META 2014, 5th International Conference on Metaheuristics and Nature Inspired Computing, in Marrakech, Morocco, October 27 - 31, 201

González Rodríguez, Inés

Palacios Alonso, Juan José

Puente Peinador, Jorge

Rodríguez Vela, María del Camino

Repositorio Institucional de la Universidad de Oviedo

Genetic Beam Search for Fuzzy Open Shop ProblemsJuan Jose´ Palacios1, Camino R. Vela1, Ine´s Gonza´lez-Rodr´ıguez2, and Jorge Puente11 Dep. of Computer Science, University of Oviedo (Spain)palaciosjuan@uniovi.es, crvela@uniovi.es, puente@uniovi.es2 Dep. of Mathematics, Statistics and Computing, University of Cantabria (Spain)gonzalezri@unican.es1 Motivation and Problem descriptionThe open shop scheduling problem (OSP) is a problem with an increasing presence in the schedulingliterature and with clear applications in industry. It consists in scheduling a set of n jobs to beprocessed on a set ofm machines. Each job consists ofm tasks, each requiring the exclusive use of adifferent machine for its whole processing time without preemption. In total, there are n×m tasks,denoted {oij , 1 ≤ i ≤ n, 1 ≤ j ≤ m}, each one with processing time pij . A solution to this problemis a schedule –an allocation of starting times for all tasks– which is feasible, in the sense that allconstraints hold, and is also optimal according to some criterion. In this paper, the objective will beminimising the makespan Cmax, that is, the time lag from the start of the first task until the end ofthe last one. For a number of machines m ≥ 3, this problem is NP -complete; in consequence, it isusually tackled via metaheuristic techniques. We highlight an ant colony optimisation hybridisedwith beam search in [2]; a particle swarm optimisation algorithm proposed in [7] and a hybridgenetic algorithm used in [1].However, in real-life applications, it is often the case that operation processing times are ill-known, and only some uncertain information is available, for instance, an interval of possibledurations or a most likely duration with a certain error. The approaches to deal with this un-certainty are diverse and, among these, fuzzy sets have been used in a wide variety of ways [4].For instance, uncertain durations are naturally modelled using a Triangular Fuzzy Number (TFN)N = (n1, n2, n3) given by an interval [n1, n3] of possible values and a modal value n2 in it.In the OSP, we essentially need two operations on fuzzy processing times, the sum and the max-imum. These are obtained by extending the operations on real numbers using the Extension Princi-ple. For any pair of TFNsM and N ,M+N = (m1+n1,m2+n2,m3+n3). However, computing themaximum is not that straightforward and the result may not be a TFN. For these reasons, it is usualto approximate the result by a TFN, so max(M,N) ≈ (max(m1, n1),max(m2, n2),max(m3, n3));the approximated and the real maximum have identical support and modal value. The membershipfunction of a fuzzy number can be interpreted as a possibility distribution on the real numbers.This allows to define its expected value, given for a TFN N by E[N ] = 14(n1 + 2n2 + n3). Itcoincides with several defuzzification indices and induces a total ordering ≤E in the set of fuzzyintervals: M ≤E N if and only if E[M ] ≤ E[N ].Here we consider the fuzzy open shop problem (FOSP) with expected makespan minimisation.Some methods have been already proposed for this problem, for instance a particle swarm algorithmin [6] or a lexicographical multiobjective approach in [5]. In the following we propose to hybridisea genetic algorithm with beam search to solve it.2 Genetic Beam SearchBeam search (BS ) is present in the most successful methods for solving the OSP. It was combinedwith a local search and embedded into an ACO algorithm in [2] and also used as part of theevaluation function of a PSO algorithm in [7]. Roughly speaking, BS are breadth-first searchalgorithms where the maximum number of nodes per level is limited by a “beam width” (bw). Thereis also a constant value ext that limits the maximum number of children that a node is allowedto generate. This naive scheme may be enhanced by adding additional features. For instance, anupper bound may be used to prune the tree, thus guiding the search to more promising areas, anda local search can be used at each step, enhancing the exploitation of the algorithm at the cost ofincreasing the computational effort.2 J. Palacios, C.R. Vela, I. Gonza´lez-Rodr´ıguez, J. PuenteFollowing [7] our BS is guided by an operation priority array. Pruning is achieved by using aformula for the lower bound of partial schedules and an upper bound. We include the search intoa genetic algorithm, using it as an evaluation function, which allows to consider different priorityarrays and evolve them. In order to have a good tradeoff between the quality of the solutions andthe computational cost, we set ext = 2 and bw = 2n, where n is the number of jobs of the problem.A critical step in the BS is the choice of a good set of candidate operations to expand a nodeat each iteration. This is not easy in the presence of uncertainty. Here we adapt to the FOSP theconcept of “conflict set” from the well-known G&T algorithm for the job shop problem. Given apartial schedule, let Ω be the set of all unscheduled operations, and for every o ∈ Ω let ESTodenote its earliest possible starting time and Co = ESTo + po its completion time. Then, theminimum completion time is computed as C∗ = {Co : E[Co] ≤ E[Co′ ]∀o, o′ ∈ Ω} and the setof candidate operations is given by ω = {o ∈ Ω : ∃i ∈ {1, 2, 3}, EST io < C∗i}. This set is thenreduced with the “ReduceToRelated” method from [2] and using the best solution found so far bythe genetic algorithm as upper bound. Finally, the node is expanded with the ext tasks having thehighest priority.3 Experimental ResultsWe test the proposed algorithm on the most challenging instances of the benchmark from [5] forthe FOSP. These are 170 fuzzy instances built from the j7 (size 7× 7) and j8 (8× 8) instances ofthe well-known benchmark for the OSP proposed in [3]. A parametric analysis indicates that thealgorithm obtains its best performance using: PMX crossover operator with probability 1, mutationby inversion with probability 0.10, 2/4 tournament between the parents and their offspring asreplacement strategy and random mating, ensuring that all individuals are mated. Regardingconvergence, 300 iterations are needed for the j7 instances and 500 for the j8 ones, with averageruntimes of 68 and 202 seconds respectively.We compare the results obtained after 30 runs of our method with those obtained with thelexicographical approach (Lex) in [6] and the PSO algorithm in [5], which are as far as we know,the best for the FOSP. The comparison is made in terms of relative errors (RE ) with respect tothe lower bounds specified in [5]. For the j7 instances, RE values are reduced in average 36% w.r.t.PSO and 34% w.r.t Lex, being these percentages 26% and 24% in the case of the j8 instances.For the sake of completeness, we run our algorithm to solve also the deterministic instances.Even though the algorithm is not specifically designed for crisp problems, it performs quite well onthem compared with the state-of-the-art algorithms: the Beam-ACO from [2], the PSO from [7]and the HGAOS from [1]. For the j7 instances, we obtain an average RE of 2.6%, being 2.5%,2.3% and 2.7% respectively for the mentioned algorithms. In the case of j8 instances, this value isreduced to 2.1% in our approach, and to 2.0%, 1.7% and 2.6% for the other methods. Althoughit is not the best, our method seems to be quite competitive. Aditionally, it has found a new bestsolution for the j8-per10-0 instance with a makespan of 1019.References1. F. Ahmadizar and M. Farahani. A novel hybrid genetic algorithm for the open shop scheduling problem.International Journal of Advanced Manufacturing Technology, 62:775–787, 2012.2. C. Blum. Beam-ACO—hybridizing ant colony optimization with beam search: an application to openshop scheduling. Computers & Operations Research, 32(6):1565–1591, 2005.3. P. Brucker, J. Hunrink, B. Jurisch, and B. Wo¨stmann. A branch & bound algorithm for the open-shopproblem. Discrete Applied Mathematics, 76:43–59, 1997.4. D. Dubois, H. Fargier, and P. Fortemps. Fuzzy scheduling: Modelling flexible constraints vs. copingwith incomplete knowledge. European Journal of Operational Research, 147:231–252, 2003.5. J.J. Palacios, I. Gonza´lez-Rodr´ıguez, C. R Vela, and J. Puente. Swarm lexicographic goal programmingfor fuzzy open shop scheduling. Journal of Intelligent Manufacturing, 2013.6. J.J. Palacios, I. Gonza´lez-Rodr´ıguez, C. R Vela, and J. Puente. Robust swarm optimisation for fuzzyopen shop scheduling. Natural Computing, 2014.7. D Y. Sha and H. Cheng-Yu. A new particle swarm optimization for the open shop scheduling problem.Computers & Operations Research, 35:3243–3261, 2008.

Genetic beam search for fuzzy open shop problems

http://digibuo.uniovi.es/dspace/bitstream/10651/33979/1/Palacios2014_META.pdf

Genetic beam search for fuzzy open shop problems

Abstract

Similar works

Full text

Available Versions

Repositorio Institucional de la Universidad de Oviedo