PSO and ACO based approach for solving combinatorial Fuzzy Job Shop Scheduling Problem

Transcription

1 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 and based approach for solving combinatorial Fuzzy Job Shop Scheduling Problem Surekha P Research Scholar PSG College of Technology, Coimbatore, India surekha_000@yahoo.com Dr.S.Sumathi Asst. Professor PSG College of Technology, Coimbatore, India ss.author@gmail.com Abstract This paper proposes a prominent approach to solve job shop scheduling problem based on Particle Swarm Optimization () and Ant Colony Optimization (). The steps to generate the solution are grouped as planning, scheduling and optimization. Initially, fuzzy logic is applied for planning and then the scheduling stage is optimized using and. The processing order of jobs for each machine is scheduled with an objective to find a feasible plan that minimizes the makespan, completion time and waiting time. The well known Fisher and Thompson 0x0 instance (FT0) and Adams, Balas, and Zawack 0x0 instance (ABZ0) problems are selected as the experimental benchmark problems. The results of the applied optimization techniques are compared with the computed parameters like makespan, waiting time, completion time and elapse time. The performance evaluation of optimization techniques are analysed for both benchmark problems and the technique is found superior. Keywords Planning, Scheduling, makespan, completion time, waiting time, particle swarm optimization, ant colony optimization. Introduction Job Shop Scheduling Problem (JSSP) is a famous combinatorial optimization problem, which is used in complex equipment manufacturing system to validate the performance of heuristic algorithms. The research on JSSP not only promotes the development of relative algorithms in the field of artificial intelligence, but also provides means of solutions and applications for complex JSSP []. JSSP can be thought of as the allocation of resources over a specified time to perform a predetermined collection of tasks. Job shop scheduling has received a large amount of attention, because it has the potential to dramatically decrease costs and increase throughput, thereby, increasing the profits in automation industries. An eminent approach based on the paradigms of computational intelligence such as Particle Swarm Optimization (), and Ant colony Optimization () for solving job shop scheduling problem is proposed to optimize the feasible schedule with minimum makespan., in contrast to evolutionary computation paradigms such as genetic algorithm, a swarm is similar to a population, while a particle is similar to an individual. As a kind of optimization algorithm based on iteration, particle swarm optimization () algorithm has some advantages (such as convergent, robust etc.) on obtaining dynamic object excellent solutions, and it is suitable for the research of the population behavior []. The Ant Colony System (ACS) algorithm is a distributed algorithm which is extensively used to solve NP-hard combinatorial optimization problems. Its original model is based on the foraging behavior of real ants who find an approximately shortest way to the food by detecting the density of pheromone deposited on the route []. Pheromone for the real ants is a chemical substance deposited by ants as they walk, but while solving optimization problems it acts as something that lures the artificial ants. The paper is organized as follows: Section gives a brief introduction about job shop scheduling problem and Section describes an application of fuzzy logic for planning and scheduling. An overview of optimization techniques such as Particle Swarm Optimization and Ant colony Optimization, need for optimization and solving job shop scheduling problem using the optimization techniques is given in Section. Section covers comparative analysis using the computational intelligence algorithms for two benchmark problems. Section deals with the conclusion and future work of solving job shop scheduling problem using computational intelligence techniques.. Job Shop Scheduling Problem Job-shop is a system that processes n number of tasks on m number of machines. The total ordering defines a set of precedence constraints, meaning that no activity can begin execution until the activity that immediately precedes it in the complete ordering has finished execution []. Each of the m activities in a single job requires exclusive use of the resources defined in the problem. No activities that require the same resource can overlap in their execution and once an activity is started it must be executed for its entire duration. In this type of environment, products are made to order in a low volume basis. Usually, these orders differ in

2 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 terms of processing requirements, materials needed, processing time, processing sequence and setup times. Jobshop problems are widely known as a NP-hard problem and commonly defined as a set of jobs whose operations are to be processed in an uninterrupted manner on a given machine for a specified length of time. There is an additional constraint that each machine can process at most one operation at a time. In the JSSP, a finite set of jobs is processed on a finite set of machines. Each job follows a predefined machine order and has a deterministic processing time. Each machine can process at most one job at a time, which cannot be interrupted until its completion. The scheduling is to arrange the processing order of jobs for each machine with an objective to find a feasible schedule that minimizes the makespan, completion time and waiting time. The job-shop scheduling problem (JSSP) [] can be described as a set of n jobs denoted by J j where j =,... n. It has to be processed on a set of m machines denoted by k where k =, m. Operation of j th job on the k th machine will be denoted by O jk with the processing time p jk. Fig.. illustrates the general flow of the job shop scheduling process. An operation that has to be performed from the job is selected for scheduling. Before scheduling the possible assignments for the operation have to be performed. This assignment includes the expected time for the operation to commence, thereby assuring the priority and the resource constraints based on the operations that have been scheduled previously. If there is no conflict in the constraints and if all the constraints are satisfied then the operation is scheduled. If there is a conflict, then the scheduling is done and the procedure goes back to the most recently scheduled operation testing a new value of its domain. Fig Flow diagram of JSSP The processing order on each machine that minimizes the corresponding cost is desired by the objectives such as minimization of process cost, makespan and flow time or maximization of throughput, systems/resource utilization and production rate. There are some assumptions made in solving job shop scheduling problem. They are A finite set of n jobs. Each job consists of a chain of operations. A finite set of m machines. Each machine can handle at most one operation at a time. Each operation needs to be processed during an uninterrupted period of a given length on a given machine. To find a reasonable schedule, the operations are allotted to machines based on minimal time interval length. Each job is composed of a sequence of operations and the operation order on the machines is pre-specified. Each operation is characterized by the required machine and the fixed processing time. The process plan specifies the routing, processing times and precedence constraints among operations of each job. There are several constraints on jobs and machines: A job does not visit the same machines twice There are no precedence constraints among operations of different jobs An operation cannot be preempted Each machine can process only one job at a time Neither release times nor due dates are specified There are no machine breakdowns throughout the scheduling process. The job-shop is static and deterministic in nature i.e., there is no randomness as the jobs, machine, processing times and all other parameters necessary for defining the job-shop problem are known and fixed.. Planning and Scheduling There are traditionally three kinds of approaches to job shop scheduling problems: priority rules, combinatorial optimization and constraint analysis. The first kind of method has the merit of being computationally very efficient and can be easily applied to real world cases. Surveys on performance analysis of priority rules can be found in the work done by Blackstone et al. (), ontazeri (0), and Grabot and Geneste (). Developing a good predictive schedule that satisfies temporal, technological and other types of constraints is basically a search problem, the solution of which requires both powerful search heuristics and adequate means of representation. However, while scheduling over extended horizons, considering temporal constraints, such as job release date and due date, as compulsory, may lead to rejecting an efficient schedule even when the violation of these constraints is insignificant with regard to the precision of the realistic limits of predictability. A large computation effort may be actually saved by avoiding failures with problems whose lack of feasibility is due to insignificant constraint violations. The computation effort can be actually saved by performing a proper planning strategy. The procedure for planning using fuzzy logic is shown as a flowchart in Fig.

3 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 time is in between 0 to then the linguistic variable is SPT, if the processing time is between to 0 then it is PT and it is LPT when it is in between to 00. For scheduling, the job is aligned in the order for machines with the data available in the benchmark instances. The time taken by a job to wait in a queue is calculated as waiting time. The sum of processing time and waiting time results in the completion time. The jobs aligned to each machines with respect to levels are evaluated to show the job sequence ordering according to the given benchmark instance. The job sequence after scheduling is optimized to rearrange the job sequence so that less waiting time and minimum makespan are obtained. Once the priority rules are evaluated and the jobs are scheduled then the system is prepared for optimizing the job sequence. Fig Planning using FL.. Fuzzy Parameters and Rule Set Fuzzy logic is used to implement the planning and scheduling of jobs. The fuzzy parameters used for assigning jobs to machine with respect to their processing time are tabulated in Table. Here numbers of input parameters are jobs and levels and output parameter is processing time. Ten jobs and ten machines is taken as test samples and processed for planning and scheduling from the benchmark instances. Table Fuzzy Parameters Parameters Value Input variables Jobs and levels Output variable Processing time No. of jobs 0 No. of levels 0 No. of machines 0 Type of membership Triangular function No. of rule base formed 00 Based on the processing time fuzzy logic is used to group the crisp values in-terms of linguistic variables like short processing time (SPT), medium processing time (PT) and long processing time (LPT). From the FT0 benchmark problem, the priority rules are formed based on the processing time as shown in Table. If the processing time is in between sec to 0sec then the linguistic variable is assigned as SPT, if the processing time is between sec to 0sec then it is PT and it is LPT when it lies in between to 00. In the case of ABZ0 (Table ), if the processing Table Fuzzy Rule Base for FT0 Levels /Jobs 0 J S L S S S S J L L S S L S J L L S L L S S L S J L L L L S L L S J S S S S S L J L S L L S S J S S S S S L S J S L L S L S S L J L L L S L S L J0 L S S L L Table Fuzzy Rule base for ABZ0 Levels /Jobs 0 J L L L L L L L J S L L L L S J L S L S S L L S J L S L S S J L L L L L S L J L L S L L S L J S L L L L L L S J L L S L L S L J L L L L S L L J0 S S L S L S L S L. Optimization Techniques This section discusses the optimization techniques based on computational intelligence techniques like Particle Swarm Optimization (), and Ant Colony Optimization () used to solve JSSP... Particle Swarm Optimization The particle swarm optimization algorithm was first proposed in by James Kennedy and Russell C. Eberhart based on the behavior of a swarm of bees for solving optimization problems [0]. is a method for optimizing hard numerical functions on metaphor of social behavior of flocks of birds and schools of fish. The original algorithm is discovered through simplified social model simulation. It was first designed to simulate birds seeking food which is defined as a cornfield vector. The bird would find food through social cooperation with other birds around it and expanded to multidimensional search.

4 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 In algorithm, each particle in the swarm represents a solution to the problem and it is defined with its position and velocity. In D-dimensional search space, the position of the ith particle can be represented by a D-dimensional vector, X i = (X i,, X id,, X id ). The velocity of the particle v i can be represented by another D-dimensional vector V i = (V i,, V id,, V id ). The best position visited by the i th particle is denoted as P i =(P i,,p id,,p id ), and P g as the index of the particle visited the best position in the swarm, then P g becomes the best solution found so far, and the velocity of the particle and its new position will be determined according to the following equations. v ( t ) wv x i j i j * cr ( x x ) cr ( x j x ) ( t ) x v ( t ) where w is called as the inertia factor, r and r are the random numbers, which are used to maintain the diversity of the population, and are uniformly distributed in the interval [0,] for the j-th dimension of the i-th particle. c is a positive constant, called as coefficient of the selfrecognition component, c is a positive constant, called as coefficient of the social component. From Equation (.), a particle decides where to move next, considering its own experience, which is the memory of its best past position, and the experience of its most successful particle in the swarm. In the particle swarm model, the particle searches the solutions in the problem space with a range [ s, s] (If the range is not symmetrical, it can be translated to the corresponding symmetrical range.) In order to guide the particles effectively in the search space, the maximum moving distance during one iteration must be clamped in between the maximum velocity [ v max, v max ] given in Equation: v sign( v )min( v, vmax ) The value of v max is p s, with 0. p.0 and is usually chosen to be s, i.e. p =. The parameter w in equation (.) is inertia weight that increases the overall performance of. It is reported that a larger value of W can favor higher ability for global search while lower value of W implies a higher ability for local research. To achieve a higher performance, the value of inertia weight W is linearly decreased over the generations to favor global research in initial generations and local research in the later generations. The linearly deceased value of inertia is based on the following equation. wmax wmin W Wmax iter * iter max where iter max is the maximum of iteration in evolution process, W max is maximum value of inertia weight, W min is the minimum value of inertia weight, and iter is current value of iteration. The algorithm of used for optimizing Job Shop Scheduling problem is summarized as follows: * Step : Initialize a population of particles (jobs) with random positions (job sequences) and velocities (processing time), where each particle contains variables (set of operations). Step : Evaluate the objective values of all particles. Let p best of each particle and its objective value be equal to its current position and objective value, and let g best and its objective value is equal to the position and objective value of the best initial particle. Step : Update the velocity and position of each particle Step : Evaluate the objective values of all particles. Step : For each particle, compare its current objective value with the objective value of its p best. If the current value is better, then update p best and its objective value with the current position and objective value. Step : Determine the best particle of the current swarm with the best objective value. If the objective value is better than the objective value of g best, then update g best and its objective value with the position and objective value of the current best particle. Step : If a stopping criterion is met, then output g best and its objective value; otherwise, go back to Step... Ant Colony Optimization The algorithm is developed using artificial ants. The artificial ants are designed based on the behavior of real ants. They lay pheromone trails on the graph edges and choose their path with respect to probabilities that depend on pheromone trails and these pheromone trails progressively decrease by evaporation. At each generation, each ant generates a complete tour by choosing the nodes according to a probabilistic state transition rule. Every ant selects the nodes in the order in which they will appear in the permutation []. For the selection of a node, an ant uses a heuristic factor as well as pheromone factor. The heuristic factor, denoted by η, and the pheromone factor, denoted by, are indicators of how good it seems to have node j at node i of the permutation. The heuristic value is generated by some problem dependent heuristics whereas the pheromone factor stems from former ants that have found good solution. The next node is chosen by an ant according to the following rule that has been called pseudo random proportional action choice rule. With probability q 0, where 0 q 0 <I is a parameter of the algorithm, the ant chooses a node from the set of nodes (s) that have not been selected so for which maximizes ( ) ( ) where α 0 and β 0 are constants that determine the relative influence of the pheromone values and the heuristic values on the decision of the ant. With probability (I q 0 ) the next node is chosen from the set S according to the probability distribution that is determined by P ( ) hs ( ) ( ) ( )

5 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 Therefore the transition probability is a trade-off between the heuristic and pheromone factor. For the heuristic factor, the close nodes (low cost of path) should be chosen with high probability, thus implementing greedy constructive heuristic. As the pheromone factor, if on an edge (i, j) there has been a lot of traffic then it is highly desirable, thus implementing the autocatalytic process. If the selection between heuristic and pheromone factor results in a P, then carry out the rule to find heuristic factor. In the heuristic factor η is computed according to the following rule,, j S F( X ) j where F(X) represents the cost function of X. The edges which are shorter posses low cost and a greater amount of pheromone deposit. amount of pheromone on the edge is modified again by applying the global updating rule ( ), where L, if this is the best tour, otherwise 0, and L indicates the length of the globally best tour. The pheromone updating rule was meant to simulate the change in the amount of pheromone due to both the addition of new pheromone deposited by ants on the visited edges and to pheromone evaporation. The algorithm stops iterating either when an ant found a solution or when a maximum number of generations have been performed. The design steps of based JSSP is shown in Fig.. Simulation Results Simulation analysis and results presents the effectiveness of any research work. In this section, the simulation results of various stages in solving the Job Shop Scheduling Problem such as scheduling and optimization is analyzed and compared. The job and machine count is considered as 0 because well known Fisher and Thompson 0x0 instance (FT0) and Adams, Balas, and Zawack 0x0 instance (ABZ0) benchmark problems are selected for the experiment analysis. The simulation was done using ATLAB R00b on Intel core Duo (.GHz), GB RA PC.. Scheduling Fig ANT Algorithm for JSSP While constructing its tour, an ant will modify the amount of pheromone on the passed edges by applying the local updating rule ( ) 0, where (t) is the amount of pheromone on the edge (i, j) at time t; ρ is a parameter governing pheromone decay such that 0 < ρ < ; and 0 is the initial value of pheromone on all edges. Once all ants have arrived at their destination, the Scheduling is the process of allocating resources to perform a collection of operations. It can also be described as a decision-making process with the goal of optimizing the objectives such as makespan, elapse time and makespan reduction efficiency. The job-shop is concerned with the simultaneous and synchronized ordering of operations on several machines. Each job should be processed through machines in a particular order which is also known as the technological constraint. Once a machine starts to process a job, no interruption is allowed. Based on each level, sequence of the jobs will vary, and hence scheduling has to be processed before optimizing the jobs. The scheduling of the jobs with respect to machine and level is shown in Tables and for FT0 and ABZ0 problems respectively. Based on this schedule, the completion time (), waiting time () and the priority levels () are computed after the job sequences on the machines are completed. The completion time is computed as the summation of waiting time of the job and processing time of the job for the corresponding machine and level. The maximum value of the completion time produces the makespan and the results of this scheduling stage serve as the basis for validation of the optimization techniques,, and. Fig shows the graphical representation of the completion and the waiting time obtained in the scheduling phase. The priority levels for both FT0 and the ABZ0 problems are also plotted in Fig.

6 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 LEVE L Table Scheduled Sequences of Jobs for FT Fig and for Scheduling Phase Table Scheduled Sequences of Jobs for ABZ0 LEVE L Fig Priority Level for Scheduling Phase. Particle Swarm Optimization The parameters of are set based on empirical study of this paper as well as referring to the experiences of other researchers from literature. The acceleration constants C and C are set to.0 and initial population of swarm varies with respect to the jobs and machines on each level. Inertia weight, W, determines the search behavior of the algorithm. Large values for W facilitate searching new locations whereas small values provide a finer search in the current area. A balance can be established between global and local exploration by decreasing the inertia weight during the execution of the algorithm. Initially the inertia weight was set to 0.. The jobs and levels are taken as the swarm particles and the algorithm iterates to determine the p best and g best to optimize the job sequences. The number of runs and the maximum number of iterations depend upon the jobs and levels. After 00 iterations the scheduling process was found to be optimized for the FT0 and ABZ0 problems using Particle swarm optimization, the waiting time, and the completion time of corresponding job is evaluated and shown in Fig. The priority levels for all the jobs satisfying the precedence constraints are shown in Fig.

7 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 waiting time and completion time are shown in Fig. The priority level of the jobs is shown in Fig. Fig and using Fig Graphical representation of and using Fig Priority Level using. Ant Colony Optimization Ant systems adapt a strategy to construct solutions that balance the pheromone trails based on the problem specific heuristic rule and they follow a method to reinforce and evaporate the pheromone trails. The number of ants was always kept equal to total number of jobs in the benchmark problem. The parameters studied were pheromone updating constant Q, Weight of pheromone trail α, Weight of heuristic information β, and the Pheromone evaporation parameter ρ. The parameters α and β decide the tradeoff between the importance of the trail intensity and visibility of an ant. The tests showed that Q had a negligible effect on the algorithm performance. The values of objective functions were observed to select the best combination of these values, and the best value was found to be α = β =. Pheromone evaporation parameter ρ is also another important parameter. A too high value of parameter ρ results in a situation called stagnation. Stagnation denotes the undesirable situation in which all ants construct the same solution over and over again making further exploration of newer paths almost impossible. While a very low value of ρ results in little information conveying from previous solutions and the algorithm becomes a randomized greedy search procedure. A value of 0. for ρ rendered minimum computation time and this was chosen the best. The algorithm for the JSSP problem was tested on both FT0 and ABZ0 problems and the makespan was computed as 0 sec and 0 sec respectively. The performance characteristics of the problems based on Fig Priority Level Vs Jobs using. Comparative Analysis The Fisher and Thompson 0x0 instance (FT0) and Adams, Balas, and Zawack 0x0 instance (ABZ0) benchmark problems were used to compare and analyze the performance of computational intelligence paradigms for solving job shop scheduling problem. Table and Table, compares the variation in the waiting and completion time and their corresponding priority level after optimizing using different optimization techniques for two benchmark instances. The maximum value of the completion time gives the makespan value. The priority value is ranked based on completion time obtained for all job sequences for all optimization techniques. For FT0, particle swarm optimization performs better since the makespan value is reduced to 0 sec while compared to. In Table, the various parameters such as akespan (S), ean Completion Time (), Least Completion Time (L), ean Waiting Time (), Least Waiting Time (L), Elapse Time (ET), S Reduction Efficiency (RE) and Algorithm Efficiency (AE) for the optimization techniques with the corresponding benchmark is compared and analyzed. Using particle swarm optimization for ABZ0, the ean Completion Time is reduced to 0. sec. For FT0, the mean completion time is reduced to. using. Comparing mean waiting time, the particle swarm optimization performs better with least (.sec) for ABZ0 and performs better with

8 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 least (. sec). The particle swarm optimization has the maximum makespan reduction efficiency of.% for FT0 and.% for ABZ0. The elapse time, S reduction efficiency and algorithm efficiency for scheduling cannot be computed, since the makespan value for scheduling is manipulated manually. The time taken for the algorithms to perform is known as the elapsed time. The elapsed time for the performance of particle swarm optimization is 0.0 sec for FT0 and 0. sec for ABZ0 which is the least elapse time when compared with methods. Hence the particle swarm optimization technique converges faster in yielding best optimized results. Comparing the optimization algorithms based on parameters such as makespan, mean completion time, mean waiting time, elapse time and makespan reduction efficiency, the particle swarm optimization performs better for optimizing both Fisher and Thompson 0x0 instance (FT0) and Adams, Balas, and Zawack 0x0 instance (ABZ0) problem with minimum makespan value, least mean completion time, condensed mean waiting time, low elapse time and maximum makespan reduction efficiency. JOB Table FT0X0 comparison J 0 00 J 0 J 0 0 J 0 0 J J 0 0 J J J J0 0 0 JOB Table ABZ0X0 comparison J J 0 J J J 0 00 J 00 0 J 0 J 0 0 J J Conclusion This paper presents a novel knowledge-based approach for the job shop scheduling problem (JSSP) by utilizing the various constituents of the computational intelligence techniques such as Particle Swarm Optimization () and Ant Colony Optimization (). The well known Fisher and Thompson 0x0 instance (FT0) and Adams, Balas, and Zawack 0x0 instance (ABZ0) problem is selected as the experimental benchmark problem and simulated using ATLAB R00b. Based on all the evaluations performed, we concluded that is the superior computational intelligence technique for solving JSSP. This research focused primarily on discovering new approaches that can match the computational intelligence techniques in solving Job Shop Scheduling problems. Significant improvements can be made by modifying the goals of this paper and adopting techniques to extend the knowledge of job shop scheduling problems. The research dealt specifically with the classical 0 x0 job shop scheduling problem with the objective of minimizing the makespan. The research can be extended to a larger size problem and analyse the performances.

9 Surekha P,Dr.S.Sumathi, Int. J. Comp. Tech. Appl., Vol (), -0 ISSN : -0 Table Comparative Analysis of Optimization Techniques Parameters Scheduling S L L Elapse time S reduction efficiency % Algorithm efficiency% FT0 0 0 ABZ0 0 0 FT0... ABZ FT0 ABZ0 FT0... ABZ0.. FT0 0 ABZ0 0 FT ABZ0-0.. FT0 -.. ABZ0 -.. FT0 -.. ABZ References [] Xiaoyu Song, Limei Sun,Qiuhong eng. "Deadlocks solving strategies in hybrid algorithm for job shop scheduling." IEEE, Vol., 00: -. [] Hesam Izakian, Behrouz Tork Ladani, Ajith Abraham, A DISCRETE PARTICLE SWAR OPTIIZATION APPROACH FOR GRID JOB EDULING, International Journal of Innovative Computing, Information and Control, vol., 00, pp. - [] Wei-Neng Chen, Jun Zhang, An ant colony optimization approach to a grid workflow scheduling problem with various QoS requirements, IEEE Transactions on Systems, an, and Cybernetics, Part C: Applications and Reviews, v. n., p.-, January 00 [] J. Christopher Beck, Patrick Prosser, Evgeny Selensky: Graph Transformations for the Vehicle Routing and Job Shop Scheduling Problems. ICGT 00: pp.0- [].R. Garey, D.S. Johnson and R. Sethi, The complexity of flowshop and jobshop scheduling. ath. Oper. Res. (), pp. [] Bo Liu, Ling Wang. "Improved Particle Swarm Optimization combined with Chaos." Chaos, Solitons and Fractals, Elsevier, 00: -. [] F. Pezzellaa, G. organtia, G. Ciaschettib. "A genetic algorithm for the Flexible Job-shop Scheduling Problem." Computers and Operations Research, Elsevier, 00. [] Qian, Hongwei Ge Wenli Du Feng. "A Hybrid Algorithm Based on Particle Swarm Optimization and Simulated Annealing for Job Shop Scheduling." IEEE Proc. Natural Computation, 00: -. [] Shyh-Chang Lin, Erik D. Goodman and William F. Punch III. "Investigating parallel genetic algorithms on job shop scheduling problems." In Evolutionary Programming IV, -. Germany: Springer-Verlag, 00. [0] 0