Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra

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1 Insttuto de Engenhara de Sstemas e Computadores de Combra Insttute of Systems Engneerng and Computers INESC - Combra Joana Das Can we really gnore tme n Smple Plant Locaton Problems? No ISSN: Insttuto de Engenhara de Sstemas e Computadores de Combra INESC - Combra Rua Antero de Quental, 199; Combra; Portugal

2 Can we really gnore tme n Smple Plant Locaton Problems? Joana Das Faculty of Economcs and INESC-Combra, Unversty of Combra, Portugal In smple plant locaton problems (SPLP), the tme dmenson s not explctly consdered, ether because there are not sgnfcant costs for relocatng facltes, or because the assgnment costs are not expected to change sgnfcantly as tme goes by. Nevertheless, locaton problems are strategc decsons by nature. In ths paper, we wll show how the explct consderaton of a plannng horzon, as well as the explct defnton of tme dependent assumptons, s essental n the defnton and applcaton of SPLPs because they can nfluence sgnfcantly the optmal decson. Keywords: locaton problems; plannng horzon; dscount rate; equvalent annual cost AMS Subject Classfcaton: 90C10; 90B80; 90C Introducton Smple plant locaton problems (SPLP) are, possbly, one of the most studed locaton problems of all tme. Consderng a set of locatons where facltes can be opened (at most one faclty at each locaton), and consderng a set of clents, the problem conssts of fndng the best set of locatons where facltes wll be opened, guaranteeng that each clent s assgned to exactly one opened faclty and mnmzng total costs: fxed costs assocated wth openng each faclty and assgnment costs related to the assgnment of each clent to an open faclty. SPLPs can be appled f a set of assumptons are fulflled, namely: a) There are no capacty constrants assocated wth the facltes. Ths means that t would be possble to have all clents assgned to one and only one opened faclty. b) Assgnment costs are not expected to change sgnfcantly durng the lfetme of the facltes, or f they do change, the change wll be of smlar order of magntude for all costs. c) Decsons regardng the locaton of facltes wll be taken at the present moment, and t s not necessary to plan now the openng or closng of facltes n the future. If assumpton a) s not fulflled, then we should consder capactated locaton problems, where each faclty has an upper lmt to the total amount of demand t can serve, or an 1

3 upper lmt to the total number of clents that can be assgned to t. If assumptons b) and c) are not fulflled, then dynamc locaton problems should be used, where tme s explctly consdered. Although SPLPs do not consder tme explctly, most of the tmes these problems consder strategc decsons, dffcult to revert and wth consequences that spread over long tme perods. In fact, when we defne fxed and assgnment costs, specal care should be taken regardng the way these costs are calculated, because they should reflect what s expected to happen durng the lfetme of the facltes. The fxed cost assocated wth openng a gven faclty should represent not only the fxed openng cost, but also the mantenance and operatonal costs durng the faclty s operatonal perod and possbly costs ncurred when the faclty s closed and ts salvage value. Assgnment costs should reflect the costs assocated wth assgnng clents to facltes durng the whole lfetme of the facltes. So, even when tme s not explctly consdered, an mplct assumpton regardng the defnton of a plannng horzon has to be consdered. In fact, ths s true not only for smple plant locaton problems but for all statc locaton problems, and t s of partcular mportance when we are dealng wth the applcaton of these mathematcal models to real world problems. If we are dealng wth facltes that show dfferent patterns of fxed and varable costs along the plannng horzon, then t s necessary to address the problem of how the overall cost wll be calculated, and the explct assumpton of a dscount rate, for nstance, s essental. We should also consder whether all facltes have the same lfespan. If not, a smple comparson of the cost flows assocated wth the facltes s mproper, and we should resort to concepts lke the equvalent annual cost. Regardng the locaton lterature, we can see that, most of the tmes, nformaton about how the objectve functon parameters should be calculated are absent, and sometmes values of completely dfferent nature and order of magntude are smply summed up together n the objectve functon. There are some few good examples. In [1], for nstance, there s an explct consderaton of annual equvalent costs assocated wth the facltes, although there s no explanaton about the assumptons made n ths calculaton. In [5], the authors explan how they have used a statc locaton problem consderng the long-run effects of the decsons and mnmzng the present value of total costs. In [3], the authors study the problem of locatng slaughterhouses under economes of scale, and carefully explan how costs are ncorporated nto the model. A locaton-routng problem appled to the locaton of ncnerators for the dsposal of sold anmal waste s studed n [4], and all costs were 2

4 calculated consderng the perod the ncnerator wll be n servce, but wthout further detals. In [6, 7], the authors descrbe the problem of locatng and decdng the capacty of plants for bottlng propane n south Inda, and consder the fxed locaton costs as beng calculated usng cash-flow patterns assocated wth each gven plant and sze when the plant s operatng at full capacty. In ths paper we ntend to show that tme should be explctly consdered n SPLPs, and that there are consequences of not takng tme nto account when applyng these models: we can end up wth a suboptmal soluton. In the next secton we defne the SPLP, and descrbe two dfferent ways of consderng tme n SPLP: usng future cash flows, dscounted at a gven dscount rate, or usng the equvalent cost concept [2]. In secton 3 we show some computatonal results. Secton 4 states the man conclusons. 2. Smple Plant Locaton Model The smple plant locaton problem can be defned as follows: ( 1 ) Mn c x f y j j I j J I Subject to: ( 2 ) Where: I set of possble locatons for facltes J set of clents y xj 1, j ( 3 ) I x y,, j 1, f a faclty s open at locaton y, I 0, otherwse 1, f clent j s assgned to faclty open at locaton xj, I, j J 0, otherwse c cost of assgnng clent j to faclty located at, I, j J j f fxed cost of openng a faclty at locaton, I j ( 4 ) 0,1, x 0,1 ( 5 ) j 3

5 The objectve functon mnmzes total cost (assgnment costs plus fxed costs assocated wth the openng of the facltes), constrants (2) guarantee that each clent wll be assgned to exactly one faclty, constrants (3) guarantee that clents wll only be assgned to opened facltes. The locaton varables varables j y are bnary. The assgnment x can be consdered as bnary varables, or x 0,1. As we are not consderng capacty constrants, n the optmal soluton each clent wll always be assgned to the faclty that has the mnmum assgnment cost, so x j wll always be 0 or 1, even f that s not explctly consdered n the model. Our attenton wll be focused on the objectve functon, manly consderng how should the values of c j and f be calculated. j 2.1. Fxed costs Let us frst consder the fxed locaton costs f. What do these costs represent? When we thnk about openng a new faclty, several dfferent stuatons can be consdered: we may have to buld the faclty, and even buld some nfrastructures; we may already own the buldng and openng a faclty wll requre the acquston of machnery, for nstance; we may be rentng a warehouse; and so on. Dfferent stuatons wll have dfferent costs assocated, but what s mportant to notce s that, n general, these fxed costs wll not be ncurred entrely at the present tme (at the tme when the decson s beng made). In general, these fxed costs wll arse n dfferent tme perods. We can magne that n the frst years the cost flows wll be greater, correspondng to the settng up of the faclty. Once the faclty s operatng at ts full potental, then there wll be fxed mantenance and operatng costs that have to be consdered. At the end of the faclty s lfetme, t s stll possble to consder a negatve cost, or a beneft, usually denomnated salvage value, that can be nterpreted as the remanng value of the asset. When talkng about facltes, salvage values can be sgnfcant due to the usual low deprecaton rate assocated. When we are consderng dfferent possble locatons for facltes, we may be facng a stuaton of comparng locatons wth completely dfferent cost flow patterns, so care must be taken when defnng f values. One way of solvng ths ssue s to resort to the concept of present value. Present value allows us to dscount future costs so that they are all n a common metrc and can then be comparable. Defnng f as the present value of all the fxed costs assocated wth openng one faclty at locaton has mplct 4

6 the assumpton of a gven plannng horzon and a gven dscount rate. The plannng horzon can be defned as the number of tme perods (let us consder years, for ease of the exposton) that the faclty s expected to be n operaton. The dscount rate can be seen as representng the tme value of money (we prefer to receve the same amount of money today than to wat, so f we are wllng to wat we should be compensated by recevng more) and also a rsk premum (we want to be compensated by the rsk we are takng wth the nvestment). In the locaton problem consdered, we are dealng wth a determnstc problem, wth no uncertanty assocated, so we can consder the dscount rate as representng the tme value of money alone. Ths means that we could consder usng a rsk-free rate as our dscount rate. Consder the followng example: There are two possble locatons for locatng warehouses, locaton A and locaton B. Locaton A has already a warehouse that we can rent by a year. Locaton B wll force us to buld the warehouse from scratch, wth an ntal cost of n the frst year, and then mantenance costs of 3000 per year. We are thnkng about usng these facltes durng 10 years. How should f and 1 f 2 be calculated? We should consder all the costs assocated wth each potental locaton n each year of the plannng horzon, as shown n Table 1. Table 1 Cost flows assocated wth two facltes Year Locaton A Locaton B PV f The present value (PV) of a flow of costs C, t 1,..., be calculated as follows 1 [2]: t T, consderng a dscount rate r can 1 In ths case we are consderng that costs are ncurred at the end of the correspondng tme perod. We could also consder that the costs would be ncurred at the begnnng of the tme perod, by consderng T 1 Ct t 0,..., T 1: PV. The present values would be slghtly changed to and t t 0 (1 r)

7 T Ct PV ( 6 ) t t 1 (1 r) Wth r equal to 2%, openng the faclty at A wll have a fxed cost of and openng a faclty at B wll have a fxed cost of But magne now that after 10 years, we would be able sell the warehouse located at B, so that we would have a beneft at the end of the warehouse s lfetme. Imagne that the beneft could be estmated n Ths value should also be taken nto account n the calculaton of the present value f 2, that would now be decreased to In the prevous example we consdered that we wll be able to use the facltes durng the whole plannng horzon. But what f we are dealng wth facltes that have dfferent lfespans? Facltes wth dfferent tme frames cannot be drectly compared, because f we calculate fxed costs as shown n (6) we wll be havng some facltes accumulatng more costs than others. Imagne, for nstance, that we want to nstall new plants, and n each potental locaton we can consder buldng a faclty that s expected to last for 5 years and/or buldng a faclty that s expected to be n operaton durng 10 years 2. Calculatng the present values assocated wth each one of the optons for a gven locaton, magne we end up wth f = and f = , where f refers to the present value assocated wth the 5-years opton, and f refers to the present value assocated wth the 10-years opton. Should these be the values to be used n the objectve functon (1)? In realty these values should not be summed up together, because they represent values n dfferent metrcs. One easer way of solvng ths problem s resortng to the concept of Equvalent Annual Cost (EAC). The dea of the EAC s to consder a cost per perod, such that f ncurred each year durng the whole plannng perod we would end up wth the same PV assocated wth the cost of the faclty tself. The EAC can be defned as T follows, where a r represents the annuty factor and T s the consdered plannng horzon [2]: PV EAC a T r ( 7 ) a T r T 1 1 r ( 8 ) r 2 The possblty of havng two dfferent facltes n operaton n the same locaton can be easly ncorporated nto the SPLP by consderng two potental fcttous locatons that correspond to the same physcal locaton. 6

8 Consderng our example, we would end up wth EAC = and EAC = These values could be used as the facltes fxed costs n SPLP. If all facltes have exactly the same lfespan, then EAC or PV are two equvalent approaches Assgnment costs Let us now consder costs c j. These costs should represent the assgnment costs: how much does t cost to assgn clent j to the faclty located at. In order to defne these costs properly, we need to defne the tme perod assocated wth these costs. Ether cj could represent the costs ncurred durng one tme perod, or t could represent the cost of assgnng clent j to faclty durng the whole plannng horzon. These assgnment costs wll be added to the total fxed costs, so care has to be taken to ensure that we are consderng coherent metrcs. If we have defned the fxed costs as beng equal to the PV of the costs flow durng the plannng horzon, then we should also consder the PV of the assgnment costs durng the plannng horzon. If these costs are constant durng the plannng horzon then the PV can be easly calculated by (9): PV c j ( 9 ) T ar If we have chosen to defne the facltes fxed costs as beng equal to EAC, then we should only consder the assgnment costs for one tme perod Example As can easly be seen, whatever the choce made by the modeler, the optmal soluton obtaned wll be dependent on two mportant model parameters: the plannng horzon and the dscount rate used. These parameters are not explctly present n the model, but wll have a determnant role n the optmal soluton calculated. Let us now llustrate these concepts wth a smple example. Consder a problem wth 5 potental locatons where we can open facltes, and 10 clents that have to be assgned to an open faclty. The spatal dstrbuton of clents and potental locatons for facltes 7

9 Clents s represented n fgure 1. For each faclty that s opened, we wll ncur n a fxed openng cost (that ncludes the operatng cost durng the frst year), and a fxed annual operatng cost. The value of each faclty wll deprecate at a rate of 20% per operatng year, allowng us to estmate ts salvage value at the end of the plannng horzon. Assgnment costs are constant throughout the plannng horzon. Tables 2 and 3 depct ths nformaton. Fgure 1 Spatal dstrbuton of clents and potental locatons for facltes Table 2 Facltes fxed and operatng costs Faclty Fxed openng cost Fxed annual operatng cost Table 3 Annual assgnment costs Facltes

10 Let us consder a plannng horzon of 10 years, and a dscount rate equal to 5%. The optmal soluton to ths problem would be to open faclty 2 only, as depcted n fgure 2. Fgure 2 Opened faclty, for T=10 and r=5% If we know consder a dscount rate equal to 10%, then the optmal soluton would be to open faclty 3 nstead (fgure 3). Fgure 3 Opened faclty, for T=10 and r=10% If we now consder a dscount rate of 10%, but wth a plannng horzon of 5 years, then the optmal soluton would be to open faclty 5 only (fgure 4). Fgure 4 Opened faclty, for T=5 and r=10% 3. Computatonal results To assess the nfluence that the defnton of dfferent plannng horzons and dscount rates could have on the optmal soluton, several smple plant locaton nstances were 9

11 randomly generated and solved. The nstances were generated accordng to the followng procedure: 1. Random generaton of (x, y) coordnates n the plane, accordng to a unform dstrbuton and consderng a square. These coordnates correspond to the locaton of clents and potental locatons for facltes. 2. Random creaton of arcs between the network nodes, consderng a probablty of 75%. 3. Creaton of arcs (not created n step 2) between nodes such that the Eucldean dstance from one another s less than 50, wth probablty of 80%. 4. Generaton of costs assocated wth arcs: for the frst perod, the costs are randomly generated accordng to a unform dstrbuton, n the nterval [100,1100]. For t >1, the cost assocated to the arc n perod t s equal to the cost n t 1 plus a changng factor randomly generated correspondng to a varaton between 10% and +10%. 5. For each tme perod, calculaton of the shortest path between each clent and each faclty, usng the Floyd Warshall algorthm. 6. For each faclty and perod t, random generaton of fxed and mantenance and operatonal costs. Facltes can be of one of two types: hgh setup costs and low mantenance and operatonal costs, or low setup costs and hgh mantenance and operatonal costs. In the frst case, fxed costs are randomly generated n the nterval [2000,10000]. In the latter, the nterval consdered s [500,3500]. Mantenance costs are calculated as a percentage of fxed costs, randomly generated usng a unform dstrbuton n the nterval [0%,10%] or [20%,75%] accordng to the type of faclty. Table 4 shows the dmenson of the randomly generated problems. In total 1620 problems were generated consdered all facltes of the same type, and another 1620 problems were generated consderng facltes of dfferent types (the choce of the type of faclty was randomly generated wth equal probabltes). Table 4 Dmenson of randomly generated nstances Number of tme perods Dscount rate Number of potental locatons for facltes Number of clents 5 0% % % The am of these computatonal results s the followng: to see f the choce of the plannng horzon and the dscount rate does or does not nfluence the optmal soluton, and how much could we lose f these two parameters were not approprately chosen. Two dfferent types of experments were carred out: 1. Consderng the dscount rate fxed, change the plannng horzon: ths wll allow us to see how much we can lose f we consder a soluton calculated wth a gven plannng horzon, but then the facltes stay n operaton durng a dfferent plannng horzon. 10

12 2. Consderng the plannng horzon fxed, change the dscount rate: ths wll allow us to see the nfluence of the dscount rate. As an example, consder that for a gven problem a dscount rate of 5% and a plannng horzon of 5 years were consdered. The optmal soluton s calculated, but after mplementng the soluton t was decded that the facltes would be operatng durng 10 years. How much are we loosng because we dd not consder a correct plannng horzon rght from the begnnng? Table 5 shows the results obtaned when we consder the dscount rate fxed and a plannng horzon of 5 years when takng the decson. We then calculate the mnmum, average and maxmum loss n the objectve functon value f, n fact, the facltes stay n operaton durng 10 or 25 years. Smlar results are presented n tables 6 and 7, for plannng horzons of 10 and 25 years. Tables 8 to 10 show smlar results, but now when we consder solvng the model wth a gven dscount rate, and then change ths dscount rate. Table 5 Plannng horzon equal to 5 years, dscount rate fxed. T=10 T=25 M N Mn Average Max Mn Average Max % 0.49% 3.69% 0.00% 3.78% 15.96% % 0.45% 4.67% 0.00% 4.06% 13.57% % 0.64% 3.01% 0.00% 3.03% 8.81% % 0.43% 1.69% 0.00% 1.70% 4.92% % 0.26% 0.78% 0.00% 0.54% 1.21% % 0.02% 0.29% 0.00% 0.03% 0.45% % 0.71% 7.53% 0.00% 5.89% 24.13% % 0.84% 4.39% 0.00% 3.18% 12.02% % 0.62% 2.66% 0.00% 3.13% 10.41% % 0.63% 2.03% 0.54% 3.31% 7.01% % 0.45% 1.19% 0.21% 2.45% 4.89% % 0.22% 0.60% 0.00% 0.79% 1.87% % 0.72% 2.44% 0.00% 4.63% 12.62% % 0.67% 1.87% 0.16% 3.80% 11.24% % 0.58% 1.90% 0.70% 3.39% 8.82% % 0.45% 0.95% 0.75% 2.48% 5.06% % 0.67% 1.85% 1.05% 4.87% 13.81% % 0.65% 2.04% 0.76% 3.81% 9.57% Table 6 Plannng horzon equal to 10 years, dscount rate fxed. T=5 T=25 M N Mn Average Max Mn Average Max % 0.38% 2.42% 0.00% 0.93% 5.72% % 0.28% 1.57% 0.00% 1.08% 3.90% % 0.56% 3.78% 0.00% 0.76% 2.72% % 0.31% 1.49% 0.00% 0.37% 1.83% % 0.21% 0.75% 0.00% 0.07% 0.41% % 0.03% 0.35% 0.00% 0.00% 0.08% % 0.50% 3.37% 0.00% 1.91% 9.01% % 0.84% 3.95% 0.00% 0.73% 3.84% % 0.43% 1.62% 0.00% 0.87% 4.26% % 0.46% 1.22% 0.04% 0.81% 2.36% % 0.35% 1.16% 0.00% 0.63% 1.48% % 0.19% 0.67% 0.00% 0.16% 0.59% % 0.56% 2.55% 0.00% 1.39% 4.45% % 0.56% 2.39% 0.00% 1.08% 4.27% % 0.52% 1.78% 0.00% 0.94% 3.03% % 0.49% 0.87% 0.12% 0.64% 1.46% % 0.62% 1.95% 0.00% 1.46% 4.90% % 0.47% 1.24% 0.08% 1.04% 2.95% 11

13 Table 7 Plannng horzon equal to 25 years, dscount rate fxed. T=5 T=25 M N Mn Average Max Mn Average Max % 2.39% 9.18% 0.00% 1.23% 7.82% % 2.12% 10.04% 0.00% 1.16% 5.46% % 2.44% 11.82% 0.00% 0.77% 3.47% % 1.45% 4.39% 0.00% 0.45% 2.01% % 0.73% 1.77% 0.00% 0.10% 0.59% % 0.08% 0.73% 0.00% 0.01% 0.22% % 2.71% 9.89% 0.00% 1.09% 8.57% % 3.10% 10.53% 0.00% 0.74% 4.30% % 2.01% 5.88% 0.00% 0.62% 2.53% % 2.31% 5.01% 0.00% 0.87% 2.86% % 1.69% 4.04% 0.00% 0.63% 1.84% % 0.76% 1.90% 0.00% 0.21% 0.52% % 2.67% 6.39% 0.00% 0.97% 4.01% % 2.46% 7.32% 0.00% 0.78% 2.60% % 2.26% 5.84% 0.00% 0.76% 2.30% % 1.92% 3.58% 0.05% 0.60% 1.59% % 2.83% 7.56% 0.00% 1.03% 2.48% % 2.35% 5.59% 0.06% 0.88% 2.80% Table 8 Dscount rate equal to 0%. Plannng horzon fxed. 5% 10% M N Mn Average Max Mn Average Max % 0.66% 5.88% 0.00% 1.81% 9.18% % 0.39% 2.08% 0.00% 1.51% 10.04% % 0.30% 1.88% 0.00% 1.23% 5.14% % 0.22% 1.13% 0.00% 0.72% 3.67% % 0.06% 0.46% 0.00% 0.34% 0.94% % 0.00% 0.00% 0.00% 0.03% 0.32% % 0.29% 1.73% 0.00% 2.01% 9.30% % 0.54% 2.63% 0.00% 1.28% 7.04% % 0.40% 1.85% 0.00% 1.12% 4.31% % 0.27% 0.95% 0.01% 1.40% 5.01% % 0.21% 1.21% 0.00% 0.85% 2.69% % 0.07% 0.28% 0.00% 0.32% 0.80% % 0.65% 3.01% 0.00% 1.68% 6.39% % 0.53% 1.75% 0.00% 1.31% 3.49% % 0.36% 1.28% 0.13% 1.13% 3.07% % 0.28% 0.61% 0.07% 0.92% 2.50% % 0.55% 1.91% 0.05% 1.52% 4.13% % 0.47% 1.82% 0.02% 1.19% 3.76% Table 9 Dscount rate equal to 5%. Plannng horzon fxed. 0% 10% M N Mn Average Max Mn Average Max % 0.45% 3.37% 0.00% 0.45% 3.41% % 0.40% 2.54% 0.00% 0.37% 2.51% % 0.42% 1.54% 0.00% 0.35% 1.41% % 0.17% 0.88% 0.00% 0.18% 1.15% % 0.05% 0.34% 0.00% 0.08% 0.44% % 0.00% 0.00% 0.00% 0.01% 0.15% % 1.17% 7.56% 0.00% 0.50% 2.73% % 0.49% 2.90% 0.00% 0.25% 1.96% % 0.47% 2.44% 0.00% 0.24% 1.29% % 0.33% 0.75% 0.00% 0.36% 1.30% % 0.22% 0.58% 0.00% 0.20% 0.94% % 0.04% 0.17% 0.00% 0.08% 0.27% % 0.76% 2.33% 0.00% 0.35% 1.42% % 0.58% 2.45% 0.00% 0.26% 1.09% % 0.48% 1.81% 0.00% 0.24% 0.69% % 0.29% 1.12% 0.00% 0.21% 0.53% % 0.75% 2.99% 0.00% 0.30% 1.16% % 0.46% 1.50% 0.00% 0.25% 1.48% 12

14 Table 10 Dscount rate equal to 10%. Plannng horzon fxed. 0% 5% M N Mn Average Max Mn Average Max % 1.81% 10.27% 0.00% 0.25% 2.80% % 1.58% 7.44% 0.00% 0.38% 3.33% % 1.37% 5.17% 0.00% 0.23% 1.97% % 0.70% 2.44% 0.00% 0.14% 0.90% % 0.27% 0.89% 0.00% 0.11% 0.41% % 0.01% 0.12% 0.00% 0.01% 0.12% % 3.28% 19.50% 0.00% 0.66% 4.83% % 1.74% 8.19% 0.00% 0.23% 2.95% % 1.59% 5.96% 0.00% 0.24% 1.10% % 1.31% 3.32% 0.00% 0.40% 1.64% % 0.91% 2.76% 0.00% 0.25% 1.08% % 0.25% 0.60% 0.00% 0.10% 0.42% % 2.52% 7.50% 0.00% 0.33% 2.14% % 1.97% 6.88% 0.00% 0.26% 0.88% % 1.59% 5.39% 0.00% 0.27% 1.14% % 1.09% 3.27% 0.00% 0.22% 0.89% % 2.41% 8.23% 0.00% 0.37% 1.40% % 1.64% 4.95% 0.00% 0.26% 1.18% From the computatonal results, we can conclude that when we consder smlar facltes, about 30% of the problems optmal solutons seem to be robust to changes n the plannng horzon or dscount rate. When we consder dssmlar facltes, ths value decreases to 13%. If all facltes have smlar cost flow patterns, then the average loss s about 0.57%, and the maxmum loss s equal 12.07%. If facltes have dfferent cost flow patterns these numbers rse to 1.32% and 24.13% respectvely. It seems that the problem s more senstve to changes n the plannng horzon than changes n the dscount rate. If we keep the number of potental locatons fxed then, when the number of clents ncreases, the senstvty regardng these parameters decreases. If we ncrease the number of potental locatons for facltes, the senstvty regardng these parameters ncreases. 4. Conclusons Although the smple plant locaton problem does not consder explctly tme n ts formulaton, tme dependent assumptons should always be explctly defned, namely what are the plannng horzon that s beng consdered and the value for the dscount rate. Ths s partcularly true f we are dealng wth facltes that have dfferent flow cost patterns, or dfferent lfespans. And f we are dealng wth real world applcatons ths s crucal. As a matter of fact, n every statc locaton model that s used to represent a locaton problem, we should defne tme dependent assumptons. In dynamc locaton models, the plannng horzon s explctly defned as beng part of the defnton of the locaton 13

15 varables, but n ths case there s the need to explctly determne the dscount rate that s used and that enables us to sum up the costs ncurred n dfferent tme perods. Consderng the nfluence that these parameters can have on the optmal soluton, t can be consdered a good practce to perform senstvty analyss consderng dscount rate and plannng horzon, to assess the senstvty of each partcular problem to tme dependent assumptons. References [1] Antunes, A.P., Locaton Analyss Helps Manage Sold Waste n Central Portugal. Interfaces, (4): p [2] Brealey, R.A. and S.C. Meyers, Prncples of Corporate Fnance. 2003: McGraw Hll. [3] Broek, J.v.d., P. Schütz, L. Stouge, and A. Tomasgard, Locaton of slaughterhouses under economes of scale. European Journal of Operatonal Research, : p [4] Caballero, R., M. González, F.M. Guerrero, J. Molna, and C. Paralera, Solvng a multobjectve locaton routng problem wth a metaheurstc based on tabu search. Applcaton to a real case n Andalusa. European Journal of Operatonal Research, : p [5] Köksalan, M. and H. Süral, Efes Beverage Group Makes Locaton and Dstrbuton Decsons for ts Malt Plants. Interfaces, (2): p [6] Sankaran, J. and N.R.S. Raghavan, Locatng and Szng Plants for Bottlng Propane n South Inda. Interfaces, (6): p [7] Sankaran, J.K., On solvng large nstances of the capactated faclty locaton problem. European Journal of Operatonal Research, : p