An improved Evolutionary Algorithm to sequence operations on an ASRS warehouse

Transcription

1 An improved Evolutionary Algorithm to sequence operations on an ASRS warehouse José A. Oliveira, João Ferreira, Guilherme A.B. Pereira and Luis S. Dias Centre ALGORITMI, University of Minho, Portugal WWW home page: Abstract. This paper describes the hybridization of an evolutionary algorithm with a greedy algorithm to solve a job-shop problem with recirculation. We model a real problem that arises within the domain of loads dispatch inside an automatic warehouse. The evolutionary algorithm is based on random key representation. It is very easy to implement and allows the use of conventional genetic operators for combinatorial optimization problems. A greedy algorithm is used to generate active schedules. This constructive algorithm reads the chromosome and decides which operation is scheduled next. This option increases the efficiency of the evolutionary algorithm. The algorithm was tested using some instances of the real problem and computational results are presented. Keywords: Evolutionary algorithms, Job shop scheduling, ASRS warehouse 1 Introduction The efficiency of an automatic storage system depends, among other factors, on the plan for loading operations on trucks. In the AS/RS (Automated Storage and Retrieval System) warehouses, where a large number of truckloads are performed on a daily basis, it is necessary to plan and execute accurately the loading procedures in order to fulfil the delivery deadlines. The issue of AS/RS warehouses has been studied since the 70s, increasing their interest in recent years [1 8]. The problem for sequencing operations in automated warehouses was presented by Oliveira [4, 5], and the problem was modelled as a variant of the job-shop scheduling problem (JSSP). Recently, Figueiredo et al.[8] modelled the problem in which the operations have general processing times and they presented a genetic algorithm to solve the problem. In this paper we describe an evolutionary algorithm that represents an improvement that represents an improvement to the work presented by Figueiredo et al.[8]. In their work the authors present a genetic algorithm to generate nondelay schedules. The authors support their option on the fact that the solution space is shorter than the solution space of active schedules. In this paper we study the hybridization of a Evolutionary Algorithm (EA) with a greedy algorithm to generate active schedules. Although the search space being larger, we show that this EA gets better results, without an increase of CPU time.

2 2 The paper is organized in the following way: next section describes the automatic warehouse type of operations; the third section presents the model adopted, some remarks about its application and also discusses some extensions to the model; the fourth section is dedicated to the characterization of the solution s methodology adopted; the fifth section presents computational results of the developed algorithm; finally the conclusions about the work are discussed. 2 The Storage System The real scheduling problem reported by Oliveira [5] occurs in a AS/SR type automatic warehouse. The AS/RS is formed by several aisles of pallets racks, with capacity about forty thousand pallets. In each aisle operates a stacker crane (also S/R machine). In the retrieval process the S/R machine moves the pallets from their storage position to the collector at the top of the aisle. Next, several forklift trucks move the pallets and place them inside the trucks. According to this process the real problem was modelled as a job-shop scheduling problem with recirculation [4] - that means a job is processed more than once in the same machine. In this case, a load for one truck could receive more than one pallet from the same aisle and the same S/R machine. For details about the warehouse functioning we address the reader to [4, 5]. In the real problem, there are about a hundred loads to dispatch per day (truck loads - also called trips ). The strategy adopted to program the trucks loading consists on defining a partition of the whole load into disjoint subsets of loads, which are processed simultaneously, called batches (blocks). This strategy guarantees that every load of a batch is finished before starting to prepare the loads for the next batch. The batches dimensioning respects the imposed limits, in order to fulfil the delivery deadlines. The number of docking bays limits the maximum number of loads in a batch. A standard workday can originate plans with batches, with 6-13 loads each [8]. Oliveira [5] demonstrated that the time spent to process the batches depends on the number of loads of the batch, and it is calculated based on the nominal values of the system s throughput. It is assumed that the storage system guarantees a nominal flow of pallets dispatch at a constant rate, and therefore it considers that the duration of the batch is proportional to the total number of batches pallets. Figueiredo et al.[8] demonstrated that time to process the batch depends also on the sequence for retrieving the pallets, which is to say, on the S/R machines operations scheduling. Due to precedence constraints between pallets, the order in which the pallets of the batches are retrieved by the S/R machines may lead to different processing times. The problem of establishing the best pallets retrieving operations sequence in each aisle is also an optimization problem, and the authors modelled it as a Job Shop Scheduling Problem (JSSP) with recirculation. In Oliveira [5] there is a detailed description of the process to retrieve the pallets - from the aisle to the truck. Part of the movement is performed by forklifts that are controlled by Radio Frequency. In his work, Oliveira [5] assumes

3 3 identical processing times to transport pallets independently of the location of the aisle and the truck. In this work, a new model considering general processing times is presented, and it takes into account the location of the truck in the docking bays and the aisle where the pallets are retrieved. This model is believed to better representing the real problem, but also turns the associated job shop problem more difficult to solve. 3 Job Shop Scheduling Problem Some Combinatorial Optimization Problems are very hard to solve and therefore require the use of heuristic procedures. One of them is the Job Shop Scheduling Problem. The use of exact methods to solve the JSSP is limited to small size instances. According to Zhang et al.[9] the Branch and Bound methods do not solve instances larger than 250 operations in a reasonable time. The heuristic methods have become very popular and have gained much success in solving job shop scheduling problems. In the last twenty years a huge number of papers has been published, presenting several metaheuristic methods. From Simulated Annealing to Particle Swarm Optimization, there are several variants of the same method class. EAs are very popular among researchers [2, 4, 5, 8, 10 12]. In 1996, Vaessens et al. [13] stated a goal for the Job Shop Problem: to achieve an average error of less than two percent within 1,000 seconds total computation time. In this work the authors presented the Genetic Algorithms as the less effective metaheuristic to solve the JSSP. A possibility to increase the efficiency and the effectiveness is an algorithm that includes specific knowledge of the problem. In the JSSP each job is formed by operations that have to be processed on a set of machines. Each job has a technological definition that determines a specific order to process the job s operations, and it is also necessary to guarantee that there is no overlap, in time, when processing different operations in the same machine; and also that there is no overlap, in time, when processing operations of the same job. The objective of the JSSP is to finish all jobs as soon as possible, that is, to minimize the makespan. The classical JSSP model considers a set of n jobs, and a set of m machines. Each job consists of a set of m operations (one operation on each machine), among which precedence relations exist and that defines a single processing sequence. Each operation is processed in one machine only, during p time units. The instant when the job is concluded is C. All operations are concluded at C max (called makespan). For the convenience of the representation, operations are numbered consecutively from 1 to N, in which N is the total number of operations. The classic model considers that all the jobs are processed once in every machine, and the total number of operations is N =n.m. In a more general model, a job can have a number of operations different from m (number of machines). The case in which a job is processed more than once in the same machine, is called a job shop model with recirculation [4]. The computational and practical significance of JSSP have motivated the attention of researchers for the last several decades.

4 4 The solutions (schedules) for the JSSP can be classified in 3 sets: semi-actives (SA), actives (A) and non-delayed (ND). Typically, these sets have the particularity that A is a proper subset of SA, and ND is a proper subset of A. In relation to the optimal solution of the problem (minimization of C max ), it is known that it is an active schedule but not necessarily a non-delayed schedule. In this work, the solutions obtained from the EA belong to the active set. This choice was taken in order to obtain better solutions. 4 Methodology In this work we adopted a method based on evolutionary algorithms. This technique s simplicity to model complex problems and its easy integration with other optimization methods were factors that were considered for its choice. Traditionally, genetic algorithms used bit string chromosomes. These chromosomes consisted of only 0s and 1s. Modern genetic algorithms more often use problem-specific chromosomes with evidence that the use of real or integer value chromosomes often outperformed bit string chromosomes. The permutation code was adequate to permutation problems. In this kind of representation, the chromosome is a literal of the operations sequence on the machines. For the classical JSSP case, Oliveira [5] presents a chromosome that is composed by m sub-chromosomes, one for each machine, each one composed by n genes, one for each operation. The i gene of the sub-chromosome corresponds to the operation processed in i place in the corresponding machine. The allele identifies the operation s index in the disjunctive graph. Nevertheless, in this work, the random key code presented by Bean [14] is used, attending the easiness to model complex systems with this type of representation, as it is the case to model the loading of trucks in an automatic warehouse. The important feature of random keys is that all offspring formed by the genetic operators are feasible solutions - this occurs when it is used as a constructive procedure based on the available operations to schedule and the priority is given by the random key allele. Another advantage of the random key representation is the possibility of using the conventional genetic operators. This characteristic allows the use of EA with other optimization problems, adapting only a few routines related to the problem. Equally easy becomes the hybridization with other heuristics when a genetic algorithm with random keys is used. Also, it is possible to hybridize with a MIP solver, once the fractional value of a variable could be used as an allele of the random key representation. A chromosome represents a solution to the problem and it is encoded as a vector of random keys (random numbers). In this work, according to Cheng et al. [10], the problem representation is indeed a mix from priority rule-based representation and random keys representation. 4.1 Constructive Algorithm The solutions represented by chromosome are decoded by an algorithm, which is based on Giffler and Thompson s algorithm (GTa)[15]. While the GTa can

5 5 generate all the active plans, the constructive algorithm only generates the plan in agreement with the chromosome. As advantages of this strategy, we pointed out the small dimension of solution space, which includes the optimum solution and the fact that it does not produce impossible or disinteresting solutions from the optimization point of view. On the other hand, since the dimensions between the representation space and the solution space are very different, this option can represent a problem because two chromosomes can represent the same solution. The constructive algorithm has N stages and in each stage an operation is scheduled. To assist the algorithm s presentation, consider the following notation for stage t: P t - the partial schedule of the (t 1) scheduled operations; S t - the set of operations schedulable at stage t, i.e. all the operations that must precede those in S t are in P t ; σ k - the earliest time that operation o k in S t could be started; φ k - the earliest time that operation o k in S t could be finished, that is φ k = σ k +p k and p k is the processing time of operation o k ; M - the selected machine where φ = min ok S t (φ k ); S - the conflict set formed by o k S t processed in M and σ k < φ ; o k - the selected operation to be scheduled at stage t. The constructive algorithm of solutions is presented in Fig. 1 in a similar format to the one used by Cheng et al. [10]. In Step 3 instead of using a priority Step 1 Let t = 1 with P 1 being null. S t will be the set of all operations with no predecessors. Step 2 Find φ = mino k S t { φk} and identify M. Form S t. Step 3 Select operation o t in S t with greatest allele value. Step 4 Move to next stage by (1) adding o t to P t, so creating P t + 1 ; (2) removing o j from S t and creating S t + 1 by adding to S t the operation that directly follows o * t in its job (unless o t completes its job); (3) incrementing t by 1. Step 5 If there are any operations left unscheduled( t < N), go to Step 2. Otherwise, stop. Fig. 1. Algorithm 1 - Constructive algorithm dispatching rule, the information given by the chromosome is used. If the maximum allele value is equal for two or more operations, one is chosen randomly. 4.2 The Evolutionary Algorithm Structure The EA has a very simple structure and can be represented in Algorithm 2, illustrated in Fig. 2. It begins with population generation and her evaluation. Attending to the fitness of the chromosomes the individuals are selected to be

6 6 parents. Comparing the fitness of the new elements and of their progenitors the former population is updated. Step 1 begin Step 2 P GenerateInitialPopulation() Step 3 Evaluate(P) Step 4 while termination conditions not meet do (1) P Select_progenitors(P) (2) P Apply_GeneticOperators(P ) (3) Evaluate(P ) (4) P Select(P P ) Step 5 end while Fig. 2. Algorithm 2 - Evolutionary algorithm On building of the offspring population we implemented an elitist strategy. Firstly, we select the better individuals of the current population to be progenitors, (10%). Then, the remaining parents are chosen by the routines of the roulette wheel (60%) and tournament (30%). So, one individual can be selected more than once to be progenitor. Secondly, we apply the genetic operators over this population according to the following partition: we use the crossover on two parents (40%); the crossover on two parents plus the mutation (30%); and the mutation (30%). The Uniform Crossover (UX) is used in this work. This genetic operator uses a new sequence of random numbers and swaps both progenitors alleles if the random key is greater than a prefixed value. This genetic operator uses a new sequence of random numbers and swaps both progenitors alleles if the random key is greater than a prefixed value. Fig.3 illustrates the UX s application on two parents (prnt1, prnt2), and swaps alleles if the random key is greater or equal to The genes 3 and 4 are changed and it originates two descendants (dscndt1, dscndt2). Descendant 1 is similar to parent 1, because it has about 75% of the genes of this parent. i prnt prnt randkey dscndt dscndt Fig. 3. The UX crossover

7 7 5 Computational Experiments Computational experiments were carried out with some representative instances of the real problem. We use the same representative instances of the real problem presented by Figueiredo et al.[8] that were generated randomly and they concern a job-shop problem with recirculation. The dimension of these instances corresponds to the defined maximum dimension of the real problem. Instances jr 13 1, jr 13 2 are constituted by 13 jobs (docking bay number) and 35 operations per job (one load s dimension), which adds up to a total of 455 operations and 11 machines (number of aisles). The 35 operations of the major part of the jobs are processed in 5 machines. This situation corresponds to the real problem. Usually, the load of a truck (35 pallets) comes from the 5 aisles closer to the docking bay. The recirculation situation happens when a job is processed more than once in the same machine. In the real problem this corresponds to collecting several pallets from the same aisle. Table 1 presents the results of computational experiences, and summarize the results obtained by Figueiredo et al. [8] with non-delay schedules and also the results for our active schedules. Globally an improvement of 1% is obtained by active schedules. Furthermore, for some instances an improvement of 2% and 3% is reached - these instances correspond to greater sizes in terms of number of operations. The algorithm was implemented in C++ and the code was compiled using GNU Compiler Collection (GCC) version The tests were run on a computer Intel i3, with 4 MB of RAM, on the Ubuntu in Windows7 using VirtualBox. The experiments were performed using two different populations - a small one with 20 individuals, and a larger one with 100 individuals. The third column of Table 1 indicates the number of jobs, and column Op. indicates the number of operations. This table presents the best value obtained from 15 runs of each configuration, and the average value of 15 runs. For all experiments 500 iterations were performed. The results are promising, while achieving better solutions in early stages of the optimization process. Also the algorithm proved to be robust and consistent, with similar performances in all experiments considered. With larger populations it is possible to achieve better results since the beginning of the optimization process, although requiring extra CPU time. Despite the code is not yet optimized for calculations speed, a run for the largest instance (jr 13 2), performing 500 iterations takes about 85 seconds with a population size of 100, and about 18 seconds with a population size of 20. Fig. 4 shows these runs - population size 20 and population size 100 for both types of schedules (non delay / active). Fig. 4 also illustrates the advantage of using a larger population (better solutions since the beginning), although this option takes more CPU time. With a population five times bigger the algorithm returns a better result within just 20% of the number of iterations. For this particular run, it is possible to verify that the generation of active schedules outperforms the non-delay schedules generation.

8 8 Table 1. Experimental results NonDelay Schedules Active Schedules Name Pop J Op Best Aver. Worst Best Aver. Worst jr_1_ jr_3_ jr_4_ jr_4_ jr_6_ jr_6_ jr_8_ jr_8_ jr_10_ jr_10_ jr_13_ jr_13_ ND 20ND 100Ac 20Ac Fig. 4. Results for different population sizes

9 9 6 Conclusions This paper presents a model for scheduling load operations in an automatic warehouse and a Evolutionary Algorithm to solve the JSSP with recirculation. The schedules are built using information given by the genetic algorithm to sequence the operations. The algorithm incorporates a constructive algorithm to build the solution from the chromosome, and it always generates feasible plans, and in particular active schedules. This constructive algorithm allows the inclusion of specific knowledge of the problem and makes the algorithm very efficient. This algorithm easily shows the solution for several daily problems that may occur in a warehouse. The algorithm was tested with success on instances of equal dimension of the real problem. The computation time suggests that it is an efficient algorithm, allowing its integration in a decision support system to evaluate different alternatives in useful time. The main advantage in generating active plans is the fact that all schedules are valid in terms of programming the S/R machines, without generating deadlocks. As future developments, we do think it would be beneficial to try using both constructive algorithms. In fact, for some instances the optimal solution is an active schedule, and for other instances it is a non-delay schedule. It is feasible to design an hybrid constructive algorithm. Since the random keys representation has no Lemarkin property it is our intention to develop a variation of random keys representation to avoid this weakness. Acknowledgments. This work was funded by the Programa Operacional Fatores de Competitividade - COMPETE and by the FCT - Fundação para a Ciência e Tecnologia in the scope of the project: FCOMP FEDER References 1. Rashid, M.M., Kasemi, B., Rahman, M.: New Automated Storage and Retrieval System (ASRS) using wireless communications th International Conference on Mechatronics: Integrated Engineering for Industrial and Societal Development, ICOM 11 (2011) 2. Crdenas, J.J., Garcia, A., Romeral, J.L., Andrade, F.: A genetic algorithm approach to optimization of power peaks in an automated warehouse. IECON Proceedings (Industrial Electronics Conference), (2009) 3. Hausman, W.H., Schwarz, L.B., Graves, S.C.: Optimal storage assignment in automatic warehousing systems. Management Science 22, (1976) 4. Oliveira, J.A.: A genetic algorithm with a quasi-local search for the job shop problem with recirculation. In: Applied Soft Computing Technologies: The Challenge of Complexity, pp Springer, Heidelberg (2006) 5. Oliveira, J.A.: Scheduling the truckload operations in automatic warehouses. European Journal of Operational Research 179, (2007) 6. Ko, M., Tiwari, A., Mehnen, J.: A review of soft computing applications in supply chain management. Applied Soft Computing 10, (2010)

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