Recent advances in mathematical programming for cell formation

Planning, Design, and Analysis of Cellular Manufacturing Systems A.K. Kamrani, H.R. Parsaei and D.H. Liles (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

R e c e n t A d v a n c e s in M a t h e m a t i c a l P r o g r a m m i n g for Cell F o r m a t i o n Chao-Hsien Chu Department of Management, College of Business, Iowa State University of Science and Technology, 300 Carver Hall, Ames, Iowa 50011, USA

In the past decade, cellular manufacturing has received considerable interest from practitioners and academicians. Cell formation, one major problem with cellular manufacturing, involves the process of grouping the parts with similar design features or processing requirements into part families and the corresponding machines into machine cells. Numerous analytical approaches to solving the problem have been introduced, among which mathematical programming models and heuristic procedures constitute the greatest part of the literature. But as yet no comprehensive study has synthesized the literature pertaining to the use of mathematical programming in cell formation. This chapter presents a state-ofthe-art review based on a systematic survey of the literature. Survey results should help answer or clarify many related questions for the cellular manufacturing community. Examples have been provided to help the interested reader use earlier studies to develop mathematical programming models.

1. I N T R O D U C T I O N

During the past decade, there has been a major shift in the design of manufacturing planning and control systems using such innovative concepts as just-intime (JIT) production, optimization production technology (OPT) (recently named the theory of constraints), flexible manufacturing systems (FMS), cellular manufacturing (CM), and group technology (GT). Cellular manufacturing in particular has received considerable interest from both practitioners and academicians because it allows small, batch-type, production to gain an economic advantage similar to that of mass production and still retaining the high degree of flexibility associated with job-shop production. The design of a CM system is quite challenging because so many strategic issues, e.g. the selection of part types suitable for manufacturing on a group of machines, the level of machine flexibility, the layout of cells, the types of material handling equipment, and the types and numbers of tools and fixtures, must be considered during design [74]. Furthermore, any meaningful cell design must be compatible with corporate tactical/operational goals such as high production rate, high product quality, high on-time delivery,

low work-in-process, low queue length at each work station, and high machine utilization. One of the first and most important problems, i.e., cell formation (CF), faced in CM practice involves the decisions surrounding the decomposition of manufacturing systems into cells. Part families and machine cells are identified such that (1) parts with similar design features, functions, materials, or processing requirements are produced in a cell sharing common resources such as machines, tools, and labor; (2) each part can be processed fully within a cell without the need for movement across cells; and (3) capital investment in resources is maintained at a level compatible with corporate strategy. Manufacturing cells can capture the inherent advantages of both mass and job-shop productions, such as reduced setup times, improved process planning, decreased lead time, reduced tool requirements, improved productivity, increased overall operational control, improved product quality, and reduced material handling costs. Common disadvantages such as lower machine and labor utilization rates and higher capital investment due to duplication of machines and tools exist, however [61]. Much effort has been directed at the cell formation problem. As a result, many procedures have been developed, among which mathematical programming models and heuristic procedures are most discussed in the literature. Notwithstanding, the cell formation problem has been proved nonpolynomial (NP); that is, finding an optimal solution becomes increasingly unlikely as problem size grows. Optimal solutions are worth pursuing for at least two reasons [71,72]: (1) they can serve as a benchmark against which to evaluate heuristics and (2) optimal algorithms and heuristics can work together. On one hand, the logic from an optimal algorithm can lead to an efficient heuristic; on the other, a heuristic solution can serve as a starting point from which to reduce computational time in the optimizing search. Although a number of studies [10,42,56] have attempted to synthesize the literature concerning the use of mathematical programming in cell formation, the scale of these studies has been rather small and the literature cited somewhat outdated. And although a number of studies [14,56,61,65,73] have provided state-of-the-art reviews of cell formation issues, problems, and techniques, the scopes of these reviews generally has seemed too broad, that is, has covered in insufficient detail the use of mathematical programming in cell formation. A comprehensive study of this topic is needed to answer many of the questions frequently asked by the CM community: 9 What kinds of mathematical programming models have been used for cell formation? 9 Which models are most popular? 9 What kinds of objective functions concerning cell formation can be modeled through mathematical programming? What have been the popular measures used in previous studies? Do these measures reflect CM practice? 9 What kinds of constraints related to cell formation can be represented by mathematical programming? What are the popular constraints used in prior research? Do they capture manufacturing reality?

9 What kinds of solution procedures or strategies have been applied to solve the mathematical models? Are they efficient or powerful enough to deal with real-world problems? 9 What types of data are needed to model mathematical programming models? Can these data be obtained easily from the shop floor? 9 What unique features are considered in current cell formation studies? Have they addressed the issues and concerns raised by earlier researchers and practitioners? 9 What kinds of computer systems and software have been used in cell formation research? The purpose of this study is twofold: (1) to examine the state of the art of mathematical programming's use in cell formation (Results from this study would help answer many of the aforementioned questions.) and (2) to illustrate how a variety of cell formation problems can be formulated by means of mathematical programming. Five examples with different objectives, constraints, and structures are provided for illustration. These examples not only represent typical cell formation problems but also can be used to demonstrate how the same scenario can be modeled through either objectives or constraints. The chapter is organized as follows. In section 2, approaches to cell formation are summarized. Section 3 discusses the most recent results concerning the use of mathematical programming models in cell formation. The review proceeds according to the questions just outlined. In section 4, examples of typical cell formation scenarios are provided. Conclusions are given in section 5, which is followed by appendices and references.

2. APPROACHES TO CELL FORMATION Extensive work has been done in the area of cell formation, and numerous approaches have been developed [61]:

* Classification and c o d i n g systems. Under this approach, users first examine the design features or manufacturing attributes of parts from blueprints and use a coding system to assign symbols (or codes) to the parts. H u m a n eyes, statistical clustering algorithms [31], or mathematical programming models [29,31,32] are then used to scrutinize the codes for similarities and part families are formed. The process of assigning codes to parts is tedious and time consuming and sometimes subjective inasmuch as it depends on h u m a n experience and judgment. These methods can be used only to identify part families. * Array based clustering methods. This approach differs from the former in t h a t it is based upon a production flow analysis [7] which uses routing sheet or process plans. A common feature of this approach is t h a t it sequentially rearranges columns and rows of the machine/part matrix according to

9

9

9

9

9

an index until diagonal blocks are generated [15]. Methods of this type have received much attention because of their simplicity. Popular methods include rank order clustering [33,34,35], direct clustering [8] and bond energy algorithm [15,24]. Common criticisms of the methods are that (1) identification of exclusive groups in a block diagram sometimes requires subjective judgment; (2) most methods consider only binary routing information and neglect other important cost and operational factors; and (3) in most cases, bottleneck machines must be removed before any machine/part groups can be identified dearly [8,15,35]. Statistical clustering algorithms. Statistical cluster algorithms have been used quite often in the decomposition of manufacturing cells [14]. In particular, use of hierarchical clustering methods such as the single and the complete linkage methods has been studied extensively [12,45]. This approach requires a calculation of similarity coefficients between each pair of parts or machines. Parts or machines with close similarity coefficients then are arranged in the same group. One study also has used a nonhierarchical clustering scheme [9]. Several problems associated with this approach remain to be solved [12,14], for instance, the selection of clustering criteria, the selection of performance measure, and the determination of the number of part families. Graphic theoretical approaches. A number of papers based upon graph theory have been published [5,14,61]. The methods described represent vertices of graphs as machines or parts and weights of arcs as similarity coefficients. The major drawback inherent in this approach is that practical issues such as production volume and alternative plans are not addressed [61]. Mathematical programming and heuristic approaches. Numerous studies of cell formation have been conducted that employ mathematical programming and heuristics to improve clustering effectiveness. These approaches are flexible enough to incorporate most objective functions and constraints in a precise format; they suffer, however, in that they consider the problem only in a static sense for purely stable manufacturing environments [61]. Additionally, none of the methods considers uncertainty or vagueness, both of which normally are presented in the information required by the models. K n o w l e d g e based and pattern recognition methods. Emerging from artificial intelligence and pattern recognition techniques, expert systems offer many new opportunities for manufacturing systems analysis and design. Yet very few papers have applied these techniques to the cell formation problem [17,61]. Developing expert systems which can capture pattern recognition, optimization, and expert cognition processes to form manufacturing cells is a promising area for exploration [61]. F u z z y clustering and modeling approaches. Most early cell formation research assumes that the information used for cell formation, such as production cost, demand, and processing time, is certain and that the objectives

and constraints considered can be formulated precisely. This early research also assumes that each part can belong to only one part family, yet parts may exist whose membership is much less evident. Only a few researchers have addressed the issues of vagueness and uncertainty in the cell formation problem [16,18]. Fuzzy modeling and clustering approaches may provide a solution in such cases. For instance, a fuzzy c-mean clustering method was used in [16] to form part families (or machine cells) such that a part (or machine) could belong to more than one family (or cell), with different degrees of membership. Recently, a fuzzy mathematical programming approach [18] has been proposed to deal with the imprecise nature of objectives and constraints. N e u r a l network approaches. Neural network is an emerging algorithmic approach that has been the subject of intent study by mathematicians, statisticians, physicists, engineers, and computer scientists. The number of studies utilizing the rapid parallel processing capability of neural networks to solve the cell formation problem has been increasing significantly [19]. Networks such as backpropagation, self-organizing map (SOM), competitive learning, adaptive resonance theory (ART), interactive activation and competition learning, and fuzzy ART, have been applied successfully to the decomposition of manufacturing cells [19].

3. STATE-OF-THE-ART REVIEWS The cell formation problem can be formulated differently depending on the mathematical programming model used, the objective functions chosen, and the constraints considered. There also have been major variations on the solution procedures used, the formation logic applied, the special features considered, and the input data involved. In this section, the state of the art, as characterized by the literature review, is summarized. Detailed information regarding individual studies (models) appears in the appendices.

3.1. Types of mathematical programming models A variety of mathematical programming approaches (See Table 1) have been applied to model and to solve the cell formation problem. The complexities of these models are limited (linear programming and 0-1 integer programming), modest (mixed integer programming and 0-1 nonlinear programming) and very great (mixed integer nonlinear programming and 0-1 nonlinear fractional programming). About 39% of prior studies use the relatively simple 0-1 integer prograrnming models; only about 5% use very complicated models. Thus, even complicated manufacturing design problems such as cell formation can be modeled with mathematical models of limited complexity.

Table 1 Summary of mathematical programming models Rank

1

Model Type

2

0-1 integer programming Mixed integer programming 0-1 integer nonlinear programming

4

Frequency Percentage Used Used (%) 23 § 10

38.98 16.95 16.95

Linear programming

5

8.47

5

Assignment model

3

5.08

5

Network Model

3

5.08 3.39

2

10 ~

7

Mixed integer nonlinear programming

2

8

0-1 nonlinear fractional programming

1

1.69

8

Dynamic programming

1

1.69

8

Branch and bound

1

1.69

Total:

59

+ Including two goal programming models. # Including five goal programming models.

3.2. Objective functions chosen The success of a mathematical programming model depends heavily upon how accurately objectives and constraints can be expressed in precise mathematical relations. Because many objective functions and constraints are considered in the cell formation problem, the challenge is not limited to the construction of equations; a more important issue is the selection of appropriate objectives and constraints that capture and reflect CM reality. There are several ways of classifying objective functions in cell formation. For instance, in [56] objectives were divided into four major categories: (1) reducing the number of setups; (2) producing parts completely within the cell; (3) minimizing investment in new equipment; and (4) maintaining acceptable utilization levels. In [14], according to the nature of objectives, performance measures were classified as either cost or noncost based and subsequently classified as either individual or aggregate. In total, 34 objectives were considered in the prior cell formation studies. In Table 2, these objectives are classified roughly as coefficient based, cost based, or operation related. The purpose of cell formation based on coefficient criteria has been either to maximize total similarity or to minimize total dissimilarity of parts or machines. These coefficients can be computed with design features or with processing requirements. Some studies even have gone so far as to consider tooling requirements between machines and parts [25,26,27,53] and similarities between machines and operators [46]. Two problems may be encountered when these

approaches are used: (1) the coefficients can be involved only one at a time; thus, the model can consider only one objective at most and (2) coefficients are routing based primarily and do not take into account other important factors such as costs and operational issues. Several studies have taken the lead in addressing this deficiency -- for instance, by considering demand, processing time, and even processing sequence in computing the coefficient [12,53], but their impacts have not been testified formally. Recent studies have deviated more and more from this course by considering a variety of costs and operational factors during model formulation. Table 2, which reflects this shift, indicating that about 26% of prior studies focus on minimizing total costs of machine investment, followed by minimizing total costs of intercell movement (24%), minimizing total intercell movement (21%), maximizing machine utilization (12%), and minimizing total processing costs (10%). No coefficient based criterion is among the top five focuses. These objectives coincide with those often used in manufacturing practice [56,74]. Also, the majority of studies (54%) still consider only a single objective during cell formation. Although models with multiple objectives can reflect manufacturing practice with comparative accuracy, they also are more difficult to develop and require longer time to solve [23,53,56,72]. One trade-off is to consider multiple related criteria in an aggregated format. About 34% of prior research has used this approach. In-depth analysis of the data from Appendix B indicates that the following groups of objectives have been used most often by prior researchers: (1) total amount of interand intra- cell movement; (2) total costs of inter- and intra- cell movement, coupled with total machine investment; (3) total costs of intercell movement and machine duplication; (4) total setup cost and inventory holding cost; and (5) total costs of machine investment, tooling investment, and processing.

3.3. Manufacturing constraints Another key component involved in constructing mathematical programming models for cell formation is to define precisely the system constraints. To be practical, constraints should capture the actual restriction (limitation) of a system. Forty-five constraints have been considered in the cell formation literature. Each of these constraints can be placed in one of four categories: (1) logical, (2) cell size, (3) physical, and (4) modeling. Logical constraints prevent models from contradicting common sense, judgment, or theoretic logic. For example, each part, machine, operation, or operator can be assigned into only one cell. Cell size constraints normally are considered to restrict the number of parts, machines, or operators allowed in each cell from exceeding an upper bound because of concerns regarding span of control, and space and capacity limitations. It also makes sense to ensure that the number of parts or machines assigned into each cell exceed a minimum. In this way, the systems are prevented from over division which may result in excessive duplication and thus waste of resources. Physical constraints such as space, budget, capacity, and number of machines available for each machine type capture another type of system restriction. Finally, there is a

10 need for modeling constraints, which provide necessary connections among decision variables, parameters, and objective functions. Table 2 S u m m a r y of objective functions Rank

Obiective Function

Code+

Frequency Percenta~ze usect Used (%)~*

Coefficient Based Measures" O1 5 8.62 Max. total similarity between parts 02 3 5.17 Max. total similarity between machines 03 3 5.17 Min. total dissimilarity between parts 04 3 5.17 Max. total compatibility between parts and machines O16 2 3.45 9 Min. total distance between parts O16 1 1.72 10 Max. total similarity between machines and operators O16 1 1.72 10 Min. total dissimilarity between machines 1 1.72 10 Min. total distance between classification codes O16 Cost Based Measures: 06 15 25.86 1 Min. total costs of intercell movement 05 14 24.14 2 Min. total costs of machine investment 08 6 10.34 5 .Min. t o t a l p r o c e s s i n g a n d m a c h i n e u t i l i z a tion costs 07 5 8.62 6 Min. total costs ofintracell movement Oll 5 8.62 6 Min. total machine duplication costs 09 3 5.17 8 Min. total costs of idle machine capacity O10 3 5.17 8 Min. tqtal setup costs due to sequence aepenaence 011 3 5.17 8 Min. total tooling and fixture costs 011 2 3.45 9 Min. total inventory or work-in-process (WIP) costs 011 1 1.72 l0 Min. total subcontracting costs 011 1 1.72 l0 Min. total space utilization costs 011 1 1.72 l0 Min. total penalty for late or early production 011 1 1.72 l0 Min. total costs to expand capacity 011 1 1.72 l0 Min. total labor costs Operation Related Measures: O13 12 20.7 3 M i n . t o t a l a m o u n t ( n u m b e r ) of i n t e r c e l l movements O15 7 12.07 4 Max. total machine utilization O14 4 6.9 7 Min. total amount (number) ofintracell movements O12 4 6.9 7 Min. n u m b e r of exceptional elements O16 3 5.17 8 Min. total setup times O16 2 3.45 9 Min. total machining hours O16 2 3.45 9 Min. total within cell load variation O16 1 1.72 l0 Max. total amount of parts produced O16 1 1.72 l0 Match each operator's skill O16 1 1.72 10 Max. average cell utilization O16 1 1.72 l0 Min. intracell load imbalance O16 1 1.72 l0 Min. intercell load imbalance + Corresponding to Appendix B. * Based upon 58 models. (One model [60] did not provide a detailed objective.) 6 8 8 8

ll Table 3 summarizes constraints actually considered in the models. The ten most used constraints are spread evenly over the first three categories and none appear in the fourth group. This is because most modeling constraints can be expressed directly in the objective function. For instance, the required number of intercell movements can be expressed in the objective function instead of as a constraint set. Examples 2, 3, 4, and 5 (of section 4) provide a contrast. Examples 2 and 3 include a constraint to determine the number of exceptional elements whereas models 4 and 5 have the relations built into the objective function. According to Table 3, some commonly used constraints are unique part assignment (52%), maximum number of machines allowed in each cell (46%), machine capacity (37%); unique machine assignment (26%); and minimum number of machines needed in each cell (24%). Clearly, all these constraints are quite realistic in terms of managing and operating manufacturing cells.

3.4. Solution procedures One major bottleneck that prevents mathematical programming models from wide use in the real world is that once problem size grows, it is inefficient to seek and almost impossible to find an optimal solution. This phenomenon is known as NP-complete or NP-hard. To overcome this deficiency, two strategies often have been used in cell formation studies. One strategy is to develop efficient heuristic procedures and the other is to decompose the model into submodels or to model the problem in multi-phases and solve each through an optimal or heuristic procedure. According to Table 4, about 29% of previous studies used the decomposition or multi-phase approach to find solutions and about 27% used heuristic procedures (including general search, simulated annealing, genetic algorithm, tabu search, and fuzzy-c clustering). At the same time, 44 % of the studies relied on optimal procedures to solve the single-phase model, and 10% used optimal procedures to solve multi-phase models. So, if we consider that almost all heuristic procedures were developed or extended from an optimal procedure, we conclude that optimal procedures seem to play an important role in cell formation research. Furthermore, interest in using search algorithm [10,64], especially simulated annealing [6,17, 66,69], genetic algorithm [28,68], or tabu search [66], to tackle the problem has been increasing. 3.5. Formation logics The cell formation literature can be divided into four categories, according to the formation logic used [74]: (1) grouping part families only; (2) forming part families and then machine cells or vice versa; (3) forming part families and machine cells simultaneously; and (4) grouping machine cells only. According to Table 5, the literature is spread very evenly over the first three formation logics, with somewhat less effort devoted to the last procedure. Several observations can be extracted from the details provided in the appendices: (1) Most coefficient based models can be used only to form part families or machine

12 Table 3 S u m m a r y of constraints Rank

Constraints

Code+

Frequency Percentage Used Used (%)*

Logical Constraints: 1

Unique part assignment- each part can be assigned to only one part family 4 Unique machine assignment - each machine can be assigned to only one machine cell 9 P a r t family formation logic - a part family m u s t b e f o r m e d b e f o r e p a r t s c a n b e ass i g n e d to t h a t f a m i l y 9 Linkage between machines & parts - to ensure that all machines needed by a part be assigned to the same cell 9 U n i q u e o p e r a t i o n s a s s i g n m e n t - e a c h ope r a t i o n c a n b e a s s i g n e d to o n l y o n e machine 12 Linkage between operations, parts, and machines 13 Unique routing selection - only one routing be selected 13 Linkage between routings and machines 14 Machine cell formation l o ~ c - a machine cell m u s t be formed before machines can be assigned to t h a t cell 14 All p a r t types assigned to p a r t families 14 Unique operator assignments - each operator can only be assigned to one machine cell 14 Layout constraints - to ensure t h a t the machines in each cell do not overlap 14 No. of cells must be less than no. of available operators 14 M i n i m u m level of machine similarity for grouping 14 M i n i m u m level of tool similarity for grouping 14 Minimum level of intercell movement for moving Cell Size Constraints: 2 M a x i m u m no. of m a c h i n e s a l l o w e d i n e a c h machine cell 5 Minimum no. of machines to be qualified as a cell 6 Maximum no. of parts allowed in each part family 10 Minimum no. of parts to be qualified as a part family 14 Exact no. of parts required for each part family 14 Maximum no. of operators allowed in each machine cell

C1

28

51.85

C2

14

25.93

C3

7

12.96

C4

7

12.96

C19

7

12.96

C 19 C 19 C 19 C 19

3 2 2 1

5.56 3.7 3.7 1.85

C 19 C19

1 1

1.85 1.85

C19

1

1.85

C 19

1

1.85

C 19

1

1.85

C 19 C 19

1 1

1.85 1.85

C5

25

46.3

C6 C7 C8

13 11 6

24.07 20.37 11.11

C 19 C 19

1 1

1.85 1.85

+ Corresponding to Appendix C. * Based on 54 models. (Five models have constraints embodied in the structure.)

13 Table 3 (Continued) S u m m a r y of c o n s t r a i n t s Rank

Constraints

Physical Constraints: 3 Mach_ine c a p a c i t y c o n s t r a i n t s - to e n s u r e t h a t t h e t o t a l o p e r a t i o n t i m e s a s s i g n e d to a cell won't exceed capacity 7 A c o n s t r a i n t to s p e c i f y t h e n u m b e r of required cells 8 C o n s t r a i n t s to c o n s i d e r t h e no. o f machine types available 9 B u d g e t c o n s t r a i n t s - to e n s u r e t h a t t h e t o t a l - c o s t o f b u y i n g m a c h i n e s , tools, a n d overhead won't exceed 11 12 14 14 14

Production r e q u i r e m e n t s constraints Space constraints - to ensure t h a t the total space of m a c h i n e s assigned to a cell can be accommodated Constraints to restrict the maximum no. of procurable machines Constraints to restrict the maximum no. of cells allowed Constraints to restrict a part from subcontracting

Code +

Frequency Percentage Used Used (%)*

CIO

19

35.19

C9

9

16.67

Cll

8

14.81

C13

7

12.96

C12 C14

4 3

7.41 5.56

C19

1

1.85

C19

1

1.85

C19

1

1.85

C15

4

7.41

C18

3

5.56

C19 C16 C17

3 2 2

5.56 3.7 3.7

C19

2

3.7

C19 C19 C19 C19 C19 C19 C19

2 1 1 1 1 1 1 212

3.7 1.85 1.85 1.85 1.85 1.85 1.85 3.93

Modeling Constraints: 11 12 12 13 13 13 13 14 14 14 14 14 14

Total:

C o n s t r a i n t s to compute the needed no. of machine types C o n s t r a i n t s to compute the total intercell movements Constraints to consider the sequence dependent setup C o n s t r a i n t s to identify the bottleneck p a r t s C o n s t r a i n t s to identify the exceptional elements C o n s t r a i n t s to compute the total skipping operations (intracell m o v e m e n t s ) Constraints to meet sequence requirements Constraints to compute the completion times Constraints to meet due date requirements Modeling constraints Constraints to estimate the amount of capacity change Constraints to restrict the undirected flows Constraims to link the stages of process

+ C o r r e s p o n d i n g to Appendix C. * Based on 54 models. (Five models have constraints e m b e d d e d in the structure.)

14 Table 4 Summary of solution procedures Rank Solution Procedure Single-Phase: 1 Optimal (O) procedure 2 Heuristic (H) procedure 4 Simulated annealing (H) 5 General search algorithm (H) 5 Branch and bound (O) 5 Network algorithm (O) 6 Genetic algorithm (H) 7 Tabu search (H) 7 Assignment algorithm (O) 7 Fuzzy-C clustering (H) Subtotal: Multi-Phase: 2 Optimal then optimal

2 Heuristic then heuristic * 3 Optimal then heuristic 7 Heuristic then optimal Subtotal:

Frequency Used

Percentage Used (%)

21 6 4 3 3 3 2 1 1 1 45

33.33 9.52 6.35 4.76 4.76 4.76 3.17 1.59 1.59 1.59 71.43

6 6 5 1 18

9.52 9.52 7.94 1.59 28.57

* One model [4] uses simulated annealing and then other heuristic.

Table 5 Summary of formation logic used Rank

Formation Logic

Frequency Used

Percentage Used (%)

1

Group part families only

17

29.82

2

Form both part families and machine cells sequentially

16

28.07

2

Form both part families and machine cells simultaneously

16

28.07

3

Group machine cells only

8

14.04

Total:

57

15 cells because these procedures consider either machine similarity or part similarity, not both, in the formation stage. (2) Most models using an operation index in the formulation, i.e., considering processing sequences, can form part families and machine cells simultaneously. (3) Because of model complexity, it is virtually impossible to use optimal procedures to group part families and machine cells sequentially or simultaneously. In fact, most models rely on a heuristic procedure or use the decomposition strategy. (4) Some studies focus attention on grouping machine cells only, these studies often assume that part families already have formed in spite of its unreality.

3.6. Required input data Many different data are needed for cell formation. The minimum requirements are routing or design feature data. Because many new models have been developed to capture manufacturing reality more faithfully, additional cost and operational data also are needed. (See Table 6.) Most of these data are readily available from the shop floor although acquiring certain information such as number of cells, machine cells size, and cell overhead and budget requires additional effort. Both binary routing and cell number are highly demanded information (80%). Thus, (1) most mathematical programming approaches rely on routing information instead of design features to form manufacturing cells and (2) users normally (80%) need to specify required cell number on a prior basis if they wish to develop mathematical models. This requirement may be controversial because in practice it is very difficult for managers to know beforehand exactly how many cells are needed.

3.7. Special features considered Numerous practical issues concerning cell formation have been raised by early researchers [14,23,61,65,74]: (1) Many of the techniques developed to date fail to capture many of the realities of cell formation. Specifically, most consider only binary routing information in forming the cells and totally or partly neglect other important cost and operational information such as production demand, processing time, machine capacity, processing sequence, machine investment cost, materials handling, and cell overhead. (2) Very few studies have considered field's stochastic nature, sequence-dependent setups, machine tool process capability, alternative process plans, or layout. (3) Efficient approaches considering a number of objectives are needed. Table 7 demonstrates that the number of studies including such special features as alternative routings (process plans) or multiple functional machines, processing sequences, tooling requirements, and sequence-dependent setups has increased significantly. A few studies even have attempted to consider layout planning [4], scheduling [1], and labor allocation [46] during cell formation. One study [32] also considers both design features and processing information in cell formation.

16 Table 6 S u m m a r y of required input data Rank

Required Input Data

Frequency Percentage Used Used (%)*

Coefficients:

Similarity coefficients between parts

6

10.17

12

Similarity coefficients between machines

4

6.78

13

Dissimilarity coefficients between parts

3

5.08

13

Compatibility between machines and parts

3

5.08

14

Distance between parts

2

3.39

15

Dissimilarity coefficients between machines

1

1.69

15

Tool similarity

1

1.69

15

Similarity coefficients between machines and operators

1

1.69

15

Similarity coefficients between classification codes

1

1.69

5

Machine Investment (M)

19

32.2

6

Intercell material handling cost (H)

14

23.73

8

Cell overhead (0) - to setup and operate cell

9

15.25

9

Budget (B)

8

13.56

11

Intracell material handling cost (H)

5

8.47

12

Setup cost (S)

4

6.78

13

Tooling and fixture cost

3

5.08

13

Inventory (WIP) holding cost

3

5.08

14

Machine idle cost

2

3.39

15

Inspection cost (I)

1

1.69

15

Subcontracting cost

1

1.69

15

Wage rate

1

1.69

15

Cost for capacity expansion

1

1.69

15

Penalty for early finish or late

1

1.69

10

Costs:

* Based upon 59 models.

17 Table 6 (Continued) S u m m a r y of required input data Rank

Required Input Data

Frequency Percentage Used Used (%)*

Operation Factors: 1

Routing (binary)

47

79.66

1

Number of cells

47

79.66

2

Demand (production volume)

31

52.54

2

Processing time (P)

31

52.54

3

Size of machine cells (M)

30

50.85

4

Machine capacity

26

44.07

7

Routing (sequence)

11

18.64

7

Size of part families (P)

11

18.64

8

Number of machines available for each machine type

9

15.25

10

Alternative routings

6

10.17

11

Tooling requirements

5

8.47

12

Setup time (S)

4

6.78

13

Total available space

3

5.08

14

Batch size

2

3.39

14

Maximum machine utilization

2

3.39

14

Design features

2

3.39

15

Inspection time (I)

1

1.69

15

Maximum no. of operators allowed in each cell (O)

1

1.69

15

Space requirements

1

1.69

15

Distance

1

1.69

15

Due date

1

1.69

15

Arrival time

1

1.69

15

Required no. of parts for each part family

1

1.69

15

Maximum no. of parts each operator can handle

1

1.69

15

Skill matching factor

1

1.69

15

Minimum level of machine similarity for grouping

1

1.69

15

Minimum level of tool similarity for grouping

1

1.69

15

Minimum level of movement for moving

1

1.69

* Based upon 59 models.

18 Table 7 Summary of special features considered in the model Rank

Special Features

Frequency Percentage Used Used (%)*

1

Consider alternative routings

12

20.34

2

Can find an appropriate n u m b e r of cells

11

18.64

3

Use operations index in the formulation

8

13.56

4

Consider processing sequences

6

10.17

4

Consider tooling requirements

6

10.17

5

Consider setup dependent sequences

5

8.47

6

Consider design features

2

3.39

6

Deal with exceptional elements after CF

2

3.39

7

Consider layout planning with CF

1

1.69

7

Consider undirected flows of parts

1

1.69

7

Consider operator assignment in CF

1

7

Integrate with scheduling

1

1.69 1.69

* Based upon 59 models. 3.8. C o m p u t e r systems and software used If undertaken manually, optimal solution of even a small mathematical programming model demands a prohibitive amount of computational effort; all practical applications of mathematical programming therefore, require the use of computer and related software. Cell formation applications are no exception. In spite of the needs, about one-third of the previous studies nevertheless neglect to mention the type of computer and software used. Based upon the data released (Table 8), LINDO is the most popular software used by cell formation researchers. ZOOM is the next most popular, followed by SAS/OR. Surprisingly, MPSX, the once popular optimization software in the operations research field, has lost its shine because of an inefficient built-in 0-1 integer algorithm and high purchasing and maintenance cost (MPSX is available only in the IBM-compatible mainframe platform). Among the programming languages, PASCAL takes the lead, followed closely by FORTRAN and C. In terms of computer systems, personal computers have replaced mainframes as the most popular systems. This trend may be due to the facts that (1) the processing capability of PCs and of corresponding software packages have been enhanced and improved significantly over the past years and (2) the problems demonstrated by most studies are relatively small and uncomplicated.

19 Table 8 Summary of software and computer used Software or Computer System

Rank

Frequency Used

Percentage Used (%)

Software: 1

Not mentioned

20

33.33

2

LINDO

11

18.33

3

8

13.33

4 5

ZOOM SAS/OR PASCAL

6

C

6 5 3

10 8.33 5

6 7

FORTRAN BASIC

8 8

MPSX RELAXT III

3 2 1

5 3.33 1.67

1

1.67

Total: C o m p u t e r Systems: 1 P e r s o n a l c o m p u t e r (PC a n d Mac) 2 Not mentioned 3 Mainframe computer 4 Mini computer 5 Unix workstation 6 Super computer Total:

60

18 18 13 7 2 2 60

30 30 21.67 11.67 3.33 3.33

4. EXAMPLES OF MODEL D E V E L O P M E N T As the foregoing survey and discussion indicate, mathematical programming has played a seminal role in cell formation history. For instance, most newly proposed heuristic procedures such as simulated annealing, genetic algorithm, tabu search, fuzzy modeling, and neural networks are based upon mathematical models. Two common problems may be encountered in the development of mathematical programming models. The first problem relates to the selection of appropriate objective functions and constraints able to capture manufacturing reality. Survey results from this study can be utilized to support and to ease this selection decision. For instance, users can refer to Tables 2 and 3 and choose their own objectives and constraints from somewhere at the top of the list, without losing much generality. The second problem involves the actual formulation of selected objectives and constraints in a precise and simplistic format. An easier way of

20 doing this is to use a building block approach, i.e., to find and then adopt or modify similar formulations from prior studies instead of developing new models from scratch. In this section, five models with different objectives, constraints, and structures are provided for illustration. Model 1 is a traditional p-median formulation based upon similarity coefficients. A unique feature of this model is t h a t it represents the required n u m b e r of cells in a constraint. Model 2 minimizes the total opport u n i t y costs of producing bottleneck parts outside the cells. This model exemplifies how to use constraints to identify bottleneck parts. Model 3 provides an example of aggregating two compatible criteria into a single objective. It also depicts yet another way of identifying exceptional elements (both parts and machines) by means of constraints. Model 4 is a typical example of the multiobjective approach to cell formation. Other features of the model are t h a t it represents exceptional elements through the objective function and considers withincell workload between machines. Model 5 illustrates how processing sequences can be considered in cell formation. The model also shows users how to use constraints to determine the required number of machines for each type and how to solve a model with a nonlinear term in the objective function. Table 9 s u m m a rizes the characteristic of these models.

4.1. N o t a t i o n s u s e d Indices:

i j k l

= = = =

Decision

Bi = D~ = Lij k M~j k uij k v~jk

= = = =

Xik = X~kz = Yjk =

p a r t index; i = 1, ..., N. machine index;j = 1, ..., M. cell index; k = 1, ..., C. operation index of part i; l = 1, ..., JiVariables:

1, if p a r t i is a bottleneck part; 0, otherwise. 1, if part i needs to be produced outside, either due to machine capacity limits or due to B~ = 1; 0 otherwise. 1, if u~jk = 1 and part i will be processed at cell k; 0, otherwise. 1, if uij k = 1 and machine j will to be duplicated at cell k; 0, otherwise. 1, if a~j = 1 and X~k = 1, but Yjk = 0; 0, otherwise. 1, if a~j= 1 and Yjk =1, but Xik = 0; 0, otherwise. 1, if part i belongs to cell k; 0, otherwise. 1, if operation l of part i is performed in cell k; 0, otherwise. 1, if machine j belongs to cell k; 0, otherwise. n u m b e r of machines of type j required in cell k.

Zjk = Parameters:

B~kz = d~ = D = D~k~ = Fj

G

a d u m m y variable used in example 5 to eliminate a nonlinear term. d e m a n d (in batches) per period for part i. n u m b e r of elements in R; i.e., D = IR I 9 a d u m m y variable used in example 5 to eliminate a nonlinear term. = the set of parts processed by machinej. = an arbitrarily large number.

r.#3

~r.D

-~=

•215

•

•

;~

•

~F~ -o

o ~ .-=~

-~~~'~1

•

>

•

=2o1~

~ ~

02o

~-=--

o o

•

o

~

i

•

A.

i

•

;~

•215

r.~

o

!o = ".= =

0

0

0~.-~I

0

0 0 -'~ ,-1=t 0r

O~

..~s ~ ~

o.~==

"~

__. E.a:= = =

.~

~-,

~, =

--%..,

"

o

i

-~ o

•215

•215

;~•215

•

~

o

i

~

o ~.~~ ,,

~ ~-~~ -

i

~::~13~0 ~I

~ i

#

~

i

='5 1,,,,-,,

"-

0

E E

~I=

+ ~'0"

l~,

'-uo

, ,~,,,

~"s~,. ,.

E

i

i

.,= .~

I

ZZ~ i

~

~

21

=~.,= ='-'~"-' E Z',-~ [-, r.r

,,17 ='-

"1=I

O~

i

o

o=

.~

I

22

Hi = cost to transport one batch of any part i between cells (bottleneck cost). = procurement cost per period of one machine of typej. m o =

mk Pi Qi R S~k

= = = = =

wij

=

average cell load for machinej induced by part i; where

Z):= M ~w o mi2 =

maximum number of machines allowed in cell k. the set of machines needed to produce part i. total number of machines in P~. set of pairs (i, j) such that a o = 1. similarity between part i and part k. to = processing time (setup plus run time) of part i on machinej. tik~ = processing time (setup plus run time) required to process one batch of part i through operation l on machine type j. Tj = available productive time (capacity) for each machine of type j per period. Uj = maximum acceptable utilization per machine of typej. workload on machinej induced by part i; where

tij x d i

wij -

Tj

9

4.2. M o d e l 1 [36,37,38] This model, called P-median, is one of the popular formulations used in early cell formation research. Its objective is to maximize total similarity between parts where the similarity between two parts can be defined in several different ways [12,58,59], of which Jaccard's similarity coefficient is used most frequent: Max

N

N

~

~,

(1)

S ik X i k

i=l k = l

Subject to: N

~, Xik

= 1

i = 1, . . . , N

(2)

k--1 N

EXkk = C

(3)

k=l

X~k -< Xkk

i,j

=

1,...,N

(4)

Constraint set (2) ensures that each part exactly belongs to one part family. Constraint (3) specifies the required number of part families. Constraint set (4) ensures that part i belongs to cell k only ff part family k has been formed. The P-median model is simple and easy to understand. The task of updating the formulation is straightforward, but the size of formulation is rather large. If N parts are to be produced, the model requires N 2 binary decision variables and N 2 + N + I constraints. Moreover, users must specify the number of part families by means of informal judgment, trial-and-error, or iteration [37]. Deciding which process to use is not an easy task and depends highly on experience, preference,

23 and judgment. The model, however, can be expanded easily to consider alternative routings or process plans [37,38]. According to [42], constraint (3) of the P-median model can be removed to limit the difficulty of determining the required number of cells. The model then will use embedded logic to find an appropriate number of part families that maximizes total similarity. And constraint set (4) can be aggregated into constraint set (5) to reduce the number of constraints and thus to improve computational efficiency [13,42]: N

Xik -< (N-1)Xkk

k = 1,...,N;i

~ k

(5)

i=l

The reduced model is more compact, consisting of only 2 N constraints. This model has been adapted and used successfully in [46] for grouping part families, machine cells, and operators simultaneously. An efficient heuristic also has been developed in [42] to improve computational efficiency further. 4.3. M o d e l 2 [71] Two major weaknesses are associated with model 1: (1) it can be used only to find part families and (2) it considers only binary routing information in the formation stage while neglecting many other important factors. Model 2 is an improvement, not only taking into consideration intercell movement cost but also forming part families and machine cells simultaneously. The purpose of the model is to minimize total cost of intercell movement, or what the authors call bottleneck costs: n

(6)

Min ]~ Hi Di i-1

Subject to: C

~,Yjk=l

] = 1, . . . , M

(7)

k = 1,..., C

(8)

k=l M

~, Yjk <- mk i=1

Xik <- E

Yjk <- QiXik

k = 1,...,C;i

= 1,...,N

(9)

JePi C

l + Bi <-~, Xik <- 1 + GBi

i - 1, . . . , N

(10)

/ - 1, ..., M

(11)

i = 1,...,N

(12)

k=l

tij di ( 1 - D i ) < Tj ieFj

Bi < D j

24 Constraint set (7) confines each machine to exactly one cell. Constraint set (8) allows only a maximum of m k machine types in cell k. Constraint set (9) ensures that all associated machines for each part are assigned to the same cell. Constraint set (10) identifies possible bottleneck parts. Constraint set (11) ensures that total machine hours assigned to a machine do not exceed its capacity. Constraint set (12) ensures that D i is set to 1 when either machine capacity is exceeded or part i is an exceptional element. If there are N parts that must use M machines and ff the number of desired cells is C, then the formulation requires (M+N)C+2N binary decision variables and 2M+C+(2C+3)N constraints. In other words, the greater the number of machines and part families, the more decision variables and constraints. Though the formulation is compact, the model has an intricate structure and thus is computationally intensive [13]. For example, updating and extending the formulation are very difficult. If the desired number of cells must be changed, most inputs of constraints- 2M+C+2(C+I)N- must be replaced. Furthermore, determinations of the number of desired cells, the maximum number of machines in each cell, and the unit cost of bottlenecking are limited by experience, acknowledgment, and preference. According to [13], if a large m k is selected, all parts might be assigned into one family; on the other hand, if too small an m k is chosen, clustering results would be unsatisfactory and computational efficiency inferior. Still, because the model could group parts into part families and form machine cells simultaneously, much time could be saved.

4.4. Model 3 [17] Taking one step farther, this model considers machine procurement (investment) cost in addition to intercell movement cost. The model's objective is to minimize the number of exceptional elements by considering a trade-off between total cost of intercell movement and machine duplication: C

Mine

C

E

IjMijk + H i

k=l (i,j)eR

(diLijk)

(13)

i = 1,...,N

(14)

] = 1, . . . , M

(15)

k = 1, ..., C

(16)

~., ~., k=l (i,j)eR

Subject to" C

Xik = 1 k=l C

~.,Yjk=l k=l M

Z Yjk < mk i=1 Yjk -Xik

+ U ijk - V i j k

L ~jk + M ~jh = u ~jk

-- 0

V(i,j) eR;k

= 1,...,C

V (i,j) e R ; k =

1,...,C

(~7) (18)

25 Constraint set (14) ensures that each part belongs to only one part family. Constraint set (15) specifies that each machine can be assigned to only one cell. Constraint set (16) prevents each cell from being assigned more than m k machine types. Constraint set (17) identifies exceptional elements. Constraints set (18) ensures that the exceptional element is either an exceptional machine (a machine needing to be duplicated) or an exceptional part (certain operations of this part needing to be transferred and processed at another cell). Constraint sets (17) and (18) can be combined into one constraint set; the decision variable uijk thus can be eliminated to decrease problem size. This model is fairly large in that it contains C(M+N+3D) binary decision variables and (N+M+C+DN) constraints. Though it can be solved using a popular mixed integer programming package such as LINDO, ZOOM, or MPSX, obtaining an optimal solution requires more intensive computation as problem size grows. A simulated annealing heuristic has been proposed in [17] to retain the model's practical value.

4.5. Model 4 [68] Although we have illustrated in the last model that one can consider jointly more than one criterion, in an aggregated format, for cell formation, the method is applicable only when the criteria under consideration are combinable (or not in conflict with each other). If criteria are quite distinct in nature (for instance, cost vs. time), then utility theory or goal programming is a more appropriate approach. In this example, a model with two distinct criteria is introduced. The first objective is to minimize total intercell movement. The second is to minimize total within-cell load variation between machines: N

Min F1 = ~ di [Xik - 1]

(19)

i=l M

Min F2 -

C

N

~ ~

Yjk ~_, (wij

,]=-1 k = l

i=1

--

mij) 2

(20)

Subject To: C

ZYjk=I

] = 1, . . . , M

(21)

k = 1,..., C

(22)

k=l M

Z Yjk > 2 i=1

Constraint set (21) ensures that each machine is assigned to only one cell. Constraint set (22) ensures that each cell contains at least two machines. The model not only can form part families and machine cells simultaneously but also takes processing time, machine capacity and part demand into consideration. Though the formulation is very compact, consisting of (N+M)C binary decision variables and (M+C) constraints, solving a 0-1 integer bi-criteria programming model is

26 much more difficult than solving a single objective 0-1 integer programming model, because as yet there is no commercial optimization software available for such a purpose. The authors have proposed a genetic algorithm heuristic to demonstrate its applicability.

4.6. Model 5 [66] Thus far, none of the above examples has considered processing sequences in the cell formation stage; neither has any directly determined the number of machines needed for each machine type. Model 5 illustrates how a 0-1 nonlinear programming model can be developed to meet such requirements. The objective function is to consider the trade-off between the total costs of intercell movement and total machine investment: Min Z Z (Ij Zjk)

"k-

Hi

]=1 k=l

Z

Z IXik(l+l) - Zikll

i=l l=l k=l

]

(23)

C

Xikt = 1

i=

1 , . . . , N ; l = 1 , . . . , J~

(24)

k=l Ji ~-~gl ~-~l Zikl tikl di

TjUj

< Zjk

] = 1, . . . , M ; l =

1, ..., Ji

(25)

M

k = 1, ..., C

Zjk <- mk

(26)

i=1

Constraint set (24) ensures that each operation of a part is completely carried out in a cell; that is, operation splitting is not allowed. Constraint set (25) computes the number of machines required in each cell. Constraint set (26) restricts the maximum number of machines allowed in each cell. Notice that the nonlinear term of the objective function can be eliminated by the addition of dummy variables Bik~and Dikz and a constraint set (27): Xik(l+l) - X i k l

= Bikl

i = 1 , . . . , N ; k = 1 , . . . , C ; l = 1. . . . ,Ji

- Dikl

(27)

As a result, the objective function can be restated as Min ]~ ]~ (IjZjk) + Hi i=l k=l

~-~ Z (Bikl - Dikl)

9

(28)

i=l l=l k=l

This model is the most complicated yet discussed. In terms of size, it consists of N C L binary and C M integer decision variables and L ( N + M + N C ) + C constraints. N

Where L = ~ Ji.

The authors, however, have developed both simulated anneal-

i=l

ing and a tabu search heuristic to avoid possible challenges.

27 5. SUMMARY AND CONCLUSIONS Studying optimal solutions for cell formation has several advantages. First, unlike many other analytical methods (for instance, array-based methods), optimal formulations are insensitive to the existence of exceptional elements; one therefore does not need to expend effort in dealing with such elements before a final clustering can be made [13]. On the other hand, certain optimal models can be developed to find and to minimize the number of exceptional elements. Another benefit of using optimal algorithms is to have a benchmark against which to compare heuristics. The analytical approaches found in the literature are predominantly heuristic. Without optimal solutions, it would be difficult to judge heuristics [72]. Still another benefit of examining optimal algorithms is to clarify the embedded logic so that efficient heuristics can be developed or an improved optimal algorithm be created. Solution of optimal cell formation problems, however, faces the "curse of the dimension" [72] because as problem size increases computation intensifies. Furthermore, many optimal models require the cell number to be specified beforehand by means of informal judgment, iterative, or trial-and-error procedures. And in practice it is very difficult for managers to make such decisions. This chapter has discussed an extensive literature search and a survey of mathematical programming's use in cell formation. Several prototypical examples considering a variety of objectives, constraints, and structures were provided. Clearly, mathematical programming is one of the most popular analytical tools used by cell formation researchers. Several other observations can be made: (1) The mathematical programming approach is flexible enough to incorporate practical objectives and constraints into a model even though it may become too complicated to solve. (2) Because the objectives and constraints that can be considered in cell formation are so numerous, selecting suitable objectives and constraints to meet individual needs becomes an important issue for model construction. (3) Most data needed for cell formation, such as routing, processing sequence, demand, machine capacity, and processing time information, can be obtained easily from the shop floor. Development of mathematical programming models for cell formation therefore does not face the same degree of difficulty in obtaining data as many other mathematical programming applications do. (4) Emerging technological breakthroughs bring genetic algorithms, simulated annealing, tabu search, expert systems, fuzzy modeling, and neural networks into the CM arena. This breakthroughs limit the potential difficulty of solving largescale optimization models. (5) The research on cell formation has experienced tremendous growth in recent years, and many studies have considered unique features such as alternative routings, processing sequences, sequence-dependent setups, and tooling requirements. Many research issues raised in early reviews remain largely unexplored. Throughout the course of this study, it has been observed that (1) the current literature on cell formation has used quite different terminologies, indices, and notations in modeling problems and (2) no suitable criterion yet can be used to

28

justify the relative performance of optimal procedures. Contributions can be sought to unify the field's terms, indices, and notations and, consequently, to lead the development of standardized building blocks of objective and constraint sets. In this chapter, we have laid the foundation for such an attempt. An extension of the project could simplify the modeling process and thereby help users focus attention on identifying and selecting appropriate objectives and constraints and on developing ever more efficient solution procedures.

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31

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