Using the Constructive Genetic Algorithm for Solving the Probabilistic Maximal Covering Location-Allocation Problem

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1 Usng the Constructve Genetc Algorthm for Solvng the Probablstc Maxmal Coverng Locaton-Allocaton Francsco de Asss Corrêa Insttuto Naconal de Pesqusas Espacas São José dos Campos - SP, Brasl fcorrea@drectnet.com.br Luz Antono Noguera Lorena Insttuto Naconal de Pesqusas Espacas São José dos Campos SP, Brasl lorena@lac.npe.br Abstract The Maxmal Coverng Locaton (MCLP) maxmzes the populaton that has a faclty wthn a maxmum travel dstance or tme. Numerous extensons have been proposed to enhance ts applcablty, lke the probablstc model for the maxmum coverng locaton-allocaton wth constrant n watng tme or queue length for congested systems, wth one or more servers per servce center. In ths paper we present one soluton procedure for that probablstc model, consderng one server per center, usng the Constructve Genetc Algorthm. The results of tests on the soluton procedure are presented. 1. Introducton The Maxmal Coverng Locaton (MCLP) has been extensvely studed n the lterature snce ts formularzaton made by Church and ReVelle (1974). The man obectve of the MCLP s to choose the locaton of facltes to maxmze the populaton that has a faclty wthn a maxmum travel dstance (or tme). Thus, a populaton s consdered covered f t s wthn a predefned servce dstance (or tme) from at least one of exstng facltes. Consderable revson of ths subect can be found n Hale and Moberg (2003), Serra and Maranov (2004) and Galvão (2004). The MCLP does not requre that all demand areas be covered, but offers attendance to the maxmum populaton, consderng the avalable resources. Some useful applcatons are extensons of ths formularzaton. In many works nvolvng locaton problems, the dstance (or tme) between demand ponts and the facltes to whch they are beng located are the factor that represents the qualty of the servces that are gven to the users. However, when servce networks are proected, as health systems or bankng, the locaton of servce centers has a strong nfluence on the congeston of each of them, and, consequently, the qualty of servces must be better defned and not only consderng the travel dstance or tme. The centers must be located to allow the users to arrve at the center n an acceptable tme, s also desrable that the watng tme for attendance s no longer than a gven tme lmt or nobody stands on lne wth more than a predetermned number of other clents. These are mportant parameters n the measure of the desred qualty [11]. Congeston happens when a center s not capable to deal, smultaneously, to all the servce requests that are made to t. Normally, the tradtonal models that deal wth congeston nclude a capacty constrant, whch forces the demand for servce, normally constant n tme and equal to an average, to be smaller than the maxmum capacty of the center all the tme. Ths s a determnstc approach to the problem, because does not consder the dynamc nature of the congeston. Dependng on how the capacty constrant s developed, ths makes that the soluton model presents dle servers, or s a system that s not capable to attend all the demand [11][12]. Maranov and Serra (1998) proposed models based on the fact that the number of requests for servces are not constant n tme, but a stochastc process, whose stochastcty of demand s explctly consdered n the capacty constrants. Instead of beng lmted to a maxmum value, the authors defne a mnmum lmt for the qualty of the servce reflected n the watng tme or the number of people watng for servce. Those researchers address the formulaton of several maxmal coverage models, wth one or more servers

2 per servce center, so that all the populaton s served wthn a standard dstance (or tme), and nobody stands on lne more than a gven tme lmt, or wth more than a gven number of other clents, wth a probablty of at least ϕ. The purpose of ths paper s to examne the Queung Maxmal Coverng Locaton-Allocaton Model (QM- CLAM) wth one server per servce center, proposed by Maranov and Serra (1998), and present a soluton usng the Constructve Genetc Algorthm (CGA). The QM-CLAM s brefly dscussed n Secton 2, the CGA s descrbed n Secton 3, computatonal results are reported n Secton 4 and conclusons are presented n Secton QM-CLAM The tradtonal Maxmum Coverng Locaton (MCLP) proposed by Church and ReVelle (1974) can not be used to deal wth the congeston constrants, because there are no allocaton varables. Then, t s mpossble to compute the requests of servces that arrve at a center, and, consequently, to determne when congeston occurs. Thus, the MCLP has been rewrtten as p-medan-lke model, modfed to accommodate the locaton and allocaton varables. The obectve s to maxmze the covered populaton, consderng a predefned number of servce centers. An nteger lnear programmng formulaton for the QM_CLAM s obtaned by ntroducng the followng varables. Let y = 1 f a center s located at a node and y = 0 otherwse; x = 1 f the users located at demand node s allocated to a center located at, and x = 0 otherwse. We consder I and N, such as I s a set of demand nodes, and N s ether a set of canddate locatons that are wthn a standard dstance from node, or the set of canddate locatons whch can be reached from node, wthn a certan standard tme. Let a be the total populaton at demand node. The QM_CLAM can be formally stated as: a x v(qm-clam) = Max, (1) Subect to x y (2), N I x z x 1 (3) b+2 µ 1 ϕ (4) or I z x = y x 1 + ln(1 ϕ) τ µ (4a) y p (5) { 0,1}, N,, (6) The obectve (1) maxmzes the populaton allocated to a center. Constrant (2) defnes that a demand pont can be allocated to a node only f there s a center n. Constrant (3) forces each demand node to be allocated to at most one servce center. Constrants (4) force to each center has no more than b people on a lne, wth a probablty of at least ϕ. Constrants (4a), make the total tme spent by a user at a center shorter than or equal to τ, wth a probablty of at least ϕ. Constrant (5) sets the number of centers to be located. Constrants (6) defne the ntegralty requrements. In order to wrte constrant (4), the authors made assumpton that requests for servce at each demand node appear accordng to a Posson process wth ntensty z. The servce requests at a center are the unon of the requests for servce of the demand nodes, and they can be descrbed as another stochastc process, equal to the sum of several Posson processes, wth ntensty λ : λ = z x (7) I Whch means that, f the varable x s one, node s allocated to center and the correspondng ntensty z wll be ncluded n computaton of λ. They have consdered the well known results for a M/M/1 queung system for each center and ts allocated users [8]. An exponentally dstrbuted servce tme, wth an average rate µ, has been consdered n those models, where µ λ, otherwse the system does not reach the equlbrum. The QM-CLAM belongs to the NP-Complete class of problems [16]. Even usng commercal solvers s not always possble to fnd the optmal soluton n a reasonable computatonal tme, due to ts classfcaton and to the problem sze. Therefore, alternatve methods are nvestgated. In the next chapter a soluton based on the Constructve Genetc Algorthm wll be presented. 3. Constructve Genetc Algorthm Genetc algorthms have been developed by John Holland, hs colleagues, and hs students at the Unversty of Mchgan. Refnements of the method have been mplemented n the followng decades. Some of them and basc nformaton can be acheved n Lacerda and Carvalho (1999) and Goldberg (1989).

3 The Constructve Genetc Algorthm (CGA) [2][9][13][14] works wth a populaton formed by schemata (ncomplete solutons) and structures (complete soluton). In ths work, structures and schemata can be generated by a strng representaton, usng the symbols 1, 2 and #. The schemata make an explct reference to the symbol # ( do not care ) and represent a populaton of partal solutons, who s a base for the constructon of a populaton wth complete better solutons, throughout the evolutonary process. The symbol 1 represents a center; the symbol 2 represents a demand pont to be allocated to a center and the do not care symbol represents a non defned pont to the problem, whch wll become a center or a demand pont durng the evolutonary process. The Fgure 1 shows an example of an ndvdual S for a problem wth 10 demand ponts and 2 centers. Constrants (2), (3) and (5) of QM_CLAM are mplctly consdered n ths representaton. The capacty constrant (4) or (4a) s consdered durng a demand pont allocaton, when addng ts ntensty z, the remanng capacty of the center s not exceeded. QM_CLAM s a clusterng problem, and any clusterng algorthm wll attempt to determne some nherent or natural groupng n the data, usng dstance or smlarty measures between ndvdual data [9]. In ths case, once a center s chosen, a cluster s determned by allocatng demand ponts to t that are wthn a coverage dstance, that satsfes ts capacty and are not allocated to any other center Fgure 1. S = (2221#1222#) (Furtado (1998)). The CGA has the tradtonal operators: selecton, recombnaton and mutaton, and dffers from a classcal GA n the way to evaluate the schemata (fgftness), n the possblty to use heurstcs to defne the ftness evaluaton functon and n the treatment of a dynamc populaton [14]. The fg-fness s a double ftness evaluaton of an ndvdual S k P α, where P α s a populaton at evolutonary nstant α. The f value represents the obectve functon, f: P α R +, expresson (1), and the g value s calculated by a heurstc, g: P α R +,such that g(s k ) f(s k ), to all S k P α. The frst evaluates the qualty of ndvdual and the second apples a problemspecfc heurstc (called tranng heurstc) to evaluate the neghborhood of ndvdual, beng the value of the best soluton found attrbuted to g. In ths work, the heurstc used for g calculaton s based on the algorthm used to mprove the prmal solutons n Perera and Lorena (2001). Ths heurstc searches for a new center n each cluster, swappng the current center wth a non-center node n the same cluster, changng the allocaton soluton. Ths change may alter both allocaton and coverng confguraton, so an algorthm for recalculatng the coverage s needed and was mplemented. The algorthm stops when swappng the current center do not mprove the value of the best reallocaton. The Constructve Genetc Algorthm works wth a dynamc populaton, ntally formed only by schemata, whch s enlarged after the use of recombnaton operators, or made smaller along the generatons, guded by an evolutonary parameter. That populaton s bult, generaton after generaton, by drectly searchng for well-adapted structures (complete soluton) and also for good schemata. The evolutonary process consders an adaptve reecton threshold, whch defnes the adaptaton of an ndvdual. Ths adaptaton s proportonal to ts rankng δ, calculated by equaton (8), d. Gmax [ g( Sk ) f ( Sk )] δ = (8) d. [ Gmax g( Sk )] that s composed by: A component concernng the adaptaton of ndvdual n relaton to the tranng heurstc (g-f). A component (G max g(s k )) that prvleges the maxmzaton of the functon g, calculatng the dstance between the ndvdual and an upper bound for all possble values for the f and g functons (G max ). A constant d, 0 d 1, to balance the components of the equaton. Thus, better ndvduals have greater ranks. The ntal populaton P 0 s made by schemata and, when the ndvduals receve ther correspondents rankng values. The ndvduals are sorted by decreasng values of δ. The populaton s then controlled n a dynamc way (see Fgure 2) by an adaptatve reecton threshold α, calculated by equaton (9), that uses the current populaton sze P, the best (δ 1 ) and the worst (δ P ) rankngs of ndvduals n current populaton, the estmated remanng number of generatons, RG, the constant ε that controls the speed of

4 evolutonary process, and l that guarantee a mnmum step n that process. ( δ1 δ P ) α t = α t + P 1 ε.. + l (9) RG The ntal α value s the worst rankng of ndvduals (δ P ) n the ntal populaton. At the end of each generaton the ndvdulas less adapted (δ(s k ) α) are elmnated from the populaton. The best ndvdual for each generaton s kept to defne, at the end of the evolutonary process, the best soluton found. P o p u l a t o n _20_0_2_ G e n e r a t o n Fgure 2. Populaton at each generaton Two structures or schemata are selected for recombnaton The frst, called base, s obtaned from the 20% better ndvduals of the populaton (S base ). The second, called gude, s randomly selected out from the whole populaton (S gude ). In the recombnaton operaton, the current labels n correspondng postons are compared. Let S new be the new offsprng after recombnaton. The structure or schema S new s obtaned by applyng the folowng operatons, based on Furtado (1998) and Olvera and Lorena (2005): S base = # and S gude = # then S new = # S base = 1 and S gude = 1 then S new = 1 S base = 2 and S gude = 2 then S new = 2 S base = 1 and S gude = # then S new = 1 S base = 2 and S gude = # then S new = 2 S base = # and S gude = 2 then S new = 2 S base = # or 2 and S gude = 1 then S new = 1 or 2, chosen randomly S base = 1 and S gude = 2 then S new = 1 ou 2, chosen randomly After a recombnaton operaton, more than p values 1 can appear n the S new. That condton s not relaxed and S new must be valdated to guarantee the constrant (5), by nsertng or removng values 1 from that offsprng. The mutaton operator changes a center wth a neghbor that t s not covered by any center. Ths ncreases the possblty of all the demand ponts to be centers. 4. Computatonal results The CGA was tested n the 30-node network provded by Maranov and Serra (1998) and n a 324-node network gotten from a geographcal data base of São José dos Campos-SP, Brasl, ncreased by fcttous populaton n each demand pont. The last one s avalable at By varyng the p, b, µ, ϕ e τ parameters, varous problems have been created. The results from CGA have been compared to the results obtaned usng the commercal solver CPLEX, verson 7.5 [7]. For the mplementaton of the QL_CLAM, the servce centers are prmary health care centers, wth one physcan at each center. Each demand pont s also a potental center locaton, and the dstances are Eucldean. To the 30-node network, t has been consdered: covered dstance equals to 1,5 mles; average servce tme (1/µ) was set at 20 mnutes; call rates were set at tmes the node populaton for the constraned queue length and 0,006 tmes the node populaton for the constraned watng tme, all defned on Maranov and Serra (1998). To the 324-node network, t has been consdered: covered dstance equals 250 meters, average servce tme (1/µ) was set at 15 mnutes; call rates were set at 0,01 tmes the node populaton for both constraned queue length and constraned watng tme. The problems have been codfed n the followng way: number of ponts, number of center, constrant type (0 for the constraned queue length and 1 for the constraned watng tme), number of people n lne or watng tme, and probablty. Example: 324_20_0_2_95, whch means 324 ponts, 20 centers, constraned queue length, maxmum of two people n lne, wth the probablty, at least, 95%. The CGA code has been wrtten n Obect Pascal. The tmes n tables are shown n seconds and have been determned n a Pentum IV 3 GHz computer, wth 1Gb of RAM memory, for the 324-node network, and n a Pentum III 800 MHz computer, wth 384 Mb of RAM, for the 30-node network. The tmes for the attanment of the CPLEX solutons was lmted n 2 hours (7200 seconds), except for the nstances marked wth one *, that t defnes a stop n the executon due to out of memory error. The results of the CGA have been gotten usng the values of the parameters shown n Table 1. Table 1. CGA parameters Parameters 30 and 324-node network G Max 1,1 tmes the sum of the populaton of all demand ponts d 0.1 ε l

5 Crossover / generaton 30 Mutaton probablty 0.20 Intal populaton 300 Maxmal number of 300 generatons The tables 2 and 3 show the results obtaned for the 324-node network and tables 4 and 5 show the results for the 30-node network. Tables 2 and 4 show the best nteger solutons found and the Gap Cplex to the 324-node and 30-node networks, respectvely. The values of Gap Cplex equal zero defne that the optmal has been acheved. Tables 3 and 5 show the results found by CGA to the 324- node and 30-node networks, respectvely. Table 2. CPLEX results for the 324-node network CPLEX CPLEX Soluton Gap Cplex (%) 324_10_0_0_ _10_0_1_ * _10_0_2_ _10_0_0_ _10_0_1_ _10_0_2_ _20_0_0_ _20_0_1_ _20_0_2_ _20_0_0_ _20_0_1_ _20_0_2_ * _10_1_40_ _10_1_41_ * _10_1_42_ _10_1_48_ _10_1_49_ _10_1_50_ _20_1_40_ * _20_1_41_ * _20_1_42_ * _20_1_48_ _20_1_49_ * _20_1_50_ The results for the CGA reflect ffty executons of each problem and are shown n four columns: the best value found (column Best soluton), the average value (column Average soluton), the average tme (column Tme) and the column Devaton, that reflects the relatve error of the average soluton for the CGA, relatve to the best found prmal soluton, and are calculated by (CPLEX soluton Average soluton)/(cplex soluton). Therefore, the negatve values of the devatons ndcate that the average soluton for the CGA was better than the CPLEX soluton. The values n boldface show the best solutons found. Table 3. CGA results for the 324-node network Best soluton Average soluton CGA Tme (s) Devaton (%) 324_10_0_0_ _10_0_1_ _10_0_2_ _10_0_0_ _10_0_1_ _10_0_2_ _20_0_0_ _20_0_1_ _20_0_2_ _20_0_0_ _20_0_1_ _20_0_2_ _10_1_40_ _10_1_41_ _10_1_42_ _10_1_48_ _10_1_49_ _10_1_50_ _20_1_40_ _20_1_41_ _20_1_42_ _20_1_48_ _20_1_49_ _20_1_50_ Table 4. CPLEX results for the 30-node network CPLEX CPLEX Soluton Gap Cplex (%) Tme (s) 30_2_0_0_ _3_0_0_ _2_0_1_ _3_0_1_ _2_0_2_ _3_0_2_ _5_0_0_ _6_0_0_ _3_0_1_ _4_0_1_ _2_0_2_ _3_0_2_ _4_1_48_ _5_1_48_ _3_1_49_ _4_1_49_ _5_1_50_ _6_1_50_ _5_1_40_ _6_1_40_ _7_1_40_ _6_1_41_ _7_1_41_ _8_1_41_ _4_1_42_ _5_1_42_ Consderng the 324-node network, the CGA has better results n 100% of the tests, n compettve tmes related to the CPLEX. For the 30-node network, the CGA suppled, n terms of average values, results equal or better than CPLEX n 58% of the tests, ncludng optmal values. Table 5 CGA results for the 30-node network CGA Best soluton Average soluton Tme (s) Devaton (%) 30_2_0_0_ _3_0_0_

6 30_2_0_1_ _3_0_1_ _2_0_2_ _3_0_2_ _5_0_0_ _6_0_0_ _3_0_1_ _4_0_1_ _2_0_2_ _3_0_2_ _4_1_48_ _5_1_48_ _3_1_49_ _4_1_49_ _5_1_50_ _6_1_50_ _5_1_40_ _6_1_40_ _7_1_40_ _6_1_41_ _7_1_41_ _8_1_41_ _4_1_42_ _5_1_42_ Conclusons Ths work presented a soluton for the probablstc maxmal coverng locaton-allocaton problem usng CGA. The results show that the CGA approach s compettve for the resoluton of ths problem n reasonable computatonal tmes. For some nstances of 30-node network, the optmal values have been found. Therefore, these results valdate the CGA applcaton to the QM-CLAM. 6. References research, Prentce Hall, Englewood Clffs, N.J:, [9] L.A.N. Lorena and J.C. Furtado, Constructve genetc algorthm for clusterng problems. Evolutonary Computaton, 2001, 9(3): [10] L.A.N. Lorena and M.A. Perera. A lagrangean/surrogate heurstc for the maxmal coverng locaton problem usng Hllsman s edton, Internatonal Journal of Industral Engneerng, 2002, 9: [11] V. Maranov and D. Serra. Probablstc maxmal coverng locaton-allocaton models for congested systems. Journal of Regonal Scence, 1998, 38(3): [12] V. Maranov and D. Serra. Herarchcal locatonallocaton models for congested systems. European Journal of Operatonal Research, 2001, 135: [13] A.C.M. Olvera and L.A.N. Lorena, Detectng promsng areas by evolutonary clusterng search. Advances n Artfcal Intellgence Seres, 2004, [14] A.C.M. Olvera and L.A.N. Lorena, Populaton tranng heurstcs. Lecture Notes n Computer Scence, 2005, 3448: [15] M.A. Perera and L.A.N. Lorena, A heurístca Lagrangeana/Surrogate aplcada ao problema de localzação de máxma cobertura. In: XXXIII Smpóso Braslero de Pesqusa Operaconal (SBPO), 2001, Campos do Jordão. Anas p [16] H. Prkul and D.A. Schllng, The maxmal coverng locaton problem wth capactes on total workload. Management Scence, 1991, 37(2): [17] D. Serra and V. Maranov, New trends n publc faclty locaton modelng. Unverstat Pompeu Fabra Economcs and Busness Workng Paper 755, Avalable at < df>. [1] R.L. Church and C. ReVelle, Maxmal coverng locaton problem. Papers of the Regonal Scence Assocaton, 1974, 32: [2] J.C. Furtado, Algortmos genétcos construtvos na otmzação de problemas combnatoras de agrupamentos p. Tese (Doutorado em Computação Aplcada) Insttuto Naconal de Pesqusas Espacas, São José dos Campos, [3] R.D. Galvão, Uncapactated faclty locaton problems: contrbutons. Pesqusa Operaconal, 2004, 24: [4] D.E. Goldberg, Genetc algorthms n search, optmzaton and machne learnng. Addson-Wesley, Readng, MA, [5] T.S. Hale and C.R. Moberg, Locaton scence revew. Annals of Operatons Research, 2003, 123: [6] ILOG CPLEX 7.5 Reference Manual 7.5v. 610p. Copyrght by ILOG, France, [7] E.G.M Lacerda and A.C.P.L.F de Carvalho, Introdução aos algorítmos genétcos. In: XIX Congresso Naconal da Socedade Braslera de Computação, 19, Ro de Janero. Anas Ro de Janero: EntreLugar, 1999, v. 2, p [8] R.C. Larson and A. R. Odon, Urban operatons