Single and multi-objective evolutionary algorithms for the coordination of serial manufacturing operations

Transcription

1 Single and multi-objective evolutionary algorithms for the coordination of serial manufacturing operations DAVID NASO, BIAGIO TURCHIANO, and CARLO MELONI Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari, Bari, Italy Phone: Fax: Abstract - This paper focuses on a typical problem arising in serial production, where two consecutive departments must sequence their internal work, each taking into account the requirements of the other department. Even if the considered problem is inherently multi-objective, to date the only heuristic approaches dealing with this problem use single-objective formulations, and also rely specific assumptions on the objective function, leaving the most general case of the problem open to innovative approaches. In this paper, we develop and compare three evolutionary algorithms for dealing with such a type of combinatorial problems. Two algorithms are designed to perform directed search by aggregating the objectives of each single department in a single fitness, while a third one is designed to search for the Pareto front of non-dominated solutions. We apply the three algorithms to considerably complex case studies derived from industrial production of furniture. Firstly, we validate the effectiveness of the proposed genetic algorithms considering a simpler case study for which information about the optimal solution is available. Then, we focus on more complex case studies, for which no a priori indication on the optimal solutions is available, and perform an extensive comparison of the various approaches. The results obtained on all the considered cases confirm the considerable potentialities of evolutionary computation and suggest many interesting directions for further investigations. Keywords: Multi-Objective Evolutionary Algorithms, Scheduling, Sequencing, Manufacturing systems

2 1 1 Introduction Modern manufacturing industry is experiencing a fast evolution toward distributed organization of production activities. Due to the various challenging problems related to collaboration, cooperation, information sharing, synchronization of logistic and production activities, research areas such as supply chains, virtual enterprises, global manufacturing networks and similar forms of aggregation have gained a prominent role in the field of industrial automation. Recent researches in this context include the problem of interstage coordination in serial production. This problem arises in all the production environments in which a series of manufacturing departments or cells (either geographically distributed or located within a single facility) have to process an ordered set of jobs having different attributes (e.g. color, shape, type of material). Often, single attributes have a specific relevance only in single departments (e.g. the color of the job is considered in the painting department, but irrelevant to the sawing department, which inversely must take into account the type of material processed). As a consequence, each department may incur in setup-related costs when processing two jobs with different values of a certain attribute (e.g. the painting department must clean the equipment when switching between different colors, whereas the sawing department must change saw whenever it has to cut different types of materials or shapes). Ideally, to minimize the individual setup costs under these assumptions, each department should sort the processed jobs according to the specific attribute of interest. On the other hand, a complex coordination problem arises when the order of the processed jobs cannot be changed between consecutive stages, forcing all the departments or cells to work on the same sequence. This fundamental restriction is quite frequent in plants processing goods of relatively large dimensions with respect to the available storage areas, as in furniture production facilities, where sorting operations cannot be performed due to the lack of sufficient space or appropriate material handling devices. Similar problems are encountered in rough mills: due to the limited capacity of sorting conveyors [Kotak et al., 21] or

3 2 storage bins [Fletcher et al., 21], the appropriate sequence of jobs must be optimized in previous stages of the production. In more general terms, the optimization of the job sequence involves not only attribute-related setup costs, but also other fundamental decision factors, such as due dates and other external priorities related to customers and logistic issues. Therefore, the interstage coordination can be viewed as a permutation problem with multiple objectives: on a periodical basis (e.g. weekly), the processing sequence of the planned jobs must be established taking into account the costs of each single department and other predefined priorities. This combinatorial problem becomes particularly complex as the number of considered jobs grows (in fact, for a problem involving the sequencing of n jobs, the size of the search space is n!). Being an inherently multi-objective problem, the interstage coordination can be attacked in two fundamental ways. A first one is to define a scalar objective function aggregating the costs of all the departments and find an efficient method to solve the resulting single-objective problem. In the case of interstage setup coordination, typical aggregation criteria are summing all the costs of the various departments and minimizing the overall production cost, or considering the maximum value between the costs of each department to keep the costs balanced across the departments. On the one hand, the aggregative strategy leads to relatively simpler models for which efficient heuristic approaches are available in technical literature [Agnetis, 21], but on the other one, finding effective forms of aggregation capable to properly incorporate and weights the multiple, often noncommensurable and conflicting, objectives is not always an easy task. Recent research in combinatorial optimization offers a second alternative to deal with the considered problem, by means of efficient meta-heuristics that are able to return a population of solutions, each representing a different compromise between the conflicting objectives. Such methods fall in the research area that is generally referred to as multi-objective optimization, and in many cases they offer practical

4 3 advantages fully counterbalancing their computational costs, which are in general higher than those associated to the aggregative approaches. In facts, as also pointed out in the recent survey by Jones et al. [Jones et al., 22], the considerable advancement of research on multi-objective search algorithms is partially due to the rapid increase in computer power, and to the development of more appropriate algorithms for coping with the various complicating factors usually neglected in traditional approaches. Jones et al. also acknowledge that the main advances in the context of multiobjective optimization have been in the area of Evolutionary Algorithms (EA s). EA's are heuristic search techniques inspired to the principles of survival-of-the-fittest in natural evolution and genetics [Michalewicz, 1996]. Every EA works iteratively on a population of candidate solutions of the problem (individuals), performing a search guided by the fitness of each solution: the higher the fitness, the more the genes of the solution will be propagated to the solutions explored in the next iterations. This Darwinian principle is emulated with specific crossover, mutation and selection operators, obtaining a stochastic search algorithm that explores solutions with progressively increasing fitness. Unlike other meta-heuristics, EA s are considerably flexible with regard to the characteristics of the objective function, as they do not rely on specific a priori hypotheses. A variety of different EA s has been developed for virtually countless single and multi-objective optimization problems. This paper focuses on a particular class of EA s technically referred to as Genetic Algorithms (GA s), which mainly differ from the rest of EA s for the peculiarities of the selection mechanisms used to replicate solutions in the successive iterations, and for the extensive use of crossover between solutions to determine new ones. While we will hereinafter assume that the reader is familiar with the basic principles of evolutionary computation, we suggest references [Fogel, 1995] or [Michalewicz, 1996] for an extensive introduction and further details on EA s classifications and nomenclature. Basically, in single-objective GA s, the value of the scalar objective function associated to each solution is directly assigned as the fitness of the solution with respect to problem objectives and constraints. Clearly, all aggregative

5 4 approaches combining multiple performance figures in a single objective function fall into this class of GA s. On the contrary, in multi-objective GA s (MOGA s) the search goals are not aggregated, but considered separately, and using special ranking and selection mechanisms the population is progressively led toward the optimal tradeoff or Pareto front, i.e. toward the set of all the solutions for which an improvement for any considered objective can only be achieved at the cost of worsening some other ones. This paper investigates the use of GA s for solving complex interstage coordination problems arising between two industrial departments working on furniture production. In order to assess the benefits and limitations of the two fundamental ways to attack the problem (either aggregative or explicitly multi-objective), we develop different algorithms suitable for both approaches, and compare the results obtained in various case studies with different characteristics, ranging from problems for which efficient heuristic approaches have recently been proposed [Agnetis et al. 21] to more complex ones that cannot be directly dealt with techniques derived from literature. In particular, while the single-objective algorithms are based on state-of-art EA s suggested by recent research [Meloni, 21, Michalewicz, 1996], the MOGA proposed in this paper is a considerably improved variant of a preliminary version [Meloni et al., 23] developed by the authors for smallsize problems (i.e. sequences of less than 3 jobs). Thanks to specific mechanisms for selection and storage of intermediate results, the MOGA proposed in this paper is able to deal with sequence of up to 3 jobs within almost the same execution times of the single-objective EA s. Moreover, as suggested by the complementary features that the various considered algorithms exhibit in the detailed experimental investigation, we also propose some hybrid combination of single and multiobjective EA s capable to combine the advantages of both approaches. The reminder of this paper is structured as follows. Section 2 overviews the related literature, underlining the original contribution of our research. Section 3 describes the main details of the

6 5 coordination problem, and illustrates the structure of the optimization algorithm. Section 4 summarizes and compares the results obtained by the proposed algorithms on case studies based on real world data derived from industrial furniture production, while Section 5 provides the conclusive remarks and draws the main directions of future research. 2 Related literature In recent years, technical literature has addressed various problems sharing similar characteristics with the interstage coordination considered in this paper. Independently from the specific objective function to be optimized, all the related problems suffer of a nearly prohibitive combinatorial complexity. In particular, to date technical literature suggests efficient heuristics only capable to deal with particular cost functions. For instance, [Agnetis et al., 21] studied the coordination between two departments with unitary setup costs (switching from a job with attribute a to one with attribute b has the same cost than switching from a to c, for any triple of attributes a b c). The reference [Agnetis et al., 21] considers two main problems, namely minimizing the sum of the cost of each department, and minimizing the maximum cost between the two departments. After proving that both problems are NP-hard, the paper proposes a heuristic procedure that is only able to deal with the minimization of the total sum of costs (or equivalently, due to the unitary cost of each changeover, the minimization of the total number of setups). Under particular cost structures, the interstage coordination can be viewed as a variant of the Traveling Salesman Problem (TSP), which is a well-known permutation problem that has been extensively considered in the last decades. Some efficient algorithms capable of finding satisfactory solutions are currently available [Gutin and Punnen, 22], although many of them assume that the visited cities correspond to cities in a Cartesian space under some standard metric [Michalewicz, 1996], or rely on other specific assumptions that impede their profitable extension to interstage

7 6 coordination problem. In particular, to authors knowledge, none of the available methods can effectively deal with all the forms of coordination problem considered in this paper, including both unitary and generic setup cost structures, e.g. non unitary (switching from a job with attribute a to one with attribute b has different cost than switching from a to c) and non-symmetrical costs (switching from attribute a to attribute b has different cost than switching from b to a). Due to the mentioned difficulties, last decades have seen an increasing interest on heuristic techniques drawing on ideas and concepts from other disciplines to help solve complex optimization problems [Jones et al., 22]. Such approaches are generally referred to as metaheuristics, and encompass various efficient and versatile search algorithms including EA s. Several versions of single-objective EA s have also been proposed for solving permutation problems as the TSP or the flow-shop scheduling, and surveys of the recent developments in these contexts can be found in [Michalewicz, 1996] and [Dimopoulos and Zalzala, 2], respectively. Recent research efforts have also been dedicated to develop efficient multi-objective EA s for permutation problems, especially in the field of flow-shop scheduling [Hishibuchi and Murata, 1998, Hishibuchi et al. 23]. As already mentioned, MOGA s simultaneously take into account two or more objectives, and aim at finding a sample of solutions as close as possible to the tradeoff front, also known as Pareto front in multi-objective literature (see e.g. [Fonseca and Fleming, 1998] for a detailed introduction). MOGA s are in general more complex and computation-demanding than normal GA's, because they must perform a higher number of comparisons to rank individuals, and because they need specific mechanisms to prevent the concentration of the search on excessively narrow segments of the Pareto front. Therefore, a substantial part of the literature on MOGA s is devoted to the development of effective selection mechanisms avoiding a premature convergence of the search. Restricting the attention to researches related to this paper, the work by Hishibuchi and Murata [Hishibuchi and Murata, 1998] is among the earliest attempts to deal with a multiobjective permutation problem. Hishibuchi and Murata develop a Multi-Objective Genetic Local

8 7 Search (MOGLS) algorithm which maintains a population of non-dominated solutions using a randomized ranking method. Namely, in a problem with n objectives, at each time two individuals must be compared, n random weights are extracted, and the weighted aggregation of the n fitness values of each individual is used for ranking and selection. The randomization of the aggregation weights produces a selective pressure that changes direction at every iteration, and progressively leads the population toward the Pareto set. The MOGLS is also equipped with a local search algorithm that cyclically explores a number of neighborhood solutions (obtained with random shifts of a job in other positions of the sequence) to achieve local improvements. In [Hishibuchi and Murata, 1998], the MOGLS is compared with other MOGA s, derived from literature on a set of benchmarks considering the simultaneous minimization of makespan and maximum tardiness (some experiments also include average flowtime as a third objective). The contribution of local search to the overall convergence is enhanced in subsequent versions of the algorithm [Hishibuchi et al., 23], where the random mechanism selecting the direction of local search is replaced by a heuristic criterion. Many recent MOGA s have been specifically devised for numerical problems in continuous realvalued domains, thus relying on operators that use a norm of the distance between solutions in the search space. Such algorithms cannot be directly applied to permutation problems, because even if in principle it is possible to define specific indices to measure the distance between two permutations of a sequence (as e.g. those overviewed in. [Landa Silva and Burke, 24]), these indices usually require an extremely large number of comparisons. Therefore, their use in MOGA s, which already suffer of a considerably heavy number of comparisons to perform non-dominate ranking, would make the total computational cost of the algorithm inadmissible. For this reasons, our attention is only focused on some state-of-art MOGA s that can be applied to permutation problems without modification. For instance, the Pareto Archived Evolution Strategy (PAES) [Knowles and Corne, 1999] is a simple (1+1) EA (see [Fogel, 1995] for details) that effectively

9 8 overcomes the difficulty of maintaining a population of non-dominated individuals. Namely, this approach maintains an archive of the non-dominated solutions progressively found during the search. The size of the archive is limited removing individuals according to an heuristic strategy based on the degree of crowding of their neighborhood, which attempts to spread the solutions in the archive on the known front. The high computational complexity of non-dominated sorting is also tackled in [Deb et al. 22] with an improved version of the Non Dominated Sorting GA (NSGA-II). The algorithm uses an improved multi-objective sorting method that needs a significantly smaller number of comparisons than in conventional non-dominated sorting (O(NM 2 ) versus O(NM 3 ), respectively, where M is the number of objectives, and N the population size). NSGA-II also uses a crowding strategy always selecting the solutions in the less crowded areas of the currently-known Pareto front. As it will be shown in the next section, in this paper we propose a MOGA for the interstage coordination that is partially inspired to a preliminary version that we have recently devised for solving small-size problems of up to 3 jobs [Meloni et al. 23]. Our algorithm also shares some similarities with the PAES and the NSGA-II, even though it is able to overcome with ad hoc mechanisms some fundamental drawbacks that both PAES and NSGA II exhibit when applied to our combinatorial problem. 3. An evolutionary approach to interstage coordination In this paper, we focus on the coordination issues between two stages of a production chain, and therefore consider only two attributes. Since our work focuses on coordination problems arising in furniture production, we will refer to the two considered stages as sawing and painting departments, and to their associated attributes as shape and color, respectively. In both departments, a changeover occurs when at least one attribute of a new part is different from those of the previously processed parts. For example, if a part must be cut with a different shape from the previous one,

10 9 cutting machinery must be reconfigured. Similarly, when a new color must be used, firstly the painting equipment must be cleaned in order to eliminate the residuals of the previous color, and then the new color must be loaded. In both cases costs are incurred in terms of time and manpower. As an obvious consequence of cost minimization, all the jobs with the same color and shape are grouped in a single batch and processed in sequence, so that when switching from a batch to the next one, at least one department will certainly incur in a setup. In general terms, the problem can be formulated as follows. Let B be a set of n batches of jobs to be sequenced on the considered time horizon, and let us refer to the sawing and painting departments as D S and D C, respectively. Furthermore Let S and C denote the sets of all shapes and colors available for production. We refer to the shapes as s i, i=1,, S and to the colors as c j, j=1,, C. For sequencing purposes, each batch can be described only specifying its pair of attributes (s i,c j ). Clearly, if batch b 2 =(s h,c k ) is processed immediately after batch S b 1 =(s i,c j ), the department D S incurs in a setup cost σ > if h i, while the department D C incurs C in a setup cost σ > if k j. More precisely, if h=i and k j (h i and k=j), only the department D C j k (D S ) pays a setup cost called local changeover, while if the two consecutive batches b 1 =(s i,c j ) and b 2 =(s h,c k ) have no attribute in common (h i and k j), both departments incur in setup costs, originating an undesirable circumstance that will be hereinafter referred to as global changeover. The interstage coordination requires finding an appropriate permutation of the batches in B taking into account the conflicting setup costs. Indicating with λ=(λ 1, λ 2,, λ n ) an ordered sequence (permutation) of the elements in B, we can compute the total cost paid by each department as the sum of the changeover costs: i h n 1 S S σ ( λ) = σ λl λ (1) l+ 1 l= 1

11 1 n 1 C C σ ( λ) = σ λl λ (2) l+ 1 l= 1 From the most general viewpoint, setups may have different costs. For instance, in the painting department, set-up times are shorter when switching from a lighter color to a darker one than viceversa. Setup may also have different costs for different departments (e.g. changing a sawing equipment only involves a setup time, whereas changing paint may involve time and waste of paint and cleaning solvents). Whenever discrepancies between the setup costs can be neglected, it is possible to obtain a simplified models amenable to efficient search algorithms. For instance, the problem with unitary setup costs is obtained if the following cost structures apply: S σ i h 1 if = if s s i i s h = s h, C σ j k 1 if = if s s j j s k = s k (3) In this case the setup costs coincide with the number of performed changeovers. This is the only case addressed in [Agnetis et. al., 21], and in particular an efficient algorithm is proposed only for the optimization of the sum of the changeovers in both departments. The extreme versatility of EA s contributes to overcome the difficulties in modeling variable setup costs, finding effective solutions to both single (aggregated) and truly multi-objective formulations of the problem. The reminder of this section will first describe the single-objective evolutionary approach, and then the multi-objective approach in two separate subsections. A. Single criterion optimization There are several ways to aggregate the cost functions of single departments in a scalar performance figure taking into account the requirements of both departments. As already mentioned, typical aggregated measures of performance for interstage coordination approaches are

12 11 the total sum of costs of changeovers across the two departments (hereinafter referred to as minsum problem, [Agnetis et al. 21]), the maximum between the costs paid by either department (the minmax problem), and the minimization of the overall number of global changeovers (that coincides with the minsum problem when changeovers have unitary costs, [Meloni et al., 23]). While the first and third problem may express the objective of maximizing the overall utility, the second form of aggregation may be used to better balance the changeover costs between the two departments. Independently from their differences, both the minsum and minmax objective functions are suitable to be used as fitness for the evolutionary optimization, and since there is no a priori reason to believe that one of them will lead to better overall results, it is interesting to test and compare GA s using both form of aggregations. Having defined a suitable objective function for the evolutionary search, the next step for the development of the GA is devising an effective encoding strategy to convert each solution (an ordered set of batches) in a string of genes on which recombination operations, such as crossover and mutation, can be easily performed. In the last decades, technical literature has proposed various encoding schemes suitable for permutation problems, which have been recently surveyed in [Michalewicz, 1996]. Among the various schemes, we choose the path representation method that was originally proposed in the context of TSP problems. In this method, a solution is directly represented by an ordered list of integers from 1 to n so that, for instance, the vector λ= [ ] describes one of the 6! solutions of the 6-job sequencing problem. We focus on the path representation scheme because it is considered the most intuitive representation for a permutation, and also for the considerable number of crossover and mutation operators compatible with this encoding scheme. Moreover, the path representation is the most common strategy for sequencing and scheduling problems in the context of manufacturing, and is also used in references [Hishibuchi and Murata, 1998, Hishibuchi et al., 23] that are tightly related to our research.

13 12 For the single-objective optimization, we developed a GA that is substantially equivalent to stateof-art algorithms based on efficient operators for selection and recombination, and successfully applied in a wide range of different problems. Figure 1 summarizes the basic structure of our GA, illustrating its main loop with a descriptive meta-language. In Figure 1, the set Pop(i) represents a set of solutions (including the associated fitness) composing the population at i-th iteration (generation). The role of each meta-function used in the description can be summarized as follows. /* Single Criterion EA */ /* Algorithm Startup */ i = 1; Pop(1) = random_pop fitness_eval(pop(1)) i = 2; /* main loop of the EA */ WHILE terminating_condition == false p_best = findbest(pop(i-1)) /*elitist preservation of the best-known individual*/ Pop(i) = select(pop(i-1),sel_ops); Pop(i) = crossover(pop(i),p_cro); Pop(i) = mutation(pop(i),p_mut); Fitness_eval(Pop(i)) Pop(i)=Pop(i) p_best i=i+1; check terminating_condition END WHILE FIGURE 1: The Single Objective GA random_pop: This function creates a random initial population. fitness_eval(pop): the fitness of each new individual (i.e. resulting from a random initialization, or from a crossover or mutation operation that produced a solution different from its parent(s)) in Pop is evaluated. Select(Pop,sel_ops): this function returns a new population of solutions selected from those in Pop with a strategy that assigns higher probability of selection to individuals with higher fitness. In

14 13 general, the effect of selection can be graded by changing one or more configuration parameters (the vector sel_ops) of the selection mechanisms. In all our single-objective GA's we use the effective tournament selection (Michalewicz, 1996) operator, and sel_ops specifies the size of each tournament. crossover(pop,p_cro): this function randomly selects couples of solutions in Pop to perform a crossover. This returns two new individuals that partially inherit characteristics of both parents. After the crossover, the resulting offspring replaces the two parents. The parameter p_cro defines the probability of each individual in Pop to be selected for crossover. Several crossover operators for the path representation/encoding scheme have been proposed. As the result of preliminary investigation of various combinations of operators, we selected the order-based crossover [Michalewicz, 1996], also known as two-point crossover [Hishibuchi et al. 23] for all the GA s proposed in this paper. Our choice is also supported by a recent comparative study [Murata et al., 1996] that suggest the use of the order-based crossover together with a specific mutation operator (see below) to obtain an effective gene recombination in scheduling problems similar to the one considered in this paper. mutation(pop,p_mut): this function randomly alters a solution to obtain a new one with partially differing characteristics. The parameter p_mut describes the probability that each individual in Pop will be subject to mutation. Among the various mutation operators compatible with path representation, we chose the insertion mutation [Michalewicz, 1996, Hishibuchi et al., 23], for its good performances when applied with order-based crossover. The proper choice of the various configuration parameters of the algorithm (sel_ops, p_cro, p_mut, etc.) is fundamental to obtain satisfactory performances. Since there are few general rules to appropriately set the values of most configuration parameters for every GA, the initial configuration of the algorithm is often based on a preliminary set of experiments. This inspection

15 14 approach has been adopted also in this paper: further details of the configuration of our algorithm will be discussed in the next section. B. Multi-criteria optimization As many optimization problems in manufacturing environments, the interstage coordination has an inherently multi-objective nature. Therefore, instead of fixing in advance a specific form of (weighted) aggregation of the various objectives, it may be more profitable to directly seek for a good approximation of the Pareto front with an ad hoc search strategy. For this purpose, we developed a MOGA considering the setup costs of each single department as a separate fitness. Figure 2 describes the main structure of our MOGA. This algorithm uses some of the functions (random_pop, crossover, mutation) already described for the single-objective GA s, and some new functions whose role and meaning can be summarized as follows. /* Multi-objective EA */ /* Algorithm Startup */ i = 1; Pop(1) = random_pop Seeds Fitness_eval(Pop(1)); Front(1) = Find_Front(Pop(1)); /* main loop of the EA */ WHILE terminating_condition==false LastPop = Pop(i) /* storing the previous population for next selection */ Pop(i) = crossover(pop(i),p_cro); Pop(i) = mutation(pop(i),p_mut); Fitness_eval(Pop(i)) Front(i+1) = Find_Front(Front(i) Pop(i)); Pop(i+1) = select(pop(i) LastPop, sel_ops); check restore_front_cond if restore_front_cond == true Pop(i+1)=restore_front(Front(i+1), Pop(i+1)) Endif check terminating_condition i=i+1; END WHILE FIGURE 2: the Multi-Objective GA

16 15 Seeds: This set encompasses some solutions that can be injected in the initial population to speed up the early progress of the algorithm. Find_Front(Pop): this function returns the non-dominated individuals found in the set of solutions Pop passed as function input. The function works in two different ways. When invoked at algorithm startup, the function must compare all the couples of different individuals in the initial population. This operation requires O(m Pop 2 ) comparisons, where m is the number of objectives (m=2 in this paper). On the contrary, when called within the main loop of the algorithm, this function compares each individual in the known front with each individual in the current population to determine if new points of the front have been discovered. The number of comparisons in this case depends also on the size of the current front, which grows as the search proceeds. Select(Pop,sel_ops): this function select the offspring solutions for the next iteration. In the MOGA, the selection must rank individuals taking into account both objective functions simultaneously. Several efficient methods to perform non-dominated ranking have been recently proposed including Non-Dominated Sorting with crowding [Deb et al, 22], tournament with random weighted aggregation [Hishibuchi, 23], to mention some recent ones. In this paper, we use an improved variant of the selection operator proposed in [Meloni et. al., 23]. It performs different operations depending on the number of non-dominated solutions f s (i) in the i-th population known at the i-th iteration. When f s is lower than the population size p s, the new population is build as the union of the f s known front solutions with p s - f s solutions randomly selected from among those laying inside a band in the objective surface delimited by the current front and its projection at a predefined distance d. When the number of known non-dominated solutions f s reaches the population size, the new population is build ranking the solutions according to their crowding distance (Deb et al., 22), which is an approximate measure of the closeness of other solutions in the front with respect to a particular individual in the same set. Solutions with smaller crowding distance are relatively closer to each other, and may not be selected for the next population. This

17 16 mechanism aims to prevent the concentration of the search in narrow segments of the known Pareto front. restore_front(front,pop): partially due to the combinatorial nature of our problem, and to the discrete fitness values, the selection mechanisms generally used for numerical optimization in continuous domains do not work properly when applied to our sequencing problem. This inefficiency arises, for instance, when using the crowding distance for ranking individuals for selection. Namely, after a certain number of iterations, when the number of individuals in the front is larger than the population size, it may occur that very good solutions in the front may be rejected from the next population because of their mutual interference, being too close to each other. In general, the frequency of this mutual interference between good solutions grows with the iterations of the GA, until the phenomenon becomes so influent to obstruct any further progress in the search for the Pareto front even for extremely long search times. To prevent the inefficiency due to the cyclical loss of good solutions from the population, (as also proposed in [Knowles and Corne, 1999], and [Meloni et al. 23]) our algorithm stores the known front in separate archive so that, whenever the lack of appreciable progresses is detected, the known front can be restored in the population using this function. restore_front_cond: this function monitors the lack of progress by counting the number of iterations spent without finding a new member of the front, so that whenever this number exceeds a given threshold (rfc_th, see Table IX), the known front is restored into the population. 4. Experimental results The algorithms described in the previous section were tested on a set of instances derived from furniture production. Each instance describes the weekly product demand, specifying the attributes of each batch of raw material that has to be processed. The weekly production consists of about 25 to nearly 3 different batches, having attributes selected from 32 different shapes and 15 different

18 17 colours. To give an idea of the contents of each single instance, table I shows one example of weekly demand, organized in a S C matrix. The number in each cell in the table corresponding to shape i and color j is the identification number of the job. Any of the n! possible permutations of the job sequence is an admissible solution to our problem. In particular, due to the particular assignment of job identification numbers as described in Table I, the ordered sequence 1,2,3,,25,251 is one of the sequences yielding the minimum setup cost for the cutting department, whereas the sequence obtained ordering the jobs by columns (e.g. 1, 12,23,, 243,2,13 ) is one of the solutions yielding the minimum painting setup cost. These two solutions are used as seeds in the multi-objective search, whereas the initial population for the singleobjective algorithms is fully random. TABLE I: A TYPICAL INSTANCE OF WEEKLY PRODUCTION IN FURNITURE INDUSTRY (EACH ENTRY IN THE MATRIX INDICATES THE ID. NUMBER OF THE JOB WITH SHAPE s i AND COLOR c j shapes Colors C1 C2 C3 C4 C5 C6 C7 C8 C9 C1 C11 S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S

19 18 We have considered three cost structures for setup operations using setup costs that are referred to as unitary, proportional and random. The case of unitary cost is equivalent to minimize the number of changeovers, and is useful for comparison purposes, since some information on the optimal solutions is available a priori. In particular, the lower bound for the objective function of the minsum problem is n-1 (this occurs when for each couple of consecutive jobs only one department incurs in a setup, which is equivalent to say that the optimal minsum is obtained by all the sequences guaranteeing the absence of global changeovers) and the lower bound for the minmax is n/2 if n is even, 1+(n-1)/2 if n is odd (the setups would be fully balanced when each department performs the 5% of the total number of setups). In principle, these theoretical bounds can be reached when the density of instances is relatively high. The density is defined as the number of combination of attributes considered in the instance with respect to the total number of possible combinations (in other words, the density is the relative number of non-empty entries in Table I). As shown in [Agnetis et al. 21], the instances coming from industrial practice and considered here are sufficiently dense to often have at least a sequence in which the optimum for the minsum problem equals the lower bound n-1. In the case of proportional costs, we assume that shapes and colors are ordered so that the costs of setup between two jobs with attributes s h and s i, (c k and c j ) is proportional to the absolute value of the difference between the indices i-h ( k-j ). This formulation is quite realistic and can model circumstances often occurring in practice as those described in the previous section (e.g. a painting setup from white to light grey has a lower cost than a setup from white to black). Finally, in the third case of random costs, the setup cost for each couple of different attributes is a random number selected in the range [1, S ] for shapes, and [1, C ] for colors. This experiment aims at investigating the ability of the algorithm in the most general case of no explicit relations between any couple of attributes.

20 19 The single-objective GA is run using either minsum or minmax as fitness function, while the MOGA uses the setup cost of the two departments. For sake of brevity these three algorithms will be hereinafter referred to as GAms, GAmm and MOGA, respectively. The next two subsections summarize the results obtained by the two versions of the GA separately, while a third one is dedicated to comparison and discussion. A. Single Objective Optimization To obtain a well-performing search algorithm, firstly a set of preliminary runs with different values for the main free parameters of the GA was performed. Independently of the type of fitness function, the best results were obtained with crossover and mutation probabilities that lay within the ranges normally suggested in technical literature. In the preliminary investigation, it was observed that the population size significantly affects the efficiency of the search algorithm, and in particular, choosing populations of relatively small sizes (3-5) provided considerably better results whereas larger population sizes (1-2) did not improve the exploration ability but rather caused the algorithm to converge to solution with lower fitness. Furthermore, we noted that in the average the algorithm converged to the final solution after 5. to 7. fitness calls. Therefore, the final configuration of the single-objective algorithm was set as indicated in Table II. TABLE II. SUMMARY OF THE GA CONFIGURATION FOR THE SINGLE-OBJECTIVE OPTIMIZATION Population Size - 5 Stopping Criterion - Number of fitness function calls =1 Probability of Crossover -.8 Probability of Mutation -.3 Selection operator - Tournament Selection with size = 3 Fitness function - Either Minmax or minsum We considered a set of five instances, and run each GA five times with different initial random populations. The results obtained by the GAms and GAmm are summarized in Table III and IV, respectively. All the results refer to the best solution found at the end of the run, and are averaged

21 2 over the five runs. The tables report separately the number of setups performed by each department, the total sum of setups, and the maximum number of setups between the two departments. Beside average values, the minimum and the maximum indices (respectively the best and the worst result over all the replications) are also provided. instance TABLE III. SUMMARY OF THE RESULTS OF GAms (MINSUM PROBLEM) FOR UNITARY SETUPS Number of jobs N. of Cutting Setups N. of Painting Setups Total Amount of Setups Maximum n. of Setups /181.4/188 62/68.6/77 25/25/25 173/181.4/ /178/185 66/73/8 251/251/ /178./ /178.8/188 63/72.2/8 251/251/ /178.8/ /179.2/188 63/71.8/8 251/251/ /179.2/ /199.6/22 79/81.4/87 281/281/ /199.6/22 It can be immediately noted in Table III that the GAms is able to find the optimal solution in all the runs and for all the instances. These results are achieved by assigning larger numbers of setup operations to the cutting department, and hence providing a quite unbalanced distribution of the costs. This should not be viewed as a flaw of the GA, but rather as a consequence of the fact that our case studies have a number of shapes that is about three times the number of available colors. It is worth underlining again that this is the only considered case for which efficient problem-specific heuristics are available in [Agnetis et al., 21], and that the proposed GA provides results of the same accuracy of the domain-specific method. TABLE IV. SUMMARY OF THE RESULTS OF GAmm (MINMAX PROBLEM) FOR UNITARY SETUPS instance Number of jobs N. of Cutting Setups N. of Painting Setups Total Amount of Setups Maximum n. of Setups /133/ /133/ /266/27 132/133/ /132.8/ /132.8/ /265.6/ /132.8/ /131/ /131/ /262/ /131/ /13.6/ /13.6/ /261.2/ /13.6/ /149.2/ /149.2/ /298.4/32 147/149.2/151 Differently from the GAms, the performance of the GAmm is slightly less repeatable. In fact, only in the third instance the GAmm ends all the runs in the same solution, whereas in the other cases

22 21 results vary of about 2-3 setups (less than 1%). However, it is worth noting the considerable ability to properly balance the costs between the two attributes (in all cases each department performs the 5% of the setups). As a further confirmation of the conflicts between the minsum and minmax objectives, it can be easily noted that the lower number of setups in the cutting department guaranteed by the GAmm comes at the expense of a significantly higher total number of changeovers with respect to the solutions found by the GAms. The results of the case of proportional setup costs are summarized in Tables V and VI. The tests were made following the same procedure described for the case of unitary cost, except for the varied cost computation in the fitness evaluation. Clearly, hereinafter the reported indices describe a setup cost rather than a number of changeovers. TABLE V. SUMMARY OF THE RESULTS OF GAms (MINSUM PROBLEM) FOR PROPORTIONAL SETUPS instance Number of jobs Total Cost of Cutting Setups Total Cost of Painting Setups Total Sum of Setup Costs Maximum Cost of Setups /163./ /145.6/ /38.6/ /163./ /159.8/166 14/151.8/166 34/311.6/ /163.8/ /163./17 148/15.6/152 32/313.6/32 152/163.4/ /164.6/ /149.8/164 37/314.4/ /165./ /178.8/194 17/179./2 348/357.8/ /187.6/2 TABLE VI. SUMMARY OF THE RESULTS OF GAmm (MINMAX PROBLEM) FOR PROPORTIONAL SETUPS instance Number of jobs Total Cost of Cutting Setups Total Cost of Painting Setups Total Sum of Setup Costs Maximum Cost of Setups /25.4/ /25.2/ /41.6/ /25.4/ /25./ /25./ /41./ /25./ /26.4/ /26.4/ /412.8/ /26.4/ /29./ /29.2/ /418.2/ /29.2/ /233./254 2/233./254 4/466./58 2/233.2/254 In the case of proportional setup costs, the performance of both single-objective GA's become less repeatable, with larger excursions between best and worst result. In particular, while the differences between the final results of GAms range from 4% to 7%, those of the GAmm are significantly higher (about 15% to 2%).

23 22 The third case considers random setup costs. The results obtained by GAms and GAmm for this case are summarized in tables VII and VIII, respectively. instance TABLE VII. SUMMARY OF THE RESULTS OF GAms (MINSUM PROBLEM) FOR RANDOM SETUPS Number of jobs Total Cost of Cutting Setups Total Cost of Painting Setups Total Sum of Setup Costs Maximum Cost of Setups / 89.2/ /1669.4/ /2559.6/ /1669.4/ / 942./ /1681.6/ /2623.6/ /1681.6/ / 921.2/ /1647.4/ /2568.6/ /1647.4/ / 88.2/ /1583.6/ /2463.8/ /1583.6/ /153./ /1781.2/ /2834.2/ /1781.2/1851 TABLE VIII. SUMMARY OF THE RESULTS OF GAmm (MINMAX PROBLEM) FOR RANDOM SETUPS instance Number of jobs Total Cost of Cutting Setups 98/ 96.4/ / 964.8/ / 989.6/ / 967.6/138 Total Cost of Painting Setups Total Sum of Setup Costs Maximum Cost of Setups 1762/1893.2/ /2853.6/ /1893.2/ /187.6/ /2835.4/ /187.6/ /1968.4/ /2958./ /1968.4/ /1843./ /281.6/ /1843./ /117.4/ /2257.6/ /3365./ /2257.6/2336 Also in this case the performance of both GA s varies in each run (6% to 1% for the GAms, and 5% to 15% for the GAmm). B. Multi-Objective Optimization Also the final configuration of the MOGA used in these experiments and summarized in table VII is the result of a preliminary comparison of various combinations of settings. Moreover, also in this case the most satisfactory behavior was obtained with standard mutation and crossover rates, and a relatively small population size. To perform a fair comparison with the single objective GA's, we used the same number of fitness function calls as stopping criterion, even though it must be noted that the MOGA has to perform a considerably higher number of comparisons for determining nondominated solutions.

24 23 TABLE IX. SUMMARY OF THE GA CONFIGURATION FOR THE MULTI-OBJECTIVE OPTIMIZATION Population Size - 5 Stopping Criterion - Number of fitness function calls =1 Probability of Crossover -.8 Probability of Mutation -.4 Selection operator - as described in section 3 Restore Front Criterion - After rfc_th=5 populations without finding new solutions in the front Fitness function - F=(Setup Cost of Cutting Dept., Setup Cost of Painting Dept.) To give an idea of the search process executed by the MOGA, Figure 3 shows the evolution of the population in a sample run, compared with the final front obtained at the end of the run. In Figure 3, it is possible to observe how the injection of seeds individual attracts part of the population toward the front even in the first generations Population at n-th iteration final front Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n Cost. of Cut.Dept. - pop. n.15 FIGURE 3: A typical evolution of the search executed by MOGA At the end of its search process, the MOGA returns two sets of solutions containing the final population and the archived front. Differently from the final population of single-objective GA's, which is almost entirely composed of copies of the best individual found, in the MOGA the final population mostly contains different individuals having different combinations of objective