Sequencing mixed-model assembly lines with genetic algorithms

Pergamon

SEQUENCING

Computers ind. Engng Vol. 30, No. 4, pp. 102%1036, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved S0360-8352(96)000~-2 0360-8352/96 $15.00+ 0.00

MIXED-MODEL ASSEMBLY GENETIC ALGORITHMS

LINES WITH

YOW-YUH LEU) LANCE A. MATHESON 2 and LOREN PAUL REES 2 'Center for High Technology Management, California State University--San Marcos, San Marcos, CA 92096-0001, U.S.A. 2Department of Management Science, The R.B. Pamplin College of Business, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. A~trsct--This research introduces the use of an artificial-intelligence based technique, genetic algorithms (GA), to solve mixed-model assembly-line sequencing problems. This paper shows how practitioners can comfortably implement this approach to solve practical problems. A substantial example is given for which GA produces a solution in just a matter of seconds that improves upon Toyota's Goal Chasing Algorithm. The new method is then investigated on a test bed of 80 problems. Results indicate GA generates an improved sequence over Goal Chasing on 50 of the problems and also shows a performance advantage of 2% across all 80 problems using Toyota's variability of parts consumption criterion. The paper concludes that further investigation to fine tune the GA methodology is warranted. It also points out that the GA approach can readily be used by practitioners to address a variety of managerial goals concurrently, such as inventory and work load equalization. Copyright © 1996 Elsevier Science Ltd

INTRODUCTION

In order to increase sales and thereby revenues, firms are under constant pressure to produce an increased number of variations or models of a basic end product. For example, an automotive company may make a 4-door sedan with and without automatic transmission, air conditioning, etc. This pressure translates directly into increased demands on the manufacturing unit of the firm. For instance, the production function may consequently be asked to operate a mixedmodel assembly line, i.e., a single line capable of making several different models at a time. Often the relative mixture of models produced on the line will vary as customers change their preferences. Mixed-model assembly lines have become popular in recent years as an integral part of Just-in-Time (JIT) production systems. This is in large part because mixed-model lines can assemble related products in small quantities if the manufacturer is able to lower sufficiently setup-time delays in changing from model to model. Producing in small quantities is important because it enables the firm to respond quickly to changes in market conditions while simultaneously reducing inventories of the specific product models [1]. Various approaches have been developed to address the design of mixed-model assembly lines, the most notable of which is Toyota's Goal Chasing Algorithm [2]. But these approaches are either restrictive in the type of real-world problems they address, or the solution procedures they utilize are limited in their search for optimality. As the monetary effects of the sequencing decision can be profound, this paper describes a procedure developed by the authors for solving the mixed-model sequencing problem. In particular, an artificial intelligence procedure, genetic algorithms, is used to improve upon feasible sequences, thereby mitigating production bottlenecks and inventory buildups, and hence reducing costs to the firm. This paper is organized as follows. In the next section the nature of the mixed-model assembly-fine sequencing problem is described. Following that, genetic algorithms are introduced, as well as key terminology and concepts. That is followed by a large-scale realistic example problem and solution. The next section presents results of a comparative study of genetic algorithms versus Toyota's Goal Chasing Algorithm on 80 randomly generated assembly line balancing problems. The paper concludes with a discussion of managerial implications. 1027

1028

Yow-Yuh Leu et al. MIXED-MODEL ASSEMBLY LINE BALANCING PROBLEM

The problem When a single assembly line is asked to produce several different models of a basic end product, as for example, an automotive line making three or four different models of a station wagon, the literature refers to the line as a mixed-model assembly line [2]. Mixed-model assembly lines, naturally, are more complicated to design than their single model counterparts because the sequence of models on the assembly line must also be specified. As will be shown, the critical issue in mixed-model design is to balance the load of models on the assembly line so that throughout the shop bottlenecks do not occur. Then the final assembly line will not experience stoppage and unnecessary inventory will not accumulate. The general mixed-model situation is depicted for a Just-in-Time manufacturer in Fig. 1. The figure shows a final assembly line, moving from left to right, which is currently assembling two model Bs and one model C, with a model A on the way. The workers on the final assembly line take subassemblies, labeled here as X~, X2, and X3, which have themselves been produced from components, and add them to the particular model going by, if appropriate. Since, by definition, different models require different subassemblies and/or different quantities of subassemblies in their assembly, it is difficult to keep all production units working evenly and to simultaneously keep all units' inventory levels low. This situation is exacerbated in a JIT environment, where parts are not allowed in general to accumulate, but rather can flow only when needed (or "demanded") by a parent production unit, i.e., a production unit closer to the final assembly line. The extra JIT difficulties emerge because JIT systems are especially tightly linked to the final assembly sequence (through control mechanisms called kanbans [2]); in effect, an invisible assembly line is set up throughout the whole factory, which is driven by the final assembly line and its sequence. In short, whether a production system can produce goods just-in-time often hinges greatly on the mixed-model sequence.

Suppliers

m mm m m mm mmm mmm

Components

;ubassemblies

Final Assembly Line Fig. 1. A closely-linked JIT production system, where necessary parts are withdrawn by production units as needed.

Sequencing m i x e d - m o d e l a s s e m b l y

1029

Table 1. Demand for the four models of a basic product Model Demand Units in cycle A B C D Total

2000 3000 4OOO 2000 11,000

2 3 4 2 11

To illustrate more specifically the difficulties faced in the mixed-model assembly-line sequencing problem, consider a basic end product available in four varieties or models that we will call A, B, C, and D. Further assume that the demand for these four models is as shown in Table 1. Miltenburg [3] has proved a result that Toyota assumed in its Goal-Chasing Algorithm: that a best solution to the mixed-model sequencing problem consists of a cyclical sequence; furthermore, the number of model units produced during each cycle should consist of a multiple of the ratio of the demand for a model to total demand for all models on the line. In our example, since the products are demanded in a 2, 3, 4, 2 ratio, a cycle may be defined wherein exactly two As, three Bs, four Cs and two Ds are built (see the last column in Table 1). A feasible sequence, not necessarily very good, would be A - A - B - B - B - C - C - C - C - D - D . The mixed-modal sequencing problem is made more difficult by different requirements for each subassembly. Table 2 shows an assumed set of bills of materials for the models of Table 1. Since each model uses a different number of each subassembly, the pattern of subassembly parts demanded during the cycle will also vary, which greatly affects shop conditions. To see this, examine the task facing the workers of Fig. 1 who make subassembly X2; to simplify the explanation, consider momentarily only end-product models A (2 required per cycle) and C (4 required per cycle). Model C requires 17 of the X2 subassemblies, while model A requires none. If we sequence (just) these two models in the order C-C-C-C-A-A, then we will produce 17,4 = 68 units of X2 all at once at the beginning of the sequence; this will be followed by a period of inactivity for the X2 workers--they will get a "break" as no more X2s are needed for the cycle (considering only these two models). Turning the sequence around to A - A - C - C - C - C creates the same problem: bursts of activity followed by periods of inactivity. Recall that with JIT, parts must only be produced when demanded by parents, "just in time" for use. Consequently, it can be seen that it is wise to balance the final assembly line so that the production and consumption of subassemblies and components is as smooth (i.e., with little or no variability) as possible. Other than re-sequencing, there are three basic "solutions" to the problem of surges in demand created by poor sequences such as those illustrated above, and none of the consequences of these solutions is particularly desirable. First, more workers can be assigned to keep up with the surge in demand. Second, more inventory can be built during relatively idle periods to be used during busy times. And third, it can be decided to do nothing at all, i.e., neither procure additional workers nor build additional inventory, in which case the pull system will degrade and perhaps even stop. As mentioned, generating a feasible sequence that consumes part X2 evenly is critical to smooth shop operation. Yet our desire is even more complicated: to spread out labor-force activity evenly and minimize inventory over all component parts--not just over component X2. Therefore, the objective we seek in our mixed-model assembly line system is a sequence of models which minimizes the overall variability in all subassembly and component parts' consumption; this in turn will minimize overall inventory and keep the work force laboring evenly. Table 2. Bills of materials and subassembly requirements for the problem of Table 1 Subassemblies Model

Cycle quantity

A B C D Subassemblies required per cycle

2 3 4 2

X~

X2

X3

10 2 1 0

0 1 17 4

5 4 0 1

30

79

24

1030

Yow-Yuh Leu et al.

Existing solution approaches Toyota was one of the first companies to use mixed-model lines with production smoothing and has done extensive research into ways of designing these assembly lines. They have developed the heuristic "Goal Chasing Algorithm" [2], whose objective is to minimize the variation in consumption of subassemblies so that the JIT system will work. GCA is widely recognized as a sequencing method, and, in fact, has been used as a benchmark in studies proposing alternative approaches [3-5]• With GCA, daily demand is first derived as a simple average from monthly demand, and then a production cycle is set based on the daily demand. Finally, the production cycle is sequenced in a manner that achieves the objective described below. Actual production repeats the production cycle over and over. GCA has as its objective the minimization of the variability in parts consumption (vpc). The vpc for a sequence of models is defined as: D

n

m

vpc = E E E (kaj - bo - flk -,j)Zxk,~,

(1)

k=li=lj=l

where i= J = k= D = n= m= ,t, = b,j =

the index for model number, the index for subassembly number, the index for the sequence position (i.e., 1 for first in the sequence, 2 for second, etc.) total demand in a production cycle; number of models in the mixed-model assembly line; number of parts needed for final products in the assembly line; number of units of model i in a cycle units of part j needed for a unit of model i;

/

~j = the ideal consumption for part j to achieve linearity of usage - - D

'

1, if model i is in position k of the sequence xk,, =

O, otherwise;

ilk-~j = actual consumption for part j over k - 1 positions; flkj = ilk- ~j + xk.~b~j, and fl0j = 0. The objective function attempts to minimize the deviation between the ideal of uniform consumption (k~j) and actual consumption (b~j + fl~_ ~j); see references [3] and [6] for details• In spite of Toyota's great success with its GCA, the approach has several limitations. First, variation in consumption is only considered at the subassembly level. That is to say, Toyota only includes the variation in subassembly parts in its vpc calculations; component parts (see Fig. 1) are ignored. Second, and perhaps more critically, GCA is a simple "greedy" algorithm in the sense that it is myopic. That is, GCA only looks one step ahead in determining the next model to sequence. Since theoretically both of these limitations have significant monetary consequences, researchers have developed alternative mixed-model assembly line balancing solution approaches. Several mathematical programming models have been suggested in the literature as solutions to the mixed-model assembly line balancing problem• Those models usually are nonlinear 0-1 integer programs, such as the one shown in (1), and hence cannot be solved in polynomial time in general unless additional assumptions are made. For example, to generate polynomial-time solutions Steiner and Yeomans [7] assume that all products require approximately the same number of mix of parts. Inman and Bulfin [8] and Kubiak and Sethi [9] assume that the unit production times of all items are the same. Researchers have returned to heuristic approaches [10] to get around the restrictive assumptions of the mathematical programming methods. These include Ding and Cheng [11], Miltenburg [3],

Sequencing mixed-model assembly

1031

Miltenburg and Sinnamon [6], and Leu et al. [12]. These approaches attempt to obviate the greedy behavior of GCA by looking at two stages of sequencing search trees (e.g., [6], or by searching multiple paths [12]). This research generates another heuristic for solving the mixed-model assembly line balancing problem, an artificial-intelligence (AI) based procedure, genetic algorithms. The procedure does not require that only the final assembly line and subassemblies be evaluated, as does Toyota's GCA; it does not require that the bill-of-materials table contain identical or at least similar numbers, or that unit processing times be identical, as do the mathematical programming approaches; and, it does not stipulate that only a few stages of sequences be evaluated at a time, as do current heuristics (including GCA). GENETIC ALGORITHMS

A genetic algorithm (GA) is, in essence, a search strategy. To implement GA, a representation of the parameters in the problem to be searched (which in this case will be straightforward to generate) is developed first. Next, several initial GA solutions (although not necessarily low-cost sequences) are formed to make an initial population of several so-called chromosomes. Then the GA operations selection, recombination, and mutation are employed to improve the search repetitively as measured by a fitness or evaluation function. The process continues until a desirable solution is obtained or allowable computer resources are exceeded. GA has been applied in a variety of fields including biology, computer science, engineering, and business (Booker et al. [13], Goldberg [14]). Leu et al. [15] have demonstrated the use of genetic algorithms for assigning tasks to workstations in assembly lines, but previously have not examined the more complicated mixed-model sequencing case. As mentioned, the genetic algorithm procedure selects and improves upon promising solutions for the problem being studied from a population of initial solutions. Each solution is just a sequence of the parameter values for the problem (chromosomes), and the individual sites on the chromosome where parameter values are stored are called genes. In the case of mixed-model line balancing, each chromosome represents a cycle of models (e.g., 2 As, 3 Bs, 4 Cs and 2 Ds in the example considered above), and each gene on the chromosome represents the particular model (e.g., A, B, C, or D) selected to be produced at that stage in the cycle. An initial population of random, feasible, mixed-model starting solutions can be generated by a straightforward computer program. Feasible initial solutions are easy to create once the number of units in a cycle is known (e.g., as in Table 1). In our example, each initial solution is merely a different permutation of the 11 units in a cycle, consisting of 2 As, 3 Bs, 4 Cs, and 2 Ds. Once the initial population is developed, an evaluation function is used to select the better performing solutions, which themselves will become candidates for improvement using the genetic operations of recombination and mutation. In general, an evaluation function for sequencing assembly lines might compute the cost, lead-time, or efficiency of a solution; any such function defined by a manager can be implemented. The particular fitness function used in this research to evaluate sequencing solutions is variability in parts consumption (vpc), as given by equation (1). Recall that this equation was developed by Toyota for their GCA, and that it has been used as a benchmark in other research. The particular strategy utilized here to select chromosomes for possible recombination and mutation is selection proportional to (vpc) fitness. With this approach, better performing chromosomes have a higher probability of being chosen than poorer ones, which is desirable for this mixed-model application, whereas poorer-performing chromosomes still have a finite chance of selection. This latter feature is important because GCA and other heuristic approaches to the mixed-model assembly line sequencing problem are known to generate solutions that tend to get "trapped" in local optima [12]. Allowing the possibility of poorer, and hence potentially quite different, chromosomes to be recombined and mutated provides hope for "jumping out" of local optima. Because of our concern of being trapped in local optima, we desire a recombination operator that maintains the merit of neighborhood search, while simultaneously allowing the exploration of the solution space far from the current, incumbent solution. We choose to implement a one

1032

Y o w - Y u h Leu et al.

cutting-point random crossover operator in our effort to assess whether GA is a viable alternative to the popular GCA and to gain some insights for future study. The particular recombination procedure used in this paper is explained in Fig. 2 (see [16], e.g., for other methods). Note that steps 1 and 2 of Fig. 2(b) are necessary to maintain feasibility of the offspring chromosomes because the GA implementation of this research starts with feasible solutions and maintains them at each step. This simple recombination method does allow for neighborhood search and the exploration of new solution regions. Specifically, if the crossover point is close to the last gene of a chromosome, the recombination operation will represent a local search, and the offspring will resemble its

a. Dividing each parent into a head and a tail at the crossover pohlt Gene Position:

1

2

3

4

5

6

7

8

9

10

11

Parent 1 H1 B

T1

.>.V-.~,~ ~.....~.~..,.:,,~:..,..~.~..........~: ~ "-~:~,. ~.-.~,...........~ ~.~,..............

~...-,..-. !..........................

~B

Parent 2

.

~

.D

...... ..7, C:.

:: . C D

T2

H2

Crossover Point

b. Creating the head of the first child Step 1: Delete the genes in the tail of Parent I (T1) from Parent 2 randomly Gene Position: Parent 1

Parent 2

Step 2: Define the head of the first child to be the genes left in Parent2 Gene Position: Child 1

1

2

~-B ~ v

"

"

3

4

5

6

7

8

9

~'-~l~A'~'~- 1]~[~ C -~ v

"

-

v

-

"

-

~%,~

Head

Tail

c. Child 1 is the head created in (b) and the tail of Parent 1 Gene Position: Child 1 .

.

v

~

w

Fig. 2. O n e - p o i n t recombination for generating

.

.

offspring.

10

11

Sequencing m i x e d - m o d e l a s s e m b l y

1033

Table 3. Bills of materials and subassembly requirements for the example Subassemblies Cycle

Model

quantity

A B C D E Subassemblies required per cycle

5 2 3 3 3

X~

X2

X3

X4

X~

X~

X~

Xs

X9

X~o

0 12 2 0 0

17 13 4 15 0

9 0 0 0 5

0 11 19 19 7

4 0 0 0 8

0 0 12 15 10

0 0 6 9 4

18 l 4 9 0

0 17 9 0 0

0 17 3 6 0

30

168

60

157

44

57

131

61

61

Ill

parents. As the crossover point is moved closer to the first gene in the string, the probability of producing a child which is very different from its parents increases. A simple mutation operator, swap head and tail, is chosen. Specifically, a mutation location is randomly selected. The two substrings are defined as a head and a tail, and then they are swapped. We choose this simple mutation over others because our overriding concern is to effectively produce a dissimilar offspring to ensure that we explore other parts of the feasible solution space. However, there are other mutation methods that might be considered. For example, Leu et al. [15] incorporate a form of inversion mutation in solving the assembly line balancing problem. This mutation method selects two positions within a chromosome at random and then regenerates the string between them. Other mutation methods include reciprocal mutation, selecting two positions at random and then swapping the genes on the positions, and insertion mutation, selecting a gene at random and inserting it in a random position.* EXAMPLE

We now demonstrate the genetic algorithm approach to solving the mixed-model assembly line sequencing problem with a realistic example problem. Table 3 shows the bills of materials for each model assuming cycle quantities of 5-2-3-3-3 for models A, B, C, D, and E, respectively; note this implies that a total of 16 units must be sequenced each cycle. Therefore, a feasible GA sequence for this problem consists of 16 genes on a chromosome, each containing one of the letters A through E, where there must be exactly 5 As, 2 Bs, 3 Cs, 3 Ds, and 3 Es on each chromosome. To enable GA to help us find a good sequence to the mixed model assembly-line problem shown in Table 3, we randomly generate five initial sequences of length 16 each with the proper 5-2-3-3-3 mix of models, and make these our initial population. Using equation (1) above to calculate the vpc for each of the initial sequences, we determine that the best of these (probably not very good) initial sequences has a vpc = 14,499. (The sequence that accomplishes this is C - A - C - D - D - E - A A-E-D-A-A-B-B-E-C.) The GA program then decides whether it will recombine or mutate. Experience on a related problem [15] plus repeated experimentation led to our setting the probability of recombination at 80% and of mutation at 20% for this problem. If recombination is chosen, then two members of the initial population are selected for recombination in proportion to their fitness; it mutation is picked, then just one member of the population is selected similarly. By "proportion to the goodness of their fitness functions" we mean that if one member of the population has a vpc that is twice as low as another member's, then the former sequence will have twice the probability of being selected as the latter. Once recombination or mutation is accomplished, the new sequences are added to the population until its size reaches a preset limit; in [15] a population size of 20 gave good results, so we used that value again. This larger population provides additional choices for recombination and mutation. After the 20-sequence limit is reached, only the best 20 chromosomes, be they children or parents, are maintained at each step. That is, if any new sequence has a better (lower) vpc than any sequence in the population, the new sequence replaces the worst sequence in the population. In this manner the population is held at size 20, and it consists of the best 20 sequences obtained thus far. Each process of (1) selection, (2) recombination or mutation, and then (3) consideration of population replacement is called an iteration. * W e w o u l d like t o e x t e n d o u r m a n y t h a n k s to the a n o n y m o u s reviewer f o r the s u g g e s t i o n .

1034

Yow-Yuh Leu et al.

After 1500 iterations of the GA computer program on a VAX 8800 mainframe, a sequence was developed with a vpc of 3073. This particular solution is impressive because it is better than the solution to this problem (vpc = 3293) obtained from using Toyota's Goal Chasing Algorithm. The particular sequence generated by GA was C - A - D - B - E - A - D - E - A - C - A - B - E - D - A - C , and it was produced in fewer than 10 sec on the VAX mainframe computer. EXPERIMENT

In order to investigate the worth of the GA approach in solving the mixed-model ALB problem, an experiment was conducted utilizing 80 test problems. These 80 problems were solved using both the GA approach and Toyota's GCA [2, Appendix 2]. The 80 problems were generated so as to provide a range of conditions across three factors important in mixed-model problems. The first factor varied (see Table 4) was the number of models produced on the assembly line; both a 5-model and a 10-model case were considered. The second factor considered was the number of subassemblies (i.e., the number of X~s) needed for each model, given that the particular subassembly was needed at all in the given end model. This factor was considered at two levels: (1) a random number between 1 and 10, chosen from a uniform distribution; and (2) a random number between 1 and 20, again chosen from a uniform distribution. The third factor chosen for variation was the degree of commonality (i.e., the percentage of non-zero entries in bill-of-materials tables such as Tables 2 and 3). This third factor was also examined at two levels: (1) 0-20% commonality, chosen randomly from a uniform distribution; and (2) 0-40% commonality, again chosen randomly from a uniform distribution. The GA approach described in the example above was utilized on all 80 test problems with a fitness function of variability of parts consumption (vpc). Further details of our vpc implementation (described in a different context) are given in [12]. In our experiment an initial population of size five was used, and an 80/20 recombination/mutation mix was adopted for all problems, as was used on the example problem in the previous section. In these problems one of the five initial solutions was the GCA solution; the purpose of the experiment was to see whether GA could improve on the Toyota result. Ten replications were run for each cell/experimental condition. Results of the experiment (Table 4) indicate that the GA method was able to generate an improvement over the GCA approach and therefore that further study refining the new method over a wider test bed of problems is warranted. In general, 50 of the 80 problems (62.5%) yielded a superior solution with GA than with GCA, where performance improvement is defined as vpcQc^ - vpc~^ × 100%

(2)

VpCacA

The average percentage improvement for the improved problems was 3.08%, whereas the overall average improvement from GA over GCA on all 80 problems was 1.94%. These results are significant in that even small percentage improvements might indicate large monetary savings, and the improvements are being made beyond Toyota's widely noted approach. It is also interesting to examine how many of the 10 cases studied in each cell were improved. These results are shown in the right side of Table 4, with at least half of the cases improved in every cell. This lends credence Table 4. Characteristics of the 80 test problems and performance of GA versus GCA Percent performance improvement (vpC~v~AVl~^ x 100%)

Number of cases improved

Number of models

//10 cases/experimental~ ~ condition )

5 Degree of commonality 0-20 0--40

Number of components 1-10 1-20 1-10 1-20

Avg 2.95 2.00 3.48 2.40

10 Max 11.51 9.05 10.59 7.39

Avg 0.55 1.46 0.79 0.90

Number of models Max 2.94 6.51 4.92 3.88

5 6 5 6 5

10 5 8 6 9

Sequencing mixed-model assembly

1035

to the notion that GA can yield improvements over GCA regardless of the type of mixed-model problem. Further study of Table 4 initially appears to suggest that GA does a better job when the number of models is low (i.e., 5). Analysis of the raw data indicates, however, that this may not be the case. The performance improvement percentages in Table 4 for the number of models = 10 column are smaller than the same percentages for the 5 models case because the denominator of equation (2) is large, not because GA is ineffective. In fact, if one examines the number-of-cases improved columns, the opposite conclusion emerges. When mixed-model assembly line balancing problems become larger, i.e., when they have 10 models and many subassemblies (1-20), then GA shows the most improvement over GCA; see the shaded cells in Table 4. From the experiment we conclude that GCA may produce satisfactory results when the problem size is small, but the effectiveness of the GA approach is not mitigated as the problem size increases. It is believed that the above experiment, as mentioned, provides justification for additional study of the GA method. Factors to be considered in such research, we believe, should include increased initial population sizes, larger numbers of total iterations, and varied percentages of recombinations and mutations (say, 70/30 and 90/10). Since in a real-world setting even small percentage improvements can yield significant monetary savings, experimentation with some of these GA parameters should be conducted. Given the ease of coding GA procedures and the relatively small computer times needed to obtain solutions, this should not be problematic for line balancers.

MANAGERIAL IMPLICATIONS

There are significant implications for practitioners of using GA to sequence mixed-model assembly lines. Properly sequenced mixed-model lines can contribute to greatly reduced inventories, smoother production flow, and more balanced work force efforts. The computer programs needed to implement GA can be quickly written in any computer language. With the current power of desktop computers many sequences can be evaluated in a matter of seconds. Furthermore, since GA chooses sequences based on an evaluation/fitness function, any function can be used in the computer program. This provides great flexibility to managers desirous of capturing their particular cost structures in their mixed-model sequences. It is also possible, and most likely highly advantageous, to combine several different objectives into a composite evaluation function or to consider several objectives as ranked goals. GA can therefore be utilized to create a sequence which satisfies several objectives at once. The GA approach to mixed-model assembly-line sequencing permits managers to focus on that particular combination of inventory, work force balance, materials handling, etc., present in his or her firm.

SUMMARY

This research introduces the use of genetic algorithms to solve mixed-model assembly-line sequencing problems. Genetic algorithms are an easily understood and easily implemented artificial intelligence technique. Practitioners can use this AI-based approach to solve practical problems and generate solutions which are also easily understood by management. In the comparative study of 80 problems we considered, we saw that GA often generated sequences in a matter of just seconds superior to those obtained from Toyota's GCA. Improved sequences can have a very large monetary impact. Moreover, it was seen that the GA approach discussed here can be used with different objective functions to address even disparate managerial goals concurrently. REFERENCES 1. R. B. Chase and N. J. Aquilano. Production and Operations Management: A Life Cycle Approach, 5th Edition. Irwin, Homewood, IL (1989). 2. Y. Monden. Toyota Production System. Industrial Engineering and Management Press, Norcross, GA (1993). 3. G. J. Miltenburg. A theoretical basis for scheduling mixed-model production lines. Mgmt Sci. 35, 192-207 (1989). 4. R. T. Sumichrast and R. S. Russell. Evaluating mixed-model assembly line sequencing heuristics for Just-in-Time production systems. J. Ops Mgmt 3, 728-735 (1990).

1036

Yow-Yuh Leu et al.

5. R. T. Sumichrast, R. S. Russell and B. W. Taylor. Comparative analysis of sequencing procedures for mixed-model assembly lines in a Just-in-Time production system. Int. J. Prod. Res. 30, 199-214 (1992). 6. G. J. Miitenburg and G. Sinnamon. Scheduling mixed-model multi-level Just-in-Time production systems, Int. J. Prod. Res. 27, 1487-1509 (1989). 7. G. Steiner and S. Yeomans. Level schedules for mixed-model Just-in-Time processes. Mgmt Sci. 39, 728-735 (1993). 8. R. R. Inman and R. Bulfin. Sequencing mixed-model assembly lines. Mgmt Sci. 37, 901-904 (1991). 9. W. Kubiak and S. Sethi. A note on level schedules for mixed-model assembly lines in Just-in-Time production systems. Mgmt Sci. 37, 121-122 (1991). 10. W. Kubiak. Minimizing variation of production rates in Just-in-Time systems: a survey. Europ. J. Opnl Res. 66, 259-271 (1993). I I. F. Y. Ding and L. Cheng. A simple sequencing algorithm for mixed-model assembly lines in Just-in-Time production systems. J. Ops Mgmt II, 45-50 (1993). 12. Y.-Y. Leu, P. Y. Huang and R. S. Russell. Using beam search techniques for sequencing mixed-model assembly line. .4nn. Ops Res. (forthcoming). 13. L. B. Booker, D. E. Goidberg and J. H. Holland. Classifier systems and genetic algorithms. Artif. Intell. 40, 235-282 (1989). 14. D. E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA (1989). 15. Y.-Y. Leu, L. A. Matheson and L. P. Rees. Assembly line balancing using genetic algorithms with heuristic-generated initial populations and multiple evaluation criteria. Decision Sci. 25, 581-606 (1994). 16. L. Davis. Applying adaptive algorithms to epistatic domains. Proc. of the Int. Joint Conf. on Artificial Intelligence, Vol. 9, pp. 162-164, Los Angeles, CA (1985).