VOL. 3, NO. 8 Aug, 2012 ISSN Journal of Emerging Trends in Computing and Information Sciences CIS Journal. All rights reserved.

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1 Job Shop Scheduling using ACO Meta-heuristic with Waiting_Time-based Pheromone Updating Elena Simona Nicoară Petroleum-Gas University of Ploieşti, Information Technology, Mathematics and Physics Department, 39 Bucharest Blvd., Ploieşti, , Romania ABSTRACT In the vast optimization field, many computer-aided techniques were proposed and tested in the last decades. The artificial intelligence meta-heuristics constitute the widest part of such techniques, which proved to be adequate to (near) optimally solve big difficult instances, as the most real optimization problems are. Among them, the agent-based techniques are the most recent ones and they reported in the literature very good results compared to many other optimization methods. Such methods are: Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Wasp Behavior Model (WBM) and negotiation techniques. In this paper a research study on ACO applicability to Job Shop Scheduling Problems (JSSP) is reported and a waiting timebased pheromone updating formula is proposed. This is tested on a simple JSSP case study using job list representation. The results show that ACO is able to optimally solve JSS optimization problems. Moreover, ACO is a meta-heuristic relatively easy to apply and has a wide optimization scope for static combinatorial optimization problems. Keywords: Ant Colony Optimization, job shop scheduling problem, optimization, meta-heuristic. 1. INTRODUCTION In order to satisfy the efficiency requirements, the fundamental activities in the most work branches (industry, farming, services etc.) are based on optimization. The vast theory developed in mathematical optimization field, starting with Fermat and Lagrange in the XVII-XVIII th centuries, was intensively used in more and more complex computer programs to solve many big optimization problems in the real world. One of the critical optimization problems (once the instance s dimensions become very big and the time-to-solve requirements become very tight) is Job Shop Scheduling Problem (JSSP). In JSSP, a list of heterogeneous jobs, formed by a number of operations, need to be optimally put on work (scheduled) on several machines; in other words, all the jobs must be sequenced such that the make span is minimum, while three cumulative constraints are satisfied: - the precedence constraint: for two consecutive operations of a job, the successor operation must be processed only after the first one ended; - the non-preemption constraint: an operation, once started, cannot be interrupted to be continued later; - the resource capacity constraint: a machine processes one and only one operation at a time and an operation is processed by at most a machine at a time. For every job, an ordered sequence of the operations is set and processing times of the operations on corresponding machines are known [1]. This optimization context is very frequent in manufacturing, packing, computer operation systems, human staff scheduling on work posts and so on. In manufacturing, for example, a production plan must be transformed in an (optimal) schedule; this means that the list of jobs must be sequenced and to each job must assign the time stamps when it begins processing by all the corresponding machine. This sequence plus all the time stamps is the result of JSSP, named schedule. The performance measure of a schedule, named make span, is computed as the total time to complete processing all the operations on the corresponding machines. A feasible schedule is a schedule that satisfies all the specified constraints, and an optimal schedule is a feasible schedule with a minimal makes span. To analyze the optimization techniques adequate to solve JSSPs, one must consider that JSSP is a combinatorial optimization problem. The combinatorial optimization handles optimization problems where the set of feasible solutions is discrete or can be reduced to such a set, each solution has associated a numeric cost and the objective is to identify the solution with minimum cost (the best solution). To solve combinatorial optimization problems in general and JSSPs in particular, many computer-based techniques were proposed and tested: exact optimization techniques ([2,3]), priority dispatch rule systems ([4,2,5]), shifting bottleneck heuristic ([6]) and many meta-heuristics: GRASP (Greedy Randomized Adaptive Search Procedure [7,8]), simulated annealing ([9,10,5]), tabu search ([11,12]), genetic algorithms ([13,14,15,16,17]), expert systems, artificial neural networks ([18]), fuzzy logic, Ant Colony Optimization([19,20,21,22]), Particle Swarm Optimization ([23]), Wasp Behavior Model ([24,21]), negotiation techniques ([5]) and some hybrid methods. Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Wasp Behavior Model and the negotiation techniques are agent-based techniques, named also multi-agent techniques, see Figure 1. Characteristic to these methods is the fact that the colony, as a whole, proves to be an intelligent functional entity even the individual agents have a 1265

2 reduced intelligence. The first three methods were inspired from nature; they simulate natural social behaviors (ants behavior, wasps behavior, birds and fish behavior) in finding an optimal route or an optimal tasks achievement. higher pheromone concentration (deposited on the routes already covered). This cooperative interaction leads to the shortest route in finding food. In other words, pheromone accumulates fast on the shortest route, which is used by the most ants. To apply ACO to a combinatorial optimization problem, first of all a specific-to-problem graph is built. Many artificiel ants (agents) simultaneously start building solutions of the problem adding step by step components (nodes of the graph) to partial solutions. The construction of the solutions is probabilistic, based on heuristic information (if available) and on pheromone trails associated to nodes already visited in the graph. The pheromone for the components is updated after each step, when all the agents added a new component to the list of visited nodes. When updates pheromone for the components of the graph, ACO takes into account the evaporation rate and the quality of the current partial solutions which selected these components. Fig 1: Multi-agent techniques applied in scheduling The agent technology become attractive to model and solve scheduling problems because the main entities in the scheduling system (machines, jobs, scheduling line etc.) can be autonomous agents able to communicate with other agents by sending messages in order to complete the initial plan. The multi-agent techniques used in scheduling are self-organizing collective systems, which operate with two types of agents: job-agents and machine-agents, and the communication between agents is either cooperative or competitive. Each of the multi-agent techniques uses specific mechanisms to identify (near) optimal solution(s). 2. ACO META-HEURISTIC APPLIED TO JSSP An ant colony is an example of natural, highly distributed, multi-agent system, composed by hundreds or thousands of simple autonomous agents, robust to loss of individual agents and to environmental changes. The colony, as a whole, works by coordinating the activities of agents (searching for food, building the nest, feeding the broods etc.) without a direct communication between the members. The only communication is based on pheromone trails and is an indirect communication. The performance of the activities proves to be very efficient, sometimes even optimal ([21]). The first optimization system in artificial ants colonies, Ant System, was proposed in 1992 by Marco Dorigo as an artificial intelligence technique and was successfully applied to traveling salesman problem. Starting with 1995, researchers as Dorigo, Gambardella, Stützle and Di Caro ([19,20,22]) extended Ant System to Ant Colony System, MAX-MIN Ant System and ACO. A comprehensive study on ant colony methods is [25]. ACO simulates the group behavior of ants regarding searching for food; when the ants decide on a search direction, they choose with higher probability the routes marked with a Pheromone trails evaporation means decreasing in time of the intensity of the pheromone on the components and is used to avoid a fast convergence towards suboptimal regions of the search space. At every step, an agent selects from the feasible neighborhood: - with a probability α the node with the highest pheromone ; - with probability 1-α a random node. The pheromone in ACO method is therefore the main guiding mechanism towards qualitative solutions, translating the experience gained by the agents in searching optimal solution. Applied to JSSP, ACO method uses a graph of jobs or a graph of operations to be scheduled. The pheromone Associates to a node is regularly determined by scheduling preferences of the job/operation associated to that node at the current position in the partial schedule [25]. The standard ACO algorithm, using a single generation, for scheduling problems is the following: 1. Initialization 1.1.all the machines are available 1.2.for every component do pheromone trail parameters setting (number of agents, evaporation rate,α) 1.4.for every agent a do schedule[a] 1.5.the best current schedule, S 0 null 2. While exist agents who did not finish the schedule do 2.1.for every agent do if agent did not finish the schedule then choose the next component of the schedule using pheromone trails 1266

3 if agent did finish the schedule and it is better than S 0 then S 0 this schedule 2.2.for every agent do update pheromone trails for the last component added 3. Return S0 The parameters for the ACO algorithm are: - Na: the number of agents (artificial ants) launched in the graph; - er: pheromone evaporation rate (er (0,1)); - α: pheromone trail influence in choosing a component in graph to visit (α (0,1)). Machine 1 Machine 2 Machine 0 Machine 3 To note that N a specifies also the number of simultaneously built solutions. 3. AN ACO MODEL FOR JSSP, WITH WAITING_TIME-BASED PHEROMONE UPDATING RULE In this section we present the waiting_time-based pheromone updating rule and ACO algorithm implementation on a JSSP case study in manufacturing. The input data of JSSP are as follows: - a finite set M of m machines; - a finite set J of N types jobs; i consists in an ordered sequence of n i - each job J operations, ( i, j), j = 1, ; n i - for each operation ( i, j) ( i J, j = 1, ni ), the machine which perform it, k, and the processing time on it k τ Z. i, j A solution of JSSP is a feasible schedule, namely an ordered sequence of the operations which satisfies all the constraints. In the considered JSSP instance three different types of products are produced in a number of batches (see Table 1), on 5 machines (see Figure 2 and Table 2); the scheduling horizon is one week. In the JSSP formulation, every batch in every type of product corresponds to a job, and the production phases to complete a batch correspond to operations of that job. Table 1: Product types distribution on batches (jobs) Poduct type No. batches (jobs) Jobs indices 1 3 1, 2, , 5, Total no. of jobs = 7 In Figure 1 the technological sequences of operations on the machines is provided (0,1,2,4 and 0,3,4). Fig 2: The technological sequences of operations on the machines Table 2: Input data for the considered JSSP instance Product type No. jobs Machine 4 No. op. Routings of the jobs on the machines -machine -processing time (min.) We use a job list representation for the schedule solution, where the following principle is used: a job, once ready to process, all its operations are executed in the imposed order on the corresponding machines, at the earliest time when these machines are available and the previous operations were ended. Therefore, any permutation of the jobs set is a valid sequence of jobs, and the start times associated to the operations in the jobs in the list are set according to the principle to the next non-processed operation, the necessary resource is assigned once it becomes available. In other words, a semi-active decoding procedure is applied, where no operation can be started earlier without modifying the processing order or violating the technological constraints [16]. To solve the JSSP instance with ACO algorithm, we need to design the graph where every agent (artificial ant) builds a schedule step by step. To each node in the graph we associate a job to be scheduled; consequently, the graph has 7 nodes and to each node we associate the pheromone value, ph i, i=1,..,7, initially 0. Two (feasible) solutions built by two different agents in this graph are: (5,3,1,2,6,4,7) and (6,3,4,2,7,1,5) depicted in Figure 3 by the continuous and interrupted lines, respectively. 1267

4 Job 3, ph 3 Job 4, ph 4 operations in jobs already in s finish processing, following the JSSP constraints. Job 1, ph 1 Job 2, ph 2 Job 6, ph 6 Job 5, ph 5 Job 7, ph 7 In relation (2), lower the makespan(s) for a given blind_sum(s), bigger the quality of the partial solution. In the used ACO implementation, an ant is forced to exit the graph, even it did not complete the list of jobs, if the makespan for the current (partial) built solution exceeds a deadline of 2400 minutes (5 days * 8 hours * 60 minutes). Therefore, the colony is informed on the poor solutions also, because the components in those routes (partial schedules) do not get extra pheromone. 4. RESULTS AND DISCUSSION The ACO algorithm was run for eight combinations of parameters values, as Table 3 shows. The best identified solutions are (5,4,2,7,1,3,6), (6,4,2,1,5,7,3), (6,5,3,7,2,1,4) and (5,6,3,7,1,2,4) with makespan 2000 minutes. Fig 3: Two solutions built by two agents in ACO model The solution of the problem is a (near) optimal schedule of jobs on the machines in the given scheduling horizon: the job sequence and the start times for the operations. At every step of the ACO algorithm, see pseudocode, every artificial ant adds to it a non-selected job in the graph, based on pheromone trails, according to the probabilistic rule mentioned in the previous section. At the end of each step, when all the agents added a component, for every component c added by one or many (partial) solution(s) s, the pheromone on that component is updated with relation: Pheromone (c)= pheromone(c)*(1-er)+quality(s), (1) where er is the evaporation rate and quality(s) is the current performance of the partial solution s. Based on this, the colony is informed about components which are part of good partial solutions, in order to guide the search of other agents in future steps. The quality of the partial solution s is determined according to an indicator of waiting time between operations in s. If the jobs already scheduled in s are waiting little time between operations, then the solution has a good quality. If the jobs are waiting big time between operations, then the solution has a poor quality. To determine this indicator for the waiting time of partial solution s, we use the relation: quality(s)= blind_sum(s) / makespan(s). (2) Here, blind_sum(s) is the total processing time of the operations of jobs in s. This is fixed, once set the input data of the problem. For example, for the partial solution s = (7,5,4), blind_sum(s) = ( ) + 2*( ) = 930 min. makespan (s) value is the time when all the Table 3: Results of eight ACO runs Parameters values Bestsolution Makespan 10,0.4,0.8 (5,2,7,4,1,6,3) i ifi 2050 ( ) 10,0.4,0.5 (5,4,2,7,1,3,6) ,0.4,0.5 (6,4,2,1,5,7,3) ,0.6,0.2 (7,2,5,3,1,4,6) ,0.6,0.2 (2,4,1,3,6,5,7) ,0.1,0.8 (6,5,3,7,2,1,4) ,0.4,0.5 (5,6,3,7,1,2,4) 2000 For example, the solution (6,5,3,7,2,1,4) is decoded as lists of start times for operations as follows: (6-0,60,100) (5-60,120,250) (3-120,180,270,400) (7-180,270,360,850) (2-240,340,560,950) (1-300,430,650,1400) (4-360,420,1850) This means that job 6 start process first operation at time 0, the second operation at time 60 and so on. Then, job 5 start process first operations at time 60, the second operation at time 120 and so on. First of all, these results indicate that the proposed ACO model is able to identify good solutions: all the obtained solutions have makespan better than 2400 minutes deadline. Better results were obtained for bigger values of α parameter. This fact proves that the pheromone information do guide in an efficient way choosing the next components to add to the partial solutions. The instance being a small one, the number of agents does not influence in a significant way the results. 1268

5 An important aspect in ACO is that the agents move independently in the graph and every agent is complex enough to find a solution (a schedule for the operations), but this is regularly poor. Good solutions emerge as a result of the collective interaction between agents, by the pheromone - based communication. This is a distributed learning process where the individual agents are not adaptive, but they adaptively modify how is perceived the process by the other agents. On the final quality of the solutions, the individual influence of an agent is not relevant, but the colony influence is major [19]. The advantages of using ACO technique are manifold. Significant are the following: - the stochastic component of the algorithm allows the agents to build various different solutions and hence, ACO explores a wider space than greedy techniques; - the heuristic information, if available, guides the agents towards the most promising solutions; - the agents experience influences building the solutions at next iterations; - using an entire colony of agents gives robustness to the solution, and the collective interaction of agents leads to efficiency. problem, Operations Research Letters, vol.8, pp , [8] R.M. Aiex et al., Parallel GRASP with path-relinking for job shop scheduling, Parallel Computing, vol. 29, pp , [9] S. Kirkpatrick et al., Optimization by simulated annealing, Science, vol. 220, pp , [10] E.H.L. Aarts et al., Job-shop scheduling by simulated annealing, Operations Research, vol. 40, pp , [11] F. Glover, Tabu Search: A Tutorial, Interfaces, vol. 20, pp , [12] E. Nowicki and C. Smutnicki, A Fast Taboo Search algorithm for the Job Shop Problem, Management Science, vol. 42, pp , [13] L. Davis, Job shop scheduling with genetic algorithms, in Proceedings of the International Conference on Genetic Algorithms and their Applications, San Mateo, 1985, Morgan Kaufmann, pp Nevertheless, when apply ACO to big complex problems, not an easy task is setting adequate values for the parameters. REFERENCES [1] A.S. Jain and S. Meeran, A State-of-the-Art Review of Job-Shop Scheduling Techniques, in European Journal of Operations Research 113, 1999, pp [2] B. Giffler and G. L. Thompson, Algorithms for Solving Production Scheduling Problems, in Operations Research 8(4), 1960, pp [3] J. Carlier, E., Pinson, An Algorithm for Solving the Job Shop Problem, in Management Science 35(29), 1989, pp [4] J.R. Jackson, Scheduling a Production Line to Minimize Maximum Tardiness, Management Sciences Research Project, UCLA, Research Report No. 43., [5] M. L. Pinedo, Scheduling. Theory, Algorithms, and Systems, 3rd ed., Springer Science-Business Media, LLC, New York, [6] J. Adams et al., The shifting bottleneck procedure for job-shop scheduling, Management Science vol. 34, pp , [7] T.A. Feo and M.G.C. Resende, A probabilistic heuristic for a computationally difficult set covering [14] R. Nakano and T. Yamada, Conventional genetic algorithms for job-shop problems, in Proceedings of the 4th International Conference on Genetic Algorithms, San Diego, California, 1991, pp [15] J. Bean, Genetics and RandomKeys for Sequencing and Optimization, ORSA Journal of Computing, vol. 6, pp , [16] C. Bierwirth, D.C. Mattfeld, Production scheduling and rescheduling with genetic algorithms, Evolutionary Computation, vol. 7, pp. 1-17, [17] E.S. Nicoară, GA-based Control of Multi-objective Flexible Job Shop Scheduling Processes (in romanian), Ph.D. Dissertation, Informatics Dept., Petroleum-Gas University of Ploieşti, Ploiesti, Romania, [18] L. Rabelo, Hybrid Artificial Neural Networks and Knowledge-Based Expert Systems Approach to Flexible Manufacturing System Scheduling, PhD. Dissertation, University of Missouri-Rolla, [19] M. Dorigo and G. Di Caro, The Ant Colony Optimization Meta-Heuristic, in New Ideas in Optimization, D. Corne, M. Dorigo and F. Glover, editors, McGraw-Hill, 1999, pp [20] M. Dorigo et al., Ant Algorithms for Discrete Optimization, Artificial Life, vol. 5, pp ,

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