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## Material Information- Title:
- Configuring cell systems to handle variable demand
- Creator:
- Schaller, Jeffrey
- Publication Date:
- 1996
- Language:
- English
- Physical Description:
- xi, 200 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Capital costs ( jstor )
Cells ( jstor ) Computer technology ( jstor ) Heuristics ( jstor ) Integers ( jstor ) Linear programming ( jstor ) Machinery ( jstor ) Objective functions ( jstor ) Production costs ( jstor ) Unit costs ( jstor ) Decision and Information Sciences thesis, Ph. D Dissertations, Academic -- Decision and Information Sciences -- UF - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1996.
- Bibliography:
- Includes bibliographical references (leaves 178-199).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Jeffrey Schaller.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 002100607 ( ALEPH )
35109039 ( OCLC ) AKT9614 ( NOTIS )
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CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND By Jeffrey Schaller A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 ACKNOWLEDGMENTS I would like to thank Dr. Selcuk Erenguc, my supervisory committee chairman, for his invaluable guidance and patient encouragement throughout my years in the program. My supervisory committee cochairman Dr. Asoo J. Vakharia has also provided invaluable guidance during the past year and I am thankful that he brought his expertise in cellular manufacturing to our department. I also would like to thank Dr. Harold Benson, Dr. Patrick Thomson, and Dr. Chung-Yee Lee for serving on my supervisory committee and for the fine instruction they provided in the classes I attended. I would also like to thank Dr. Antal Majthay for the direction he provided while supervising my assistantship assignments. Finally, and most importantly, I want to thank my mother and father for the love and support they have provided my entire life. Anything good that I have ever accomplished is a result of having the best parents. TABLE OF CONTENTS ACKNOWLEDGMENTS. ii LIST OF TABLES vii LIST OF FIGURES. ix Abstract x Chapters 1 INTRODUCTION 1 A Brief Description of Cellular Layouts 1 Product Line Layout 1 Process Layout. 3 Cellular Layouts 4 Cell Formation 7 Early Cell Formation Techniques 8 Later Cell Formation Techniques 9 The Impact of Dynamic Demand on Cell Formation Decisions 11 Dynamic Demand's Impact on Cellular Layouts 11 Strategies to Handle Dynamic Demand while using a Cellular layout 16 Current Practices 18 Automotive Parts Manufacturer. 18 Enclosure Manufacturer 20 The Problem and its Terminology. 23 The Organization of this Proposal 24 2 REVIEW OF CELL FORMATION LITERATURE 26 Introduction 26 Procedures Based solely on a Part-Machine Incidence Matrix 30 Objectives and Constraints. 31 Frequently used objectives 31 iii Frequently used constraints. 34 Other objectives 35 Methods 37 Manual Identification. .. 38 Matrix Manipulation 39 Clustering using Similarity Coefficients 40 Graph Theory. 42 Mathematical Models 43 Other Methods 44 Evaluation of Procedures 44 Procedures Which use a Part-Machine Incidence Matrix and Incorporate Data to Provide Weights for Parts or Machines. 46 Objectives and Constraints. 47 Methods 48 Routing Based Procedures Without Machine Capacities 49 Operations Sequences. 50 Alternative Routings. 51 Procedures Which Incorporate Setup Information. 52 Procedures Which Incorporate Machine Capacities 55 Model 1 55 Model 2 57 Methods 59 Procedures That Consider Multiple Scenarios for Part Volumes. 60 Other Procedures 63 Summary 64 3 MODELS 66 Introduction 66 Model I. 67 Description of Model I 67 Model I Formulation 69 Description of Model I Equations. 71 An Example of Model I 74 Model II 77 Description of Model II. 77 Model II Formulation. 79 Description of Model II Equations 81 An Example of Model II 83 4 SOLUTION PROCEDURES. Introduction . Model I Heuristic . Lower Bounds for Model II. . Linear Programming Relaxation. Lagrangean Relaxation . Linear Programming Relaxation Tightened Adding Valid Inequalities Valid Inequality Set 1 . Valid Inequality Set 2 . Valid Inequality Set 3 . Heuristics for Model II . Heuristic 1. . Description of Heuristic 1 . Formal Procedure for Heuristic 1 Discussion of Heuristic 1 . Heuristic 2. . Heuristic 3A . Description of Heuristic 3A. Formal Procedure . Discussion of Heuristic 3A . Heuristic 3B . 5 DESCRIPTION OF EXPERIMENTS AND DATA S 87 88 91 91 92 by 94 94 94 96 S 97 98 98 S 98 100 101 S 101 S 105 .105 .107 S 110 111 114 Introduction 114 Experiments 115 Experiment 1 Data 116 Part-Machine Incidence Matrices 117 Processing Times 119 Part Demand. 120 Inventory and Backorder Costs. 120 Machine Capacities and Number of Machines of Each Type 121 List of Potential Cells. 122 Cell Production and Intercell Production Costs 123 Summary 124 Experiment 2 Data 125 6 RESULTS. 127 Results of Experiment 1 127 Lower Bounds for Model II Heuristic Results for Model Discussion of Results Lower Bounds. . Heuristics . Results of Experiment 2 . 7 CONCLUSIONS AND FUTURE RESEARCH. Conclusions . Future Research . APPENDICES A FORMULATION OF MODEL I EXAMPLE B FORMULATION OF MODEL II EXAMPLE C PROBLEM 1 DATA AND SOLUTION Problem 1 Data . Problem 1 Solution . LIST OF REFERENCES. . BIOGRAPHICAL SKETCH . S 127 I 130 . 131 131 S 136 . 138 S 142 S 142 . 144 147 S 149 S 151 . 151 S 171 S 178 S 200 LIST OF TABLES TABLE pace 1. Part Routing. 5 2. Cellular Layout. 5 3. An Example of a Part-Machine Incidence Matrix. 8 4. Capacity Consideration 10 5. Part Family Machine Requirements.. 12 6. Load on Machine Type by Period. 13 7. Average Load. 13 8. Part-Machine Incidence Matrix for Model I Example. 75 9. By Part Demand.. 75 10. Machine Data. 75 11. Potential Cells. 84 12. Production Costs 85 13. Part-Machine Data Sets 118 14. Summary of Experimental Problems 124 15. L.P. and Lagrangean Relaxations 128 16. Valid Inequality Relaxations 129 17. Heuristics 1 and 2. 132 18. Heuristics 3A and 3B 133 19. Problem 1 Results 140 20. Problem 2 Results 140 21. Problem 1 Selected Cells 172 22. Part/Cell Processing 173 LIST OF FIGURES FIGURE page 1. Product Line Layout 2 2. Process Layout 3 3. A Process Layout. 6 4. A Cellular Layout.. 6 5. Family 1 Load by Period.. .. 14 6. Family 2 Load By Period.. 14 7. Total Load by Period.. 15 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND By Jeffrey Schaller May 1996 Chairperson: Dr. S. Selcuk Erenguc Major Department: Decision and Information Sciences This research addresses the problem of configuring cell systems for environments that experience variable demand. Past cell formation models consider only average product demand to determine configurations of cells. Short term fluctuations in demand may cause a cell configuration that would appear to be good, when considering only average demand, to perform poorly because during some periods machines of a certain type may be under utilized in one cell while machines in another cell may not be able to produce the required products. For these reasons, a methodology for configuring cell systems in environments that experience variable demand is needed. In this dissertation, two mathematical models are formulated. The first model is used to develop a list of potential cells, and the second model is used to select cells to include in a shop. Since both models are computationally complex, this research develops heuristic procedures to solve the models. Given that other approaches can also be used to develop a list of potential cells, the primary focus of this research is on the second cell selection model. Lower bounding procedures are proposed for this model. In order to test the effectiveness of the lower bounding and heuristic procedures for the second model, fifteen problems from the literature were modified and solved optimally. The lower bounding and heuristic procedures were then used on these problems, and the lower bounds and objective function values are compared to the optimal objective function values. A second experiment was conducted, in order to test the effectiveness of the lower bounding and heuristic procedures on larger problems. Using two larger published data sets the results are compared to the results obtained using the lower bounding and heuristic procedures. CHAPTER 1 INTRODUCTION 1.1 A Brief Description of Cellular Layouts Cellular Manufacturing is a manufacturing application of the group technology philosophy where groups of processes are dedicated to sets of products or parts. The associated layout (referred to as cellular layout) is a form of production layout that combines features of two traditional forms of plant layout: product line layout and process layout. 1.1.1 Product Line Layout In a plant with a product line layout, machinery and equipment are dedicated to a specific product, and thus, machines needed to manufacture each product are laid out in a way that facilitates efficient material flow. The advantages of product line layout are fast throughput times, efficient material handling, simplified scheduling, and control and low work-in-process inventories. The disadvantages of product line layout are that it requires a high level of investment in specialized machinery and equipment, and it lacks flexibility because only one type of product can be manufactured without incurring a high changeover cost. This type of layout is typically used for high volume, standardized products. Examples are automobiles, television sets and appliances. An example of a product line layout is shown in Figure 1. ,- Product Flow [] Machines S-B6 > Product 1 )I.> Product 2 J A- N .-> Product3 Figure 1. Product Line Layout. 1.1.2 Process Lavout A process layout groups machines which perform similar functions into departments. Many different types of products may use a specific machine type and each product may visit different machine types in various sequences. An example of a process layout is shown in Figure 2. --+ Product Flow l Machines Product 3 SProduct 2 Product 2 Figure 2. Process Layout. The products in this example have the same routings as the products in Figure 1 but notice that the material flows are much more jumbled than in Figure 1. The advantage of a process layout is that many different types of products can be manufactured on the same set of machines which leads to flexibility in the utilization of machinery. The disadvantages of a process layout are that material handling costs are high, material flows are complex which causes planning and control to be difficult, flow time is long, setup costs are high and work-in-process inventory is high. Typically low volume or nonstandardized products are produced using a process layout. Examples include (Hyer 1984): aerospace, agricultural machinery, business machines, control devices, diesel engine assemblies, machine tools, and machined parts. 1.1.3 Cellular Layouts Cellular layouts are hybrid layouts which combine features of both product and process layout. The objective of implementing a cellular layout is to enable a manufacturer of low volume specialized or custom products to obtain some of the benefits of product line layout while maintaining some of the flexibility of a process layout. This is done by forming product families which consist of products that require the same types of machines (preferably in the same sequence). A machine of each machine type (or more than one if required) required to produce the products in a family is put into a cell and is dedicated to producing a family of products. As an example consider the products shown in Table 1. Table 1. Part Routing. Product 1 2 3 4 5 6 Table 2. Machine Sequence 1 4 5 5 4 1 2 3 5 2 3 4 5 1 2 3 4 1 3 2 4 Cellular Layout. Machines 1,4,5 2,3,4,5 1,2,3,4 Products 1,2 3,4 5,6 Figure 3 (on the following page) shows the process layout for producing these products. Figure 4 shows the Cell 1 2 3 resulting product flows if the three cell layout in table 2 is used. Process Layout --p Product Flow 1 Machines Product 2 - Product 3 - Product 4 > Product 5 Product 6 Figure 3. A Process Layout. Cellular Layout *-0 Product Flow ] Machines Figure 4. A Cellular Layout. 7 Notice the simpler product flows in the cellular layout vs the process layout. Creating cells in this manner enables material handling costs to be reduced and materials handling flows to be simplified. This in turn can lead to reduced flow time, reduced work-in-process inventories and simplified planning and control. Setup time and cost for parts manufactured in a cell can typically be reduced by developing specialized procedures for setups required in the cell. Quality improvement is also usually a benefit because cell operators have more experience manufacturing a specific set of products. 1.2 Cell Formation In order to implement cellular manufacturing cells must be formed. The cell formation problem includes the following decisions. How many cells should there be? Which machine types should be included in a cell? How many machines of each type should be included in a cell? Which parts should a cell produce? 1.2.1 Early Cell Formation Techniques Most of the early cell formation algorithms used some form of part-machine incidence matrix. In a part-machine incidence matrix machines are represented as rows and parts are represented as columns in the matrix (or vice versa). Each element in the matrix is '1' if part j requires machine i for processing and '0' otherwise. Table 3. An example Machine Type A B C D a part-machine incidence matrix Table 3 shows an example of a part-machine incidence matrix. In this example part 1 requires machine types A and C but not B and D, and part 5 requires machine types A, B, and D but not C. The algorithms use the part-machine incidence matrix as input and then use various techniques to break up the parts and machines into cells. These techniques include manual identification (Burbidge 1963); clustering using measures of similarity (McAuley 1972, Carrie 1973); graph theory using measures of similarity (Rajagopalan and Batra 1975, Chandrasekharan and Rajagopalan 1986, Vannelli and Kumar 1986, and Kumar, Kusiak, and Vannelli 1986); and matrix manipulation (King 1980, King and Nakornchai 1982, Chan and Milner 1981, and Chandrasekharan and Rajagopalan 1986). 1.2.2 Later Cell Formation Techniques Many of the algorithms that were developed later tend to be more sophisticated and are based on mathematical models that include operational considerations in the objectives and constraints. These models also usually include some sort of integrality constraint. Operational considerations that have been incorporated into algorithms include operations sequences, machine setup similarities between parts, part volumes, and machine capacities. A typical integrality constraint is that the number of machines of a machine type that are assigned to a cell is required to be integer. Table 4. Capacity Consideration Part Hours required Family on machine type 1 1 2700 2 2500 Total 5200 # of type 1 machines available 3 Capacity per type 1 machine 2000 hrs Total capacity 6000 hrs For an example of why capacity, volume, and machine integrality are important considerations when making cell formation decisions, consider Table 4. In this example two part families have been formed and both families have parts which require processing on machine type 1. The table shows the hours required on machine type 1 for each family, the number of type 1 machines available, and the capacity per type 1 machine. If we could assign 1.5 type 1 machines to each family, we would have ample capacity to produce both families and everything would be fine. However, we cannot dedicate fractions of machines to families. If we assign two machines to family 1 and one machine to family 2, then there 11 will be idle capacity dedicated to family 1 while we do not have enough capacity to produce family 2. If we assign two machines to family 2 and one machine to family 1, a similar situation arises. One alternative to this problem is to combine both families and create a larger cell by assigning all the machines to this single cell. Using this alternative there is adequate capacity to produce both part families but, because the cell is larger and the parts that are produced in the cell are more diverse, some of the benefits of cellular manufacturing are likely to be lost. 1.3 The Impact of Dynamic Demand on Cell Formation Decisions 1.3.1 Dynamic Demand's Impact on Cellular Layouts Demand for a product is said to be dynamic if it varies from period to period. Past cell formation models consider only average product demand to determine the configurations of cells and which products to produce in a cell. Short term fluctuations in demand may cause a cell configuration that would appear to be good, when considering only average demand, to perform poorly. During some periods machines of a certain type may be underutilized in one cell while machines in another cell may not be able to produce the required Table 5. Part Family Machine Requirements. Part Family Required Machines 1 2 6 7 2 1 5 6 quantities of products. For an example consider the part families in Table 5. Suppose there are 2 machines of type 6 available and the capacity of a type 6 machine is 2500 units per period. The load on machine type 6 by period for each family is shown in Table 6. Table 7 (on the following page) shows the average load on machine type 6 for each period. If one machine of type 6 was dedicated to producing family 1 and the other was dedicated to producing family 2 there would be enough capacity in each cell to meet the average load. If you look at the by-period requirements in Table 6, however, you can see that insufficient capacity would exist to produce family 2 while capacity would be underutilized in family l's cell during the first three periods. During the last three periods the situation is reversed. When looking at the load of both families in aggregate there is enough Table 6. Load on Machine Type by Period. Load on machine type 6 by period Period 1 2 3 4 5 6 Family 1 1480 2068 1256 2676 2831 3360 Family 2 2856 2914 3144 2154 738 674 Total 4336 4982 4400 4830 3569 4034 Table 7. Average Load Average Family 1 2278.5 Family 2 2080.5 Total 4358.5 Capacity 2500 2500 5000 capacity to produce both families during every period. Figures 5, 6, and 7 show the load variations graphically. Figure 5 (on the following page) shows the load caused by family 1, Figure 6 the load caused by family 2, and Figure 7 shows the load in aggregate. The aggregate load is much more stable than the individual family loads. This illustrates an advantage of a process layout when there is dynamic demand. By combining all the machines of a given type into a 3500 Family 1 Load by period ...ii I -. 500 1 2 3 4 5 e PklWN Figure 5. Family 1 Load by Period. Family 2 Load by period Figure 6. Family 2 Load by Period. Y PloM Total Load by period WM- 1000/ 1 2 3 4 5 6 Figure 7. Total Load by Period. department and allowing any product that requires that machine type to use any machine of that type flexibility is provided so that the peaks and valleys of a product's demand can be offset by other products. Studies have shown the adverse effects of variable demand on a cellular layout. Flynn and Jacobs (1986, 1987) conducted simulation studies which show that when demand variability is high a process layout is favored over a cellular layout in terms of mean flowtime. Garza and Smunt (1991) showed that demand variability can cause intercell flows in a cellular layout and this results in less stable flow time performance as compared to a process layout. 1.3.2 Strategies to Handle Dynamic Demand While Using a Cellular Layout There are strategies which can be used to offset the effects of demand variability in a cellular manufacturing system. These include combining cells to smooth out load variations, routing parts through alternative cells, let parts move between cells (intercell movement), hold inventories of parts, and incur backorders for parts. Combining cells to form larger cells allows the large cell to produce a greater variety of parts than the individual small cells. Since there are more parts it is likely that machine loads will be more stable in the large cell. The large number of parts produced by the cell will also cause more complex materials flows and higher material handling cost. Therefore, there is a tradeoff that must be considered when choosing a cell's size; stable loads in large cells vs lower material handling cost in small cells. Cells can also be formed so that there is more than one cell that can produce a part. Hence, if there is insufficient capacity to produce the part in its preferred cell, it may be possible to produce the part in a different cell. By having alternative cells, load variations can be 17 smoothed out while meeting product demand. The disadvantage of this approach is that it may be more costly to produce parts in their alternative cells and/or there may be a need to duplicate more machines to develop these types of configurations. If there is insufficient capacity for a machine type to produce all the parts required, then some parts can be moved to another cell which has that machine type for processing. This is called an intercell transfer. Ang and Willey (1984) showed that low levels of intercell transfer can greatly improve the flow time performance of a cellular layout. The disadvantages of intercell transfer are that this requires materials handling capability between cells, there is increased paper work required to track part movement, and quality could be adversely affected because of the loss of responsibility by a single cell for the part's quality. A final strategy that could be used is that during periods of low demand, inventories of parts can be produced which can be used to meet demands for parts during periods when insufficient capacity exists in the cells to produce all the parts required. Alternatively parts could be backordered when capacity is not available in a cell and then produced during a later period when there is adequate capacity in the cell. Using inventories and backorders will level out the load requirements in the cells, but the cost of holding inventories or backordering parts must be incurred. 1.4 Current Practices In order to find out how cellular layout is used in industry, we visited two plants that utilize a cellular layout. The first plant produces parts for automotive manufacturers. The second plant produces steel enclosures that are used for industrial applications to hold electrical circuitry. A brief description of how each plant uses a cellular layout and how they deal with demand variability presented next. 1.4.1 Automotive Parts Manufacturer This plant produces parts that are major components in the assembly of automobiles and some service parts. It uses product line, process, and cellular layout. Major factors in the choice of layout are maintaining machine utilization and efficient use of space. The plant has over 20 cells that are 19 used for production. In some cases the cells are production lines that produce one part. One cell is a small job shop that produces approximately 150 parts. The other cells are of varying sizes in terms of the number of machines they contain and the number of products they produce. There are certain processes in the plant which are strictly process departments. An example of this is the paint department. All parts that require painting go to the paint department (there are no paint facilities in the cells). The reason why this department was chosen to be a process layout is that one oven was sufficient to handle the entire plant's requirements. The plant also has 20 high speed roll mills. Fourteen of the mills are grouped together in a department and six have been moved to cells. The roll mills which have not been moved are large mills which perform complicated processing. It was decided that it would be too expensive and require too much space to duplicate these machines in multiple cells. The mills that were moved to cells are smaller and the capacity of the mills matches the requirements demanded by their cells. Some of the cells were created by combining production lines. This enabled the plant to better utilize equipment and save space. The plant experiences demand for products that is variable. This is particularly true with service demand which is very lumpy. A variety of strategies are used to handle this variability. Many of the low volume parts, including service parts, are produced in the large job shop cell. The varying demands of the individual parts tend to offset each other and help stabilize the load in the cell. Sometimes a large order is produced partially in advance of its due date and parts are inventoried. There are instances when orders are rescheduled to a later date. There are also times when production capacities in other cells are used to perform some operations and then intercell transfers of parts occur. When this happens, it is a disruption in the plant because all the presses for the job must be setup at the same time to maintain quality. The primary benefits obtained by using a cellular layout in this plant are reduced equipment and space, reduced material handling cost, and reduced setups. 1.4.2 Enclosure Manufacturer Approximately 2800 standard end items are produced in this plant. The plant also produces some custom jobs. The standard end items are produced and then sent to regional warehouses. Roughly 200 items account for 90% of the units produced by the plant. This plant originally used a process layout. The current layout slowly evolved to a hybrid cellular process layout. Currently there are three cells and plans for more cells are being formulated. The cells are designed to produce high volume products very efficiently and are setup as flow lines. Currently 75% of the unit volume and 40% of the dollar volume is produced using cells. High volume production in cells is undertaken to justify the investment in dedicated equipment and tooling. The demand for standard end items is variable and seasonal while demand for custom items tends to be lumpy. The plant responds to demand variability using a variety of methods. A portion of the plant is still using a process layout. This portion of the plant is generally used to produce low volume standard items and custom items. The demand for these items tend to be more erratic, but by producing many items on the same equipment, the load generated is fairly stable. High volume items are produced in the cells and the demand for these items tends to be fairly stable. There are 22 times, however, when some facilities in the plant are idle while other facilities become bottlenecks. The plant also had to invest in additional equipment to build cells that could meet the seasonality of demand. Sometimes custom jobs will be processed in cells if the equipment in the cell is underutilized. At other times, a product normally produced in a cell will be produced in the process portion of the plant if the cell capacity is exceeded. Inventory is also held and customer orders are backordered to help deal with demand variability. The following benefits have been attributed to the use of cellular layout in the plant. 1) Material handling cost has been significantly reduced because conveyors are used to move materials in the cells. 2) Setups have been significantly reduced because tools have been dedicated to products and designed for quick setup. 3) Productivity has increased. Labor costs have been cut in half. 4) Leadtimes have been reduced from 4 weeks to 2 weeks. 5) WIP inventory has been dramatically reduced. 6) Quality has been improved due to the fast feedback provided by the cellular layout. 7) The plant has become a safer place to work because there is 23 less material handling and simpler setups. 8) The fill rate on customer orders has increased from 50% to 80%. 1.5 The Problem and its Terminology Based on our experience and industry visits, the problem we address is as follows. Given a set of parts, a finite planning horizon with T periods, and a set of machine types, the problem is to specify a set of cells to be included in the shop and a production and inventory plan that will minimize total cost. For each cell the number of machines of each type must be specified. The set of cells to be included in the shop can contain cells of varying sizes in terms of the total number of machines included in each cell. For example the shop could include several small cells and a large "remainder cell." A remainder cell is a portion of the shop that typically is organized as a process layout. The production and inventory plan must specify the quantity produced of each part in each cell during each period, and the quantities of each part produced using intercell movement during each period. Each part has associated with it the following: (a) a set of machine types that are required to produce the part and the processing time required by the part on each machine type; (b) a demand deterministicc) for each period during the planning horizon; (c) a cost to hold one unit of inventory for one period (unit inventory holding cost); (d) a cost to hold one unit on backorder for one period (unit backorder cost); (e) costs to produce a unit of the part in various size cells (in terms of the number of machines contained in the cell). (These costs are referred to as cell production costs. It is assumed that larger cells have a higher cost); (f) a cost to produce a unit of the part in two or more cells (intercell movement cost). Each machine type has associated with it the following: (a) the number of machines of that type that are available and (b) the capacity of a single machine of that type. 1.6 The Organization of this Dissertation The remainder of this dissertation is organized as follows. Chapter 2 contains a survey of cell formation literature. Chapter 3 introduces the two models that can be 25 used to develop cellular configurations, and Chapter 4 describes solution procedures for these models. Chapter 5 describes the experimental data used to test the solution procedures described in Chapter 4, and Chapter 6 presents computational results of these tests. Finally Chapter 7 proposes future research directions. CHAPTER 2 REVIEW OF CELL FORMATION LITERATURE 2.1 Introduction The cell formation problem has attracted more academic interest than any other aspect of cellular manufacturing. The reason research has focused so heavily on cell formation problem is due to the complex nature of the cell problem. In order to design a cellular system decisions must be made as to how many cells there should be, which machine types should be included in a cell, and which parts should a cell produce. Arriving at these decisions constitutes a solution to the cell formation problem. Many aspects of the effectiveness and efficiency of an operation will be affected by the cell formation decisions. Examples are labor costs, investment, leadtimes for products, supervision of personnel, quality, and amount of paperwork needed to control the operation. This leads to the consideration of many possible objectives in arriving at a solution. The 26 problem is also complex because an entire machine usually must be assigned to a cell. Because of this most mathematical models of the problem are integer programming models which are difficult to solve. The procedures that have been developed to solve the cell formation problem are diverse with respect to 1) the objectives and constraints considered by the procedure, 2) the data required as input for the procedure, and 3) the techniques used by the procedure to solve the problem. The first two items are somewhat related. Usually, to incorporate an objective or a constraint into a problem, data are required to support that objective or constraint. For example, if a constraint of the problem is to use only existing equipment to configure cells, then data by machine type of the number of machines available are required. Many procedures use objectives that are surrogates for other objectives which if incorporated would require additional data and cause the problem to become more complex. For example, a possible objective when designing a cellular system is to maintain a minimum level of machine utilization for each machine. This objective would require data on part volumes, part processing times, and machine capacities. Many cell formation procedures use a surrogate measure for machine utilization based on the percentage of parts to be produced by the cell that uses a machine. This measure is used in the hope that insight into machine utilization will be gained without explicitly considering the data required to determine machine utilization. Of course, surrogate measures can sometimes turn out to be a poor representation for the desired objective or constraint. The diversity of the cell formation procedures with respect to the items mentioned above causes the development of a framework for reviewing the procedures to be difficult. The most comprehensive review of cell formation procedures to date was done by Wemmerl6v and Hyer (1986). Over 70 procedures are categorized in Wemmerl6v and Hyer's paper. The authors classify cell formation procedures based on which of four general approaches was used in the procedure. In all four approaches parts and machines must be selected for possible inclusion in the cellular manufacturing system. An evaluation step of the proposed cells is also required. The four approaches are 1) identify part families without the help of machine routings (including procedures that use group technology codes and those that do not); 2) identify 29 machining groups; 3) identify part families using routings; 4) identify part families and machine groups simultaneously. The advantages of this method of classification are that it is inclusive and provides classifications for procedures which perform similar functions. Within each of the last three approaches, however, the procedures can vary greatly with respect to the objectives and constraints considered explicitly by the procedure as well as the data required for the procedure. In this literature review we propose six categories which are based on the data used, by a procedure, to classify procedures. There are two reasons for classifying procedures this way. 1) Identifying data used by a procedure provides insight into factors that are explicitly considered by the procedure and helps determine the strengths and limitations of the procedure, and 2) procedures that use similar data can be compared with one another. The six categories that were chosen are 1) procedures that are based solely on data that can be represented by a part-machine incidence matrix, 2) procedures that use a part-machine incidence matrix and incorporate data which can be used to provide weights for parts or machines, 3) procedures that incorporate operations sequences, 4) procedures that consider part setup times or costs, 5) procedures that include part volumes, part processing times, and machine capacities, and 6) procedures that consider multiple scenarios for part volumes. Some procedures may fall into two or more categories. There, also, are a few procedures which use data that do not fall into any of the categories, therefore, a seventh category called "other" is included. For each category discussions of the objectives and constraints considered by procedures, and methods used by procedures to develop a solution are included. 2.2 Procedures Based solely on a Part-Machine Incidence Matrix These are the most common of the cell formation procedures. The concept of a part-machine incidence matrix was described in chapter 1. As a brief refresher, in a part- machine incidence matrix machines are represented as rows and parts are represented as columns in the matrix (or vice versa). An element in the matrix is '1' if part j requires machine i for processing and '0' otherwise. The advantages of these procedures are that they have low data requirements (all the data can be obtained from part routings) and they tend to be relatively simple and efficient. The disadvantage of these procedures is they do not consider factors such as operations sequences, part volumes, machine capacities, and demand variability. When a solution is developed using one of these procedures the cell designer must somehow factor in other important considerations to arrive at a final solution. There are several objectives and constraints considered by procedures in this category and many techniques have been used to develop solutions. 2.2.1 Obiectives and Constraints The most common objectives used in these procedures are to minimize exceptional elements, minimize duplicate machines, and maximize a measure of similarity between parts or machines assigned to a cell. The two most common constraints are maximum cell size and measure of similarity threshold. 2.2.1.1 Frequently used objectives Exceptional elements. Many procedures (Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and Milner 1982, 32 King and Nakornchai 1982, Waghodekar and Sahu 1984, Boctor 1989, Chow and Hawaleshka 1993, Khator and Irani 1987, and Boctor 1991) have as an objective the minimization of exceptional elements. An exceptional element occurs when a part requires two or more cells for processing. This objective is important because exceptional elements would require increased material handling cost and paperwork cost, or subcontracting cost to remove the exceptional element from the plant. If many parts are exceptional elements when a cellular system is designed the benefits of the system will be greatly diminished. Machine duplication. Minimization of machine duplication is also an objective used by many procedures (Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and Milner 1982, King and Nakornchai 1982, Khator and Irani 1987, and Al-Qattan 1990). Machine duplication (also sometimes referred to as a bottleneck machine) occurs when a machine type is required in two or more cells. When a machine type is required in two or more cells additional machines may be required which increases machine investment and may cause machine utilization to be lower vs a process layout. This objective is usually used in conjunction with the exceptional element constraint. Most procedures trade off exceptional elements against machine duplication. Vannelli and Kumar's (1986) procedure attempts to minimize the number of machine types that must be duplicated while allowing no exceptional elements. Measures of similarity. Several procedures attempt to maximize measures of similarity between pairs of parts or pairs of machines (Rajagopalan and Batra 1975, Carrie 1973, Mcauley 1972, Kusiak 1987, De Witte 1980, Chandrasekharan and Rajagopalan 1986a, Luong 1993, Askin et al. 1991, Srinivasan et al. 1990, Gunasingh and Lashkari 1989b, and Wei and Kern 1991). Measures of similarity between pairs of parts are based on how similar the processing requirements are for the parts. Measures of similarity between pairs of machines are based on which parts require the machines. Creating cells that produce parts that have similar processing requirements and machines that process similar parts is one of the main objectives of cellular manufacturing, therefore, this is an important objective. Most of the measures of similarity that are used in cell formation are based on some sort of similarity coefficient (the concept of a similarity coefficient will be described 34 in the next section) which can take on a value between 0 and 1. 2.2.1.2 Frequently used constraints Cell size. The most popular constraint used by these procedures is a cell size constraint (Chandrasekharan and Rajagopalan 1986, Boctor 1989, Luong 1993, Stanfel 1985, Boctor 1991, Wei and Kern 1989, Al-Qattan 1990, and Vannelli and Kumar 1986). The cell size constraint limits the number of machine types allowed in a cell. The constraint is imposed to limit the physical size of cells. It is generally assumed that intracell material handling cost will be insignificant but this assumption loses validity as cell size grows. This constraint is used with each of the objectives described above. Without this constraint, a cell consisting of all machine types could be created which would have no exceptional elements and would not require any duplicate machines. Similarity threshold. A second constraint which is sometimes imposed is a measure of similarity threshold (Carrie 1973, McAuley 1972, and Chandrasekharan and Rajagopalan). This constraint is usually used in conjunction with the objective that maximizes some similarity measure. 35 The similarity threshold constraint does not allow a machine or a part into a cell unless the cell's similarity measure is above the threshold. The threshold can be applied to a single pair of items (single linkage) in a cell, any pair of items (complete linkage) in the cell, or the average similarity between items (average linkage) in the cell. If the threshold constraint is not applied then a single large cell would result and the shop would be a process layout. If a high threshold is applied to a measure of similarity between machines the resulting layout would have small cells and possibly many exceptional elements. If a high threshold is applied to a measure of similarity between parts then the result would be many part families and to create cells for each family could require many machine types to be duplicated. Picking a similarity threshold requires judgement by the cell designer. Usually the procedures would be run with several similarity thresholds and then a part- machine structure would be selected. 2.2.1.3 Other objectives Chandrasekharan and Rajagopalan (1986) and Kumar and Chandrasekharan (1990), and Stanfel (1985) have developed three measures which are slightly more sophisticated and use these measures as objectives. Grouping efficiency. The first measure is called grouping efficiency (Chandrasekharan and Rajagopalan 1986). This measure is a convex combination of within-group utilization and inter-cell efficiency. Within-group utilization is a measure of how heavily parts visit machines in a cell (summation of the number of machine types used by each part in each cell divided by the summation of the number of machine types times the number of parts in each cell). Inter-cell efficiency equals the fraction of exceptional elements divided by the number of non- exceptional elements. This measure serves as a surrogate for a tradeoff between machine investment and materials handling cost. It is speculated that cell layouts with a high within- group utilization will result in a shop in which machine types have high utilization because most of the parts in the cell use each machine type in the cell. This should result in a lower required machine investment than a shop that has a low within-group utilization. A cell layout with a high inter-cell efficiency should have a low material handling cost because there are relatively few exceptional elements. Grouping efficacy. The second measure is called grouping efficacy (Kumar and Chandrasekharan 1990). Grouping efficacy is similar to grouping efficiency but is adjusted to take into consideration the density of the part-machine incidence matrix (the density of the part-machine incidence matrix is equal to the 1 entries in the matrix divided by total entries). Weighted transfers. Stanfel (1985) uses an objective that considers both inter-cell and intra-cell transfers. Intra-cell transfers for a part are based on the number of machine types that are in a cell that are not required to process the part. The objective weights both inter-cell and intra-cell transfers (intra-cell transfers usually have a lower weight). This objective serves as a surrogate for material handling cost and also considers cell size. Cells with more machine types are likely to be larger and require a higher intra-cell movement cost. 2.2.2 Methods A wide variety of methods have been used to solve the cell formation problem with only a part-machine incidence matrix as input. The most popular methods have been manual identification, matrix manipulation, clustering using similarity coefficients, graph theory, and solving mathematical models. 2.2.2.1 Manual identification Probably the earliest procedure to be used for cell formation is a manual procedure called production flow analysis developed by Burbidge (1963). This procedure uses part routings to analyze inter and intra departmental flows. There are two stages in production flow analysis: 1) factory flow analysis and 2) departmental flow analysis. Factory Flow analysis studies the basic routes in the factory and then eliminates exceptions. Departmental analysis is broken into group analysis and line analysis. In group analysis parts are divided into families so groups of machines can be put together so that the parts in a family can be completely manufactured by the group. The main criterion is that the division into groups should not necessitate an increase in the number of machines required. El-Essawy and Torrance (1972) also develop a manual procedure called component flow analysis. This procedure is similar to production flow analysis. Since production flow analysis and component flow analysis are manual procedures they become cumbersome to apply as the problem size becomes large. 2.2.2.2 Matrix manipulation This method attempts to create a block diagonal structure, in which almost all l's occupy the diagonal submatices, and almost all zeros occupy the off-diagonal submatrices, by manipulating the data in the part-machine incidence matrix. King (1980) was the first to use this approach. His procedure is called rank order clustering (ROC). Each row and column are considered as binary words, and rows and columns are alternately sorted until a block diagonal structure is obtained. King and Nakornchai (1982) revised the procedure so it would be more efficient to run on a computer (if there are many rows or columns in the part-machine incidence matrix the binary word approach cannot be used) and used a relaxation if there are bottleneck machines (bottleneck machines are machines which process many parts). Chan and Milner (1981) developed a procedure called the direct clustering algorithm (DCA). The advantage of this procedure is that it can start with any matrix and arrive at the same result (if the order of parts or machines is changed the ROC procedure may obtain a different result). The authors also show that DCA will converge to a solution in a limited number of iterations. 40 Chandrasekharan and Rajagopalan (1986) developed a procedure called modified rank ordering clustering algorithm (MODROC). This procedure builds on the ROC procedure by taking ROC output and using clustering to create cells. Khator and Irani (1987) developed a procedure called the occupancy value method (OV). The OV method eliminates limitations of other methods (ROC, DCA) by building up clusters along the diagonal using small selected sections of the larger matrix. 2.2.2.3 Clustering using similarity coefficients Clustering methods using similarity coefficients have been used by many researchers to develop procedures to solve the cell formation problem (Carrie 1973, McAuley 1972, Chandrasekharan and Rajagopalan 1987, Waghodekar and Sahu 1984, DeWitte 1980, Chow and Hawaleshka 1993, Luong 1993, Shafer and Rogers 1993, Askin et al. 1991, Srinivasan and Narendran 1991, and Wei and Kern 1989). These procedures use some measure of similarity between pairs or groups of parts (or pairs or groups of machines) based on the part-machine incidence matrix. A clustering procedure is then used to group similar items into clusters which become the basis for forming cells. The earliest clustering procedure for cell formation was developed by McAuley (1972). He used the Jaccard similarity coefficient. The Jaccard similarity coefficient for a pair of machines is defined as the number of parts processed by both machines divided by the number of parts processed by either machine (the similarity coefficient ranges between 0 and 1). McAuley then used single linkage cluster analysis to form groups of machines. Single linkage cluster analysis uses the most similar pairs of items between two groups to define the similarity between two groups. Other clustering methods have also been used. Two common methods are complete linkage cluster analysis and average linkage cluster analysis. Complete linkage cluster analysis defines the similarity between two groups by the least similar pair of items between the groups and average linkage cluster analysis uses the average similarity between the items in the groups. Seifoddini (1988, 1989a) compares single linkage clustering and average linkage clustering for cell formation and outlines the advantages and disadvantages of each method. Many similarity coefficients and distance measures have been used in clustering procedures for cell formation. Shafer and Rogers (1993a) did a comprehensive survey of these measures. 2.2.2.4 Graph theory Several procedures use graph theory to solve the cell formation problem. Rajagopalan and Batra (1975) were the first to use a graph-theoretic approach. In their procedure machines are considered vertices, and edges are relationships between the machines. The Jaccard similarity coefficient is computed for each pair of machines and an edge is included if the similarity coefficient is above a threshold value. The weight of an edge is equal to the similarity coefficient. A graph partitioning approach is then used to create machine groups and parts are allocated to machine groups to create manufacturing cells. Chandrasekharan and Rajagopalan (1986a) show that the problem can be represented as a bipartite graph in which one set of vertices represents machines and the other set of vertices represents parts. An edge exists between a machine vertex and a part vertex if the part uses the machine. Vannelli and Kumar (1986), and Kumar, Kusiak and Vannelli (1986) also use this approach and propose heuristics to find the minimal cut nodes needed to create the required number of cells. A cut represents intercell movement. 2.2.2.5 Mathematical models Several researchers have formulated models that require only the information provided by a part-machine incidence matrix. These models can then be solved optimally; or if the problem is too large a lower bound can be obtained, and heuristic procedures can be evaluated against the lower bound. Kusiak (1987) formulated the problem using the p-median model. The objective is to maximize the similarities between parts that are assigned to the same part family (similarities refer to a similarity coefficient). The constraints in this model assign each part to exactly one family, specifies the number of part families, ensure that part i belongs to family j only if family j is formed, and ensure integrality. Shtub (1989) shows that Kusiak's model can be formulated as a generalized assignment problem. Srinivasan, Narendran, and Mahaderan (1990) formulate the problem as an assignment model which differs from Kusiak's model in that the number of groups to be created is not required as an input for the model. Boctor (1989, 1991) formulated a 0-1 integer program that has as its objective the minimization of intercell transfers (a machine type is assigned to a cell that is different from that of a part that requires processing on it). Constraint sets that assign each part and each machine type to only one cell are included and a constraint set for cell size is included. The author proves that the integrality constraints for part assignment can be dropped and an integer solution will still be obtained. 2.2.2.6 Other methods Many other methods have been used to develop procedures for cell formation. Our review of the literature found the following set theoretic-Purcheck (1975), Sundaram and Fu (1987), and Purcheck (1985); string manipulation-Wu, Venugopal, and Barash (1986); polyhedral dynamics-Robinson and Duckstein (1986); neural network-Chu (1993), and Karparthi and Surresh (1993); Hamiltonian path-Askin et al. (1991); branch and bound-Al-Qattan (1990); and simulated annealing-Boctor (1991). 2.2.3 Evaluation of Procedures Considering the amount of research that has been done to develop cell formation procedures, based on part-machine incidence matrices, there has been comparatively little research that compares the effectiveness of various procedures. The majority of research that has been done in comparing cell formation procedures is a comparison of various similarity measures and clustering procedures. Harrigan and Mosier (1988) tested four similarity coefficients used in clustering procedures. The objective of their study was to asses the appropriateness of cellular manufacturing within a particular manufacturing environment. Mosier (1989) tested similarity coefficients and clustering procedures using 30 generated problems which varied by cluster definition and block diagonal density. Shafer and Rogers (1993b) tested 16 similarity measures and four clustering procedures and compared the procedures based on various performance measures. Miltenburg and Zhang (1991) tested 9 procedures that used a variety of methods on 8 well known problems from the literature and 60 generated problems. Chu and Tsai (1990) tested three procedures that used matrix manipulation on 11 problems from the literature. Shafer and Meredith (1990) conducted a study that used actual data from three companies. Cell layouts were developed using 7 procedures. Two of the procedures used matrix manipulation, four procedures used clustering and one procedure was based on operations sequences. A simulation model was then built and the performance of each of the cell layouts generated by the procedures was tested. The performance measures used were average distance traveled, number of extra-cellular operations, average WIP, maximum WIP, and longest average queue. 2.3 Procedures Which Use a Part-Machine Incidence Matrix and Incorporate Data to Provide Weights for Parts or Machines These procedures are similar to the procedures of section 2.2 with the exception that these procedures recognize that some parts are more important than others, and some machine types are more important than others. Data is incorporated into these procedures which allows parts and machine types to have various weights of importance. Part weights are usually based on the volume of units required for a part times a cost per unit. The cost per unit is usually based on the cost of intercell movement or an incremental subcontracting cost to remove the part from the system. Machine type weights are usually based on the cost of a single machine of a given type. 2.3.1 Objectives and Constraints The objectives used by these procedures are 1) minimize the cost of exceptional elements (Kusiak and Chow 1987, Seifoddini 1989c, Seifoddini and Wolfe 1987, and Kumar and Vannelli 1987); 2) minimize the cost to duplicate machines (Seifoddini 1989c, and Sundaran and Fu 1987); 3) maximize a measure of similarity (Steudel and Ballakur 1987, Balasubramanian and Panneerselvam 1993, Okogbaa et al. 1992, and Gupta and Seifoddini 1990). The two most popular constraints are cell size and measure of similarity threshold. The first objective is sometimes used by itself subject to the constraint that a machine type can only appear in one cell and a cell size constraint (Kusiak and Chow 1987, Kumar and Vannelli 1987); or in conjunction with the second objective (Seifoddini 1989). Some of the similarity measures used by these procedures incorporate weights. Steudel and Ballakur (1987) use a similarity measure called cell bond strength (CBS) which is based on the processing times of parts. Balasubramanian and Panneerselvam's (1993) measure of similarity is based on the number of excess moves a part requires in each cell. Okogbaa et al. (1992) base their measure on flows between machine types. Gupta and Seifoddini (1990) incorporate part volumes and processing times into a similarity coefficient. 2.3.2 Methods Several methods are used by these procedures. Seifoddini and Wolfe (1987) cluster using several similarity thresholds and then check the intercell movement cost for each threshold and choose the best to use as a layout. Gupta and Seifoddini (1990) cluster using complete linkage with a threshold constraint. Steudel and Ballakur (1987) use a dynamic programming algorithm to maximize their similarity coefficient. Balasubramanian and Panneerselvam (1993) first create a set of potential cells by creating a cell for each part and then creating additional potential cells using the rank ordering clustering procedure. They then select cells based on a warehouse covering algorithm. Kumar and Vannelli (1987) use a procedure which starts with part seeds or machine seeds. Part seeds are parts that do not use any common machine types and machine seeds are machine types which do not process any common parts. The procedure then uses objective function criteria to add parts and machines to cells. 2.4 Routing Based Procedures Without Machine Capacities These procedures require part routings to be used as opposed to a part-machine incidence matrix. The part routings provide additional information about a part's sequence of operations and possible alternative processing plans. The procedures in this category also can incorporate cost and volume information, but do not incorporate any information about machine capacities. The procedures are further categorized into two groups: 1) procedures which use operations sequences, and 2) procedures which use alternative routings. Procedures which consider operations sequences focus on efficient materials flows when forming cells. Procedures which consider alternate routings focus on equipment utilization and flexibility when forming cells. 2.4.1 Operations Sequences Vakharia and Wemmerl6v (1990), Selvam and Balasubramanian (1985), Tam (1990), and Logendran (1991) have developed procedures which incorporate operations sequences. The first three procedures use similarity coefficients. Vakharia and Wemmerl6v (1990) attempt to create cells that do not require parts to backtrack (An attempt is being made to create flow-line cells in which there is a unidirectional flow thru the machines in the cell. If a part must move in the opposite direction of the flow then a backtrack has occurred). A similarity coefficient, that compares the number of machines in two groups that are used in the same sequence to the total number of machines in the groups, is used to cluster groups. Selvam and Balasubramanian (1985) develop a set of potential cells using a similarity coefficient based on operations sequences, and then select the desired number of cells by using a covering algorithm. The objective is to minimize material handling cost. Tam (1990) also develops a similarity coefficient based on operations sequences, and then uses a clustering method called k-nearest-neighbor (KNN) to form cells. Logendran aggregates machine types in an attempt to minimize weighted intra-cell and inter-cell moves based on operations sequences. 2.4.2 Alternative Routings Kusiak (1987), Shtub (1989), and Sankaran and Kasilingam (1990) have developed models which incorporate the possibility of alternate routings. Kusiak (1987) formulates the problem as a p-median model. The objective is to maximize the similarities between part routings assigned to the same family (similarities are based on a comparison of process plans for pairs of parts). The constraints in this model are only one routing for a part is selected, each part is assigned to one family, only N part families are formed where N is an input parameter, part i belongs to family j only if family j is formed, and integrality. Shtub (1989) shows that Kusiak's (1987) formulation is equivalent to the generalized assignment problem (GAP). Sankaran and Kasilingam (1990) formulate a model that maximizes the number of routings that can be completed in a cell. The purpose of this model is to design a cellular system that is flexible (parts will have alternate cells that can be used for processing). The constraints of the model are at least one routing is chosen for each part, cell size limit, a budget limit that can be spent on machines, and if a routing is assigned to a cell then all the machine types required by the routing are assigned to the cell. 2.5 Procedures Which Incorporate Setup Information Several procedures incorporate setup objectives into the cell formation problem. Setup is an important consideration in cell formation because by combining items with similar setups and dedicating equipment to producing them setup time can be substantially reduced which reduces labor costs and increases plant capacity. Shafer and Rogers (1991) formulated models which include minimizing setup time as an objective (minimizing intercellular movements and investment in machines are also objectives). Shafer and Roger's models also consider sequence dependent setups in the cell formation problem. The models combine a p-median formulation with the travelling salesman problem (TSP). A two stage heuristic is used to solve the problem. In stage I a p-median model is solved to form part families and cells. Stage II then solves a travelling salesman problem to determine sequences for each family. Chakravarty and Shtub (1984) developed a procedure to form cells that considers inventory and setup costs associated with batch sizes as well as materials handling costs. The authors develop a cost function for the total setup and inventory holding cost for all components at a machine and then derive an optimal production cycle time for a machine (lotsize). The authors then show how to assign a machine, that could be assigned to more than one group (based on processing sequences), to one of the groups based on the cost function. Askin and Subramanian (1987) constructed an economic model of manufacturing costs as a function of cell configuration. The model includes the following costs: setup, variable production, production cycle inventory, WIP, material handling, and fixed machine. Setup costs include both product and family setup costs. A heuristic, that is an extension of the approach of Boucher (1984) is used to 54 develop product and family cycle times and considers family setup costs, WIP costs and finished goods inventory costs. The procedure used to create cells consists of three sequential stages. ROC is used in the first stage to cluster the part-machine incidence matrix. Stage two attempts to combine adjacent groups based on an economic evaluation. Stage three aggregates groups based on machine capacities and further economic evaluation. Rajamani, Singh, and Aneja (1992) formulate a mathematical model for cell formation in environments where sequence dependent setups exist. The model considers the tradeoff between setup costs and machine investment (by dedicating machines to produce certain parts setup cost is reduced but machine investment increases). The model's objective function is to minimize the sum of total discounted cost of machines assigned to all the cells, and setup costs incurred due to sequence dependence of parts in each cell. The constraints in the model ensure each part is produced in a cell, each part has a place in a sequence, a sequence is defined, relationships between pairs of parts, and capacity constraints (these include processing time and setup time). 2.6 Procedures Which Incorporate Machine Capacities These procedures recognize that it may not be possible to process all parts in a desired cell due to limited machine capacity. These procedures require for each part a demand requirement and the processing time on each machine type, and for each machine type the time available for processing. The procedures in this category are based on mathematical models which require an integer number of machines of each type assigned to a cell. Two common models can be identified, although, there are several variations to each of these models. 2.6.1 Model 1 Shafer, Kern and Wei (1992), Shafer and Rogers (1991), Askin and Chin (1990), Choobineh (1988), Askin and Subramanian (1987), Vakharia, Chang and Selim (1993), Rajamani, Singh and Aneja (1990), Sankaran (1990), Sule (1991), and Kamrani and Pansaei (1993) use models which are variations of this basic model. These models include at least two items in the objective function. The first item deals with the cost of parts which cannot be produced in a single cell. Shafer, Kern and Wei (1992) consider a subcontracting cost to remove the part, the other models consider the cost to move parts between cells. The second item is the cost to use additional machines. If there is an infinite number of machines available and there is no cost to use machines a cell could be created for each part and no intercell transfers or subcontracting would be required. The models recognize that there is a tradeoff between the two objectives that were described. The cost of machines can be decreased at the expense of intercell material handling cost or subcontracting, and the reverse is true. These models contain constraints which do not allow the processing time of a machine type in a cell exceed the time available of a machine type in a cell. The models, with one exception, also contain constraints which restrict the size of a cell to k machine types where k is a required input parameter, or require cells defined by machine types as input (Shafer, Kern and Wei 1992, Askin and Subramanian 1987, Sule 1991, and Choobineh 1988). The exception is Sankaran (1990) who uses a constraint which requires a minimum level of similarity (based on parts which are processed) between machines in a cell. These models, with the exception of Sule (1991), require that a part is assigned to a single cell (a part's production requirement cannot be split among cells). There are several variations to this model. Askin and Chin (1990), Choobineh (1988), Vakharia, Chang and Selim (1993), and Sule (1991) incorporate operations sequences into the model. Choobineh (1988) and Rajamani, Singh and Aneja (1990) consider multiple process plans or alternate routings in their formulation. Shafer and Rogers (1991) include setup time in the objective function; and Askin and Subramanian (1987) include setup costs and inventory costs in the objective function. Kamrani and Pansaei (1993) do not include a cost for parts that cannot be produced in a cell in the objective function but include constraints which do not allow intercell transfers. 2.6.2 Model 2 Wei and Gaither (1990a, 1990b), Dahel and Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and Logendran (1990, 1993) use models which are variations of this basic model. 58 These models include the cost of parts which cannot be produced in a single cell. Wei and Gaither (1990a, 1990b) consider a subcontracting cost to remove the part, Dahel and Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and Logendran (1990, 1993) consider the cost to move parts between cells. These models also contain constraints which do not allow the processing time on a machine type in a cell exceed the time available of the machine type in the cell. These models contain two other sets of constraints: one that limits the number of machines that can be assigned to a cell to a prespecified number; and one that limits the number of machines that are available of each machine type. The second constraint is important because if an unlimited number of machines are available for each type then a cell could be created for each part and there would be no need for subcontracting or intercell transfers. Dahel and Smith (1991, 1993) incorporate part operations detail into their model. Nagi, Harhalakis, and Proth (1990) consider alternate routings in their formulation and allow a part's production requirement to be spread over several routings so machines can be better utilized. Wei and Gaither (1990b) include, along with the cost of bottleneck parts, average cell utilization, intracell load imbalances, and intercell load imbalances in the objective function. Each of the items is given a weight and the objective is to minimize the weighted sum of the items. Logendran (1990, 1993) includes both intracell and intercell moves in the objective. Weights are given to both types of moves (moves are based on operations sequences and a lower weight is usually given to intracell moves) and the objective is to minimize weighted moves. Logendran (1990) includes constraints that do not allow a machine's utilization to fall below a specified level; and Logendran (1993) includes machine utilization in the objective function. 2.6.3 Methods Since the models contain integer variables, reasonable size problems can be computationally difficult to solve, therefore, many of the procedures used are heuristics which may not solve the problem optimally. As in other categories a variety of methods are used in the procedures. Several procedures (Shafer and Rogers 1991, Askin and Chin 1990, Askin and Subramanian 1987, Logendran 1990, Nagi, Harhalakis, and Proth 1990, and Sule 1991) break the problem into stages and solve each stage sequentially. Some of the procedures solve the subproblems in each stage optimally. Shafer and Rogers (1991) break their formulation into two subproblems, a p-median problem and a travelling salesman problem, and solve the subproblems optimally. Nagi, Harhalakis, and Proth (1990) create two subproblems and then iterate between the two subproblems until convergence to a local minimum is achieved. Other methods are also used. Wei and Gaither (1990) fix seed machines and then form cells by solving the integer program. Vakharia, Chang and Selim (1993) use simulated annealing and tabu search. Dahel and Smith (1991, 1993) use the constraint method to solve their multi objective model. 2.7 Procedures that Consider Multiple Scenarios for Part Volumes Procedures in this category recognize that demand for parts may not remain the same over time, or that demand for parts may be stochastic. Previous procedures, which considered part volumes, were based on annual or average demand and were considered to be deterministic. If demand varies from period to period or estimates for demand turn out to be inaccurate a cell layout may not perform as well as expected. Cell layouts created by procedures in this category should have the flexibility to perform well even if there are expected volume swings or volume differs from what was estimated. Little work has been done in this area. Vakharia and Kaku (1988, 1993) developed a procedure to handle long-term demand changes. Seifoddini (1990) developed a procedure for probabilistic demand estimates. Vakharia and Kaku (1988, 1993) focus on a strategy of part reallocation to respond to demand changes as opposed to partial cell system redesign or complete cell system redesign. This strategy is chosen because it has the lowest cost of change (if cell design is robust changes can be handled by this strategy). Vakharia and Kaku proposed a model. The objective is to minimize material handling cost and additional machine investment costs. The constraints of the model are each operation for each part is assigned to a cell, determination of the number of new machines required based on machine capacity, a cell is visited if an operation is performed in a cell, integrality, and nonnegativity. Additional constraints can be added to limit cell size to a specified number of machines. A heuristic is used to solve the model. Seifoddini's (1990) procedure incorporates the uncertainty of product mix into the final solution. The procedure has three major steps: 1) the product mix and the associated part-machine incidence matrix are expressed in the form of a probability function, 2) a set of alternate solutions are developed based on all possible product mixes, 3) a measure of effectiveness is developed and used as a criterion for evaluating different solutions for the purpose of choosing the best solution. This procedure assumes there is a discrete distribution of product mixes and part-machine incidence matrices. The procedure uses similarity coefficients and a clustering algorithm to develop a cell layout for each product mix that minimizes intercell material handling cost. The intercell material handling cost is then calculated for each product mix and each layout, and an expected intercell material handling cost is calculated 63 for each layout. The layout with the lowest expected cost is chosen. 2.8 Other Procedures Procedures in this category incorporate data that is not included in the previous categories. Gunasingh and Lashkari (1989b) formulate an integer programming model that groups machines based on machine tooling availability and parts tooling requirements. The authors define a compatibility index between a machine and a part. The index is based on the number of common tools between part i and machine j divided by the minimum of the number of tools part i requires or machine j has available. The objective of the model is to maximize the compatibility index. Constraints of the model include a cell size restriction, allocation of each machine to at least one cell, and integrality. Irani, Cavalier and Cohen (1993) consider both the layout of cells and machines in order to minimize the effect of intercell movement. The authors use a from-to travel 64 chart to layout machines within a cell and layout cells with respect to each other. The procedure also considers options such as placing a machine that is common to two cells, that are adjacent to each other, between the two cells so that greater flexibility of machine utilization can be achieved. Min and Shin (1991) formulated a model that incorporates human factors into cell formation. The authors believe that, in addition to the normal functions, cell formation should also allocate operators to cells, assign machines to each operator, and determine the job scope for each operator. The model includes many sets of constraints and is a multiobjective goal program. Some of the constraints and objectives that stem from incorporating human factors are maximum number of operators allowed in a cell, maximum number of parts that operator p can handle, assignment of parts to operators, wage rates of operators, and skill matching of an operator with a part. 2.9 Summary The review of procedures in the previous sections illustrates how diverse cell formation procedures are with 65 respect to factors considered and data required, as well as the methods used. A single procedure probably cannot be developed which considers all the possible factors that could impact on the performance of a cell layout. When choosing a cell formation procedure to apply to a particular situation selection should depend on the factors that appear to be most important. In many instances it may be appropriate to use several procedures and then compare the layouts that are generated. The review of the procedures in section seven indicates that more work could be done to develop procedures that create good cell layouts in situations that experience demand variability or demand uncertainty. Cellular layouts usually tradeoff strengths and weaknesses of product line layouts and process layouts. A key strength of a process layout is its flexibility to handle volume changes or uncertainty, product line layouts tend to be less flexible in these environments. Procedures which can evaluate whether a shop should be organized with small efficient cells dedicated to a few parts or larger less efficient cells which can be used to produce many parts, or a mixture of large and small cells should be of aid to the shops which exist in the environment described above. CHAPTER 3 MODELS 3.1 Introduction This chapter presents two models which can be used for identifying cellular configurations in the presence of dynamic demand. Model I attempts to form cells that do not exceed a predetermined cell size in terms of the number of machines contained in each cell. This model considers materials handling costs as well as inventory and backordering costs to select a solution. Model II develops a cell configuration for a shop by selecting cells from a set of potential cells, which could be of varying sizes. To develop a cell configuration, first several runs of model I could be made with varying cell size parameters as input. The cells that were formed in each of the runs can be included in the set of potential cells. Model II would then be run using this set of potential cells to create a cell configuration. 3.2 Model I 3.2.1 Description of Model I This model attempts to form cells that do not exceed a cell size constraint which is specified in terms of the number of machines a cell can contain. The model can also form a remainder cell which has no cell size constraint. The model uses a planning horizon which consists of T periods. The parts to be produced during the planning horizon are known. The demand for each part is known and can vary from period to period. Each part requires a set of machine types for production, and the processing time per unit for each part on each machine type, that is required by the part, is a known parameter. There are a limited number of machines of each type that can be used to produce parts. Each machine has a limited capacity available during each period in the planning horizon. The capacity of the machine types within each cell is an important constraint in this model. The total processing time required of a machine type within a cell during any period cannot exceed the total capacity of that machine type within that cell. In order to effectively utilize machine capacity the model allows for inventory planning decisions. The number of units produced of a part during a period can exceed the demand requirement for the period. The excess production can be stored in inventory and used to satisfy demand requirements in later periods. A part's demand requirement, for a period, may also be partially or completely backordered and excess production of the part during a later period may be used to fill backorders. If an inventory of a part is held then there is a cost associated with the inventory. If a backorder is incurred there is a cost associated with the backorder. Each part has a unit inventory holding cost and a unit backorder cost associated with it. These costs are on a per period basis. The unit cost of producing a part depends on where in the shop the unit is produced. If the unit is produced entirely in a cell that meets the cell size constraint then the production cost is zero. If the unit is produced entirely in the remainder cell there is a cost associated (by part) with producing the unit. If the unit is produced 69 using machines that reside in two or more cells there is an intercellular movement cost associated (by part) with producing the unit. Each part has a unit remainder cell production cost and a unit intercellular movement production cost associated with it. 3.2.2 Model I Formulation The formulation of model I is shown below. I T Min Z= E (MMRiXitr+MMIiXiti+hilit+biBit) i-I t-I Subject to K Iit-l-Bit-1+1: Xitk+Xiti-Iit+Bit=dit k-I PijXitk CjYjk i-1 for i= .,I; (2) t=l,...,T. for for for I K 1 jE PijXitk+ Pij (Xiti+Xitr) < CjAj for - k-i -1 for k=l, ...,K & r; j=l, ... ,J; t=l,...,T. j=l, .. .,J; t=l, .,T for j=l,...,J. K E YiA3 Yk- J Yjk for k=l,...,K Yjk is integer for all j & k. (7) All variables are a 0 (8) where I=The total number of parts. T=The total number of periods. J=The total number of machine types. K=The total number of cells not including a remainder cell. i=part index (i=1,...,I). t=period index (t=l,...,T). j=machine type index (j=l,...,J). k=cell index (k=l,...,K & r). MMRi=The cost to produce 1 unit of part i in the remainder cell. MMIi=The cost to produce 1 unit of part i using two or more cells (uses intercell movement). hi=The cost to hold 1 unit of part i for 1 period. bi=The cost to backorder 1 unit of part i for 1 period. dit=The demand for part i in period t. Pij=The processing time for part i on machine type j. Cj=The capacity in hours of 1 machine of type j. Aj=The number of machines available of type j. CS=The maximum number of machines allowed in a cell excluding the remainder cell. Xik=The number of units of part i produced in cell k during period t. Xi,,=The number of units of part i produced using intercell movement during period t. Iit=The inventory of part i at the end of period t. Bit=The number of units on backorder of part i at the end of period t. Yjk=The number of machines of type j assigned to cell k. 3.2.3 Description of Model I Equations (1) Objective function: The objective function is the sum over all parts and all periods of four terms for each part during each period. The first term is the cost of producing a unit of part i in the remainder cell times the number of units produced of part i in the remainder cell during period t. The second term is the cost of producing a unit of part i using intercell movement times the number of units produced of part i using intercell movement during period t. The third term is the cost to hold one unit of part i in inventory for one period times the number of units of part i that are in inventory at the end of period t. The last term is the cost to hold one unit of part i on backorder for one period times the number of units on backorder at the end of period t. (2) This set of equations represents material balance constraints. The first two terms on the left side of the equality (Iit_--Bit-,) represent the net inventory position of part i at the beginning of period t (or the end of period t-l) and the last two terms on the left side of the equality (-Iit+Bi,) represent the negative of the net inventory position of part i at the end of period t. The middle two K terms on the left side of the equality ( Xltk+Xiti) equals production in units of part i during period t. Therefore, the net inventory position of part i at the beginning of period t plus production of part i during period t minus the net inventory position of part i at the end of period t must equal the demand for part i during period t (the right side of the equality). There is a constraint for each part during each period. 73 (3) These constraints ensure that the total processing time required by parts produced in cell k during period t does not violate the capacity of machine type j in cell k. There is a constraint for each machine type, in each cell during each period. (4) These constraints ensure that the total processing time required by parts produced during period t does not violate the capacity of machine type j. There is a constraint for each machine type during each period. (5) These constraints ensure that the total number of machines of type j that are assigned does not exceed the number of machines of type j that are available. There is a constraint for each machine type. (6) These constraints require that cell k not contain more than a total of CS (the cell size parameter) machines. There is a constraint for every cell with the exception of the remainder cell. (7) These constraints require that the number of type j machines assigned to cell k be integer. There is a constraint for each machine type in each cell. (8) These constraints require all the production, inventory, and backorder variables to be nonnegative. 3.2.4 An Example of Model I The following example shows how model I is applied to a problem with 2 parts, 3 machine types, and two periods of demand. The part-machine incidence matrix for this example is shown in table 8. Part 1 requires machine types 2 and 7, and part 2 requires machine types 1 and 2. The cell size limit used in this example is two machines. A remainder cell can also be formed. Table 9 shows the demand by period, the unit inventory cost, and the unit backorder cost for each part. Table 10 shows the number of machines of each type that are available and the capacity in units per machine of each type. The cost to manufacture either part in a cell that meets the cell size restriction is 0, the cost to manufacture either part in the remainder cell is 1, and the cost to manufacture either part using intercell movement is 3. In the formulation of this problem we allow for two cells that meet the cell size constraint and a remainder cell. The objective for this problem is Min Z=Xllr+Xi2r+X21r+X22r+3Xii+3X12i+3X21i+3X22I+4B11+4B12+4B21 +4B22+I +112+I21 +I22 Table 8 Part-Machine Incidence Matrix for Model I Example. Machine Type Part 1 2 7 1 1 1 2 1 1 Table 9 By Part Demand. Period Unit Unit Part 1 2 Inv. Cost B/o Cost 1 450 0 1 4 2 0 450 1 4 Table 10 Machine Data. Machine Type # Available Capacity/Machine 1 1 450 2 2 225 7 1 450 The full formulation for this problem is shown in appendix A. Examples of each of the different types of constraints will be shown below. The demand constraint for part 1, period 1 is Xiii+Xi2+Xinr+XII-I,+B,,=450 The capacity constraint for machine type 2 in cell 1 during period 1 is X,,,+X211225Y21 The shop capacity constraint for machine type 2 during period 2 is X121+Xi22+X12r+X121+X221+X222+X22r+X22,i450 The machine availability constraint for machine type 7 is Y,7+Y72+Y7r:1 The cell size constraint for cell 1 is Y11+Y21+Y,712 The solution to this model is to form one cell with all the machines in it (a process layout), produce 450 units of part 1 during period 1, and produce 450 units of part 2 during period 2. With this solution no inventory or backorders would be required, and the objective function value equals 900. It is interesting to see how the solution would change if we had assumed a constant demand of 225 units for each part during each period. With this assumption, the model forms two cells. The first cell would have a type 2 machine and a type 7 machine. This cell would produce 225 units of part 1 in each period. The second cell would have a type 1 machine and a type 2 machine. This cell would produce 225 units of part 2 in each period. The objective function value would equal 0. If this plan were implemented and the demand turned out to be the same as in table 9 the cost would be 1125. 3.3 Model II 3.3.1 Description of Model This model selects cells, which will be formed to produce parts, from a set of potential cells. The objective of the model is to minimize production, inventory, and backordering costs. The model uses a planning horizon which consists of T periods. The parts to be produced during the planning horizon are known. The demand for each part can vary from period to period. Each part requires a set of machine types for production, and the processing time per unit for each part on each machine type, that is required by the part, is a known parameter. There are several potential cells that are being considered for inclusion in the shop. All the cells cannot be included because there are not enough machines available. There are several different types of machines in the shop and the number of machines available of each type must be specified as an input parameter for the model. The configuration of each potential cell in terms of the number of machines of each type required by the cell is known. Each machine has a limited capacity available during each period in the planning horizon. The capacity of the machine types within each cell is an important constraint in this model. The total processing time required of a machine type within a cell during any period cannot exceed the total capacity of that machine type within that cell. In order to effectively utilize machine capacity the model allows for inventory planning decisions. The number of units produced of a part during a period can exceed the demand requirement for the period. The excess production can be stored in inventory and used to satisfy demand requirements in later periods. A part's demand requirement, for a period, may also be partially or completely backordered and excess production of the part during a later period may be used to fill backorders. If an inventory of a part is held then there is a cost associated with the inventory. If a backorder is incurred there is a cost associated with the backorder. Each part has a unit inventory holding cost and a unit backorder cost associated with it. These costs are on a per period basis. The unit cost of producing a part varies depending on which cell the unit is produced in. Each part has a cost 79 assigned for each cell. These costs can reflect the material handling and other costs associated with producing a unit of the part in a particular cell. A unit of a part can also be produced by using machines that reside in two or more cells. If this occurs the unit will incur costs associated with intercell material handling and other costs, such as the paperwork required for intercell movement and tracking. 3.3.2 Model II Formulation The formulation of model II is shown below. 1 T K I T Min Z= (CiiXit,+hiit+biBiJt + (CikXik) i-I t-1 k-1 i-I t-I Subject to K Iit-1-Bit-i+ E Xitk+XitI- Ii+Bit=dit I PijXitk< CjZkMjk i-1 SPijXitk+ pij (Xitz) CA i-I k-I i-I K E MjZk Aj k-I for i=l, ..,I; (10) t=1,...,T. for k=l,...,K & r; (11) for j=1, ...,J; for t=l,...,T. for j=l, ...,J; for t=l, ...,T (12) for j=1,...,J. (13) 80 Zk, {0,1} for k=l,...,K. (14) All other variables a 0 (15) where I=The total number of parts. T=The total number of periods. J=The total number of machine types. K=The total number of cells. i=part index (i=l,...,I). t=period index (t=l,...,T). j=machine type index (j=1,...,J). k=cell index (k-l,...,K). Cik=The cost to produce 1 unit of part i in cell k. Cii=The cost to produce 1 unit of part i using two or more cells (uses intercell movement). hi=The cost to hold 1 unit of part i for 1 period. bi=The cost to backorder 1 unit of part i for 1 period. dit=The demand for part i in period t. Pij=The processing time for part i on machine type j. Cj=The capacity in hours of 1 machine of type j. Aj=The number of machines available of type j. Mjk=The number of machines of type j required to form cell k. Xilk=The number of units of part i produced in cell k during period t. Xi,,=The number of units of part i produced using intercell movement during period t. Iit=The inventory of part i at the end of period t. Bit=The number of units on backorder of part i at the end of period t. Zk=1 if cell k is formed; 0 otherwise. 3.3.3 Description of Model II Equations (9) Objective function: The objective function sums production, inventory, and backorder costs for all parts over all periods. There are four terms in the objective function. The first term sums across all parts, cells, and periods the cost to produce a unit of part i in cell k times the number of units produced of part i produced in cell k during period t. The second term sums the cost to produce a unit of part i using intercell movement times the number of units produced of part i using intercell movement during period t across all parts and all periods. The third term is the cost to hold one unit of part i in inventory for one period times the number of units of part i that are in inventory at the end of period t summed across all parts and all periods. The last term is the cost to hold one unit of part i on backorder for one period times the number of units on backorder at the end of period t summed across all parts and all periods. (10) This set of equations represents material balance constraints. The first two terms on the left side of the equality (Iit-_-Bit_1) represent the net inventory position of part i at the beginning of period t (or the end of period t-l) and the last two terms on the left side of the equality (-Iit+Bit) represent the negative of the net inventory position of part i at the end of period t. The middle two terms on the left side of the equality (F Xit+Xiti) equals k-i production in units of part i during period t. Therefore, the net inventory position of part i at the beginning of period t plus production of part i during period t minus the net inventory position of part i at the end of period t must equal the demand for part i during period t (the right side of the equality). There is a constraint for each part during each period. 83 (11) These constraints ensure that the total processing time required by parts produced in cell k during period t does not violate the capacity of machine type j in cell k. There is a constraint for each machine type, in each cell during each period. (12) These constraints ensure that the total processing time required by parts produced during period t does not violate the capacity of machine type j. There is a constraint for each machine type during each period. (13) These constraints ensure that the cells selected for inclusion in the shop do not require more type j machines than are available. There is a constraint for each machine type. (14) These constraints require that cell k is either selected (Zk=l) or not selected (Zk=0) for inclusion in the shop. (15) These constraints require all the production, inventory, and backorder variables to be nonnegative. 3.3.4 An Example of Model II This example uses the data presented in tables 8 thru 10 of section 3.2.4. Table 8 shows the part-machine incidence matrix; table 9 shows the demand by period, the unit inventory cost, and the unit backorder cost for each part; and table 10 shows the number of machines of each type that are available and the capacity in units per machine of each type. Three potential cells are considered for inclusion in the shop. Table 11 shows the configuration of each cell in terms of the number of machines of each type that would be included in the cell if the cell is selected. Table 11 Potential Cells Machine Type Cell 1 2 7 1 0 1 1 2 1 1 0 3 1 2 1 Table 12 shows unit production costs for each part in each cell that the part can be completely manufactured in. Table 12 also shows unit production costs using intercell movement. The objective for this problem is Min Z=X13+X+X12 +X213+X223+3X1I+3X12I+3X21I+3X22I+4B11+4B12+4B21 +4B22+I11+I1+I1+I22 Table 12: Production Costs Part cell 1 cell 2 cell 3 Intercell 1 0 n/a 1 3 2 n/a 0 1 3 The full formulation for this problem is shown in appendix B. Examples of each of the different types of constraints will be shown below. The demand constraint for part 1, period 1 is Xiii+X13,+XI -In+B11=450 The capacity constraint for machine type 2 in cell 1 during period 1 is X,111225Z, The shop capacity constraint for machine type 2 during period 2 is X121+X222+X123 +X12+X223+X22I 450 The machine availability constraint for machine type 2 is: Zi+Z2+2Z32 The solution to this model is to select only cell 3. Cell 3 contains all the machines in the shop so this solution is a Process layout. 450 units of part 1 should be produced in period 1, and 450 units of part 2 should be 86 produced in period 2. This solution requires no inventory or backorders, and the objective function value equals 900. It is interesting to see how the solution would change if we had assumed a constant demand of 225 units for each part during each period. With this assumption, the model selects cells 1 and 2 for inclusion in the shop. 225 units of part 1 would be produced in cell 1 during each period, and 225 units of part 2 would be produced in cell 2 during each period. The objective function value would equal 0. If this plan were implemented and the demand turned out to be the same as table 9 the cost would be 1125. CHAPTER 4 SOLUTION PROCEDURES 4.1 Introduction In this chapter, solution procedures for the models are proposed. Since model II develops a shop configuration from a list of potential cells, that may have been created by solving model I several times with different cell size parameters, the focus of the procedures proposed is on model II. A heuristic for solving model I is proposed in section 4.2. This heuristic can then be run, using several cell size parameters, to create a list of potential cells. Section 4.3 proposes lower bounding schemes for model II. Heuristics for solving model II are proposed in section 4.4 The lower bounding schemes in section 4.3 can be used, for large problems, to evaluate the heuristics of section 4.4. 4.2 Model I Heuristic This heuristic requires several iterations to develop a solution for model I. During each iteration an attempt is made to form one cell that meets the cell size constraint. If a cell is formed (that meets the cell size constraint) then the integer variables for that cell are fixed, and then the next iteration of the heuristic attempts to form another cell. If a cell is not formed (that meets the cell size constraint) the heuristic stops and a solution has been developed. The heuristic also stops if all the machines have been used to form cells that meet the cell size constraint. In addition to the notation of section 3.2.2, the following notation is required. Let n=The iteration index for the heuristic. Let Mjk=The number of type j machines required by the cell formed in iteration n of the heuristic. The following steps are required: Step 0. Initialization. set n=l; Step 1. Solve problem P1 (described below). Step 2. a. set a=1; for j=l to J do if Yjk>0 then a=0; b. set b=l; for j=l to J do K if Aj- Mjk>0 then b=O; k-i c. if a+b>0 then stop else n=n+l and repeat step 1. Problem P1 is as follows: I T Min Z=- E (MMRiXitr+MMIiXiti+hilit+biBit) i-1 t-1 Subject to Iit-l-Bit-1+ Xitk +Xitr+Xit- Iit+Bit=dit k-1 I PijXitk CjMjk Pi XinC 7 i~tn j~ Ep=!.Y (16) for i=l, ...,I; (17) t=1,...,T. for k=l, ...,n-l; for j=l, ...,J; for t=l,...,T. for j=l, ...,J; for t=l, ...,T. (18) (19) |

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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E44K2LWY4_XNIGET INGEST_TIME 2012-08-13T14:50:05Z PACKAGE AA00011763_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND By Jeffrey Schaller A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 PAGE 2 ACKNOWLEDGMENTS I would like to thank Dr. Selcuk Erenguc, my supervisory committee chairman, for his invaluable guidance and patient encouragement throughout my years in the program. My supervisory committee cochairman Dr. Asoo J Vakharia has also provided invaluable guidance during the past year and I am thankful that he brought his expertise in cellular manufacturing to our department. I also would like to thank Dr Harold Benson, Dr. Patrick Thomson, and Dr Chung-Yee Lee for serving on my supervisory committee and for the fine instruction they provided in the classes I attended I would also like to thank Dr. Antal Majthay for the direction he provided while supervising my assistantship assignments. Finally, and most importantly I want to thank my mother and father for the love and support they have provided my entire life Anything good that I have ever accomplished is a result of having the best parents ii PAGE 3 TABLE OF CONTENTS ACKNOWLEDGMENTS. LIST OF TABLES LIST OF FIGURES. Abstract Chapters 1 INTRODUCTION A Brief Description of Cellular Layouts Product Line Layout Process Layout. Cellular Layouts Cell Formation Early Cell Formation Techniques Later Cell Formation Techniques The Impact of Dynamic Demand on Cell Formation ii vii ix X 1 1 1 3 4 7 8 9 Decisions 11 Dynamic Demand's Impact on Cellular Layouts 11 Strategies to Handle Dynamic Demand while using a Cellular layout 16 Current Practices 18 Automotive Parts Manufacturer 18 Enclosure Manufacturer The Problem and its Terminology. The Organization of this Proposal 2 REVIEW OF CELL FORMATION LITERATURE 20 23 24 26 Introduction. 26 Procedures Based solely on a Part-Machine Incidence Matrix 30 Objectives and Constraints. 31 Frequently used objectives 31 lll PAGE 4 Frequently used constraints. 34 Other objectives 35 Methods 37 Manual Identification. 38 Matrix Manipulation 39 Clustering using Similarity Coefficients 40 Graph Theory. 42 Mathematical Models 43 Other Methods 44 Evaluation of Procedures 44 Procedures Which use a Part-Machine Incidence Matrix and Incorporate Data to Provide Weights for Parts or Machines. 46 Objectives and Constraints. 47 Methods 48 Routing Based Procedures Without Machine Capacities 49 Operations Sequences. 50 Alternative Routings. 51 Procedures Which Incorporate Setup Information 52 Procedures Which Incorporate Machine Capacities 55 Model 1 55 Model 2 57 Methods 59 Procedures That Consider Multiple Scenarios for Part Volumes Other Procedures Summary 3 MODELS Introduction Model I. Description of Model I Model I Formulation. Description of Model I Equations An Example of Model I Model II Description of Model II. Model II Formulation. Description of Model II Equations An Example of Model II iv 60 63 64 66 66 67 67 69 71 74 77 77 79 81 83 PAGE 5 4 SOLUTION PROCEDURES. 87 Introduction. Model I Heuristic Lower Bounds for Model II. Linear Programming Relaxation. Lagrangean Relaxation Linear Programming Relaxation Tightened Adding Valid Inequalities Valid Inequality Set 1 Valid Inequality Set 2 Valid Inequality Set 3 Heuristics for Model II Heuristic 1. Description of Heuristic 1 Formal Procedure for Heuristic 1 Discussion of Heuristic 1 Heuristic 2. Heuristic 3A Description of Heuristic 3A Formal Procedure Discussion of Heuristic 3A Heuristic 3B 87 88 91 91 92 by 94 94 96 97 98 98 98 100 101 101 105 105 107 110 111 5 DESCRIPTION OF EXPERIMENTS AND DATA 114 6 Introduction Experiments Experiment 1 Data Part-Machine Incidence Matrices Processing Times Part Demand. 114 115 116 117 119 120 Inventory and Backorder Costs. 120 Machine Capacities and Number of Machines of Each Type 121 List of Potential Cells. 122 Cell Production and Intercell Production Costs 123 Summary 124 Experiment 2 Data 125 RESULTS 127 Results of Experiment 1 127 V PAGE 6 Lower Bounds for Model II Heuristic Results for Model II Discussion of Results Lower Bounds. Heuristics Results of Experiment 2 7 CONCLUSIONS AND FUTURE RESEARCH. Conclusions Future Research APPENDICES A FORMULATION OF MODEL I EXAMPLE B FORMULATION OF MODEL II EXAMPLE C PROBLEM 1 DATA AND SOLUTION Problem 1 Data Problem 1 Solution LIST OF REFERENCES BIOGRAPHICAL SKETCH vi 127 130 131 131 136 138 142 142 144 147 149 151 151 171 178 200 PAGE 7 LIST OF TABLES TABLE 1. Part Routing. 5 2. Cellular Layout. 5 3. An Example of a Part-Machine Incidence Matrix. 8 4. Capacity Consideration 10 5. Part Family Machine Requirements. 12 6. Load on Machine Type by Period. 13 7. Average Load. 13 8. Part-Machine Incidence Matrix for Model I Example. 75 9. By Part Demand.. 75 10 Machine Data 75 11 Potential Cells. 84 12 Production Costs 85 13. Part-Machine Data Sets 118 14 Summary of Experimental Problems 124 15. L.P and Lagrangean Relaxations 128 16 Valid Inequality Relaxations 129 17 Heuristics 1 and 2. 132 18. Heuristics 3A and 3B 133 vu PAGE 8 19. Problem 1 Results 20. Problem 2 Results 21. Problem 1 Selected Cells 22. Part/Cell Processing viii 140 140 172 173 PAGE 9 LIST OF FIGURES FIGURE 1. Product Line Layout 2 2. Process Layout 3 3. A Process Layout. 6 4 A Cellular Layout .. 6 5 Family 1 Load by Period .. 1 4 6. Family 2 Load By Period .. 1 4 7 Total Load by Period .. 15 IX PAGE 10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CONFIGURING CELL SYSTEMS TO HANDLE VARIABLE DEMAND By Jeffrey Schaller May 1996 Chairperson: Dr. S. Selcuk Erenguc Major Department: Decision and Information Sciences This research addresses the problem of configuring cell systems for environments that experience variable demand Past cell formation models consider only average product demand to determine configurations of cells. Short term fluctuations in demand may cause a cell configuration that would appear to be good, when considering only average demand, to perform poorly because during some periods machines of a certain type may be under utilized in one cell while machines in another cell may not be able to produce the required products. For these reasons, a methodology for configuring cell systems in environments that experience variable demand is needed. X PAGE 11 In this dissertation, two mathematical models are formulated. The first model is used to develop a list of potential cells, and the second model is used to select cells to include in a shop. Since both models are computationally complex, this research develops heuristic procedures to solve the models. Given that other approaches can also be used to develop a list of potential cells, the primary focus of this research is on the second cell selection model. Lower bounding procedures are proposed for this model In order to test the effectiveness of the lower bounding and heuristic procedures for the second model fifteen problems from the literature were modified and solved optimally The lower bounding and heuristic procedures were then used on these problems, and the lower bounds and objective function values are compared to the optimal objective function values. A second experiment was conducted, in order to test the effectiveness of the lower bounding and heuristic procedures on larger problems. Using two larger published data sets the results are compared to the results obtained using the lower bounding and heuristic procedures xi PAGE 12 CHAPTER 1 INTRODUCTION 1.1 A Brief Description of Cellular Layouts Cellular Manufacturing is a manufacturing application of the group technology philosophy where groups of processes are dedicated to sets of products or parts. The associated layout (referred to as cellular layout) is a form of production layout that combines features of two traditional forms of plant layout: product line layout and process layout. 1.1 1 Product Line Layout In a plant with a product line layout, machinery and equipment are dedicated to a specific product, and thus, machines needed to manufacture each product are laid out in a way that facilitates efficient material flow. The advantages of product line layout are fast throughput times, efficient material handling, simplified scheduling, and 1 PAGE 13 2 control and low work-in-process inventories. The disadvantages of product line layout are that it requires a high level of investment in specialized machinery and equipment, and it lacks flexibility because only one type of product can be manufactured without incurring a high changeover cost. This type of layout is typically used for high volume, standardized products. Examples are automobiles, television sets and appliances. An example of a product line layout is shown in Figure 1. Product Flow D Machines tJ Product I Product2 Product3 Figure 1. Product Line Layout PAGE 14 3 1.1.2 Process Layout A process layout groups machines which perform similar functions into departments. Many different types of products may use a specific machine type and each product may visit different machine types in various sequences. An example of a process layout is shown in Figure 2. Product Flow D Machines Product 3 -------;~r--r-----;1"'MIB-A--Product2 Produ t I ---t-=--'f'-,f---+----t-~ J Product,------',;:-"'--:-!------\----->s::""--::,"---, Figure 2. Process Layout. The products in this example have the same routings as the products in Figure 1 but notice that the material flows are much more jumbled than in Figure 1 The advantage of a process layout is that many different types of products can PAGE 15 4 be manufactured on the same set of machines which leads to flexibility in the utilization of machinery. The disadvantages of a process layout are that material handling costs are high, material flows are complex which causes planning and control to be difficult, flow time is long, setup costs are high and work-in-process inventory is high. Typically low volume or nonstandardized products are produced using a process layout. Examples include (Hyer 1984): aerospace, agricultural machinery, business machines, control devices, diesel engine assemblies, machine tools, and machined parts. 1.1.3 Cellular Layouts Cellular layouts are hybrid layouts which combine features of both product and process layout. The objective of implementing a cellular layout is to enable a manufacturer of low volume specialized or custom products to obtain some of the benefits of product line layout while maintaining some of the flexibility of a process layout This is done by forming product families which consist of products that require the same types of machines (preferably in the same sequence). A machine of each machine type (or PAGE 16 5 more than one if required) required to produce the products in a family is put into a cell and is dedicated to producing a family of products. As an example consider the products shown in Table 1. Table 1. Part Routing. Product 1 2 3 4 5 6 Table 2. Cellular Layout. Cell 1 2 3 Machines 1,4,5 2,3,4,5 1,2,3,4 Machine Sequence 1 4 5 5 4 1 2 3 5 2 3 4 5 1 2 3 4 1 3 2 4 Products 1,2 3,4 5,6 Figure 3 (on the following page) shows the process layout for producing these products. Figure 4 shows the PAGE 17 resulting product flows if the three cell layout in table 2 is used. Process Layout ___. Product Flow 0 Machines Produ ---+----------Product3 Product-+~~-----l-r==-=,...,,..,~ ~~......,._-Product4 Product 6 Figure 3. A Process Layout. Cellular Layout _____. Product Flow D Machines Cell I Produc ~ I Product I Ill Product 2 Product 2 4 Cell 2 Product 4 Product 4 Cell 3 roduct 3 Product ""'-4-_ Product6 Product 5 Figure 4. A Cellular Layout. 6 PAGE 18 7 Notice the simpler product flows in the cellular layout vs the process layout. Creating cells in this manner enables material handling costs to be reduced and materials handling flows to be simplified. This in turn can lead to reduced flow time, reduced work-in-process inventories and simplified planning and control. Setup time and cost for parts manufactured in a cell can typically be reduced by developing specialized procedures for setups required in the cell. Quality improvement is also usually a benefit because cell operators have more experience manufacturing a specific set of products. 1.2 Cell Formation In order to implement cellular manufacturing cells must be formed. The cell formation problem includes the following decisions. How many cells should there be? Which machine types should be included in a cell? How many machines of each type should be included in a cell? Which parts should a cell produce? PAGE 19 1.2 1 Early Cell Formation Techniques Most of the early cell formation algorithms used some form of part-machine incidence matrix. In a part-machine incidence matrix machines are represented as rows and parts are represented as columns in the matrix (or vice versa ) 8 Each element in the matrix is '1' if part j requires machine i for processing and '0' otherwise. Table 3. An example of a part-machine incidence matrix Machine p a r t s Type 1 2 3 4 5 A 1 0 1 0 1 B 0 0 1 1 1 C 1 1 1 0 0 D 0 1 0 1 1 Table 3 shows an example of a part-machine incidence matrix. In this example part 1 requires machine types A and C but not Band D, and part 5 requires machine types A, B, and D but not C The algorithms use the part-machine incidence matri x as input and then use v arious techniques to break up the parts PAGE 20 9 and machines into cells. These techniques include manual identification (Burbidge 1963); clustering using measures of similarity (McAuley 1972, Carrie 1973); graph theory using measures of similarity (Rajagopalan and Batra 1975, Chandrasekharan and Rajagopalan 1986, Vannelli and Kumar 1986, and Kumar, Kusiak, and Vannelli 1986); and matrix manipulation (King 1980, King and Nakornchai 1982, Chan and Milner 1981, and Chandrasekharan and Rajagopalan 1986). 1.2.2 Later Cell Formation Techniques Many of the algorithms that were developed later tend to be more sophisticated and are based on mathematical models that include operational considerations in the objectives and constraints. These models also usually include some sort of integrality constraint. Operational considerations that have been incorporated into algorithms include operations sequences, machine setup similarities between parts, part volumes, and machine capacities. A typical integrality constraint is that the number of machines of a machine type that are assigned to a cell is required to be integer PAGE 21 10 Table 4. Capacity Consideration Part Hours required Family on machine type 1 1 2700 2 2500 Total 5200 # of type 1 machines available 3 Capacity per type 1 machine 2000 hrs Total capacity 6000 hrs For an example of why capacity, volume, and machine integrality are important considerations when making cell formation decisions, consider Table 4. In this example two part families have been formed and both families have parts which require processing on machine type 1. The table shows the hours required on machine type 1 for each family, the number of type 1 machines available, and the capacity per type 1 machine. If we could assign 1.5 type 1 machines to each family, we would have ample capacity to produce both families and everything would be fine. However, we cannot dedicate fractions of machines to families If we assign two machines to family 1 and one machine to family 2, then there PAGE 22 11 will be idle capacity dedicated to family 1 while we do not have enough capacity to produce family 2. If we assign two machines to family 2 and one machine to family 1, a similar situation arises. One alternative to this problem is to combine both families and create a larger cell by assigning all the machines to this single cell Using this alternative there is adequate capacity to produce both part families but, because the cell is larger and the parts that are produced in the cell are more diverse, some of the benefits of cellular manufacturing are likely to be lost. 1.3 The Impact of Dynamic Demand on Cell Formation Decisions 1.3.1 Dynamic Demand's Impact on Cellular Layouts Demand for a product is said to be dynamic if it varies from period to period. Past cell formation models consider only average product demand to determine the configurations of cells and which products to produce in a cell. Short term fluctuations in demand may cause a cell configuration that would appear to be good, when considering only average demand, to perform poorly During some periods machines of a certain type may be underutilized in one cell while machines in another cell may not be able to produce the required PAGE 23 Table 5. Part Family Machine Requirements. Part Family 1 2 Required Machines 2 1 6 5 7 6 12 quantities of products. For an example consider the part families in Table 5. Suppose there are 2 machines of type 6 available and the capacity of a type 6 machine is 2500 units per period. The load on machine type 6 by period for each family is shown in Table 6. Table 7 (on the following page) shows the average load on machine type 6 for each period. If one machine of type 6 was dedicated to producing family 1 and the other was dedicated to producing family 2 there would be enough capacity in each cell to meet the average load. If you look at the by-period requirements in Table 6, however, you can see that insufficient capacity would exist to produce family 2 while capacity would be underutilized in family l's cell during the first three periods. During the last three periods the situation is reversed. When looking at the load of both families in aggregate there is enough PAGE 24 13 Table 6. Load on Machine Type by Period. Load on machine type 6 by period Period 1 2 3 4 5 6 Family 1 1480 2068 1256 2676 2831 3360 Family 2 2856 2914 3144 2154 738 674 Total 4336 4982 4400 4830 3569 4034 Table 7 Average Load Average Capacity Family 1 2278.5 2500 Family 2 2080 5 2500 Total 4358 5 5000 capacity to produce both families during every period Figures 5, 6, and 7 show the load variations graphically. Figure 5 (on the following page) shows the load caused by family 1, Figure 6 the load caused by family 2, and Figure 7 shows the load in aggregate The aggregate load is much more stable than the individual family loads. This illustrates an advantage of a process layout when there is dynamic demand. By combining all the machines of a given type into a PAGE 25 14 Family 1 Load by period 3500 sooo 2500 ..,2000 0 .. uoo 1000 500 0 1 e Ptlbl Figure 5. Family 1 Load by Period. Family 2 Load by period 3 500 3000 2500 .,2000 0 .J1 500 1 000 500 Ptr10d Figure 6. Family 2 Load by Period PAGE 26 15 Total Load by period Figure 7. Total Load by Period. department and allowing any product that requires that machine type to use any machine of that type flexibility is provided so that the peaks and valleys of a product's demand can be offset by other products. Studies have shown the adverse effects of variable demand on a cellular layout. Flynn and Jacobs (1986, 1987) conducted simulation studies which show that when demand variability is high a process layout is favored over a cellular layout in terms of mean flowtime. Garza and Smunt (1991) showed that demand variability can cause intercell flows in a cellular layout and this results in less stable flow time performance as compared to a process layout. PAGE 27 1.3.2 Strategies to Handle Dynamic Demand While Using a Cellular Layout There are strategies which can be used to offset the effects of demand variability in a cellular manufacturing system. These include combining cells to smooth out load variations, routing parts through alternative cells, let parts move between cells (intercell movement), hold inventories of parts, and incur backorders for parts. Combining cells to form larger cells allows the large cell to produce a greater variety of parts than the individual small cells. Since there are more parts it is likely that machine loads will be more stable in the large cell. The large number of parts produced by the cell will 16 also cause more complex materials flows and higher material handling cost. Therefore, there is a tradeoff that must be considered when choosing a cell's size; stable loads in large cells vs lower material handling cost in small cells. Cells can also be formed so that there is more than one cell that can produce a part. Hence, if there is insufficient capacity to produce the part in its preferred cell, it may be possible to produce the part in a different cell By having alternative cells, load variations can be PAGE 28 17 smoothed out while meeting product demand. The disadvantage of this approach is that it may be more costly to produce parts in their alternative cells and/or there may be a need to duplicate more machines to develop these types of configurations. If there is insufficient capacity for a machine type to produce all the parts required, then some parts can be moved to another cell which has that machine type for processing. This is called an intercell transfer. Ang and Willey (1984) showed that low levels of intercell transfer can greatly improve the flow time performance of a cellular layout. The disadvantages of intercell transfer are that this requires materials handling capability between cells, there is increased paper work required to track part movement, and quality could be adversely affected because of the loss of responsibility by a single cell for the part's quality. A final strategy that could be used is that during periods of low demand, inventories of parts can be produced which can be used to meet demands for parts during periods when insufficient capacity exists in the cells to produce all the parts required. Alternatively parts could be backordered when capacity is not available in a cell and PAGE 29 18 then produced during a later period when there is adequate capacity in the cell. Using inventories and backorders will level out the load requirements in the cells, but the cost of holding inventories or backordering parts must be incurred. 1.4 Current Practices In order to find out how cellular layout is used in industry, we visited two plants that utilize a cellular layout. The first plant produces parts for automotive manufacturers. The second plant produces steel enclosures that are used for industrial applications to hold electrical circuitry. A brief description of how each plant uses a cellular layout and how they deal with demand variability presented next. 1.4.1 Automotive Parts Manufacturer This plant produces parts that are major components in the assembly of automobiles and some service parts. It uses product line, process, and cellular layout. Major factors in the choice of layout are maintaining machine utilization and efficient use of space. The plant has over 20 cells that are PAGE 30 19 used for production. In some cases the cells are production lines that produce one part. One cell is a small job shop that produces approximately 150 parts. The other cells are of varying sizes in terms of the number of machines they contain and the number of products they produce. There are certain processes in the plant which are strictly process departments. An example of this is the paint department. All parts that require painting go to the paint department (there are no paint facilities in the cells) The reason why this department was chosen to be a process layout is that one oven was sufficient to handle the entire plant's requirements. The plant also has 20 high speed roll mills. Fourteen of the mills are grouped together in a department and six have been moved to cells The roll mills which have not been moved are large mills which perform complicated processing. It was decided that it would be too expensive and require too much space to duplicate these machines in multiple cells The mills that were moved to cells are smaller and the capacity of the mills matches the requirements demanded by their cells. Some of the cells were created by combining production lines This enabled the plant to better utilize equipment and save space. PAGE 31 20 The plant experiences demand for products that is variable. This is particularly true with service demand which is very lumpy. A variety of strategies are used to handle this variability. Many of the low volume parts, including service parts, are produced in the large job shop cell. The varying demands of the individual parts tend to offset each other and help stabilize the load in the cell. Sometimes a large order is produced partially in advance of its due date and parts are inventoried. There are instances when orders are rescheduled to a later date. There are also times when production capacities in other cells are used to perform some operations and then intercell transfers of parts occur. When this happens, it is a disruption in the plant because all the presses for the job must be setup at the same time to maintain quality. The primary benefits obtained by using a cellular layout in this plant are reduced equipment and space, reduced material handling cost, and reduced setups. 1.4.2 Enclosure Manufacturer Approximately 2800 standard end items are produced in this plant. The plant also produces some custom jobs. The standard end items are produced and then sent to regional PAGE 32 warehouses. Roughly 200 items account for 90% of the units produced by the plant. 21 This plant originally used a process layout The current layout slowly evolved to a hybrid cellular process layout. Currently there are three cells and plans for more cells are being formulated. The cells are designed to produce high volume products very efficiently and are setup as flow lines. Currently 75% of the unit volume and 40% of the dollar volume is produced using cells. High volume production in cells is undertaken to justify the investment in dedicated equipment and tooling. The demand for standard end items is variable and seasonal while demand for custom items tends to be lumpy. The plant responds to demand variability using a variety of methods. A portion of the plant is still using a process layout This portion of the plant is generally used to produce low volume standard items and custom items. The demand for these items tend to be more erratic, but by producing many items on the same equipment, the load generated is fairly stable. High volume items are produced in the cells and the demand for these items tends to be fairly stable. There are PAGE 33 22 times, however, when some facilities in the plant are idle while other facilities become bottlenecks. The plant also had to invest in additional equipment to build cells that could meet the seasonality of demand. Sometimes custom jobs will be processed in cells if the equipment in the cell is underutilized. At other times, a product normally produced in a cell will be produced in the process portion of the plant if the cell capacity is exceeded. Inventory is also held and customer orders are backordered to help deal with demand variability. The following benefits have been attributed to the use of cellular layout in the plant. 1) Material handling cost has been significantly reduced because conveyors are used to move materials in the cells. 2) Setups have been significantly reduced because tools have been dedicated to products and designed for quick setup. 3) Productivity has increased. Labor costs have been cut in half. 4) Leadtimes have been reduced from 4 weeks to 2 weeks. 5) WIP inventory has been dramatically reduced. 6) Quality has been improved due to the fast feedback provided by the cellular layout. 7) The plant has become a safer place to work because there is PAGE 34 23 less material handling and simpler setups. 8) The fill rate on customer orders has increased from 50% to 80%. 1.5 The Problem and its Terminology Based on our experience and industry visits, the problem we address is as follows. Given a set of parts, a finite planning horizon with T periods, and a set of machine types, the problem is to specify a set of cells to be included in the shop and a production and inventory plan that will minimize total cost. For each cell the number of machines of each type must be specified. The set of cells to be included in the shop can contain cells of varying sizes in terms of the total number of machines included in each cell. For example the shop could include several small cells and a large ''remainder cell." A remainder cell is a portion of the shop that typically is organized as a process layout. The production and inventory plan must specify the quantity produced of each part in each cell during each period, and the quantities of each part produced using intercell movement during each period. PAGE 35 24 Each part has associated with it the following: (a) a set of machine types that are required to produce the part and the processing time required by the part on each machine type; (b) a demand (deterministic) for each period during the planning horizon; (c) a cost to hold one unit of inventory for one period (unit inventory holding cost); (d) a cost to hold one unit on backorder for one period (unit backorder cost); (e) costs to produce a unit of the part in various size cells (in terms of the number of machines contained in the cell). (These costs are referred to as cell production costs. It is assumed that larger cells have a higher cost); (f) a cost to produce a unit of the part in two or more cells (intercell movement cost). Each machine type has associated with it the following: (a) the number of machines of that type that are available and (b) the capacity of a single machine of that type. 1.6 The Organization of this Dissertation The remainder of this dissertation is organized as follows. Chapter 2 contains a survey of cell formation literature. Chapter 3 introduces the two models that can be PAGE 36 used to develop cellular configurations, and Chapter 4 describes solution procedures for these models. Chapter 5 describes the experimental data used to test the solution procedures described in Chapter 4, and Chapter 6 presents computational results of these tests. Finally Chapter 7 proposes future research directions. 25 PAGE 37 CHAPTER 2 REVIEW OF CELL FORMATION LITERATURE 2.1 Introduction The cell formation problem has attracted more academic interest than any other aspect of cellular manufacturing. The reason research has focused so heavily on cell formation problem is due to the complex nature of the cell problem In order to design a cellular system decisions must be made as to how many cells there should be, which machine types should be included in a cell, and which parts should a cell produce. Arriving at these decisions constitutes a solution to the cell formation problem. Many aspects of the effectiveness and efficiency of an operation will be affected by the cell formation decisions. Examples are labor costs, investment, leadtimes for products, supervision of personnel, quality, and amount of paperwork needed to control the operation. This leads to the consideration of many possible objectives in arriving at a solution. The 26 PAGE 38 problem is also complex because an entire machine usually must be assigned to a cell. Because of this most mathematical models of the problem are integer programming models which are difficult to solve. 27 The procedures that have been developed to solve the cell formation problem are diverse with respect to 1) the objectives and constraints considered by the procedure, 2) the data required as input for the procedure, and 3) the techniques used by the procedure to solve the problem. The first two items are somewhat related. Usually, to incorporate an objective or a constraint into a problem, data are required to support that objective or constraint. For example, if a constraint of the problem is to use onl y existing equipment to configure cells, then data by machine type of the number of machines available are required. Many procedures use objectives that are surrogates for other objectives which if incorporated would require additional data and cause the problem to become more complex For example, a possible objective when designing a cellular system is to maintain a minimum level of machine utilizat i on for each machine. This objective would require data on part volumes, part processing times, and machine capacities Many PAGE 39 28 cell formation procedures use a surrogate measure for machine utilization based on the percentage of parts to be produced by the cell that uses a machine. This measure is used in the hope that insight into machine utilization will be gained without explicitly considering the data required to determine machine utilization. Of course, surrogate measures can sometimes turn out to be a poor representation for the desired objective or constraint The diversity of the cell formation procedures with respect to the items mentioned above causes the development of a framework for reviewing the procedures to be difficult. The most comprehensive review of cell formation procedures to date was done by Wemmerlov and Hyer (1986). Over 70 procedures are categorized in Wemmerlov and Hyer's paper The authors classify cell formation procedures based on which of four general approaches was used in the procedure In all four approaches parts and machines must be selected for possible inclusion in the cellular manufacturing system. An evaluation step of the proposed cells is also required. The four approaches are 1) identify part families without the help of machine routings (including procedures that use group technology codes and those that do not); 2) identify PAGE 40 29 machining groups; 3) identify part families using routings; 4) identify part families and machine groups simultaneously The advantages of this method of classification are that it is inclusive and provides classifications for procedures which perform similar functions. Within each of the last three approaches, however, the procedures can vary greatly with respect to the objectives and constraints considered explicitly by the procedure as well as the data required for the procedure. In this literature review we propose six categories which are based on the data used, by a procedure, to classify procedures There are two reasons for classifying procedures this way. 1) Identifying data used by a procedure provides insight into factors that are explicitly considered by the procedure and helps determine the strengths and limitations of the procedure, and 2) procedures that use similar data can be compared with one another. The six categories that were chosen are 1) procedures that are based solely on data that can be represented by a part-machine incidence matrix, 2) procedures that use a part-machine incidence matrix and incorporate data which can be used to provide weights for parts or machines, 3) procedures that PAGE 41 30 incorporate operations sequences, 4) procedures that consider part setup times or costs, 5) procedures that include part volumes, part processing times, and machine capacities, and 6) procedures that consider multiple scenarios for part volumes. Some procedures may fall into two or more categories. There, also, are a few procedures which use data that do not fall into any of the categories, therefore, a seventh category called "other" is included For each category discussions of the objectives and constraints considered by procedures, and methods used by procedures to develop a solution are included. 2.2 Procedures Based solely on a Part-Machine Incidence Matrix These are the most common of the cell formation procedures The concept of a part-machine incidence matrix was described in chapter 1. As a brief refresher, in a partmachine incidence matrix machines are represented as rows and parts are represented as columns in the matrix (or vice versa) An element in the matrix is '1' if part j requires machine i for processing and '0' otherwise. The advantages of these procedures are that they have low data requirements PAGE 42 31 (all the data can be obtained from part routings) and they tend to be relatively simple and efficient The disadvantage of these procedures is they do not consider factors such as operations sequences, part volumes, machine capacities, and demand variability. When a solution is developed using one of these procedures the cell designer must somehow factor in other important considerations to arrive at a final solution. There are several objectives and constraints considered by procedures in this category and many techniques have been used to develop solutions. 2.2.1 Objectives and Constraints The most common objectives used in these procedures are to minimize exceptional elements, minimize duplicate machines, and maximize a measure of similarity between parts or machines assigned to a cell. The two most common constraints are maximum cell size and measure of similarity threshold. 2.2.1.1 Frequently used objectives Exceptional elements. Many procedures (Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and Milner 1982, PAGE 43 32 King and Nakornchai 1982, Waghodekar and Sahu 1984, Boctor 1989, Chow and Hawaleshka 1993, Khator and Irani 1987, and Boctor 1991) have as an objective the minimization of exceptional elements. An exceptional element occurs when a part requires two or more cells for processing. This objective is important because exceptional elements would require increased material handling cost and paperwork cost, or subcontracting cost to remove the exceptional element from the plant. If many parts are exceptional elements when a cellular system is designed the benefits of the system will be greatly diminished. Machine duplication. Minimization of machine duplication is also an objective used by many procedures (Burbidge 1963, Vannelli and Kumar 1986, King 1980, Chan and Milner 1982, King and Nakornchai 1982, Khator and Irani 1987, and Al-Qattan 1990). Machine duplication (also sometimes referred to as a bottleneck machine) occurs when a machine type is required in two or more cells. When a machine type is required in two or more cells additional machines may be required which increases machine investment and may cause machine utilization to be lower vs a process layout. This objective is usually used in conjunction with PAGE 44 the exceptional element constraint. Most procedures trade off exceptional elements against machine duplication. Vannelli and Kumar's (1986) procedure attempts to minimize the number of machine types that must be duplicated while allowing no exceptional elements. 33 Measures of similarity. Several procedures attempt to maximize measures of similarity between pairs of parts or pairs of machines (Rajagopalan and Batra 1975, Carrie 1973, Mcauley 1972, Kusiak 1987, De Witte 1980, Chandrasekharan and Rajagopalan 1986a, Luong 1993, Askin et al. 1991, Srinivasan et al. 1990, Gunasingh and Lashkari 1989b, and Wei and Kern 1991). Measures of similarity between pairs of parts are based on how similar the processing requirements are for the parts. Measures of similarity between pairs of machines are based on which parts require the machines. Creating cells that produce parts that have similar processing requirements and machines that process similar parts is one of the main objectives of cellular manufacturing, therefore, this is an important objective Most of the measures of similarity that are used in cell formation are based on some sort of similarity coefficient (the concept of a similarity coefficient will be described PAGE 45 34 in the next section) which can take on a value between O and 1. 2.2.1.2 Frequently used constraints Cell size. The most popular constraint used by these procedures is a cell size constraint (Chandrasekharan and Rajagopalan 1986, Boctor 1989, Luong 1993, Stanfel 1985, Boctor 1991, Wei and Kern 1989, Al-Qattan 1990, and Vannelli and Kumar 1986). The cell size constraint limits the number of machine types allowed in a cell. The constraint is imposed to limit the physical size of cells. It is generally assumed that intracell material handling cost will be insignificant but this assumption loses validity as cell size grows This constraint is used with each of the objectives described above. Without this constraint, a cell consisting of all machine types could be created which would have no exceptional elements and would not require any duplicate machines Similarity threshold. A second constraint which is sometimes imposed is a measure of similarity threshold (Carrie 1973, McAuley 1972, and Chandrasekharan and Rajagopalan). This constraint is usually used in conjunction with the objective that maximizes some similarity measure PAGE 46 35 The similarity threshold constraint does not allow a machine or a part into a cell unless the cell's similarity measure is above the threshold. The threshold can be applied to a single pair of items (single linkage) in a cell, any pair of items (complete linkage) in the cell, or the average similarity between items (average linkage) in the cell. If the threshold constraint is not applied then a single large cell would result and the shop would be a process layout. If a high threshold is applied to a measure of similarity between machines the resulting layout would have small cells and possibly many exceptional elements. If a high threshold is applied to a measure of similarity between parts then the result would be many part families and to create cells for each family could require many machine types to be duplicated. Picking a similarity threshold requires judgement by the cell designer. Usually the procedures would be run with several similarity thresholds and then a part machine structure would be selected. 2 2.1.3 Other objectives Chandrasekharan and Rajagopalan (1986) and Kumar and Chandrasekharan (1990), and Stanfel (1985) have developed three measures which are slightly more sophisticated and use these measures as objectives. PAGE 47 36 Grouping efficiency. The first measure is called grouping efficiency (Chandrasekharan and Rajagopalan 1986). This measure is a convex combination of within-group utilization and inter-cell efficiency. Within-group utilization is a measure of how heavily parts visit machines in a cell (summation of the number of machine types used by each part in each cell divided by the summation of the number of machine types times the number of parts in each cell). Inter-cell efficiency equals the fraction of exceptional elements divided by the number of non exceptional elements. This measure serves as a surrogate for a tradeoff between machine investment and materials handling cost. It is speculated that cell layouts with a high within group utilization will result in a shop in which machine types have high utilization because most of the parts in the cell use each machine type in the cell. This should result in a lower required machine investment than a shop that has a low within-group utilization. A cell layout with a high inter-cell efficiency should have a low material handling cost because there are relatively few exceptional elements. Grouping efficacy. The second measure is called grouping efficacy (Kumar and Chandrasekharan 1990) Grouping PAGE 48 37 efficacy is similar to grouping efficiency but is adjusted to take into consideration the density of the part-machine incidence matrix (the density of the part-machine incidence matrix is equal to the 1 entries in the matrix divided by total entries). Weighted transfers. Stanfel (1985) uses an objective that considers both inter-cell and intra-cell transfers. Intra-cell transfers for a part are based on the number of machine types that are in a cell that are not required to process the part. The objective weights both inter-cell and intra-cell transfers (intra-cell transfers usually have a lower weight). This objective serves as a surrogate for material handling cost and also considers cell size. Cells with more machine types are likely to be larger and require a higher intra-cell movement cost. 2.2.2 Methods A wide variety of methods have been used to solve the cell formation problem with only a part-machine incidence matrix as input The most popular methods have been manual identification, matrix manipulation, clustering using similarity coefficients, graph theory, and solving mathematical models. PAGE 49 38 2.2.2.1 Manual identification Probably the earliest procedure to be used for cell formation is a manual procedure called production flow analysis developed by Burbidge (1963). This procedure uses part routings to analyze inter and intra departmental flows. There are two stages in production flow analysis : 1) factory flow analysis and 2) departmental flow analysis Factory Flow analysis studies the basic routes in the factory and then eliminates exceptions. Departmental analysis is broken into group analysis and line analysis In group analysis parts are divided into families so groups of machines can be put together so that the parts in a family can be completely manufactured by the group. The main criterion is that the division into groups should not necessitate an increase in the number of machines required. El-Essawy and Torrance (1972) also develop a manual procedure called component flow analysis. This procedure is similar to production flow analysis. Since production flow analysis and component flow analysis are manual procedures they become cumbersome to apply as the problem size becomes large. PAGE 50 39 2.2.2.2 Matrix manipulation This method attempts to create a block diagonal structure, in which almost all l's occupy the diagonal submatices, and almost all zeros occupy the off-diagonal submatrices, by manipulating the data in the part-machine incidence matrix. King (1980) was the first to use this approach. His procedure is called rank order clustering (ROC). Each row and column are considered as binary words, and rows and columns are alternately sorted until a block diagonal structure is obtained. King and Nakornchai (1982) revised the procedure so it would be more efficient to run on a computer (if there are many rows or columns in the part-machine incidence matrix the binary word approach cannot be used) and used a relaxation if there are bottleneck machines (bottleneck machines are machines which process many parts) Chan and Milner (1981) developed a procedure called the direct clustering algorithm (DCA). The advantage of this procedure is that it can start with any matrix and arrive at the same result (if the order of parts or machines is changed the ROC procedure may obtain a different result). The authors also show that DCA will converge to a solution in a limited number of iterations. PAGE 51 40 Chandrasekharan and Rajagopalan (1986) developed a procedure called modified rank ordering clustering algorithm (MODROC). This procedure builds on the ROC procedure by taking ROC output and using clustering to create cells. Khator and Irani (1987) developed a procedure called the occupancy value method (OV). The OV method eliminates limitations of other methods (ROC, DCA) by building up clusters along the diagonal using small selected sections of the larger matrix. 2.2.2.3 Clustering using similarity coefficients Clustering methods using similarity coefficients have been used by many researchers to develop procedures to solve the cell formation problem (Carrie 1973, McAuley 1972, Chandrasekharan and Rajagopalan 1987, Waghodekar and Sahu 1984, Dewitte 1980, Chow and Hawaleshka 1993, Luong 1993, Shafer and Rogers 1993, Askin et al. 1991, Srinivasan and Narendran 1991, and Wei and Kern 1989). These procedures use some measure of similarity between pairs or groups of parts (or pairs or groups of machines) based on the part-machine incidence matrix A clustering procedure is then used to group similar items into clusters which become the basis for forming cells PAGE 52 41 The earliest clustering procedure for cell formation was developed by McAuley (1972) He used the Jaccard similarity coefficient The Jaccard similarity coefficient for a pair of machines is defined as the number of parts processed by both machines divided by the number of parts processed by either machine (the similarity coefficient ranges between O and 1). McAuley then used single linkage cluster analysis to form groups of machines Single linkage cluster analysis uses the most similar pairs of items between two groups to define the similarity between two groups. Other clustering methods have also been used. Two common methods are complete linkage cluster analysis and average linkage cluster analysis. Complete linkage cluster analysis defines the similarity between two groups by the least similar pair of items between the groups and average linkage cluster analysis uses the average similarity between the items in the groups. Seifoddini (1988, 1989a) compares single linkage clustering and average linkage clustering for cell formation and outlines the advantages and disadvantages of each method Many similarity coefficients and distance measures have been used in clustering procedures for cell formation PAGE 53 Shafer and Rogers (1993a) did a comprehensive survey of these measures. 2.2.2.4 Graph theory 42 Several procedures use graph theory to solve the cell formation problem. Rajagopalan and Batra (1975) were the first to use a graph-theoretic approach. In their procedure machines are considered vertices, and edges are relationships between the machines. The Jaccard similarity coefficient is computed for each pair of machines and an edge is included if the similarity coefficient is above a threshold value. The weight of an edge is equal to the similarity coefficient. A graph partitioning approach is then used to create machine groups and parts are allocated to machine groups to create manufacturing cells. Chandrasekharan and Rajagopalan (1986a) show that the problem can be represented as a bipartite graph in which one set of vertices represents machines and the other set of vertices represents parts. An edge exists between a machine vertex and a part vertex if the part uses the machine. Vannelli and Kumar (1986), and Kumar, Kusiak and Vannelli (1986) also use this approach and propose heuristics to find the minimal cut nodes needed to create the required number of cells. A cut represents intercell movement PAGE 54 43 2.2.2.5 Mathematical models Several researchers have formulated models that require only the information provided by a part-machine incidence matrix. These models can then be solved optimally; or if the problem is too large a lower bound can be obtained, and heuristic procedures can be evaluated against the lower bound. Kusiak (1987) formulated the problem using the p-median model. The objective is to maximize the similarities between parts that are assigned to the same part family (similarities refer to a similarity coefficient). The constraints in this model assign each part to exactly one family, specifies the number of part families, ensure that part i belongs to family j only if family j is formed, and ensure integrality. Shtub (1989) shows that Kusiak's model can be formulated as a generalized assignment problem Srinivasan, Narendran, and Mahaderan (1990) formulate the problem as an assignment model which differs from Kusiak's model in that the number of groups to be created is not required as an input for the model. Boctor (1989, 1991) formulated a 0-1 integer program that has as its objective the minimization of intercell PAGE 55 44 transfers (a machine type is assigned to a cell that is different from that of a part that requires processing on it) Constraint sets that assign each part and each machine type to only one cell are included and a constraint set for cell size is included. The author proves that the integrality constraints for part assignment can be dropped and an integer solution will still be obtained. 2.2.2.6 Other methods Many other methods have been used to develop procedures for cell formation. Our review of the literature found the following set theoretic-Purcheck (1975), Sundaram and Fu (1987), and Purcheck (1985); string manipulation-Wu, Venugopal, and Barash (1986); polyhedral dynamics-Robinson and Duckstein (1986); neural network-Chu (1993), and Karparthi and Surresh (1993); Hamiltonian path-Askin et al. (1991); branch and bound-Al-Qattan (1990); and simulated annealing-Boctor (1991). 2 2 3 Evaluation of Procedures Considering the amount of research that has been done to develop cell formation procedures, based on part-machine incidence matrices, there has been comparatively little PAGE 56 research that compares the effectiveness of various procedures. 45 The majority of research that has been done in comparing cell formation procedures is a comparison of various similarity measures and clustering procedures Harrigan and Mosier (1988) tested four similarity coefficients used in clustering procedures. The objective of their study was to asses the appropriateness of cellular manufacturing within a particular manufacturing environment. Mosier (1989) tested similarity coefficients and clustering procedures using 30 generated problems which varied by cluster definition and block diagonal density. Shafer and Rogers (1993b) tested 16 similarity measures and four clustering procedures and compared the procedures based on various performance measures. Miltenburg and Zhang (1991) tested 9 procedures that used a variety of methods on 8 well known problems from the literature and 60 generated problems Chu and Tsai (1990 ) tested three procedures that used matrix manipulation on 11 problems from the literature. Shafer and Meredith ( 1990) conducted a study that used actual data from three companies Cell layouts were PAGE 57 46 developed using 7 procedures. Two of the procedures used matrix manipulation, four procedures used clustering and one procedure was based on operations sequences. A simulation model was then built and the performance of each of the cell layouts generated by the procedures was tested. The performance measures used were average distance traveled, number of extra-cellular operations, average WIP, maximum WIP, and longest average queue. 2.3 Procedures Which Use a Part-Machine Incidence Matrix and Incorporate Data to Provide Weights for Parts or Machines These procedures are similar to the procedures of section 2.2 with the exception that these procedures recognize that some parts are more important than others, and some machine types are more important than others. Data is incorporated into these procedures which allows parts and machine types to have various weights of importance. Part weights are usually based on the volume of units required for a part times a cost per unit. The cost per unit is usually based on the cost of intercell movement or an incremental subcontracting cost to remove the part from the PAGE 58 system. Machine type weights are usually based on the cost of a single machine of a given type. 2 3.1 Objectives and Constraints 47 The objectives used by these procedures are 1) minimize the cost of exceptional elements (Kusiak and Chow 1987, Seifoddini 1989c, Seifoddini and Wolfe 1987, and Kumar and Vannelli 1987); 2) minimize the cost to duplicate machines (Seifoddini 1989c, and Sundaran and Fu 1987); 3) maximize a measure of similarity (Steudel and Ballakur 1987, Balasubramanian and Panneerselvam 1993, Okogbaa et al. 1992, and Gupta and Seifoddini 1990). The two most popular constraints are cell size and measure of similarity threshold. The first objective is sometimes used by itself subject to the constraint that a machine type can only appear in one cell and a cell size constraint (Kusiak and Chow 1987, Kumar and Vannelli 1987 ) ; or in conjunction with the second objective (Seifoddini 1989) Some of the similarity measures used by these procedures incorporate weights. Steudel and Ballakur ( 1987) use a similarity measure called cell bond strength ( CBS ) which is based on the processing times of parts. PAGE 59 48 Balasubramanian and Panneerselvam's (1993) measure of similarity is based on the number of excess moves a part requires in each cell. Okogbaa et al. (1992) base their measure on flows between machine types. Gupta and Seifoddini (1990) incorporate part volumes and processing times into a similarity coefficient. 2.3.2 Methods Several methods are used by these procedures. Seifoddini and Wolfe (1987) cluster using several similarity thresholds and then check the intercell movement cost for each threshold and choose the best to use as a layout. Gupta and Seifoddini (1990) cluster using complete linkage with a threshold constraint Steudel and Ballakur (1987) use a dynamic programming algorithm to maximize their similarity coefficient Balasubramanian and Panneerselvam (1993) first create a set of potential cells by creating a cell for each part and then creating additional potential cells using the rank ordering clustering procedure. They then select cells based on a warehouse covering algorithm Kumar and Vannelli (1987) use a procedure which starts with part seeds or machine seeds Part seeds are parts that do not use any PAGE 60 common machine types and machine seeds are machine types which do not process any common parts. The procedure then uses objective function criteria to add parts and machines to cells. 2 4 Routing Based Procedures Without Machine Capacities 49 These procedures require part routings to be used as opposed to a part-machine incidence matrix. The part routings provide additional information about a part's sequence of operations and possible alternative processing plans. The procedures in this category also can incorporate cost and volume information, but do not incorporate any information about machine capacities. The procedures are further categorized into two groups: 1) procedures which use operations sequences, and 2) procedures which use alternative routings. Procedures which consider operations sequences focus on efficient materials flows when forming cells. Procedures which consider alternate routings focus on equipment utilization and flexibility when forming cells. PAGE 61 50 2.4.1 Operations Sequences Vakharia and Wemmerlov (1990), Selvam and Balasubramanian (1985), Tam (1990), and Logendran (1991) have developed procedures which incorporate operations sequences. The first three procedures use similarity coefficients. Vakharia and Wemmerlov (1990) attempt to create cells that do not require parts to backtrack (An attempt is being made to create flow-line cells in which there is a unidirectional flow thru the machines in the cell. If a part must move in the opposite direction of the flow then a backtrack has occurred). A similarity coefficient, that compares the number of machines in two groups that are used in the same sequence to the total number of machines in the groups, is used to cluster groups Selvam and Balasubramanian (1985) develop a set of potential cells using a similarity coefficient based on operations sequences, and then select the desired number of cells by using a covering algorithm. The objective is to minimize material handling cost. Tam (1990) also develops a similarity coefficient based on operations sequences, and then uses a clustering method called k-nearest-neighbor PAGE 62 (KNN) to form cells. Logendran aggregates machine types in an attempt to minimize weighted intra-cell and inter-cell moves based on operations sequences. 2.4.2 Alternative Routings 51 Kusiak (1987), Shtub (1989), and Sankaran and Kasilingam (1990) have developed models which incorporate the possibility of alternate routings. Kusiak (1987) formulates the problem asap-median model The objective is to maximize the similarities between part routings assigned to the same family (similarities are based on a comparison of process plans for pairs of parts). The constraints in this model are only one routing for a part is selected, each part is assigned to one family, only N part families are formed where N is an input parameter, part i belongs to family j only if family j is formed, and integrality. Shtub (1989) shows that Kusiak's (1987) formulation is equivalent to the generalized assignment problem (GAP) Sankaran and Kasilingam (1990) formulate a model that maximizes the number of routings that can be completed in a cell. The purpose of this model is to design a cellular system that is PAGE 63 5 2 flexible (parts will have alternate cells that can be used for processing). The constraints of the model are at least one routing is chosen for each part, cell size limit, a budget limit that can be spent on machines, and if a routing is assigned to a cell then all the machine types required by the routing are assigned to the cell 2 5 Procedures Which Incorporate Setup Information Several procedures incorporate setup objectives into the cell formation problem. Setup is an important consideration in cell formation because by combining items with similar setups and dedicating equipment to producing them setup time can be substantially reduced which reduces labor costs and increases plant capacity Shafer and Rogers (1991) formulated models which include minimizing setup time as an objective ( minimizing intercellular movements and investment in machines are also objectives ) Shafer and Roger's models also consider sequence dependent setups in the cell formation problem. The models combine a p-median formulation with the travel l ing PAGE 64 53 salesman problem (TSP). A two stage heuristic is used to solve the problem. In stage I a p-median model is solved to form part families and cells. Stage II then solves a travelling salesman problem to determine sequences for each family. Chakravarty and Shtub (1984) developed a procedure to form cells that considers inventory and setup costs associated with batch sizes as well as materials handling costs. The authors develop a cost function for the total setup and inventory holding cost for all components at a machine and then derive an optimal production cycle time for a machine (lotsize). The authors then show how to assign a machine, that could be assigned to more than one group (based on processing sequences), to one of the groups based on the cost function. Askin and Subramanian (1987) constructed an economic model of manufacturing costs as a function of cell configuration. The model includes the following costs: setup, variable production, production cycle inventory, WIP, material handling, and fixed machine. Setup costs include both product and family setup costs. A heuristic, that is an extension of the approach of Boucher (1984) is used to PAGE 65 54 develop product and family cycle times and considers family setup costs, WIP costs and finished goods inventory costs. The procedure used to create cells consists of three sequential stages. ROC is used in the first stage to cluster the part-machine incidence matrix. Stage two attempts to combine adjacent groups based on an economic evaluation. Stage three aggregates groups based on machine capacities and further economic evaluation. Rajamani, Singh, and Aneja (1992) formulate a mathematical model for cell formation in environments where sequence dependent setups exist. The model considers the tradeoff between setup costs and machine investment (by dedicating machines to produce certain parts setup cost is reduced but machine investment increases). The model's objective function is to minimize the sum of total discounted cost of machines assigned to all the cells, and setup costs incurred due to sequence dependence of parts in each cell. The constraints in the model ensure each part is produced in a cell, each part has a place in a sequence, a sequence is defined, relationships between pairs of parts, and capacity constraints (these include processing time and setup time). PAGE 66 55 2.6 Procedures Which Incorporate Machine Capacities These procedures recognize that it may not be possible to process all parts in a desired cell due to limited machine capacity. These procedures require for each part a demand requirement and the processing time on each machine type, and for each machine type the time available for processing. The procedures in this category are based on mathematical models which require an integer number of machines of each type assigned to a cell. Two common models can be identified, although, there are several variations to each of these models. 2.6.1 Model 1 Shafer, Kern and Wei (1992), Shafer and Rogers (1991), Askin and Chin (1990), Choobineh (1988), Askin and Subramanian (1987), Vakharia, Chang and Selim (1993), Rajamani, Singh and Aneja (1990), Sankaran (1990), Sule (1991), and Kamrani and Pansaei (1993) use models which are variations of this basic model. These models include at least two items in the objective function. The first item deals with the cost of PAGE 67 56 parts which cannot be produced in a single cell. Shafer, Kern and Wei (1992) consider a subcontracting cost to remove the part, the other models consider the cost to move parts between cells. The second item is the cost to use additional machines. If there is an infinite number of machines available and there is no cost to use machines a cell could be created for each part and no intercell transfers or subcontracting would be required. The models recognize that there is a tradeoff between the two objectives that were described. The cost of machines can be decreased at the expense of intercell material handling cost or subcontracting, and the reverse is true. These models contain constraints which do not allow the processing time of a machine type in a cell exceed the time available of a machine type in a cell. The models, with one exception, also contain constraints which restrict the size of a cell to k machine types where k is a required input parameter, or require cells defined by machine types as input (Shafer, Kern and Wei 1992, Askin and Subramanian 1987, Sule 1991, and Choobineh 1988). The exception is Sankaran (1990) who uses a constraint which requires a minimum level of similarity (based on parts which are PAGE 68 processed) between machines in a cell. These models, with the exception of Sule (1991), require that a part is assigned to a single cell (a part's production requirement cannot be split among cells). There are several variations to this model. Askin and 57 Chin (1990), Choobineh (1988), Vakharia, Chang and Selim (1993), and Sule (1991) incorporate operations sequences into the model. Choobineh (1988) and Rajamani, Singh and Aneja (1990) consider multiple process plans or alternate routings in their formulation. Shafer and Rogers (1991) include setup time in the objective function; and Askin and Subramanian (1987) include setup costs and inventory costs in the objective function. Kamrani and Pansaei (1993) do not include a cost for parts that cannot be produced in a cell in the objective function but include constraints which do not allow intercell transfers. 2.6.2 Model 2 Wei and Gaither (1990a, 1990b), Dahel and Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and Logendran (1990, 1993) use models which are variations of this basic model. PAGE 69 58 These models include the cost of parts which cannot be produced in a single cell. Wei and Gaither (1990a, 1990b) consider a subcontracting cost to remove the part, Dahel and Smith (1991, 1993), Nagi, Harhalakis, and Proth (1990), and Logendran (1990, 1993) consider the cost to move parts between cells. These models also contain constraints which do not allow the processing time on a machine type in a cell exceed the time available of the machine type in the cell These models contain two other sets of constraints: one that limits the number of machines that can be assigned to a cell to a prespecified number; and one that limits the number of machines that are available of each machine type. The second constraint is important because if an unlimited number of machines are available for each type then a cell could be created for each part and there would be no need for subcontracting or intercell transfers. Dahel and Smith (1991, 1993) incorporate part operations detail into their model. Nagi, Harhalakis, and Proth (1990) consider alternate routings in their formulation and allow a part's production requirement to be spread over several routings so machines can be better PAGE 70 59 utilized. Wei and Gaither (1990b) include, along with the cost of bottleneck parts, average cell utilization, intracell load imbalances, and intercell load imbalances in the objective function. Each of the items is given a weight and the objective is to minimize the weighted sum of the items Logendran (1990, 1993) includes both intracell and intercell moves in the objective. Weights are given to both types of moves (moves are based on operations sequences and a lower weight is usually given to intracell moves) and the objective is to minimize weighted moves. Logendran (1990) includes constraints that do not allow a machine's utilization to fall below a specified level; and Logendran (1993) includes machine utilization in the objective function. 2.6.3 Methods Since the models contain integer variables, reasonable size problems can be computationally difficult to solve, therefore, many of the procedures used are heuristics which may not solve the problem optimally. As in other categories a variety of methods are used in the procedures. PAGE 71 60 Several procedures (Shafer and Rogers 1991, Askin and Chin 1990, Askin and Subramanian 1987, Logendran 1990, Nagi, Harhalakis, and Proth 1990, and Sule 1991 ) break the problem into stages and solve each stage sequentially. Some of the procedures solve the subproblems in each stage optimally. Shafer and Rogers (1991) break their formulation into two subproblems, a p-median problem and a travelling salesman problem, and solve the subproblems optimally. Nagi, Harhalakis, and Proth (1990) create two subproblems and then iterate between the two subproblems until convergence to a local minimum is achieved. Other methods are also used Wei and Gaither (1990) fix seed machines and then form cells by solving the integer program. Vakharia, Chang and Selim (1993) use simulated annealing and tabu search. Dahel and Smith (1991, 1993) use the constraint method to solve their multi objective model 2.7 Procedures that Consider Multiple Scenarios for Part Volumes Procedures in this category recognize that demand for parts may not remain the same over time, or that demand for PAGE 72 61 parts may be stochastic. Previous procedures, which considered part volumes, were based on annual or average demand and were considered to be deterministic. If demand varies from period to period or estimates for demand turn out to be inaccurate a cell layout may not perform as well as expected. Cell layouts created by procedures in this category should have the flexibility to perform well even if there are expected volume swings or volume differs from what was estimated. Little work has been done in this area. Vakharia and Kaku (1988, 1993) developed a procedure to handle long-term demand changes. Seifoddini (1990) developed a procedure for probabilistic demand estimates. Vakharia and Kaku (1988, 1993) focus on a strategy of part reallocation to respond to demand changes as opposed to partial cell system redesign or complete cell system redesign. This strategy is chosen because it has the lowest cost of change (if cell design is robust changes can be handled by this strategy). Vakharia and Kaku proposed a model. The objective is to minimize material handling cost and additional machine investment costs. The constraints of the model are each operation for each part is assigned to a PAGE 73 62 cell, determination of the number of new machines required based on machine capacity, a cell is visited if an operation is performed in a cell, integrality, and nonnegativity Additional constraints can be added to limit cell size to a specified number of machines. A heuristic is used to solve the model. Seifoddini's (1990) procedure incorporates the uncertainty of product mix into the final solution. The procedure has three major steps: 1) the product mix and the associated part-machine incidence matrix are expressed in the form of a probability function, 2) a set of alternate solutions are developed based on all possible product mixes, 3) a measure of effectiveness is developed and used as a criterion for evaluating different solutions for the purpose of choosing the best solution. This procedure assumes there is a discrete distribution of product mixes and part-machine incidence matrices. The procedure uses similarity coefficients and a clustering algorithm to develop a cell layout for each product mix that minimizes intercell material handling cost. The intercell material handling cost is then calculated for each product mix and each layout, and an expected intercell material handling cost is calculated PAGE 74 63 for each layout. The layout with the lowest expected cost is chosen. 2.8 Other Procedures Procedures in this category incorporate data that is not included in the previous categories. Gunasingh and Lashkari (1989b) formulate an integer programming model that groups machines based on machine tooling availability and parts tooling requirements The authors define a compatibility index between a machine and a part. The index is based on the number of common tools between part i and machine j divided by the minimum of the number of tools part i requires or machine j has available. The objective of the model is to maximize the compatibility index. Constraints of the model include a cell size restriction allocation of each machine to at least one cell, and integrality. Irani, Cavalier and Cohen (1993) consider both the layout of cells and machines in order to minimize the effect of intercell movement The authors use a from-to travel PAGE 75 64 chart to layout machines within a cell and layout cells with respect to each other. The procedure also considers options such as placing a machine that is common to two cells, that are adjacent to each other, between the two cells so that greater flexibility of machine utilization can be achieved. Min and Shin (1991) formulated a model that incorporates human factors into cell formation. The authors believe that, in addition to the normal functions, cell formation should also allocate operators to cells, assign machines to each operator, and determine the job scope for each operator The model includes many sets of constraints and is a multiobjective goal program. Some of the constraints and objectives that stem from incorporating human factors are maximum number of operators allowed in a cell, maximum number of parts that operator p can handle, assignment of parts to operators, wage rates of operators, and skill matching of an operator with a part. 2.9 Summary The review of procedures in the previous sections illustrates how diverse cell formation procedures are with PAGE 76 65 respect to factors considered and data required, as well as the methods used. A single procedure probably cannot be developed which considers all the possible factors that could impact on the performance of a cell layout. When choosing a cell formation procedure to apply to a particular situation selection should depend on the factors that appear to be most important. In many instances it may be appropriate to use several procedures and then compare the layouts that are generated. The review of the procedures in section seven indicates that more work could be done to develop procedures that create good cell layouts in situations that experience demand variability or demand uncertainty. Cellular layouts usually tradeoff strengths and weaknesses of product line layouts and process layouts. A key strength of a process layout is its flexibility to handle volume changes or uncertainty. product line layouts tend to be less flexible in these environments. Procedures which can evaluate whether a shop should be organized with small efficient cells dedicated to a few parts or larger less efficient cells which can be used to produce many parts, or a mixture of large and small cells should be of aid to the shops which exist in the environment described above PAGE 77 CHAPTER 3 MODELS 3.1 Introduction This chapter presents two models which can be used for identifying cellular configurations in the presence of dynamic demand. Model I attempts to form cells that do not exceed a predetermined cell size in terms of the number of machines contained in each cell. This model considers materials handling costs as well as inventory and backordering costs to select a solution. Model II develops a cell configuration for a shop by selecting cells from a set of potential cells, which could be of varying sizes. To develop a cell configuration, first several runs of model I could be made with varying cell size parameters as input. The cells that were formed in each of the runs can be included in the set of potential cells. Model II would then be run using this set of potential cells to create a cell configuration. 66 PAGE 78 67 3.2 Model I 3.2.1 Description of Model I This model attempts to form cells that do not exceed a cell size constraint which is specified in terms of the number of machines a cell can contain. The model can also form a remainder cell which has no cell size constraint The model uses a planning horizon which consists of T periods. The parts to be produced during the planning horizon are known The demand for each part is known and can vary from period to period. Each part requires a set of machine types for production, and the processing time per unit for each part on each machine type, that is required by the part, is a known parameter There are a limited number of machines of each type that can be used to produce parts. Each machine has a limited capacity available during each period in the planning horizon. The capacity of the machine types within each cell is an important constraint in this model. The total processing time required of a machine type within a PAGE 79 cell during any period cannot exceed the total capacity of that machine type within that cell. 68 In order to effectively utilize machine capacity the model allows for inventory planning decisions. The number of units produced of a part during a period can exceed the demand requirement for the period. The excess production can be stored in inventory and used to satisfy demand requirements in later periods. A part's demand requirement, for a period, may also be partially or completely backordered and excess production of the part during a later period may be used to fill backorders. If an inventory of a part is held then there is a cost associated with the inventory. If a backorder is incurred there is a cost associated with the backorder. Each part has a unit inventory holding cost and a unit backorder cost associated with it. These costs are on a per period basis The unit cost of producing a part depends on where in the shop the unit is produced. If the unit is produced entirely in a cell that meets the cell size constraint then the production cost is zero. If the unit is produced entirely in the remainder cell there is a cost associated ( by part ) with producing the unit. If the unit is produced PAGE 80 69 using machines that reside in two or more cells there is an intercellular movement cost associated (by part) with producing the unit. Each part has a unit remainder cell production cost and a unit intercellular movement production cost associated with it. 3.2.2 Model I Formulation The formulation of model I is shown below. I T Min Z= L L (MMRiXitr+MMiiXitI+hiiit+biBit) il t-1 (1) Subject to K rit -1 -Bit -1 + L xitk+xitrrit+Bit=d it c-1 I L Pi jxitk~ c j yjk rl I K I LL pijxitk+ L pij (XitI+Xitr) CjAj il kl il K I: fori=l, ... ,I; (2) t=l, ... T. for k=l, I K & r; ( 3) for j =1, I J j for t=l, ... ,T. for j =1, I J j ( 4) for t=l, ... ,T for j=l, ... ,J. (5) for k=l, ... K ( 6) PAGE 81 Y jk is integer for all j & k. ( 7) All variables are ~ O ( 8) where I=The total number of parts. T=The total number of periods J=The total number of machine types. K=The total number of cells not including a remainder cell. i=part index (i=l, ... ,I). t=period index (t=l, ... ,T). j=machine type index (j=l, ... ,J). k=cell index (k=l, ... ,K & r). MMR i =The cost to produce 1 unit of part i in the remainder cell. 70 MMi i =The cost to produce 1 unit of part i using two or more cells (uses intercell movement ) h i =The cost to hold 1 unit of part i for 1 period. b i =The cost to backorder 1 unit of part i for 1 period d it =The demand for part i in period t. P ij= The processing time for part ion machine type j. PAGE 82 C j =The capacity in hours of 1 machine of type j. A j =The number of machines available of type j. CS=The maximum number of machines allowed in a cell excluding the remainder cell X i tk=The number of units of part i produced in cell k during period t. X i tr=The number of units of part i produced using intercell movement during period t. I it =The inventory of part i at the end of period t. Bit=The number of units on backorder of part i at the end of period t. Y j k=The number of machines of type j assigned to cell k. 3.2.3 Description of Model I Equations 71 (1) Objective function: The objective function is the sum over all parts and all periods of four terms for each part during each period. The first term is the cost of producing a unit of part i in the remainder cell times the number of units produced of part i in the remainder cell during period t The second term is the cost of producing a unit of part i using intercell movement times the number of units produced PAGE 83 72 of part i using intercell movement during period t. The third term is the cost to hold one unit of part i in inventory for one period times the number of units of part i that are in inventory at the end of period t. The last term is the cost to hold one unit of part ion backorder for one period times the number of units on backorder at the end of period t. (2) This set of equations represents material balance constraints. The first two terms on the left side of the equality (Iit -iBit -i ) represent the net inventory position of part i at the beginning of period t (or the end of period t-1) and the last two terms on the left side of the equality ( Iit+Bit) represent the negative of the net inventory position of part i at the end of period t. The middle two K terms on the left side of the equality ( L X itk +Xitr) equals 1-1 production in units of part i during period t. Therefore, the net inventory position of part i at the beginning of period t plus production of part i during period t minus the net inventory position of part i at the end of period t must equal the demand for part i during period t (the right side of the equality). There is a constraint for each part during each period. PAGE 84 73 (3) These constraints ensure that the total processing time required by parts produced in cell k during period t does not violate the capacity of machine type j in cell k. There is a constraint for each machine type, in each cell during each period. (4) These constraints ensure that the total processing time required by parts produced during period t does not violate the capacity of machine type j. There is a constraint for each machine type during each period. (5) These constraints ensure that the total number of machines of type j that are assigned does not exceed the number of machines of type j that are available. There is a constraint for each machine type. (6) These constraints require that cell knot contain more than a total of CS (the cell size parameter) machines. There is a constraint for every cell with the exception of the remainder cell. (7) These constraints require that the number of type j machines assigned to cell k be integer. There is a constraint for each machine type in each cell. (8) These constraints require all the production, inventory, and backorder variables to be nonnegative. PAGE 85 74 3.2.4 An Example of Model I The following example shows how model I is applied to a problem with 2 parts, 3 machine types, and two periods of demand The part-machine incidence matrix for this example is shown in table 8. Part 1 requires machine types 2 and 7, and part 2 requires machine types 1 and 2. The cell size limit used in this example is two machines. A remainder cell can also be formed. Table 9 shows the demand by period, the unit inventory cost, and the unit backorder cost for each part. Table 10 shows the number of machines of each type that are available and the capacity in units per machine of each type. The cost to manufacture either part in a cell that meets the cell size restriction is 0, the cost to manufacture either part in the remainder cell is 1, and the cost to manufacture either part using intercell movement is 3. In the formulation of this problem we allow for two cells that meet the cell size constraint and a remainder cell. The objective for this problem is PAGE 86 Table 8 Part-Machine Incidence Matrix for Model I Example. Machine Type Part 1. 2. ]_ 1 1 1 2 1 1 Table 9 By Part Demand. Part 1 2 Period _1 ___ 2_ 450 0 0 450 Unit Unit Inv. Cost B/o Cost 1 4 1 4 Table 10 Machine Data. Machine Type # Available Capacity/Machine 1 1 450 2 2 225 7 1 450 The full formulation for this problem is shown in appendix A. Examples of each of the different types of constraints will be shown below. The demand constraint for part 1, period 1 is 75 The capacity constraint for machine type 2 in cell 1 during period 1 is PAGE 87 The shop capacity constraint for machine type 2 during period 2 is The machine availability constraint for machine type 7 is Y n + Y12+Y 1r~ l The cell size constraint for cell 1 is 76 The solution to this model is to form one cell with all the machines in it (a process layout), produce 450 units of part 1 during period 1, and produce 450 units of part 2 during period 2. With this solution no inventory or backorders would be required, and the objective function value equals 900 It is interesting to see how the solution would change if we had assumed a constant demand of 225 units for each part during each period. With this assumption, the model forms two cells. The first cell would have a type 2 machine and a type 7 machine. This cell wou l d produce 225 units of part 1 in each period. The second cell would have a type 1 machine and a type 2 machine. This cell would produce 225 units of part 2 in each period The objective function value would equal 0. I f this plan were implemented and the demand turned out to be the same as in table 9 the cost would be 1125. PAGE 88 77 3.3 Model II 3.3.1 Description of Model This model selects cells, which will be formed to produce parts, from a set of potential cells. The objective of the model is to minimize production, inventory, and backordering costs. The model uses a planning horizon which consists of T periods. The parts to be produced during the planning horizon are known. The demand for each part can vary from period to period. Each part requires a set of machine types for production, and the processing time per unit for each part on each machine type, that is required by the part, is a known parameter. There are several potential cells that are being considered for inclusion in the shop. All the cells cannot be included because there are not enough machines available There are several different types of machines in the shop and the number of machines available of each type must be specified as an input parameter for the model. The configuration of each potential cell in terms of the number of machines of each type required by the cell is known PAGE 89 78 Each machine has a limited capacity available during each period in the planning horizon. The capacity of the machine types within each cell is an important constraint in this model. The total processing time required of a machine type within a cell during any period cannot exceed the total capacity of that machine type within that cell. In order to effectively utilize machine capacity the model allows for inventory planning decisions. The number of units produced of a part during a period can exceed the demand requirement for the period. The excess production can be stored in inventory and used to satisfy demand requirements in later periods. A part's demand requirement, for a period, may also be partially or completely backordered and excess production of the part during a later period may be used to fill backorders. If an inventory of a part is held then there is a cost associated with the inventory. If a backorder is incurred there is a cost associated with the backorder Each part has a unit inventory holding cost and a unit backorder cost associated with it. These costs are on a per period basis The unit cost of producing a part varies depending on which cell the unit is produced in. Each part has a cost PAGE 90 79 assigned for each cell. These costs can reflect the material handling and other costs associated with producing a unit of the part in a particular cell. A unit of a part can also be produced by using machines that reside in two or more cells. If this occurs the unit will incur costs associated with intercell material handling and other costs, such as the paperwork required for intercell movement and tracking. 3.3.2 Model II Formulation The formulation of model II is shown below. I T K I T Min Z= L L (CirXitr+hiiit+biBit) + L L L (Cikxitk) ( 9) tl t-1 k-l il tl Subject to K Iit-1 -Bit -1 + L xitk+x itr Iit+Bit=di t k-l I L Pi jxitk cjzkMjk t l I K I LL P ijxitk+ L P ij (Xitr> C jA:i t l k-l il K I: for i=l, ... I; ( 10) t=l, ... IT. fork=l, ... ,K&r; (11) for j =l, ... J; for t=l, ... T. for j=l, ... ,J; (12) for t=l, ... T for j =l, ... J. ( 13) PAGE 91 where All other variables~ O I=The total number of parts. T=The total number of periods. for k=l, ... ,K. J=The total number of machine types. K=The total number of cells. i=part index (i=l, ... ,I). t=period index (t=l, ... ,T). j=machine type index (j=l, ... ,J) k=cell index (k-1, .. ,K). 80 (14) (15) Cik=The cost to produce 1 unit of part i in cell k. Cil=The cost to produce 1 unit of part i using two or more cells (uses intercell movement). hi=The cost to hold 1 unit of part i for 1 period. b i =The cost to backorder 1 unit of part i for 1 period. dit=The demand for part i in period t. Pi j =The processing time for part ion machine type j. C j =The capacity in hours of 1 machine of type j. A j =The number of machines available of type j. PAGE 92 Mjk=The number of machines of type j required to form cell k. Xitk=The number of units of part i produced in cell k during period t. Xitr=The number of units of part i produced using intercell movement during period t. Iit=The inventory of part i at the end of period t. Bit=The number of units on backorder of part i at the end of period t. Zk=l if cell k is formed; O otherwise. 3.3.3 Description of Model II Equations 81 (9) Objective function: The objective function sums production, inventory, and backorder costs for all parts over all periods. There are four terms in the objective function. The first term sums across all parts, cells, and periods the cost to produce a unit of part i in cell k times the number of units produced of part i produced in cell k during period t. The second term sums the cost to produce a unit of part i using intercell movement times the number of units produced of part i using intercell movement during period t across all parts and all periods. The third term is PAGE 93 82 the cost to hold one unit of part i in inventory for one period times the number of units of part i that are in inventory at the end of period t summed across all parts and all periods. The last term is the cost to hold one unit of part ion backorder for one period times the number of units on backorder at the end of period t summed across all parts and all periods (10) This set of equations represents material balance constraints. The first two terms on the left side of the equality (Iit-i-Bit-i) represent the net inventory position of part i at the beginning of period t (or the end of period t-1) and the last two terms on the left side of the equality ( Iit+Bit) represent the negative of the net inventory position of part i at the end of period t. The middle two K terms on the left side of the equality ( L X itk +Xitr) equals 1--1 production in units of part i during period t. Therefore, the net inventory position of part i at the beginning of period t plus production of part i during period t minus the net inventory position of part i at the end of period t must equal the demand for part i during period t (the right side of the equality). There is a constraint for each part during each period. PAGE 94 83 (11) These constraints ensure that the total processing time required by parts produced in cell k during period t does not violate the capacity of machine type j in cell k. There is a constraint for each machine type, in each cell during each period. (12) These constraints ensure that the total processing time required by parts produced during period t does not violate the capacity of machine type j. There is a constraint for each machine type during each period. (13) These constraints ensure that the cells selected for inclusion in the shop do not require more type j machines than are available. There is a constraint for each machine type. (14) These constraints require that cell k is either selected (Zk=l) or not selected (Zk=O) for inclusion in the shop. (15) These constraints require all the production, inventory, and backorder variables to be nonnegative 3 3.4 An Example of Model II This example uses the data presented in tables 8 thru 10 of section 3.2.4. Table 8 shows the part-machine PAGE 95 incidence matrix; table 9 shows the demand by period, the unit inventory cost, and the unit backorder cost for each 84 part; and table 10 shows the number of machines of each type that are available and the capacity in units per machine of each type. Three potential cells are considered for inclusion in the shop. Table 11 shows the configuration of each cell in terms of the number of machines of each type that would be included in the cell if the cell is selected Table 11 Potential Cells Machine Type Cell l. 2. ]_ 1 0 1 1 2 1 1 0 3 1 2 1 Table 12 shows unit production costs for each part in each cell that the part can be completely manufactured in. Table 12 also shows unit production costs using intercell movement. The objective for this problem is PAGE 96 Table 12: Production Costs Part 1 2 cell 1 0 n/a cell 2 n/a 0 cell 3 1 1 Intercell 3 3 The full formulation for this problem is shown in appendix B. Examples of each of the different types of constraints will be shown below. The demand constraint for part 1, period 1 is 85 The capacity constraint for machine type 2 in cell 1 during period 1 is The shop capacity constraint for machine type 2 during period 2 is The machine availability constraint for machine type 2 is: The solution to this model is to select only cell 3. Cell 3 contains all the machines in the shop so this solution is a Process layout. 450 units of part 1 should be produced in period 1, and 450 units of part 2 should be PAGE 97 86 produced in period 2. This solution requires no inventory or backorders, and the objective function value equals 900. It is interesting to see how the solution would change if we had assumed a constant demand of 225 units for each part during each period. With this assumption, the model selects cells 1 and 2 for inclusion in the shop. 225 units of part 1 would be produced in cell 1 during each period, and 225 units of part 2 would be produced in cell 2 during each period. The objective function value would equal 0. If this plan were implemented and the demand turned out to be the same as table 9 the cost would be 1125 PAGE 98 CHAPTER 4 SOLUTION PROCEDURES 4.1 Introduction In this chapter, solution procedures for the models are proposed. Since model II develops a shop configuration from a list of potential cells, that may have been created by solving model I several times with different cell size parameters, the focus of the procedures proposed is on model II. A heuristic for solving model I is proposed in section 4 2 This heuristic can then be run, using several cell size parameters, to create a list of potential cells. Section 4.3 proposes lower bounding schemes for model II. Heuristics for solving model II are proposed in section 4.4 The lower bounding schemes in section 4.3 can be used, for large problems, to evaluate the heuristics of section 4.4. 87 PAGE 99 88 4.2 Model I Heuristic This heuristic requires several iterations to develop a solution for model I. During each iteration an attempt is made to form one cell that meets the cell size constraint If a cell is formed (that meets the cell size constraint) then the integer variables for that cell are fixed, and then the next iteration of the heuristic attempts to form another cell. If a cell is not formed (that meets the cell size constraint) the heuristic stops and a solution has been developed. The heuristic also stops if all the machines have been used to form cells that meet the cell size constraint. In addition to the notation of section 3.2.2, the following notation is required. Let n=The iteration index for the heuristic. Let M jk= The number of type j machines required by the cell formed in iteration n of the heuristic. The following steps are required: Step 0. Initialization set n=l; Step 1. Solve problem Pl (described below). PAGE 100 Step 2. a. set a=l; for j=l to J do if yj k >O then a=O; b. set b=l; for j =1 to J do K if ~L Mjk>O then b=O; kl C. if a+b>O then stop else n=n+l and repeat step 1. Problem Pl is as follows: I T Min Z= L L (MMRiXitr+MMiiXitr+hii i t+biB it) il tl Subject to Iit-1-B it-l + L x itk +Xitr+XitlIit+B it =d it k-1 I L Pijx itk !s cjMjk for ,-1 for for I L pijx itn!s CjYjn for 1 for 89 (16) for i = 1, ... I; ( 1 7) t=l, ... ,T. k=l, I n-1 j (18) j =1, J j t=l, ... IT. j =1, I J j (19) t=l, IT. PAGE 101 I E pijxitr ~ cj Yjr 1 I n I for j =1, ... J; for t=l, ... T. 90 ( 2 0) EE Pijxitk+E Pij(Xitr+XitR) ~ CjA:i for j=l, ... ,J; (21 ) il c-1 1-1 for t=l, ... T n-1 E Mjk+Yjn+Yjr~ Aj kl Yjn, Yjr are integer All variables are ~ O for j =1, ... J for all j. ( 2 2) (23) ( 24) ( 2 5) Model I requires J*(K+l) integer variables. Problem Pl reduces the number of integer variables to 2J; problem Pl, however, may have to be solved K times. For large problems, the computation time for the heuristic should be significantly less than the computation time required to solve model I optimally. The heuristic is a "greedy" heuristic in that at each iteration the best possible cell is formed given the cells formed during previous iterations. This heuristic should develop good solutions for model I, but not necessarily optimal solutions. PAGE 102 91 4.3 Lower Bounds for Model II This section describes methods that can be used to obtain a lower bound for model II without incurring the computational time, for large problems, that would be required to solve model II optimally. Three methods for obtaining a lower bound, for model II, are proposed in this section: linear programming relaxation, lagrangian relaxation, and linear programming relaxation tightened with valid inequalities. Three versions of the last method are presented. 4.3.1 Linear Programming Relaxation The linear programming relaxation for model II is formed by replacing (14) with for all k ( 2 6 ) The linear programming relaxation may then be solved using the simplex method If the optimal solution to the linear program has z k values which are all integer then it is also the optimal solution for the problem If the optimal solution to the linear program contains one or more non integer z k values then the corresponding objective function PAGE 103 92 value can be used as a lower bound on the optimal objective function of the problem. 4.3.2 Lagrangian Relaxation A Lagrangian relaxation for model II can be formed by dropping constraint set (11) from the problem and adding the following term to the objective function. J T K I LL L (Ujtk ( L Pijxitk-cjMjkzk) ) jl tl .t-1 tl where Ujtk=The Lagrangian multiplier for machine type j in cell k during period t. The resulting relaxed problem can be separated into two subproblems. Subproblem 1 (SPl) is the following linear program. I T K J Min Zepl =LL L (Xitk ( cik+ L ujtkpij) ) i l tI 1 k ,jl +LL (XitrCu+hiiit+biBit) i l tl Subject to (10), (12), and (15). Subproblem 2 (SP2) is the following integer program. J T K Max zep2= LL L ujtkMjkcjzk jl tl k l Subject to (13) and (14). ( 2 7) ( 2 8) Even though subproblem 2 is an integer program it has far fewer constraints (subproblem 2 has J constraints) than the original problem (model II can have as many as IT+JTK+JT+J PAGE 104 constraints) and should be able to be solved relatively easily. 93 For a given set of Ujtk multipliers, subproblems 1 and 2 can be solved independently, and Z 0 P 1 Z 0 P 2 is a lower bound on the objective function of model II. This Lagrangian relaxation does not have the integrality property so the lower bound obtained may be better than the lower bound obtained from the linear programming relaxation. If the solution to the subproblems meets all the constraints of (11) then the solution can be put into the objective function of model II and represents an upper bound on the optimal objective function for model II. If the solution to the subproblems meets all the constraints of (11) and there is no slack in any of the constraints of (11) then the solution is an optimal solution to model II. An attempt can be made to find the optimal solution or a better lower bound by using subgradient optimization to adjust the Ujt k multipliers, and repeatedly solve the two subproblems. PAGE 105 4.3.3 Linear Programming Relaxation Tightened by Adding Valid Inequalities Additional constraints can be added to the linear programming relaxation in an attempt to tighten the formulation of the problem and improve the lower bound on model II's objective function. The constraints that are added are redundant to the original integer program, but eliminate part of the linear programming relaxation's feasible region. These constraints are called "valid inequalities." Three formulations will be proposed in this section. 4.3.3.1 Valid inequality set 1 94 It is possible that there may be a cell in the list of potential cells that can produce one part or a few parts at a lower cost than any other potential cell. If the parts that would utilize this cell have very low capacity requirements, relative to the capacity of this cell, then the linear programming relaxation of model II would select a fraction of this cell for inclusion in the shop. This could lead to many fractional cells being selected for inclusion in the shop which is not possible to accomplish. If many fractional cells are selected, there will probably be a PAGE 106 large gap between the lower bound produced by the linear programming relaxation and the optimal objective function value for model II. 95 If any units of any part are produced in cell k then cell k must be selected and Zk=l. To tighten the linear programming relaxation, constraints which force zk to be larger than the fraction of any part's total requirement that is produced in cell k are added. It is likely that there is a part i that has a preferred cell kin which we would like to produce part i's entire requirement (over the planning horizon). If this is the case the additional constraints described above would either force z k to 1 or we would have to produce part of part i's requirement in a different cell and incur a higher production cost. This will cause the linear programming relaxation's objective function value to rise which improves the lower bound If we assume that the cost to hold a unit of inventory for one period of any part is greater than or equal to zero, there will be an optimal solution to model II in which total production of any part i will not exceed the total demand requirement of part i To see that this is so, suppose there is a solution in which the total units produced of a part were greater PAGE 107 96 than the part's total production requirement This implies the part would have inventory at the end of the planning horizon. The only effect of reducing production of this part in the final period (or preceding periods if necessary) so that the total production of the part equals the total demand requirement is to reduce the inventory held of the part at the end of the planning horizon to zero. Since the cost to hold inventory is assumed to be nonnegative, the objective value will not increase. Therefore, these additional constraints will not eliminate at least one optimal solution to model II and the optimal solution to this tightened linear programming relaxation will be a lower bound on model !I's objective function value. Formally let D i =part i's total demand requirement T (D i = L d i t ) Then we add the following constraints to the tl linear programming relaxation. T L x itk ~ D i z k 1-1 for all i & k This formulation is referred to as VIl 4.3.3.2 Valid inequality set 2 ( 29 ) To further tighten the linear programming relaxation, the following additional constraints can be added. Let l ~ d ~ T-1 then PAGE 108 d d L xit k rid ~ L ditzk tl tl for all i, d & k 97 ( 3 0) These constraints force zk to be greater than or equal to the fraction of part i's cumulative demand, from period 1 to period d, that is produced in cell k minus the fraction of part i's cumulative demand that is in inventory at the end of period d. Suppose there is a part i that does not have its entire requirement produced in cell k, but there exists a period din which part i's cumulative requirement till period dis produced in cell k. If only constraints (29) are added it is possible that Zk PAGE 109 constraints tighten the formulation because they deal with individual periods. Suppose there exists an i, t & kin which X itk =d it and Ii t =O, and Bit -i =O. Then these constraints force Zk=l. With only constraint sets (29) and (30) it is possible that Zk PAGE 110 99 I can be solved several times using various cell size parameters, the heuristic will check several sets of cells. The set of cells with the lowest objective function value will be selected for inclusion in the shop. To obtain model II's objective function value for each set of cells, the following linear program (LPSET) is run. (LPSET) I T K I T Min Zsetm= L L ( ciixitI+hiiit+biBit) + L L L (Cik~tk) ( 3 2) il tl kl i l tl Subject to x. r it-1 -Bit -1 + L xitk+xitrrit+Bit=dit kl fori=l, ... ,I; (33) I L Pijxitk cjMjk il I K. I LL P ij xitk+ L Pij (Xitr) cjAj il kl i l All other variables~ 0 t=l, ... ,T. for k=l, ... K.n; (34) for j=l, ... ,J; for t=l, ... T. for j=l, ... ,J; (35) for t=l, . T ( 3 6) where K.n equals the number of cells in solution set m. The rest of the notation used in LPSET is consistent with the PAGE 111 100 notation of model II. It is likely that the number of cells included in each run of LPSET is much smaller than the complete list of potential cells. Therefore, there will be fewer constraints and variables in LPSET than in the linear programming relaxation of model II. The number of times LPSET is run depends on the number of cell size parameters used to create solutions to model I. In any case, this should not be a prohibitively large number of runs. 4.4.1.2 Formal procedure for heuristic 1 The following notation is required to formally state the heuristic. Let m=set index. Let Zmin=the best objective function value. Let Bestset=the best solution set found. Let nsol=the number of solution sets. The following steps are required for heuristic 1. Step 0. Initialization. Set m=l; Step 1. a. Run LPSET using cell set m; Bestset=m; PAGE 112 Step 2. If m=nsol then stop else m=m+l and repeat step 1; 4.4.1.3 Discussion of heuristic 1 101 If the optimal solution to model II contains several cells that are of similar size, in terms of the number of machines in each cell, with possibly one larger "remainder cell" then this heuristic should provide a reasonably good solution If, however, the optimal solution to model II contains cells of varying sizes (cells come from a variety of model I solutions) then the solution selected by this heuristic may be a poor solution, relative to the optimal solution. 4.4.2 Heuristic 2 A run of the VIl linear programming relaxation (VIl is described in section 4.3.3.1) is made to start this heuristic. The heuristic then performs K iterations, where K is the number of potential cells. During each iteration the heuristic attempts to select the cell, that has not been selected yet, that has the highest selection value from the VIl run. Finally a linear program is run to obtain the objective function value of the solution. PAGE 113 The following notation is required for heuristic 2. J=The number of machine types. K=The number of potential cells. 102 A[j]=The number of machines of type j that are available. nml[j]=The number of machines of type j that are available after subtracting machines required by the cells that have been selected. M[j,k]=The number of machines of type j that are required by cell k. Z[k]=The selection value for cell k. fix[k] indicates whether Z[k] has been fixed. numfix=The number of cells with fixed Z[k] values. Zmax=The maximum Z[k] value among cells that have not had their Z[k] values fixed. max=The cell with the maximum Z[k] value among cells that have not had their Z[k] values fixed. feas indicates whether the cell being tested for selection will require more machines of a type than are still available The following steps are required for heuristic 2. Step 0. Initialization For j=l to J do set nml [ j ] =A [ j ] ; PAGE 114 For k=l to K do set fix[k]=O; Set numfix=O; Step 1. Run VIl; For k=l to K do set Z[k] to the zk value from the VIl run; Step 2. Set Zmax=-1; For k=l to K do Step 3. if fix[k]=O and Z[k]>Zmax then begin Zmax=Z[k]; max=k; end; set feas=O; For j=l to J do if M[j,max]>nml[j] then feas=l; if feas=l then begin set Z[k]=O; set fix[rnax]=l; end 103 PAGE 115 else begin end; for j=l to J do nml[j]=nml[j]-M[j,max]; set Z[max]=l; set fix[max]=l; set numfix=numfix+l; Step 4. If numfix PAGE 116 105 provide a poor solution compared to the optimal solution to model II. 4.4.3 Heuristic 3A 4.4.3.1 Description of heuristic 3A Heuristic 3A starts with a solution obtained by using heuristic 2 Heuristic 3A then attempts to improve the solution. If the solution improvement phase is successful, the solution is updated and then the solution improvement phase is repeated using the new solution. The solution improvement phase iteratively fixes cells to be included or excluded from a solution. This phase performs an iteration for each cell until either an improved solution is found or an iteration for each cell has been completed. During each iteration a trial solution is developed, and the trial solution's objective function value is compared to the current solution's objective function value. The trial solution is formed based on one of the following cases 1) the cell being examined is in the current solution or 2) the cell being examined is not in the current solution. If case 1) applies, the cell being examined is excluded from the trial solution, the other cells included PAGE 117 106 in the current solution are included in the trial solution, and an integer program may select some of the cells, not included in the current solution, to be included in the trial solution. If case 2) applies the cell being examined is included in the trial solution, the other cells that are not included in the current solution are excluded from the trial solution, and an integer program may select some of the cells, that were included in the current solution, to be included in the trial solution. In either case, the integer program used to select cells has a 0-1 variable for each cell to indicate whether or not the cell is selected (some variables are fixed based on which case applies), and constraints which do not allow the cells selected to require more machines of a type than are available. The objective function coefficient for each variable is the cell selection value obtained from the VIl linear programming relaxation. The objective of the integer program is to maximize the sum of the cell selection values of the cells that are selected to be included in the trial solution. The trial solution's objective function value is then obtained by running LPSET If the trial solution's objective function value is less than the current solution's objective function value then PAGE 118 107 the trial solution becomes the current solution and the improvement phase is repeated, otherwise the trial solution is discarded and the next cell (if there are cells which have not been examined) is examined. 4.4.3 2 Formal procedure The following notation is required for heuristic 3A K=The number of cells. J=The number of machine types. k=The cell index m=The iteration index. Z k =Cell selection variable used in VIl. Obj [k]=An objective function coefficient for cell k used in an integer program. Z[k]=A cell selection indicator for cell k Solval[k ] =Cell selection indicator for cell k for the current solution incum=The objective function value of the current solution. Change=An indicator used to decide if the improvement phase should be repeated. Mjk =The number of type j machines required for cell k. A j =The number of type j machines available Heuristic 3A requires the following steps. PAGE 119 Step 0. Initialization. a. Run VIl; For k=l to K do set Obj [k] =Zk; b. Run heuristic 2 to obtain the initial Z[k] values; For k=l to K do Solval[k]=Z[k]; 108 set incum=objective function value of heuristic 2's solution; Step 1 a. set m=l; b Run procedure fixvalues (described below); c. Run procedure Intprog (described below); d. Run LPSET using cells with Z[k]=l; If the objective function value PAGE 120 Step 2. If Change=l then repeat step 1; If change=O and m PAGE 121 Zk=l Zk=0 ZkE{ 0, 1} for all k such that Z[k]=l. for all k such that Z[k]=0. for all k such that Z[k]=-1. Step b. For k=l to K do set z [k] =Zk; 4.4.3.3 Discussion of heuristic 3A 110 ( 3 9) ( 40) ( 41) The objective function value of the final solution obtained using heuristic 3A will be at least as good as the objective function value of the solution obtained by using heuristic 2. The reason for this is that heuristic 3A starts with a solution obtained by using heuristic 2, and only modifies a solution if a new solution will be an improvement. For problems that heuristic 2 performs poorly, the improvement obtained by running heuristic 3A may be significant. A potential drawback of using heuristic 3A is that this heuristic requires a 0-1 integer program to be solved in step le For most problems the integer program should be fairly easy to solve, however, there may be large problems which would cause the integer program to be computationally prohibitive to solve. Therefore, heuristic 3B, which does not require an integer program to be solved, is proposed in the next section PAGE 122 111 4.4.4 Heuristic 3B Heuristic 3B is very similar to heuristic 3A. The difference between heuristic 3B and 3A is that steps le and ld in heuristic 3B replace step le in heuristic 3A. Step le in heuristic 3B uses the linear programming relaxation of the 0-1 integer program used in step le of heuristic 3A. Step ld of heuristic 3B then uses heuristic 2 to create an integer trial solution. Heuristic 3B requires the following steps. Step 0. Initialization. This step is the same as step 0 of heuristic 3A. Step 1. a. set m=l; b. Run procedure Fixvalues (described in section 4.4.3); c. Run procedure Linprog (described below); d. Run heuristic 2; e. Run LPSET using cells with Z[k]=l; If the objective function value PAGE 123 For k=l to K do set Solval[k]=Z[k]; set Change=l; end; Step 2. This step is the same as step 2 of heuristic 3A. Procedure Linprog; Step a. Run the following linear program; Max Z= (37) Subject to (38), (39), (40) and 112 for all k such that Z[k]=-1 (42) step b. For k=l to K do set z [k] =Zk; Since heuristic 3B starts with a solution created by heuristic 2 and attempts to improve on it, the objective function value obtained using the final solution will be at least as good as the objective function value of the solution obtained using heuristic 2. The performance of heuristic 3B, in terms of objective function value, should be similar to heuristic 3A's performance. The advantage of using heuristic 3B instead of heuristic 3A is that heuristic PAGE 124 113 3A could have a long run time on some large problems because an integer program must be solved many times. PAGE 125 CHAPTER 5 DESCRIPTION OF EXPERIMENTS AND DATA 5.1 Introduction The purpose of the experiments, described in this chapter, is to test the effectiveness of lower bounding and heuristic procedures for model II. Using the heuristic (described in the previous chapter) to solve model I or other cell formation procedures, a reasonably good list of potential cells could be created for most problems. Selecting the proper cells from a large list of potential cells that will perform well in an environment that experiences dynamic demand is a challenging task. In order to configure a cell system that will perform well, questions such as the following need to be answered Should cells that are of similar size or cells of a variety of sizes be created? Should the shop be completely broken into cells or should a few cells be created with a large remainder cell? 114 PAGE 126 115 Since model II attempts to answer these types of questions, that is why it was decided to test procedures for model II. 5.2 Experiments Two experiments were conducted during this research. Fifteen small problems were created and used in the first experiment. Two larger problems were used in the second experiment to see the effect of problem size on the performance of the procedures presented in chapter 4. For each problem (in both experiments) model I was solved, using the heuristic described in chapter 4, for several cell size parameters to create a list of potential cells. Model II was then solved optimally for each problem, and the optimal objective function value for each problem was recorded. For each problem in experiment 1, the lower bounding procedures described in chapter 4 were used to obtain a lower bound on model II's optimal objective function value and the lower bound was compared to the optimal objective function value (The procedure that uses the Lagrangian relaxation to obtain a lower bound, uses subgradient optimization and performs 2000 iterations. The best lower PAGE 127 116 bound obtained is compared to the optimal objective function value). Each of the heuristic procedures described in chapter 4, for solving model II, were also used to obtain a solution for each problem and the objective function value for each solution was compared to the optimal objective function value. The following lower bounding procedures were used to obtain a lower bound on model II's optimal objective function value for each problem used in experiment 2 linear programming relaxation, VIl linear programming relaxation, and VI3 linear programming relaxation. The lower bounds obtained were compared to the optimal objective function value. Each of the heuristic procedures described in chapter 4, for solving model II, were also used to obtain a solution for each experiment 2 problem and the objective function value for each solution was compared to the optimal objective function value. 5.3 Experiment 1 Data This section describes how the data for the 15 problems used in experiment 1 were obtained or created. For each PAGE 128 117 problem the following data is required a part-machine incidence matrix, the processing time required by each part on each machine type, demand by period for each part, inventory and backorder costs for each part, per machine capacity for each machine type, the number of machines available of each type, a list of potential cells specifying how many machines of each type are required by each cell, unit production costs for each part for each cell the part can be manufactured in, and the unit production cost using intercell movement for each part. 5.3.1 Part-Machine Incidence Matrices The 15 part-machine incidence matrices used in the experimental problems were obtained from the literature. Several of the original part-machine matrices were modified. This was done to ensure the resulting problems were small enough to solve model II optimally. Table 13 shows the reference for each problem's original part-machine matrix. The table also shows the number of parts(!) and the number of machine types (J) in each matrix. The original matrices for problems 8 thru 15 were modified for this study. The modifications to these PAGE 129 118 Table 13: Part-Machine Data Sets Number Number of of Machine Problem Reference Parts (I) Types (J) 1 Logendran (1990) 14 7 2 Selim (1994) 19 10 3 Askin and Subramanian 24 14 (1987) 4 Srinivasan, et al (1990) 20 10 5 Seifoddini (1989) 22 11 6 Chandrasekharan and 20 8 Rajagopalan (1986) 7 Dewitte (1980) 19 12 8 Stanfel (1985) 24 7 9 Stanfel (1985) 18 13 10 Stanfel (1985) 20 10 11 Stanfel (1985) 12 7 12 Carrie (1973) 18 8 13 Carrie (1973) 35 10 14 Carrie (1973) 28 11 15 Srinivasan, et al (1990) 30 12 problems are described below. Problem 8 : The original matrix was 24X14. Seven machine types were created for problem 8 by pairing machine types and creating a single machine type for each pair. PAGE 130 119 Problems 9 thru 11: The original matrix was 50X30. This matrix can be decomposed into three independent matrices. Each of the three matrices is used in the experiments. Problem 12: The original matrix was 18X24. The machine types were divided into 8 groups with 3 machine types in each group. A single machine type was created for each group to create the matrix used in this problem. Problem 13: The original matrix was 30X20. Ten machine types were created by pairing machine types and creating a single machine type for each pair. Problem 14: The original matrix was 46X28. The only parts used in this study were parts that required machine types 19 thru 28, and only machine types 1 thru 12 are used in this study Problem 15: The original matrix was 30X16. Machine types 13 thru 16 were paired with machine types 3 thru 6 and a single machine type was created for each pair. 5.3.2 Processing Times The time to process a unit for each part is required for each machine type. If a part does not require a machine type, its processing time is zero on that machine type. For PAGE 131 120 each machine type required by each part a processing time was generated randomly. This was done by selecting a number from a uniform distribution with parameters [1,10] and rounding to the nearest integer. 5.3.3 Part Demand Demand for each part for each period in the planning horizon is required. There are six periods in the planning horizon, for all problems, in this study. Period demand for each part was obtained in two steps. First a total demand for each part was generated, and then each part's total demand was broken into period demand. The total demand for each part was generated randomly by selecting a number from a uniform distribution with parameters [100,4000] and rounding to the nearest hundred. To create the period demand for each part, an index was generated for each period (by part) using a uniform distribution with parameters [0,1] The index for each part was then normalized, and the normalized index was multiplied by total part demand to obtain each period's demand 5 3.4 Inventory and Backorder Cost A per unit inventory cost and a per unit backorder cost is required for each part The per unit inventory cost for PAGE 132 121 each part is generated randomly. For each part, either 1 or 2 is selected, with equal probability, as the inventory cost To obtain a backorder cost for each part, the following is done. A multiplier for each part is generated randomly. Each part's multiplier is either 3,4, or 5. Each of the three values is equally likely to be chosen. Each part's multiplier is multiplied by the part's inventory cost and the result is the part's backorder cost. 5.3.5 Machine Capacities and Number of Machines of Each Type For each machine type, the number of machines available and the amount of time, per period, each machine is available (capacity) are required. For each machine type a target utilization rate is generated randomly. The target utilization rate selected will be 70, 80, or 90 percent. Each value has an equal probability of being selected. To determine the number of machines available for a type three steps are performed. First the total processing time, over the entire planning horizon, required by all parts on each machine type is calculated. The total processing time for each machine type is then divided by the machine type's target utilization rate. The final step is to divide the PAGE 133 122 result of the previous step by a constant (The constant is 20000, 40000, or 60000. The actual constant chosen controls the total number of machines) and round to the next highest integer The capacity per machine is then set, for each machine type, so that the average utilization over the planning horizon will be identical to the utilization target. 5.3 6 List of Potential Cells To create the list of potential cells for each problem, model I was solved, using the heuristic described in chapter 4, for several cell size parameters. The cells obtained for each solution were added to the list of potential cells A cell is also included, in the list of potential cells, that contains all the machines (a Process layout) for that problem. The number of cell size parameters and the cell size for each parameter were picked in a way that would ensure the resulting list of potential cells for each problem would not exceed 30. A unit remainder cell production cost and a unit intercell production cost for each part was arbitrarily set equal to the part's inventory cost, and the unit intercell PAGE 134 production cost for each part was set equal to the part's unit remainder cell production cost plus two. 5.3.7 Cell Production and Intercell Production Costs 123 For each part, the cost to manufacture a unit of the part in each cell, that can manufacture the part, is required. A formula is used to assign these costs The formula assigns a higher cost to larger, in terms of number of machines contained in the cell, cells that can manufacture a given part compared to smaller cells. The following notation is required for the formula Let Cik=The unit cost to produce part i in cell k Let nmp i =The number of machine types required by part i Let nmc k =The number of machines contained in cell k. The formula is ( 4 2 ) For each part, the unit cost to produce the part using intercell movement is required. This cost was set equal to the cost of producing the part in the cell that contains all the machines ( for each problem a cell is included that contains all the machines) plus two. PAGE 135 124 5.3.8 Summary Table 14 summarizes some of the results of the data creation procedures. For each problem, the table shows the total number of machines in the shop, the minimum number of machines available of any type, the maximum number of machines available for any type, and the number of potential cells. Table 14: Summary of Experimental Problems Number Range of of Total# # of Machines Potential Problem of Machines of a Type Cells 1 11 1-2 16 2 23 1-3 23 3 28 1-4 29 4 23 1-4 26 5 21 1-3 22 6 16 1-4 21 7 30 1-4 28 8 28 2-6 29 9 20 1-2 19 10 21 1-3 23 11 16 1-3 18 12 22 2-4 20 PAGE 136 125 Table 14: Summary of Experimental Problems Number Range of of Total# # of Machines Potential Problem of Machines of a Type Cells 13 35 2-5 29 14 24 1-5 22 15 27 1-4 28 5.4 Experiment 2 Data The same items, required in experiment 1, are required for each problem that is used in experiment 2. The two part-machine incidence matrices used in this experiment were obtained from the literature. The first matrix is from Chandrasekharan and Rajagopalan (1987). This matrix is based on routings obtained from a medium size production system, and consists of 100 parts and 40 machine types The second matrix is from Burbidge (1975) This matrix is based on routings obtained from the Black and Decker company and consists of 90 parts and 30 machine types PAGE 137 126 The data for processing times, part demand, inventory and backorder cost, machine capacities, number of machines available of each type, and cell production (including intercell) costs were created using the methods that were used to create the data for the problems used in experiment 1. To create the list of potential cells, for each problem, the same procedure that was used in experiment 1 was used except that the resulting list of potential cells for each problem was allowed to exceed 30. The first problem included a total of 114 machines and 99 potential cells. The maximum number of machines of a specific type was 6. Listings for the data used in this problem are included in appendix C. The second problem included a total of 91 machines and 102 potential cells. The maximum number of machines of a specific type was 11. PAGE 138 CHAPTER 6 RESULTS 6.1 Results of Experiment 1 6.1.1 Lower Bounds for Model II Table 15 shows the results for the linear programming and Lagrangian relaxations; and table 16 shows the results for the linear programming relaxations that are tightened with valid inequalities. For each problem, the optimal objective function value is shown in the second column of each table, and the lower bounds obtained by the relaxations are shown under the columns titled "value." The lower bound obtained by each relaxation is also compared to the optimal objective function value. This is shown, for each relaxation, in a column titled "Gap g.. 0 The Gap % is calculated as follows: 100-(Lower bound/Optimal objective function value*l00) The average Gap% is shown for each relaxation in the last row of each table. 127 PAGE 139 128 Table 15 : L.P. and Lagrangian Relaxations Optimal L P Relaxation Lagrangian Relax. Obj Fune. Gap Gap Problem Value Value % Value g.. 0 1 40460 28948 29 30130 26 2 88987 77596 13 76883 14 3 81552 67577 17 67502 17 4 41281 39293 5 39153 5 5 37719 27060 28 26601 30 6 42566 30256 29 29483 31 7 76799 62871 18 62142 19 8 37227 33300 11 34208 8 9 100525 80579 20 80155 20 10 59390 42589 28 44499 25 1 1 52635 34894 34 35121 33 1 2 64464 48499 25 48344 25 13 104579 75640 28 74840 28 14 101178 72602 28 69946 31 15 125028 87035 30 87041 30 Average 23 22 PAGE 140 129 Table 16: Valid Inequality Relaxations Opt. VIl VI2 VI3 Obj. Relaxation Relaxation Relaxation Prob. Fune. Value Gap Gap Gap Value % Value !!-0 Value % 1 40460 38049 6.0 39147 3 .2 39590 2.2 2 88987 85766 3 6 86944 2.3 87321 1. 9 3 81552 74072 9.2 77492 5.0 77787 4 6 4 41 281 39396 4.6 39 628 4.0 39821 3.5 5 37719 3687 2 2.2 37422 0.8 37431 0 8 6 42566 40419 5.0 41292 3.0 41420 2.7 7 76799 74316 3.2 76100 0.9 76494 0.4 8 37227 34868 6.3 35714 4.1 36175 2.8 9 100525 99806 0.7 100500 0.0 100525 0.0 10 59390 49 738 16 .3 51398 13. 5 52165 12.2 11 52635 45544 13.5 47048 10.6 47506 9 7 12 64464 59453 7 .8 61781 4.2 62839 2.5 13 104579 89485 14 4 91410 12.6 92614 11 .4 14 101178 97560 3.6 99620 1.5 100227 0 9 15 125028 113810 9.0 115427 7 .7 117372 6 1 Ave 7.0 4.9 4.1 PAGE 141 130 6.1.2 Heuristic Results for Model II Table 17 shows the results for heuristics 1 and 2; and table 18 shows the results for heuristics 3A and 3B. For each problem, the optimal objective function value is shown in the second column of each table, and the lower bound obtained using the VIl linear programming relaxation is shown in the third column of each table. The objective function value of the solution obtained using each heuristic is shown in the columns titled "value" for each problem. The objective function value obtained by each heuristic is compared to the optimal objective function value and to the lower bound obtained using the VIl linear programming relaxation. For each heuristic, the columns titled "Gap vs opt % 11 show the comparison between the objective function value obtained using the heuristic and the optimal objective function value The Gap vs opt% is calculated as follows: Let ObjH=The objective function value of the heuristic Let Optobj=The optimal objective function value Gap vs opt %=(ObjH/Optobj*100)-100 For each heuristic, the columns titled "Gap vs VIl % 11 show the comparison between the objective function value obtained PAGE 142 131 using the heuristic and the lower bound obtained using the VIl linear programming relaxation. The Gap vs VIl % is calculated as follows: Let LbVIl=The lower bound using VIl. Gap vs VIl %=(ObjH/LbVI1*100)-100 The average Gap vs opt% and Gap vs VIl % is shown for each heuristic in the last row of each table. 6.1.3 Discussion of Results 6.1 3.1 Lower Bounds Linear Programming Relaxation. The results indicate that the linear programming relaxation of model II provides a weak lower bound on the optimal objective function value of model II. A gap of less than 10 percent was obtained for only one of the 15 problems; and the gap was greater than or equal to 20 percent for 10 of the 15 problems. The average gap was 23 percent. Lagrangean Relaxation The results of the Lagrangean relaxation, proposed for model II, were disappointing. There are cases (5 of 15) where the Lagrangean relaxation improved on the lower bound provided by the linear programming PAGE 143 13 2 Table 17 : Heur i stics 1 and 2 Heuristic 1 Heuristic 2 Opt. Obj. VIl Fune. L ow er G ap v s G ap vs Gap v s Gap vs P rob. Value Boun d V a l u e Op t % V Il % Value Opt % VIl % 1 40460 38049 40460 0 0 6.3 40460 0 0 6 .3 2 88987 85766 942 4 6 5 .9 9 9 88987 0.0 3.8 3 8 155 2 7 40 72 941 20 15. 4 2 7 .1 83928 2.9 13.3 4 41281 39396 49996 2 1. 1 26 9 41281 0.0 4.8 5 37719 368 72 42703 13 .2 15 8 37719 0 0 2 2 6 42566 40419 4289 8 0.8 6.1 42566 0 0 5 3 7 76799 74316 957 91 24. 7 28 9 76799 0.0 3 .3 8 37227 3 48 6 8 4 89 2 3 31 4 40 3 37227 0 0 6 8 9 100525 99806 100 52 5 0. 0 0 7 100525 0.0 0. 7 10 59390 49 7 38 75 2 6 4 2 6. 7 5 1. 3 92719 56 1 8 6 .4 1 1 52635 45544 52635 0 0 15 6 71935 3 6 .6 57.9 12 64464 59453 6 4 464 0.0 8 4 7 2489 12.4 21. 9 13 1045 7 9 8948 5 144 4 17 38.1 61.4 104579 0 0 1 6. 9 14 101178 97560 101178 0.0 3.7 101178 0 0 3 .7 15 125028 113810 147730 18 2 29.8 140850 12.7 23 8 Ave. 13.0 22.1 8 .0 1 7.1 PAGE 144 133 Table 18: Heuristics 3A and 3B jieuristic 3A Heurist i c 3B Opt O b j. V Il Fune Lo w er G ap vs G ap VS G ap vs Gap vs P ro b V a l u e Bou nd V al u e Op t % VIl % Val u e O pt % VIl % 1 40460 38049 40460 0 0 6.3 40460 0.0 6.3 2 889 8 7 85 766 88 9 87 0.0 3. 8 8 8 98 7 0.0 3.8 3 81 55 2 7 4 07 2 81553 0.0 10 1 8 2263 0 9 11 1 4 412 8 1 39396 41 281 0. 0 4.8 41281 0.0 4.8 5 37719 3 6872 3 771 9 0 .0 2.2 37719 0.0 2 2 6 42566 40419 4 25 66 0 .0 5 3 42566 0 0 5 3 7 7 6 7 99 7431 6 76799 0 0 3 3 7 6799 0 0 3.3 8 37227 3 4 8 6 8 3722 7 0.0 6.8 3 7 22 7 0 0 6.8 9 100 5 25 99 8 06 100 525 0 .0 0.7 100525 0.0 0 7 10 59390 4 9738 66306 11 6 33.3 66306 11. 6 33.3 11 52635 4 5 544 5 3903 2 .4 18. 4 5390 3 2 4 18.7 12 64464 5 9453 65 7 90 2.1 10.7 65 7 90 2.1 10.7 13 104579 89485 104 5 79 0 0 16 9 104579 0 0 16 9 14 101178 97560 101178 0 0 3 7 101178 0 0 3.7 15 125028 113810 125028 0 0 9.9 125028 0 .0 9.9 Ave 1.1 9 1 1.1 9.1 relaxation; but the improvements were not that significant ( The best improvement was for problem 10; the gap decreased from 2 8% to 2 5% ) and the overall average gap% was 22 percent compared to 23 percent for the linear programm in g r elaxat i o n. PAGE 145 134 VIl linear programming relaxation. For 14 of the 15 problems, the lower bound obtained by using the VIl linear programming relaxation was significantly better than the lower bounds obtained by using either the linear programming relaxation or the Lagrangian relaxation. The exception was problem 4. There was a slight improvement for this problem The average gap% for the VIl linear programming relaxation was 7.0 percent (compared to 23 percent for the linear programming relaxation and 22 percent for the Lagrangian relaxation), and for 12 of the 15 problems the gap was less than 10 percent (for 7 of the 15 problems the gap was less than or equal to 5 percent). VI2 and VI3 linear programming relaxations. The lower bound using VI2 will be at least as good as the lower bound using VIl. The reason why this is so is that VI2 includes all the constraints included in VIl. For all 15 problems there was an improvement in the lower bound when the additional constraints are included. The average gap percent is reduced from 7.0 percent, using VIl, to 4.9 percent, using VI2. For problems that VIl had a gap percent greater than 10 percent, VI2 also had a gap percent greater than 10 percent. This is some what disappointing because these are PAGE 146 the problems where an improvement in the lower bound is needed. VI2 also may need as many as K*I*(T-1) more constraints than VIl does, which causes the computational time for VI2 to be significantly longer than for VIl. 135 Since VI3 includes all the constraints included in VI2, the lower bound using VI3 will be at least as good as the lower bound using VI2. For all 15 problems, there was an improvement in the lower bound when the additional constraints that are in VI3 are included. The improvement for most problems, however, was small; and the average gap percent only improved from 4.9 percent for VI2 to 4.1 percent for VI3. VI3 also may need as many as K*I*(T-1) more constraints than VI2 does, so the computational time for VI3 will be significantly longer than the computational time for VI2 or VIl. Summary. The results indicate that there can be a significant improvement in the lower bound if the additional valid inequality constraints are included in the linear programming relaxation of model II. The results also indicate that there is only a small potential improvement in the lower bound by using the Lagrangian relaxation procedure described. Since there appears to be a diminishing PAGE 147 136 improvement in the lower bound as the constraints in VI2 and VI3 are added to the VIl linear programming relaxation, and the VIl relaxation performed significantly better than the linear programming and Lagrangian relaxations (in terms of lower bounds) it is proposed that VIl be used to evaluate heuristics that solve model II. This is why columns have been added to tables 17 and 18 that compare the objective function values obtained by the heuristics to the lower bound obtained by VIl. 6.1.3.2 Heuristics Heuristic 1. Heuristic 1 solved 5 of the 15 problems optimally, there were, however, 5 problems in which the objective function value obtained by using heuristic 1 was at least 20 percent greater than the optimal solution. The average gap percent vs the optimal objective function value was 13.0 percent, and the average gap percent vs the VIl lower bound was 22.1 percent For two of the 15 problems the gap vs VIl lower bound was over 50 percent and there was a third problem in which the gap was over 40 percent. These results indicate that heuristic 1 may be unreliable Based on the procedure used by heuristic 1 these results are not unexpected PAGE 148 137 Heuristic 2. Heuristic 2 performed better than heuristic 1, in terms of gap percent. Heuristic 2 solved 10 of the 15 problems optimally, and the average gap percent vs optimal objective function value was 8.0 percent for heuristic 2. For two of the 15 problems, however, the gap vs opt. percent was greater than 30 percent, and the gap vs VIl percent was greater than 50 percent. These results indicate that heuristic 2 can also be unreliable, but performs well most of the time (the gap vs opt percent was less than 3 percent for 11 of the 15 problems). Heuristics 3A and 3B. Heuristic 3A solved 12 of the 15 problems optimally, and heuristic 3B solved 11 of the 15 problems optimally. The average gap vs opt percent was 1.1 for both heuristics, and the gap vs opt. percent was less than 3 percent for 14 of the 15 problems using either heuristic For both heuristics, the average gap vs VIl percent was 9.1 percent, and for only 1 of the 15 problems the gap vs VIl percent was over 20 percent. These results indicate that heuristics 3A and 3B are much more reliable than heuristics 1 and 2, and that these heuristics can greatly improve the solution when heuristic 2 performs poorly. The results indicate that heuristic 3B performs PAGE 149 138 almost as well as heuristic 3A, therefore, heuristic 3B can be run instead of 3A to insure that computational time does not become excessive, without sacrificing performance in terms of objective function value. Summary. Since the results indicate that heuristics 1 and 2 may be unreliable, it is proposed that heuristic 3B is used (heuristic 3B's performance was very similar to 3A's performance, and the computational time to run heuristic 3A may be excessive on some problems). A second alternative is to run both heuristics 1 and 2, and choose the solution that has the better objective function value. If this procedure is used on the experimental problems, 12 of the 15 problems are solved optimally, the average gap vs opt percent is 2.8, and the average gap vs VIl percent is 14.1. Also using this procedure, only 1 of the 15 problems has a gap vs opt percent greater than 20 percent, and only 1 of the 15 problems has a gap vs VIl percent greater than 30 percent 6.2 Results of Experiment 2 Table 19 shows the results for problem 1 and table 20 shows the results for problem 2. For each problem, the optimal objective function value is shown in the first row; PAGE 150 139 the lower bounds obtained using the linear programming relaxation, VIl linear programming relaxation, and VI3 linear programming relaxation are shown in rows 2 thru 4; and the objective function value obtained using each heuristic are shown in rows 5 thru 8. The column titled "Gap %" in each table is calculated as follows for the lower bounding procedures: 100-(Lower bound/Optimal objective function value*l00) For the heuristic procedures the "Gap % 11 is calculated as follows: Let ObjH=The objective function value of the heuristic. Let Optobj=The optimal objective function value. Gap%= (ObjH/Optobj*l00) Detailed information about the optimal solution for problem 1 can be found in appendix C. The results indicate that the linear programming relaxation provides a weak lower bound, and the VIl and VI3 linear programming relaxations provide tighter lower bounds than the linear programming relaxation. The performance of the VIl linear programming relaxation, on the two large PAGE 151 140 Table 19. Problem 1 Results. Procedure Value Gap% Optimal Solution 554186 0.0 LP relaxation 403686 27.2 VIl relaxation 497449 10.2 VI3 relaxation 530909 4.2 Heuristic 1 814092 46.9 Heuristic 2 562607 1.5 Heuristic 3A 554186 0.0 Heuristic 3B 554186 0.0 Table 20 Problem 2 Results. Procedure Value Gap% Optimal Solution 517829 0.0 LP relaxation 337387 34.9 VIl relaxation 445362 14.0 VI3 relaxation 479635 7.4 Heuristic 1 632128 22.1 Heuristic 2 798346 54.2 Heuristic 3A 564945 9.1 Heuristic 3B 566236 9 3 problems used in experiment 2, was good but not as good as the average performance on the small problems used in PAGE 152 experiment 1. The VI3 linear programming relaxation significantly improved the lower bound on both problems This indicates that it may be worth the additional processing time needed by this tighter relaxation. 141 Heuristics 1 and 2 performed inconsistently on the problems in this experiment. Heuristic 1 performed poorly on the first problem and its performance was mediocre on the second problem Heuristic 2 performed very well on the first problem, but performed poorly on the second problem Heuristics 3A and 3B significantly improved on the solution found by heuristic 2 on the second problem, and found the optimal solution for the first problem. These results indicate that it is worthwhile to run either heuristic 3A or 3B when there is a large gap between the objective value of the solution found using heuristic 2 and the lower bound found. The results also show that the performance of heuristic 3B is similar to that of 3A. PAGE 153 CHAPTER 7 CONCLUSIONS AND FUTURE RESEARCH 7.1 Conclusions Two models were presented in this proposal. These models address the possible effects that variable demand can have on cell configuration. Model I is used to develop a list of potential cells. Model I can be solved optimally for small problems; or a heuristic presented in this proposal can be used to solve model I for large problems. Model II is used to select cells, from a list of potential cells, that will be included in the cell configuration. Model II can be solved for small problems; or a heuristic procedure can be used to solve model II for large problems. Four heuristic procedures were presented in this proposal that can obtain solutions for model II. Three methods that obtain lower bounds on model II's optimal objective function value were also presented. These methods can be used, for large problems, to estimate the gap between the objective function 142 PAGE 154 143 value of a solution generated by a heuristic and the optimal objective function value for model II. The lower bounding methods for model II, and the heuristic procedures for model II were tested on 15 small problems in experiment 1, and on two larger problems in experiment 2. The results of the experiments lead to the following conclusions. 1) The linear programming relaxation of model II provides a weak lower bound on model !I's optimal objective function value. 2) The Lagrangian relaxation presented for model II also provides a weak lower bound for model !I's optimal objective function value. 3) The lower bound on model !I's optimal objective function value can be significantly tightened by adding valid inequality constraints to the linear programming relaxation of model II. 4) Heuristic procedures 3A and 3B provide solutions that have objective function values that are very close to the optimal objective function value of model II. When heuristic 2 fails to generate a solution which has an objective function value that is close to optimal, heuristic procedures 3A and 3B can significantly improve upon the solution. Heuristic 1 provides solutions that are inconsistent in terms of the gap between the objective function value and the optimal objective function value. PAGE 155 144 7.2 Future Research Future research should be conducted to help answer the following questions. 1) How can solutions obtained by heuristics 3A or 3B for difficult problems be improved? 2) What are the practical implications of using the models described in this research to develop cell configurations? 3) How can the models described in this research be extended to include multi-level environments? Heuristics 3A and 3B obtained solutions for problem 10 in experiment 1 and problem 2 in experiment 2 which had an objective function value at least 9% greater than the optimal objective function value. For many problems a gap of 9% or greater may represent a significant monetary value. These problems should be examined in detail to find out why these problems were difficult to solve. Based on this examination a procedure could be developed that could improve the solutions for problems which exhibit characteristics of difficult problems Working with a company to develop a cellular configuration would provide insights that would help to PAGE 156 145 answer question 2 posed above. Data required by the models could be collected for a company that has a manufacturing facility (which is a candidate for a cellular layout), and then the models could be solved so that a cell configuration can be developed. If the facility is organized in a Cellular layout, the cell configuration developed using the models can be compared to the actual cell configuration used in the facility. If there are differences between the configuration developed using the models and the actual configuration then the reasons for these differences can be identified and may provide feedback that can be used to modify the models. There are two benefits that can be obtained by performing this exercise: 1) Possible modifications that can be made to the models could be identified. These modifications include enhancements to or simplification of the models to fit various environments that could be encountered. 2) Problems companies may have when trying to use the models could be identified. Questions such as: "How should the model be used?" and "How should the data be obtained?" could be answered. Many manufacturing facilities produce products which require activities which occur at various levels of a PAGE 157 146 product's structure. For example, a manufacturing facility may contain a fabrication area, a subassembly area, and a final assembly area. Components which are produced in the fabrication area are then used to build subassemblies in the subassembly area, these subassemblies are then used in the final assembly area to build the finished product Each of the three areas; fabrication, subassembly, and final assembly; could use various degrees of cellular layout When determining the cellular configuration for each area it is important to consider the demand that will be generated by the production requirements of higher levels of the product structure, and the effect of the layout on production planning which will drive demand for production at lower levels of the product structure. Therefore, the cell configuration decisions for each area are not independent of each other The models presented in this research could be extended to include the multi-level environment and solution procedures for solving these models could be developed PAGE 158 APPENDIX A FORMULATION OF MODEL I EXAMPLE Min Z=X11r+X12r+X21r+X 22r +3X1u+3X12r+3X21r+3X22r+4B11 +4B12+4B21 +4B22+I11 +I12+I21 +I22 Subject to X122 +X 222~2 2 SY 22 147 PAGE 159 X 111 ~4S0Y 71 X 121 ~ 4S0Y 71 X 112 ~4S0Y 72 X 122 ~4S0Y 72 X11r~4SOY7r Yu+ Y12+ Y1r ~l All variables are ~ o 148 PAGE 160 APPENDIX B FORMULATION OF MODEL II EXAMPLE Min Z=X1n+X123+X213+X223+3X111+3X121+3X211+3X221+4B11 +4B12+4B21 +4B22+I11 +I12+I21 +I22 Subject to 149 PAGE 161 Z 1 +Z 3 ~ 1 Z 1 +Z 2 +2Z 3 ~ 2 Z 2 +Z 3 ~ 1 Z 1 Z 2 Z 3 are 0-1 variables All variables are ~o 150 PAGE 162 APPENDIX C PROBLEM 1 DATA AND SOLUTION C.1 Problem 1 Data Listings of the data used in problem 1 of experiment 2 are provided below. The first listing is of the processing times for each part on each machine type. Each part has two rows; the first row has the processing times for machine types 1 thru 20 and the second row has the processing times for machine types 21 thru 40. If a part does not require a machine type then the processing time is shown as a zero. 40 0 0 0 0 0 0 0 0 0 0 10 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 151 PAGE 163 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 6 0 9 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 3 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 6 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O O O O O O 5 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 210 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 152 PAGE 164 0 0 0 2 0 0 3 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 9 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 5 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 6 0 3 5 0 2 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 5 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 010 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 010 0 0 7 0 2 0 0 0 0 0 0 0 0 0 0 0 5 0 6 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 000000110 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 010 0 0 2 6 2 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 010 0 4 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 9 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 9 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 4 0 0 153 PAGE 165 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 5 0 3 0 6 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 9 0 2 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 2 010 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 1 0 010 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 9 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 5 0 0 3 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 4 0 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 9 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 010 0 0 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 3 0 010 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 9 0 0 0 0 0 0 7 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 010 0 8 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 7 0 4 0 0 0 0 1 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 0 3 0 0 0 0 0 0 0 0 8 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 01010 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 2 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 9 0 0 0 0 0 0 2 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 2 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 O O O O O O O O O O O O 6 1 0 154 PAGE 166 0 0 0 0 0 0 0 0 0 0 0 0 6 7 0 3 0 0 0 0 0 0 0 0 0 5 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 0 0 0 0 4 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 3 0 0 0 9 0 0 0 0 010 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 9 0 6 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 2 0 0 0 0 5 0 4 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 8 0 0 0 0 0 7 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 3 0 0 0 0 0 0 0 8 0 0 0 0 0 3 0 0 0 0 9 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 7 010 0 0 0 0 0 5 0 0 0 0 0 9 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 0 4 0 0 0 0 2 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 4 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 5 0 0 0 0 10 0 0 0 0 0 0 0 0 9 3 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 4 3 0 0 0 0 0 0 0 0 0 0 0 0 6 7 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 010 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 0 0 0 0 0 010 010 0 0 0 0 0 0 2 010 0 0 010 0 0 0 7 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 3 5 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 o o 3 0 9 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 4 0 0 0 0 8 0 0 0 0 6 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 2 7 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 7 0 0 0 0 0 9 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 910 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 6 0 0 0 0 0 0 0 0 0 0 0 2 0 0 810 010 0 0 0 0 0 0 0 0 5 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 1 0 0 6 0 0 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 3 0 0 0 0 0 0 0 0 0 6 0 0 0 010 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 9 0 0 0 0 0 0 0 0 0 0 0 0 810 0 1 0 155 PAGE 167 156 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 9 1 0 0 0 0 3 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 3 9 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 7 0 0 0 0 0 10 0 0 2 0 0 0 7 0 0 0 0 0 0 0 0 0 9 5 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 3 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 10 6 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 0 0 0 0 7 0 0 6 0 0 0 0 0 0 0 0 0 0 7 0 0 9 4 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 10 0 0 0 0 0 6 0 0 7 0 0 0 0 0 0 0 0 0 3 0 0 0 7 2 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 7 0 The following listing shows demand by period for each part. The last two columns show the inventory carrying cost per unit per period and the unit backorder cost for each part 100 6 161 385 243 443 349 118 2 6 427 214 89 451 332 487 1 5 58 5 33 28 39 37 1 4 76 61 57 81 79 46 1 4 140 157 7 184 52 160 1 3 41 271 109 40 325 14 2 8 636 742 919 618 294 490 2 8 345 109 282 50 220 94 1 4 104 44 40 94 97 22 2 10 9 37 157 10 53 34 1 5 162 612 1044 925 106 251 1 5 706 311 104 295 585 799 1 3 130 81 134 131 110 114 2 10 432 202 535 542 172 117 2 6 93 293 418 107 394 195 2 6 475 585 13 865 246 116 2 10 201 128 144 184 155 287 1 5 217 446 425 196 622 893 2 6 181 1057 370 595 187 10 2 8 323 412 85 30 298 252 1 3 PAGE 168 157 280 504 644 205 42 125 1 3 305 479 688 726 30 372 2 10 183 211 183 50 139 34 2 8 467 432 476 95 93 636 2 6 489 926 225 836 402 523 1 4 914 505 719 135 714 914 1 5 238 194 43 75 212 138 2 6 78 101 126 24 239 132 1 4 87 243 229 275 388 378 1 5 147 259 218 328 75 73 1 3 125 87 210 203 54 122 2 8 233 553 236 501 508 468 1 5 687 464 661 292 458 138 1 4 406 907 739 191 687 270 2 6 51 124 76 93 29 127 1 3 144 93 214 213 131 6 2 10 395 384 434 39 140 409 1 3 304 312 78 362 296 149 2 6 170 244 257 157 19 254 1 3 730 646 42 558 33 491 1 3 673 506 464 628 316 714 2 8 870 578 442 170 277 863 2 6 65 754 487 869 150 274 2 8 9 90 7 152 159 183 1 5 196 4 217 123 239 121 1 4 209 644 311 340 3 693 1 4 144 395 355 124 559 924 1 4 727 713 315 110 226 209 1 3 215 15 93 29 139 209 2 6 326 246 448 186 229 464 2 8 273 202 153 132 20 20 1 3 103 476 318 429 135 338 1 5 611 451 599 230 651 458 2 6 320 580 341 90 218 50 1 3 18 516 320 256 520 570 2 6 100 133 106 18 112 31 1 4 215 888 541 560 580 316 2 6 388 587 585 353 580 707 1 5 732 686 691 953 165 573 1 3 288 529 667 747 11 359 2 10 396 214 658 53 907 172 2 10 295 1273 721 167 813 531 2 10 679 697 1037 290 967 329 2 8 PAGE 169 158 193 79 367 99 228 134 1 3 645 189 752 850 897 267 1 3 859 271 643 1019 297 511 2 6 609 364 308 143 82 595 1 3 508 68 559 186 537 343 2 10 646 745 86 702 416 6 1 3 355 298 605 963 514 266 2 6 624 134 669 681 11 581 1 5 327 999 308 543 163 361 2 8 29 2 18 16 18 18 2 8 532 439 835 357 113 25 2 10 300 147 216 102 90 244 2 10 478 158 640 28 630 66 2 8 289 476 195 354 796 490 2 8 297 54 149 162 258 180 1 3 76 81 574 76 344 449 2 10 169 31 165 182 32 721 2 10 618 1107 156 1013 778 228 1 4 853 54 588 36 422 246 2 8 94 183 56 157 239 170 1 3 751 399 669 693 342 746 1 4 114 55 102 27 61 41 1 3 735 211 682 748 235 89 1 4 302 291 332 184 429 161 1 4 67 236 9 299 221 267 1 5 184 445 849 161 723 1038 2 6 423 240 381 8 331 116 1 5 149 630 533 442 713 432 1 4 335 775 4 7 901 878 2 6 170 172 333 371 462 491 2 10 196 40 366 369 159 370 2 10 429 315 379 319 323 436 1 4 483 453 725 201 300 37 2 6 252 191 170 300 248 140 2 8 202 195 199 99 59 47 1 4 7 26 13 9 20 26 1 3 479 665 143 346 323 244 2 6 The following listing shows the composition of each potential cell. There is a row for each potential cell. Each PAGE 170 159 machine type has a column which shows the number of machines of that type required by each cell If a cell does not require a machine type it is shown as a zero. 0000000000101000000000000000000000000000 0000001000101000000000000000000000000000 0000000000000000000000010010100000000000 0001000010000000000100000000000000000000 1010001000000000000000000000000000000000 0000000000000100100000000000000000100000 0000000000100100000000000000000000100000 0001000010000000000100000000010000000000 0000000100000000000000010010100000000000 1010001000000000000000000000000100000000 0000001000201000000000000000000000000000 0000000000000000001000001001010000000000 1010001000000000000000000000000100000~00 0000100100000000000000000000000000001010 0100000001000000000010000000001000000000 0000010000000000000000000100000000000101 0000000000000100100000000000000000200000 0000010000010000000000000100000000000100 0000000000000000000000010011100000000000 0000000000000000001000001001010000000000 0010000000101010000000000000000000000000 0000000000000100010000000000000001010000 0000010000010000000000000100000000000101 1020001000000000000000000000000100000000 0000000000000000001000001002010000000000 0000000100000000000000010011100000000000 0000010000010000000000000100000000000101 1010001000010000000000000000000100000000 0100000001000001000010000000001000000000 0000100100000000000001000000000000001010 0000000000100100100000000000000000200000 0010001000000000000000000010000100000000 0000001000101100000000000000000000000000 0000000000000010010000000000000011010000 0000001000101000000000010010100000000000 PAGE 171 1020002000000000000000000000000100000000 0000100100000000000001100000000000001010 0000100100000000000001100000000000001010 0000000000101100100000000000000000200000 0000100100000000000001100000000000001010 0010001000000000000000010010100100000000 1010002000101000000000000000000100000000 0001000010000000001100001001010000000000 0000100200000000000001100000000000001010 1010001000000000000000010010100100000000 0000010000010000000000000200000000000201 0000200100000000000001100000000000001010 0000010100010001000000000100000000000101 0000100100000000010001100000000000001010 0001000010101000000100010010100000000000 1010001000000000001000001001010100000000 0000200200000000000001100000000000001010 0000020000010000000000000200000000000201 0000100200000000000001200000000000001010 0000001000101100100000000000000000200000 0001000010000000001100001002010000000000 1010002000000000000000010010100100000000 0000100100000000010001100000000000001010 1010002000101000000000010010100100000000 0000010000010000001000001102010000000101 0001100210000000000101100000000000001010 0100010001010001000010000100001000000101 1010101100000000000001100000000100001010 0000200100000000000001110010100000001010 0000000000101100101000001001010000200000 0001000010101000001100011011110000000000 1020012000010000000000000100000200000101 0000110200010000000001100100000000001111 0100100101000001000011100000002000001010 1010001000000000001000011012110100000000 0000101100101100100001100000000000101010 0100010101010001000010000100001000010101 0001100110000000010101100000000000001010 1011001010101000001100011011110100000000 0100110101010001000011100100001000001111 1010112100010000000001100100000100001111 0010001000101100101000011012110100100000 0001210210010000000101100100000000001111 160 PAGE 172 0100100101000001010011100000002000011010 1021002010101000001100011012110200000000 0100110202010001000011110110101000001111 0000010000020000000000000100000000000101 0001000010000000000200000000000000000000 0000100100000000000000010010100000001010 0000010000020000000000000200000000000101 0000000000101200100000000000000000200000 0000010000020000000000000200000000000201 1121422411241211121113411302022112223432 0010000100120110010000110101010012110200 0010100000120110010000110101010012110200 0121111211130111020111210201012112021311 0120110101140111010011110201012112021312 0120111101130110010011220111112112021211 1111210311130111020111210201012112021311 0100410401121001010013520200101000013331 0110411401040002010014510201002111013332 1231532612151012022124522414133312024543 2242534622251212122224532424233312224543 2242534622352212122224532424233312224543 The following listing shows the cell production costs for each part including the intercell production costs. Each part has several rows associated with it. The number of 161 cells the part can be processed in is provided, followed by the cells, and then followed by the costs. 99 1 23 1 2 11 21 33 35 39 42 50 55 59 65 66 71 74 77 80 86 88 95 97 98 99 100 0.00 0.45 0.83 0 83 0.83 1.46 1.46 1.74 2 00 1.74 2.47 2.47 2.90 2.90 3.66 3.66 4.32 1.74 10 17 7.27 11.86 12.97 13 10 16.49 2 23 1 2 11 21 33 35 39 42 50 55 59 65 66 71 74 77 80 86 88 95 97 98 99 100 0.00 0.45 0 83 0.83 0.83 1.46 1.46 1.74 2 00 1.74 2.47 2.47 PAGE 173 2.90 2.90 3 66 3 66 4 32 1.74 10.17 7.27 11.86 12.97 13.10 16.49 3 23 1 2 11 21 33 35 39 42 50 55 59 65 66 71 74 77 80 86 88 95 97 98 99 100 0 00 0.45 0.83 0.83 0.83 1.46 1.46 1.74 2.00 1.74 2.47 2.47 2 90 2 90 3 66 3 66 4 32 1.74 10.17 7.27 11.86 12 97 13.10 16.49 4 23 1 2 11 21 33 35 39 42 50 55 59 65 66 71 74 77 80 86 88 95 97 98 99 100 0 00 0.45 0.83 0 83 0.83 1.46 1.46 1.74 2.00 1.74 2.47 2.47 2 90 2 90 3 66 3 66 4 32 1. 74 10 17 7 27 11.86 12.97 13.10 16.49 5 23 1 2 11 21 33 35 39 42 50 55 59 65 66 71 74 77 80 86 88 95 97 98 99 100 0.00 0.45 0 83 0.83 0 83 1.46 1.46 1.74 2.00 1.74 2.47 2.47 2.90 2.90 3.66 3.66 4.32 1.74 10.17 7 27 11.86 12.97 13.10 16.49 615 6 717313955657177868889909899 100 0.45 0.45 0 83 1.16 1.46 1.74 2.47 2.90 3.66 1.74 10 17 4.32 4.32 12.97 13.10 16.49 7 23 1 2 11 21 33 35 39 42 50 55 59 65 66 71 74 77 80 86 88 95 97 98 99 100 0.00 0.45 0 83 0 83 0.83 1.46 1.46 1.74 2.00 1.74 2.47 2.47 2 90 2.90 3 66 3 66 4.32 1. 74 10 17 7.27 11.86 12 97 13.10 16.49 8 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 0.00 0.46 1.58 1.90 1.90 2.48 3 00 2.48 3.93 3.93 4 75 0.46 11.90 8.49 8.62 13 97 15.33 15.49 18 52 9 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80 81 84 93 97 98 99 100 0.00 0.46 0.46 0 87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3.00 3 00 3 93 3.93 4 75 4 94 1.58 7 82 13.97 15 33 15.49 18 52 10 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0.46 0.46 0 87 0.87 1.24 1.58 1.58 1.90 1.90 2.48 2.48 3 00 3 00 3 93 3 93 4 75 11.90 13 97 15 33 15.49 18.52 11 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 0.00 0.46 1.58 1.90 1.90 2.48 3.00 2.48 3.93 3.93 4 75 0.46 11.90 8.49 8 62 13 97 15.33 15.49 18 52 12 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80818493979899100 0.00 0.46 0.46 0.87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3 00 3 00 3 93 3 93 4.75 4 94 1.58 7 82 13.97 15 33 15.49 18.52 13 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 162 PAGE 174 0.00 0.46 1.58 1.90 1.90 2.48 3 00 2.48 3 93 3 93 4.75 0.46 11.90 8.49 8.62 13.97 15.33 15.49 18.52 14 12 8 43 56 66 74 80 88 91 94 97 98 99 100 0.46 1.58 1.90 3.00 3 93 4. 75 11.90 8.49 8 62 13 97 15 33 15.49 18 52 15 12 6 17 31 39 55 65 71 77 86 88 98 99 100 0 00 0.46 0.87 1.24 1.58 2.48 3.00 3 93 1.58 11.90 15 33 15.49 18.52 16 12 6 17 31 39 55 65 71 77 86 88 98 99 100 0.00 0.46 0 87 1.24 1.58 2.48 3 00 3.93 1.58 11.90 15 33 15.49 18 52 17 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80818493979899100 0.00 0.46 0.46 0.87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3 00 3.00 3 93 3 93 4.75 4.94 1.58 7.82 13.97 15.33 15.49 18 52 18 21 5 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0 00 0.46 0.46 0.87 0 87 1.24 1.58 1.58 1.90 1.90 2.48 2.48 3.00 3.00 3 93 3.93 4.75 11.90 13 97 15 33 15.49 18 52 19 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80 81 84 93 97 98 99 100 0.00 0.46 0.46 0.87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3.00 3.00 3.93 3.93 4 75 4.94 1.58 7 82 13.97 15 33 15.49 18 52 20 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 0.00 0.46 1.58 1.90 1.90 2.48 3 00 2.48 3 93 3 93 4 75 0.46 11.90 8.49 8 62 13 97 15.33 15.49 18 52 21 9 39 55 65 71 77 86 88 98 99 100 1.24 1.58 2.48 3 00 3.93 1.58 11.90 15.33 15.49 18 52 22 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 0 00 0.46 1.58 1.90 1.90 2.48 3.00 2.48 3 93 3 93 4.75 0.46 11.90 8.49 8.62 13 97 15 33 15.49 18 52 23 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 0 00 0.46 1.58 1.90 1.90 2.48 3.00 2.48 3 93 3 93 4.75 0.46 11.90 8.49 8 62 13 97 15.33 15.49 18.52 24 12 6 17 31 39 55 65 71 77 86 88 98 99 100 0 00 0.46 0 87 1.24 1.58 2.48 3 00 3 93 1.58 11.90 15 33 15.49 18.52 25 15 2 11 33 35 42 55 59 71 74 77 80 88 97 98 99 100 0 00 0.46 0.46 1.24 1.58 1.58 2.48 3.00 3.93 3 93 4 75 11.90 163 PAGE 175 13 97 15.33 15.49 18 52 26 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80 81 84 93 97 98 99 100 0 00 0.46 0.46 0.87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3 00 3 00 3.93 3 93 4.75 4.94 1.58 7 82 13.97 15.33 15.49 18.52 27 13 7 31 39 55 65 71 77 86 88 89 90 98 99 100 0 00 0 87 1.24 1.58 2.48 3 00 3.93 1.58 11.90 4 75 4.75 15 33 15.49 18.52 28 10 33 39 55 65 71 77 86 88 98 99 100 0.46 1.24 1.58 2.48 3 00 3.93 1.58 11.90 15 33 15.49 18.52 2921101324283642455157596367707476 80 88 94 97 98 99 100 0.46 0.46 0 87 0 87 1.24 1.58 1.58 1.90 1.90 2.48 2.48 3.00 3 00 3 93 3 93 4 75 11.90 8 62 13 97 15 33 15.49 18.52 30 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80 81 84 93 97 98 99 100 0 00 0.46 0.46 0 87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3.00 3 00 3 93 3.93 4 75 4 94 1.58 7 82 13 97 15.33 15.49 18 52 31 22 3 9 19 26 35 41 45 50 57 59 64 66 70 74 77 80818493979899100 0 00 0.46 0.46 0 87 1.24 1.24 1.58 1.90 1.90 2.48 2.48 3 00 3 00 3 93 3 93 4 75 4.94 1.58 7 82 13.97 15.33 15.49 18 52 32 18 4 8 43 50 56 61 66 73 74 78 80 83 88 91 94 97 98 99 100 0.00 0.46 1.58 1.90 1.90 2.48 3.00 2.48 3 93 3 93 4.75 0.46 11.90 8.49 8.62 13.97 15 33 15.49 18 52 33 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0.00 0 00 0.47 0.47 0 90 1.29 1.29 1.66 1.66 2.32 2 32 2 93 2 93 4 00 4 00 4.94 13.20 15.60 17 17 17.35 20.08 34 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0.00 0.00 0.47 0.47 0.90 1.29 1.29 1.66 1.66 2 32 2 32 2.93 2 93 4 00 4.00 4 94 13.20 15 60 17 17 17 35 20 08 35 16 29 62 69 72 75 79 81 88 91 92 94 95 96 97 98 99 100 0.47 2 32 2 93 2.93 4. 00 3.48 5 17 13 20 9 27 8.49 9.42 9.11 10 28 15 60 17.17 17 35 20.08 36 27 23 27 46 48 53 60 62 67 68 72 75 76 78 81 82 85 87 88 91 92 93 94 95 96 97 98 99 100 0.47 0 .4 7 1.29 1. 29 1.66 2 32 2 32 2 93 2 93 2 93 4 00 4 00 4 00 5 17 0 90 1. 29 1.66 13 20 9 27 8.49 8.49 9.42 9 11 10.28 15 60 17 17 17 35 20.08 164 PAGE 176 37 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0.00 0.00 0.47 0.47 0 90 1.29 1.29 1.66 1.66 2 32 2.32 2 93 2 93 4.00 4 00 4 94 13 20 15.60 17 17 17 35 20 08 38 28 18 23 27 46 48 53 60 62 67 68 72 75 76 78 81 82 85 87 88 91 92 93 94 95 96 97 98 99 100 0 00 0.47 0 .4 7 1. 29 1.29 1.66 2.32 2.32 2.93 2.93 2 93 4 00 4 00 4 00 5.17 0.90 1.29 1.66 13.20 9 27 8.49 8.49 9.42 9 11 10 28 15.60 17 17 17 35 20.08 39 10 22 88 89 90 91 92 93 94 98 99 100 0.00 13 20 4 94 4 94 9.27 8.49 8.49 9.42 17 17 17 35 20.08 40 9 9 26 64 81 84 93 97 98 99 100 0 00 0.47 2 32 5 17 1.29 8.4915 6017.1717.3520 08 41 13 32 41 45 57 59 70 74 77 80 93 97 98 99 100 0 00 0 90 1.29 1.66 2.32 2.93 4.00 4.00 4.94 8.49 15 60 17 17 17 35 20.08 4228162327464853606267687275767881 82858788919293949596979899100 0.00 0 47 0.47 1.29 1.29 1.66 2.32 2.32 2 93 2 93 2.93 4.00 4 00 4 00 5 17 0 90 1.29 1.66 13.20 9 27 8.49 8.49 9.42 9 11 10.28 15 60 17 17 17 35 20 08 43 11 35 50 59 66 74 77 80 93 97 98 99 100 0.90 1.66 2 32 2.93 4.00 4.00 4.94 8.49 15 60 17.17 17.35 20 08 44 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0 00 0 00 0.47 0.47 0 90 1.29 1.29 1.66 1.66 2.32 2.32 2 93 2 93 4.00 4 00 4 94 13 20 15 60 17 17 17.35 20.08 45 11 34 88 89 90 91 92 93 94 97 98 99 100 0.47 13 20 4.94 4 94 9.27 8.49 8.49 9.42 15 60 17.17 17 35 20 08 46 11 19 26 66 70 74 77 80 93 97 98 99 100 0 00 0.47 2 93 2 93 4.00 4 00 4 94 8.49 15 60 17 17 17 35 20.08 4716296269727579818891929495969798 99 100 0.47 2 32 2 93 2.93 4.00 3.48 5.17 13 20 9 27 8.49 9.42 9 11 10 28 15.60 17.17 17 35 20 08 48 5 21 88 97 98 99 1 00 0 00 13 20 15 60 17 17 17.35 20 08 49 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0 00 0 00 0 .4 7 0.47 0 90 1.29 1.29 1.66 1.66 2 32 2.32 2 93 2 93 4. 00 4.00 4 94 13.20 15.60 17.17 17 35 20 08 50 8 28 67 76 88 94 97 98 99 100 0.47 2 93 4 00 13.20 9.42 15 60 17 17 17.35 20.08 165 PAGE 177 512818 23 27 46 48 53 60 62 67 68 72 75 76 78 81 82 85 87 88 91 92 93 94 95 96 97 98 99 100 0.00 0.47 0.47 1.29 1.29 1.66 2.32 2 32 2 93 2.93 2.93 4.00 4.00 4 00 5.17 0.90 1.29 1.66 13 20 9.27 8.49 8.49 9.42 9.11 10.28 15.60 17.17 17.35 20 08 5227232746485360626768727576788182 858788919293949596979899100 0.47 0.47 1.29 1.29 1.66 2.32 2.32 2.93 2 93 2 93 4.00 4 00 4 00 5 17 0.90 1.29 1.66 13.20 9.27 8.49 8.49 9.42 9 11 10.28 15.60 17 17 17 35 20 08 53 18 15 29 62 69 72 75 79 81 88 91 92 93 94 95 96 97 98 99 100 0 00 0.47 2.32 2.93 2 93 4 00 3.48 5.17 13.20 9 27 8.49 8.49 9.42 9 11 10.28 15 60 17.17 17 35 20.08 54 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0.00 0.00 0.47 0.47 0.90 1.29 1.29 1.66 1.66 2.32 2.32 2 93 2.93 4.00 4.00 4.94 13.20 15.60 17.17 17 35 20 08 55 20 10 13 24 28 36 42 45 51 57 59 63 67 70 74 76 80 88 97 98 99 100 0.00 0.00 0.47 0.47 0.90 1.29 1.29 1.66 1.66 2 32 2.32 2.93 2.93 4.00 4.00 4 94 13 20 15.60 17.17 17.35 20.08 56 17 12 20 25 43 51 56 60 65 66 70 74 77 80 88 97 98 99 100 0 00 0.00 0.47 1.29 1.66 1.66 2 32 2.32 2.93 2 93 4 00 4.00 4.94 13.20 15 60 17 17 17 35 20 08 57 17 12 20 25 43 51 56 60 65 66 70 74 77 80 88 97 98 99 100 0.00 0 00 0.47 1.29 1.66 1.66 2 32 2 32 2 93 2 93 4.00 4 00 4 94 13.20 15 60 17 17 17 35 20 08 58 17 12 20 25 43 51 56 60 65 66 70 74 77 80 88 97 98 99 100 0.00 0 00 0.47 1.29 1.66 1.66 2 32 2.32 2.93 2 93 4 00 4.00 4 94 13 20 15 60 17 17 17 35 20 08 59 17 12 20 25 43 51 56 60 65 66 70 74 77 80 88 97 98 99 100 0 00 0.00 0.47 1.29 1.66 1.66 2.32 2 32 2.93 2.93 4.00 4 00 4 94 13 20 15.60 17.17 1 7 .35 20 08 60 9 39 55 65 71 77 86 88 98 99 100 0.90 1.29 2.32 2.93 4.00 1.29 13.20 17.17 17 35 20.08 61 17 12 20 25 43 51 56 60 65 66 70 74 77 80 88 97 98 99 100 0 00 0 00 0.47 1.29 1.66 1.66 2.32 2.32 2 93 2 93 4 00 4.00 166 PAGE 178 4 94 13 20 15 60 17.17 17.35 20 08 62 17 12 20 25 43 51 56 60 65 66 70 74 77 80 88 97 98 99 100 0 00 0 00 0.47 1.29 1.66 1.66 2.32 2.32 2.93 2 93 4.00 4 00 4 94 13 20 15.60 17.17 17 35 20 08 63 34 14 30 37 38 40 44 47 49 52 54 58 61 63 64 68 697173757678798184889192939495 96 97 98 99 100 0.00 0.47 0 90 0 90 0 90 1.29 1.29 1.29 1.66 1.66 1.29 2 32 2.32 2 32 2.93 2 93 2.93 2 32 4 00 4.00 4.00 3.48 5 17 1.29 13.20 9 27 8.49 8.49 9.42 9 11 10.28 15.60 17 17 17.35 20.08 6427232746485360626768727576788182 858788919293949596979899100 0 00 0.00 0.92 0 92 1.32 2 07 2.07 2 75 2 75 2 75 3 94 3.94 3 94 5 25 0.48 0 92 1.32 14.24 9 83 8.96 8 96 10.00 9.66 10 97 16 91 18 66 18 87 21.33 65 14 48 62 72 75 81 88 91 92 94 95 96 97 98 99 100 0 92 2.07 2 75 3.94 5 25 14 24 9.83 8 96 10.00 9 66 10.97 16 91 18.66 18 87 21.33 66 32 30 37 38 40 44 47 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0 00 0.48 0 .4 8 0.48 0.92 0.92 0 92 1.32 1.32 0 92 2 07 2.07 2 07 2 75 2 75 2 75 2.07 3.94 3.94 3.94 3 37 5 25 14 24 9.83 8.96 8.96 10.00 9 66 10 97 16 91 18 66 18.87 21.33 67 11 34 88 89 90 91 92 93 94 97 98 99 100 0 00 14 24 5.00 5 00 9 83 8 96 8 96 10 00 16.91 18.66 18 87 21.33 68 31 37 38 40 44 47 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.48 0.48 0 48 0 92 0 92 0 92 1.32 1.32 0 92 2.07 2 07 2.07 2.75 2.75 2.75 2 07 3.94 3 94 3.94 3 37 5 25 14 24 9 83 8 96 8 96 10 00 9.66 10 97 16 91 18.66 18 87 21.33 69 31 37 38 4 0 44 4 7 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.48 0.48 0.48 0 92 0.92 0.92 1.32 1.32 0.92 2 07 2 07 2 07 2 75 2.75 2.75 2 07 3 94 3 94 3 94 3.37 5 25 14 24 9 83 8 96 8.96 10 00 9 66 10.97 16.91 18 66 18 87 21.33 70 27 23 27 46 48 53 60 62 67 68 72 75 76 78 81 82 85 87 88 91 92 93 94 95 96 97 98 99 100 0.00 0 00 0 92 0 9 2 1.32 2 07 2.07 2.75 2 75 2 75 3 9 4 3 94 3.94 5 .2 5 0 .4 8 0 92 1.32 14 24 9 83 8 96 8 96 10 00 9 66 10.97 167 PAGE 179 16.91 18 66 18.87 21.33 71 4 88 97 98 99 100 14.24 16.91 18.66 18 87 21.33 72146272758188919293949596979899100 2.07 2 75 3.94 5 25 14.24 9 83 8.96 8 96 10 00 9.66 10 97 16 91 18.66 18.87 21.33 73 4 88 97 98 99 100 14.24 16 91 18.66 18 87 21.33 7427232746485360626768727576788182 858788919293949596979899100 0.00 0.00 0.92 0.92 1.32 2.07 2.07 2.75 2.75 2.75 3.94 3 94 3 94 5 25 0.48 0 92 1.32 14 24 9.83 8 96 8 96 10 00 9 66 10.97 16 91 18.66 18.87 21.33 75 11 72 88 91 92 93 94 95 96 97 98 99 100 2 75 14.24 9.83 8 96 8 96 10.00 9.66 10 97 16 9118.66 18 87 21.33 76 27 23 27 46 48 53 60 62 67 68 72 75 76 78 81 82 858788919293949596979899100 0 00 0 00 0.92 0 92 1.32 2.07 2.07 2 75 2 75 2.75 3 94 3 94 3 94 5 25 0.48 0.92 1.32 14.24 9.83 8 96 8 96 10 00 9 66 10.97 16 91 18 66 18 87 21.33 771369757981889192949596979899100 2. 75 3 94 3 37 5 25 14.24 9 83 8 96 10.00 9.66 10 97 16 91 18 66 18.87 21.33 78 13 62 72 75 81 88 91 92 94 95 96 97 98 99 100 2.07 2.75 3 94 5 25 14 24 9 83 8 96 10 00 9 66 10 97 16 91 18 66 18.87 21.33 7916296269727579818891929495969798 99 100 0 00 2 07 2 75 2.75 3 94 3.37 5.25 14 24 9 83 8.96 10.00 9 66 10 97 16 91 18 66 18 87 21.33 80 27 23 27 46 48 53 60 62 67 68 72 75 76 78 81 82 85 87 88 91 92 93 94 95 96 97 98 99 100 0 00 0.00 0.92 0.92 1.32 2.07 2.07 2.75 2.75 2 75 3 94 3.94 3 94 5 25 0.48 0 92 1.32 14 24 9 83 8 96 8.96 10 00 9 66 10 97 16 91 18.66 18.87 21.33 81 3 88 98 99 100 15 07 19 92 20 15 22 37 82 31 37 38 40 44 47 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.00 0.00 0 00 0.48 0.48 0.48 0 93 0.93 0.48 1.75 1.75 1.75 2.49 2 .4 9 2.49 1. 75 3.80 3 80 3 80 3.17 5.22 15 07 10.25 9 30 9 30 10.43 10 06 11.49 18.00 19 92 20 15 22.37 168 PAGE 180 83 7 88 91 92 94 97 98 99 100 15 07 10 25 9 30 10.43 18.00 19 92 20 15 22.37 84 9 61 73 78 88 91 94 97 98 99 100 1.75 1.75 3.80 15.07 10 25 10.43 18.00 19.92 20 15 22 37 85 313738 40 44 47 49 52 54 58 61 63 64 68 69 71 737576787981889192939495969798 99 100 0 00 0.00 0.00 0.48 0.48 0.48 0 93 0.93 0.48 1.75 1.75 1.75 2.49 2.49 2 49 1. 75 3 80 3.80 3 80 3 17 5.22 15.07 10.25 9.30 9 30 10.43 10.06 11.49 18.00 19.92 20 15 22.37 86 13 48 72 75 81 88 91 92 94 95 96 97 98 99 100 0.48 2.49 3.80 5.22 15 07 10 25 9.30 10.43 10.06 11.49 18.00 19 92 20 15 22.37 87 15 68 75 76 78 81 88 91 92 93 94 95 96 97 98 99 100 2.49 3.80 3.80 3.80 5.22 15.07 10.25 9.30 9.30 10.43 10.06 11.49 18.00 19.92 20 15 22 37 88 13 62 72 75 81 88 91 92 94 95 96 97 98 99 100 1.75 2.49 3 80 5.22 15 07 10 25 9.30 10.43 10.06 11.49 18.00 19.92 20.15 22.37 89 31 37 38 40 44 47 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0 00 0.00 0.00 0.48 0.48 0.48 0 93 0.93 0.48 1.75 1.75 1.75 2.49 2.49 2.49 1.75 3 80 3.80 3 80 3.17 5 22 15 07 10 25 9.30 9 30 10.43 10 06 11.49 18.00 19 92 20.15 22.37 90 31 37 38 40 44 47 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.00 0.00 0.00 0.48 0.48 0.48 0.93 0.93 0.48 1.75 1.75 1.75 2.49 2.49 2.49 1. 75 3.80 3.80 3.80 3.17 5 22 15 07 10.25 9 30 9 30 10.43 10 06 11.49 18.00 19.92 20.15 22 37 91 10 88 89 90 91 92 93 94 97 98 99 100 15.07 4 95 4.95 10 25 9.30 9.30 10.43 18 00 19.92 20 15 22.37 92 31 37 38 40 44 47 49 52 54 58 61 63 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.00 0.00 0.00 0.48 0.48 0.48 0 93 0 93 0.48 1.75 1.75 1.75 2.49 2.49 2.49 1. 75 3.80 3 80 3 80 3.17 5 22 15 07 10.25 9 30 9.30 10.43 10.06 11.49 18.00 19 92 20 15 22.37 93 11 72 79 88 91 92 94 95 96 97 98 99 100 2.49 3 17 15 07 10 25 9.30 10.43 10 06 11.49 18.00 19 92 20 15 22.37 94 11 79 88 91 92 93 94 95 96 97 98 99 100 169 PAGE 181 3 17 15 07 10 25 9.30 9.30 10.43 10 06 11.49 18.00 19.92 20 15 22 37 95 15 68 75 76 78 81 88 91 92 93 94 95 96 97 98 99 100 2.49 3.80 3.80 3.80 5 22 15.07 10.25 9.30 9.30 10.43 10.06 11.49 18.00 19.92 20 15 22 37 96 313738 40 44 47 49 52 54 58 6163 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.00 0.00 0.00 0.48 0.48 0.48 0 93 0 93 0.48 1.75 1.75 1.75 2.49 2.49 2.49 1. 75 3.80 3.80 3 80 3 17 5.22 15 07 10 25 9.30 9.30 10.43 10.06 11.49 18.00 19 92 20 15 22 37 97 15 68 75 76 78 81 88 91 92 93 94 95 96 97 98 99 100 2.49 3 80 3.80 3.80 5 22 15.07 10.25 9.30 9.30 10.43 10 06 11.49 18.00 19 92 20 15 22 37 98 3137 38 40 44 47 49 52 54 58 6163 64 68 69 71 73 75 76 78 79 81 88 91 92 93 94 95 96 97 98 99 100 0.00 0.00 0.00 0.48 0.48 0.48 0.93 0.93 0.48 1.75 1.75 1.75 2.49 2.49 2.49 1.75 3 80 3 80 3 80 3 17 5 22 15 07 10 25 9 30 9.30 10.43 10 06 11.49 18.00 19.92 20 15 22.37 99 4 88 97 98 99 100 15.76 18 92 21.00 21.25 23.24 100 14 49 58 73 79 88 91 92 93 94 95 96 97 98 99 100 0.00 0 00 1.37 2 90 15.76 10.55 9 52 9.52 10.75 10.35 11.89 18.92 21.00 21.25 23.24 170 The following listing shows the number of machines available for each type and the processing time available for each machine per period. 2 7675 2 6435 4 8946 2 7145 3 9734 4 8214 4 7615 PAGE 182 6 9559 2 8010 2 9879 3 7209 3 9561 3 7476 2 9730 2 6155 2 8708 2 5715 2 5427 2 7890 2 9297 3 7360 4 9132 5 8146 3 8904 2 7594 4 8448 2 9704 1 1468 2 7131 1 3788 2 9231 2 6339 1 7459 2 9716 2 6115 2 7740 4 9933 4 9907 3 6807 3 9370 C.2 Problem 1 Solution 171 For each cell selected by the solution for problem 1, table 21 shows the machine types included in the cell and PAGE 183 172 the number of parts that will have their entire requirements, over the planning horizon, processed in the cell. Table 21 Problem 1 Selected Cells # of parts completely Cell Machines processed in cell 2 7 11 13 4 8 4 9 20 30 2 12 19 25 28 30 1 13 1 3 7 32 4 25 19 25 28 28 30 0 26 8 24 27 28 29 2 28 1 3 7 12 32 1 34 15 18 33 34 36 2 37 5 8 22 23 37 39 0 40 5 8 22 23 37 39 0 41 3 7 24 27 29 32 1 72 2 6 8 10 12 16 21 26 31 36 38 4 40 73 4 5 8 9 18 20 22 23 37 39 2 75 2 5 6 8 10 12 16 21 22 23 26 5 31 37 38 39 40 85 6 12 12 26 26 38 40 5 86 11 13 14 14 17 35 35 8 PAGE 184 173 For each part, table 22 shows the cells that the part is processed in If a part is processed in more than one cell then the percentage of units processed in each cell is shown in parenthesis. If a part requires intercell processing this is shown with a "I". Table 22 Part/Cell Processing. Part Cells 1 2 (78) I 86(22) 2 2 3 2 4 2 5 2 (75) I 86(25) 6 86 7 2(87), 86(13) 8 8 (27) I 73 ( 73) 9 26 (29), 41(71) 10 13 11 8 ( 72) I 73(28) 12 26 (21) / 41(79) 13 8 (51) I 73(49) 14 8 15 86 16 86 17 26(74), 41(26) PAGE 185 174 Table 22. Part/Cell Processing. 18 13(72), 28(28) 19 26 (80) I 41(20) 20 8 ( 83) I 73(17) 21 86 22 8 23 8 (45) I 73(55) 24 86 25 2 26 26 (58) 1 41(42) 27 86 28 86 29 13 30 26 (93) I 41 ( 7) 31 26 (11) I 41(89) 32 8 ( 60) I 73(40) 33 13 34 13 ( 8) I 28(92) 35 72 36 85 37 13(53), 28(47) 38 85 39 I 40 26 41 41 42 72(47), 75 (27) I 85(26) PAGE 186 175 Table 22. Part/Cell Processing. 43 I 44 13 45 34 46 26 47 72(82), 75(18) 48 I 49 13 (20), 28 (80) 50 28 51 85 52 85 53 72(83), 75(17) 54 13(62), 28 (38) 55 13 (51), 28(49) 56 12 57 12(39), 25(61) 58 12 (81), 25(19) 59 12 (46) / 25(54) 60 86 61 12 (75), 25(25) 62 12(22), 25(78) 63 37 (26) / 40 (26), 75(48) 64 75 65 72 (71), 75(29) 66 37 (25), 40 (21), 73(30), 75(24) 67 34 PAGE 187 176 Table 22. Part/Cell Processing. 68 37 (49) I 40(43), 73(2), 75(6) 69 37 (47), 40(38), 75(15) 70 72 (19), 75(8), 85(73) 71 I 72 72 73 I 74 85 75 72 76 72 ( 8) 75 (51), 85 (41) 77 75 78 72 (84) I 75(16) 79 72(33), 75(67) 80 72 (19), 75 (15), 85(66) 81 I 82 37(57), 40 (25), 73(18) 83 I 84 73 85 37(62), 40(38) 86 72 (75) I 75 (25) 87 75 88 72 (27) I 75(73) 89 37 (26) / 40 (41) 1 73 (11) / 75(22) 90 37(48), 40 (46) 1 7 5 ( 6) 91 I 92 37(52), 40(48) PAGE 188 177 Table 22. Part/Cell Processing. 93 72 94 I 95 75 96 37 (30), 40 (46), 73(24) 97 75 98 37 (19) I 40 (40) I 73(5), 75(36) 99 I 100 73 PAGE 189 LIST OF REFERENCES Al-Qattan, I "Designing Flexible Manufacturing Cells Using a Branch and Bound Method," International Journal of Production Research, vol 28 no. 2, pp. 325-336, 1990. Aneke, N. A.G. and A. S. Carrie, "A Comprehensive Flow line Classification Scheme," International Journal of Production Research, vol. 22, no 2, pp. 281-297, 1984. Aneke, N.A.G., and A S. 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In 1991, Jeff started in the doctoral program in the Department of Decision and Information Sciences at the University of Florida. His special area of study is operations management On completion of his doctorate he will be taking up a career in university research and teaching. 200 PAGE 212 I certify that I have read this study and that i n my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and qua l ity as a dissertation for the degree of Doctor Philosophy c, Chairman and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ;; A o Vakharia, Cochairman A sistant Professor of Decision and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sherman Bai Assistant Professor of Industrial & Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality as a dissertation for the degree of Doc o f Philosophy ck Thompson Assistant Professor of Decision and Information Sciences PAGE 213 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. B~w&~~ Harold Benson Professor of Decision and Information Sciences This dissertation was submitted to the Graduate Faculty of the Decision and Information Sciences Department in the College of Business and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May, 1996 Dean, Graduate School PAGE 214 L D 1780 199 6 5c1_qgUNIVERSITY OF FLORIDA II I II IIIII I Ill I l l ll l ll l l lll I I I II I II I I II II l l llll 1 11 1 1111 1 1 1 11111 1 3 1262 08557 0108 |