Linear programming embedded particle swarm optimization for solving an extended model of dynamic virtual cellular manufacturing systems


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1 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems H. Rezazadeh*, M. Ghazanfari, S. J. Sadjadi, Mir.B. Aryanezhad, A. Makui Deparmen of Indusrial Engineering, Iran Universiy of Science and Technology, Tehran, Iran Tel.: ; Fax: ; h ABSTRACT The concep of virual cellular manufacuring sysem (VCMS) is finding accepance among researchers as an exension o group echnology. In fac, in order o realize benefis of cellular manufacuring sysem in he funcional layou, he VCMS creaes provisional groups of resources (machines, pars and workers) in he producion planning and conrol sysem. This paper develops a mahemaical model o design he VCMS under a dynamic environmen wih a more inegraed approach where producion planning, sysem reconfiguraion and workforce requiremens decisions are incorporaed. The advanages of he proposed model are as follows: considering he operaions sequence, alernaive process plans for par ypes, machine imecapaciy, worker ime capaciy, cross raining, lo spliing, maximal cell size, balanced workload for cells and workers. An efficien linear programming embedded paricle swarm opimizaion algorihm is used o solve he proposed model. The algorihm searches over he 0 1 ineger variables and for each 0 1 ineger soluion visied; corresponding values of ineger variables are deermined by solving a linear programming sub problem using he simplex algorihm. Numerical examples show ha he proposed mehod is efficien and effecive in searching for near opimal soluions. RESUMEN El concepo de sisema de manufacura celular virual (SMCV) esá siendo acepado enre los invesigadores como una exensión de la ecnología de grupos. De hecho, para hacer realidad los beneficios del sisema de manufacura celular en el diseño funcional, el SMCV crea grupos provisionales de recursos (máquinas, pares y rabajadores) en la planificación de la producción y el sisema de conrol. En el presene rabajo se describe el desarrollo de un modelo maemáico para diseñar el SMCV en el marco de un enorno dinámico con un enfoque más inegrado en donde se incorporan la planificación de la producción, la reconfiguración del sisema y las decisiones relacionadas con los requisios de la fuerza de rabajo. Las venajas del modelo propueso son las siguienes: considera la secuencia de operaciones, planes de proceso alernaivos según los ipos de pares, iempo de rabajo de la máquina, iempo de rabajo del rabajador, capaciación mixa, división del rabajo, amaño máximo de la célula y carga de rabajo balanceada para las células y rabajadores. Para resolver el modelo propueso se usa un algorimo eficiene de opimización por enjambre de parículas embebidas de programación lineal. El algorimo busca en las variables eneras 0 1 y cada variable enera 0 1 visiada; los valores correspondienes de las variables eneras se deerminan resolviendo una pare de un problema de programación lineal por medio del algorimo simple. Mediane ejemplos numéricos se demuesra que el méodo propueso es eficiene y efecivo en la búsqueda de soluciones casi ópimas. Keywords: Dynamic virual cellular manufacuring sysem; producion planning; paricle swarm opimizaion; linear programming 1. Inroducion The Virual cellular manufacuring sysem (VCMS) belongs o he family of modern producion mehods, which many indusrial secors have used beneficially in recen years. A VCMS is aimed a increasing he efficiency of producion and sysem flexibiliy by uilizing he producion conrol sysem. Idenifying logical groups of resources wihin he producion conrol sysem, offers he possibiliy of achieving advanages of cellular manufacuring in siuaions where radiional cellular manufacuring sysems may no be feasible. Resuling advanages may include he Journal of Applied Research and Technology 83
2 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, improved flow performance, higher efficiency, simplified producion conrol and beer qualiy. A schema of VCMS including machine and worker sharing beween virual cells for wo consecuive periods is shown in Fig.1. In his figure, i is assumed ha nine machines and six workers exis in layou. Because of he processing requiremens, he logical grouping of machines and workers (virual cells) is changed from period 1 o period 2. Research on VCMS has gained momenum during he las decade. Recen sudies on VCMS have focused on improvemens in queue relaed performance measure of he job shop. The emphasis has been on he performance evaluaion of VCMS compared o radiional funcional and cellular layou. There has been lile research unil now on he design of VCMS forming he subjec of his paper. The objecive of his paper is o propose a new mahemaical model for inegraed virual cellular manufacuring sysem designing wih producion planning, dynamic reconfiguraion and workforce requiremens. The remainder of his paper is organized as follows: In Secion 2, we review relevan lieraure on he VCMS. Secion 3 presens he mahemaical formulaion for he VCMS. In Secion 4, we inroduce a brief review of paricle swarm opimizaion. Implemenaion of linear programming embedded paricle swarm opimizaion is described in Secion 5. Compuaional resuls are repored in Secion 6 and he conclusion is given in Secion Lieraure review The concep of VCMS was inroduced a he Naional Bureau of Sandards (NBS) o address he specific conrol problems found in he design phase of he auomaed manufacuring baches of machined pars (Simpson e al. 1982). Monreuil e al (1992) inroduced he idea ha he logical sysem can be separaed from he physical sysem, i.e., i is no necessary o have a funcional organizaion if a process layou is in place and ha a produc organizaion is no exclusively Period 1 Period 2 M1 M2 M3 M1 M2 M3 W1 W2 W2 W6 W2 W2 M6 M5 M4 M6 M5 M4 W4 W3 W3 W1 W3 W4 M7 M8 M9 M7 M8 M9 W5 W5 W6 W5 W5 W1 Figure 1. A schema of changing virual cells in relaion o changing producion needs 84 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
3 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, associaed wih a produc layou. Oher paper in his sream presens a link wih he evenual abiliy o move resources o accommodae o changing manufacuring requiremens, i.e. he possibiliy o use dynamic cells (Rheaul e al. (1995), Drole e al. (1996)). In his sense, a VCMS is o be associaed wih a specific parameer range of he dynamic faciliy layou problem (see, for example, Balakrishnan and Chang (1998)) i.e., when he produc mix and volumes change so much in relaion o he relocaion equipmen coss so ha changing he faciliy layou is never worh he effor. Vakharia e al. (1999) compared he performance of virual cells and muli sage flow shops hrough analyical approximaions. Some advanages of his sudy were he number of processing sages, he number of machines per processing sage, he bach size and raio of seup o run ime per bach for he implemenaion of he virual cells. Rachev (2001) proposed a four phase procedure for he virual cell formaion. In he firs phase, processing alernaives are generaed; in he second phase, he capabiliy of boundaries of he virual cell is defined; in he hird phase, machine ools are seleced, and finally, in he fourh phase, he performance of he sysem is evaluaed. Sarker and Li (2001) suggesed an approach for virual cell formaion wih special emphasis on job rouing and scheduling raher han on cell sharing. The basic feaure of heir approach lies in he idenificaion of a sequence of machines o minimize a job hroughou ime in a mulisage producion sysem where here are muliple idenified machines per sage and a job can only be assigned o one machine per sage. Thomalla (2000) addressed he same problem, bu wih he objecive of minimizing ardiness. In his work, he problem is solved by using a Lagrangian relaxaion approach. Irani e al. (1993) proposed a wo sage procedure which was a combinaion of he graph heoreic approach and he mahemaical programming approach for forming virual manufacuring cells. Subash e al. (2000) proposed a framework for virual cell formaion. In essence, he auhors esed several clusering algorihms for he formaion of virual cells. Saad e al. (2002) also presened an inegraed framework in a hree sep approach for producion planning and cell formaion. They sudied he possibiliy of using virual cells as a reconfiguraion sraegy. Besides he above issues, here is a number of papers published on virual cell formaion. Mos of hem are conrolled or simulaion oriened. Furhermore, hose papers ha specially address he formaion of virual cells are dedicaed o special par families. On he oher hand, he aspecs of shared cell formulaion have no received much aenion. Mak and Wang (2002) proposed a new geneic based scheduling algorihm o minimize he oal maerial and componen raveling disance incurred when manufacuring he produc wih he review o se up virual manufacuring cells and o formulae feasible producion schedules for all manufacuring operaions. The proposed algorihm differs from he convenional geneic algorihms in ha he populaions of he candidae soluions consis of individuals from various age groups, and each individual is incorporaed wih an age aribue o enable is birh and survival raes o be governed by predefined ageing paerns. In 2005, Mak and e al. improved heir mehodology by adding anoher objecive of minimizing he sum of ardiness of all producs. Baykasoglu (2003) proposed a simulaed annealing algorihm for developing a disribued layou for a virual manufacuring cell. Nomden e al. (2006) classified he virual cell formaion procedures ino hree main classes: design, operaion and empirical. A comprehensive axonomy and review of prior research in he area of VCMS can be found in heir sudy. Nomden e al. (2008) sudied parallel machine shops ha implemened he concep of VCMS for producion conrol. They srived o have a more comprehensive sudy on he relevance of rouing configuraion in VCMS. Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 85
4 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Some auhors proposed ha workforce requiremens should be aken ino accoun a he cell design sage. Min and Shin (1993) and Suresh and Slomp (2001) proposed cell design procedures in which he complex cell formaion problem is solved in wo or more phases. The las phase in boh procedures concerns workforce requiremens. A basic assumpion in he problem formulaion of Min and Shin (1993) is ha workers are linked wih he various pars by means of socalled skill maching facors. A skill maching facor indicaes o wha exen a worker is able o produce a par. These facors are used for he opimizaion of he worker assignmen problem. Cross raining issues were no considered in his work. Suresh and Slomp (2001), in he las phase of heir procedure, address various workforce requiremens such as he pariioning of funcionally specialized worker pools and he required addiional raining of workers. The need for cross raining is predeermined in heir approach by seing minimum and maximum levels for he muli funcionaliy of workers and he redundancy of machines. They do no deermine he need for cross raining analyically. Suer (1996) presened a wo phase hierarchical mehodology for operaor assignmen and cell loading in worker inensive manufacuring cells. Here, he major concern is deerminaion of he number of workers in each cell and he assignmen of workers o specific operaions in such a way ha worker produciviy is maximal. A funcional arrangemen of asks was assumed in each cell wihou considering raining and mulifuncionaliy problems. Askin and Huang (2001) focused on he relocaion of workers ino cells and he raining needed for effecive cellular manufacuring. They proposed a mixed ineger, goal programming model for guiding he worker assignmen and raining process. The model inegraes psychological, organizaional, and echnical facors. They presened greedy heurisics as means o solve he problem. Askin and Huang (2001) assumed ha he required skills are cell dependen and ha workers may need some addiional raining, again wihou considering cross raining issues. Norman e al. (2002) presened a mixed ineger programming formulaion for he assignmen of workers o operaions in a manufacuring cell. Their formulaion permis he abiliy o change he skill levels of workers by providing hem wih addiional raining and raining decisions aken in order o balance he produciviy and oupu qualiy of a manufacuring cell and he raining coss. Slomp e al. (2004, 2005) presened a framework for he design of VMCS, specifically accouning for he limied availabiliy of workers and worker skills. They propose a goal programming formulaion ha firs groups jobs and machines and hen assigns workers o he groups o form VMCS. The objecive is o use he capaciy as efficienly as possible, bu also o have VMCS in places ha are as independen as possible. 3. Problem formulaion In his secion, we develop a new mixed ineger programming model o design he VCMS under a dynamic environmen wih a more inegraed approach where producion planning, sysem reconfiguraion and workforce requiremens decisions are incorporaed. Figure 2 presens a graphical descripion of he model. The model is formulaed under he following assumpion. 3.1 Assumpions 1. Each par ype has a number of operaions ha mus be processed respecively as numbered. 2. The processing ime for all operaions of a par ype on differen machine ypes are known and deerminisic. 86 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
5 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, User Inpus Period 1 Period H Faciliy Layou Informaion Virual Cell Creaion Virual Cell Creaion Workforce Informaion Workforce Grouping Par Family Creaion Workforce Grouping Par Family Creaion Producion Daa Producion Planning Producion Planning Figure 2. Srucure of he model 3. The demand for each par ype in each period is known and deerminisic. 4. The capabiliies and ime capaciy of each machine ype are known and consan over he planning horizon. 5. The skills and ime capaciy of each worker are known and consan over he planning horizon. 6. All workers are assumed o be muli funcional. Thus, each worker can be able o operae a leas wo machines. 7. The abiliy of each worker for raining on individual machines is known and consan over he planning horizon. 8. The manufacuring cos of each machine ype is known. The manufacuring cos implies he operaing cos ha is independen of he workload allocaed o machine. 9. The disance beween wo machine locaions in layou is firs known as a prior. 10. All machine ypes are assumed o be mulipurpose. Thus, each par ype can have several alernaive process rouing wih differen processing imes. 11. Backorders are no allowed. All demands mus be saisfied in he given period. 12. Finished pars invenory is allowed in he producion sysem. 3.2 Noaion Indexes h index for ime periods (h=1,,h) c index for virual cells (c=1,,c) w index for workers (w=1,,w) m index for machine ypes (m=1,,m) p index for par ypes (p=1,,p) j index for operaions belong o par ype p (j=1,,k p ) c jp index for virual cell used o process operaion j of par ype p m jp index for machine ype used o process operaion j of par ype p Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 87
6 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Inpu parameers H C W M P K p D ph α m jpm γ p d mn s p i p β p r m T m T w q 1 q 2 UB a wmh b wm number of ime periods number of virual cells number of workers number of machine ypes number of par ypes number of operaions for par ype p demand for par p in period h operaing cos per uni ime per machine ype m processing ime required o perform operaion j of par ype p on machine ype m uni cos o move par ype p beween machines disance beween machine locaions m,n subconracing cos per par ype invenory holding cos per par ype p per ime period inernal producion cos per par ype p cos of raining a worker for machine ype m ime capaciy of machine ype m in each period ime capaciy of worker w in each period facor of workload balancing beween virual cells facor of workload balancing beween workers Maximal virual cell size = 1 if worker w has abiliy o operaing on machine ype m in each period = 1 if worker w has capabiliy o raining on machine ype m 3.4 Decision variables X jpmwch =1 if operaion j of par ype p is done on machine ype m by worker of w in virual cell c in period h Q pmh = he quaniy of pars of ype p processed on machine ype m in period h SC ph = he quaniy of pars of ype p subconraced in period h IH ph = he quaniy of invenory of par ype p kep in period h and carried over o period h Mahemaical model By using he above noaion, he nonlinear mahemaical formulaion for he VCMS is presened as follows: H C W M P Kp H C W M P min Z = α..q.x + β.q.x m jpm pmh jpmwch p pmh jpmwch h=1 c=1 w=1 m=1 p=1 j=1 h=1 c=1 w=1 m=1 p=1 j=1 + γ.d.q.x.x p (m jp )(m j+ 1,p ) pmh jpmjpwcjph j+ 1,pmj+ 1,pwcj+ 1,ph p ph p h=1 w=1 p=1 j=1 h=1 p=1 h=1 p=1 Kp H W P Kp 1 H P H P + i. IH + s. (1) H C W M P Kp + b. (1 a ). r. X h=1 c=1 w=1 m=1 p=1 j=1 wm wmh m jpmwch 88 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
7 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Subjec o: K p C W P c1 = w1 = p=1 j1 = C M P K p.q.x T m,h jpm pmh jpmwch m.q.x T w,h jpm pmh jpmwch w c1 = m1 = p=1 j1 = C W M Q pmh.x jpmwch +IH p,h1+scph IH ph =D ph j,p,h c=1 w= 1 m= 1 W M P Kp C W M P Kp q1 jpm pmh jpmwch jpm pmh jpmwch w= 1 m= 1 p=1 j= 1 C c=1 w= 1 m= 1 p=1 j= 1.Q.X.Q.X c,h C M P Kp C W M P Kp q2 jpm pmh jpmwch jpm pmh jpmwch c= 1 m= 1 p=1 j= 1 W c=1 w= 1 m= 1 p=1 j= 1 M m = 1 X jpmwch.q.x.q.x w,h UB j,p,w,c,h a = b. X.(1 a ) + a j,p,m,w,c,h 2 wmh wm jpmwc, h 1 wm, h 1 wm, h 1 X = a + b a. b j,p,m,w,c,h jpmwch wmh wm wmh wm { } X 0,1 j,p,m,w,c,h jpmwch Qpmh 0 and ineger. p,m,h SCph 0 and ineger. p,h IH ph 0 and ineger. p,h (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) The objecive funcion given in Eq. (1) is o minimize he oal sum of he manufacuring cos, maerial handling cos, subconracing cos, invenory holding cos, inernal producion cos and needed cos of cross raining for workers over he planning horizon. The firs erm represens he manufacuring cos of all machines in all virual cells over he planning horizon. I is he sum of he produc of he ime workload allocaed o each machine ype and heir associaed cos. This erm causes a balance beween he workload assigned o machines a each virual cell. The hird erm compues he oal disance raveled by he maerials and componen pars for manufacuring all he incoming demand in each period. So, he par demand can be saisfied in each period hrough inernal producion, subconracing or invenory carried over from he previous period, he second, fourh and fifh erms compues his coss. Finally, he sixh erm calculaes he oal needed coss of cross raining for workers. Eqs. (2),(3) show how machine and worker capaciy consrains are respeced. Eq. (4) is he relaionship beween inernal producion, invenory holding and subconracing levels in each period over he planning horizon. Eqs. (5),(6) enforce workload balance among virual cells and workers, respecively, where he facors of q,q 1 2 [0,1), in each inequaliy, are used o deermine he exen of he workload balance. Eq. (7) ensures he maximal virual cell size is no violaed. Eq. (8) updaes he skill marix of workers in he beginning of each period. Eq. (9) ensures ha he worker assigned o he machine has he needed skill in each period. Finally, consrain ses of (10) (13) represen he logical binary and non negaiviy ineger requiremens on he decision variables. Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 89
8 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Linearizaion of he proposed model Eq. (1) is a non linear ineger equaion. The ransformaion of he non linear erms of he objecive funcion ino linear erms can be performed by using he procedures given below. Linearizaion of he firs and second erms: he firs and second erms can be linearized by inroducing a non negaive variable Y jpmwch. The ransformaion equaion is as follows: Afer, hese hree erms are linearized; he objecive funcion of he ineger programming model includes linear erms only. All consrains in he model are also linear. The number of variables and number of consrains in he linearized model are presened in Table 1, based on he variable indices. Y jpmwch = Q pmh. X jpmwch (14) Where below consrain mus be added o he original model. Y jpmwch M. X jpmwch j,p,m,w,c,h (15) Where, M is a large posiive value. Linearizaion of he hird erm: he hird erm of he objecive funcion can be linearized by inroducing a non negaive variable W jpmnwckh, and a binary variable Z jpmnwckh. The ransformaion equaion is as follows: W jpmnwckh = Z jpmnwckh. Q pmh, where Z jpmnwckh = X jpmwch. X j+1,pnwkh, under he following ses of consrains: Z jpmnwckh X jpmwch + X j+1,pnwkh 1 j,p,m,n,w,c,k,h (16) W jpmnwckh M. X jpmwch j,p,m,n,w,c,k,h (17) Variable name Variable ype Variable coun Consrain Toal coun X jpmwch Binary KP M W C H (2) M H Z jpmnwckh Binary KP M 2 W C 2 H (3) W H Q pmh Ineger P M H (4) KP W H Y jpmwch Ineger KP M W C H (5) C H W jpmnwckh Ineger KP M 2 W C 2 H (6) W H IH ph Ineger P H (7) KP W C H SC ph Ineger P H (8) KP M W C (H1) (9) KP M W C H (15) KP M W C H (16) KP M 2 W C 2 H (17) KP M 2 W C 2 H KP: Toal number of operaions in all of he pars. Table 1 Number of variables and consrains in he linearized model 90 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
9 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Paricle swarm opimizaion (PSO) 4.1 Brief review of paricle swarm opimizaion The paricle swarm opimizaion (PSO) algorihm was firs proposed by Kennedy and Eberhar (Kennedy and Eberhar, 1995) and had exhibied many successful applicaions, ranging from evolving weighs and srucure for arificial neural neworks (Eberhar and Shi, 1998), manufacure end milling (Tandon, 2000), reacive power and volage conrol (Yoshida e al., 1999), o sae esimaion for elecric power disribuion sysems (Shigenori, 2003). The convergence and parameerizaion aspecs of he PSO have also been discussed horoughly (Clerc and Kennedy, 2002). The PSO is inspired by observaions of birds flocking and fish schooling. Birds/fish flock synchronously, change direcion suddenly, and scaer and regroup ogeher. Each individual, called a paricle, benefis from he hisorical experience of is own and ha of he oher members of he swarm during he search for food. The PSO models he social dynamics of birds/fish and serves as an opimizer for nonlinear funcions. 4.2 Discree paricle swarm opimizaion In he discussion above, he PSO is resriced in real number space. However, many opimizaion problems are se in a space feauring discree or qualiaive disincions beween variables. To mee he need, Kennedy and Eberhar (Kennedy and Eberhar, 1997) developed a discree version of PSO. The discree PSO essenially differs from he original (or coninuous) PSO in wo characerisics: Firs, he paricle is composed of he binary variable; second, he velociy mus be ransformed ino he change of probabiliy, which is he chance of he binary variable aking value one. Le X i =(x i1, x i2,..., x id), x id {0, 1} be paricle i wih D bis a ieraion, where X being reaed i as a poenial soluion has a rae of change called velociy. Denoe he velociy as V i =(v i1,v i2,..., v id), v id R. Le P i =(p i1, p i2,..., p id) be he bes soluion ha paricle i has obained unil ieraion, and P g= (p g1, p g2,..., p gd) be he bes soluion obained from P i in he populaion (gbes) or local neighborhood (lbes) a ieraion. As in coninuous PSO, each paricle adjuss is velociy according o he cogniion par and he social par. Mahemaically, we have: v = v + c r (p  x ) + c r (p  x ), (18) 1 id id 1 1 id id 2 2 gd id Where c 1 is he cogniion learning facor, c2 is he social learning facor, and r and 1 r2 are random numbers uniformly disribued in [0, 1]. Eq. (6) specifies ha he velociy of a paricle a ieraion is deermined by he previous velociy of he paricle, he cogniion par, and he social par. Values c.r 1 1, c.r 2 2 deermine he weighs of he wo pars, where heir sum is usually limied o 4 (Kennedy and Eberhar, 2001). By Eq. (6), each paricle moves according o is new velociy. Recall ha paricles are represened by binary variables. For he velociy value of each bi in a paricle, Kennedy and Eberhar (Kennedy and Eberhar, 1997) claim ha a higher value is more likely o choose 1, while a lower value favors he 0 choice. Furhermore, hey consrain he velociy value o he inerval [0, 1] by using he following sigmoid funcion: 1 s(v id ) =, (19) 1 + exp(v id ) Where s(v id ) denoes he probabiliy of bi x id aking 1. To avoid s(v id ) approaching 0 or 1, a consan V max is used o limi he range of v id. In pracice, Vmax is ofen se a 4, i.e.,. v id [ V max, + V max ]. Kennedy e al. [19] gave he pseudo code of discree PSO as follows (for maximizaion problem): Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 91
10 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Loop For i = 1 o Np If G(X i )>G(P i ) hen For d = 1 o D bis p id = x id Nex d End if g = i j g // G( ) evaluaes objecive funcion //p is bes so far //arbirary For j=indices of neighbors (or populaion) If G(P )>G(P ) hen g = j Nex j For d = 1 o D v = v + c r ( p  x ) + c r (p. x ) v [V,+V ] 1 id id 1 1 id id 2 2 gd id id max max 1 id 1 + exp(v id ) id id id s(v ) =, id //g is index of bes performer in neighborhood (or populaion) If random number < s(v ) hen x = 1; else x = 0 Nex d Nex i Unil crierion 5. Linear programming embedded paricle swarm opimizaion In his secion, we develop a linear programming embedded paricle swarm opimizaion algorihm (LPEPSO) in order o solve he model presened in Secion 3 efficienly for large daa se. For a given soluion poin, he value of 0 1 binary decision variables (X jpmwch ) is obained by decoding he soluion represenaion. To compue he corresponding values of he ineger variables and he value of he objecive funcion, a LP subproblem is solved using he simplex algorihm in lingo 8.0 sofware. The main idea of embedding a simplex algorihm in a mea heurisic is similar o ha presened in Teghem e al. (1995). The advanage of embedding an LP sub problem in he paricle swarm opimizaion algorihm can be explained as follows: For a given soluion of 0 1 binary decision variables, here may be infinie combinaions of he values for he ineger variables.; however, by solving he LP subproblem, values ha opimally correspond o he ineger soluion can be obained easily. I is also imporan o noe ha he soluion of he LP subproblem saisfies several consrains having ineger variables which oherwise may be difficul o saisfy by using paricle swarm opimizaion search alone. The seps of LPEPSO are represened in he flow char given in Fig Encoding The mos imporan issue in applying PSO successfully is o develop an effecive 'problem mapping' mechanism. The soluion encoding of he proposed model involves he 0 1 binary decision variables X jpmwch enabling a randomly generaed soluion. Fig. 4 illusraes a paricle srucure assuming P par ype (P 1 P P ) are o be processed on M machines (M 1 M M ) by W workers (W 1 W W ) in C cells (C 1 C C ) during H planning period. A segmen corresponding o a given ime period has hree sub segmens: he firs subsegmen, labeled Machines, represens he operaion assignmen of he pars o various machines, he second sub segmen, labeled Workers, represens he operaion assignmen of he pars o various workers and he hree subsegmen, labeled Cells, represens he machine configuraions. In his figure, P P in he sub segmens of Machines, Workers and cells in period 1 are shown in deail: 92 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
11 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Figure 3. The seps of LPEPSO Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 93
12 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Figure 4. Soluion represenaion 5.2 Definiion of discree paricle We define paricle i a ieraion as X = (X,X,...,X ), as X = (x,...,x, i i1 i2 ih ih ih111 ihk P PM ih111 ihk PPW ih111 ihk PPC ihjpm ihjpw { 01} where y,...,y, z,...,z ), x ihjpm x,y,z,, h=1,...,h, equal 1 if operaion j of par ype p of paricle i is assigned o machine ype m in period h and 0 oherwise, y ihjpw equal 1 if operaion j of par ype p of paricle i is assigned o worker w in period h and 0 oherwise and Z equal 1 if operaion j of par ype p of paricle i is assigned o cell c in period h and 0 oherwise. The value of decision variable X jpmwch equals 1 if x ihjpm, y ihjpw and Z equal 1 and 0 oherwise. For example, suppose he sequence of X i is {(111,1,1,1), (111,2,2,2), (112,2,2,2), (121,2,2,2), (122,1,1,1), (211,1,2,1), (212,1,1,1), (221,2,2,2), (222,2,2,2)}, ((hpj,m,w,c) denoes ha operaion j of par ype p in period h is assigned o mh machine, wh worker and ch cell). By his definiion, we have X =1, X =1, X =1, X =1, X =1, X =1, X =1, X =1, X =1. (see Fig. 5). 5.3 A linear programming sub problem The values of all he 0 1 binary decision variables obained by decoding a paricle as explained in he previous secion. The corresponding values of ineger variables Q pmh, SC ph, IH ph deermined by solving a linear programming sub problem given below. This LP sub problem is o minimize he oal sum of he manufacuring cos, maerial handling cos, subconracing cos, invenory holding cos and inernal producion cos, subjec o he consrains in Eqs. (2) (6). In his LP subproblem, hese consrains are renumbered as Eqs. (21) (25). Figure 5. Definiion of paricle X i 94 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
13 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, H C W M P Kp H C W M P min Z = α..q.x + β.q.x LP m jpm pmh jpmwch p pmh jpmwch h=1 c=1 w=1 m=1 p=1 j=1 h=1 c=1 w=1 m=1 p=1 j=1 Kp Subjec o H W P Kp 1 H P H P + γ.d.q.x.x + i. IH + s. SC p (m jp )(m j+ 1,p ) pmh jpmjpwcjph j+ 1,pmj + 1,pwcj + 1,ph p ph p ph h=1 w=1 p=1 j=1 h=1 p=1 h=1 p=1 (20) C W P K p c1 = w1 = p=1 j1 = C M P K p c1 = m1 = p=1 j1 = C W M c=1 w= 1 m= 1.Q.X T m,h jpm pmh jpmwch m.q.x T w,h jpm pmh jpmwch w Q.X +IH +SC IH =D j,p,h pmh jpmwch p,h1 ph ph ph W M P Kp C W M P Kp q1 jpm pmh jpmwch jpm pmh jpmwch w= 1 m= 1 p=1 j= 1 C c=1 w= 1 m= 1 p=1 j= 1.Q.X.Q.X c,h C M P Kp C W M P Kp q2 jpm pmh jpmwch jpm pmh jpmwch c1 = m1 = p=1 j1 = W c=1 w1 = m1 = p=1 j1 =.Q.X.Q.X w,h (21) (22) (23) (24) (25) 5.4. The finess funcion The purpose of he finess funcion is o measure he finess of he candidae soluions in he populaion wih respec o he objecive and consrain funcions of he model. For a given soluion, is finess obained by Eq. (26) as he sum of he objecive funcion of he model (Eq. (1)) and he penaly erms of consrain violaions. The value of he model objecive funcion is he sum of he objecive funcion of he LP subproblem, needed coss of cross raining for F = Model Objecive Funcion H C W P kp M + f cs. max 0, X jpmwch UB h= 1 c= 1 w= 1 p= 1 j= 1 m= 1 % kp H C M W P { + } +f. X a b a. b ws jpmwch wmh wm wmh wm h= 1 c= 1 m= 1w= 1 p= 1 j= 1 1 ; if Fmax = F min, F F F F 0.1 ; oherwise. max max F = ; > 0.1, Fmax Fmin Fmax Fmin workers over he planning horizon. The penaly erms are o enforce he cell size and worker skill consrains. Facors f cs and f ws are used for scaling hese penaly erms. In his sudy, hese facors are deermined by rial and error where saisfacory values were obained wih lile effor. Finally, for a minimizaion problem, he raw finess score F needs o be ransformed so ha he minimum raw finess will correspond o he maximum ransformed finess. This is achieved by using Eq. (27) where F % is he ransformed finess funcion. (26) (27) Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 95
14 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Velociy rail Afer a period seleced, o move a paricle o a new sequence, we define he velociy of par of paricle i a ieraion as V = (vx ih ih111 ihk PPM ih111 ihk PPW ih111 ihk PPC vx, vy,...,vy, vz,...,vz ),,...,v vx ihjpm, vy ihjpw, vz R, where vx ihjpm is he velociy value for operaion j of par ype p of paricle i assigned o machine ype m in period h a ieraion, vy ihjpw is he velociy value for operaion j of par ype p of paricle i assigned o worker w in period h a ieraion and vz is he velociy value for operaion j of par ype p of paricle i is assigned o cell c in period h a ieraion. Velociy V, called velociy rail, is ih inspired by he frequency based memory (Onwubolu, 2002). The frequency based memory records he number of imes ha an operaion of pars visis a paricular machine, worker or cells, and i is ofen used in combinaorial opimizaion, e.g., he long erm memory of abu search, o provide useful informaion ha faciliaes choosing preferred moves. Here, we make use of he similar concep o design he velociy rail. A higher value of vx ihjpm in he rail indicaes ha operaion j of par ype p is more likely o be processed on machine ype m, while a lower value favors assigning operaion j of par p ou of he mh machine; value of vy ihjpw in he rail indicaes ha operaion j of par ype p is more likely o be assigned o worker w, while a lower value favors assigning operaion j of par p ou of he wh worker and value of vz in he rail indicaes ha operaion j of par ype p is more likely o be assigned o cell c, while a lower value favors assigning operaion j of par p ou of he ch cell. The paricle s new velociy rail is updaed by he following equaions: vx =w. vx + c r (p  x ) + c r (p  x ) 1 ihjpm ihjpm 11 ihjpm ihjpm 2 2 ghjpm ihjpm vy =w. vy + c r (p  x ) + c r (p  x ) (28) 1 ihjpw ihjpw 11 ihjpw ihjpw 2 2 ghjpw ihjpw 1 vz =w. vz + c11 r (p  x ) + c2r 2(p ghjpc  x ) Here, P = (P,P,...,P ), as P = (px,...,px, i i1 i2 ih ih ih111 ihk P PM ih111 ihk PPW ih111 ihk PPC ihjpm ihjpw { } denoes py,...,py, pz,...,pz ), px,py,pz 0,1, h=1,...,h, he bes soluion ha paricle i has obained unil ieraion, P = (px,...,px, gh gh111 ghk P PM gh111 ghk PPW gh111 ghk PPC ghjpm ghjpw ghjpc { } py,...,py, pz,...,pz ), px,py,pz 0,1, h=1,...,h, denoes he bes soluion obained from paricles in he populaion a ieraion and w is he ineria weigh proposed by Shi and Eberhar (1998). A consan V use o limi he range max ofvx,vy and vz,i.e., ihjpm ihjpw vx ihjpm, vy ihjpw, vz [ V max, + V max ]. We now explain he meaning of velociy rail. For simpliciy, suppose here exis only he social par in Eqs. (28) and c 2 =r 2 =1. The sequence of X i is assumed o be {(111,1,1,1), 111,2,2,2), 112,2,2,2), 121,2,2,2), 122,1,1,1), 211,1,2,1), 212,1,1,1), 221,2,2,2), 222,2,2,2)}, he firs period is seleced randomly and he sequence of P g1 be {(111,1,1,1), 112,2,1,2), 121,1,1,1), 122,1,2,2)}. I is clear ha Vx i1jpm=p g1jpm X i1jpm=1,0, 1,Vy i1jpw=p g1jpx i1jpw=1,0, 1 and Vz i1jpc=p g1jpc X i1jpc=1,0, 1 (see Fig. 6). Values 1 inensify he assignmens of operaion j of par p in he mh machine, wh worker and ch cell, respecively, whereas, values 1 diversify such assignmens. In he calculaion, we can simply add P g1jpm=1, P g1jpw=1 and P g1jpc=1 o he corresponding Vx i1jpm, Vy i1jpw and Vz i1jpc, subrac X i1jpm=1, X i1jpw=1 and X i1jpc=1 from Vx i1jpm, Vy i1jpw and Vz i1jpc, respecively, and leave ohers unchanged. If each one of he Vx i1jpm, Vy i1jpw and Vz i1jpc is smaller han V max, hen se i wih V max ; if each one of he Vx i1jpm, Vy i1jpw and Vz i1jpc is greaer han +V max, hen se i wih +V max. 96 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
15 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Figure 6. The resuling values of The above example and Eq. (28) demonsrae ha he velociy rail is gradually accumulaed by he individual s own experience and individual s companions experience. This social behavior of sharing useful informaion among individuals in searching for he opimal soluion is he meri of PSO over more classical mea heurisics. As in discree PSO, he velociy rail values need o be convered from real numbers o he changes of probabiliies by he following sigmoid funcions: 1 jpm 0 if s(vx ihjpm ) = 1 + exp(vx ihjpm ) 0 if = jpm 0 1 s(vy ihjpw ) = 1 + exp(vy ) ihjpw 1 s(vz ) = 1 + exp(vz ) Where, s(vx ihjpm) represens he probabiliy of X ihjpm aking value 1, s(vy ihjpw) represens he probabiliy of X ihjpw aking value 1 and s(vz ) represens he probabiliy of X aking value 1. For example s(vx i1111)=0.2 in Fig. 7 represens ha here is a 20% chance ha operaion 1 of par ype 1 of paricle i will be assigned o he firs machine a firs period. 5.6 Consrucion of a paricle sequence (neighborhood soluion) (29) In he proposed algorihm, each paricle consrucs is new sequence based on is changes of probabiliies from he velociy rail. In he convenional approach, paricle i sars wih a null sequence in he seleced period and assigns an operaion of par according o he following probabiliies: s(vx ) qx ih (jp, m) =, s(vx ) ju ju ihjpm ihjpm s(vy ) qy ih (jp, w) =, s(vy ) qz (jp, c) = ih ju ihjpw ihjpw s(vz ) s(vz ) (30) Where U is he se of all operaions of pars. The operaions of pars are appended successively o he parial sequence unil a complee sequence is consruced. To reduce he compuaional effor, we replace U by a smaller se of operaions of pars in our algorihm. The basic idea of his approach is o ake he informaion of he bes sequence ino consideraion and reduce he compuaional effor. We employ parameers f ha are deermined by experimens. Based on he experimens in our VCMS, he use of he smaller se no only reduces he compuaion ime, bu also improves he soluion qualiy. The new probabiliies are as follows: Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 97
16 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, s(vx ) qx ih (jp, m) =, s(vx ) j F jf ihjpm ihjpm s(vy ) qy ih (jp, w) =, s(vy ) qz (jp, c) = ih jf ihjpw ihjpw s(vz ) s(vz ) Where F is he se among he f randomly seleced of operaions of pars as presen in he bes sequence (B) obained so far. For example, suppose firs period is seleced randomly, f=10, B={(111,1,1,1),(112,2,2,2),(121,2,2,2),(122,1,1,1)} and s(vx i1jpm), s(vy i1jpw) and s(vz i1jpc) are as given in Fig. 7. We sar wih he null sequence a firs period and selec en operaions of pars randomly. By Eq. (31), we calculae he probabiliies of seleced operaions of pars and generae a random number for each one of hem, drawn from a uniform disribuion in (0, 1). Then, among seleced operaions of pars, each one has probabiliy greaer han is random number, is bi ges value 1 (see Fig. 7) Varian of he gbes model (31) For he neighborhood srucure of paricles in he social par, we inroduce he gbes model bu modify he approach of searching for Pgk in our algorihm. In he original PSO approach, Pgk is obained from P ik ( i = 1, 2,..., Np). Based on our compuaional experimens in he virual cellular manufacuring sysem problem, we find ha he approach which obains P gk from he curren paricles X ik ( i = 1, 2,..., Np) performs beer. Alhough our approach spends more compuaion ime on converging, i increases he probabiliy of leaving a local opimum. 6. Numerical examples 6.1. Model analysis In his secion, we presen a numerical example showing some of he basic feaures of he proposed model and illusraing he need of an inegraed approach in manufacuring sysem analysis. The considered example consiss of seven par ypes, six machine ypes and wo periods in which each par ype is assumed o have wo operaion ha mus be processed respecively; each operaion being able o be performed on wo alernaive machines. Thus, each par ype has 2 2=4 process plans and here are 4 7 combinaions o selec a process plan for each par ype in each period. For he numerical example, we assume ha he upper bound for he virual cell sizes is 6, workload balancing facors Figure 7. The consrucion of a paricle 98 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
17 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, q 1, q 2 are 0.9 and uni maerial raveling cos is one dollar (i.e. γ=1 $). Afer linearizaion, he proposed model consiss of variables and 8316 consrains under he considered example. Tables 2 4 show relaed daa o he considered example. Table 2 shows disances beween he machines. Table 3 provides informaion abou he workforce such as iniial skills, abiliy and inabiliy of workers for raining on individual machines. Also able 3 presens cos of cross raining on each machine and ime capaciy of each worker in each period. Table 4 presens he producion daa (ime capaciy and operaing cos of each machine ype, quaniy of demand for each par ype in each period, processing ime, invenory holding cos, subconracing cos and inernal producion cos). Wih he daa given in Tables 2 4, he proposed model was solved using he general branch and cu algorihm in LINGO where he soluion generaed by he proposed paricle swarm opimizaion algorihm was used as a saring incumben soluion. Decisions regarding virual cell configuraion, inernal producion, subconracing and invenory level are given in Tables 5 and 6. These ables show some of he characerisics and advanages of he proposed model. The demand for par ypes 1 is enirely saisfied by subconracing and he demand for some of he par ypes, such as par ypes 3, 6 during period 1 and par ypes 5, 6 during period 2, is saisfied parially by inernal producion and parially by subconracing. Par 3 is enirely processed in virual cell 1. This is indicaed in Table 5 In Table 7 is he convergence hisory of LINGO and LPEPSO in solving his problem. As can be seen from Table 7, he lower bound (F bound ) and he bes objecive funcion value (F bes ) for he problem were and 92114, respecively, found by LINGO afer 46 hours of compuaion. A his poin of he compuaion, he opimaliy gap was ( )/ =4.55%. From his able, i can also be seen ha saring from he firs 26 seconds of compuaion ime, LPEPSO found soluions beer han hose generaed using LINGO in 48 hours. The opimaliy gap of he final soluion found using LPEPSO wih reference o he LINGO lower bound was 0.58%. An improved lower bound for he problem was also deermined by solving i o opimaliy afer relaxing he consrains in Eq. (7) and seing he number of virual cells o 1. The improved lower bound was and he opimaliy gap of he final soluion found using LPEPSO wih reference o his improved lower bound was 0.52 %. This suggesed ha he opimaliy gap of he LPEPSO soluion wih reference o an opimal soluion of he original problem is less han 0.52%. From\To M1 M2 M3 M4 M5 M6 M M M M M M Table 2 Traveling disances (meers) among he machine locaions Worker Machine T w (hours) M1 M2 M3 M4 M5 M6 W1 S UA UA UA S A 600 W2 A A S A UA S 600 W3 UA S UA A UA S 600 W4 A UA A UA S UA 600 W5 UA A A S A UA 600 r m ($) S means ha he worker had he skill required for operaing an individual machine A means ha he worker is able o rain on an individual machine Table 3 Workforce informaion Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 99
18 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Par O p Machine Periodic demand Coss relaed o M1 M2 M3 M4 M5 M6 h=1 h=2 s p i p β p γ p P P P P P P P T m (hours) α m Table 4 Producion daa for he numerical example Par Oper. Machine Inernal producion Invenory holding M1 M5 M6 M2 M3 M4 P P /w /w4 P /w4 106/w /w1 P /w4 175/w /w3 P /w /w 5 P /w /w Subconracing Table 5 The producion planning of pars for period Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
19 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Par Oper. Machine Inernal producion Invenory holding M1 M5 M6 M2 M3 M4 P P /w /w3 475/w5 P /w /w4 376/w3 P /w /w1 P /w /w1 P /w /w5 P /w /w3 Subconracing Table 6 The producion planning of pars for period 2 Time F bound F BEST LPEPSO 00:00: :00: :00: :00: :01: :05: :15: :31: :03: * 05:07: * 10:03: * 14:21: * 19:57: * 25:16: * 31:34: * 39:18: * 44:36: * 48:41: * *A erminaion crierion was me. Table 7 Comparison of LPEPSO wih LINGO for he example problem 6.2 Compuaion performance In addiion o he example problem discussed above, several oher example problems were developed o evaluae he compuaional efficiency of he developed paricle swarm opimizaion algorihm. These problems generaed randomly based on consideraion of similar daa in he lieraure. Also, problems are solved under condiions discussed in Secion 3.7. For simpliciy, we assume ha he capaciy of he machines are independen of heir ype, bu depends on he lengh of he planning horizon. For his purpose, we assume ha he planning horizon is a hree monhs period or one season. Also, each period includes 75 workdays and each workday includes 8 hours. Therefore, each period Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 101
20 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, is equal o 75 8 = 600 hours. Consequenly, by aking ino accoun he conrollable and unconrollable reasons for inerruping producion aciviies, we consider a 500 hour effecive capaciy for all machine ypes. Each problem is allowed 7200 seconds (2 hours). However, because of he compuaional complexiy, he proposed model canno be opimally solved wihin 7200 seconds or even more ime for medium and large sized insances. Thus, o solve he small and medium sized problems, we consider a possible inerval for he opimum value of objecive funcion (F * ) ha are consruced by he F bound and F bes values ha are inroduced by bound * bes Lingo sofware where F F F. According o he Lingo sofware s documens, The F bes indicaes he bes feasible objecive funcion value (OFV) found so far. F bound indicaes he bound on he objecive funcion values. This bound is a limi on how far he solver will be able o improve he objecive. A some poin, hese wo values may become very close. Given ha he bes objecive value can never exceed he bound, he fac ha hese wo values are close indicaes ha Lingo s curren bes soluion is eiher he opimal soluion or very close o i. A such a poin, he user may choose o inerrup he solver and go wih he curren bes soluion in he ineres of saving on addiional compuaion ime. As menioned earlier, we inerrup he solver wihin 7200 seconds LPEPSO resuls In his secion, he performance of LPEPSO developed in Secion 5 will be verified. In he preliminary experimens, he following ranges of parameer values from he PSO lieraure were esed N p =[5,60], c k =[1,4], w=(0.8,1.2), V max =[3,20], f=[0.2 k p P (M+W+C), 0.8 k p P (M+W+C)]. Based on experimenal resuls, he bes PSO parameer seings are shown in Table 8. By aenion o wheher he mean and bes OFV found by LPEPSO lie in inerval [F bound,f bes ] or no, six measures for judgmen on he effeciveness of LPEPSO are defined as 1. G mean = Gap beween F bes and Z mean. We assume ha if Z mean < F bes hen he associaed gap will be a negaive number. 2. G bes = Gap beween F bes and Z bes. We assume ha if Z bes < F bes hen he associaed gap will be a negaive number. 3. Avg ( G bes G mean ) = average of he absolue difference beween G mean and G bes. 4. N mean = he number of problems of heir corresponding Z mean lies in inerval [F bound,f bes ]. 5. N bes = he number of problems of heir corresponding Z bes lies in inerval [F bound,f bes ]. 6. CPU ime. Parameer N P w C 1 C 2 V max F Value k p P (M+W+C) Table 8 PSO parameer seings In measures 1 4, F bes is a base o evaluae he performance. Thus, a negaive value implicaes a beer performance. Table 9 indicaes he comparison beween he resuls obained from B&B and he LPEPSO algorihm corresponding o he 20 problems. The wo las columns of Table 9 show he values of G mean and G bes resuled from LPEPSO, respecively. In general, he smaller values of G mean, G bes and Avg ( G bes G mean ) are more favorable and implicae soluions wih a higher qualiy. Obviously, measure Avg ( G bes G mean ) is direcly dependen o he sandard deviaion of OFV, in 20 imes run. As shown in he las row of Table 9, he average values of G mean, G bes and Avg ( G bes G mean ) resuled from LPEPSO are obained as 0.77%, 1.19% and 4.15%, respecively. In oher words, Z mean is worse average 0.77% han F bes while Z bes is beer average 1.19% han F bes and also here is average 4.15% difference beween Z mean and Z bes. 102 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
21 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Moreover, we have N mean = 6 and N bes = 11, hus, in 30% of problems, Z mean lies in inerval [F bound,f bes ] and in 55% of problems, Z bes lies in inerval [F bound,f bes ]. The average of CPU ime is obained as seconds ha is a promising resul in comparison o he CPU ime repored for B&B. The average degree of infeasibiliy is obained as 0.45 ha is negligible. Fig. 8a and b shows he behavior of Z mean and Z bes values obained by LPEPSO versus F bes according o he informaion provided in Table 9. In hese figures, he problems are arranged by erms of F bes values. The obained resuls show ha he LPEPSO is a relaively suiable approach o solve he large size problems. No. Problem info. B&B LPEPSO Gap P M W C H UB F bes F bound T B&B Z bes Z mean T LPEPSO G bes (%) G mean (%) G bes G mean (%) ,625 25, ,625 25, ,686 61, ,686 61, ,591 64, ,591 65, ,455 98, , ,145 86,379 > , , , ,967 > , , ,648 92,349 > ,954 94, , ,365 > , , , ,658 > , , , ,544 > , , , ,697 > , , , ,249 > , , , ,568 > , , , ,549 > , , , ,246 > , , , ,456 > , , , ,426 > , , , ,102 > , , , ,125 > , , , ,625 > , , Average Table 9 Comparison beween B&B and LPEPSO runs Figure 8. Comparison beween he B&B and LPEPSO resuls (Table 9): (a) Z mean found by LPEPSO vs. F bes and (b) Z bes found by LPEPSO vs. F bes. Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 103
22 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Conclusion In his paper a comprehensive mahemaical model of a dynamic virual cellular manufacuring sysem (DVCMS) is inroduced. The advanages of he proposed model are as follows: simulaneous considering dynamic sysem configuraion, operaion sequence, alernaive process plans for par ypes, machine and worker capaciy, workload balancing, cell size limi and lo spliing. The objecive is o minimize he oal sum of he manufacuring cos, maerial handling cos, subconracing cos, invenory holding cos, inernal producion cos and needed cos of crossraining for workers over he planning horizon. The proposed model is NP hard and may no be solved o opimaliy or near opimaliy using ofhe shelf opimizaion packages. To his end, we developed a heurisic mehod based on he Paricle swarm opimizaion algorihm so called LPEPSO o solve he proposed model. During he course of he search, he Paricle swarm opimizaion algorihm uses he simplex algorihm ineracively o solve a linear programming subproblem corresponding o each ineger soluion visied in he search process. The obained resuls show ha LPEPSO can provide a good soluion in a negligible ime where he average gap beween he qualiy of he soluion found by LPEPSO and he bes soluion found by he branch and bound (B&B) mehod is nearly 0.77%. The formulaed mahemaical is sill open for fuure research wih considering oher issues such as incorporaing virual cellular manufacuring ino supply chain design, considering individual learning and forgeing characerisics in workforce grouping for improving sysem produciviy and considering produc qualiy in he design of VCMS. References [1] Askin, R. G., Huang, Y., (2001). Forming effecive worker eams for cellular manufacuring. Inernaional Journal of Producion Research, 39(11), [2] Balakrishnan, J., cheng, C.H., (2005). Dynamic cellular manufacuring under muli period planning horizons. Journal of manufacuring echnology managemen 16 (5), [3] Baykasoglu, A., (2003). capabiliy based disribued layou approach for virual manufacuring cells. Inernaional journal of producion research 41(11), [4] Clerc, M., Kennedy, J., (2002). The paricle swarm explosion, sabiliy, and convergence in a mulidimensional complex space, IEEE Transacion on Evoluionary Compuaion 6, [5] Drole, J., Abdulnour G., Rheaul M., The cellular manufacuring evoluion. Compuers & indusrial engineering 31(1), [6] Eberhar, R.C., Shi, Y., Evolving arificial neural neworks, in: Proceedings of he Inernaional Conference on Neural Neworks and Brain, (1998), pp. PL5 PL13. [7] Irani, S.A., Cavalier, T.M., Cohen, P.H., (1993). Virual manufacuring cells: exploiing layou design and iner cell flows for he machine sharing problem. Inernaional journal of producion research 31(4), [8] Kennedy, J., Eberhar, R.C., (1995). Paricle swarm opimizaion, in: Proceedings of he IEEE Inernaional Conference on neural Neworks, vol. IV, 1995, pp [9] Kennedy, J., Eberhar, R.C., (1997). A discree binary version of he paricle swarm algorihm. Proc world muli conference on Sysemics, Cyberneics and Informaics. NJ: Piscaawary, [10] Kennedy, J., Eberhar, R.C., Shi, Y., (2001). Swarm inelligence. San Francisco, CA: Morgan Kaufmann [11] Mak, K.L., Lau, J.S.K., Wang, X.X., (2005). A geneic scheduling mehodology for virual cellular manufacuring sysems: an indusrial applicaion. Inernaional Journal of producion research 43(12), Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
23 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, [12] Mak, K.L., Wang, X.X., (2002). Producion scheduling and cell formaion for virual cellular manufacuring sysems. Inernaional Journal of Advanced Manufacuring Technology 20(2), [13] Min, H., Shin, D., (1993). Simulaneous formaion of machine and human cells in group echnology: A muliple objecive approach. Inernaional Journal of Producion Research, 31(10), [14] Monreuil, B., Drole, J., Lefrancois, P., (1992). The design and managemen of virual cellular manufacuring sysems. In Proceedings of American Producion & Invenory Conrol Sociey Conference, 4 5 Ocober, Quebec, Monreal, pp [15] Nomden, G., Slomp, J., Suresh, N.C., (2006). Virual manufacuring cells: a axonomy of pas research and idenificaion of fuure research. Inernaional journal of flexible manufacuring sysems. 17, [16] Nomden, G., Van der zee, D J., (2008). virual cellular manufacuring: configuring rouing flexibiliy. Inernaional journal of producion economics, doi: /j.ijpe. [17] Norman, B. A., Tharmmaphornphilas, W., Needy, K. L., Bidanda, B., Warner, R. C., (2002). Worker assignmen in cellular manufacuring considering echnical and human skills. Inernaional Journal of Producion Research, 40(6), [18] Onwubolu, G.C., (2002). Emerging opimizaion echniques in producion planning and conrol. London: Imperial College Press [19] Rachev, S.M., (2001). Concurren process and faciliy prooyping for formaion of virual manufacuring cells. Inegraed Manufacuring Sysems 12(4), , [20] Rheaul, M., Drole, J.R., Abdulnour, G., (1995). Physically reconfigurable virual cells: a dynamic model for a highly dynamic environmen. Compuers & Indusrial Engineering 29(4), [21] Saad, S.M., Baykasoglu, A., Gindy, N.N.Z., (2002). An inegraed framework for reconfiguraion of cellular manufacuring sysems using virual cells. Producion Planning & Conrol 13(4), [22] Sarker, B.R., Li, Z., Job rouing and operaions scheduling: a nework based virual cell formaion approach. Journal of he Operaional Research Sociey 52(6), [23] Shigenori, N., Takamu, G., Toshiku, Y., Yoshikazu, F., (2003). A hybrid paricle swarm opimizaion for disribuion sae simaion, IEEE Transacion on Power Sysems 18 (2003) [24] Shi, Y., Eberhar, R.C., (1998). A modified paricle swarm opimizer. Proc IEEE Congress on Evoluionary Compuaion, NJ: Piscaaway, [25] Simpson, J.A., Hocken, R.J., Albus, J.S., (1982). The Auomaed Manufacuring Research Faciliy of he Naional Bureau of Sandards. Journal of manufacuring sysems, 1 (1), [26] Slomp, J., Chowdary, B.V., Suresh, N.C., (2004). Design and operaion of virual manufacuring cells. In Proceedings of FAIM Conference, Tampa, FL. [27] Slomp, J., Chowdary, B.V., Suresh, N.C., (2005). Design of virual manufacuring cells: a mahemaical programming approach. Roboics and Compuer Inegraed Manufacuring 21(3), [28] Slomp, J., Bokhors, J.A.C., Molleman, E., (2005). Cross raining in a cellular manufacuring environmen. Compuers & Indusrial engineering, 48(3), ) Subash, B.A., Nandurkar, K.N., Thomas, A., Developmen of virual cellular manufacuring sysems for SMEs. Logisics Informaion Managemen 13(4), [30] Suer, G. A., (1996). Opimal operaor assignmen and cell loading in labor inensive manufacuring cells. Compuers and Indusrial Engineering, 26(4), [31] Suresh, N. C., Slomp, J., (2001). Labor assignmen and grouping in cellular manufacuring: A muliobjecive mehodology. Inernaional Journal of Producion Research, 39(18), Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 105
24 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, [32] Tandon, V., (2000). Closing he gap beween CAD/CAM and opimized CNC end milling, Maser hesis, Purdue School of Engineering and Technology, Indiana Universiy, Purdue Universiy, Indianapolis. [33] Teghem, J., Pirlo, M., Anoniadis, C., (1995). Embedding of linear programming in a simulaed annealing algorihm for solving a mixed ineger producion planning problem. Journal of Compuaional and Applied Mahemaics 64, [34] Thomalla CS (2000) Formaion of virual cells in manufacuring sysems. In Proc. Group Technology/Cellular Manufacuring World Symposium, San Juan, Puero Rico, pp [35] Vakharia, A.J., Moily, J.P., Huang, Y., (1999). Evaluaing virual cells and mulisage flow shops: An analyical approach. Inernaional Journal of Flexible Manufacuring Sysems 11(3), [36] Yoshida, H., Kawaa, K., Fukuyama, Y., Nakanishi, Y., (1999). A paricle swarm opimizaion for reacive power and volage conrol considering volage sabiliy, in: Proceedings of he Inernaional Conference on Inelligen Sysem Applicaion o Power Sysems, 1999, pp Vol.7 No. 1 April 2009, Journal of Applied Research and Technology
25 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Auhors Biography Hassan Rezazadeh He received his PhD in indusrial engineering from Iran Universiy of Science & Technology, Iran in He holds an MSc in Indusrial Engineering and Sysems from Iran Universiy of Science & Technology, Iran and a BSc in Applied Mahemaical from Tabriz Universiy, Iran. Currenly, he serves as a par ime lecurer a Tabriz Universiy. His research and eaching ineress are in applicaion of mea heurisics, in paricular evoluionary algorihms in design and operaion of producion sysems, producion scheduling and supply chain managemen. Mehdi Ghazanfari He Is an Associae Professor of indusrial engineering a Iran Universiy of Science & Technology, Iran. He received his PhD in indusrial engineering from Universiy of New Souh Wales, Ausralia, in He also holds an MSc in indusrial engineering from Indusrial Engineering a Iran Universiy of Science & Technology, Iran. His research and eaching ineress are in producion and operaions managemen wih emphasis on JIT and manufacuring sraegy, operaions research and supply chain managemen. Seyed Jafar Sadjadi He is an Associae Professor of Indusrial Engineering a Iran Universiy of Science & Technology, Iran. He received his PhD in indusrial engineering from Universiy of Waerloo, Canada, in He also holds an MSc in Indusrial Engineering from Indusrial Engineering a Iran Universiy of Science & Technology, Iran. His research and eaching ineress are in producion planning, operaions research, invesmen and finance, operaions research modeling, prevenive mehods, mainenance engineering, faciliy planning and economerics. Vol.7 No. 1 April 2009, Journal of Applied Research and Technology 107
26 Linear programming embedded paricle swarm opimizaion for solving an exended model of dynamic virual cellular manufacuring sysems, H. Rezazadeh e al, Mir Bahador Goli, Arianezhad He is an Associae Professor of Indusrial Engineering a Iran Universiy of Science & Technology, Iran. He received his PhD in indusrial engineering from Universiy of Waerloo, Canada, in He also holds an MSc in Indusrial Engineering from Indusrial Engineering a Iran Universiy of Science & Technology, Iran. His research and eaching ineress are in producion planning, operaions research, invesmen and finance, operaions research modeling, prevenive mehods, mainenance engineering, faciliy planning and economerics. Ahmad Makui He is an indusrial engineering Associae Professor a Iran Universiy of Science & Technology, Iran. He received his PhD in indusrial engineering from Iran Universiy of Science & Technology, Iran in He also holds an MSc in indusrial engineering from Indusrial Engineering a Iran Universiy of Science & Technology, Iran. His research and eaching ineress are in operaions research, producion planning and decision analysis. 108 Vol.7 No. 1 April 2009, Journal of Applied Research and Technology