Application of PSO with Different Typical Neighbor Structure to Complex Job Shop Scheduling Problem

Transcription

1 Appl. Math. If. Sci. 7, No. 2L, (2013) 499 Applied Mathematics & Iformatio Scieces A Iteratioal Joural Applicatio of PSO with Differet Typical Neighbor Structure to Complex Job Shop Schedulig Problem Fuqig Zhao 1,2, Jiaxi Tag 1, Jizhe Wag 1, Jubiao Wag 2 ad Jorialdi 3 1 School of Computer ad Commuicatio, Lazhou Uiversity of Techology, Gasu,Chia 2 Key Laboratory of Cotemporary Desig & Itegrated Maufacturig Techology,Miistry of Educatio, Northwester Polytechical Uiversity, Shaxi, Chia 3 School of Egieerig, Computer Sciece ad mathematics,uiversity of Exeter,EX4 4QF,Uited Kigdom Received: 10 Oct. 2012, Revised: 22 Ja. 2013, Accepted: 25 Ja Published olie: 1 Ju Abstract: Job shop schedulig is to schedule a set of jobs o a set of machies, which is subject to the costrait that each machie ca process at most oe job at a give time ad the fact that each job has a specified processig order through the machies. It is ot oly a NP-hard problems, it also has the well-eared reputatio of beig oe of the strog combiatorial problems i maufacturig systems. I this paper, the job-shop schedulig problem (JSSP), with the optimizatio goal of the schedulig problem is miimum of total process time C max, was modeled. A improved particle swarm optimizatio with acceleratio factor (AFPSO) is proposed to improve the ability of particles to explore the global ad local optimizatio solutios, ad to reduce the probability of beig trapped ito the local optima. The eighbor structure of differet particle cadidate was studied to improve the iformatio exchage speed i optimizig process. Simulatio results show that the proposed model ad algorithm are effective to task evaluatio ad implemetatio. Keywords: Job Shop Schedulig Problem, Particle Swarm Optimizatio, Topology Structure 1 Itroductio Job shop schedulig [1,2] is to schedule a set of jobs o a set of machies, which is subject to the costrait that each machie ca process at most oe job at a give time ad the fact that each job has a specified processig order through the machies. It is ot oly a NP-hard problems, it also has the well-eared reputatio of beig oe of the strog combiatorial problems i maufacturig systems. I the previously study, JSSP has bee primarily treated by mathematics methods [3], brach ad boud methods [4] ad heuristics based o priority rules [5]. Over the past two decades, meta-heuristics have gaied wide research attetio, icludig such topics as simulated aealig (SA) [6], tabu search [7], geetic algorithm (GA) [8], particle swarm optimizatio(pso) [9], ad scatter search(ss) [10]. Owig to the good performace of covergece speed i cotiuous ad discrete problems, PSO has bee adopted as a optimizatio method for JSSP for decades. By modifyig the particle positio represetatio, particle movemet, ad particle velocity, Sha [11] costructed a particle swarm optimizatio (PSO) for a elaborate multi-objective job-shop schedulig problem. A Geeral PSO (GPSO) model, which utilized differet typical topology, is adopted by particles to exchage iformatio ad search radomly i cadidate space. Optimizatio of the JSP, with a search space divisio scheme ad the meta-heuristic method of PSO, by assigig each machie i a JSP with a idepedet swarm of particles, was obtaied by multiple idepedet particle swarms [12]. Ivers et al. examies the optimizatio of the Job Shop Schedulig Problem (JSP) by a search space divisio scheme ad use of the meta-heuristic method of Particle Swarm Optimizatio (PSO) to solve it [13]. Besides the velocity represetatio ad the guide equatios used by the simple PSO, eight extesios focused o the relatioship betwee the particles ad the ijectio of kowledge of the problem was developed to JSP [14]. May studies have bee cosequetly carried out to prevet premature covergece ad to balace the exploratio ad exploitatio abilities [15]. Correspodig author fzhao2000@hotmail.com Natural Scieces Publishig Cor.

2 500 FQ. Zhao, JX. Tag, JZ Wag: Applicatio of PSO with Differet Structure to Job Shop Schedulig Problem 2 Job Shop Schedulig Problem There are m distict machies to process jobs that have their specific processig routies. Each job s operatio has its precedece ad takes up a determiistic time period at a specific machie. At oe time, there is oly oe operatio at a machie ad the job does ot leave this machie util the operatio is completed. Schedulig model is the correspodig evolutio process to the iitial schedulig model, so the iitial problem modelig is as follows. mimax{c is i I} s.t.: s i j+1 s i j+1 + p i j,i I,J {1,...,s 1} (1) (m i1 j m i2 j) (s i1 j c i1 j s i2 j c i2 j) (2) s i1 0,s i j i 1,i 2 I,i 1 i 2, j J (3) c i j = s i j + p i j,i I, j J (4) j 1 k=1 p ik,i I, j {2,...,s} (5) m i j R j = {r jl,...,r jl j J},i I (6) Where, i is the workpiece umber ad i I, I = {1, 2,, s}, j is the level umber ad j J,J = {1,2,,s}, r jl is the machie umber, s i j is the start time of iitial schedulig, m i j is the start machie, p i j is the processig time of the workpiece. dyamic evolutioary equatios ca be simply described as follows: v(t + 1) = ωv(t) + c 1 r 1 (p i (t) x(t)) + c 2 r 2 (p g (t) x(t)) + lr 3 x(t + 1) = x(t) + v(t + 1) (8) The typical particle topology structure of eighbor domai of particle cadidate i PSO has followig performace. Star structure: every particle has direct coectio with other members. I this structure, the iformatio exchage rate betwee particles is very fast, so the covergece speed is fast as well. However, it has the property of the tedecy to trap ito local optimizatio i the later evolutios. The topology structure of star structure is show as Figure 1(a). Rig structure: all cadidates form a rig structure. Each cadidate iteracts with the eighbor cadidate directly to guaratee diversity of the populatio. The iformatio trasfer is slower tha that of other structures. The topology structure of rig structure is show as Figure 1(b). Rotate structure: each eighbor domai is ot formed by 3 cadidates. It is formed by a rotate structure with oe i ceter ad the rest coected with the oe i ceter. The topology structure of rotate structure is show as Figure 1(c). Vo Neuma structure: each cadidate directly coect with the four cadidate circled with it to form a et structure. The topology structure of rotate structure is show as Figure 1(d). 3 The Improved PSO with a Acceleratio Factor ad Its Performace with Differet Topology Structure I this paper, a improved PSO algorithm, based o stadard PSO algorithm, improves the velocity updatig formula by addig a acceleratio factor. The modified algorithm ca be described as follows: v t+1 id = ωv t id + c 1r 1 (p id x t id ) + c 2r 2 (p gd x t id ) + lr 3 x t+1 id = x t id + vt+1 id (7) Where, l = d 1 (x d 2 ), which is a liear declie fuctio cotrolled by parameters d 1 ad d 2. The variable r 3 is a radom positive umber, drawig from the uiform distributio [0,1]. The mathematical model of the Fig. 1 The eighbor structure of cadidate The followig 5 test fuctios were selected as the test case to the PSO algorithm. I the experimet setup, the Natural Scieces Publishig Cor.

3 Appl. Math. If. Sci. 7, No. 2L, (2013) / populatio size, dimesio ad iteratio umber are set 20, 30 ad 1000, respectively. f 1 (X) = f 2 (X) = f 3 (X) = [x 2 i 10cos(2πx i ) + 10], 10 < x i < 10 x 2 i cos( x i i ) + 1, 600 < x i < 600 [100(x 2 i+1 x i ) 2 + (1 x i ) 2 ], 100 < x i < 100 f 4 (X) = f 5 (X) = ( i j=1 x 2 i, 100 < x i < 100 x 2 i ), 100 < x i < 100 Table 1 The Optimizatio Result Test Topology Mea value Variace Time 4*f1 Star Rig Rotate Vo Neuma *f2 Star Rig Rotate Vo Neuma *f3 Star Rig Rotate Vo Neuma *f4 Star Rig Rotate Vo Neuma *f5 Star Rig Rotate Vo Neuma From the Table 1, f1 fuctio has best optimizatio effect usig star topology, but has best stability with star topology. It eeds loger ruig time tha that of obtaied by other topology because of the eighbor structure. For fuctio f2, the optimizatio performace ad stability was obtaied uder vo Neuma structure. For fuctio f3, the best optimizatio was got by star structure. Uder star structure, rotate structure ad vo Neuma structure, f3 ca obtai good performace as well. So the eighbor structure has importat ifluece to optimizatio performace. The optimizatio result ad covergece curves of f1 ad f2 are show as Figure 2 (a), (b). Fig. 2 Covergece curves 4 Numerical Test ad Comparisos To test the performace of the proposed AFPSO, a computatioal simulatio is carried out with some well studied bechmarks. For the problems used as test case, two regular criteria, the makespa Cmax(s) ad the maximum tardiess T max(s),are used as criteria. We set the two objectives as follows (s is the job solutio): Miimize f 1 (s) = Cmax(s) f 2 (s) = T max(s) The performaces used i the parameters tuig are Best, Mea, Mi, ad Best rate, where the Best, Mea ad Mi stad for the best oe, the mea oe ad the miimum aspects of the objective values achieved i 100 rus. The Best rate is the rate to reach the best value. The algorithm was ru 100 times with differet radom seeds for each parameter settig to test the radom effect o the solutio. From the results show i Table 2, we ca see that all the test cases obtaied the best value. I order to testify the effective ad efficiecy of AFPSO, the results reported i Xia ad Wu [16], Sha ad Hsu [17] were compared with the results obtaied uder AFPSO, as show i Table 2. Natural Scieces Publishig Cor.

4 502 FQ. Zhao, JX. Tag, JZ Wag: Applicatio of PSO with Differet Structure to Job Shop Schedulig Problem Table 2 Result by usig AFPSO 2*Test 2*xm 2*C* AFPSO Best Mea Mi Time FT06 6x FT10 10x FT20 20x LA01 10x LA06 15x LA11 20x LA16 10x LA21 15x LA26 20x LA31 30x LA36 15x Coclusios A modified PSO with a acceleratio factor based o basic particle swarm optimizatio was preseted i this paper. The modified algorithm effectively improved the deficiecies of the proe to local optimizatio solutio ad slow covergece by addig a acceleratio factor to the velocity updatig formula of the evolutioary computig algorithm. Theoretical aalysis, which is based o stochastic processes, proves that the trajectory of particle is a Markov processes ad AFPSO algorithm coverges to the global optimal solutio with mea square merit. Experimetal simulatios show that the AFPSO algorithm has better performace o the covergece, ad achieves better solutios i shorter time for typical bechmark fuctios tha the stadard PSO. Ackowledgemet This work is fiacially supported by the Natioal Natural Sciece Foudatio of Chia uder Grat No ad Ad it was also supported by Chia Postdoctoral Sciece Foudatio, Sciece Foudatio for The Excellet Youth Scholars of Lazhou Uiversity of Techology, ad Educatioal Commissio of Gasu Provice of Chia uder Grat No , 1014ZCX017 ad 1014ZTC090, respectively. Refereces [1] HC Cheg, TC Chiag, LC Fu. A two-stage hybrid memetic algorithm for multiobjective job shop schedulig, Expert Systems with Applicatios (2011). [2] R. Tavakkoli-Moghaddam, M. Azarkish, A. Sadeghejad- Barkousaraie. A ew hybrid multi-objective Pareto archive PSO algorithm for a bi-objective job shop schedulig problem, Expert Systems with Applicatios (2011). [3] L. F. Gelders. Sequecig ad schedulig - a itroductio to the mathematics of the job-shop, Europea Joural of Operatioal Research textbf (1982). [4] R. Bouabda, B. Jarboui, M. Eddaly, A. Rebai. A brach ad boud ehaced geetic algorithm for schedulig a flowlie maufacturig cell with sequece depedet family setup times, Computers & Operatios Research (2011). [5] A. Allahverdi, H. Aydilek. Heuristics for the twomachie flowshop schedulig problem to miimise makespa with bouded processig times, Iteratioal Joural of Productio Research (2009). [6] A. Elmi, M. Solimapur, S. Topaloglu. A simulated aealig algorithm for the job shop cell schedulig problem with itercellular moves ad reetrat parts, Computers & Idustrial Egieerig (2011). [7] JQ Li, QK Pa, YC Liag. A effective hybrid tabu search algorithm for multi-objective flexible job-shop schedulig problems, Computers & Idustrial Egieerig (2010). [8] V. Kachitvichyaukul, S. Sitthitham. A two-stage geetic algorithm for multi-objective job shop schedulig problems, Joural of Itelliget Maufacturig (2011). [9] ZG Lia. A Local ad Global Search Combie Particle Swarm Optimizatio Algorithm for Job-Shop Schedulig to Miimize Makespa, Discrete Dyamics i Nature ad Society (2010). [10] M. M. Nasiri, F. Kiafar. A hybrid scatter search for the partial job shop schedulig problem, Iteratioal Joural of Advaced Maufacturig Techology (2011). [11] D. Y. Sha, L. Hsig-Hug. A multi-objective PSO for jobshop schedulig problems, Expert Systems with Applicatios (2010). [12] B Liu, L Wag, YH Ji. A effective hybrid PSO-based algorithm for flow shop schedulig with limited buffers, Computers & Operatios Research (2008). [13] B. Ivers, G. G. Ye. Job Shop Optimizatio through multiple idepedet particle swarms, 2007 IEEE Cogress o Evolutioary Computatio (2007). [14] K. Bulbul. A hybrid shiftig bottleeck-tabu search heuristic for the job shop total weighted tardiess problem, Computers & Operatios Research (2011). [15] HW Ge, L Su, YC Liag, F Qia. A effective PSO ad AIS-based hybrid itelliget algorithm for job-shop schedulig, IEEE Trasactios o Systems, Ma, ad Cyberetics-Part A (Systems & Humas) (2008). [16] WJ Xia, ZM Wu. A hybrid particle swarm optimizatio approach for the job-shop schedulig problem, Iteratioal Joural of Advaced Maufacturig Techology (2006). [17] DY Sha, CY Hsu. A hybrid particle swarm optimizatio for job shop schedulig problem, Computers ad Idustrial Egieerig (2006). Fuqig Zhao, bor i 1977, Ph.D of Lazhou Uiversity of Techology. His research iterests iclude theory ad applicatio of patter recogitio, computatioal itelligece ad its applicatio. Correspodig author of this paper. Natural Scieces Publishig Cor.

5 Appl. Math. If. Sci. 7, No. 2L, (2013) / Jiaxi Tag, bor i 1985, graduate studet of Lazhou Uiversity of Techology. His research iterest is the theory ad applicatio of patter recogitio ad artificial itelligece. Jizhe Wag, bor i 1986, graduate studet of Lazhou Uiversity of Techology. His research iterest is the theory ad applicatio of patter recogitio ad artificial itelligece. Jubiao Wag, bor i 1968, professor of Northwester Polytechical Uiversity. His research iterest icluded itegrated maufacturig techology ad artificial itelligece. Jorialdi, bor i 1977, professor of Uiversity of Exeter. His research iterest icluded the swarm itelligece ad itegrated maufacturig techology. Natural Scieces Publishig Cor.