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1 Internatonal Journal of Industral Eng. & roducton Research (2008) pp Volume 19, Number 4, 2008 Internatonal Journal of Industral Engneerng & roducton Research Journal Webste: Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 Effcent Soluton rocedure to Develop Maxmal Coverng Locaton roblem Under Uncertanty (Usng GA and Smulaton) K. Shahanagh & V.R. Ghezavat K. Shahanagh, Department of Industral Engneerng, Iran Unversty Of Scence and Technology, Tehran, Iran V.R. Ghezavat, hd student at the same Department, Tehran, Iran Keywords maxmal coverng; locaton; genetc algorthm; smulaton; uncertanty modelng ABSTRACT In ths paper, we present the stochastc verson of Maxmal Coverng Locaton roblem whch optmzes both locaton and allocaton decsons, concurrently. It s assumed that travelng tme between customers and dstrbuton centers (DCs) s uncertan and descrbed by normal dstrbuton functon and f ths tme s less than coverage tme, the customer can be allocated to DC. In classcal models, travelng tme between customers and facltes s assumed to be n a determnstc way and a customer s assumed to be covered completely f located wthn the crtcal coverage of the faclty and not covered at all outsde of the crtcal coverage. Indeed, solutons obtaned are so senstve to the determned travelng tme. Therefore, we consder coverng or not coverng for customers n a probablstc way and not certan whch yelds more flexblty and practcablty for results and model. Consderng ths assumpton, we maxmze the total expected demand whch s covered. To solve such a stochastc nonlnear model effcently, smulaton and genetc algorthm are ntegrated to produce a hybrd ntellgent algorthm. Fnally, some numercal examples are presented to llustrate the effectveness of the proposed algorthm IUST ublcaton, All rghts reserved. Vol. 19, No Introducton * Ths paper examnes a class of maxmal coverng locaton problem whch has wde real-world applcaton. The maxmal coverng locaton problem (M.C.L.) was orgnally developed to determne a set of faclty locatons whch would maxmze the total customers demand servced by the facltes wthn a predetermned crtcal servce crteron. Obvously, the model can be drectly useful to most faclty locaton plannng such as warehouses, health-care centers, fre statons, recreaton centers, emergency centers, etc. In addton, ths model can be appled to data abstracton problems, to portfolo formaton [1]. Ths model maxmzes the number of demand ponts covered wthn a specfed crtcal coverage or tme by a fxed number of facltes. It does not need that all demand Correspondng author: K. Shahanagh E-mal: shahanagh@ust.ac.r E-mal: Ghezavat@ust.ac.r aper frst receved Feb. 22, 2007, and n revsed form Jan. 24, ponts be covered. The M.C.L. and ts many developments compose a sgnfcant class of problems n locaton lterature. Brandeau and Chu [2] presented a survey of representatve problems that have been studed n locaton research. They recognzed more than 50 problem types and ndcated how those problem types relate to one another. Galvao et al. [3] evaluated the performance of two heurstc methods named LaGrange an and surrogate relaxatons to solve maxmal coverng locaton problem. The frst method had the ntegralty property and the surrogate relaxed problem they resolved was the lnear programmng relaxaton of the orgnal 0 1 knapsack problem. The performances of these two methods were compared wth 331 test problems n the lterature. The error of these heurstcs were very low and dd not dffer sgnfcantly among themselves. Galvão and ReVelle [4] proposed a LaGrange an heurstc for the maxmal coverng locaton plannng. In ths algorthm, the upper bound was specfed by a vertex addton and

2 22 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 another heurstc and lower bounds were generated through a sub gradent optmzaton method. Ths algorthm was valdated by solvng 150 nstances. For large szed problems a dualty gap was computed at the end of the algorthm. Church and ReVelle [5] were the frst who ntroduced the MCL. In ther model, the am was to maxmze the populaton covered wthn a desred servce dstance by optmally locatng a fxed number of facltes. ReVelle et al. [6] used some met heurstcs methods to solve large scale problems n maxmal coverng area. They beleved that exact methods may be too unweldy for real-world applcatons, and so heurstcs can allocate faster soluton tmes wth sub-optmal results. In ther nstances exact algorthms are provded by lnear programmng and branch and bound. Karasakal et al. [7] developed maxmal coverng locaton problem as a partal coverage so that coverage can change from full coverng to no coverng. They also, ntroduced soluton procedure based on LaGrange an relaxaton method and compared t wth two classcal approaches. Rchard L. Church et al. [8] developed a notaton of a model based on the premse that reserve selecton or ste prortzaton can be defned as a classcal coverng problem usually appled n many locaton problems. Specally, they developed a structure of the maxmal coverng locaton model to classfy sets of stes whch represent the maxmum possble representaton of specfc speces. Boffey and Narula [9] survey the mult obectve coverng and routng problems. Karasakal [10] formulated the MCL n the presence of partal coverage, and developed a soluton procedure based on LaGrange an relaxaton and showed the result of the approach on the optmal soluton by comparng t wth the classcal methods. Yupo Chan and M. Mahan [11] suggested a varant of the maxmal coverng locaton problem to locate up to p sgnalrecevng statons. In ther study the demands, called geolocatons, to be covered by these statons are dstress sgnals and/or transmssons from any targets. Araz et al., [12] proposed a mult-obectve MCL based on an emergency vehcle locaton. They consdered the maxmzaton of the covered customers and mnmzaton of the total travel dstance from the emergency servces. Younes and George [13] ntroduced a zero one mxed nteger formulaton for a maxmal coverng problem where ponts were covered by nclned parallelograms n a plane. A verson of maxmal expected coverng locaton problem was proposed by Daskn [14] as an extenson of the maxmal coverng locaton problem (MCL) formulated by Church and ReVelle [5], to account for the possblty of server unavalablty due to a congested system. Ths model relates the problem of optmally locatng servers so as to maxmze the expected coverage of demand, consderng the possblty of server unavalablty when a s call receved by the server. When formulatng ths problem the author makes 3 smplfyng assumptons: servers operate ndependently, each server has the same busy probablty and server busy probabltes are nvarant wth respect to ther locaton. Lus Gonzalo Espeo et al. [15] proposed a 2-level herarchcal development of the MCL and also a combned Lagrangean surrogate (L S) relaxaton to solve the model. Ther results were compared wth exact results obtaned usng CLEX software. Berman and Krass [16] consdered a generalzaton of the maxmal cover locaton problem whch allowed for partal coverage of customers, wth the degree of coverage beng a non-ncreasng step functon of the dstance to the nearest faclty. John R. Current and Davd A. Schllng [17] proposed two b-crteron routng problems ncludng the medan tour problem and the maxmal coverng tour problem. In both problems the tour must select only p of the n faclty on the network. Indeed, both problems have as one of ther obectves the mnmzaton of total travel dstance. The second obectve n both problems s to maxmze access to the tour for the facltes not drectly on t. The rest of ths paper s as follows: Secton 2 presents a new mathematcal model for the gven problem. Secton 3 proposes the soluton procedure based on smulaton and genetc algorthms. The computatonal results are llustrated and dscussed n Secton 4. Fnally, the remarkng concluson s gven n Secton Model Formulaton In ths secton, we descrbe n more detal the probablty maxmal locaton problem n whch we are nterested. There s a set of demand ponts N, at whch requests are generated, and a set of locatons M, where facltes may be opened. We assume that the requests at a demand pont D are generated ndependent of the processes at other demand ponts n N. We develop a formulaton based on the classcal p- medan formulaton. However, nstead of mnmzng the total dstance, ths model maxmzes the expected coverage of the demand ponts by determnng one of the selected faclty stes, whch ensures maxmum coverage confdence for each demand pont. 2-1 arameters: d : The demand for customer. θ : Coverage tme for DC. f : Random varable denotng travelng tme between customer and DC whch has normal dstrbuton functon wth the mean µ and varanceσ. Cap : Maxmum capacty for DC. 2

3 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng 23 Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 : Total number of DCs to be located Decson Varables: If demand of customer s satsfed by 1 DC. X Otherwse. U 1 If a DC s located at ste. Otherwse. p : Total probablty of coverage for customer. p : robablty of coverage customer by Dc. A notable pont n ths model s that all dstrbuton centers can not servce all customers because each DC has specal coverage tme and f a customer sn t n ths coverage, the DC can not servce that customer. In ths model we consder ths parameter that leads to the model that wll be more realstc. We assume that ths travelng tme s probablty and then coverng or not coverng a customer by a DC wll be probablstc. So, to compute the obectve functon, we measure expected value for t. (Amount of demand robablty of coverage). So, ths constrant must be added to compute probablty of coverage for each customer per each DC: r( f ( θ 0 σ θ 1 e 2π ) = 0 x µ σ d ) = 0 x (1) X = 0 = X (2) 2-3. Model Descrpton MaxZ = d (3) Subect to: X U M, N (4) X 1 M (5) X d cap N (6) U = p (7) x µ θ 1 2 σ ( e 0 σ 2π d x ) = 0 M (8) X = 0 M (9) = 0,1), U = (0,1), 0, X ( 0 (10) The obectve functon maxmzes total expected demands whch are covered. Set constran (4) says that a customer can allocate to DC when that DC s opened. Set constran (5) ensures that a customer cannot be allocated to more than one DC. Set constran (6) forces that total demand assgned to DC must be less that ts capacty. Set constran (7) guarantees that total number of DCs to be located should be equal to p. Set constran (8) computes probablty of coverage customer per each DC (Based on normal dstrbuton). Set constrant (9) shows total probablty of coverage for each customer by all DCs. Set constrans (10) determnes type of varables. 3. Soluton rocedure Because ths modelng s nonlnear and software such lngo gves us local optmum soluton and capactated locaton problems are N-hard, we use hybrd met heurstcs to solve ths problem. So, n order to solve general stochastc programmng models for capactated network desgn problem, we use GA and smulaton to produce a hybrd algorthm. Contrastng the general GA, ths algorthm wll reduce the computaton greatly, whch makes t possble to deal wth problems of qute large sze by the algorthm Smulaton rocess Because of the complexty of the proposed model, we desgn a smulaton model to compute uncertan functons coverng or not coverng customers by dstrbuton centers. Frst, we experment the smulaton process to generate a 0-1 matrx, namely Z. Also, we defne a lower bound for probablty of coverage. If probablty of coverage s greater than the lower bound, we assume that the customer can be covered by dstrbuton center completely. For nstance, f probablty of coverage customer 5 by dstrbuton center 8 s 0.78 and we defne the lower bound as 0.70, we assume that Z 5, 8 =1 and t means that Customer 5 can be covered by dstrbuton center 8 [18]. Z 1 f rob( f θ otherwse ) r (11) Due to the complexty, we desgn some stochastc smulatons to calculate uncertan functons shown n Fgure 1. (Uncertan functon s coverng or not coverng customers by DCs). After the mentoned smulaton process, we can obtan Z matrx. Output of ths smulaton s nput for genetc algorthm. (Output of smulaton s Z matrx).

4 24 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng After smulaton Z can be defned and the model wll be changed as follows: U = p (16) Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 Z 1 If DC at ste can cover customer. Otherwse Model Lnearzaton MaxZ = X d Z (12) Subect to: X U Z M, N (13) X 1 M (14) X d cap N (15) For =1 to customers For =1 to DCs Step 1) set V k = 0 X ( 0,1), U = (0,1) (17) = Set constrant (13) says that customer can be allocated to DC f that DC s opened and the customer can be covered by DC. The defntons of the other constrants have been sad before Genetc Algorthm GA s a stochastc search and heurstc optmzaton technque, whch has been wdely adopted by many researchers to solve varous problems. Ths algorthm was frst developed by Holland [19]. It mmcs the mechansm of genetc evoluton n the bologcal nature and conssts of a populaton of chromosomes (strngs or ndvduals) that are composed of genes. These genes represent a number of values, called alleles. Each chromosome (genotype) represents one potental soluton (phenotype). Step 2) Generate random numbers Z by repeatng the followng procedure for 100 tmes: ) Generate random number µ, µ 1 2 unformly from [0, 1]. ) Let ϕ1 = 2 Ln( µ 1) Sn(2πµ 2 ) ) Let ϕ ϕ σ + µ 2 = 1 v) If ϕ2 θ then V k +1 V k Step 3) f V k 75 then (In defnton of Z matrx we assume r = 0.75) Z =1 Else Z =0 End f Next Next Fg. 1. seudo code of smulaton process The process of genetc operators (.e., crossover and mutaton) s carred out n the pool; after that, an evoluton s completed by creatng new chromosomes (offsprng). Ths offsprng s expected to be stronger than the parents, but ths may not always be true Representaton of Chromosome In our GA-based approach, each chromosome or bt strng (.e., an example soluton) conssts of a matrx whch s made of 0 and 1. The dmenson of ths matrx s equal to: GA does not rely on analytcal propertes of the functon to be optmzed (Goldberg [20]). In short, GA has two maor processes: 1) GA s teratvely and randomly generatng new solutons; and 2) these solutons are checked for the optmalty accordng to predefned ftness functons. Ths becomes the most powerful prncple of GA. It makes them wdely sutable for fndng an optmal soluton n many complex problems, such as the travelng salesman problem (TS) and any forms of schedulng problems. For more detals fgure 2 llustrated classcal genetc algorthms.

5 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng 25 The number of customers n the network number of DCs. The matrx represents the assgnments of the customers to DCs and gets the value 1 n element [] f Intal Soluton customer s assgned to DC. sample of chromosome structure has shown n fgure 3. Select chromosome for cross over Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 ool Calcuatng obectve functon for chromosomes Dong cross over and mutaton operators k k Step1. Intalze pop_sze chromosomes X = [ x ] k = 1,2,.., op _ Sze from the potental regon {( X) g ( x ) 0, x = 1,2,..., n} randomly. 1 [ x k ] If DC at ste covers customer n chromosome k. Otherwse. At frst p DCs whch should be opened among all potental DCs randomly are selected and then a customer s selected randomly. In the second phase, for each customer the number of DCs whch can cover t s determned. Then for ths customer, a random DC from those DCs determned prevously s selected randomly and the customer n chromosome k s allocated to t consderng the capacty of DCs. (Because DCs have capacty, some customers cannot be allocated to DCs). Step 2. Compute the ftness of all chromosomes V k, k = 1,2,..., op _ Sze. The rank-based evaluaton functon s defned as the obectve functon for chromosome k. select chromosomes for next generaton Fg. 2. Structure of genetc algorthms Fg. 3. Sample of chromosome structure Select gens for mutaton checkng stoppng rule Step 3. It s very mportant to create new chromosomes (.e., offsprng) from the selected chromosomes (called parents) wth the current populaton. Ths process s carred out by the use of genetc operators, namely crossover and mutaton. Renew the chromosomes V k, k= 1,2,..., op_ Sze by crossover operaton. In order to determne the parents for crossover operaton, we repeat the followng process from k = 1 to pop sze: generatng a random real number r from the nterval [0, 1], the chromosome V k wll be selected as a parent provded that r < c; where the parameter c s the probablty of crossover. Then we group the selected parents V V,,... to the 1, 2 V3 pars, ( V 1, V2 ),( V3, V4 ),... wthout loss of generalty; let us llustrate the crossover operator on each par by ( V 1, V 2). At frst, we make a matrx wth 2 dmenson. (= customers and = DCs). In other words, we accrete parent matrxes and make one matrx. Then n the new matrx, n each row, f number of X wth value 1 s more than one, select one of them randomly and set t to be 1 and set the other to be 0. By ths, a customer s not allocated to dfferent DCs. In the new matrx we have at most 2 p opened DCs. For

6 26 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 generatng offsprng we select p DC from opened DCs n the new matrx, randomly. In fact we select p columns from the new matrx. (The note n ths procedure s, f column r (r ) s selected then column r + must not be selected and f column r (r ) s selected then column r - must not be selected). By dong the mentoned procedure we can get offsprng by crossover operator. An example of desgned crossover s llustrated n fgure 4. Step 4. Update the chromosomes V k, k= 1,2,..., op_ Sze by mutaton operaton. Smlar to the proves of selectng parents for the crossover operaton, we repeat the followng steps from k =1 to op Sze: generatng a random real number r from the nterval [0, 1], then chromosome Vk wll be selected as a parent provded that r < m; where the parameter m s the probablty k of mutaton. For each selected parent: X = x ], we [ k mutate t n the followng way. Select randomly from opened DCs and name t and then select randomly from closed DCs and name t. Then wth below procedure close DC and open DC. Frst for customers allocated to DC, we k k set X = [ x ] = 0. Then start to allocate unassgned customers to DC consderng coverage radus and capacty constrant, randomly. Example of desgned mutaton s llustrated n fgure 5. Step 5. Compute obectve functon for all chromosomes where the chromosomes V, V are assumed to 1 2,..., V op _ Sze have been rearranged from good to bad accordng to ther obectve values. Step 6. Select the chromosomes for a new populaton. The selecton process s based on selectng 50% from the best chromosomes and 50% randomly. Thus we obtan pop sze copes of chromosomes, denoted also byv ; k Step 7. Repeat the second to seventh steps for a gven number of cycles. Step 8. Report the best chromosome X = [ x * ] as * the optmal soluton. A notable pont n the mentoned GA s operators s to generate feasble offsprng accordng to the capacty and the other constrants. So, n ths process, t needn t to repar or reect offsprng snce they are generated feasble. Fg.3. Fg. 4. Sample of crossover operator n GA 4. Computatonal Results 4-1. Valdaton Genetc Algorthm In order to llustrate the effectveness of the genetc algorthm, we gve some random numercal examples that are performed on a personal computer. All algorthms consdered n ths study were coded n Vsual Basc 6 and run on a entum IV C wth 1.5 GHz CU and 256MB RAM. For ths purpose we solve some examples and compare our solutons wth global solutons obtaned from lngo 8 software. In ths secton, we ust try to valdate genetc algorthms alone wthout usng smulaton. In other words, n ths secton, genetc algorthm s appled to solve the second model and we assume that the Z matrx s a determnstc parameter n order to measure ust the effectveness of the genetc algorthm. In the next secton we wll measure robustness of hybrd genetc. Consder a manufacturer decdes to locate new dstrbuton centers. Assume that there are 25 customers. Indeed, suppose that a decson maker

7 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng 27 needs to select dstrbuton centers from 6 potental dstrbuton centers to cover customers. In ths study, we consder 30 populaton szes and 200 generatons for each test problem solved by the proposed algorthm. Customers demand has been generated unformly from nterval [10, 20]. Also, DC capacty has been generated unformly from nterval [ 0.3 d, 0.5 d ] where d s the total demand of all customers. Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 Customers DCs Fg Sample of mutaton operator n GA Tab. 1. Comparson between the GA and global solutons. op_sze m c Iteraton Opened DC Obectve Functon Global Obectve Functon Error CU Tme (Seconds) ,3, % ,2, % ,3, % ,3, % % % ,3, % 9 In Table 1, we compare solutons when dfferent problems are taken wth the same generatons as a stoppng rule. It appears that all the obectve functon and global optmum dffer lttle from each other. In order to account for t, we present a parameter, called the percent error,.e. (global optmum obectve functon)/global optmum, where the global obectve value s obtaned from lngo 8 software. The last column named by error n table 1 ndcates ths parameter. It follows from table 1 that the percent error does not exceed 2.31 % when dfferent problems are selected, whch mples that the genetc algorthm s effectve enough to solve model Senstvty Analyss for Smulaton To llustrate the effectveness of the hybrd genetc algorthm, we solve a specal problem for eght tmes wth numerous dfferent runs for the smulaton model. We show that ths hybrd algorthm s also robust to the smulaton settngs. Also set parameterng on GA and smulaton was shown n table 2. Consder a manufacturer who wants to locate new dstrbuton centers. Assume that there are 45 customers. Suppose that a decson maker needs to select 5 dstrbuton centers from 12, potental dstrbuton centers to serve customers. In Table 2, we compare solutons when dfferent smulaton runs are taken wth the same generatons as a stoppng rule. It appears that all the maxmal covered demand dffer lttle from each other. In order to account for t, we present a parameter, called the percent error,.e. (the best obectve value - obectve value)/the best obectve value, where the best obectve value s the

8 28 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng maxmum of all the ten maxmal covered demand obtaned above. The last column named by error n Table 2 s ust ths parameter. It follows from Table 2 that the percent error does not exceed 2.18 % when dfferent smulatons are selected, whch mples that the hybrd genetc algorthm s robust to the smulaton runs. In other words, dfferences n smulaton runs don t have any maor mpact on the fnal soluton. Thus, the hybrd method s robust to the smulaton settngs and therefore s also effectve to solve the model. Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 Customers Tab. 2. Measurng the robustness of the HGA to the smulaton run DCs op_sze Iteraton Opened DC Obectve Functon The Best Obectve Error ,4,8, % ,3,8, % ,3,4, % ,4,8, % ,6,8, % ,3,4, % ,3,6, % ,7,8, % ,4,8, % ,2,4, % 5. Concluson In ths paper, we defned notaton of locaton allocaton problem whle we have stochastc travelng tme between servce centers and customers ths tme s stochastc wth normal dstrbuton functon. In the proposed model, we assumed that servce centers have coverage radus restrcton. We formulated the problem as a non-lnear nteger programmng. The advantages of the proposed study are as follows: consderng probablstc coverage radus whch yelds more flexblty for the model, presentng hybrd method ncludng smulaton for realzng uncertanty and genetc algorthm to solve the model. erformance of the soluton procedure was verfed by randomly generated test problems. These experments show that the proposed algorthm s effcent to generate hgh qualty solutons n a short perod of tme. We suggest some new areas to develop the mentoned model: By usng ths contrbuton n the ntegrated supply chan systems, t assumes that such nventory and transportaton can be a sutable development as future research for the presented model. Also, consderng servce level constrant to cover each customer can be an nterestng future studes. Because n ths model we had travelng tme between servce centers and customers, consderng the tme wndow constrant for customers to get servce s another sutable development. References [1] Chen-Cua Chung: Recent Applcaton of the Maxmal Coverng Locaton lannng Model, Journal of operatons research socety. Vol.37, No.8, 1986, pp [2] Brandeau, M.L., Chu, S.S., An Overvew of Representatve roblems n Locaton Research. Management Scence. 35, 1986, pp [3] Roberto, D., Galvão, Lus Gonzalo Acosta Espeo and Bran Boffey, A comparson of Lagrangean and surrogate relaxatons for the maxmal coverng locaton problem, European Journal of Operatonal Research Vol. 124, Issue 2, 16 July 2000, pp [4] Roberto Déguez Galväo, Charles ReVelle, A Lagrangean Heurstc for the Maxmal Coverng Locaton roblem: European Journal of Operatonal Research Vol. 88, 1996, , Locaton Scence, Volume 5, Issue 1, May 1997,. 72. [5] Church, R.L., ReVelle, C.S., The Maxmal Coverng Locaton roblem. apers of the Regonal Scence Assocaton; 32:101 18, [6] Charles, ReVelle, Mchelle Scholssberg, Justn, Solvng the Maxmal Coverng Locaton roblem Wth Heurstc Concentraton, Computers & Operatons Research, Vol. 35, Issue 2, Feb. 2008, pp [7] Orhan Karasakal, Esra, K., Karasakal, A Maxmal Coverng Locaton Model n the resence of artal Coverage, Computers & Operatons Research, Vol. 31, Issue 9, August 2004, pp [8] Rchard, L., Church, Davd, M., Stoms, Frank, W., Davs: Reserve Selecton as a Maxmal Coverng Locaton roblem, Bologcal Conservaton, Vol. 76, Issue 2, 1996, pp [9] Boffey, B., Narula, S.C., Mult Obectve Coverng and Routng roblems. In: Karwan MH, Spronk J, Wallenus J. edtors. Essays n decson makng: a volume n honour of Stanley Zonts. Berln: Sprnger, 1997.

9 K. Shahanagh & V.R. Ghezavat / Effcent Soluton rocedure to Develop Maxmal Coverng 29 [10] Karasakal, O., Esra, K., A Maxmal Coverng Locaton Model n the resence of artal Coverage. Computers & Operatons Research. 31, 2004, pp Downloaded from epr.ust.ac.r at 5:13 IRST on Saturday December 15th 2018 [11] Chan, Y., Mahan, J.M., Chrsss, J.W., Drake, D.A, Wang, D., Herarchcal Maxmal-Coverage Locaton Allocaton: Case of Generalzed Search-and-Rescue. Computers & Operatons Research. 35, 2008, [12] Araz, C., Selm, H., Ozkarahan, I., A Fuzzy Mult- Obectve Coverng-Based Vehcle Locaton Model for Emergency Servces. Computers & Operatons Research. 34, 2007, [13] Younes, H., Wesolowsky, G.O., A Mxed Integer Formulaton for Maxmal Coverng by Inclned arallelograms. Eur. J. of Operatonal Research, 159, 2004, pp [14] Daskn, M.S., A Maxmal Expected Coverng Locaton Model: Formulaton, ropertes and Heurstc Soluton. Transportaton Scence, 17, 1983, pp [15] Lus Gonzalo Acosta Espeo, Roberto, D., Galvão, Bran Boffey, Dual-Based Heurstcs for a Herarchcal Coverng Locaton roblem- Computers & Operatons Research, Vol. 30, Issue 1, Jan. 2003, pp [16] Berman, O., Krass, D., The Generalzed Maxmal Coverng Locaton roblem. Computers & Operatons Research, 29, 2002, pp [17] John, R., Current, Davd, A., Schllng: The Medan Tour and Maxmal Coverng Tour roblems: Formulatons and Heurstcs, European Journal of Operatonal Research, Vol. 73, Issue 1, 24 Feb.1994, pp [18] Hwang, H.S., Desgn of Supply Chan Logstc System Consderng Servce Level. Computers & Industral Engneerng. 43, 2002, pp [19] Holland, J.H., Adaptaton n Natural and Artfcal Systems: An Introductory Analyss wth Applcatons to Bology, Control, and Artfcal Intellgence, 2nd Ed., MIT ress, Cambrdge, [20] Goldberg, D.E., Genetc Algorthms n Search, Optmzaton and Machne Learnng. Addson-Wley, MA, 1989.