A Factorial Representation of Permutations and Its Application to Flow-Shop Scheduling

Transcription

1 Systems and Computers in Japan, Vol. 38, No. 1, 2007 Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J85-D-I, No. 5, May 2002, pp A Factorial Representation of Permutations and Its Application to Flow-Shop Scheduling Kazunori Watase Nagasaki Institute of Applied Science, Nagasaki, Japan SUMMARY The first objective of the current research was to incorporate the NEH search method, which is the classical heuristic algorithm for the flow-shop scheduling problem, into a genetic algorithm to improve search performance. To achieve this objective, the author used factorial numbers to represent permutations. Since chromosome representations according to factorial numbers have a one-to-one correspondence with permutations, there is no redundancy. Since no lethal gene is produced by crossover, uniform crossover can also be applied, not just one-point or two-point crossover. In addition, it was apparent that the NEH concept could be naturally introduced into the genetic algorithm search process by arranging n jobs in ascending order of total work times as the basic permutation that is used when associating permutations and factorial numbers. Factorial numbers can also be used to represent certain types of constraints. The second objective of the current research was to verify the effectiveness of the factorial number representation in order-constrained permutation searches. To accomplish this, the author performed numerical experiments and obtained superior results than were obtained by conventional methods Wiley Periodicals, Inc. Syst Comp Jpn, 38(1): 73 86, 2007; Published online in Wiley InterScience ( DOI /scj Key words: factorial numbers; NEH; genetic algorithm. 1. Introduction Solutions of the traveling salesman problem or flowshop scheduling problem are often represented as permutations. Recently, attempts have been made to search for solutions of these kinds of problems by using genetic algorithms and other methods that are collectively referred to as metaheuristics. When attempts were made to apply genetic algorithms to the traveling salesman problem, it was necessary to suppress lethal genes that may result from crossover. To solve this problem, Grefenstette and colleagues [4] proposed a permutation representation that used an ordered representation. Also, even when permutations are directly used as chromosomes, lethal genes can be suppressed by devising the proper crossover methods. For example, Goldberg and Lingle [3] proposed a crossover method called partially mapped crossover (PMX) and showed that lethal genes could be suppressed. However, although these methods succeeded in suppressing lethal genes, their search performance was not very high. Therefore, Kobayashi and colleagues [18] proposed a crossover method that preserves new traits. In addition, Maekawa and colleagues [8] tried to use macro genes to preserve connection relationships that are contained in the parents. These methods show good search results compared with conventional crossover methods that only took into consideration lethal gene suppression [14]. Since solutions for the flow-shop scheduling problem are also represented as permutations, crossover methods similar to those that were developed for the traveling salesman problem can be applied. Murata and colleagues [9] Wiley Periodicals, Inc.

2 compared several crossover methods and showed that twopoint ordered crossover was the best. However, they also reported at the same time that when the genetic algorithm was used independently, the search performance was lower than that of simulated annealing or a tabu search. Therefore, an attempt was made to improve the solution by incorporating simulated annealing within the genetic algorithm. Applying a method of causing the inheritance of traits that were effective in the traveling salesman problem to the flow-shop scheduling problem can also be considered. However, the fact that it is difficult for a new solution to be generated by the method of preserving traits has also been pointed out [19]. Since the research of Johnson [7], various methods have been proposed for the n job m machine flow-shop scheduling problem. These include several methods that are referred to as constructive heuristics such as Campbell, Dudek, and Smith (CDS) [1]; Dannerbring (DES) [2]; and Nawaz, Enscore, and Ham (NEH) [10]. Turner and Booth [16] performed numerical experiments to compare these three methods (CDS, DES, and NEH) and showed that NEH was the best. As described above, to obtain approximate solutions of the flow-shop scheduling problem, research has been conducted on methods referred to as constructive heuristics such as NEH and methods referred to as metaheuristics such as genetic algorithms. However, I believe that these two approximation methods have not as yet been combined. Therefore, in the current research, my first objective is to incorporate the NEH search method, which is an example of constructive heuristics, into a genetic algorithm to improve search performance. To achieve this objective, I decided to use factorial numbers to represent permutations. Although the use of a factorial representation of permutations in order-constrained permutation searches has been reported [17], there is no redundancy for chromosome representations since a one-to-one correspondence with permutations is also guaranteed. Also, since no lethal gene is produced by crossover, uniform crossover can also be applied, not just one- or two-point crossover. In addition, the NEH concept can be naturally introduced into the genetic algorithm search process by properly devising the permutations that are to be used (I will refer to these as basic permutations) when associating permutations with factorial numbers. For these reasons, I decided to represent permutations as factorial numbers in the current research. Also, I decided to conduct numerical experiments to compare the proposed method with conventional methods. 2. NEH Method In the current research, I consider the flow-shop scheduling problem with the objective of minimizing the makespan when n jobs that are processed at m machines are assigned in the same order. For this flow-shop scheduling problem, the NEH method is known as a method that generates an excellent approximate solution quickly. The outline of the NEH method is extremely simple as shown below. 1) For each job, obtain the sum total of the work times at all machines. 2) Rearrange the jobs in descending order of total work times. 3) Take the two jobs having the largest total work times. Determine the optimum job order when only these two jobs are assigned. This is obtained by comparing the makespans for the two possible orderings. The relative order of the two jobs that was determined here is preserved throughout the following procedure. 4) Take the job with the n-th largest work time and consider inserting it in the appropriate position in the job order that was determined up to then. Since the relative order relationship of the n 1 jobs that had been determined up to then is maintained, there are n possible permutations. Calculate the makespans for these n permutations to determine the insertion position for which the makespan is minimized. 5) For all jobs, determine the insertion position according to step 4). Since the job order for the job with the largest total work time is fixed, NEH can be considered to be an algorithm that performs a depth-first search in order from the job with the largest total work time. 3. Permutation Representation According to Factorial Numbers In the current research, I let the n-digit factorial number C n C n 1 C 2 C 1 represent the number where each digit coefficient C j is a number in the range 0 C j < j (j = n,..., 1). Although the last term is unnecessary since C 1 is always 0, I deal with factorial numbers that consist of n terms for convenience in the explanation. For example, the largest number that can be represented by a 4-digit factorial number is 3! 3 + 2! 2 +1! 1 + 0! 0 = 4! 1. Therefore, 4! cases can be represented, including 0. A method of using factorial numbers to count all permutations without omission was proposed long ago [20]. However, in that method, the factorial numbers were only used as simple counters. Therefore, let us first inves- 74

3 tigate the correspondence between factorial numbers and permutations. To do so, we define the relative positional relationship of elements that are included in the permutation as follows. [Definition] Subsequent position Consider a specific permutation consisting of n different elements (hereafter, this will be called the basic permutation). Let the sequential element positions from the beginning be referred to as position 0, position 1,..., position n 1. When element x p at position p and x q at position q in the basic permutation are taken, if p < q, then element x q is said to be at a subsequent position from element x p. Consider a conversion from a permutation to a factorial number. In the basic permutation consisting of n different elements, let x k denote the element at position k (k = 0,..., n 1). Also, let C n C n 1 C 2 C 1 denote an n-digit factorial number. When an arbitrary permutation consisting of the n elements that are contained in the basic permutation is given, the coefficients C j (j = n,..., 1) of the corresponding factorial number are determined by using the following rule. [Rule 1] (Conversion from a permutation to a factorial number) 1) Take the subpermutation up to element x k in the arbitrary given permutation. 2) Let the factorial number coefficient C n k denote the number of elements that are in subsequent positions from element x k within the subpermutation. For example, let us consider converting the permutation bdac of size 4 (n = 4) to a 4-digit factorial number. Let the alphabetical order permutation abcd, for example, denote the basic permutation. We will try to determine the corresponding coefficients sequentially starting from the first element of the basic permutation. In this case, we will determine the coefficients beginning with the coefficient corresponding to element a at position 0. 1) Take the subpermutation bda up to element a from the permutation bdac. Since the basic permutation is abcd, the number of elements that are in subsequent positions from element a in the subpermutation is 2 or, in other words, C 4 = 2. 2) In the subpermutation b up to element b, the number of elements that are in subsequent positions to element b is 0 or, in other words, C 3 = 0. 3) In the subpermutation bdac up to element c, the number of elements that are in subsequent positions to element c is 1 or, in other words, C 2 = 1. 4) In the subpermutation bd up to element d, the number of elements that are in subsequent positions to element d is 0 or, in other words, C 1 = 0. As a result, the factorial number corresponding to the permutation bdac is encoded as Next, we will determine the rule for converting an n-digit factorial number to a permutation of size n. Assume that the basic permutation x k (k = 0,..., n 1) consisting of n different elements and the n-digit factorial number C n C n 1 C 2 C 1 are given. At this time, the following rule is used to create the permutation of size n from the n-digit factorial number. [Rule 2] (Conversion from a factorial number to a permutation) 1) Consider an array of size n and let the positions from the left be sequentially denoted by position 0, position 1,..., position n 1. 2) Let k = 0. 3) Store element x k in position C n k. 4) Increase the value of k by 1. 5) If k = n, stop. Otherwise, excluding the positions at which elements are already stored in the array of size n, let the positions from the left be sequentially denoted by position 0, position 1,..., position n k 1. 6) Repeat steps 3) to 5). For example, we will generate a permutation of size 4 from the 4-digit factorial number As before, we denote the basic permutation by the alphabetic order permutation abcd. We will determine the positions within the permutation sequentially starting from the first element of the basic permutation. 1) Since C 4 = 2, store element a in position 2. Also, excluding the location where element a was stored, reassign the position numbers. 2) Since C 3 = 0, store element b in position 0 and reassign the position numbers. 3) Since C 2 = 1, store element c in position 1 and reassign the position numbers. 4) Since C 1 = 0, store element d in position 0. As a result, the permutation bdac is generated from factorial number Let us now look at differences between the proposed representation and order representation [4]. When a permutation is generated by using order representation, information indicating which position among the remaining elements is to be used is stored in the chromosome. Therefore, when handling a permutation of size n, the maximum values that genes can take sequentially from the beginning 75

4 are n 1, n 2,..., 2, 1, 0. As a result, the order representation can also be viewed as a representation using factorial numbers. These representations differ in the rules determining the correspondence between the factorial number and permutation. While the order representation method determines the order beginning from the first element of the permutation that is generated, the method used in the current research determines the position at which a job should be inserted sequentially starting from the first element of the basic permutation. In other words, with the correspondence determined by the current research, the chromosome can be considered to represent the position information of each element that is contained in the basic permutation. As a result, the method used in the current research can be referred to as a position representation in contrast to the order representation. Since the correspondence rules differ, different permutations may also be generated from the same factorial number. The permutation that is generated according to the order representation rule from the factorial number 2010, which was used in the above example, is cadb. 4. Sample Application to a Genetic Algorithm In this section, we describe the method of using factorial numbers to represent permutations and apply this method to a search using a genetic algorithm. An arbitrary permutation can be used for the basic permutation that is required to associate the factorial numbers and permutations. Therefore, as the basic permutation, we use the permutation that is obtained when n jobs are arranged in ascending order of total work times. In the preliminary example for 5 jobs and 4 machines shown in Fig. 1, the basic permutation is acbde. Next, we use random numbers to generate the factorial number C n C n 1 C 2 C 1 so that the coefficients C j ( j = n,..., 1) satisfy the conditions 0 C j < j. We repeat this operation to create the four factorial numbers shown in Fig. 2. This is the initial group of parents. Also, we use rule 2 to convert these factorial numbers to permutations and calculate their makespans. The makespans and permutations are Fig. 2. Initial group of parents. also shown together with the factorial numbers in Fig. 2. Note that the chromosomes (factorial numbers) are rearranged in descending order of fitness (ascending order of makespans). Let us focus on the two last jobs d and e of the basic permutation acbde. Since the basic permutation has been rearranged in ascending order of total work times, these jobs are the two jobs having the largest total work times. It is apparent that the factorial number coefficients for these jobs are always 0 for job e, which has the largest work time, and only 0 or 1 for job d, which has the next largest work time. Since the ranges of numbers that can be taken by these coefficients are limited, all possible combinations are expected to be able to be generated even when random numbers are used to create the factorial numbers. The two possible combinations 0, 1 and 0, 0 actually appear in the initial group shown in Fig. 2. Also, even when the permutation up to job b, which has the third largest work time, is considered, it is apparent that 3 of the possible 6 combinations appear. In this way, it is apparent that when an ascending order basic permutation is used and permutations are represented by factorial numbers, then the larger the total work time is, the greater the proportion of relative order relationships can be found. To perform crossover processing, one set of parents must be selected. Although a selection method based on the roulette method is widely used to select the parents for which crossover is to be performed, this method is hard to apply to a minimization problem and scaling is also required. Therefore, in the current research, I decided to select the parents based on rank. In other words, when SIZE denotes the size of the group, the probability p(i) that a high fitness chromosome is selected for the i-th member (i = 0,..., SIZE 1) is given as follows: Fig. 1. An example for 5 jobs and 4 machines. One advantage due to the use of factorial numbers to represent permutations is that a lethal gene is not produced by crossover. For two factorial numbers C n C n 1 C 2 C 1 and C n g g C n 1 C 2 g C 1 g, assume that the new child g C n C n 1 C p C p 1 C 2 g C 1 g is generated by one-point crossover, where the crossover position is p. Since C j < j and C j g < j (j = n,..., 1), the child that is created also 76

5 satisfies the condition for the factorial number coefficients. In other words, a lethal gene is not produced by crossover. It is also clear that a lethal gene is not produced by two-point crossover or uniform crossover. When uniform crossover is used as the crossover method, roughly only half of the genes of a parent are inherited. As a result, it is possible for a variety of children who differ greatly from the parent to be created. Therefore, in the current research, I used uniform crossover to generate diverse children and prevent premature convergence. Mutation also can be implemented by varying coefficient values. For each chromosome, we use a random number to determine the gene locus for performing mutation processing. For example, let us assume that the k-th (k = 0,..., n 1) gene locus counting from the beginning is subject to mutation. From the factorial number coefficient condition, the k-th gene locus value must be less than n k. Therefore, we create a random number from 0 to n k 1 and let it be a new gene. From rule 2 for permutation generation, the position of element x k can be changed by changing the factorial number coefficients. Note that we decided not to perform a mutation operation for the most excellent members of the parent group. This is to preserve elite members. After mutation is performed, the factorial number is converted to a permutation and the makespan is calculated. Then the members are rearranged in ascending order of makespan values, and the required number of chromosomes are retained as the next parents. 5. Genetic Algorithm Numerical Experiments I performed numerical experiments to verify the search performance of the proposed method. In the current research, I set 20, 50, 100, and 200 as the number of jobs n. Also, I set 5, 10, and 20 as the number of machines m. Therefore, there are 12 possible combinations of numbers of jobs and numbers of machines. The following six methods are used for comparison with the proposed method. 1) NEH The proposed method incorporates the concepts of NEH, which is the classical heuristic algorithm. NEH is added to the numerical experiments to verify whether or not the proposed method is effective in improving the NEH solution to some degree. 2) One-point order crossover (abbreviated as one-point ) This is the crossover method used by Reeves [12]. Permutations are used to represent chromosomes, and mutation is a shift operator. Note that although Reeves includes the solution that is obtained by NEH within the initial group, to compare search performance, random numbers are used to generate the entire initial group. 3) Middle part order crossover (abbreviated as middle ) This method showed the best results for genetic algorithm crossover processing in the numerical experiments of Murata and colleagues [9], which dealt with flow-shop scheduling. As with one-point order crossover, permutations are used to represent chromosomes, and mutation is a shift operator. Two crossover positions are set, and both ends are directly inherited from one parent. The middle part is inherited from the other parent in the appearance order of its genes. Note that since this method is in the two-point order crossover category, like the next method, I decided to refer to it as middle part order crossover. 4) Both end part order crossover (abbreviated as both end ) This is the method used by Iima and Sannomiya [5, 6] in searches using order-constrained permutation genetic algorithms. Two cutting points are set, and the genes between the cutting points are directly inherited from one parent, the part to the left of the first cutting point is inherited from the other parent in the appearance order of its genes, and although the part to the right of the second cutting point is also inherited from the other parent, this is achieved by an operation of assigning the genes so that they are in the same order when viewed from the right. I decided to refer to this kind of two-point order crossover as both end part order crossover. Mutation also is a shift operation. 5) Order representation (abbreviated as order ) This is the chromosome representation method that was proposed by Grefenstette and colleagues [4]. It is added for comparison since it is also considered as one of the factorial number representation methods. Uniform crossover is used for the crossover method. Also, like the proposed method, mutation is performed by replacing specific genetic locus values with numbers that satisfy a condition. 6) Proposed method (abbreviated as proposed ) As described in the previous section, chromosomes are represented by using factorial numbers. Factorial numbers and permutations are associated according to rules 1 and 2 from Section 3. Also, uniform crossover is used for the crossover method. Among these methods, 2), 3), 4), 5), and 6) obtain the solution according to a genetic algorithm. Except for the crossover and mutation operation parts, the same procedure is used for all of them. This procedure is outlined below. 77

6 The group size SIZE is set to n or, in other words, the number of jobs. To create the initial group, random numbers are used to generate SIZE factorial numbers. Methods 5) and 6) use those factorial numbers directly, and methods 2), 3), and 4) convert them to permutations. Therefore, the initial group starts from the same members in the numerical value experiment for each method. The initial group that is obtained is rearranged in descending order of fitness (ascending order of makespan values). As described earlier, crossover differs in each method. Crossover is performed based on rank. A SIZE/2 set of parents is created to generate SIZE children. Mutation is executed for the SIZE + (SIZE 1) chromosomes excluding those of the SIZE children and the parent having the highest fitness among the group of parents. In methods 2), 3), and 4), mutation is executed by a shift operator. Also, in methods 5) and 6), mutation is executed by replacing the factorial number coefficient of the gene locus that is selected according to a random number by another value. After crossover and mutation operations are performed, the chromosomes of the 2 SIZE total number of parents and children are rearranged in descending order of fitness, and the highest SIZE chromosomes are retained as the next parents. Generation changes are performed for up to 1000 generations, for which the solution is generally believed to be stable. 6. Test Data In past numerical experiments such as those of Osman and Potts [11], for example, the work time at each machine for each job was assigned according to a random number from 1 to 100. Therefore, in the current research, I created a hypothetical example in which the work times were also decided by using uniform random numbers from 1 to 100. I refer to the example that was created by using uniform random numbers from 1 to 100 as the Type A hypothetical example. However, it has been pointed out that it is actually quite rare for the work times to be random. Rinnooy Kan [13] assumed two patterns that appear among work times. One is that the mean work time differs according to the machine, and the other is that the work time of each machine tends to differ according to the job. Although the solution search is simple for the first pattern because the work time at a specific machine regulates the overall makespan, obtaining better approximate solutions is known to be difficult for the second pattern [12]. Therefore, in the current research, I decided to create a hypothetical example in which the mean work times also differ according to the job. Following Rinnooy Kan [13], the processing time at each machine for each job is determined according to a random number in the following range. Note that ε i is a number that is randomly selected from among 1, 2, 3, 4, and 5 for each job. I refer to the example that was created in this way as the Type B hypothetical example. For both the Type A and Type B hypothetical examples, I created 30 examples each for the 12 possible combinations of numbers of jobs and numbers of machines. Therefore, the total number of hypothetical examples that were created is Results of Numerical Experiments Using Genetic Algorithms To judge the relative merits of each algorithm, I calculated two indices. The first index counts the number of times that an algorithm obtained better results than the other algorithms for 30 numerical examples. Note that since there may be cases in which makespans are the same, the total will not necessarily be 30. The second index is the t value that is obtained as a result of pair comparison. Pair comparison is a method of statistically verifying whether the difference between the observation results obtained from a pair of experiments can be considered to equal zero. The official procedure is summarized as follows. (Step 1) Let µ denote the population mean of the difference between the two statistics and set the null hypothesis H 0 as follows. H 0 : µ = 0 (Step 2) Set the alternative hypothesis H 1 as follows. H 1 : µ 0 _ (Step 3) When X denotes the mean of the difference X = X 1 X 2 (where X 1 represents the solution obtained from the method that is the comparison target and X 2 represents the solution obtained from the proposed method) for a sample of size n (30 in the current research) and S 2 denotes the sample variance, calculate the following test statistic T: (Step 4) Based on the null hypothesis H 0, use the fact that the test statistic T obeys a t distribution with n 1 degrees of freedom. In the current research, I set the significance level to 1% at both sides. Also, since the number of samples is 30 in every test, the critical region is as follows: T < or T > This is shown together with the test results in the table. When the proposed method is thought to be superior to the method that is the comparison target, H is entered, and when it is thought to be inferior, / is entered. 78

7 Table 1. Number of times best solution was found for Type A data Table 1 shows the experimental results for Type A test data using uniform random numbers from 1 to 100 as work times. It is apparent that the proposed method improves the NEH solution regardless of the number of jobs or number of machines. It is also apparent that it is superior in all cases when compared with order representation, which is the same factorial number representation. In addition, it is apparent that when the number of jobs and number of machines are large (n = 100 and m = 20 or n = 200 and m = 20), the proposed method is superior to all three methods that use order crossover. Table 2 shows experimental results for Type B hypothetical examples in which the mean of the work times differed according to the job. As is clear from Table 2, the proposed method shows better results than other methods in almost all cases. In particular, the superiority of the proposed method is striking in problems of a scale in which the number of jobs is at least 50 and the number of machines is at least 10. The proposed method is better than the other methods both in terms of the number of times the best solution was found and the statistical test results. For Type A in which test data is assigned using random numbers, the individual work times tend to mutually offset each other so that the difference between the total work times tends to be smaller. This is a disadvantage for the proposed method, which incorporated NEH concepts. On the other hand, with Type B, since the mean of the work times differs for each job, this condition is unlikely to occur. I think that this is the reason that the proposed method showed better results for Type B test data than for Type A test data. The proposed method shows excellent results in large-scale problems (n = 100, m = 20 or n = 200, m = 20) that also include Type A test data. From this, we can conclude that the proposed method is effective when the mean of the work times differs for each job and in problems that are relatively large-scale. 8. Investigation of Results Using the Amount of Information From the results of the numerical experiments, it is apparent that the proposed method is better than the conventional methods. In this section, we investigate the reasons for this based on the amount of information. Table 2. Number of times best solution was found for Type B data 79

8 When C n C n 1 C 2 C 1 denotes an n-digit factorial number, then from the hypothesis in Section 3, coefficient C j satisfies the condition 0 C j < j (j = n,..., 1). Now, let us focus on coefficient C 2. From the condition, C 2 takes the value of either 0 or 1. Since C 1 is always 0, C 2 = 1, C 1 = 0 and C 2 = 0, C 1 = 0 represent the possible permutations of the two jobs having the largest total work times. Since coefficient values are determined by using random numbers in the initial group, 1 and 0 should be included nearly the same number of times. In other words, the entropy of coefficient C 2 should be a value near 1. However, from the search results, the entropy of coefficient C 2 seems to be smaller. This means that the results converged to a specific one of the two possible permutations. In other words, by calculating the entropy of the coefficients, we should be able to measure the degree to which the results converged to a specific permutation. The entropy can also be calculated for other coefficients. Coefficient C j can take j values. If we assume that n j of each coefficient had been included in a group of parents of size n, then the entropy E(C j ) of coefficient C j can be calculated by using the following equation: Table 3. Entropy of factorial numbers after the search In the same way as described for coefficient C 2, we should be able to measure the degree to which the results converged to a specific permutation among the possible permutations by calculating the entropy of the coefficients. Therefore, I calculated the entropy of coefficients for the cases n = 100, m = 20 and n = 200, m = 20 for which the proposed method is thought to be particularly superior. The coefficients that I targeted were C 2, C 5, C 10, C 15, C 20, C 25, and C 30. Note that for a method that represents chromosomes by using permutations, I used rule 1 in Section 3 to convert to factorial numbers and then calculated the entropy. Also, for the order representation, after first converting to a permutation, I converted it to a factorial number to calculate the entropy. Table 3 shows the results. The initial value of the entropy of coefficient C 2 is close to 1 in every case. After a 1000-generation search, it gets smaller regardless of the method. From this, it is apparent that the relative order of the two jobs with the largest total work time converges to one alternative. Although the entropy also tends to get smaller for other coefficients, the degree differs according to the design of the genetic algorithm. With coefficients C 2 or C 5, cases in which order crossover is used are sometimes smaller than the proposed method. However, for coefficients C 15 and above, the proposed method is clearly smaller. In other words, it is apparent that the relative order relationship has converged together with the search for jobs with small total work times, not just for jobs with large total work times. This indicates that among the possible orders, the searches have become fixed at a small number of orders. This matches the feature of NEH that the job order is fixed in descending order of total work times. Although a genetic algorithm is excellent for global searches, its local search capability is considered to be low. Therefore, Murata and colleagues [9] proposed a method of performing a local search that uses simulated annealing after performing a global search according to a genetic algorithm. However, in the current research, by using factorial numbers to represent permutations, I was able to perform a depth-first search, which is a feature of NEH. A depth-first search can be considered to be a local search in which the relative positional relationships of jobs with the largest total work times are fixed. As we learned from the experiments, as the generational changes proceeded, the entropy of each variable was reduced, and the depth-first tendency intensified. In other words, the proposed method can be considered to be a search algorithm that gradually switches from a global search to a local search in which the 80

9 relative order relationships among the jobs with the largest total work times are fixed. I believe that this is the reason why it can obtain better solutions than the conventional methods. 9. Order-Constrained Permutation Search Certain types of order constraints can be easily represented by using factorial numbers to represent permutations. In this section, we investigate an application of the proposed method for a search of order-constrained permutations represented by factorial numbers. If a job that belongs to a certain group must be processed before a job that belongs to another group, this is considered to be a serial constraint. This occurs when a job that was ordered from a specific customer within a certain period must be processed before a job that was ordered later. Since we can generally consider that jobs from many other customers are being processed, scheduling must be performed that simultaneously takes into consideration jobs in which order constraints are assigned and jobs having no order constraints. Figure 3 shows an example having this kind of constraint. Figure 3 represents a situation in which jobs c, d, and e must be processed before jobs f and g, and jobs f and g must be processed before jobs h and i. However, there is no order constraint between jobs a and b and jobs c, d, and e. The number of permutations that satisfy the constraint shown in Fig. 3 is as follows: Let us try to use factorial numbers to represent permutations having this kind of constraint. We create the basic permutation first. The basic permutation is created by first arranging the jobs with no constraints, and then arranging them in an order in which jobs having serial constraints go first. For the example in Fig. 3, permutations such as abcdefghi or badcegfih, for example, can be taken as the basic permutation. We will let the permutation abcdefghi be the basic permutation here. From the factorial number representation rule, the values that can be taken by coefficient C n j of the j-th Fig. 3. Hypothetical example of preceding succeeding relationships. Fig. 4. Arrangement of variables. element x j in the basic permutation are 0 C j < j. Therefore, the values that can be taken by coefficient C 9, which corresponds to element a, are numbers less than 9. For example, let 4 be taken as the value of coefficient C 9. Similarly, coefficient C 8, which corresponds to element b, can take any number less than 8. For example, let C 8 = 5. Next, we consider coefficient C 7, which corresponds to element c. From the factorial number condition, C 7 can take a number less than 7. However, since there is an order constraint, there are several numbers it cannot take. Element c has elements f, g, h, and i as elements that follow it. As a result, the position of element c must be determined while reserving storage locations for these following elements. A factorial number coefficient determines which sequential position among the empty positions it is to be stored in. Therefore, the upper bound is reduced by the number of following variables (Fig. 4). In other words, when there is a serial constraint, the conditions that each coefficient of the factorial number must satisfy can be consolidated as follows. (Condition for factorial number coefficients) If the j-th element x j in the basic permutation has fw j following variables, the values that coefficient C j can take must satisfy the relationship The range of each coefficient can be determined based on this condition. Table 4 shows the results. The factorial number satisfies the Table 4 conditions. Also, decfagbih can be obtained by converting this to a permutation, and this permutation satisfies the constraints in Fig. 3. In addition, it is apparent from Table 4 that the number of possible combinations of coefficients is 81

10 Table 4. Conditions for factorial numbers and that this matches the number of permutations that satisfy the constraint. If we consider this together with the fact that different permutations are generated from different factorial numbers, it is apparent that there is a one-to-one correspondence between factorial numbers and permutations that satisfy the constraint. Let the factorial number coefficients satisfy the condition 0 C j < j fw j. In this case, even if crossover is performed, the coefficients of the newly created factorial numbers will still satisfy the condition 0 C j < j fw j. Therefore, permutations that are obtained from factorial numbers will satisfy the order constraint. In other words, for a serial constraint, by using factorial numbers to represent permutations, you can suppress permutations in which the same element appears multiple times and permutations that do not satisfy the order constraint. This means that in a search that uses a genetic algorithm, the solution need not be corrected and no penalty need be assigned. 10. Numeric Experiments for Constrained Permutation Searches When permutations are used to represent chromosomes, even if lethal genes that include the same element multiple times are suppressed during crossover, it is possible that a permutation that does not satisfy the constraint may end up being generated. Two methods of dealing with this problem have been investigated [5, 6]. One method corrects the permutation so that the constraint is satisfied, and the other method imposes a penalty according to the degree to which the constraint is not satisfied. In the current research, I decided to refer to these as the correction method and the penalty method. The specific procedures for these methods are shown below. Note that these tasks are necessary when permutations are used to represent chromosomes, but they are not necessary when factorial numbers are used to represent chromosomes as with the proposed method. When a chromosome that does not satisfy the order constraint appears during the search process that uses a genetic algorithm, this method obtains the makespan by making small corrections to some permutations so that the constraint is satisfied. In the current research, I made corrections by starting with the first element of the permutation and comparing it with other elements and replacing it if the constraint was not satisfied. Therefore, when n denotes the number of jobs, a computational complexity of O(n 2 ) is required. Note that although I also considered a method of replacing the original permutation with the corrected permutation, I decided not to make this kind of replacement after referring to the research of Iima and Sannomiya [5, 6]. 2) Penalty method When a chromosome that does not satisfy the constraint appears during the search process, this method attempts to assign some kind of penalty in the fitness calculation. In other words, the fitness (makespan) is calculated as follows: Fitness = z + θg where z is the makespan that is obtained without taking the order constraint into consideration, and G is the number of chromosomes (number of combinations of elements) for which the order constraint is not satisfied. This is obtained by checking sequentially starting from the first element of the permutation and counting the cases in which the order constraint is not satisfied. Therefore, the computational complexity when n denotes the number of jobs is O(n 2 ). The greatest problem with the penalty method is deciding how large to make the coefficient θ. From trial experiments, the variation in makespan was at most approximately several dozen. Therefore, I decided to set the penalty parameter in intervals of 10. Specifically, I set the coefficient in 12 ways (0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, and ) to execute the genetic algorithm and used the one for which the best solution was obtained. Therefore, 12 experiments were required to solve one problem. In the current research, I performed numerical experiments for the following four types of problems. I set these problems since they show differences according to the scale of the problem or strength of the constraint. Figure 5 shows 1) Correction method Fig. 5. Four types of hypothetical examples. 82

11 four hypothetical examples. Note that the numbers in the figure indicate job numbers. The basic permutation consists of jobs in order of these numbers or, in other words, in ascending order. In the first type of hypothetical example, which is called Type 1, there are 30 jobs of which 10 have no constraints. Also, 10 jobs must be processed before the remaining 10 jobs. In the Type 2 hypothetical example, there are 50 jobs of which 20 have no order constraints. The other 30 jobs consist of 10 jobs that must be processed first, 10 jobs that must be processed next, and 10 jobs that must be processed after these 20 jobs. The Type 3 and Type 4 hypothetical examples both have 100 jobs. The Type 3 example has weaker constraints with no order constraints for 70% of the jobs. Conversely, order constraints are assigned for 70% of the jobs in the Type 4 example. In both cases, the jobs for which constraints are assigned consist of groups of 10 jobs each, and serial constraints are assigned to these groups. The following five algorithms are compared. 1) Both end part order crossover + correction method 2) Both end part order crossover + penalty method 3) Middle part order crossover + correction method 4) Middle part order crossover + penalty method 5) Proposed method In algorithms 1) to 4), permutations are used to represent chromosomes. Also, mutation is a shift operator. In algorithm 5) for the proposed method, factorial numbers are used to represent chromosomes. Also, mutation is performed by changing coefficients. I applied these algorithms to the four types of problems described in Fig. 5. At that time, I created 30 different numerical value examples for each type of problem. The processing time at each machine for each job was determined by using uniform random numbers from 1 to 100. Also, the jobs that were contained in each group were rearranged in ascending order of total work times. Random numbers were used to create factorial numbers so that the coefficient condition in Section 9 was satisfied. In the proposed method, these were used directly as the initial set of parents. In the other methods, these were converted to permutations, which were used as the initial set of parents. Table 5 summarizes the experimental results. To judge the relative merits of each algorithm, I calculated three indices. The first index counts the number of times that an algorithm obtained better results than the other algorithms for 30 numerical examples. Note that since there may be cases in which makespans are the same, the total will not necessarily be 30. The second index is the t value that is obtained as a result of pair comparison. Like in Section 7, I set the critical region to 1% at both ends. Therefore, the solution that is obtained from the proposed method is judged to be significantly superior when the absolute value of the t value is greater than The third index is the computation time. In the experiments, I used a Dell PowerEdge 300 Pentium III 600 MHz CPU with Windows 95 as the operating system and C as the programming language. The values shown in Table 5 are computa- Table 5. Results of numerical experiments 83

12 tion times (in seconds) per numerical value example. For the penalty method, it is the total computation time when the time was calculated for 12 types of penalties. For the Type 1 problem in which the proportion of 30 jobs with no constraints was 1/3, the best solution was obtained the greatest number of times (11) by the middle part order crossover and penalty method and the proposed method. However, from the results of the t test, no significant difference was observed, and no relative merit was seen in small-scale problems. From the experimental results for the Type 2 hypothetical example in which there were 50 jobs and constraints were set among three groups, the superiority of the proposed method over the other methods could be verified. In other words, the proposed method obtained the best solution 23 times out of 30 trials. Although the middle part order crossover and correction method showed the next best results, it obtained the best solution only 6 times. The t value was positive in all comparisons and was also greater than In other words, the mean of the makespans of the solutions obtained from the proposed method was smaller than for every other method. The effectiveness of the proposed method can be verified from the numerical value experiments for the Type 3 hypothetical example in which there were 100 jobs. From the experiment for the Type 3 hypothetical example, for 30% of the total number of jobs, the proposed method obtained the best solution 25 out of 30 times. The middle part order crossover and correction method obtained the best solution 5 times, and the both end part crossover methods never obtained the best solution. All t values were greater than It is also apparent from this that the proposed method obtains good solutions. From the experimental results for the Type 4 hypothetical example in which serial constraints were assigned for 70% of the total number of jobs, the superiority of the proposed method was remarkably apparent. The proposed method obtained the best solution 29 times out of 30 trials. From the t values, it was also apparent that the makespans of the solutions obtained from the proposed method were smaller than the makespans of the solutions obtained from the other methods. Through these experiments, it was apparent that the proposed method showed superior results as the scale grew larger or the proportion of jobs to which constraints were assigned increased. Let us now try to compare processing times. When there are 30 jobs, which is a relatively small number, the difference between the computation times for the correction method and proposed method is not very great. The execution time for one numerical value example is approximately 2 or 3 seconds. However, as the number of jobs increases, the difference between the computation times also increases. For example, when there are 100 jobs, the computation time for the proposed method is approximately 20 seconds, while computation times roughly 2 to 4 times greater are required for the correction method when ordered crossover is used. Even greater computation times are required for the penalty method. This is because when permutations are used to represent chromosomes, the permutations for which the order constraint is not satisfied must be counted or corrected. When factorial numbers are used, these kinds of operations are unnecessary since lethal genes are suppressed. This is the reason for the difference between the computation times. In other words, even when viewed from a computation time perspective, it is apparent that the proposed method is superior. Note that the computation times for both end part crossover are greater than the computation times for middle part crossover because the genes must be rearranged twice in the order of one of the parents when a child is created. 11. Conclusions In the current research, I investigated the use of factorial numbers to represent permutations and applied this to the flow-shop scheduling problem. By using permutations arranged in ascending order of total work times, I designed a genetic algorithm for carefully searching for relative positional relationships among jobs with large total work times. This concept is also used by NEH, which is one of the methods known as constructive heuristics. I also used the fact that no lethal gene is produced from crossover of chromosomes that are represented by factorial numbers and decided to use uniform crossover. This has the effect of preventing the group from converging to a local solution within a small number of generation changes. As a result, better solutions can be obtained than with genetic algorithms that had conventionally been proposed. In addition, in the current research, I used a flow-shop scheduling problem with so-called serial order constraints to compare methods that use permutations to represent chromosomes with a method that uses factorial numbers. When factorial numbers are used, not only are there no chromosomes in which the same element appears multiple times, but chromosomes that do not satisfy the order constraint can also automatically be excluded. Therefore, I expected there would be the benefit that computation times would be shorter, and this was verified through experiments. In addition, when the number of jobs was increased to 50 or 100, it was apparent that the makespans of the solutions that were obtained were also shorter than for the other methods. Permutations that satisfy order constraints are a subset of the possible permutations. As a result, when permutations are used to represent chromosomes, it is difficult to make sure that no lethal genes are produced by crossover or mutation. Although they dealt with different constraints 84

13 than the ones used in the current research, namely, the fact that jobs could not be inserted consecutively, Iima and Sannomiya [5, 6] were able to suppress the occurrence of lethal genes since permutation representations were used. Even if the problem had the so-called serial constraints that were dealt with in the current research, it would probably be difficult to suppress lethal genes when permutations were used to represent chromosomes. As described in Section 9, it is apparent that by using factorial numbers, lethal genes can easily be suppressed when serial constraints are used. However, factorial numbers cannot be used to deal with all order constraints. An investigation of the order constraints for which factorial number representations are effective is a remaining topic of research. Acknowledgments. The successful compiling of this research depended on the valuable advice received from Professors Toshikazu Nishimura and Shoko Takahashi of the Systems Integration Engineering Program in the Department of Science and Engineering at Keio University. I express my sincere gratitude. Also, my research committee members investigated thoroughly and offered helpful advice. I express my gratitude to them also. REFERENCES 1. Campbell HG, Dudek RA, Smith ML. A heuristic algorithm for the n Job m machine sequencing problem. Manage Sci 1970;16:B Dannenbring DG. An evaluation of flow shop sequencing heuristics. Manage Sci 1977;23: Goldberg DE, Lingle R Jr. Alleles, loci, and the traveling salesman problem. Proc 1st ICGA, Grefenstette J, Gopal R, Rosmaita B, Van Gucht D. Genetic algorithms for the traveling salesman problem. Proc 1st ICGA, Iima H, Sannomiya N. GA behavior for scheduling problems with difficult constraints. Proc 20th SICE Systems Symposium, , Iima H, Sannomiya N. Effect of lethal genes on genetic algorithm operation. Trans Soc Instrum Control Eng 1995;31: Johnson SM. Optimal two- and three-stage production schedules with setup times included. Nav Res Logistics Q 1954;1: Maekawa K, Tamaki H, Kita H, Nishikawa Y. A solution method for the traveling salesman problem based on genetic algorithms. Trans Soc Instrum Control Eng 1995;31: Murata T, Ishibuchi H, Tanaka H. Flowshop scheduling by genetic algorithm and its application to multiobject problems. Trans Soc Instrum Control Eng 1995;31: Nawaz M, Enscore EE Jr, Ham I. A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. OMEGA 1983;11: Osman I, Potts C. Simulated annealing for permutation flow-shop scheduling. OMEGA 1989;17: Reeves CR. A genetic algorithm for flowshop sequencing. Comput Oper Res 1995;22: Rinnooy Kan AHG. Machine scheduling problems: Classification, complexity and computations. Martinus Nijhoff; Sannomiya N, Kita H, Tamaki H, Iwamoto S. Genetic algorithms and optimization. Asakura Shoten; Taillard E. Some efficient heuristic methods for the flow shop sequencing problem. Eur J Oper Res 1990;47: Turner S, Booth D. Comparison of heuristics for flow shop sequencing. OMEGA 1987;15: Watase K. Obtaining a causal model using factorial numbers. Tech J IEICE, AI99-19, Kobayashi S, Ono I, Yamamura M. Character-preserving genetic algorithms for traveling salesman problem. J Japan Soc Artif Intell 1992;7: Yagiuri M, Nagamochi H, Ibaraki T. Two comments on the subtour exchange crossover operator. J Japan Soc Artif Intell 1995;10: Yoneda N. Generation of permutations by computer. Bit 1969;1:

14 AUTHOR Kazunori Watase (member) graduated in 1974 from the Systems Integration Engineering Program in the Department of Engineering at Keio University. In 1979, he left Keio University after completing his doctoral course in the Graduate School of Engineering. Currently, he is a professor in the Management Systems Engineering Section of the Department of Engineering at Nagasaki Institute of Applied Science. His chief areas of research are financial statements analysis, analysis of causal relationships, and knowledge acquisition. 86