An Adaptive Multiphase Approach for Large Unconditional and Conditional p-median Problems

Transcription

1 An Adaptive Multiphase Appoach fo Lage Unconditional and Conditional p-median oblems Chanda Ade Iawan Said Salhi Maia aola Scapaa Cente fo Logistics & Heuistic Optimization (CLHO), Kent Buess School, Univesit of Kent, Cantebu, Kent CT 7E, UK Abstact A multiphase appoach that incopoates demand points aggegation, Vaiable Neighbouhood Seach (VNS) and an eact method is poposed fo the solution of lage-scale unconditional and conditional p-median poblems. The method consists of fou phases. In the fist phase seveal aggegated poblems ae solved with a Local Seach with Shaking pocedue to geneate pomig facilit sites which ae then used to solve a educed poblem in hase ug VNS o an eact method. The new solution is then fed into an iteative leaning pocess which tackles the aggegated poblem (hase 3). hase 4 is a post optimisation phase applied to the oiginal (disaggegated) poblem. Fo the p-median poblem, the method is tested on thee tpes of datasets which consist of up to 89,600 demand points. The fist two datasets ae the BIRCH and the TS datasets wheeas the thid is ou newl geometicall constucted dataset that has guaanteed optimal solutions. The computational epeiments show that the poposed appoach poduces ve competitive esults. The poposed appoach is also adapted to cate fo the conditional p-median poblem with inteesting esults. Kewods Vaiable neighbouhood seach, eact method, aggegation, lage p-median poblems, adaptive leaning 1. Intoduction The p-median poblem is a discete location poblem whee the objective is to find the location of p facilities among n discete potential sites in such a wa to minimise the sum of the weighted distances between customes and thei neaest facilities. The p-median poblem becomes the conditional poblem when some (sa q) facilities alead eist in the stud aea and the aim is to locate p new facilities given the eisting q facilities. This poblem is also known as the (p, q) median poblem. A custome can be seved b one of the eisting o the new open facilities whicheve is the closest to the custome. When q = 0, the poblem 1

2 educes to the unconditional poblem (the p-median poblem fo shot). A futhe but bief desciption elated to the conditional p-median poblem will be pesented in Section 6 whee some esults ae also given. The p-median poblem is categoized as N-had (Kaiv and Hakimi, 1969). Fo elativel lage poblems, optimal solutions ma not be found and hence heuistic o metaheuistic methods ae usuall consideed to be the best wa fowad fo solving such poblems. Mladenovic et al. (007) povided an ecellent eview on the p-median poblem focug on metaheuistic methods. The p-median poblem was oiginall fomulated b ReVelle and Swain (1970). Howeve, Rog et al. (1979) enhanced the p-median poblem fomulation to educe its solution time. In thei model, the futhest p-1 assignments associated with each demand point ae ignoed. This eduction scheme is based on the obsevation that in the wost case, a demand point i is seved b its (n-p+1) th closest site. The enhanced p-median fomulation is fomulated as follows: Minimise Subject to Whee ii j F i jj Y ij X j jf Y ij i w d( i, Y (1) i ij 1 i I () X p (3) j X 0, i, j F (4) j i { 0,1} j, j J (5) Y { 0,1} i, j (6) ij F i (I,J): set of customes ( i I {1,..., n}) and set of potential sites ( j J {1,..., M}) (i.e. : n I and M J ) espectivel w i : demand o weight of custome i; d ( i, : distance between custome i and potential site j (Euclidian distance is used hee); p : the equied numbe of facilities to locate; Y ij = 1, if custome i is full seved b a facilit at site j and = 0 othewise; X j = 1, if a facilit is opened at potential site j and = 0 othewise; F i : set of all sites ecept the p 1 futhest sites fom demand point i. The objective function (1) minimises the total demand-weighted distance. Constaints () guaantee that each custome i is assigned to one facilit onl. Constaint (3) states that the

3 numbe of facilities to be located is p. Constaints (4) ensue that custome i can onl be allocated to facilit j (i.e., Y ij = 1) if a facilit is opened at site j (i.e., X j = 1). The use of the sets F i in constaints (), (4) and (6) ields a moe compact fomulation, equiing a fewe numbe of vaiables and constaints than the classical fomulation. In some applications, p-median poblems ma involve a lage numbe of demand points and potential facilit sites. These poblems aise, fo eample, in uban o egional aeas whee the demand points ae individual pivate esidences. Fancis et al. (009) stated that it ma be impossible and time consuming to solve location poblems consisting of a lage numbe of demand points. To simplif the poblem, it is quite common to aggegate demand points (and/o potential facilit sites) when solving lage-scale location poblems. In othe wods, the numbe of demand points (and/o potential facilit sites) can be educed fom n to m points (m << n) so that the appoimated poblem can be solved within a easonable amount of computing time. Howeve, aggegation intoduces eos in the data as well as in the models output, thus esulting in less accuate esults. The main contibutions of this pape include: (i) a novel multiphase appoach that incopoates aggegation, Vaiable Neighbouhood Seach (VNS) and an eact method fo solving lage p-median poblems, (ii) new best solutions fo some benchmak poblems, (iii) the constuction of a new lage dataset fo p-median poblems with guaanteed optimalit, and (iv) an adaptation of the poposed appoach fo the conditional p-median poblem. The pape is oganized as follows. Section pesents a bief eview of the past effots at solving lage p-median poblems. Section 3 descibes the ingedients that make up ou method as well as the oveall algoithm. Detailed eplanations of the main steps and the Local Seach with Shaking pocedue ae descibed in Section 4. Computational esults ae pesented in Section 5 ug lage datasets including the one with guaanteed optimal solutions which we constucted. Section 6 pesents a bief eview on the conditional p- median poblem followed b the adaptation and the implementation of ou appoach fo this elated poblem. The last section povides a summa of ou findings and highlights some avenues fo futue eseach.. ast effots at solving lage p-median poblems This section pesents an oveview of past effots at solving lage p-median poblems (see Fancis et al., 009, fo an ecellent eview). Hillsman and Rhoda (1978) intoduced a 3

4 classification of aggegation eos ug thee tpes, namel souce A, B, and C eos. Souce A eo occus when the distance between an Aggegate Spatial Unit (ASU) and a facilit is utilized in the model, instead of the tue distance between a Basic Spatial Unit (BSU) and a facilit. Souce B eo eists in the special case when a facilit is located at an ASU wheeas souce C eo appeas when a BSU is assigned to the wong facilit. Goodchild (1979) stated that aggegation tends to poduce moe damatic effects on location than on the values of the objective function while also noting that thee is no aggegation scheme without a possible esulting eo. Bach (1981) mentioned that the level of aggegation eets a stong influence on the optimal locational pattens as well as on the values of the locational citeia. Michandani and Reill (1986) eamined the effect of eplacing distances to demand points (BSUs) in a egion b the distance to a gle point (ASU) epesenting that egion. Cuent and Schilling (1987) poposed a method fo eliminating souce A and souce B eos. The intoduced a novel wa of measuing aggegated weighted tavel distances fo p- median poblems. Let d ( i, denote the distance between the i th and the j th ~ BSUs and d ( k, the distance between the epesentative point of the k th ASU and the j th BSU. The distance between the k th ASU and the j th facilit is taditionall defined as: ˆ ~ d ( k, = W k d ( k, (7) whee W k i A w k i with A k being the set of aggegated BSUs at the k th ASU. To eliminate souce A and B eos, the distance poposed in Cuent and Schilling (1987) is set as: d ˆ( k, = i A w k id( i, (8) Howeve, this method is not able to eliminate souce C eos. Casillas (1987) intoduced two measues to assess the accuac of aggegated models. These include the t eo ( ce f ( F': C) f ( F: C)) and the optimalit eo ( oe f ( F : C) f ( F: C)) whee F and F epesent the optimal locations of the p facilities found with the oiginal and the aggegated models espectivel, while C and C denote the list of BSUs and ASUs. The objective functions f ( F : C), f ( F : C) and f ( F : C) epesent the objective function evaluated ug F and C, F and C, and F and C espectivel. Oshawa et al (1991) studied the location eo and the t eo due to ounding in the unweighted 1-median and 1-cente poblems in the one-dimensional continuous space. 4

5 Aggegation eo bounds fo the median and the cente poblems wee developed b Fancis and Lowe (199). A Geogaphical Infomation Sstem (GIS) method fo eliminating souce C eo was poposed b Hodgson and Neuman (1993). Tanspot ting eos fo the median poblems wee investigated b Ballou (1994) who demonstated that t eos incease with p but decease with m. An investigation b Fotheingham et al (1995) suggested that the level of aggegation affects the location eo moe significantl than the objective function value. Fancis et al. (1996) intoduced a median ow-column aggegation method to find an aggegation which gives a small eo bound. In addition to the A, B, and C eos, Hodgson et al. (1997) intoduced souce D eo which aises when the BSU locations act as potential sites. Mua and Gottsegen (1997) investigated the influence of data aggegation on the stabilit of facilit locations and the objective function fo the plana p-median model. Demand point aggegation pocedues fo the p-median and the p-cente netwok location models wee studied b Andesson et al. (1998). Hodgson and Salhi (1998) poposed a quadtee-based technique to eliminate souce A, B, and C eos in the allocation pocess. Boweman et al. (1999) investigated the demand patitioning method fo educing souce A, B, and C aggegation eos in p-median poblems. Ekut and Bozkaa (1999) povided a eview of aggegation eos fo the p-median poblem. Fancis et al. (000) computed eo bounds fo seveal location models. lastia (001) investigated how to minimise aggegation eos when selecting the ASUs location at which to aggegate given goups of BSUs. Hodgson (00) intoduced data suogation eo in the p-median poblem which appeas when an oiginal population s demand is substituted b inappopiate values. To solve lage p-median poblems without aggegation, Chuch (003) put fowad an enhanced Mied Intege Linea ogamming fomulation called COBRA. He also poved that thee ae edundant assignment vaiables that can be consolidated if the satisf some equivalent assignment conditions. These conditions ae based on the ode of closeness of facilit sites with espect to pais of demand points. This popet leads to a eduction that can be up to 80% of the oiginal numbe of vaiables. An enhanced model fomulation efeed to as Both Eact and Appoimate Model Repesentation (BEAMR) was late poposed b Chuch (008). Hansen et al. (009) intoduced a pimal-dual VNS metaheuistic fo lage p-median clusteing poblems whee a Reduced VNS is used to get good initial solutions which ae then fed into a VNS with decomposition. The wost-case analsis of demand point aggegation fo the Euclidean p-median poblem on the plane was 5

6 investigated b Qi and Shen (010). An altenative coveing based fomulation which has a small subset of constaints and vaiables is studied b Gacia et al. (010). This method is elativel moe efficient when p is lage. Avella et al. (01) designed an aggegation heuistic based on Lagangean elaation. The poposed thee main pocedues, namel a sub-gadient column geneation, a coe heuistic, and an aggegation heuistic. The fist pocedue solves the Lagangean elaation b combining subgadient optimisation with column geneation. The coe heuistic is defined b a subset of the most pomig vaiables found accoding to the Lagangean educed ts associated with the open facilities as well as those associated with the allocation vaiables. An aggegation heuistic is then intoduced to tackle the poblem when the value of p is elativel small. Ve ecentl, Iawan and Salhi (013) intoduced an appoach ug demand points aggegation and vaiable neighbouhood seach fo solving lage-scale p- median poblems. Thei method used a multi-batch methodolog whee a leaning pocess that feeds infomation fom one batch to anothe is utilised. A batch consists of aggegated poblems. In this pape, we popose a multiphase appoach instead of a multi-batch appoach. The fist batch of the method b Iawan and Salhi (013) is simila to ou hase 1 ecept that a moe efficient implementation of the local seach is adopted. Subsequent phases ae also designed to guide the seach in eploing new aeas while etaining pomig egions. Moeove, we also enhance the method used to solve the aggegated p-median poblem; we pesent an efficient wa in aggegating the demand points; and we put fowad an effective implementation of the local seach that is used to solve the disaggegated (oiginal) poblem. 3. An Adaptive Appoach (AA) fo solving lage p-median poblems We popose an adaptive appoach which consists of fou phases. The main steps of these phases ae depicted in Figue 1 but a bief oveview is given below. Moeove, a visualisation of ou methodolog is pesented in Appendi A wheeas a detailed desciption of the main steps is given in the net section. In this stud, fo simplicit we conside potential facilit sites as custome sites (i.e. M=n). In the fist phase a leaning pocess is conducted. Hee, a clusteing pocedue is used to aggegate n BSUs into m ASUs, with m << n. As each custome site acts as a potential facilit site, the aggegated poblem educes to having m customes and m potential facilit sites. This phase consists of solving a numbe of aggegated poblems of m ASUs ug a Local Seach with Shaking pocedue with the aim of choog p facilit locations. This will 6

7 be descibed futhe in subsection 4.3. Let L denote a list of distinct facilities obtained fom the solutions of the aggegated poblems. Initialization Define the paametes m, itema, L ma, T and set L = Ø. hase 1 Do T times the following steps (t = 1,, T) (i) Aggegate n BSUs into m ASUs and constuct m clustes b allocating all BSUs to thei neaest ASUs. (ii) Calculate the distance d ˆ( k,, k=1,..,m; j=1,,m. (iii) Solve the t th aggegated p-median poblem ug Local Seach with Shaking (m, m, p), hase t t t t let Ft ( 1,,..., p ) be the obtained facilit locations with i denoting the i th facilit at iteation t and set L L Ft. (i) Constuct L clustes aound these L pomig facilities, and compute the distance d ˆ( k,, k=1,.., L ; j=1,, L. (ii) If L L ma solve the p-median poblem with CLEX ( L, L, p), othewise appl a VNS ( L, L, p). Let Fˆ best be the obtained facilit configuation. (iii) Solve the p-median poblem with Local Seach with Shaking (n, L, p) ug the initial facilit configuation. Let F be the obtained set of open facilities and f ( F : C) its coesponding t. (iv) Set Fˆ best = F and f best f ( F: C). Let Fˆ best and ite = 0. hase 3 (i) Aggegate n into m potential sites b including the facilit locations in E. (ii) Solve the p-median poblem ug Local Seach with Shaking (n,m,p) ug (iii) If hase 4 Figue 1. An Adaptive Appoach (AA) Fˆ best as Fˆ best as the initial solution. Let F be the obtained facilit configuation and f ( F : C) its t. f Set ) ( F : C fbest then Fˆ best = F, f best f ( F : C) and best hase 3. Fˆ. Set ite = 0 and go to Step (i) of Else set ite = ite + 1. If ite itema go to hase 4, else go to Step (i) of hase 3. Solve the p-median poblem ug Local Seach (n,n,p) ug Fˆ best as the initial solution. 7

8 In hase, L facilities ae consideed as the pomig facilities to set up an aggegated p-median poblem which is then solved with a VNS o with CLEX, depending on the size of the augmented poblem. Namel, if L is elativel small then the poblem is solved b CLEX ( L, L, p), othewise a VNS ( L, L, p) is adopted whee Method (g, s, p) efes to the pocedue Method fo locating p facilities, ug s potential sites and seving g customes. When the VNS is applied, the best solution found in hase 1 is used as the initial solution. To get a feasible solution to the p-median poblem with the oiginal customes set, the Local Seach with Shaking is then used with L potential facilit sites stating fom the solution etuned b VNS o CLEX in the pevious step. We efe to this pocedue as the Local Seach with Shaking (n, L, p). The best solution in this phase is then fed into the net phase (hase 3). The thid phase is an iteative pocess that incopoates potential facilit sites aggegation and the use of the Local Seach with Shaking. Unlike hase 1, the aggegation hee includes the pomig sites found in the pevious iteation. This set of pomig sites is denoted b E. The esulting aggegated poblem with n customes and m potential facilit sites is then solved b Local Seach with Shaking (n, m, p). The obtained solution is then used as an initial solution fo the net iteation and the pocess is epeated until a stopping citeion is met. In ou stud, the pocess stops when thee is no impovement afte a pescibed numbe of consecutive iteations which we denote b itema. In the final phase (hase 4), a post optimisation is caied out. Hee, a local seach is used to solve the oiginal poblem (without aggegation) stating fom the best solution obtained in the pevious phase. To speed up the seach, a eduction scheme, which is descibed in subsection 4.5, is also incopoated into the seach. 4. Desciption of the main phases of the Adaptive Appoach In this section we pesent the aggegation scheme and the distance calculation method. These ae followed b the desciption of the Local Seach with Shaking, the VNS, the eact method, and the local seach which is used in the oiginal poblem The aggegation methodolog (hases 1(i) and 3(i)) This subsection descibes the pocedue to aggegate n BSUs into m ASUs used in hases 1(i) and 3(i). The set of the m ASUs includes the followings: 8

9 the pomig facilit locations obtained fom pevious iteations in hase 3 (i.e. the set E). Note that in hase 1, E = Ø. (mγ) pseudo andoml geneated points, whee γ is a paamete (γ > 0). (m- E -mγ) andoml geneated points. Fistl, the method includes the pomig facilit locations (E) as pat of the aggegated points. We assume the use of these points ma incease the pobabilit of obtaining a good solution. Secondl the Basic Cell Appoach (BCA), as shown in Figue and biefl descibed below, is used to geneate the subsequent mγ aggegated points. All demand points ae coveed b squae cells and the cell infomation is used fo detemining m ASUs. This scheme ovecomes the weaknesses of a simple andom pocess when dealing with clusteed demand points. In addition, it ensues that the geneated ASUs ae not too close to each othe. This is achieved b impog that the distance between an pai of ASU points is lage than a cetain theshold which is based on the side of the cell. Finall, some andoml geneated points ae added to the set of ASUs to incease the divesit of the solutions. The BCA method is adapted fom the appoach given in Iawan and Salhi (013) which is oiginall based on the one b Salhi and Gamal (003) fo the multisouce Webe poblem. An illustation of the BCA is shown in Figue. The main steps of the method ae fomall given in Figue 3. Figue. The basic cell appoach (BCA) 9

10 Step 1 Constuct c L c W squae cells to get appoimatel m cells of length which will cove all demand points. Set paametes ρ, γ, and ς. Step Recod the numbe of demand points in each cell z as G z and detemine its coesponding pobabilit distibution, sa z = G z / n, z = 1,,( c c ). Step 3 Define C = E as the set of initial pomig sites (in hase 1, E = Ø). Step 4 Set count = 0. Step 5 Geneate andoml (0,1) and choose ~ z st ~ z F 1 ( z ) ( ) with z) L W F( v z 1v. Step 6 Choose andoml a demand point k in the cell z ~ and calculate d (the distance between demand point k and the neaest point in C ). Step 7 If d ρ set C = C {k}. If C = mγ stop the seach Othewise Set count = count + 1. If count = ς then set ρ = ρ/ and go to Step 4 else go to Step 5 Figue 3. The main steps of the BCA Initiall, ( c c ) squae cells ae constucted. We set the numbe of cells to be L W appoimatel m. Let denote the length of the side of the cell which is given b: ( ma min ) / m ma ma min min whee ma and min efe to the maimum and the minimum coodinate of the points espectivel. Similal, ma and min efe to the maimum and the minimum coodinate espectivel. A cell is identified b its bottom-left cone. In othe wods, the coodinates of the bottom-left cone of the z th cell is denoted b (X z, Y z ), z = 1,,( c c ). The bottom-left cone of cell 1 is X, Y and successive cells ae defined as follow: 1 1 min, X Y ( z mod c ), ( z mod c ) min z, z min L min W The numbe of demand points in each cell, G z, is then ecoded and its coesponding pobabilit distibution is calculated as z = G z / n, z = 1,,( c c ). Net, a cell is chosen in a pseudo andom manne based on the cumulative pobabilit distibution. In othe wods, we geneate andoml (0,1) and choose ~ ~ 1 z st z F ( ) with F z) v z 1v L W L W ( z) (. Fo instance as an illustation, ~ z 3 in Figue 4. A demand point (sa point a) is then chosen andoml in the cell ~ z as long as it satisfies the theshold distance sepaation citeion d Min d( a, with ρ being a paamete whose value is dnamicall a jc deceased if no aggegated point is found afte a numbe of attempts being made (sa ς, in ou 10

11 stud ς = m). The selection of a cell and of a point within the cell is epeated until a pescibed numbe of ASUs is eached. The centoid of the points in a cell was also attempted but a pelimina stud showed that the qualit of the solution was found to be slightl infeio. Cumulative obabilit (F (z) ) z th cell c L c W Figue 4. The illustation of detemining ~ z 4.. The distance calculation method (hases 1(ii) and (i)) When ug the Local Seach with Shaking, the VNS, o CLEX to solve the aggegated p-median poblem, the distance mati between points in C (ASUs) has to be detemined fist. The wa this is pefomed depends on the tpe of aggegated p-median poblems used. The aggegated poblems can be categoised as (m, m, p), (n, m, p), o ( L, L, p) whee (a,b,p) efes to solving the p-median with a customes and b potential sites. In hase 1(iii), the (m, m, p) aggegated poblem is solved. Hee, the pocedue to calculate the distance between points in C is pefomed fist b constucting m clustes, and allocating all demand points to the neaest points in C. Secondl, the total weight of each cluste W k, k = 1,,m is computed. Finall, the appoimate distance between each pai of points in C, denoted b d ˆ( k,, is calculated ug (7). In hase (ii), the ( L, L, p) aggegated poblem is solved instead. In this case, we calculate the distance between each pai of points in L ug (8) which is pactical in this case as the ( L, L, p) aggegated poblem is solved onl once. In hases (iii) and 3(ii), whee no clusteing is needed, the tue distance between the i th BSU and the j th facilit, d ( i,, is used fo both the (n, L, p) and the (n, m, p) aggegated poblems. 11

12 4.3. The Local Seach with Shaking fo the aggegated p-median poblem (hases 1(iii), (iii), and 3(ii)) We use a Local Seach with Shaking to speed up the seach. This choice is due to the fact that ou method is an iteative-based appoach and theefoe finding solutions to the aggegated poblems with the VNS o CLEX would be too time consuming. The method utilises one shaking and one call to the local seach onl. We eploe the following two appoaches: (a) Onl the fist neighbouhood (N 1 ) is used. This shaking pocess can be consideed as a petubation. This is then enhanced b a local seach. (b) Simila to (a), but instead of ug the fist neighbouhood (N 1 ), the k th neighbouhood (N k ) is andoml geneated whee k (1, k ma ). We efe to (a) and (b) as Va1 and Va espectivel. We caied out some pelimina epeiments to test the pefomance of these two vaiants. The esults, epoted in the computational esults section, show that Va is elativel supeio. The shaking pocess adapted hee applies the shaking algoithm used b Hansen and Mladenovic (1997). Let X denote the facilit configuation of the cuent solution and H the set of potential facilit sites. The k th neighbouhood stuctue N k is defined as: N k (X) = use of N 1 (X) k times with k = 1,,k ma and N 1 ( X ) X whee is chosen andoml in H-X and X is selected to ield the best impovement. The Local Seach with Shaking is used to solve the (m, m, p), the (n, L, p), and the (n, m, p) p-median poblems. In both hases and 3, this pocedue takes the best solution fom the pevious steps as the initial solution when solving the (n, L, p), and the (n, m, p) p- median poblems. Fo the (m, m, p) p-median poblem solved in hase 1, p andoml chosen points ae consideed instead. Fo the (m, m, p) poblem, the local seach pocess uses the pocedue FindBestCustome poposed b Iawan and Salhi (013) combined with the use of an efficient data stuctue initiall pesented b Resende and Weneck (007). The latte ecods intemediate calculations so to eliminate an unnecessa ecomputations. A simila data stuctue was also successfull implemented b Osman and Salhi (1996) when solving the vehicle fleet mi poblem. We efe to this local seach as IS-RW. This pocedue is based on the fast intechange heuistic intoduced b Whitake (1983) which is then adapted to incease the computational speed at the epense of a small loss in qualit. The pocedue 1

13 identifies a point (sa point i) among the potential facilit sites to be inseted and a facilit (sa facilit j, jx) that ields the highest saving to be emoved. The pocedue esticts the seach as follow: it consides point i S j with S j being the set of potential sites that ae neae to facilit j than the othe open facilities in the cuent solution ( S j < H -p). Fo the (n, L, p) and the (n, m, p) poblems, we use the well-known fast swap-based local seach pocedue of Resende and Weneck (007). We efe to this local seach as RW. We can affod to use this local seach hee without the eduction scheme used in IS-RW as good solutions ae usuall obtained fom hase 1 which ae then used as initial solutions fo the (n, L, p) and the (n, m, p) poblems.. Figue 5 pesents the Local Seach with Shaking when solving the aggegated p- median poblems with the use of Va. Step 1 Initialization Set the initial solution (X). Choose p points andoml fo hase 1 whilst fo hases and 3 take the best solution fom the pevious steps. Define N k (X), k = 1,,k ma with k ma = p. Step Shaking Geneate k (1,k ma ) andoml and detemine X N k (X). Step 3 Local Seach Fo the (m, m, p) poblem: Appl the local seach IS-RW on X. Fo the (n, L, p) and the (n, m, p) poblems: Appl the local seach RW on X. Figue 5 The Local Seach with Shaking (Va) fo solving the aggegated poblems We conducted pelimina epeiments to test the pefomance of these two local seaches namel IS-RW and RW. The esults ae epoted in the computational esults section The VNS and CLEX (hase (iii)) Vaiable Neighbouhood Seach (VNS) is a metaheuistic fist intoduced b Bimbeg and Mladenovic (1996) fo solving continuous location-allocation poblems. Hansen and Mladenovic (1997) fomall fomulated this heuistic and applied it to solve the p-median poblem. VNS combines both local seach and neighbouhood seach. The fist seach looks fo local optimalit, while the latte aims to escape fom these local optima b sstematicall ug a lage neighbouhood if an impovement is not found and then evets back to the smalle one othewise. Initial VNS implementations ae given in Hansen and Mladenovic 13

14 (001), but newe vaiants of VNS and successful applications can be found in Hansen et al. (010). The VNS o CLEX is utilized to solve the augmented ( L, L, p) p-median poblems. The use of these elativel moe intensive methods is acceptable as one un of VNS/CLEX is needed onl. Moeove, the size of the p-median poblem ( L, L, p) is still elativel small. When the VNS is used, we limit the computing time fo solving the poblem to T vns 0.5 seconds. In this stud, we set T vns 10n p m / 1000 which is found based on a pelimina stud. The algoithm of the VNS is based on the one b Hansen and Mladenovic (1997) incopoating the fast swap-based local seach (RW). The enhanced fomulation (1) (6) of the p-median poblem, as given b Rog et al. (1979), is used in the CLEX implementation The Local Seach fo the oiginal p-median poblem (hase 4) An additional post optimisation step to solve the disaggegated poblem (oiginal poblem) stating fom the best solution found in the pevious phase is intoduced. The main steps ae simila to the ones poposed b Iawan and Salhi (013) ecept hee we adopt a moe efficient implementation of the pocedue FindBestCustome whee we estict the seach even futhe b impog that the substituted location must lie within a cetain coveing adius (). The value of is based on the aveage of the longest distances fom the facilities to thei associated potential sites. In othe wods, we set Min o, MinR p whee R j ma d( i, and o ( / p) R j. This setting is used to ensue that the seach is j j 1 is moe estictive while emaining within each R j, j=1,,p. is a coection paamete which we set, in ou epeiments, to 0.5. This was found empiicall ug a small sample poblem. B ug this estiction, the numbe of potential sites used in the pocedue FindBestCustome is dasticall deceased which led to a massive eduction in the computing time of the local seach without a significant loss in solution qualit. Hee, we do not appl the data stuctue of Resende and Weneck as the numbe of potential facilit sites is elativel lage (n) which ceates an ecessive memo poblem due to the use of a two dimensional mati as pat of its data stuctue. jx j 14

15 5. Computational Results To assess the pefomance of ou solution method, we caied out an etensive computational stud. The code was witten in C++.Net 010 and used the IBM ILOG CLEX vesion 1.5 Concet Liba. The tests wee un on a C with an Intel Coe i5 CU 650@ 3.0GHz pocesso, 4.00 GB of RAM and unde Windows 7(3bit). In ou computational epeiments, we used two eisting datasets fom the liteatue and a new dataset with guaanteed optimal solutions which we constucted. We fist povide pelimina computational esults fo the two local seaches and the two vaiants. Full computational epeiments on the p-median poblem ae given net. The eisting datasets These consist of the BIRCH and the TS datasets. The BIRCH dataset is kindl povided b Avella et al. (01) in wheeas the TS dataset can be downloaded fom The newl geneated dataset This is constucted ug a well-defined though tivial geometic stuctue so to guaantee optimalit when the value of p is equal to the numbe of goups/clustes in the dataset. We efe to the new dataset as the Cicle dataset. Two eamples of this dataset (n = 500, p = 4 and n = 0000, p = 100) ae shown in Figue 6. The algoithm fo geneating the Cicle dataset is given in Appendi B and its poof of optimalit is povided in Appendi C. This is based on the following two items: (i) the p-median poblem educes to p 1-median poblems; (ii) the optimal cente of each cluste educes to the same optimal cente of each ing which is the point as defined in ou constuction. Diffeent instances of this dataset can be downloaded fom the CLHO (013) website ( Some notation The esults of ou epeiments ae pesented in seveal tables. The notation in the tables is as follows: n: numbe of demand points p: numbe of medians 15

16 Z: objective function value Time: computational time Deviation(%): this is the pecent gap fom the best known solution and is computed as: Z c Zb Deviation 100, whee Z c and Z b coespond to the Z value obtained with method Zb c and the best Z value espectivel. This is equivalent to the optimalit eo as defined b Casillas (1987). Bold values in the table efe to the best solutions. n = 500, p =4 n = 0000, p = 100 Figue 6. Two eamples of the Cicle Dataset (CLHO, 013) 5.1. elimina Empiical Compaison In this subsection we conduct small epeiments to assess the pefomance of the two local seaches as well as the two vaiants which we discussed in the ealie section. a) The two local seaches (RW vs IS-RW) Small epeiments on 3 TS datasets (mu1979, tz6117, and m7663) wee conducted to compae the pefomance of RW and IS-RW. Values of p vaing fom 10 to 100 with an incement of 10 ae used. Each instance was eecuted 10 times fo both local seaches stating fom the same initial solution. Table 1 shows the pefomance of both local seaches whee Saving (%) efes to the pecentage saving in CU time of IS-RW ove RW. Fom this table, it can be noted that IS-RW uns appoimatel 15-36% faste than RW. Howeve, the qualit of the solution obtained b IS-RW is, as epected, slightl affected with a 16

17 deteioation between % and 11%. We use IS-RW when solving the aggegated poblems (hase 1) as the obtained facilit locations of the aggegated poblem do not necessa ield the coesponding best solution of the oiginal poblem. In othe wods, the main objective in hase 1 is to identif the pomig facilit locations while consuming a smalle amount of computing time. Table 1 The aveage of CU time of RW and IS-RW on the TS data (in seconds) p mu1979 (n = 1979) tz6117 (n = 6117) m7663 (n = 7663) RW IS+RW Saving (%) RW IS+RW Saving (%) RW IS+RW Saving (%) Aveage b) The two Vaiants (Va1 and Va) We also tested Va1 and Va ug IS-RW as a local seach on the TS dataset vaing in size fom n = 734 to 9,976. We incease the value of p with n. Each instance was eecuted 10 times and eve un in both appoaches used the same initial solution. The summa esults ae pesented in Table which shows the aveage of the objective function (Z), the deviation (%), the aveage total CU and the aveage shaking time. As the Local Seach with Shaking is used onl once fo each subpoblem, efeing to the aveage behaviou athe than the best is, in ou view, moe eliable. In geneal, Va geneates bette esults than Va1 as it poduces both a highe numbe of smalle aveage objective values and a smalle deviation ( %). This means that conducting the shaking pocess k times does impove the qualit of the solution. To ou supise, the table also shows that Va uns faste than Va1. This could be due to the fact that the shaking pocess does not onl affect the pefomance of the local seach but makes the task of the local seach elativel easie. Moeove, the shaking time is found to be negligible when compaed to the total CU time (appo. 1.89% eta CU time onl fo 17

18 Va in the wost case, see fo eample instance ei846). These esults suppot ou choice of ug Va within the Local Seach with Shaking. File Name n Table Compaison between the Va1 and the Va Methods p Z Aveage Total CU Aveage Deviation Time Shaking Time (%) (milliseconds) (milliseconds) Va1 Va Va1 Va Va1 Va Va1 Va u ,14 07, zi ,00 09, mu ,885 30, ca ,885,883 6,817, ,878.80, tz ,41,69,371, , , eg ,010,761 1,003, , , m ,430,669 1,416, , , ei ,35,810 1,41, , , ja ,365,986 3,311, , , g ,869,116 1,859, , , kz ,378,764 6,340, , , Aveage ,43.5 5, Epeiments on lage p-median poblems In ou computational stud, we set the paametes as follows: m = 0.1n, T = 10, L ma = 300, and itema = 5. Those paametes wee chosen based on a small pelimina stud. The numbe of aggegated points is onl 10% of the numbe of demand points. The value of m affects the qualit of the solution. The highe the value of m, the highe is the chance of getting a bette solution. Howeve, the computing time also inceases with inceag values of m. The numbe of iteations (T) in hase 1 also influences the qualit of the obtained solution. The likelihood of getting a good solution inceases when T is high, which also inceases divesification, but at the epense of a longe computing time. We set L ma = 300 as CLEX uns elativel long when the size of the poblem eceeds this value. The method teminates when thee is no impovement in five consecutive iteations (itema = 5). In addition, we fied the seed fo the andom geneato to a constant, sa m, so the esults can be epoducible if need be. Diffeent settings wee consideed fo the paamete ρ used to detemine the theshold distance in the BCA method ( = ρ). Based on some pelimina tests, we set ρ = 0 fo the clusteed datasets and ρ = 0.5 fo the non-clusteed ones. The BIRCH and Cicle datasets belong to the clusteed dataset catego wheeas the TS fits the non-clusteed catego. This 18

19 choice of the paamete ρ implies that the theshold distance is elativel small fo the clusteed dataset as the demand points ae spead in a cetain aea. Fo the non-clusteed dataset, we set the numbe of aggegated points geneated fom the pomig facilit locations based on the BCA method to be 75% (γ = 0.75), wheeas the emaining 5% wee geneated andoml. Fo the clusteed dataset, we set γ = Epeiments on the eisting datasets The BIRCH dataset includes the lagest instances tested in the liteatue (n anges fom 5,000 to 89,600). Fo the TS dataset, we use the Ital, Sweden, Buma, and China instances (n anges fom 16,86 to 71,009). Case 1: BIRCH Dataset The esults of ou epeiments on the BIRCH dataset ae compaed with the ones obtained b Iawan and Salhi (013), Avella et al. (01), and Hansen et al. (009). We efe to these 3 methods as IS, AV, and VNSH espectivel. The computational esults of the AV and VNSH methods ae taken fom Avella et al. (01). The value of p anges between 5 and 64. The specification of the compute used to eecute IS is the same as the one used hee. Computational epeiments fo AV and VNSH wee caied out b Avella et al. (01) on an Intel Coe Quad CU.6 GHz, 4.00 GB of RAM and unde Windows X64. Dongaa s tansfomation is used to povide a fai compaison in tems of CU time. The fomulation of Nf1 this tansfomation is as follow: T T1, whee T 1 denotes the epoted time in Machine 1 Nf and T the estimated time in Machine. Nf 1 and Nf epesent the numbe of Mflops in Machines 1 and espectivel. The softwae used to ecod the values of Nf 1 and Nf can be downloaded fom In that softwae, we ecod the value of 3 bit SSE MFLOS. As we could not obtain pecisel the numbe of Mflops of the compute used b Avella et al. (01), we povide an appoimation based on a slightl slowe but simila compute available to us, namel a C Intel Coe Duo.6GHz, 4 GB of RAM. The computational esults fo ou method (AA) on the BIRCH dataset ae pesented in Table 3 whee the summa esults of the fou methods (AA, IS, AV, and VNSH) ae shown: the best known objective function (Z), the deviation (%), and the time (in seconds). In these 19

20 epeiments, we used two tpes of BIRCH instances, namel BIRCH instances of tpe 1 and BIRCH instances of tpe 3. Table 3 Computational Results fo the AA method on the BIRCH dataset File Z Best Deviation (%) Time (seconds) n p Name Known VNSH AV IS AA VNSH AV IS AA BIRCH instances of tpe 1 1 5, , , , , , , , , ,57,58 1, , , , , , , , ,454 1, , , ,931,834 1,510 1, , , , , ,185 1, , , ,787,441 1, , , ,678 4,501,069 1,487 # best Aveage , , BIRCH instances of tpe 3 1 5, , , , , , , , , ,64 1, , , , , , , , , , , ,393 1,89 1, , , , , , , , ,556 1, , , ,779,189 1,688 # best Aveage T1, Mflops 3, , , T 1, On the BIRCH instances of tpe 1, the AA s esults ae simila to the ones of IS but AA is appoimatel 15% faste than IS. AA povides bette solutions compaed to AV. Compaed to VNSH, it poduces simila objective function values, while ielding a slightl smalle deviation (0.0113%). On the BIRCH instances of tpe 3, the uppe bound of VNSH was not povided b Avella et al. (01). AA outpefoms IS and AV whee AA found 10 best solutions and ielded the smallest deviation (0.0014%). As stated in Hansen et al. (009), ou epeiments also show that the BIRCH instances of tpe 3 ae hade to solve compaed to tpe 1 instances. 0

21 Compaed to AV, accoding to Dongaa's tansfomation, the specification of the compute used to eecute ou method (AA) is appoimatel 5% faste than the one used b Avella et al. (011). Table 3 also pesents the tansfomed computing time (T ). Note that AA is faste than both IS and AV while geneating elativel bette esults. Oveall, AA found 5 new best solutions fo this difficult tpe of instances. Case : TS Dataset The computational esults fo the TS dataset ae given in Table 4. Thee ae 4 instances (Ital, Sweden, Buma, and China), whee each instance is solved with p vaing fom 5 to 100 with an incement of 5, totalling 16 instances. Ou esults ae compaed with the ones of IS. Fo p 100, we set γ = 0.90 and ρ = Table 4 Computational Results fo the AA method on the TS dataset Desciption Ital Data (n=16,86) Sweden Data (n=4,978) Buma Data (n=33,708) China Data (n=71,009) p The Best Known Z Deviation (%) Time (seconds) IS AA IS AA 5 7,406, ,100, ,087, ,490, ,098, ,665, ,783, ,677, ,6, ,597, ,187, ,731, , ,794, ,7.85, ,56, , , ,786, , , ,865, ,030.75, #Best 3 13 Aveage , In geneal, ou method (AA) poduces bette esults than the ones of IS. Hee, AA ields a highe numbe of best solutions and a smalle deviation ( %). The poposed appoach also poduces 13 new best solutions which can be used fo futhe benchmaking. Based on the aveage computing time, AA uns appoimatel 1% faste than IS. 1

22 5... Epeiments on the newl constucted dataset with guaanteed optimalit B conducting epeiments on the Cicle dataset, the optimal eos can be obtained. The numbe of demand points (n) anges fom 0,000 to 60,000 with an incement of 10,000 whilst the numbe of opened facilities (p) is equal to 0.5%n and 1%n. Thee ae 10 instances denoted b C1 to C10. Table 5 pesents the esults of these epeiments. Z* efes to the optimal objective function value. Table 5 shows that AA poduces the optimal solutions fo all instances. Oveall, it seems that the Cicle dataset can be solved quite easil b ou method. With espect to the computing time, solving the poblems with lage p equies moe time than the one with smalle p. This could be patl due to the fact that in the Local Seach with Shaking, k ma is set to p and as k 1, k ), consequentl a lage value of p will obviousl incease the ( ma computational buden. It is woth noting that although these instances ae visuall tivial, the wee constucted puposel this wa to assess the abilit of ou appoach to find optimal solutions. It is not that obvious fo the algoithm to find the solutions as no eta infomation is fed into the seach. Table 5 Computational Results fo the AA method on the Cicle dataset File Name n p Z* Z AA Deviation (%) Time (Seconds) C1 0, ,648, ,648, C 0, , , C3 30, ,47,878.81,47, C4 30, ,34, ,34, C5 40, ,97, ,97, C6 40, ,645, ,645, C7 50, ,14, ,14, C8 50, ,056,991.9,056, , C9 60, ,945, ,945, ,50.56 C10 60, ,468,389.55,468, , Aveage The conditional p-median poblem In this section we eview some papes focug on the conditional p-median poblem followed b the adaptation of ou appoach to this elated poblem and a summa of some computational esults.

23 6.1. A bief eview of the conditional p-median poblem The conditional location poblem was fist fomall intoduced b Minieka (1980) whee conditional centes and medians on a gaph wee investigated. Chen (1990) developed a method fo solving minisum and minima conditional location-allocation poblems with p1. An algoithm that equies the one-time solution of an unconditional (p+1) cente o (p+1) median fo solving the conditional (p+1) cente o (p+1) median on netwoks was suggested b Beman and Simchi-Levi (1990). Dezne (1995) poposed a geneal heuistic fo the conditional p-median poblem on both the netwok and the plane. In his pape, the tem (p,q) median poblem was intoduced. Let Q denote the set of eisting facilities whee Q J. The objective function fo the p-median poblem, equation (1), can be modified as follow (see Dezne, 1995): Z wi MinMin d( i,, Min d ( i, (9) ii jq jj, jq As D Mind ( i, i can be computed fo each i I befoehand, equation (9) can be jq ewitten as : Z wi MinDi, Min d ( i, (10) ii jj, jq The use of equation (10) is computationall moe efficient as it avoids unnecessa calculations. Beman and Dezne (008) suggested a method fo solving both the conditional p-median and p-cente poblems. The method needs the one-time solution of an unconditional p-median and p-cente poblem ug the shotest distance mati. A hbidization appoach combining a hamon seach and a geed heuistic fo solving p-median poblems was ecentl poposed b Kaveh and Esfahani (01). 6.. The adaptation of AA fo the lage (p,q) median poblems Ou poposed method (AA) which is designed to solve lage p-median poblems can easil be adapted fo tackling lage (p,q) median poblems. Ou evised appoach, which is efeed to as AAq, contains the following mino modifications. 3

24 a) The aggegation method The q eisting facilit locations ae alwas included in the pomig facilit locations (E) which ae then used to aggegate the points. These locations ae also used in hase 1, meaning that E Ø in hase 1 but E = Q. b) The Local Seach with Shaking In both the shaking and the local seach, the eisting facilities ae alwas etained open in the solutions. In othe wods, the eisting facilities ae not even checked fo possible emoval. The shaking When finding the best facilit to be emoved (sa facilit fom the cuent solution, facilit j is not one of the eisting facilities (i.e. j Q). The local seach The implementation of the best impovement stateg does not include the eisting facilities as these locations ae alwas pat of the solution. c) The eact method The implementation of the eact method with CLEX is still ug equations (1) (6). Howeve, constaints (11) ae added to ensue that the eisting facilities ae alwas in the solution. X j 1 j Q (11) The intoduction of such constaints (11) into the p-median fomulation makes the poblem elativel much easie to solve. In ou stud, we ae now able to incease the value of L ma fom 300 to 1100 while ug a simila amount of computational time. d) The post-optimisation (the local seach on the oiginal poblem) The modification in the post-optimisation is quite simila to the local seach in the Local Seach with Shaking descibed ealie in pat (b) Computational esults on the (p,q) median poblem The configuation of the paametes used fo solving the (p,q) median poblem is simila to the one fo the p-median poblem ecept we set L ma = 1,100. We test ou modified method on the TS dataset (Ital, Sweden, Buma, and China) that has alead been tested on the unconditional p-median poblem. We opted fo this dataset so that we could use the solutions obtained b solving the p-median poblem to set the eisting q facilities in the (p,q) median poblem. Namel, the q eisting facilit locations ae obtained fom the solution of the p- 4

25 median poblem solved in the pevious section. Fo eample fo the (p=5, q=5) median poblem, the eisting 5 facilit locations ae fom the solution of the (p=5) median poblem. The objective function of the (p=5, q=5) median poblem is then compaed to the one of the unconditional (p=50) median poblem which is taken as a lowe bound. In this case, the objective function value of the (p=5, q=5) median poblem should be wose than o equal to the one of the (p=50) median poblem. Note that such a claim is onl valid if an eact method is used instead of a heuistic. The computational esults of AAq on the TS dataset ae given in Table 6. The deviation (%) is the gap between the objective function value found b the AAq and the one b AA. The esults show that solving (p,q) ug the AAq equies almost a thid less amount of computing time than AA. This is quite epected as in the local seach, the eisting facilities ae alead fied and consequentl the numbe of combinations in the swapping pocedue deceases dasticall. Table 6 Computational Results fo the (p,q) median poblem on the TS dataset Desciption Ital Data (n=16,86) Sweden Data (n=4,978) Buma Data (n=33,708) China Data (n=71,009) p p-median poblem (AA) Z Time (seconds) (p, q)-median poblem (AAq) p q Z Deviation (%) Time (seconds) 50 5,100, ,407, ,087, ,490, ,7, ,30, ,698, ,605, ,665, ,163, ,783, ,677, ,098, ,155, ,00, ,898, ,597, ,099, ,187, ,731, ,557, ,58, ,155, ,05, ,56, , ,437, ,786, , ,865,606.7, ,448, , ,367, ,96, , ,649, Aveage 1,