average CV of 0.31). Poisson distribution had the highest total annual cost. In order to see if the shape of the distribution or the variability has a greater impact on the total annual cost, we ran the simulation model with larger variations in Normal and uniform distributions. The standard deviation for Normal distribution is shown in Table 5.7, the coefficient of variation is approximately 0.28. The range of uniform distribution is increased to the values shown in Table 5.8. The coefficient of variation for uniform distribution in this case is approximately 0.33. These values of CV for Normal and uniform are now comparable to Poisson distribution. Table 5.7: Mean and Standard Deviation for Normal Distribution Product Mean Std. Deviation A 14 4 B 7 2 C 7 2 D 21 6 32
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Table 5.8: Mean and Range for Uniform Distribution Product Range A 14 8 B 7 4 C 7 4 D 21 12 5.4.1 Results and Analysis of Comparable C.V. The simulation was run for 50 cycles. The run was replicated 100 times. The performance measure is the total annual cost. The backorder cost as earlier is assumed to be twice the inventory cost. Figure 5.4 shows the analysis of variance. The detailed results are shown in Appendix D. From Figure 5.4 the F ratio = 5.23 obtained are compared to an appropriate upper tail percentage point of the F 2, 297 distribution. We get F 0.05, 2, 297 = 3.05. Since 5.23 (F ratio) >3.05 we conclude that the means are significantly different in this case. The distributions with higher variability significantly affect the total cost. We observe that as the variability was increased the values of total cost increased significantly in comparison to the case with demand having low variability. Figure 5.5 shows the box plot. Oneway ANOVA: Uniform, Normal, Poisson Analysis of Variance Source DF SS MS F P Factor 2 20759 10380 5.23 0.006 Error 297 589387 1984 Total 299 610146 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +++Uniform 100 78.46 51.60 (*) Normal 100 66.74 42.52 (*) Poisson 100 58.16 38.51 (*) +++Pooled StDev = 44.55 60 72 84 Figure 5.4: Analysis of Variance for Demand with Higher Variability 33
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PoissonNormalUniform 3002001000 Boxplots of Uniform Poisson(means are indicated by solid circles) Figure 5.5: Box Plot Showing Mean and Variability for Demand Distributions Uniform distribution has the highest total annual cost in this case. It can be interpreted that the variability associated with the distribution has a greater effect on the total annual cost than the shape of the distribution. 5.5 Variable Processing Times The effect of processing time variability is analyzed. The mean processing times at each stage is shown in Table 4.5 in the previous chapter. The levels of the variability are shown in Table 5.9 Table 5.9: Levels of Processing Time Variability Level of Uncertainty Coefficient of Variation None 0.00 Low 0.05 High 0.10 34
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5.5.1 Performance Measure for Variable Processing Times The performance measure is the time taken to complete 50 cycles (Annual Production Time). 5.5.2 Results and Analysis of Variable Processing Times The simulation is run for 50 cycles and is replicated 100 times. The effect of the variation in the processing time on the production time can be seen in the Table 5.10. The results for each replication are shown in Appendix E. Table 5.10: Mean and Standard Deviation in Hours Annual Production Time (Hours) Levels (C.V. of Processing Time) Mean Std. Deviation 0.00 8589.58 0.00 0.05 8590.77 1.05 0.10 8591.96 1.86 A test of hypothesis of no differences in treatment means is performed. The analysis of variance is summarized in Figure 5.6 and the box plot is shown in Figure 5.7. Analysis of Variance Source DF SS MS F P Factor 2 282.18 141.09 92.43 0.000 Error 297 453.35 1.53 Total 299 735.53 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +++0.00 100 8589.58 0.00 (*) 0.05 100 8590.77 1.05 (*) 0.10 100 8591.96 1.86 (*) +++Pooled StDev = 1.24 8590.0 8591.0 8592.0 Figure 5.6: Analysis of Variance for Variation in Processing Times 35
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0.100.050.00 859885938588 Boxplots of 0.00 0.10(means are indicated by solid circles) Figure 5.7: Box Plot of Levels From Figure 5.7 we see that the mean square (MS) of the factor is larger than the error mean square. We get F 0.05, 2, 297 = 3.05. Since the F ratio (92.43) > 3.05 we reject the hypothesis that all the means are equal and conclude that the means differ. We observe that even when the variation in means is small they have statistically significant effect on the annual production time. The box plot shows the mean and standard deviation for the different variation in processing times. 5.6 Comparison with Constant Production Schedule Model The variable schedule model is compared with the constant production schedule model for the same example problem. The motivation for this comparison was given in Section 4.7. Both the models have uncertainty in demand. Both models are run for 50 cycles and replicated 10 times (limitations of the student version of the software). The performance measure is the total annual cost. 36
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5.6.1 Results and Analysis of Comparison A two factorial design was performed to study the effect of the type of production schedule and the demand distribution on the total annual cost. The types of production schedule tested are the variable production schedule and the constant production schedule. Normal and Poisson distribution for demand arrival are used in both the production schedules. The results are shown in the Appendix F. Figure 5.8 shows the analysis. Factor Type Levels Values Demand D fixed 2 1 2 Model Type fixed 2 1 2 Analysis of Variance for Total Co, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Demand D 1 17703 17703 17703 22.91 0.000 Model Ty 1 31688 31688 31688 41.01 0.000 Error 37 28590 28590 773 Total 39 77982 Figure 5.8: Analysis of Variance for Total Cost From the Figure we see that for factor demand distribution F ratio (22.91) > F 0.05, 1, 37 (4.02) and for factor model type F ratio (41.01) > F 0.05, 1, 37 = 4.02. Thus we conclude that the main effects of the factors demand distribution and model type are significant. The F ratio also indicates that model type has a higher effect than the demand distribution. Table 5.11 shows the mean values for total cost when both the production schedule models are replicated 100 times. Table 5.11: Total Cost in Thousands of Dollars for Production Schedules Distribution Variable Prod. Model Constant Prod. Model Normal 30.89 61.27 Poisson 58.13 119.25 37
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From Table 5.11 we observe that in the constant production schedule the uncertainty in demand causes higher backorder and inventory accumulation compared to the variable schedule model, thus resulting in higher total annual cost. The total cost is 98 % and 105 % higher when Normal and Poisson are used respectively as demand distributions. Thus costs can be significantly reduced when demand uncertainties are taken into account in the production schedule. 5.7 Summary Several experiments were conducted to see the impact of demand uncertainty and processing time variability. In the case of demand uncertainty the variability of the distribution has a greater impact than the shape of the distribution on the total annual cost comprising of backorder and inventory costs. The increase in annual production time with higher variability was small but it was found to be statistically significant. Finally the model with constant production schedule underestimates the average annual backorder and inventory. The total annual cost in the constant production schedule was considerably larger than the variable production schedule. The next chapter summarizes the entire thesis and states the scope for future research. 38
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CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 6.1 Summary and Conclusions A simulation model for a pharmaceutical batch plant producing four products was developed using Arena simulation software. All the products required the same processing steps. The production process had five stages, with parallel equipment in the first and fourth stage. The bottleneck stage dictated the cycle time; therefore parallel units were placed at the longest processing stages to minimize the cycle time. Zero wait transfer policy and single product campaign with optimal sequence were assumed. The batch sizes of all the products were assumed to be the same. A variable production schedule was developed by incorporating uncertainties in demand and processing time variability. Demand was assumed to arrive at the beginning of a planning period and satisfied at the end of the period. Excess capacity in a planning period was utilized for making inventory of products. If the demand could not be satisfied in a planning period the excess demand was backordered. Attempts to solve a similar problem (unpaced transfer line) using Markov chain were reported for only simplified cases based on restrictive assumptions. An example problem was modeled using variable production schedule with simulation as the tool. Cost was assigned to the average annual backorder and inventory and was treated as a performance measure for the model. Most of the literature for scheduling of multiproduct batch plant does not include demand uncertainties and processing time variability. A simulation model was also developed for the same example problem where the production schedule is fixed. The constant production schedule model had uncertain demand arrival, but the schedule was independent of demand variations. If excess demand arrived in a planning period then it 39
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was considered as backordered and if demand was insufficient compared to the constant schedule then the product was made for inventory. Total cost for inventory and backorder was set as the performance measure. The effect of demand uncertainty on the total annual cost for the variable schedule model was analyzed with different demand distributions. Analysis was performed for low and high demand variability in order to see if the shape of the curve had a greater impact or the variability. It was found that variability of the distribution had a significant effect on the total annual cost. Higher the variability the higher was the cost. Sensitivity analysis was performed to see the effect of different backorder to inventory cost ratios. The difference in total annual costs persisted with different backorder to inventory cost ratios for all the distributions. Variability in processing times was analyzed with the annual production time as the performance measure. The increase in annual production time with higher variability was small but it was found to be statistically significant. A two factorial design was performed to study the effect of the type of production schedule and the demand distribution on the total annual. It was found that the main effects of the factors demand distribution and model type are significant. The constant production schedule in previous research underestimates the average annual inventory and backorder. The total cost in variable schedule was found to be considerably lower in comparison to the constant production schedule. Demand uncertainties have a significant effect on the total annual cost and should be incorporated in the production schedule. 6.1 Future Research There are several extensions which have a scope for future research 1. In this research effect of demand uncertainty and processing time variability were analyzed separately. A model can be developed where both; the demand and processing time variability can be incorporated. 40
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2. Zero wait policy was modeled in this research; however, it would be interesting to analyze the effect of demand uncertainty and processing time variability on inventory accumulation in a finite intermediate storage policy. Comparisons can be made between zero wait policy and finite intermediate storage policy in terms of increase in production capacity versus work in process requirements. 3. It is well known that use of inprocess storage would reduce the cycle time and increase the total production. A model can be developed for such a case to see the increase in production in comparison to zerowait policy when demand is uncertain and processing times are variable. 41
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REFERENCES [1] Askin, R.G. and Standridge C.R. (1993) Modeling and Analysis of Manufacturing Systems. John Wiley and Sons, Inc. [2] Balasubramanian, J. and Grossmann, I.E. (2002) A Novel Branch and Bound Algorithm for Scheduling Flowshop Plants with Uncertain Processing Times. Comp. Chem. Eng., vol.26, pp.41. [3] Balasubramanian, J. and Grossmann, I.E. (2000) Scheduling to minimize expected completion time in flowshop plants with uncertain processing times. Proc. ESCAPE10, Elsevier, pp.79. [4] Biegler, W.L.T., Grossmann, I.E. and Westerberg, A.W. (1997) Systematic Methods of Chemical Process Design Prentice Hall. [5] Birewar, D.B. and Grossmann, I.E. (1990) Simultaneous Synthesis, Sizing and Scheduling of Multiproduct Batch Plants. Paper presented at AIChE Annual meeting, San Francisco, Ind. Eng. Chem. Res., vol. 29, pp. 22422251. [6] Birewar, D.B. and Grossmann, I.E. (1989) Incorporating Scheduling in the Optimal Design of Multiproducts Batch Plants. Comp. Chem. Eng., vol.13, pp. 307315. [7] Birewar, D.B. and Grossmann, I.E. (1989) Efficient Optimization Algorithms for ZeroWait Scheduling of Multiproduct Batch Plants. Ind. Eng. Chem. Res., vol. 28, pp.13331345. [8] Conway, R., Maxwell, W., McClain, J.O. and Thomas, L.J. (1988) The Role of Workin Process Inventory in Serial Production Lines. Operations Research, vol. 36(2), pp. 229241. [9] Fruit, W.M., Reklaitis, G.V. and Woods, J.M. (1974) Simulation of Multiproduct Batch Chemical Processes The Chemical Engineering Journal, vol 8, pp. 199211. [10] Graham, R.L., Lawler, E.L., Lenstra J.K. and Rinnooy Kan. A.H.G. (1979) Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey. Ann. Discrete Math., vol.5, pp. 287326. 42
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[11] Honkomp, S.J., Mockus, L. and Reklaitis, G.V. (1997) Robust Scheduling with Processing Time Uncertainty. Comp. Chem. Engg. vol. 21, pp.1055. [12] Ierapetritou, M.G. and Pistikopoulos E.N. (1995) Design of Multiproduct Batch Plants with Uncertain Demands Comp. Chem. Eng., vol.19, pp 627 632. [13] Ivanescu, C.V., Fransoo, J.C. and Bertrand, J.W.M. (2002) Makespan Estimation and Order Acceptance in Batch Process Industries When Process Times are Uncertain. OR Spectrum, vol. 24, pp. 467495. [14] Karimi, I.A. and Reklaitis, G.V. (1985) Deterministic Variability Analysis for Intermediate Storage in Non Continuous Processes. AIChE J., pp. 79. [15] Kelton, W.D., Sadowski R.P and Sadowski, D.A. (1998) Simualtion with arena Mcgraw Hill. [16] Knopf, F.C., Okos, M.R. and Reklaitis, G.V. (1982) Optimal Design of Batch/Semi continuous Process. Ind. Eng. Chem. Process. Des. Dev, vol. 1 pp. 7686. [17] Kuriyan, K. and Reklaitis, G.V. (1989) Scheduling Network Flowshops so as to Minimize Makespan. Comp. Chem. Eng., vol. 13, pp. 187200. [18] Kuriyan, K., Joglekar, G. and Reklaitis, G.V. (1987) Multiproduct Plant Scheduling Studies using BOSS. Ind. Eng. Chem. Res., vol. 26, pp.15511558. [19] Miller, D.L. and Pekny, J.F. (1991) Exact Solutions of Large Asymmetric Traveling Salesman Problem. Science, vol. 225, pp. 754761. [20] Modi, A.K. and Karimi, I.A. (1989) Design of Multiproduct Batch Processes with Finite Intermediate Storage. Comp. Chem. Eng., vol. 13, pp. 127139. [21] Petkov, S.B and Maranas, C.D. (1998) Design of SingleProduct Campaign Batch Plants under Demand Uncertainty. AIChE Journal, Vol. 44, pp. 896911. [22] Petkov, S.B and Maranas, C.D. (1998) Design of Multiproduct Batch Plants under Demand Uncertainty with Staged Capacity Expansions Comp. Chem. Eng., vol.22, pp 789 792. [23] Pistikopoulos, E.N., Thomaidis, T.V., Melin, A. and Ierapetritou, M.G. (1996) Flexibility, Reliability and Maintenance Considerations in Batch Plant Design under Uncertainty. Comp. Chem. Eng., vol. 16, pp. S1209S1214. [24] Reklaitis, G.V., Sunol, A.K., Rippin, D.W.T. and Hortacsu, O. (1992) Batch Processing Systems Engineering. Computer and Systems Sciences, vol. 143, Springerverlag, New York. 43
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[25] Rippin, D.W.T. (1983) Simulation of Single and Multiproduct Batch Chemical Plants for Optimal Design and Operation. Comp. Chem. Eng., vol. 7, pp 137 156. [26] Sahinidis, N.V. and Grossmann, I.E. (1991) MINLP Formulation for Cyclic Multiproduct Scheduling on Continuous Parallel Lines. Comp. Chem. Eng., vol. 5(2), pp. 85103. [27] Straub, D.A. and Grossmann, I.E. (1992) Evaluation and optimization of stochastic flexibility in multiproduct batch plants. Comp. Chem. Eng., vol.16, pp 6987. [28] Voudouris, V.T. and Grossmann, I.E. (1991) Mixed Integer Linear Programming Reformulations for Batch Process Design with Discrete Equipment Sizes. Ind. Eng. Chem. Res., vol. 30, pp. 671688. [29] Voudouris, V.T. and Grossmann, I.E. (1993) Optimal Synthesis of Multiproduct Batch Plants with Cyclic Scheduling and Inventory Considerations. Ind. Eng. Chem. Res., vol.32, pp.19621980. [30] Wellons, M.C. and Reklaitis, G.V. (1989) Optimal Schedule Generation for A Single Product Production Line1 Problem Formulation. Comp. Chem. Eng., vol. 13(1/3), pp. 201212. [31] Yeh, N.C. and Reklaitis, G.V. (1987) Synthesis and Sizing of Batch Semi Continuous Processes. Comp. Chem. Eng., vol.6, pp. 639654. 44
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APPENDICES 45
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Appendix A. Variable Schedule Model Demand A Demand B Demand C Demand D Product A Product B Product C Product D TimesReassigning New Cycle?TrueFalse week doneTill Previous 0 0 0 0 0 0 Figure A.1: Demand Arrival Sub Model 46
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Appendix A (Continued) Inventory on hand?TrueFalse Inventory?Can demand be met byTrueFalse Backorder Need Update Inventory Not Product D?TrueFalse to ZeroReset Next Cycle Time Remaining?TrueFalse Is it First Cycle?TrueFalse ?Inventory or BackorderTrueFalse Prodction TimeExpected Production TimeExpected RemainingAmount of Time OvertimeHow much RemainingAmount Time Time Over 0 0 0 0 0 0 0 0 0 0 0 0 Figure A.2: Evaluation Sub Model 47
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Appendix A (Continued) and DInventory for A Ratio of D?Produce Min DemandTrueFalse Make D ElseWhat to make? Just D ProductsInventory for All and BInventory for D A Time ?TrueFalse of BatchAssign Number Can make A?TrueFalse Inv a batch of A Route 3 Route 4 Station 2 0 0 0 0 0 0 Figure A.3: Inventory Sub Model 48
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Appendix A (Continued) Backorder?TrueFalse batch of ABackorder a Batch of D?TrueFalse ElseWhat To Backorder Back D and DBackorder for A Backorder only D A and BBackorder for D ProductsBackorder for All of BatchesAssign Number Backorder A?TrueFalse Station 1 Route 1 Route 2 0 0 0 0 0 0 Figure A.4: Backorder Sub Model 49
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Appendix A (Continued) NumberIncrease NumAssign Batch 1 Duplicate Is Batch Num 1?TrueFalse total?Batch Num less than Batch Number==Number of Batch (Product Type)Batch Number
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Appendix A (Continued) Reaction Recovery Purify Crystallize Centrifuge Satisfied Demand ReadWrite 1 recovererHold for reactorRelease Hold for purifier recovererRelease crystallizerHold for Release purifier CentrifugerHold for CrystalizerRelease Hold for reactor 0 0 0 0 0 0 Figure A.6: Production Sub Model 51
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Appendix B. Constant Production Model DemandsDifference in TrueFalseProd A? Inventory A Backorders A Prod B?TrueFalse Backorders B Inventory B Prod C?TrueFalse Backorder C Inventory C Prod D?TrueFalse Backorder D Inventory D 0 0 0 0 0 0 0 0 Figure B.1: Inventory and Backorder Sub Model 52
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Appendix C. Results for Total Cost with Low Demand Variability Table C.1: Total Costs in Thousands of Dollars for Low Demand Variability Uniform Normal Poisson 1 18.22 21.14 33.47 2 49.11 49.19 103.17 3 40.14 27.48 53.23 4 34.02 27.76 48.03 5 14.35 11.76 17.72 6 24.10 18.30 33.51 7 30.98 29.87 52.62 8 61.11 47.46 99.08 9 14.85 11.41 19.11 10 23.58 21.95 39.46 11 16.56 20.93 19.91 12 28.48 30.88 50.92 13 15.88 12.88 28.48 14 15.61 15.11 30.46 15 10.38 12.34 19.48 16 17.41 19.55 40.47 17 17.65 20.86 29.34 18 22.70 15.58 35.54 19 29.30 32.24 48.69 20 26.58 21.94 49.95 21 14.34 12.76 22.65 22 52.04 57.89 75.71 23 39.56 32.48 68.57 24 32.61 35.19 53.02 25 28.90 22.73 43.03 Uniform Normal Poisson 26 9.22 16.13 12.66 27 14.58 20.25 24.29 28 36.66 33.62 91.93 29 19.87 15.45 34.77 30 109.37 115.07 181.03 31 17.04 12.65 17.36 32 21.5 11.74 22.16 33 43.00 43.67 103.44 34 39.94 28.23 67.52 35 18.02 12.37 33.66 36 31.51 17.1 52.72 37 27.09 19.9 39.36 38 64.52 87.42 97.91 39 7.42 4.6 8.92 40 51.3 28.00 65.18 41 57.6 49.03 83.46 42 35.33 33.72 58.22 43 47.48 27.38 76.67 44 29.58 22.18 48.71 45 34.2 20.58 50.77 46 54.05 37.78 83.9 47 50.23 29.64 71.35 48 44.83 25.4 67.48 49 68.12 36.57 108.12 50 30.24 14.84 45.56 53
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Appendix C. (Continued) Table C.1: (Continued) Uniform Normal Poisson 51 40.22 32.6 77.00 52 85.11 45.1 114.69 53 23.88 8.77 39.14 54 76.12 77.44 175.95 55 30.34 32.8 59.22 56 74.33 60.58 113.9 57 24.7 16.2 20.21 58 96.77 105.5 134.87 59 109.52 80.03 156.75 60 22.08 17.8 28.92 61 51.73 62.05 96.85 62 11.3 9.63 15.01 63 64.28 64.3 121.81 64 28.6 27.76 42.52 65 49.5 42.11 96.25 66 26.44 16.9 32.46 67 101.93 98.6 195.66 68 19.31 13.52 25.36 69 15.62 12.35 25.83 70 52.53 35.19 75.78 71 31.64 10.93 18.78 72 66.12 39.6 80.55 73 24.09 18.36 36.81 74 8.45 7.85 10.14 75 26.65 21.21 43.02 Uniform Normal Poisson 76 36.01 32.34 57.97 77 17.84 14.00 27.16 78 36.14 40.46 44.92 79 18.77 25.49 27.98 80 50.18 40.45 70.49 81 65.64 45.14 105.22 82 12.17 10.27 14.58 83 19.6 18.13 21.26 84 34.3 20.25 40.17 85 26.07 20.44 42.19 86 68.40 57.66 104.57 87 38.54 33.1 57.03 88 21.8 22.36 31.12 89 60.03 50.07 100.35 90 47.6 44.45 87.96 91 29.51 20.58 42.59 92 37.17 15.12 44.2 93 32.95 30.42 46.11 94 43.6 30.55 48.35 95 27.75 18.55 31.13 96 33.22 26.01 45.34 97 26.78 24.6 32.00 98 78.21 64.3 104.31 99 11.77 10.25 15.68 100 26.54 24.02 70.43 54
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Appendix D. Results for Total Cost with High Demand Variability Table D.1: Total Costs in Thousands of Dollars for High Demand Variability Uniform Normal Poisson 1 36.77 35.92 33.48 2 103.68 74.26 103.17 3 79.47 93.03 53.23 4 69.21 57.50 48.00 5 26.90 26.12 17.72 6 44.73 37.52 33.50 7 62.39 58.93 52.62 8 129.71 113.34 99.08 9 31.75 24.14 19.11 10 52.88 51.31 39.46 11 41.90 36.22 19.91 12 65.80 68.94 50.93 13 36.94 29.76 28.48 14 35.66 39.42 30.46 15 22.29 23.83 19.50 16 34.50 43.75 40.73 17 36.66 37.82 29.34 18 50.36 40.00 35.50 19 59.01 61.48 48.70 20 66.76 58.16 50.00 21 31.45 22.60 22.65 22 142.06 97.96 75.72 23 64.73 75.20 68.57 24 68.60 61.65 53.00 25 71.94 45.05 43.00 Uniform Normal Poisson 26 28.73 32.00 12.66 27 32.79 41.97 24.29 28 87.39 83.26 91.93 29 45.51 36.08 34.77 30 270.79 211.70 181.03 31 33.26 22.50 17.36 32 26.00 24.12 22.16 33 107.93 92.93 103.44 34 73.27 57.10 67.52 35 34.55 26.12 33.66 36 56.54 52.25 52.72 37 55.19 47.41 39.36 38 166.46 155.34 97.91 39 13.81 13.26 8.72 40 99.00 76.00 65.18 41 111.65 117.00 83.46 42 76.34 66.75 58.22 43 82.00 65.64 76.67 44 47.30 42.08 48.71 45 68.69 44.53 50.77 46 97.81 72.75 83.90 47 116.66 71.76 71.35 48 77.28 84.47 67.50 49 136.75 91.30 108.12 50 57.09 43.76 45.56 55
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Appendix D. (Continued) Table D.1: (Continued) Uniform Normal Poisson 51 67.64 58.23 77.00 52 199.79 162.96 114.70 53 48.74 30.35 39.14 54 167.13 150.42 175.95 55 60.75 66.74 59.22 56 139.69 126.20 113.90 57 44.00 46.02 20.21 58 187.33 192.00 134.87 59 259.11 181.50 156.75 60 42.52 36.43 28.92 61 104.92 117.00 96.85 62 17.81 16.74 15.01 63 138.23 127.86 121.81 64 66.60 45.09 42.52 65 120.07 101.49 96.25 66 49.73 35.73 32.46 67 241.50 189.95 195.66 68 40.00 36.27 25.36 69 34.32 27.63 25.83 70 109.68 73.32 75.80 71 53.00 34.74 18.78 72 125.64 86.58 80.55 73 59.57 42.35 36.81 74 18.35 13.54 10.14 75 54.69 49.47 43.00 Uniform Normal Poisson 76 77.26 65.76 58.00 77 35.53 30.29 27.16 78 80.57 86.00 44.92 79 41.15 40.95 27.98 80 111.86 88.11 70.50 81 143.14 117.50 105.22 82 30.44 29.00 14.58 83 40.93 40.58 21.26 84 68.04 49.00 40.17 85 64.93 53.00 42.19 86 151.08 126.00 104.57 87 80.26 69.44 57.00 88 53.26 44.44 31.12 89 137.90 119.00 100.35 90 97.34 99.64 87.96 91 67.07 42.14 42.60 92 64.55 33.33 44.20 93 65.31 58.70 46.11 94 85.73 59.50 48.35 95 62.79 42.30 34.13 96 68.66 78.00 45.34 97 51.50 44.46 32.00 98 172.60 145.13 104.31 99 23.46 17.95 15.68 100 50.87 59.33 70.43 56
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Appendix E. Result for Variable Processing Times Table E.1: Annual Production Time in Hours CV=0.00 CV=0.05 CV=0.10 1 8589.58 8590.60 8588.66 2 8589.58 8591.19 8591.13 3 8589.58 8590.09 8589.97 4 8589.58 8589.33 8590.17 5 8589.58 8590.67 8592.63 6 8589.58 8590.41 8591.24 7 8589.58 8591.41 8593.51 8 8589.58 8591.13 8591.89 9 8589.58 8591.14 8591.19 10 8589.58 8590.72 8591.04 11 8589.58 8590.93 8591.23 12 8589.58 8591.66 8593.09 13 8589.58 8592.53 8593.02 14 8589.58 8589.64 8588.91 15 8589.58 8589.24 8588.10 16 8589.58 8590.57 8591.47 17 8589.58 8589.57 8588.94 18 8589.58 8589.92 8590.31 19 8589.58 8593.99 8594.68 20 8589.58 8590.97 8590.27 21 8589.58 8591.88 8594.90 22 8589.58 8591.19 8594.50 23 8589.58 8589.69 8589.65 24 8589.58 8592.14 8594.13 25 8589.58 8590.26 8593.92 CV=0.00 CV=0.05 CV=0.10 26 8589.58 8589.58 27 8589.58 8591.95 8592.94 28 8589.58 8590.56 8590.89 29 8589.58 8591.01 8592.44 30 8589.58 8590.30 8591.03 31 8589.58 8589.37 8593.09 32 8589.58 8589.43 8588.65 33 8589.58 8589.74 8591.28 34 8589.58 8591.05 8592.52 35 8589.58 8591.55 8592.97 36 8589.58 8590.88 8591.22 37 8589.58 8591.49 8591.06 38 8589.58 8592.25 8594.34 39 8589.58 8591.32 8591.89 40 8589.58 8590.64 8594.59 41 8589.58 8591.35 8590.88 42 8589.58 8589.60 8589.03 43 8589.58 8591.12 8593.04 44 8589.58 8589.97 8590.43 45 8589.58 8592.81 8594.86 46 8589.58 8589.16 8593.81 47 8589.58 8592.76 8594.79 48 8589.58 8590.73 8593.84 49 8589.58 8593.53 8597.49 50 8589.58 8592.44 8591.73 8589.58 57
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Appendix E. (Continued) Table E.1: (Continued) CV=0.00 CV=0.05 CV=0.10 51 8589.58 8592.62 8591.53 52 8589.58 8590.59 8593.84 53 8589.58 8589.90 8591.03 54 8589.58 8590.17 8591.44 55 8589.58 8588.79 8588.68 56 8589.58 8589.73 8591.14 57 8589.58 8590.29 8591.18 58 8589.58 8591.32 8591.48 59 8589.58 8591.87 8592.63 60 8589.58 8591.04 8594.30 61 8589.58 8590.44 8591.30 62 8589.58 8589.65 8594.88 63 8589.58 8591.08 8593.00 64 8589.58 8589.43 8589.48 65 8589.58 8588.92 8590.85 66 8589.58 8590.04 8590.63 67 8589.58 8590.09 8590.22 68 8589.58 8590.74 8595.41 69 8589.58 8590.65 8592.25 70 8589.58 8590.49 8592.40 71 8589.58 8591.10 8592.81 72 8589.58 8590.61 8592.23 73 8589.58 8592.27 8594.47 74 8589.58 8590.44 8591.77 75 8589.58 8589.91 8592.82 CV=0.00 CV=0.05 CV=0.10 76 8589.58 8590.35 8591.74 77 8589.58 8589.68 8589.99 78 8589.58 8591.87 8593.55 79 8589.58 8591.02 8591.72 80 8589.58 8590.89 8590.73 81 8589.58 8590.01 8591.77 82 8589.58 8589.93 8589.05 83 8589.58 8589.14 8590.74 84 8589.58 8590.77 8591.65 85 8589.58 8590.28 8591.07 86 8589.58 8591.46 8592.73 87 8589.58 8590.66 8591.51 88 8589.58 8590.98 8591.30 89 8589.58 8590.99 8590.89 90 8589.58 8588.81 8592.89 91 8589.58 8592.95 8595.26 92 8589.58 8590.34 8591.18 93 8589.58 8590.74 8594.96 94 8589.58 8591.52 8591.30 95 8589.58 8590.49 8591.09 96 8589.58 8592.55 8594.31 97 8589.58 8590.13 8590.55 98 8589.58 8591.33 8593.21 99 8589.58 8592.52 8594.90 100 8589.58 8590.26 8588.76 58
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Appendix F. Results for Two Factorial Design Table F.1: Total Cost in Thousands of Dollars for Comparison Run Order Blocks Demand Distribution Model Type Total Cost 1 1 2 2 131.30 2 1 1 1 21.14 3 1 2 2 154.51 4 1 2 2 117.54 5 1 1 1 49.19 6 1 2 1 33.48 7 1 1 2 71.46 8 1 1 1 27.48 9 1 2 1 103.17 10 1 2 1 53.23 11 1 1 1 27.76 12 1 1 2 78.92 13 1 2 1 48.03 14 1 1 2 58.01 15 1 1 2 77.40 16 1 2 1 17.72 17 1 1 1 11.76 18 1 2 1 33.51 19 1 1 1 18.30 20 1 2 2 161.48 21 1 1 2 30.34 22 1 2 2 52.45 23 1 1 1 29.87 24 1 2 1 52.62 25 1 2 1 99.08 59
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Appendix F (Continued) Table F.1 (Continued) RunOrder Blocks Demand Distribution Model Type Total Cost 26 1 2 2 113.29 27 1 2 2 168.40 28 1 2 1 19.11 29 1 1 2 56.40 30 1 2 2 156.98 31 1 1 1 47.46 32 1 1 2 95.47 33 1 1 2 78.04 34 1 2 2 102.46 35 1 1 1 11.41 36 1 1 1 21.95 37 1 2 1 39.46 38 1 2 2 91.58 39 1 1 2 46.48 40 1 1 2 49.07 60
