A Multiobjective Programming for Transportation Problem with the Consideration of both Depot to Customer and Customer to Customer Relationships

Transcription

1 Proceedngs of the Internatonal ultconference of Engneers and Computer Scentsts 2009 Vol II A ultobectve Programmng for Transportaton Problem wth the Consderaton of both to and to Relatonshps Wuttnan unkaew and Busaba Phruksaphanrat Abstract A multobectve programmng for transportaton model wth the consderaton of both depot to customer and customer to customer relatonshps s proposed n ths research. The obectves are to mnmze the total transportaton cost whch s the baselne obectve and to mnmze the overall ndependence value. A Lexcographc Goal Programmng (LGP) s appled to the proposed model. A mnmzaton of the total transportaton cost s set to the frst goal and a mnmzaton of the overall ndependence value s set to the second goal of the proposed model. Ths model can obtan the better result than a sngle obectve transportaton model wth a mnmzaton of the total transportaton cost obectve. That s because the depot to customer relatonshp s consdered n the frst prorty to get the lowest cost and the vcnty of customers n the same depot s concerned n the second prorty to group near by customers to be served n the same depot f the capacty of the depot s suffcent. oreover, each customer can be served by only one depot. These advantages are more compatble to the realty than conventonal transportaton model. Index Terms to relatonshp, Lexcographc Goal Programmng, ultobectve programmng, Transportaton Problem I. ITRODUCTIO In general, dstrbuton of product from depot to customer s called Transportaton Problem (TP) whch frst developed by F. L. Htchcock snce 1941 [1], [2]. It usually ams to mnmze the total transportaton cost [3]-[7]. Other obectves that can be set are a mnmzaton of the total delvery tme, a maxmzaton of the proft, etc [8]-[11]. From the nvestgaton, the entre exstng obectves n sngle obectve transportaton models are represented by quanttatve nformaton. Ths may cause the neglgence of some crucal ponts whch can not be descrbed by quanttatve data [12], [13]. In realty, consderng only one obectve s not suffcent because t may not lead to the practcal optmal soluton. Thus, anuscrpt receved on December 30, W. unkaew s a Graduate Student of Industral Engneerng Department, Faculty of Engneerng, Thammasat Unversty, Rangst campus, Klongluang, Pathum-than, 12120, Thaland. (correspondng author; e-mal: zeta_engtu@hotmal.com, lbusaba@engr.tu.ac.th) B. Phruksaphanrat s an Assstant Professor of Industral Engneerng Department, Faculty of Engneerng, Thammasat Unversty, Rangst campus, Klongluang, Pathum-than, 12120, Thaland (e-mal: lbusaba@engr.tu.ac.th). the Decson akers (Ds) are rather to pay attenton on several obectves at the same tme. Ths s a characterstc of a multobectve transportaton model [1], [8], [14]-[16]. The multobectve transportaton model s set to solve the transportaton problem smultaneously assocated wth several obectves. ormally, exstng multobectve transportaton models use a mnmzaton of the total cost obectve as one of ther obectves. The other obectves may concern about quantty of goods delvered, underused capacty, energy consumpton, total delvery tme, etc [8], [14], [17], [18]. These obectves consder manly on depot to customer relatonshp. An effcent method for the alternatve warehouse network evaluaton and supply chan desgn was proposed by J. Korpela et al. [13], [19], [20]. Ths method bases on Analytc Herarchy Process (AHP) and xed Integer Programmng (IP) ntegraton. The maxmzaton obectve of the total customer s preference value based on the customer s vewpont s appled nstead of a mnmzaton obectve of the total cost. Ths condton refers to customer to depot relatonshp consderaton. However, the consderaton of the relatonshp between customer and customer s also crtcal because n fact vehcle route for each depot does not move from depot to customer and returns back from customer to depot as n the transportaton model, t moves from depot to customer and moves forward to the other customers. So, t needs also to consder customer to customer relatonshp. oreover, the qualtatve nformaton about transportaton problem should also be consdered. We can evaluate these qualtatve data by several methods such as the parwse comparson, scale evaluaton, and usng lngustc varables [12], [21]-[23]. In ths research, we propose a multobectve programmng for transportaton problem wth the depot to customer and the customer to customer relatonshp consderatons that contans both quanttatve and qualtatve data. The remander of ths paper s organzed as follows. The conventonal transportaton problem and ts mathematcal model are dscussed n Secton II. Then, t s followed by the model formulaton n Secton III. Detal dscusson of customer to customer relatonshp s contaned n Secton IV. ext, a numercal example s llustrated n Secton V. Results and dscussons are n Secton VI. Fnally, the concluson of ths research s provded n Secton VII of ths research. ISB: IECS 2009

2 Proceedngs of the Internatonal ultconference of Engneers and Computer Scentsts 2009 Vol II II. TRASPORTATIO PROBLE The Transportaton Problem (TP) was frst developed and proposed by F. L. Htchcock snce The classcal transportaton problem s referred to a specal case of Lnear Programmng (LP) problem and ts model s appled to determne an optmal soluton of delvery avalable amount of satsfed demand n whch the total transportaton cost s mnmzed. The transportaton problem network form can be shown as n Fg.1. The followng notaton s used. Index sets: ndex for depot, for all =1,2,,. ndex for customer, for all =1,2,,. l ndex for customer, for all l=1,2,,. k ndex for obectve, for all k=1,2,,k. t ndex for goal, for all t=1,2,,t. Decson varables: x s 1 f customer s served by depot and 0, otherwse. ρ t s the postve devaton or overachevement of goal t. η t s the negatve devaton or underachevement of goal t. Parameters: a s the capacty of depot. b s the demand of each customer. c s the unt transportaton cost delvered from depot to customer. y s an amount of demand transported from depot to customer. τ s the specfed target for goal t. t R s the relatonshp value between customer l and. l Rmax s the maxmum scale of the relatonshp value whch s assgned to 9. R s the ndependence value between customer l and. l Rl = Rmax - Rl. The baselne model for transportaton problem can be shown as follows [2], [9], [], [24]-[27], subect to mn f(y) = c y, (1) y a, for all. (2) y = b, for all. (3) a = b, (4) y 0, for all and. (5) Equaton (1) s the obectve functon of the transportaton model that s to mnmze the total transportaton cost. The total served demand of each depot must be less than or equal to the avalable supply as shown n (2). Equaton (3) represents that the sum of receved demand of each customer must be equal to ts demand. Equaton (4) shows that the sum of avalable supply of all depots must be equal to the sum of all demand. Equaton (5) represents non-negatvty constrant. Fg.1 The Transportaton Problem etwork The earler presented model s a sngle obectve transportaton problem, whch s extensvely used. For the problem assocated wth more than one obectve, the decson maker need to smultaneously take other obectves apart from the mnmzaton obectve of transportaton cost.the other obectves for transportaton problem may related to delvery tme, quantty of goods delvered, unfulflled demand, underused capacty, relablty of delvery, safety of delvery, etc. The multobectve transportaton model wth k obectves can be represented as [8] subect to k mn f (y)= c y, 1 K K mn f (y) = c y, y a, for all. y = b, for all. a = b, y 0, for all and. where c represent the coeffcents related to y varable for obectve k. There are many researchers adaptng ths model for ther computatonal researches [1], [4], [14], [17], [18], [28]. However, n these exstng transportaton models the demand for a customer may be served by multple depots, whch s not reasonable and not satsfed by customers. Thus, the zero-one nteger programmng should be ntegrated nto the transportaton model for enforcng that each customer can solely receve all requested demand from only one depot f the capacty s suffcent. oreover, both exstng sngle and multple obectve models focus only on the depot to customer relatonshp consderaton. Ths knd of relatonshp can be derved n a quanttatve form. In the research work of J. Korpela et al. [13], [19], [20], the total customer s preference value n a warehouse network and supply chan desgn obectve s maxmzed, nstead of a mnmzaton obectve of the total cost, usng the ntegraton of AHP and IP. A preference value for each alternatve 1 ISB: IECS 2009

3 Proceedngs of the Internatonal ultconference of Engneers and Computer Scentsts 2009 Vol II depot s obtaned from each customer s perspectve. It s derved n quanttatve and qualtatve forms based on customer s vewpont and refers to customer to depot relatonshp consderaton. In case of two or more customers that need to be served by the same depot e.g., customer A and B are subsdary companes of the same headquarter, the exstng models cannot support ths stuaton. It s a result of lackng of customer to customer relatonshp consderaton. For llustraton, Fg.2 (a) depcts the locaton layout of the supposed system. In depot to customer relatonshp consderaton of a transportaton problem e.g., mnmzaton of the total transportaton cost, customers A and B certanly are closer to depot than depot. Smlarly, customers E, F, G, and H are closer to depot than depot. It can properly assgn the customers to be served by each depot as shown n Fg.2 (b). C s possble to be assgned to depot or depot. There s no sgnfcant dfference between assgnng customer C to depot and depot for cost aspect because they are assumed to have the same dstance. But, we can clearly observe that customer C should be assgned to depot because customer C s n the vcnty of customers A and B whch are assgned to be served by depot. It means that customer to customer relatonshp consderaton s also necessary for a transportaton problem. Fg.2 (c) shows both consderatons of depot to customer and customer to customer relatonshps for the transportaton problem. These two relatonshps lead to the better and more approprate soluton. Hence, n ths research, both determnatons of depot to customer and customer to customer relatonshps are n concern. Fg.2 (a) The locaton layout of the supposed system Fg.2 (b) to customer relatonshp consderaton Fg.2 (c) to customer and customer to customer relatonshp consderaton ext, we wll demonstrate the way to develop a model whch supports customer to customer consderaton but stll cover the conventonal approach whch based on depot to customer relatonshp consderaton. III. ODEL FORULATIO ost of exstng research works of a transportaton problem has consdered depot to customer relatonshp. However, the relatonshp between customer and customer s also crtcal because n fact vehcle route for each depot does not move from depot to customer and returns back from customer to depot as n the transportaton model, t moves from depot to customer and moves forward to the other customers. So, t needs also to consder customer to customer relatonshp to obtan the neghborhood customers. Then, two obectves are concerned. The frst obectve s to mnmze the total transportaton cost whch s the baselne obectve for all transportaton models. It s the depot to customer relatonshp consderaton usng quanttatve data. The second obectve s to mnmze the overall ndependence value between customer and customer, whch means the consderaton of customer to customer relatonshp. Ths problem s called multobectve problem. To solve a multobectve problem, there are several methods used n general e.g., Goal Programmng (GP) [3], [8], [29]-[34], Fuzzy Lnear Programmng (FLP) [1], []-[18], [28], [], [36], Compromse Programmng (CP) [8], [32], etc. In ths research, goal programmng s chosen because of ts smplcty, popularty and ease to understand. The goal programmng comprses of two well-known methods.e., Weghted Goal Programmng (WGP) and Lexcographc Goal Programmng (LGP). For WGP, the dffculty s how to assgn approprate weght to each goal. In the LGP method, the goals are satsfed accordng to a lexcographc order, the hghest preemptve prorty goal wll be satsfed frst. Then, the remanng prorty wll be optmzed accordngly. Thus, the LGP has been chosen to formulate ths problem. oreover, n a transportaton problem the demand for a customer should be served by only one depot f the capacty of one depot s suffcent but the conventonal model does not consdered ths condton. So t s ncluded n the constrant of the proposed method. A. Obectve functons The Frst Obectve Functon: To mnmze the total transportaton cost mn f (x)= c x b. (6) 1 Ths frst obectve functon s smlar to a conventonal transportaton model that s to mnmze the total transportaton cost, whch reflects the depot to customer relatonshp consderaton. A zero-one nteger programmng s ntegrated nto transportaton model for enforcng that each customer can solely receve all demand from only one depot. The Second Obectve Functon: To mnmze the overall ndependence value between customer and customer R s the relatonshp value of customer to customer. It l ISB: IECS 2009

4 Proceedngs of the Internatonal ultconference of Engneers and Computer Scentsts 2009 Vol II means the value that ndcates the nterrelatonshp between two customers. It may be derved from qualtatve or quanttatve values such as the dstance, busness relatons and manageral convenence. to customer relatonshp can be evaluated by usng the parwse comparson matrx. Afterwards, ths matrx s converted to the ndependence value for each par of customers. The detal of ths approach s explaned n the followng secton. Then, R can be calculated from R l = R max - R l. The overall ndependence value of customer l wth the other customers, =1, 2,, n depot can be represented by Q = x R, for all and l. (7) l l Then, the overall ndependence value for all customers wth the other customers, =1, 2,, n the same depot can be represented by Q x, where l=. l So, the second obectve functon s mn f (x)= Q x. (8) 2 l Equaton (8) s developed on the bass of the adacency score that commonly use for defnng the relatonshp value between departments n a faclty plannng problem whch was presented by J. A. Tompkns et al. (2003) [37], [38]. B. Goal programmng model Accordng to two obectves above, two goal functons can be derved as follows: f (x) ρ + η = τ, (9) f (x) ρ + η = τ. () Equaton (9) s the goal functon of the frst obectve and () s the goal functon of the second obectve [32]-[34]. The obectve functon of the lexcographc goal programmng n order to mnmze the devaton of the target transportaton cost and the devaton of the target overall ndependence value can be shown as, ( ρ + η ) ( ρ + η ) lex mn= 1 1, 2 2, (11) subect to c x b = 1 ρ η τ, (12) Ql x ρ2 + η2 = τ 2, where l=. (13) x = 1, for all. (14) x b a, for all. () x, ρt, ηt 0, for all,, and t. (16) Constrants (12) and (13) are goal constrants. Constrant (14) s added to ensure that each customer must be served by only one depot. Constrant () ensures that the capacty of each depot s not exceeded, whereas (16) s a non-negatve constrant. l IV. A CUSTOER TO CUSTOER RELATIOSHIP COSIDERATIO Such mentoned prevously, a customer to customer relatonshp s necessary to be consdered n the transportaton model. The customer to customer relatonshp ( R ) between customer l and s allocated based on 1-9 Saaty s scale [22], [23] n the parwse comparson matrx. Table I shows the modfed 1-9 Saaty s scale, used for assgnng customer to customer relatonshp value. In general, 1-9 Saaty s scale usng n the Analytc Herarchy Process (AHP) s used to defne the comparatve prorty. It s an effectve decson tool and applcable for both quanttatve and qualtatve data. The upper trangular matrx and the lower trangular matrx n AHP are recprocal value 1 that s R l =. But n ths research, we emphasze on a R l relatonshp value not a prorty value, so R l = R l l s assgned. The elements n a dagonal of a matrx are relatonshp values of comparng tself, so t s denoted by the maxmum scale of the relatonshp value ( R ). An example of relatonshp max ratng n a parwse comparson matrx s gven as Table II. In the next secton, a numercal example wll be llustrated. V. A UERICAL EXAPLE In order to demonstrate the applcaton of the proposed model, a smple problem wth two depots and ten customers s gven on the assumpton that each customer must be served all demand by only one depot. oreover, a depot s capacty s suffcent to serve a customer. The lst of the basc data for the partcular example s shown n Table III. Fg.3 depcts the locaton map, whch we can presume the antcpated soluton by quanttatve data (the dstance between depot and customer) n depot to customer relatonshp consderaton that customers C1, C2, C3, and C4 should be assgned by depot and customer C7, C8, C9, and C should be assgned by depot, whereas customer C5 and C6 may be assgned by depot or. Table I The modfed 1-9 Saaty s scale Scale Low Relaton edum Low Relaton edum Relaton Demonstrated Relaton Extreme Relaton Compromse Value R l ,4,6,8 Table II An example of relatonshp ratng C1 C2 C3 C4 C5 C C C C C ISB: IECS 2009

5 Proceedngs of the Internatonal ultconference of Engneers and Computer Scentsts 2009 Vol II Fg.3 The locaton map of the partcular example Table III Transportaton cost per unt (n U.S. dollar) and customer demand Demand c (transportaton cost per unt) b (unt) C1 C2 C3 C4 C5 C6 C7 C8 C9 C Avalable supply a (unt) Table IV A parwse comparson matrx of the customer to customer relatonshp value C1 C2 C3 C4 C5 C6 C7 C8 C9 C C C C C C C C C C C To apply the proposed model n ths example, we frstly evaluate the relatonshp between customer and customer by ratng scale as n Table IV. The mathematcal expresson for ths problem can be shown as follows, ( ρ + η ) ( ρ + η ) lex mn= 1 1, 2 2, subect to c x b ρ + η = 65,200, Ql x ρ2 + η 2 =84, where l=. 2 x = 1, for all. x b a, for all. x, ρt, ηt 0, for all,, and t. In ths example, we set the goal targets ( τ 1, τ 2 ) by usng aspraton level of each obectve functon. The possble results of the proposed model when the frst prorty s optmzed are shown n Tables V (a)-v (d). There are four solutons. These solutons are dentcal wth the sngle obectve optmzaton problem that has a mnmzaton of total transportaton cost as an obectve. Obtaned results of the total transportaton cost are the same, whch s $65,200. After optmzng the second prorty, the optmal soluton s ganed as shown n Table VI wth the total transportaton of $65,200 and the overall ndependence value of 84. Ths s the best soluton from all possble solutons. It can be reached by usng our proposed model. It s an assgnment of each customer to each depot n order to satsfy the both goals. Table VII shows all of results for both goals at each stage. The postve devatons of both goals are zero resultng from mnmzaton of LGP whch means that both targets can be reached. VI. RESULTS AD DISCUSSIOS From llustraton, there are four possble solutons from the sngle obectve transportaton problem wth a mnmzaton obectve of the total transportaton cost. These solutons have Table V (a) A possble soluton of the frst prorty: case a C1, C2, C3, C4 C5, C6, C7, C8, C9, C Table V (b) A possble soluton of the frst prorty: case b C1, C2, C3, C4, C5 C6, C7, C8, C9, C Table V (c) A possble soluton of the frst prorty: case c C1, C2, C3, C4, C6 C5, C7, C8, C9, C Table V (d) A possble soluton of the frst prorty: case d C1, C2, C3, C4, C5, C6 C7, C8, C9, C Table VI The soluton after optmzng the second prorty C1, C2, C3, C4, C5, C6 C7, C8, C9, C Table VII Results from LGP of the proposed model Goal 1 2 Target 84 Obectve value n the frst prorty Obectve value after optmzng the second prorty 160 (case a) 116 (case b) 128 (case c) 84 (case d) 84 Postve devaton of each goal 0 0 ISB: IECS 2009

6 Proceedngs of the Internatonal ultconference of Engneers and Computer Scentsts 2009 Vol II the same as the results of the frst prorty of our proposed model. Some solutons may have hgh ndependence values (less relatonshp among customers whch should be served by the same depot), whch means that the customers for each depot may not locate n the same area. The sngle obectve optmzaton contans only quanttatve data. It can serve only depot to customer relatonshp consderaton but omt customer to customer relatonshp consderaton. eanwhle, after performng the second prorty optmzaton of our proposed model, the lowest cost and the lowest ndependence value alternatve can be obtaned. That means the lowest total transportaton cost and the nearest vcnty of customers are determned. oreover, each customer can be served by only one depot f the capacty of the depot s suffcent. The proposed model combnes consderaton of both depot to customer and customer to customer relatonshps. Both quanttatve and qualtatve data are ncluded n the model. VII. COCLUSIO Owng to lack of qualtatve or ntangble consderaton especally the customer to customer relatonshp n a conventonal transportaton problem, the multobectve programmng for transportaton problem wth the consderaton of depot to customer and customer to customer relatonshps s developed. LGP s chosen to solve the multobectve transportaton problem wth a mnmzaton of the total transportaton cost and the overall ndependence value. The proposed model can obtan the effcent reasonable soluton that satsfed both consderatons of depot to customer and customer to customer relatonshp that means the lowest total transportaton cost and the nearest vcnty of customers are determned. oreover, each customer can be served by only one depot f the capacty of the depot s suffcent. These advantages are more compatble to the realty than conventonal transportaton model. For further researches, the proposed model should be appled to the practcal real world applcatons and t should be mproved for the remanng assumptons. REFERECES [1] L. L and K. K. La, "A fuzzy approach to the multobectve transportaton problem," Computers & Operatons Research vol. 27, pp , [2] K. C. Rao and S. L. shra, Operatons research: Alpha Scence Internatonal Ltd., [3] J. P. Ignzo, Lnear programmng n sngle- & multple-obectve system: Prentce-Hall, Inc., [4] S. T. Lu and C. Kao, "Solvng fuzzy transportaton problems based on extenson prncple," European Journal of Operatonal Research, vol. 3, pp , [5]. Sakawa, I. shsak, and Y. Uemura, "Fuzzy programmng and proft and cost allocaton for a producton and transportaton problem," European Journal of Operatonal Research, vol. 131, pp. 1-, [6] S. Chanas and D. Kuchta, "A concept of the optmal soluton of the transportaton problem wth fuzzy cost coeffcents," Fuzzy Sets and Systems, vol. 82, p , [7] S. Chanas and D. Kuchta, "Fuzzy nteger transportaton problem," Fuzzy Sets and Systems, vol. 98, pp , [8]. Zeleny, ultple crtera decson makng: cgraw-hll Book Company, [9]. Semens, C. H. artng, and F. Greenwood, Operatons research: plannng, operatng, and nformaton systems: The Free Press: A Dvson of acmllan Publshng Co., Inc, [] S. C. Lttlechld and. F. Shutler, Operaton research n management: Prentce Hall, [11] J. Korpela, A. Lehmusvaara, and J. sonen, "Warehouse operator selecton by combnng AHP and DEA methodologes," Internatonal Journal of Producton Economcs, vol. 8, pp , [12] C. T. Chen, "A fuzzy approach to select the locaton of the dstrbuton center," Fuzzy Sets and Systems, vol. 118, pp , [13] J. Korpela, A. Lehmusvaara, and. Tuomnen, " servce based desgn of the supply chan," Internatonal Journal of Producton Economcs, vol. 69, pp , [14] E. E. Ammar and E. A. Youness, "Study on multobectve transportaton problem wth fuzzy numbers," Appled athematcs and Computaton, vol. 166, pp , [] W. F. A. El-Wahed, "A mult-obectve transportaton problem under fuzzness," Fuzzy Sets and Systems vol. 117, pp , [16] W. F. A. El-Wahed and S.. Lee, "Interactve fuzzy goal programmng for mult-obectve transportaton problems," Omega, vol. 34, pp , [17] T. F. Lang, "Dstrbuton plannng decsons usng nteractve fuzzy mult-obectve lnear programmng," Fuzzy Sets and Systems vol. 7, pp , [18] T. F. Lang, "Integratng producton-transportaton plannng decson wth fuzzy multple goals n supply chans," Internatonal Journal of Producton Research, vol. 46, pp , [19] J. Korpela, K. Kylaheko, A. Lehmusvaara, and. Tuomnen, "An analytc approach to producton capacty allocaton and supply chan desgn," Internatonal Journal of Producton Economcs, vol. 78, pp , [20] J. Korpela and A. Lehmusvaara, "A customer orented approach to warehouse network evaluaton and desgn," Internatonal Journal of Producton Economcs, vol. 59, pp , [21] S. J. Chen and C. L. Hwang, Fuzzy ultple Attrbute Decson akng: Sprnger-Verlag Berln Hedelberg, [22] T. L. Saaty, Fundamentals of decson makng and prorty theory wth the analytc herachy process, 1 ed. vol. VI: RWS Publcatons, [23] T. L. Saaty and L. G. Vargas, odels, ethods, Concepts & Applcatons of the Analytc Herarchy Process: Kluwer Academc Publshers, [24] K. J. Hastngs, Introducton to the mathematcs of operatons research wth mathematca, 2 ed.: Chapman & Hall/CRC, [25] H. A. Taha, Operatons research : an ntroducton, 5 ed.: acmllan Publshng Company, [26] A. Ravndran, D. T. Phllps, and J. J. Solberg, Operaton research: prncples and practce, 2 ed.: John Wley & Sons, [27] J. J. oder and S. E. Elmaghraby, Handbook of operatons research: foundatons and fundamentals: Ltton Educaton Publshng, Inc., [28] T. F. Lang, "Interactve mult-obectve transportaton plannng decsons usng fuzzy lnear programmng," Asa-Pacfc Journal of Operatonal Research, vol. 25, pp , [29] I. Gannkos, "A multobectve programmng model for locatng treatment stes and routng hazardous wastes," European Journal of Operatonal Research, vol. 4, pp , [30] W. Ho and A. Emrouznead, "ult-crtera logstcs dstrbuton network desgn usng SAS/OR," Expert Systems wth Applcatons, vol. In Press, [31] J. E. Hotvedt, "Applcaton of lnear goal programmng to forest harvest schedulng " Southern Journal of Agrcultural Economcs, [32] C. Romero, Handbook of Crtcal Issues n Goal Programmng: Pergamon Press, [33] R. E. Steuer, ultple Crtera Optmzaton: theory, computaton and applcaton Wley, [34]. T. Tabucanon, utple crtera decson makng n ndustry: Elsever Scence Publshng Company, Inc., [] H. J. Zmmermann, Fuzzy sets and systems 1: orth Holland Publshng Company, [36] H. J. Zmmermann, Fuzzy Set Theory and Its Applcatons, 2 ed.: Kluwer Academc Publshers, [37] R. L. Francs, Faclty layout and locaton: an analytcal approach, 2nd ed.: Prentce Hall, [38] J. A. Tompkns, J. A. Whte, Y. A. Bozer, and J.. A. Tanchoco, Facltes Plannng, 3rd ed.: John Wley & Sons, Inc., ISB: IECS 2009