CAT SWARM OPTIMIZATION FOR SINGLE STAGE SUPPLY CHAIN DISTRIBUTION SYSTEM WITH FIXED CHARGES

CAT SWARM OPTIMIZATION FOR SINGLE STAGE SUPPLY CHAIN DISTRIBUTION SYSTEM WITH FIXED CHARGES P. Maiara 1 ad V. Selladurai 2 1 Departet of Mechaical Egieerig, Kaaraj College of Egieerig ad Techology, Idia E-ail: drpaiara@gail.co 2 Departet of Mechaical Egieerig, Coibatore Istitute of Techology, Idia E-ail: profvsdcit@yahoo.co Abstract I this paper, Cat Swar Optiizatio (CSO) Algorith is proposed for sigle stage supply chai distributio syste with a fixed cost. This paper cosiders two kids of cost: a cotiuous cost, that liearly icreases with the aout trasported betwee a supplier ad a custoer ad a fixed cost, which is icurred wheever a o-zero quatity is trasported betwee a supplier ad a custoer ad it is idepedet of the aout trasported. The ai of this paper is to deterie the quatities to be distributed to satisfy the custoer dead with iiu cost. Sice fixed costs results discotiuities i the objective fuctio, solutio procedures are becoe ore difficult ad are kow to be o-deteriistic polyoial (NP) hard. I this paper Cat Swar Optiizatio (CSO) Algorith is proposed for the optiizatio of sigle stage supply chai proble to provide optial or ear optial solutio. The results of the proposed odel of this paper have bee copared with a spaig tree-based Geetic Algorith ad biary coded Geetic Algorith. Coputatioal results show the superiority of CSO algorith over other algoriths. Keywords: Cat Swar Optiizatio (CSO) Algorith, Sigle Stage, Supply Chai Proble, Fixed Cost 1. INTRODUCTION I today s world, idustries have to cope with the growig arkets ad with the icreasig custoer expectatios. Because of the high level of custoer expectatios about acquirig the products at the right tie ad right quatity at iiu cost ad besides this, the iproveets agaist the risks created by the sudde fluctuatios i local ad global arkets, copaies eed to exaie their workig techiques. Today, the success proportio for the idustries are thought as iiu costs, lesser productio tie, shorteig product life cycle, less reserve, larger product rage, ore reliable delivery tie, better custoer services, higher quality, ad providig the effective coordiatio aog dead, supply ad productio. The supply chai distributio proble with fixed charge is a special case of the fixed cost liear prograig proble, previously itroduced i the origis of the Operatios Research [1]. I the presece of oetie costs like fixed costs, the trasportatio proble is called fixed cost trasportatio cost (FCTP). May practical trasportatio ad distributio probles ca be odeled as FCTP i which the trasportatio cost cosists of a cotiuous cost, that liearly icreases with the aout trasported betwee a supplier ad a custoer ad a fixed cost, which is icurred wheever a o-zero quatity is trasported betwee a supplier ad a custoer ad it is idepedet of the aout trasported. Exaples of these fixed costs are the toll fee paid o the highways, ladig fee at the airport, perit fee or property tax, the reward give to the driver, set-up costs i productio syste or cost of buildig roads i trasportatio systes, etc [2]. This proble is characterized etirely by the presece of a trasportatio etwork structure. It is used i a wide rage of practical busiess, coerce, ad idustrial applicatios ad siultaeously received so thorough a theoretical developet, i such a short period of tie. The cost of distributio accouts for about 30% of the total cost of the product ad plays a vital role i the deteriatio of its price. The objective is to fid the cobiatio of routes that iiizes the total variable ad fixed costs while satisfyig the supply ad dead requireets of each origi ad destiatio. While siilar to the trasportatio proble, the FCTP is ore difficult to solve due to the fixed costs that result i discotiuities i the objective fuctio ad reders it usolvable by the direct applicatio of the trasportatio algorith. It has bee show that this FCTP is Nodeteriistic Polyoial-tie hard (NP-hard) proble ad that is difficult to solve usig geeral solvig ethodology [1-5] Sice the proble is NP-hard, the coputatioal tie to obtai a exact solutio icreases i a polyoial fashio ad, as the size of the proble icreases, quickly becoes difficult ad log. FCTP is ofte forulated ad solved as a ixed iteger etwork-prograig proble. Due to the excessive aout of coputatio tie required, exact solutio algoriths are ot very useful whe a proble reaches a certai level of coplexity. Therefore, ay heuristic ethods have bee proposed ad developed [4-7]. Although they are usually coputatioally efficiet, oe ajor proble with heuristics ethods is the possibility of teriatig at a local optial solutio that ay have a objective fuctio value that is uch worse tha that of a global optiu [8]. Ge Li ad Ida [9] proposed spaig treebased Geetic Algoriths (GA) for FCTP usig Prüfer uber ecodig. Syarif et al. [10] preseted spaig tree-based GA usig Prufer uber represetatio to study the choice of facilities to be opeed ad the distributio etwork desiged to satisfy the custoer dead with iiu cost. Adlakha ad Kowalski [7] proposed a siple heuristic algorith for solvig sall FCTP. However, it is stated that the proposed ethod is ore tie cosuig tha the algoriths for solvig a regular trasportatio proble. Jo et al. [5] preseted the spaig tree-based Geetic Algorith approach for solvig o-liear FCTP. Kowalski ad Lev [11] cosidered step FCTP i which the fixed cost is i the for of a step fuctio depedet o the load i a give route, ad developed a coputatioally siple heuristic algorith for solvig sall two-step FCTP. Jawahar ad Balaji [12] used GA 687

P MANIMARAN AND V SELLADURAI: CAT SWARM OPTIMIZATION FOR SINGLE STAGE SUPPLY CHAIN DISTRIBUTION SYSTEM WITH FIXED CHARGES for the two stage supply chai distributio proble associated with a fixed charge i which the capacity of the distributio ceter is assued as very large. Ki ad Ki [13] developed a ixed iteger prograig odel ad a three-phase heuristic algorith to solve the FCTP. Hajiaghaei-Keshteli et al. [14] preseted GA based o spaig tree for FCTP ad desig a chroosoe that does ot eed a repairig procedure for feasibility. Maiara et al. [15] developed a biary coded GA for sigle stage supply chai etwork associated with fixed cost. Molla- Alizadeh- Zavardehi et al. [16] proposed artificial iue ad geetic algoriths with a Prüfer uber represetatio to solve a capacitated two-stage FCTP, by cosiderig uit trasportatio cost, fixed cost associated with each route, fixed cost for opeig potetial distributio ceters, ad capacitated distributio ceters or warehouses. Otha et al, [17] applied two Geetic Algorith for FCTP ad also two fuzzy logic cotrollers are developed to autoatically tue two critical paraeters (Pc ad P) of oe of these two GA s. Hajiaghaei-Keshteli [18] developed GA ad artificial iue algorith for selectig soe potetial places as distributio ceters i order to supply deads of all custoers. Atoy Arokia Durairaj ad Rajedra [19] represeted the two stage FCTP as a sigle stage FCTP ad developed GA to solve the proble. Chu ad Tsai [20] itroduced the cat swar optiizatio (CSO) algorith i 2007. This optiizatio algorith was proposed based o ispectig the behavior of the cat. The strog curiosity of ovig objects ad the outstadig hutig skill are two distictive features of a cat. These two behavioral traits of cats are odeled for CSO: seekig ode ad tracig ode. Each cat has the followig attributes: its ow positio coposed of K diesios, velocity for each diesio, a fitess value that delieates the accoodatio of the cat to the fitess fuctio, ad a flag to idetify whether the cat is i seekig ode or tracig ode. The fial solutio of CSO algorith is to fid the cat that has the best fitess value. Tsai et al. [21] preseted a parallel cat swar optiizatio (PCSO) ethod based o the fraework of parallelizig the structure of the CSO ethod. Xu ad Hu [22] preseted a CSO algorith for resource costraied project schedulig proble. CSO based schee for the RCPSP has three ai stages: first radoly iitialize the paraeters of cats, the update the positio i iteratio ad calculate the fitess through serial SGS ethod, fially teriate the process if the coditio is satisfied. The IIR syste idetificatio task is forulated as a optiizatio proble ad a recetly itroduced Modified Cat Swar Optiizatio (MCSO) is used to develop a ew populatio based learig rule for the odel [23]. Liu ad She [24] itroduced a recet etaheuristic ethod CSO to fid the proper clusterig of data sets. Two clusterig approaches based o cat swar optiizatio called Cat Swar Optiizatio Clusterig (CSOC) ad K-haroic eas Cat Swar Optiizatio Clusterig (KCSOC) are proposed. IIR syste idetificatio task is forulated as a optiizatio proble ad a recetly itroduced cat swar optiizatio (CSO) is used to develop a ew populatio based learig rule for the odel [25]. Carle, Martel ad Zufferey [26] proposed a aget-based etaheuristic to solve large-scale ulti-period supply chai etwork desig probles. The geeric desig odel forulated covers the etire supply chai, fro vedor selectio, to productio distributio sites cofiguratio, trasportatio optios ad arketig policy choices. Furtherore Wag, Chag ad Li [27] adopted the CSO strategy to obtai the optial or ear optial solutio of the stego-iage quality proble. The CSO strategy is geerated by observig the behavior of cats, which has bee proved to achieve better perforace o fidig the best global solutios. CSO algorith is utilised as the traiig algorith ad the Optial Brai Daage ethod as the pruig algorith used for Optiizig Artificial Neural Networks [28]. For the past two decades, eta-heuristics have replaced the aalytical ethods. These eta-heuristics icludes Geetic Algorith, Artificial Bee Coloy Algorith Siulated Aealig, At Coloy Optiizatio, Particle Swar Optiizatio, Scatter search, Tabu search etc. O class of solutio procedures that is receivig reewed attetio, ad cosidered i this study is the Cat Swar Optiizatio (CSO) Algorith. This literature shows that several researchers have ade valuable cotributios i developig odels ad efficiet algoriths for liear FCTP. To the best of our kowledge, there is o published work for solvig FCTP by usig CSO Algorith. Hece this paper proposes CSO Algorith to provide optial or ear optial solutio for the sigle stage FCTP. The orgaizatio of the paper is as follows: Sectio 2 presets a atheatical forulatio of the sigle stage FCTP. Sectio 3, describes various odules of the proposed ABC Algorith. Sectio 4 provides uerical illustratio ad sectio 5, presets coclusio ad future work. 2. SINGLE STAGE SUPPLY CHAIN PROBLEM The Supply chai proble ca be stated as a distributio proble i which there are suppliers (warehouses or plats) ad custoers (destiatios or deads). Each of the suppliers ca ship to ay of the custoers at a shippig cost per uit c (uit cost for shippig fro supplier i to custoer j) plus a fixed cost f, assued for opeig this route. Each supplier i = 1; 2; : : : ; has a i uits of supply ad each custoer j = 1; 2; : : : ; has a dead of b j uits. The objective is to deterie which routes are to be opeed ad the size of the shipet o those routes, so that the total cost of eetig dead, give the supply costraits, is iiized. The various assuptios ivolved i this paper are described below: 1. Sigle product is delivered. 2. Capacities of suppliers ad deads of custoers are deteriistic ad are kow i advace. 3. Each custoer will be served by oe or ore facility. 4. Shortages are ot allowed. The atheatical odel for the sigle stage Supply chai proble is as follows. c x f iiize Z (1) i1 j1 688

j1 x S, for i 1,2,... (supplier costrait) (2) i x D j, for j 1,2,... (custoer costrait) (3) i1 x i1 0 otherwise S i D 0 j1 0, for i 1,2,..., ad j 1,2,... 1if x The Eq.(1) iiizes the total cost Z. The Eq.(2), iplies that the product shipped fro the suppliers does ot exceed its capacity. The Eq.(3) idicates that the total quatity of products shipped fro the plats to the custoers is to satisfy dead of the custoer. Fially, Eq.(4) describes that the total capacities of the plat ust be greater tha or equal to total dead of the custoers. 3. CAT SWARM OPTIMIZATION ALGORITHM Chu ad Tsai have proposed a ew optiizatio algorith which iitates the atural behavior of cats. Cats have a strog curiosity towards ovig objects ad possess good hutig skill. Eve though cats sped ost of their tie i restig, they always reai alert ad ove very slowly. Whe the presece of a prey is sesed, they chase it very quickly spedig large aout of eergy. These two characteristics of restig with slow oveet ad chasig with high speed are represeted by seekig ad tracig, respectively. I CSO these two odes of operatios are atheatically odeled for solvig coplex optiizatio probles. 3.1 SEEKING MODE The seekig ode correspods to a global search techique i the search space of the optiizatio proble. A ter used i this ode is seekig eory pool (SMP). It is the uber of copies of a cat produced i seekig ode. The steps ivolved i this ode are: 1. Create T (= SMP) copies of j th cat i.e. Y kd where (1 6 k 6 T) ad (1 6 d 6 D). D is the total uber of diesios. 2. Apply a utatio operator to Y k. 3. Evaluate the fitess of all utated copies. 4. Update the cotets of the archive with the positio of those utated copies which represet odoiated solutios. 5. Pick a cadidate radoly fro T copies ad place it at the positio of j th cat. 3.2 TRACING MODE The tracig ode correspods to a local search techique for the optiizatio proble. I this ode, the cat traces the target j (4) while spedig high eergy. The rapid chase of the cat is atheatically odeled as a large chage i its positio. Defie positio ad velocity of i th cat i the D-diesioal space as X i = (X i1, X i2,..., X id ) ad V i = (V i1, V i2,..., V id ) where d (1 6 d 6 D) represets the diesio. The global best positio of the cat swar is represeted as, X g = (X g1, X g2,..., X gd ). The steps ivolved i tracig ode are: 1. Copute the ew velocity of i th cat usig Eq.(5), Vid w* Vid c* r * X gd Xid (5) where, w is the iertia weight, c is the acceleratio costat ad r is a rado uber uiforly distributed i the rage [0, 1]. The global best X g is selected radoly fro the exteral archive. 2. Copute the ew positio of i th cat usig Eq.(6). Xid Xid Vid (6) 3. If the ew positio of i th cat correspodig to ay diesio goes beyod the search space, the the correspodig boudary value is assiged to that diesio ad the velocity correspodig to that diesio is ultiplied by _1 to cotiue the search i the opposite directio. 4. Evaluate the fitess of the cats. 5. Update the cotets of the archive with the positio of those cats which represet o-doiated vectors. 4. NUMERICAL ILLUSTRATION To evaluate the perforace of the preseted algorith, two previously addressed probles by Jo et al. with differet sizes are solved ad coparig with the solutio preseted by the, Maiara et al. [15] usig biary coded Geetic Algorith ad with solutio fro LINGO. The sizes of the probles are 4 5 ad 5 10 respectively. The fixed cost, capacity of the plats, deads of the custoer ad trasportatio cost are show i Table.1 for the proble 1 ad Table.2 & Table.3 shows for the proble 2. Table.1. Fixed cost ad uit trasportatio cost for Proble 1 Fixed costs Capacity Trasportatio cost Custoers 1 2 3 4 5 1 2 3 4 5 1 60 88 95 76 97 57 8 4 3 5 8 2 51 72 65 87 76 93 3 6 4 8 5 3 67 89 99 89 100 50 8 4 5 3 4 4 86 84 70 92 88 75 4 6 8 3 3 Dead 88 57 24 73 33 275 689

P MANIMARAN AND V SELLADURAI: CAT SWARM OPTIMIZATION FOR SINGLE STAGE SUPPLY CHAIN DISTRIBUTION SYSTEM WITH FIXED CHARGES Table.2. Fixed cost for Proble 2 Custoers 1 2 3 4 5 6 7 8 9 10 Capacity 1 160 488 295 376 297 360 199 292 481 162 157 2 451 172 265 487 176 260 280 300 354 201 293 3 167 250 499 189 340 216 177 495 170 414 150 4 386 184 370 292 188 206 340 205 465 273 575 5 156 244 460 382 270 180 235 355 276 190 310 Dead 225 150 90 215 130 88 57 124 273 133 1485 Table.3.Uit trasportatio cost for Proble 2 Custoers 1 2 3 4 5 6 7 8 9 10 1 8 4 3 5 2 1 3 5 2 6 2 3 3 4 8 5 3 5 1 4 5 3 7 4 5 3 4 2 4 3 7 3 4 1 2 8 1 3 1 4 6 8 2 5 4 5 6 3 3 4 2 1 2 1 The CSO algorith reaches its optial solutio usig two groups of cats, i.e. oe group cotaiig cats i seekig ode ad other group cotaiig cats i tracig ode. The two groups cobie to solve the optiizatio proble. A ixture ratio (MR) is used which defies the ratio of uber of cats i tracig ode to that of uber of cats i seekig ode (Chu ad Tsai). The coputatioal procedure of the basic CSO algorith is described as follows: 1. Radoly iitialize the positio of cats i D-diesioal space for the populatio, i.e. X id represetig positio of i th cat i d th diesio. 2. Radoly iitialize the velocity for cats, i.e. V id. 3. Evaluate the fitess of each cat ad store the positio of the cat with best fitess as P g where = 1, 2,..., D. 4. Accordig to MR, cats are radoly picked fro the populatio ad their flag is set to seekig ode, ad for others the flag is set to tracig ode. 5. If the flag of i th cat is seekig ode, apply the cat to the seekig ode process, otherwise apply it to the tracig ode process. The steps of the correspodig odes are followed. 6. Evaluate the fitess of each cat ad store the positio of the cat with best fitess as P l where = 1, 2,..., D. 7. Copare the fitess of P g ad P l ad update P g. 8. Check the teriatio coditio, if satisfied, teriate the progra. Otherwise repeat steps 4 7. The copariso is show i Table.4. Sl. No. Proble Size Table.4. Copariso of Results Jo et al. st-ga Maiara et al. GA LINGO Proposed CSO 1 4 5 1,642 1,484 1,544 1,484 2 5 10 6,696 6,467 6,719 6,290 5. CONCLUSION AND FUTURE WORK I this paper, a atheatical odel ad solutio procedure for sigle stage Supply chai proble usig CSO Algorith is proposed to fid the iiu trasportatio cost. To validate the efficiecy of the developed algorith, the results are copared with spaig tree based GA ad biary coded GA. The proposed ethod is ore efficiet cocerig the total cost. The structure of the proposed ethod is very siple ad we believe that this ethod will be efficiet to solve Supply chai probles. The work ay be exteded for ulti products, Custoer dead is assued as ot deteriistic ad for ulti objective tie ad custoer service level etc. This work ay be attepted with other heuristics such as Geetic Algorith, Particle Swar Optiizatio, Siulated Aealig Algorith, Tabu search, Scatter Search, Firefly Algorith etc., ACKNOWLEDGEMENT The authors are grateful to the reviewers for their valuable suggestios to iprove the quality of the paper. They thak the Maageets of Kaaraj College of Egieerig ad Techology ad Coibatore Istitute of Techology for the cooperatio ad ecourageet exteded with all facilities to carry out this work. REFERENCES [1] W. M. Hirsch ad G. B. Datzig, The fixed charge proble, Naval Research logistics Quarterly, Vol. 15, pp. 413-424, 1968. [2] U. S. Palekar, M. H. Karwa ad S. Ziots, A brach ad boud ethod for the fixed charge trasportatio proble, Maageet sciece, Vol. 36, No. 9, pp. 1092-1105, 1990. [3] V. Adlakha, K. Kowalski, R.R. Veugati ad B. Lev, More-for-less algorith for fixed-charge trasportatio probles, Oega, Vol. 35, No. 1, pp. 116-127, 2007. [4] J. Jo, Y. Li ad M. Ge, Noliear fixed charge trasportatio proble by spaig tree based geetic algorith, Coputers ad Idustrial Egieerig, Vol. 53, No. 2, pp. 290-298, 2007. [5] V. Adlakha ad K. Kowalski, O the Fixed-Charge Trasportatio proble, Oega, Vol. 27, No. 3, pp. 381-388, 1999. [6] Katta G. Murty, Solvig the fixed charge proble by rakig the extree poits, Operatios Research, Vol. 16, No. 2, pp. 268 279, 1968. 690

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