Parallel Artificial Bee Colony (PABC) for Job Shop Scheduling Problems Sudip Kumar Sahana, Indrajit Mukherjee, Prabhat Kumar Mahanti

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1 1 Sudip Kumar Sahana, 2 Indrajit Mukherjee, 3 Prabhat Kumar Mahanti 1, First Author Dept. of Computer Science & Engineering, Birla Institute of Technology, Mesra., 1 sudipsahana@bitmesra.ac.in 2 Dept. of Computer Science & Engineering, Birla Institute of Technology, Mesra., 2 imukherjee@bitmesra.ac.in * 3, Corresponding Author, Dept. of Computer Science, University of New Brunswick, Canada, 3 pmahanti@unb.ca Abstract Artificial Bee Colony (ABC) algorithm has proved to be useful for solving various complex problems such as assignment problems, scheduling problems, transportation problems etc. Job shop scheduling is well-known as one of the hardest combinatorial optimization problems and has been demonstrated to be NP-hard problem. In the past decades, several researchers have devoted their efforts to develop evolutionary algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) for job shop scheduling problem. In this paper a noble Parallel Artificial Bee Colony () Algorithm has been proposed and applied to solve Job Shop Scheduling Problem efficiently. We have considered large search spaces and obtained better quality of results using parallelization of ABC algorithm. Different swarms are assigned separate threads and different types of communications among these swarms are proposed and examined. Standard benchmark problems have been tested and satisfactory results have been observed. Keywords: Parallel artificial bee colony, job shop scheduling, parallel algorithm 1. Introduction Scheduling problems play a pivotal role in many of the fields today, such as manufacturing systems, industrial process, production etc. The job-shop scheduling problem has been categorized as a nondeterministic NP hard problem [5]. Hence, it is very difficult to find a better algorithm which can give an optimal solution under a given time-constraint and the job becomes more difficult to obtain when it is done for large-sized scheduling problems. A scheduling can be generally characterized as a set of job, where each job can have more than one operation to perform. The operations for a given jobs must be performed in a specified sequence on a given specific machines. The main aim of the scheduling would be to determine the scheduling of the jobs that would minimize a measure of performance. A common measure of performance for dealing with the problems of job-shop scheduling is the make-span [4] of a schedule. Technique for solving the job-shop scheduling problem [1,13] ranges from simple First-In-First-Out (FIFO) and Shortest Processing Time (SPT), to more elaborate techniques such as Branch and Bound (BB), Tabu Search (TS), Ant Colony Optimization (ACO), Genetic Algorithm (GA), Artificial Bee Colony (ABC) and many others. Out of these, Bee Colony, Ant Colony and Genetic Algorithm have high demand in the research community nowadays. Artificial Bee Colony algorithm is a new branch of swarm intelligence, which is nature inspired algorithms that are focused on insect behavior of food finding in order to develop some meta-heuristics which can mimic the insect problem solving abilities. It is an evolutionary algorithm first introduced in 2005 by Karaboga. Ant Colony Optimization algorithm is a member of Swarm Intelligence (SI) and was initially proposed by Marco Dorigo in Ants initially wander randomly and upon finding the food they return their colony while laying down pheromone trails. This trail is then used by other to follow that path instead of wandering randomly, and they also leave the pheromone trail while returning. As the pheromone will also evaporate over time, so the path that will have shortest distance will automatically have more rather than large distance. ACO has been successfully implemented for NP hard problems [18-21]. 1

2 Genetic algorithm is a search-heuristics that mimics the process of natural selection. In a genetic algorithm, a population of the candidate solutions to an optimization problem is evolved toward better solution. Each candidate solution has a set of properties which can be mutated and altered. Generally, the solutions are represented as 0 s and 1 s of a binary string. The evolution usually starts from a population of randomly generated individuals. The fitness function is evaluated for each individual, and more fit individual is selected, and each individual s genome is modified to form a new generation. The new generation is then used for the next iteration. The algorithm terminates either when the maximum number of generations has been produced or a satisfactory fitness level is achieved. There are few literatures available which make use of meta-heuristic algorithms [7] to solve the jobshop scheduling problem, such as Particle Swarm Optimization [8,14], Ant Colony Algorithm [16,17], Artificial Neural Network [15,3], Genetic Algorithm [11,6 2], Simulated Annealing [12]. A lot of works has been done recently which found that the skilled combination of two or more techniques often improve the performance especially for the large-scale problems. Although there is no fixed way of choosing the techniques to hybridize, we generally try to combine the advantages provided by different techniques in a way that they complement each other, so that an optimal solution can be achieved. Some literature for the hybridized techniques is: Hybrid Particle Swarm optimization [9], Hybridized Genetic Algorithm [10] and others. In this paper, we have used the modified version of Artificial Bee Colony algorithm that is Parallel Artificial Bee Colony () Algorithm for solving the Job Shop Scheduling Problem (JSSP). The paper is organized in the following way. In section 2 and section 3 we have introduced artificial bee colony and JSSP respectively. In section 4 the implementation details have been discussed. Section 5 deals with the experimental results on some benchmark problems and a comparative study with other standard techniques. Finally, in Section 6 conclusions has been drawn future scope has been informed. 2. Artificial Bee Colony Artificial Bee Colony tries to model natural behavior of real honey bees in food foraging. The algorithm consists of three groups of bees: employed bees, onlooker bees and scout bees. The colony is divided in two parts, where the first part consists of the employed bees while the other consists of onlookers. For every food source present, there will be a single employed bee. So, the total number of bees is equal to the total number of food sources available. Scout bees are formed, when an employed bee abandons a food source. The employed bees are responsible for exploiting the nectar source which is exploited before, and then they share this information with the onlooker bees. The onlooker bees select a particular food source by the help of the waggle dance that is performed by the employed bees. By performing the waggle dance the employed bees share information such as direction and distance to the food source. Main steps of the ABC algorithm which imitates these behaviors can be listed as: STEP 1: Begin STEP 2: Initialize STEP 3: Repeat STEP 3.1: Move the employed bees onto their food sources and calculate their nectar amount. STEP 3.2: Place the onlooker bees on the food sources and calculate their nectar amount. STEP 3.3: Send the scout bees to search area for discovering new food sources. STEP 3.4: The best food source found so far is registered. STEP 4: UNTIL (requirements are met). In ABC, the position of a food source represents a possible solution to the optimization problem and the nectar amount of a food source corresponds to the quality (fitness) of the associated solution. A food source stands for a potential solution for the problem that needs to be optimized. ABC algorithm is an iterative algorithm which starts by associating all employed bees with randomly generated food solutions. The initial population of the solutions is filled with S n number of D dimensions, which are generated randomly. Let Xi = {x i1, x i2,.,x id} represent the i th food source in the population, SN be the number of food source which is equal to number of employed bees. The following formula can generate the food source: 2

3 x ij = LB j + (UB j LB j ) x r (1) Where i є {1, 2,..., SN }, j є {1,2,...D} are randomly chosen indexes, r is the uniform random number in the range [0,1], LBj and UBj are lower and upper bounds for the dimension j respectively. Then each employed bee x ij generates a new food source v ij the neighborhood of its present position as follows: v ij = x ij + Φ ij (x ij x kj ) (2) k = int (rand * SN) + 1; Where Φij= (rand-0.5) 2, is a uniformly distributed real random number within the range [-1,1], i є {1,2,..., SN }, k є {1,2,..., SN } and k i, j є {1,2,...,n} are randomly chosen indexes. After producing the new solution v i, this new solution will be evaluated and compared to the x i. If the objective fitness value of v i is smaller than the fitness of x i, the former will be accepted as the new basic solution, otherwise the latter would be obtained. The employed bee exploits for the better solution. After the completion of this process by all the employed bees, the onlooker bees can collect the information from all the employed bees about the food sources. And they can choose a food source depending on the probability value associated by that food source which can be obtained using the following expression: Where fitness i is the fitness value of the solution i evaluated by its employed bee, which is proportional to the maximum value of the food source in the position i. It is conclusive from the equation 3 that as the maximal value of the food source decreased; the probability with the preferred source by the onlooker bee also decreases. Then a new source will be selected by the onlooker bees from equation (2). The nectar amount of the new food source will be compared with the primary food solution and would be selected if it is better. After all onlookers have completed this process, sources are checked whether they are abandoned or not. Food sources are abandoned if it is not improved through determined number of the trails limit, then the employed bee abandons the food source and becomes a scout. The expression that determines how for a scout to search for a food source randomly is as follows: x ij = LB j + (UB j LB j ) r (3) Where r is a uniform random number in the range [0, 1] After the new source is produced, another iteration of ABC algorithm begins, and the whole process is repeated until the termination condition is met. 2.1 Parallel Artificial Bee Colony Algorithm () ABC is a novel heuristic intelligent search method that are widely used to find an acceptable approximate solution to the NP-hard problems such as Travelling Salesman Problem (TSP), Job Shop Schedule Problem (JSSP) and so on within a reasonable amount of time. However, for TSP or JSSP for higher number of cities or jobs respectively, the time required to find the solution increases drastically. In order to reduce the time, we have applied some kind of parallelism in the artificial bee colony algorithm that enhances the speed of the algorithm to obtain a better solution in a considerable amount of time Parallel Strategy for The entire colony of bees are divided into sub-colonies. The number of sub-colonies are proportional to the number of processor nodes that we have. Each processor is assigned a sub-colony and need to find the global best solutions independently. There is a communication between each sub-colony that 3

4 allows them to get rid of the worst solution and at the end of the iteration a best solution so far can be obtained. We use the master-slave model, where the master node is responsible for result display, parameter setting and other computational resource allocation. And the rest of the nodes are responsible for engaging in the calculation. The master node is also responsible for calculation apart from the other jobs discussed above. The equation that determines the parameter relation between the entire colony and sub-colony is: sub_sn (4) (5) Sub_LIMIT = LIMIT. (6) Equation (5) shows the population relationship between the entire colony and the sub-colony. Num refers to total number of the population, while sub_sn refers the colony size of each sub-colony and SN is the total size of the colony. The iteration in each sub-colony is referred by sub_iteration which is shown in equation (6), derived from the total_iteration of the whole colony and the times of communication between each sub-colony i.e. communi_num. The parameter LIMIT defines the limited value for the whole colony and is equal to the limited value for each sub-colony too. 3. Job Shop Scheduling Problem Job shop scheduling problem (JSSP) concerns for finding a sequential allocation of the competing resources that can optimize a particular objective function. The JSSP in this paper consists of a set of jobs on a set of machines with the objective of minimizing the make span. A job in the JSSP can be handled by only one machine at a time. We consider that one job consists of m operations that are to be processed on a specified sequence. Once a job has started to be processed on a particular machine, it cannot be interrupted in between. The main objective of the job scheduling is to find a schedule that will minimize the total time of the completion of the jobs. A good schedule will be the one that minimizes the idle time of the machines. In general, a n m job-shop scheduling problem can be stated as follows. Let J = {1, 2,...n} denote the set of operations to be scheduled, M = {1,2,...m} represent the set of machines, and O ={0,1,...,n m,n m+1} be operation. 0 and n m+1 are dummy operation and denote the initial and the last operation respectively. The operations are interrelated by two kinds of constraints. The precedence constraints are used to make sure that each operation j to schedule after all the predecessor operation Pj is finished. Moreover, operations j can only be scheduled if the machine that it requires is idle. Further let dj and Fj denote the fixed duration time and the finish time of the operation j. Let A(t) be the set of operations being processed at time t, and let ζjm =1, if operation j requires machine m to be processed and ζjm =0 otherwise. According the conclusion of the above stated facts, the model is as follows: Minimize F n m+1 (7) F k F j - d j ; j = 1,..., n m+1;k є P j (8) (9) F j 0, J=1, 2,...,n m+1. (10) 4

5 The objective function (8) minimizes the finish time of the last operation and is known as make span. The precedence constraint can be achieved through equation (9) while equation (10) ensures that the only one machine can perform one operation time at a given time. The last expression ensures that the make span to be non-negative. A common representation of job shop scheduling is a disjunctive graph. In the graph there is a node for each operation, and there is a source operation O=0 and a sink operation O = n m+1. The positive weight for each of the operation is equivalent to the processing time of that particular operation. The starting and completion time of the source and sink respectively will define the start and completion time of our job schedule. A set of directed disjunctive arcs represents precedence constraint of each job whereas a set of disjunctive arcs is used to represent the capacity constraint to ensure that no two operations processed by the same machine can be executed simultaneously. An example of a 3 3 disjunctive is shown in figure 1. The dotted lines represent disjunctive arcs and solid line represents conjunctive arc. The set of operations on each machine is given by: M 1 = {O 11, O 22, O 32}, M 2 = {O 12, O 23, O 33} and M 3 = {O 13, O 21, O 31}. Where O ij represent j th operation of the job J i. Solutions to a job shop scheduling problem can be obtained by finding a permutation of the machines to make the bidirectional arcs to become unidirectional. The make span of our given problem can be formed by finding the length of longest directed graph which is calculated by summing the processing time of all the operation on that path. A feasible solution is represented in figure 2. Figure 1. A Disjunctive representation of a 3 3 instances Figure 2. A feasible solution 5

6 A visualization of the feasible solution as depicted from Fig. 2 gives make span from the operations {O 23, O 33, O 31, O 32} which is Implementational Details A list of elite solutions is used to denote foragers that are currently performing the waggle dance, and the duration of that waggle dance is related to the number of iteration that a solution can stay in the list. Each data structure of these solutions contains the forager s path, its make span, the maximum number of iterations allowed and the iteration number when a solution is added to the list. In our implementation process, the path is stored in a list. The operations which are connected are present in the list. So, for the solution in figure 3, the edges are {O, O 11}, {O 11, O 21}, {O 21, O 31}, {O 31, O 22}, {O 22, O 12}, {O 12, O 32}, {O 32, O 23}, {O 23, O 13}, {O 13, O 33}, and {O 33, *}. O and * are source and the sink respectively. In the implementation process, as the neighbor search is done by the current bee and the other bee in the same colony. As there will lesser number of bees in sub-colony due to the division of the whole population among small sub-colony. So, if we continue to do the local search as in the traditional ABC, the algorithm will be bound to trap in a local best solution. So, to overcome this situation, we devise a new strategy to conduct neighbor search. The algorithm picks two parameters randomly in the entire parameter to generate the new possible solution for the neighbor search. We have simulated the parallel algorithm a single computer using counting the time required for completion of all the sub-colony as a unit time. Proposed algorithm for Job shop scheduling has four phases namely: initialization phase, employed bees phase and scout bee phase. The algorithm devised can be shown below: [Initialization Phase] i. Initialize parameters of in a master node. It includes population of entire bees, iterations times, limitation value and times of communication during the computation. ii. Calculate parameter for sub-colony. It includes population of sub-colony; times of sub-iteration and those points executed data communications. iii. Every sub-colony receives the data which is sending from the master node. iv. Every sub-colony generates a series of random solutions and calculates their fitness functions according to parameters which are send from the mater node. Find out the minimum and maximum value of fitness functions in each sub-colony and replace the variable global best solution in every sub-colony with their maximum fitness functions. [Employed bee phase] i. Each sub-colony executes the model of employed bees. ii. Every sub-colony calculates the probability which depends on fitness function. [Onlooker bee phase] i. Onlookers make decisions based on the probability obtained in employed bee phase. And then, it conducts the neighbor search following the employed bee which it chooses. ii. Every sub-colony finds out the current best solution and current worst solution, updates the global best solution in each sub-colony. [Scout bee phase] i. Every sub-colony carries out the model of scouts. Judge variables which records times of neighbor search of every employed bee. If some variables exceed a value we set in the beginning of the algorithm, the variable return to zero and corresponding employed bees of this variable is replaced by a new randomly generating solution. Otherwise, the variables add one. ii. Every sub-colony judges the iteration times, if the iteration time doesn t reach the limit times, go to step employed bee phase. If the iteration times reach the limit times, go to next step. 6

7 iii. Every sub-colony communicates the best fitness function with each other according to the ring topological structure. Replace the worst fitness function by using the best fitness function receiving from other colonies and the variable which records times of neighbour search of every employed bee returns to zero. iv. Every sub-colony judges the communication times, if the communication times don t reach the limit times we set before, go to employed bee phase. If the iteration time reaches the limit times, go to next step. v. End. 5. Experiments and Results A simulation has been performed on Intel core i3 machine with 2.7 GHz Processor, 4GB RAM on Windows 7 platform in Code-block IDE on some benchmark problems. Results has been analyzed and a comparative study has been performed with Genetic Algorithm and Ant Colony Algorithm. 5.1 Problem Instances The standard instances for JSP contains a set of 82 test instances contributed to the OR-Library by Dirk C. Mattfeld and Rob J.M. Vaessens in the website Each instance consists of a line of description, a line containing the number of jobs and the number of machines, and then one line for each job, listing the machine number and processing time for each step of the job. The problems considered here are given below. i. 3 problems from Fisher & Thompson (1963), referred as ft06, ft10 and ft20 ii. 10 problems from Lawrence (1984), referred as la01 la10 iii. 10 problems due to Storer et al. (1992), referred as swv11 swv20 iv. 3 problems formulated by Adams et al. (1988), referred as abz7 abz9 The sizes of these problems range from 6 to 50 jobs and 5 to 20 machines. Larger sizes of shop problems are not considered in the study as past results have concluded that the size of the shop does not affect the relative performance of dispatching rules, and valid conclusions could be drawn from experiment with relatively small shops. The experimental results on different benchmark problems are compared with other standard algorithms and are listed in table 1 with the entries Name of the Problem, size of the problem, Best Known Solution (BKS) and solutions using GA, ACO, ABC and proposed. Another two columns Time (without parallelization) for ABC and Time (with Parallelization) for ABC i.e. proposed are shown to observe the effect of parallelization on runtime of the algorithm. The entries in bold and italics (ACO partial, ABC, and Time without/with Parallelization) are generated from our experimental results whereas the other entries (BKS, GA and ACO partial) are collected from the available resources. The make span time of Swv11-20 problems for ACO are not available in previous literature. 7

8 makespan time (in sec) Problem name Table 1. Comparison for the GA, Ant colony and Parallel ABC. Size BKS GA AC O AB C Propos ed Time(ms) (without Parallelizatio n) ABC Time (ms) (with Parallelization) Ft Ft Ft La La La La La La La La La La Swv Swv Swv Swv Swv Swv Swv Swv Swv Swv Abz Abz Abz Ft06 Ft10 Ft20 problem instance BKS GA ACO ABC Figure 3. Comparison between the make span for 3 algorithms GA, ACO and for ft06, ft10.ft20 instances. A comparison for make span between GA, ACO, ABC and the proposed is shown in figure 3 with the Best-Known Solution (BKS), for the 3 standard problem instances ft06, ft10 and ft20. Figure 4, 5 and 6 shows the make span time for La0 to La10, Swv12 to Swv20 and Abz7 to Abz9 respectively. 8

9 makespan time (in sec) makespan time (in sec) makespan time(in sec) La01La02La03La04La05La06La07La08La09La10 problem instance BKS GA ACO ABC Figure 4. Comparison between the make span for 3 algorithms GA, ACO and for la01-la10 instances problem instance BKS GA ACO ABC Figure 5. Comparison between the make span for 3 algorithms GA, ACO and for swv11-swv 20 instances Abz7 Abz8 Abz9 problem instance BKS GA ACO ABC Figure 6. Comparison between the make span for 3 algorithms GA, ACO and for Abz7-Abz9 instances. 9

10 Figure 7. Comparison between the processing times for proposed with parallelization and without parallelization. A comparative analysis for the execution time without parallelization for Artificial Bee Colony (ABC) and with parallelization for the proposed Parallel Artificial Bee Colony Algorithm () has been performed as shown in figure 7. It is observed that for the large size problems shows its efficiency in terms of time consumed. 6. Conclusion The novelty of this work is to implement job-shop schedule problems using a variation of Artificial Bee Colony termed as Parallel Artificial Bee Colony () which is equipped with performance comparable to other popular heuristics algorithms like GA and ACO and able to achieve the Best- Known Solutions so far. It is observed that performs much better that ABC in terms of accuracy and time consumed. A variety of test cases has been considered with different size network and it is found that for increase on the network size improves the results in comparison to other considered algorithms. 10. References 1. Adams, J., Balas, E. and Zawack, D., The shifting bottleneck procedure for job shop scheduling, Management Science, vol. 34(1), pp , Amirthagadeswaran KS, Arunachalam VP, Improved solutions for job shop scheduling problems through genetic algorithm with a different method of schedule deduction, Int. J. Adv. Manuf. Technol., vol. 28, pp , Fisher, H. and Thompson G. L., Probabilistic learning combination of local job shop scheduling rules, Industrial Scheduling, vol. 1963, pp , Ganesan,V.K., Sivakumar A.I., and Srinivasan G, Hierarchical minimization of completion time variance and makespan in job shops, Computers and Operation research, vol. 22,pp Gary MR,Johnson DS, Sethi R, The complexity of flow shop and job scheduling. Math, Oper.Res., 1, pp , Hajri S, Liouane N, Hammadi S, Borne P, A controlled genetic algorithm by fuzzy logic and belief functions for job shop scheduling, IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol 30(5), pp , Lawrence, S., Resource constrained project scheduling: an experimental investigation of heuristic scheduling techniques (supplement), Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburg, Technical Report,

11 8. Lian ZG, Jiao B,Gu XS, A similar particle swarm optimization algorithm for job-shop scheduling problem, Eur.J.Oper.Res., Vol. 113,pp 123_136, Lin TL, Horng SJ, Kao TW, Chen YH, Run RS, Chen RJ, Lai JL, Kuo IH, An efficient job-shop scheduling algorithm based on particle swarm optimization, Exp. Syst. Appl., Vol. 37, pp , Goncalves JF, Mendes JJDM, Resende MGC, A hybrid genetic algorithm for the job shop scheduling problem. Eur. J. Oper. Res., Vol.167, pp 77-95, Liu TK, Tsai JT, Chou JH, Improved genetic algorithm for the job-shop scheduling problem, Int. J. Adv. Manuf. Technol., 27, pp , Low C, Yeh JY, Huang KI, A robust simulated annealing heuristic for flow shop scheduling problems, Int. J. Adv. Manuf. Technol., Vol. 23, pp , Storer, R. H., Wu, S. D. and Vaccari, R., New search spaces for sequencing problems with application to job shop scheduling, Management Science, Vol. 38(1), pp , Xia WJ, Wu ZM, A hybrid particle swarm optimization approach for the job-shop scheduling problem, Int. J. Adv. Manuf. Technol., 29, pp , Yu HB, Liang W, Neural network and genetic algorithm-based hybrid approach to expanded job-shop scheduling, Comput. Ind. Eng., Vol. 39, pp , Zhang J, Hu XM, Tan X, Zhong JH, Huang Q, Implementation of an ant colony optimization technique for job shop scheduling problem, Trans. Inst. Meas. Control., Vol 28, pp , Adhikari, A. and Sahana, S.K., Job Shop Scheduling Based on Ant Colony Optimization, In: Proceedings of 2 nd International Conference ICACM-2013 by Elsevier, 2-3 August 2013,Hyderabad, pp-13-19, Srivastava, S., Sahana, S, K., Nested hybrid evolutionary model for traffic signal optimization", Applied Intelligence, Springer, Volume 45(146), 1-11, ISSN X(print), (online), Sahana,S,K., AL-Fayoumi, M and Mahanti,P.K., Application of Modified Ant Colony Optimization (MACO) for Multicast Routing Problem, I.J. Intelligent Systems and Applications, Vol-8(4), pp43-48, Sahana,S,K., Jain,A and Mahanti,P.K., Ant Colony Optimization for Train Scheduling: An Analysis, I.J. Intelligent Systems and Applications, Vol-6(2), pp29-36, Sahana, S.K., and Jain, A., An Improved Modular Hybrid Ant Colony Approach for Solving Traveling Salesman Problem, International Journal on Computing (JoC), Vol-1(2), pp ,