scheduling (SMS) we can recognize one of the ... Finally, we use a fitness landscape analysis to ..... Dobrasinovic, who developed the software to generate.
Fitness landscapes for the Total Tardiness Single Machine Scheduling problem Alexandre S. Mendes, Paulo M. Franca and Pablo Moscato Abstract We address several issues related to the approximate solution of the Single Machine Scheduling problem with sequence-dependent setup times using metaheuristic methods. Instances with known optimal solution are solved using a Memetic Algorithm and a Multiple Start approach. A fitness landscape analysis is also conducted on a subset of the instances to understand the different performances of the two approaches. We present a novel way to create instances with known optimal solutions from optimally solved ATSP instances. Finally we argue for the test set of instances to be used in future works as a convenient performance benchmark.
Consider that all jobs are available for processing at time zero. We have a list {p1, ..., pn} of processing times for each job and a list {d1, ..., dn} of due dates. We also know a matrix {sij} of setup times where sij is the time required to set up job j after job i has just finished. It is assumed that sij needs not to be equal to sji. We are also given a list {s01, s02, ..., s0n} of setup times where soi is the setup time required by job i if it is the first one in the sequence. Task: Find a permutation that minimizes the total tardiness of the schedule (Ttotal), which is calculated as: n
T total = ∑ max {0, c k − d k }
(1)
k =1
1. Introduction Under the generic denomination of single machine scheduling (SMS) we can recognize one of the first studied class of problems in the scheduling area, as the survey paper by Graves [6] shows. The problem has many variants. In particular, the SMS problem with sequence-dependent setup times is an important issue in Management Science and Operations Research since it is present in many industrial manufacturing systems. In this case, the jobs require a setup time that depends on the predecessor job. In this paper we will address a problem with an extra constraint: each job has also an associated due date. The goal is to minimize the total tardiness of the production schedule. We will formally define this concept as follows. The SMS problem we address in this paper can be described as:
where ck is the completion time of job k. We can also reference this problem as 1/sds/Ttotal, according to the classification system proposed by Graham et al [5].
Input: Let n be the number of jobs, indexed with 1, 2, ..., n, to be processed in one machine.
In this paper we propose the use of a Memetic Algorithm (MA) [3] for solving the 1/sds/Ttotal. The MA utilized is a variation of a Genetic Algorithm that applies a local search procedure in every new individual generated. The MA has several refinements, like hierarchically structured population and hybrid neighborhoods for local search. Details of the implementation can be found in ref. [4]. For comparison we used a Multi Start (MS) procedure. The MS is a simple algorithm that creates an initial random solution and then applies a local search procedure on it. By doing this repetitively, it makes a fairly good exploration of the solutions space. The MS is frequently used as a benchmark for scheduling problems.
Faculdade de Engenharia Elétrica e de Computação – FEEC Universidade Estadual de Campinas – UNICAMP C.P. 6101 – 13081-970 – Campinas – SP – Brazil e-mail:{smendes, franca, moscato}@densis.fee.unicamp.br
The test set used for the experimental analysis consists of instances that vary from 17 to 100 jobs. This set is constructed from Asymmetric Travelling Salesman Problem (ATSP) instances
with known optimal solutions available from the literature. We show how, using some structural modifications, is it possible to transform ATSP instances into SMS instances while keeping the optimality property of the optimum ATSP permutation. All ATSP instances used in this work, as well as their optimal solutions, are available at the TSPLIB web site [11]. Finally, we use a fitness landscape analysis to figure out the reason for the different performances of the MA and the Multiple Start (MS) procedure for this problem. 2. Asymmetric Travelling Salesman Problem Instances The optimization version of the ATSP (or min ATSP) can be defined as:
Step 1: Generation of the {s0j} list Let [π(k), π(k+1)] be the pair of adjacent arcs of the ATSP optimal tour with the largest value of d. Let s0,π(k+1) be equal to the cost of this arc, dπ(k),π(k+1). Let all other soj, j ≠ π(k+1) be greater than so,π(k+1). This guarantees that the sum of all setup times in the optimal SMS sequence will be equal to the ATSP tour length, thus preserving the optimal tour as the optimal sequence. Moreover, the initial job will be π(k+1). Step 2: Generation of the {pi} list Construct the list { pπ(1), pπ(2), ..., pπ(n) } in such a way that pπ(1) < pπ(2) < ... < pπ(n). Step 3: Generation of the {di} list Construct the due dates in such a way that: dπ(1) < dπ(2) < ... < dπ(n) dπ(i) ∈ [ cπ(i) - pπ(i) , cπ(i) ] ∀ π(i): i = 1, ..., n
Input: Let n be the number of cities, indexed 1, 2, ..., n. We are given a matrix {dij} of distances where dij is the distance from city i to city j (dij needs not to be equal to dji).
where cπ(i) is the completion time of job π(i) in the ATSP optimal sequence.
Task: Find a permutation of the cities [ t(1), t(2), ... , t(n) ] that minimizes the total distance of the tour.
Although the literature of Evolutionary Computation refers to many early pioneers before 1970, it is widely considered that the introduction of John Holland's book entitled “Adaptation in Natural and Artificial Systems” was the significant event that let Genetic Algorithms, and the larger field of Evolutionary Computation, be established as new research areas. Since then, applications of GAs became increasingly conspicuous in many engineering fields and in Artificial Intelligence problems. In the mid 80's, a new class of “knowledge-augmented GA”, sometimes called “hybrid GAs”, started to appear in the literature. These methods stressed the idea of including all possibly knowledge about the problem at hand. As a consequence, the resulting algorithms had little resemblance with biological evolution. Recognizing important differences and similarities with other population-based approaches, some of them were categorized as Memetic Algorithms (MAs) in 1989 [8]. The field of cultural evolution was suggested as being more relevant, as a working metaphor, to understand the performance and find inspiration sources to improve these new methods.
3. From solved ATSP instances to solved SMS instances: how to construct them From the definitions given in sections 1 and 2, it is clear that the ATSP and the SMS have some similarities. The most important is the one concerning setup times and distances between cities. This similarity suggests a transformation procedure to create SMS instances, and also to construct their optimal solutions, using an arbitrary ATSP instance with known optimal solution. We propose a three-step procedure to generate such SMS instances. An SMS instance includes a matrix of setup times and three lists of n integers. In the transformation we propose, the setup times matrix needs to be equal to the distance matrix of the ATSP instance. We will show next, a procedure to generate the three lists required to complete the SMS instance. Let suppose that π = [ π(1), π(2), ..., π(n) ] represents the sequence of cities in the optimal tour for the ATSP instance, then:
4. Memetic/Genetic algorithm approach
4.1. Population Structure It is illustrative to show how some MAs resemble more the cooperative problem solving techniques that can be found in some organizations. For instance, in our approach we use a hierarchically structured population based on a complete ternary tree. In contrast with a non-structured population, the complete ternary tree can also be understood as a set of overlapping subpopulations (that we will call clusters). The use of this population structure for interactions restricts the selection scheme used for recombination. Leader
fragment of the chromosome from one of them is randomly selected and copied into the offspring. In the second phase, the offspring's empty positions are sequentially filled according to the chromosome of the other parent. The procedure tends to perpetuate the relative order of the jobs, although some alterations might appear. In each generation we create 13 new individuals, that is the number of individuals present in the population. There are some acceptance rules for the addition of new individuals and, as a consequence, some of the offspring may be discarded – the insertion of new solutions in the population will be discussed later. 4.3. Mutation
Cluster Supporters
Figure 1: Population structure In Figure 1, we can see that each cluster consists of one single leader and three supporter agents. Note that the leader agents in any intermediate layer have both leader and supporter roles. The leader agent always contains the best solution – considering the total tardiness – of all agents in the cluster. The number of agents in the population is equal to the number of nodes in the ternary tree, i.e., we need 13 individuals to make a ternary tree with 3 levels and 40 individuals to have 4 levels. For this work, we fixed the population size to be 13, after conducting thorough tests with 40 and 121 individuals. Larger populations generally might work better for small number of jobs, but our tests showed no noticeable improvement for the larger instances when we have a timeexpiration criterion for stopping the algorithm. 4.2. Representation and Crossover For the SMS problem the representation we have chosen is quite intuitive, with a solution being represented as a chromosome with the alleles assuming different integer values in the [1, n] interval, where n is the number of jobs. The crossover tested is the well-known Order Crossover (OX). After choosing two parents, a
In our implementation a traditional mutation strategy based on swapping of jobs was implemented. Two positions are uniformly at random selected and we swap their values. This mutation procedure is applied (on the average) to 10% of all new individuals every generation. We also implemented a heavy mutation procedure [2]. It executes the job swap move 10.n times in each individual, except the best one. This procedure is executed every time the population diversity is considered to be low, i.e., it has converged to individuals that are too similar (See section 4.6). 4.4. Local Search Local search algorithms for combinatorial optimization problems rely on a neighborhood definition that establishes a relationship between solutions in the configuration space. In this work, two neighborhood definitions were chosen and they were both used in the MS and in the MA approaches. One of the neighborhoods implemented was the “all-pairs”. It consists in all possible swaps of pairs of jobs from a given solution. A hill-climbing algorithm can be defined by reference to this neighborhood; i.e., starting with an initial permutation of all jobs, every time a proposed swap reduces the total tardiness it is confirmed and another cycle of swaps takes place, until no further improvement is achieved. The second neighborhood implemented was the “insertion” neighborhood. It consists in removing
a job from one position and inserting it in another one. The hill-climbing iterative procedure is the same regardless of the neighborhood definition.
1. The offspring is inserted into the population replacing the supporter that took part in the recombination that generated it.
Given the large size of the all-pairs and insertion neighborhoods, and the computational complexity required to calculate the total tardiness for each solution, the implementation of some reduction policies is mandatory. Given a certain neighborhood definition, we can restrict the search using some rules that specify which moves might be worth testing. A well-designed reduction scheme should reduce the number of evaluations as much as possible without loosing the strength of the search. That is, only movements that may return good results should belong to the reduced neighborhood and be evaluated. Analogously, bad moves must be promptly identified and discarded. Some authors refer to these reduction schemes as candidate list strategies.
2. The replacement occurs only if the fitness of the new individual is better than the supporter.
We based our reduction rules on the setup times' values. We have observed that most good schedules have a common characteristic: the setup times between jobs in these solutions are very small. This is reasonable, since schedules with small setup times between jobs will be less lengthy, generating few delays. The reduced neighborhoods were only used in the MA approach and are explained in detail in ref. [4]. 4.5. Selection to Recombination The recombination of solutions in the hierarchically structured population can only be made between a leader and one of its supporters within the same cluster. The recombination procedure selects any leader uniformly at random and then it chooses – also uniformly at random – one of the three supporters. An intensification strategy was also implemented. It makes the best individual take part in approximately 10% more recombinations than the others.
If during the recombination phase no individual was accepted for insertion, we conclude that the population has converged and apply the heavy mutation procedure. Finally, after the recombination phase, the population is restructured. The fitness of the leader of a group must be lower than the fitness of the leader of the group just above it. Following this policy, the higher subgroups will have leaders with better fitness than the lower groups and the best solution will be the leader of the root subgroup. The adjustment is done comparing the supporters of each subgroup with the leader. If any supporter turns out to be better than its respective leader, they swap their places. 5. Multiple Start The MS algorithm implemented consists of the iterative generation of an initial random solution followed by a hill-climbing procedure that uses both the all-pairs and the insertion neighborhood. This procedure is the same used in the MA but we did not implement any reduction schemes. As there are no evolutionary processes involved, the search is more powerful with the complete neighborhoods. The MS is one of the simplest methods and uses no adaptive procedure at all; exception made to the local search procedure. Despite of this fact, MS implementations have shown good performances for sequencing problems and are often used as benchmarks.
4.6. Offspring Insertion in Population
6. Computational results
Once the leader and one supporter are selected, the recombination, mutation and local search take place and an offspring is generated. The acceptance of the offspring follows two rules:
In this section we describe how the test instances were generated and the results obtained by the MA and the MS. The processing times and the due dates were generated according to two classifications:
LOW ⇒ pk ∈ 0 ,
( )
1 ⋅ max sij 4 i , j
Table 1: Number of jobs (cities) and setup times (distance between cities) of ATSP instances ∀ k; k=1, ..., n
HIGH ⇒ pk ∈ 0 , 2 ⋅ max(sij ) i, j ∀ k; k=1, ..., n
The LOW policy makes the setup times a more critical aspect in the problem, emphasizing the “ATSP character” of the problem. The HIGH policy makes the processing times be more relevant in the schedule. The HIGH instances were easier to solve for both MA and MS. The due dates generation also followed two policies: HARD ⇒ dk = ck SOFT ⇒ dk ∈ [ ck - pk , ck ]
∀ k; k = 1, ..., n ∀ k; k = 1, ..., n
The HARD policy generates instances in which the total tardiness of the optimal schedule equals to zero. Nevertheless, it is not a trivial problem, since only one of the n! solutions will have a total tardiness value equal to zero. The SOFT instances have optimal total tardiness different from zero and are better suited to make relative comparisons, where percentage deviations from optimality are numerically necessary. No relevant variations were observed in the behavior of the algorithms for the HARD and the SOFT cases. They appear to change very little the fitness landscape and the performance of the MA and MS. As a matter of fact, both HARD and SOFT policies generate due dates well distributed along the production horizon. Given earlier results in ref. [4], if the due dates were concentrated over a narrower interval, the performance would certainly be different. The construction procedure presented in section 3, however, does not allow such type of due dates. Following we show a table with some characteristics of the original ATSP instances such as number of jobs and setup interval – minimum and maximum sij.
Instance br17.atsp ftv33.atsp ftv55.atsp ftv70.atsp kro124p.atsp
n 17 34 56 71 100
Min sij 0 7 6 5 81
Max sij 74 332 324 348 4545
As one can see, the instances tried to cover a wide range of number of jobs. Moreover, the setup ranges are very different and that was a critical component on the performance of the MA and the MS. The Table 2 shows all the results obtained for the set of instances created. The name of the instance is in the first column and is divided in two parts. The initial part is the name of the original ATSP instance. Then comes two capital letters that indicate if the instance is LOW or HIGH and HARD or SOFT. The second column indicates the number of jobs and the third column brings the optimal tardiness. MA average indicates the tardiness average considering 10 runs. The column labeled “# Opt” indicates the number of times the optimal solution was found. The CPU time limit was fixed at 120 seconds and when the optimal solution is found, or the limit is reached, the processing stops. In the column labeled “Average CPUtime” it is shown the average time for the 10 runs. The same applies to the last three columns, which are related to the MS performance. The results for the MA show an average performance much better than the MS. While the MA could not find the optimal sequence only for two of the kro124p instances, the MS method started to miss the optimal in the ftv55 instances. The average tardiness found by the MA is also always smaller than the MS, as it is the average CPU time All HH and HS variants – with large processing times – appear to be easier to solve than the LH and the LS ones. The due dates variation showed little influence on the hardness of the instances.
Table 2: Computational tests for the SMS problem Instance
br17LH br17LS br17HH br17HS ftv33LH ftv33LS ftv33HH ftv33HS ftv55LH ftv55LS ftv55HH ftv55HS ftv70LH ftv70LS ftv70HH ftv70HS kro124pLH kro124pLS kro124pHH kro124pHS
n
Optimal Tardiness
17 17 17 17 34 34 34 34 56 56 56 56 71 71 71 71 100 100 100 100
0 52 0 547 0 664 0 5324 0 1170 0 8515 0 1506 0 12368 0 26111 0 223890
MA # Average average Opt CPUtime tardiness 0 10 0.1 52 10 0.1 0 10 0.1 547 10 0.1 0 10 0.9 664 10 2.2 0 10 0.4 5324 10 0.3 189.8 9 32.9 1170 10 43.1 0 10 1.7 8515 10 4.7 4167.4 4 100.8 4685.9 4 103.8 138.8 9 37.4 12489 8 51.5 97344.7 0 120.0 136611.2 0 120.0 12154.5 6 80.9 233074.3 6 85.8
The ftv70LH.sms and the kro124pLH.sms returned very high average tardiness results. The optimal is zero and the MA values were 4167.4 and 97344.7, respectively. It is important to note that these instances are very hard to solve given their size and their large setup times intervals. Moreover, one characteristic of the tardiness function is that a single job in the wrong position may lead to a cumulative tardiness in all subsequent jobs, generating extremely large values. In the next chapter we will address this difficulty in a different way, using fitness landscapes to explain such results. The software language and the computer utilized were Java JDK 1.3 and a PENTIUM II-266 Mhz, respectively. 7. Fitness landscape analysis Fitness Landscape studies can be traced back to 1932 with the work of Sewall Wright, one of the early pioneers of population genetics [12]. In the earlier vision of this idea, genotypes or genes have some “fitness” as a function of their frequency in the population. Here we use a simpler idea, for each genotype sequence of jobs (i.e. for each individual i) we assign the value given by its total tardiness.
MS average tardiness
# Opt
Average CPUtime
0 52 0 547 0 765.9 0 5324 5768.5 6676.1 0 8515 19179.5 21246.6 340.8 12544.7 292631.3 371382.9 24856.1 271183.8
10 10 10 10 10 7 10 10 0 0 10 10 0 0 6 2 0 0 0 0
0.2 0.2 0.2 0.2 24.6 56.5 1.1 0.7 120 120 2.2 4.7 120 120 71.5 108.7 120 120 120 120
The main goal of this study in our case is to evaluate the instances in terms of difficulty for the MA we have implemented. Methodologically, this approach can be traced back to ref. [8] where several results by S. A. Kauffman and S. Levin, as well as S. Kirkpatrick and G. Toulouse were highlighted as relevant to understand the performance of MAs in optimization problems. The discussion on correlation of local optima, as the working hypothesis that can explain the results obtained with MAs was further investigated in ref. [9]. Among later works that discuss fitness landscapes for combinatorial optimization problems we can cite Boese [1], Reeves [10] and Merz and Freisleben [7]. Initially, we expect that the difficulty of finding the optimal solution would increase with the number of jobs. We also expect the performance to be dependent on the way the processing times were generated. Both factors are important since they change how the local minima are distributed on the space of solutions. The MA works with an initial population of solutions that is expected to converge – according to a certain distance measure – to the optimal solution after a given number of generations. Since the evolutionary process uses the fitness concept to guide the search, it is
important to have a high correlation between how far the objective function value of a solution is from the optimal one and the metric distance between them. The role of a distance metric is to measure how similar two sequences are. A high correlation would indicate that when we approach the optimal solution in terms of the objective function value, we also approach in terms of the metric distance, and vice-versa. This is the best working scenario for an MA. It is also possible to visualize this correlation in a graphical way, by putting these two values in a XY two-dimensional graphic, where the optimal solution is placed at the origin. Generally, the higher is the correlation, the narrower is the region generated by the local minima in the graphic. Moreover, a large number of local minima near the origin is an indicative that the problem is easy for the specific method and, similarly, if the local minima are concentrated far from the origin, the instance is hard to solve to optimality. All the works cited in this section used the XY graphic visualization for the fitness landscape. For the y-axis – the fitness axis – we adopted the objective function value itself, i.e., the total tardiness. For the metric distance – the x-axis – it is possible to define several suitable schemes. Nevertheless we decided to adopt a distance measure – namely Accumulated Distance (AD) – that apparently fits to the SMS problem. The AD has great appeal for sequencing problems, since it measures not only if a given job is located at the same position in the sequence for both solutions, but also, if it is not, how far is one position from the other. According to this, the AD can be defined as:
graphics with the fitness landscapes obtained for a subset of the instances in Table 2. 8. Fitness landscapes for MA and MS The graphics presented next are the fitness landscapes generated for 10 instances from Table 2. The y-axis represents the total tardiness of the solution and the xaxis the AD of the solution to the optimal one. All points are local minima obtained during the MA or MS search. The optimal solution is always located at the origin. All graphics were drawn based on a list of at least 3000 local minima, but fewer points are plotted since during the search process, the algorithms may reach the same local minimum more than once. MA
MS
Figure 2: Fitness landscapes for ftv33HS.sms instance. The ftv33HS.sms instance is very easy to solve for both algorithms, presenting a high correlation. Notice that with only 34 jobs the number of local minima is small, especially for the MS, which used a non-reduced neighborhood. That made the search process faster and easier. Both methods reached the optimal 10 times. MA
MS
n
dist AB = ∑ pos A (i ) − pos B (i ) i =1
Where posA(i) is the position of allele i in the solution A; analogously for posB(i). This is illustrated with the example below, with two sequences of 10 jobs. Table 3: AD example for two sequences of 10 jobs 1 Position Solution A 4 Solution B 7 8 Distance 16 AD
2 3 3 0
3 7 1 2
4 9 9 0
5 2 2 0
6 1 6 3
7 10 10 0
8 8 8 0
9 6 4 3
10 5 5 0
In Table 3, job 4 occupies the first position in solution A and the ninth position in solution B, what results in a distance of 8 for this job only. After adding up all jobs, the final AD is 16. The next section shows a series of
Figure 3: Fitness landscapes for ftv33LS.sms instance. Figure 3 shows landscapes with less correlation than Figure 2. In general, the HARD instances led to higher correlation features. For the MA there is a clear and well-defined path from the initial solutions, located at the top-right area, to the origin, where the optimal is located. The MS does not present such behavior, with most of the points lying farther than AD = 100. That may be the reason why the MS failed to find the optimal three times in 10 tries.
MA
MS
Figure 4: Fitness landscapes for ftv55HS.sms instance. The ftv55HS.sms instance is apparently easier than the previous one, although it has a larger number of jobs. The MA shows a high correlation, what makes this instance easy for it. The MS also presents similar characteristics. The large number of local minima near the origin is also an indication of the reason the algorithm performs so well. MA
This instance showed a nice correlation for the MA with a strong concentration of points near the optimal solution. The area covered by the points is very narrow, helping the MA find the optimal 9 times out of 10 runs. This is the same performance obtained for the 56-job ftv55LH.sms instance. The fitness landscape for the MS is fair, with a small concentration of points near the origin. The algorithm found the optimal sequence 6 times, making this instance one of the easiest considering its large size. MA
MS
MS Figure 7: Fitness landscapes for ftv70LH.sms instance.
Figure 5: Fitness landscapes for ftv55LH.sms instance. In Figure 5, the path-to-optimal is present only in the MA landscape and correlation is medium for both cases. Notice the strong concentration of points below Ttotal = 5000 for the MA and a dangerous “dead-end” feature situated around AD = 50 and Ttotal = 2500. This disconnection in the path may cause the algorithm to miss the optimal sometimes, by trapping the population in this region. For the MS the landscape reveals an extremely difficult instance, with almost all local minima located at Ttotal = 10000+ and AD = 200+. Finding the optimal for such an instance through MS is probably very difficult. MA
MS
Figure 6: Fitness landscapes for ftv70HH.sms instance.
The MA landscape presents a low correlation and a clear evolution path with several disconnected regions with high concentrations of points. This disconnected concentration of points far from the origin can act like “traps”, just as the “dead-end” feature in instance ftv55LH.sms. These traps might divert the search process making the MA miss the optimal sequence, what actually occurred six times. The MS landscape is very poor, with no points below Ttotal = 10000. In this case, we can say that finding the optimal in a reasonably CPU time is extremely improbable. MA
MS
Figure 8: Fitness landscapes for kro124pLH.sms instance. The kro124p-based instances gave the most impressive results, perhaps because of its high values for the sij elements – the interval is [81, 4545]. The LH variant has an optimal tardiness of zero but the MA best tardiness was Ttotal ≈ 40000. That appears to be a very bad result, but considering the initial local minima at Ttotal ≈ 1800000, we can conclude that the MA evolution was eloquent. The correlation is good and the
evolution path is clearly visible, but the optimal solution appears to be out of reach for the method. For the MS the results are even worse. We remark the fact that the scale factor of the tardiness axis is 106 and not 105 as for the MA. Therefore, the concentration of points is above Ttotal ≈ 300000 and the best solution found was Ttotal ≈ 170000. Correlation is good, but again it is virtually impossible to find the optimal using MS. MA
MS
Figure 9: Fitness landscapes for kro124pLS.sms instance. Results for both methods are very similar to those obtained in the LH variant. For the MA the concentration of points in the lower part of the graphic is extremely high. The best solution is around Ttotal ≈ 60000 and the optimal for this instance is 26111. The optimal solution again seems to be out of reach for the method. The MS returned as the best solution Ttotal ≈ 300000. The worst local minima are located at Ttotal ≈ 2200000. MS is also not likely to find the optimal in any reasonable time. Thus, we conclude that the kro124pLH and kro124pLS instances are the most difficult ones and an open challenge for the algorithms presented. 9.
complexity of the methods. The MS is just a local search procedure applied on random initial solutions. The MA, on the other hand, uses the same local search, but also includes all the knowledge of the GAs. In fact, it is very difficult, if not impossible, to make a MS that performs better than a MA in a large variety of instances. And even if it happens, in this case, probably the parameters of the MA are not well tuned. In order to compare the performance for the two methods, a fitness landscape analysis on a subset of the instances was proposed. The fitness landscape sometimes gives an insight on why the algorithms failed and raised two open problems: the LOW-type kro124p problems. For these problems, the optimal solution in known but is virtually unreachable for both methods. Acknowledgments This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq – Brazil) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP – Brazil). The authors wish to thank Luciana Buriol and Shara Amin who provided the original ATSP instances as well as the optimal ATSP tours. We are also grateful to Natasa Dobrasinovic, who developed the software to generate the fitness landscapes figures. References [1]
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Conclusions [2]
This paper treated the Single Machine Scheduling problem with sequence-dependent setup times and due dates. The objective was to minimize the total tardiness. Two approaches were used: Memetic Algorithms and Multiple Start. The SMS instances were generated from ATSP instances with known optimal solutions. Using this procedure, we could know, in advance, the optimal schedule and consequently the optimal tardiness.
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[4]
The computational results show that the MA has a very good performance, reaching the optimal solution in problems up to 100 jobs. The MS, on the other hand, usually fails to find the optimum for more than 56 jobs. This performance gap is a direct consequence of the
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P. Moscato, “On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms”, Caltech Concurrent Computation Program, C3P Report 826, 1989.
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P. Moscato, “An introduction to population approaches for optimization and hierarchical objective functions: a discussion on the role of tabu search”, Annals of Operations Research (41)/1-4, pp. 85-121, 1993.
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[11] TSPLIB web page, available at the URL http://www.crpc.rice.edu/softlib/tsplib/ [12] S. Wright, “The Roles of Mutation, Inbreeding, Crossbreeding and Selection in Evolution”, Proceedings of the 6th International Congress on Genetics (1), pp. 356-366, 1932.
