Extensions of the Psuedo-Boolean Representation of the p-median Problem

Transcription

1 Extensons of the Psuedo-Boolean Representaton of the p-medan Problem Gabrel Sloggy Advsed by Professor Rchard Church, Department of Geography Abstract The p-medan problem (PMP) s nherently a graph theoretc problem wth applcatons n locaton theory and optmalzaton. Due to the p-medan problem beng NPhard, most research n t s reducton based, generally usng reducton tests and boundng, or explorng specfc solvable scenaros. Ths paper examnes and summarzes the pseudo- Boolean formulaton of the problem developed by AlBdaw et al and apples t to a dfferent formulaton of the problem by Elloum, creatng an alternate formulaton of Elloum s model. In dong so, we demonstrate that the model developed by AlBdaw et al. s dentcal to the model developed by Cornuejols et al., albet n a pseduo-boolean formulaton. We then provde a computatonal comparson of the dfferent models. Ultmately, we offer a pseduo- Boolean formulaton of the p-medan problem that s more nteger-frendly than that of AlBdaw. In comparng computatonal models, we show that the pseudo-boolean Cornuejols model s slghtly faster than the AlBdaw model, and thus assumedly, so s the pseudo- Boolean Elloum model. 1 Introducton The p-medan problem s an NP-hard problem concernng the mnmal weghted dstance between a subset of ponts on a graph. We defne the problem n terms of facltes and users as follows: Gven I, a set of n nodes, each of whch has a partcular demand J as set of m nodes, each of whch serves as a possble faclty ste. Notce that I and J are not necessarly dsjont, and n some cases may be dentcal. d j : IxJ R +, a dstance functon between node and node j a, a demand or weghtng varable for a demand or clent p m, some subset of facltes that we want to open, we want to determne whch subset of p facltes to open n order to mnmze the sum of the demand-weghted dstances from each user to ts nearest open faclty. We assume that each faclty has unlmted producton capacty and that there are no restrctons on ste placement. 1

2 1.1 Hstory of the p-medan Problem The p-medan problem s a generalzaton of a problem frst posed by Fermat: gven three dstnct ponts on a plane, fnd a medan pont on the plane that mnmzes the sum of the dstances from each of these ponts to the medan pont. Weber (1909) generalzed ths to n weghted ponts on the plane, mnmzng the sum of the weghted dstance from each of these ponts to the medan pont. The Weber problem was reformulated by Hakm (1964) to apply to a graph by ntroducng the notons of the absolute center and the absolute medan. Hakm frst defnes a dstance functon d(x, y) on the graph G by the length of the shortest path n G between the ponts x and y, where the length of a path s the sum of the weghts of the branches of that path. He then defnes the absolute center to be the pont x 0 on an element of a weghted n-vertex of G f for all vertces v wth weght h, and every pont x on the graph G, max 1 n h d(v, x 0 ) max 1 n h d(v, x). That s, the maxmal weghted dstance from x 0 to any vertex s less than than that of any other pont on the graph. Smlarly, y 0 s defned as the absolute medan f for each pont y n G n n h d(v, y 0 ) h d(v, y) So the sum of the weghted dstances from y 0 to each of the vertces v s less than that of any other pont on G. Hakm goes on to show that an absolute medan of a graph s always at a vertex of a graph, thus reducng the set of possble solutons to only the vertces of the graph. In a later paper (1965), he defnes a dstance functon between a vertex of G, v, and a set of p ponts, X p, as d(v, X p ) = mnd(v, x 1 ), d(v, x 2 ),..., d(v, x p )} He s then fnally able to defne the p-medan problem: A set of ponts X p s a p-medan of G f for every X p on G n h d(v, Xp) n h d(v, X p ) The classc formulaton of the p-medan problem was developed by Revelle and Swan n 1970, and s gven below. The Classc formulaton (RF) n m mn Z = a d j x j (1) m x j = 1 for = 1, 2,..., n (2) m x jj = p (3) 2

3 x j x jj for = 1, 2,..., n; j = 1, 2,..., m wth j (4) x j 0, 1} for = 1, 2,..., n j = 1, 2,..., m. (5) Here, the objectve functon mnmzes the total weghted dstance accrued by assgnng each demand node to ts closest user faclty. Constrant (2) ensures that every demand node s satsfed by exactly one faclty. Constrant (3) ensures that p nodes assgn themselves as the faclty supplyng ther demand; that s, p potental stes are assgned as facltes. Constrant (4) ensures that a ste can only assgn to a ste j f ste j s assgned as a faclty. Constrant (5) ndcates the bnary nature of the x j varables n that faclty supples clent j or not. In partcular, 1 f demand at assgns to faclty j x j = 0 otherwse 1 f a faclty s sted at j and meets the demand at j as well x jj = 0 otherwse Ths model s used as the startng pont for varous reducton technques, as n many of the papers mentoned below. 1.2 Reducton Technques Snce the p-medan problem s NP-hard, most research nto the problem explores methods that reduce the number of varables or constrants, and hence computaton tme, or attempt to substtute certan steps of the problem wth polynomally solvable algorthms. Varous hybrd models have been suggested, such as Rosng and Revelle s heurstc concentraton, a combnaton of a heurstc approach wth lnear programmng usng the branch-and-bound algorthm (1997). AlBdaw et al. (2011) analyze the problem through pseudo-boolean polynomals, combnng the work of Hammer (1968) and Beresnev (1973) n an attempt to elmnate redundant varables through an algebrac representaton of the elements of the cost matrx. Church (2003) develops a model called COBRA that reduces the number of varables and constrants by elmnatng redundant varables n the model. Elloum (2010) provdes a mxed-nteger formulaton of the problem by adaptng a smple plant locaton problem developed by Cornuejols et a.l n Goals Ths paper summarzes the models developed by AlBdaw et al. and Elloum respectvely and provdes a bref worked example for each of them. We then apply AlBdaw s pseduo-boolean formulaton to Elloum s model, obtanng a more nteger-frendly model than that of AlBdaw et al. but wth comparable processng speed. We then compare the dfferent models wth ther dfferent reducton technques and analyze ther effcacy n reducng the problems szes of varous benchmark problems. The paper ends wth conclusons and potental for future research. 3

4 2 The Pseudo-Boolean Formulaton Defnton 1. A pseudo-boolean functon f s a functon of the form f : 0, 1} n R for n N as By ths defnton, notce that any pseudo-boolean functon f can be unquely wrtten f(x) = a + a x + <j a j x x j + <j<k a jk x x j x k +... (6) The 2011 paper by AlBdaw et al. begns wth the cost matrx of a p-medan problem and manpulates t so that the objectve functon can then be wrtten as a pseduo-boolean functon. We summarze ths formulaton and provde a bref computatonal example for clarty n how the algorthm works. 2.1 The AlBdaw formulaton The formulaton developed by AlBdaw et al. s penalty based. That s, the model begns at the closest ste to a demand source. If the demand s not met at that ste, a penalty n the form of the weghted dstance between that ste and the next closest ste s accrued, and the next closest ste s consdered. Ths process s repeated, accummulatng penaltes, untl a ste s reached that wll meet the ntal ste s demand. In order to formulate ths concept, we must nsttute both an orderng based upon dstance and a penalty based upon the dfference n weghted dstance between stes of adjacent nearness. Let C = [c j ] be a cost matrx that defnes the cost of supplyng a demand at node from a faclty at node j. For clent defne Π = (π j,..., π nj ) to be an orderng of 1,..., n such that c πj c πk for all, k 1,..., n}. That s, Π s a permutaton of 1,..., n such that n row of C, the cost n column j s less than the cost n column k f j < k for all, k 1,..., n}. We also defne = (δ 1,..., δ n ) where δ 1 = c π1 and δ r = c πr c π(r 1) for r = 2,..., n. That s, s a vector of length n whose frst entry s the least weghted dstance from ste, and entry r beng the dfference n weghted dstance between the r and r 1 farthest stes from. Fnally, let y = (y 1,..., y n ) be such that y = 0 f a plant s open at locaton and 1 otherwse. So y s a vector representng whch stes have facltes located at them. Gven ths notaton, we can now defne the cost of satsfyng a demand at as f (y) = δ 1 + n k 1 δ k y πr (7) k=2 In order to aggregate the Π and vectors, we defne the orderng matrx Π = [π j ] and the dfference matrx = [δ j ]. Notce that whle s dstnct for a gven cost matrx, Π s not necessarly so n the case that two stes have equal weghted dstance from a thrd. Wth r=1 4

5 ths notaton, we can now represent the objectve functon of the p-medan problem as } n n m k 1 B(y) = f (y) = δ 1 + δ k y πr k=2 r=1 (8) But now the model s formulated strctly n terms of y j = 1 x jj. We can make ths substtuton for x jj n constrants (3) and (4) n the orgnal formulaton to form constrant (10), but we stll must consder how else the constrants on the x j terms translate to constrants on the y j terms. To do ths, when presented wth a mult-part monomal c 1 y a y b y c... n the objectve functon, we replace t wth c 1 z k for some k. By dong ths, we transform the objectve functon nto a lnear combnaton of sngle-part monomals. For ease of notaton, we defne S k to be the set of ndces of the y terms correspondng to z k. For example, f z 1 = y 1 y 2 y 4 y 7, then S 1 = 1, 2, 4, 7}. In applcaton, z k s an ndcator for the t closest stes to some ste j where t = S k. So z k = 0 f any y has a plant located at t and z k = 1 f no y j has a plant located at t for j S k. So z k j S k y j S k + 1, whch gves us constrant (11). And thus the statement of the whole formulaton s: The AlBdaw formulaton (AF) } n m k 1 mn B(y) = δ 1 + δ k y πr (9) k=2 r=1 n y = n p (10) y z k S k 1 for each S k correspondng to each z k (11) S k y 0, 1} for = 1, 2,..., n (12) z k 0 for each k constructed. (13) 2.2 Reductons n the pseudo-boolean formulaton As mentoned n the ntroducton, much of the research on the p-medan problem focuses on elmnatng varables and constrants to make runtme more effcent. Ths pseudo-boolean formulaton provded by AlBdaw et al. has two prncple methods of reducton: truncaton and combnng lke terms n the objectve functon. A bref argument for truncaton, also called choppng, s as follows: Suppose we are on a network wth n nodes and have to place p facltes. Consder a node. In creatng Π we order the stes by weghted dstance from. In the worst case scenaro, all p stes are the farthest possble stes from : those at Π [n p],..., Π [n]. wll then assgn to the closest one: the ste (n p) farthest from. But n ths case, wll never assgn to a ste farther than (n p), so we may elmnate the farthest (p 1) stes from. 5

6 The paper by Rosng, Revelle, and Rosng (1979) gves more detals. In relaton to the pseudo-boolean formulaton, truncaton s effected by replacng the largest p elements of row n C wth c π(n p+1). That s, we replace the largest p values n each column of the cost matrx C wth the (p+1) largest value n the respectve column. In formulatng the pseudo-boolean functon, we have nstances n whch the terms of a monomal are dentcal and we may combne them. For example, gven 4y 1 y 3 y 6 y 8 + 7y 3 y 8 y 1 y 6, we may combne them to get 11y 1 y 3 y 6 y 8. Combnng smlar varables n the objectve functon allows us the potental of removng varables and constrants, as n the gven example, we effectvely replaced the y 3 y 8 y 1 y 6 varable wth y 1 y 3 y 6 y 8. More concretely, what occurs n ths scenaro s that a ste has the same k closest stes as a ste j. Then the decson of where to assgn a faclty s the same for j as t s for, and we may substtute the correspondng varables of n for j. In order to llustrate the process of formng the orderng and dfference matrces, as well as the reducton process, the followng computatonal example s gven. Ths example does not reduce by truncaton, as such a reducton s not unque to ths formulaton, and may confuse the process. 2.3 A computatonal example Consder a p-medan problem wth m = 5 potental faclty stes, n = 5 clents and p = 2 facltes to place, wth cost matrx C = A possble orderng matrx for C s Π = whch has correspondng dfference matrx = We can then formulate the objectve functon n terms of the pseudo-boolean polynomal: B(y) = 3 + 2y 3 + 2y 1 y 3 + 2y 1 y 2 y 3 + 3y 1 y 2 y 3 y y 3 + 2y 1 y 3 + 0y 1 y 2 y 3 + 3y 1 y 2 y 3 y y 3 + 1y 2 y 3 + 2y 2 y 3 y 4 + 1y 2 y 3 y 4 y 5 6

7 Combnng smlar terms, we get y 5 + 3y 4 y 5 + 2y 2 y 4 y 5 + 2y 1 y 2 y 4 y y 4 + 2y 4 y 5 + 3y 2 y 4 y 5 + 1y 2 y 3 y 4 y 5 B(y) = y 3 + 1y 4 + 4y 1 y 3 + 1y 2 y 3 + 5y 4 y 5 + 2y 1 y 2 y 3 + 2y 2 y 3 y 4 + 5y 2 y 4 y 5 + 6y 1 y 2 y 3 y 4 + 2y 1 y 2 y 4 y 5 + 2y 2 y 3 y 4 y 5 = y 3 + 1y 4 + 4z 1 + 1z 2 + 5z 3 + 2z 4 + 2z 5 + 5z 6 + 6z 7 + 2z 8 + 2z 9 Notce that whle the orgnal objectve functon had 23 non-zero monomals, the reduced objectve functon has only 12. We can now formulate the model as: mn B(y) = y 3 + 1y 4 + 4z 1 + 1z 2 + 5z 3 + 2z 4 + 2z 5 + 5z 6 + 6z 7 + 2z 8 + 2z 9 5 y j = 3 y j z k S k 1 for each S k correspondng to each z k j S k y j 0, 1} for j = 1, 2,..., 5 z k 0 for each k constructed. 3 The Elloum formulaton The formulaton of the p-medan problem developed by Elloum (2008) s also a penalty-based formulaton, and as we wll see, s very smlar to the pseudo-boolean model developed by AlBdaw et al. The Elloum model s based upon a formulaton developed by Cornuejols et al. n 1980, whch we wll show s dentcal to the AlBdaw formulaton when translated nto a pseudo-boolean form. For ease of notaton, hereafter we wll refer to the dfferent formulatons by ther ntals: the classcal model developed by Revelle and Swan s RF, the pseudo-boolean model developed by AlBdaw et al. s AF, Elloum s formulaton s EF, and Cornuejol et al. s formulaton s CF. We now ntroduce notaton common to CF and EF. For a ste, let K be the number of dfferrent dstances from ste to any faclty. Notce that K does not necessarly equal p as some facltes may be equdstant from, so K p. Let D 1 < D 2 <... < D K be these dstances n ncreasng order. Let V k = j : d j D k }. That s, V k s the neghborhood composed of the k closest facltes to, for k = 1, 2,..., K. Wth these defntons, V 1 s the equdstant set of stes closest to, and V K s the set of all faclty stes. Then n an optmal soluton, a clent wll assgn to the smallest neghborhood contanng an open faclty. V k 7

8 3.1 The Cornuejols formulaton In formulatng CF, Cornuejols et al. defne y j equal to the x jj varables used n RF. That s, y j = 1 f a faclty s open and 0 f closed. Notce that ths defnton of y j s opposte that of the AF defnton of y j. Whle t s confusng, we ntentonally keep ths dscrepancy, as ts resoluton dentfes the two models. The CF also ntroduces bnary varables of the form z. In partcular, z k = 1 f and only f every faclty n V k s closed, and 0 otherwse. Alternately, z k = 0 f any faclty n V k s open, and 1 otherwse. Usng these z k and y we can now present Cornuejols et al. s formulaton of the p-medan problem. The Cornuejols formulaton (CF) } n K 1 mn Z(z, y) = D 1 + D k )z k k=1 (D k+1 (14) m y j = p (15) z k + j:d j D k y j 1 for = 1, 2,..., n; k = 1, 2,..., K (16) z K = 0 for = 1, 2,..., n (17) z k 0 for = 1, 2,..., n; k = 1, 2,..., K (18) y j 0, 1} for j = 1, 2,..., m (19) Constrant (15) s the same as constrant (3) n RF; that s, t ensures that exactly p stes wll assgn facltes to be opened. Constrant (16) ensures that for a gven clent, ether z k = 1 or at least one faclty n V k s open. Constrant (17) ensures that for a gven clent, at least one faclty n the whole neghborhood V K s open. Ths constrant s redundant, as gven constrant (15), p stes wll be assgned, so z K wll certanly be 0. We wll heretofore gnore ths contrant n our own developments of the model. Constrants (18) and (19) are postvty and bnary constrants for the respectve varables. The objectve functon (14) sums the total weghted dstance from each clent to the faclty ste chosen to provde for t. Remark 1. AF and CF are functonally dentcal. We restate the formulaton of AlBdaw et al. for ease of comparson: The AlBdaw formulaton (AF) } n m k 1 mn B(y) = δ 1 + δ k y πr n y = n p k=2 S k y z k S k 1 for each S k correspondng to each z k 8 r=1

9 y 0, 1} for j = 1, 2,..., n z k 0 for each k constructed. The mmedate dfference n the two models s that they have swtched the defntons of and j. That s, n the AlBdaw et al. model, represents facltes and j demand, whle n most other models represents demand and j facltes. To mtgate ths, we present an altered formulaton of CF, usng the AF defnton of y j. Hence we replace every y j n CF wth ȳ j = 1 y j and z k wth z k = 1 z k. The altered Cornuejols formulaton (CF ) } n K 1 mn Z(z, y) = D 1 + D k )z k k=1 (D k+1 (20) n ȳ j = m p (21) j=m j:d j D k Computatons: Let ȳ j = 1 y j, z k = 1 z k m m (21) ȳ j = (1 y j ) = m (22)z k + z k + j:d j D k j:d j D k z k + D k z k j:d j D k j:d j D k y j 1 ȳ j z k D k 1 for = 1, 2,..., n; k = 1, 2,..., K (22) z K = 1 for = 1, 2,..., n (23) ȳ j 0, 1} for j = 1, 2,..., m (24) z k 0 for = 1, 2,..., n; k = 1, 2,..., K (25) (1 ȳ j ) 1 j:d j D k ȳ j 1 ȳ j 1 D k ȳ j z k D k 1 m y j = m p (23)z K = 0 1 z K = 1 K z = 1 (24) Ths follows mmedately because by defnton y 0, 1} ȳ 0, 1} (25) z k 0 because by defnton z k 0, 1} z k 0, 1} We leave t to the reader to determne that D 1 = δ 1, that for k > 1, (D k+1 D k ) = δ k, and that z k = y πr. The results follow from examnng the defntons of the relevant terms. 9

10 Upon ths substtuton, we see that the models are nearly dentcal, wth the excepton beng the extra constrant (23) n CF. 3.2 The Elloum formulaton The formulaton that Elloum develops s nearly dentcal to CF, except that constrant (16) s separated nto two dstnct contrants based upon a recursve defnton of z. Notce how ths effects the summaton n each of the relevant constrants: n CF, constrant (16) sums over all nodes wthn dstance k or less, whle n EF, constrants (28) and (29) sum over only those nodes at the exact dstance of k. EF effectvely adds another set of constrants n order to make the constrants smaller and easer to manpulate. EF uses the exact same notaton as CF; the followng recursve statement s manpulaton not defnton. = (1 y j ), for = 1, 2,..., n; k = 1, 2,..., K z k j V k whch mples the followng recursve defnton: z k = z k 1 z 1 = (1 y j ), for = 1,..., n j:d j =D k j:d j =D 1 (1 y j ), for = 1,..., n; k = 2, 3,..., K. Ths motvates the formulaton of the Elloum model: The Elloum formulaton (EF) } n K 1 mn Z(z, y) = D 1 + D k )z k k=1 (D k+1 (26) m y j = p (27) z 1 + y j 1 for = 1, 2,..., n (28) z k + j:d j =D 1 y j z k 1 for = 1, 2,..., n; k = 1, 2,..., K (29) j:d j =D k z K = 0 for = 1, 2,..., n (30) z k 0 for = 1, 2,..., n; k = 1, 2,..., K (31) y j 0, 1} for j = 1, 2,..., m (32) As mentoned prevously, the only dfference between CF and EF s that constrant (16) n CF s broken up nto constrants (28) and (29) n EF. Another way of conceptualzng ths s to consder the sets over whch the summatons occur. CF has a flled rng surroundng the relevant z j contanng entres of dstance D k or less; whle EF has a hollow rng surroundng the relevant z j, only contanng entres precsely D k away. Elloum goes on to show that f CF and EF are the LP-relaxatons of CF and EF 10

11 respectvely, then 1. The feasble soluton set of EF s ncluded n the feasble soluton set of CF, wth the ncluson possbly beng strct. 2. CF and EF have the same optmal values. 3. Gven an optmal soluton to one of CF or EF, we can deduce an optmal soluton of the other wth the same optmal value. So the Elloum model s at least equvalent to the Cornuejols model, wth the possble advantage of havng a tghter feasble regon. 3.3 Reductons n the Elloum formulaton Elloum offers three reducton steps nherent n the defnton of the z varables, though one s trval, so only two are presented. 1. For any clent, f V 1 s a sngleton y a then z 1 = 1 y a for any feasble soluton. Then the varable z 1 can be replaced by 1 y a, and the constrant defnng z 1, z 1 + y a 1 may be elmnated. In applcaton, for a clent, f there s only one ste the closest dstance away, then we may replace the z varable representng that ste wth 1 y a and elmnate ts correspondng constrant. 2. For any two clents and, f V k = V k for some k, k, then z k = z k for any feasble soluton. We may then replace the varable z k wth zk, and elmnate the constrant defnng z k : zk + j:d j=d y k j z k 1. That s, f gven two clents and, f the sets of the k and k respectve closest faclty stes are dentcal, then we may replace z k wth zk and elmnate the constrant representng z k. Notce that ths reducton s dentcal to the combnng of terms that occurs n AF. 3.4 A computatonal example We use the same scenaro and cost matrx as wth the AlBdaw model to llustrate the smlartes and dfferences between the dfferrent formulatons. Consder a p-medan problem wth m = 5 potental faclty stes, n = 5 clents and p = 2 facltes to place, wth cost matrx C = Let each column represent a faclty F and each row a clent C for = 1, 2,..., 5. 11

12 V 1 1 = F 3 } V 2 1 = F 1, F 3 } V 3 1 = F 1, F 3, F 4 } V 4 1 = F 1, F 2, F 3, F 4 } V 5 1 = V K V 1 2 = F 3 } V 2 2 = F 2, F 3 } V 3 2 = F 2, F 3, F 4 } V 4 2 = F 1, F 2, F 3, F 4 } V 5 2 = V K V 1 3 = F 3 } V 2 3 = F 3, F 4 } V 3 3 = F 2, F 3, F 4 } V 4 3 = F 2, F 3, F 4, F 5 } V 5 3 = V K V 1 4 = F 4, F 5 } V 2 4 = F 2, F 4, F 5 } V 4 3 = F 2, F 3, F 4, F 5 } V 4 4 = V K V 1 5 = F 4 } V 2 5 = F 4, F 5 } V 5 3 = F 2, F 4, F 5 } V 4 5 = F 2, F 3, F 4, F 5 } V 5 5 = V K Table 1: Neghborhoods of the computatonal example We can now formulate the objectve functon of the opmzaton problem: Z(z, y) = 3 + 2z z z z z z z z z z z z z z z z z z 4 5 = z z z z z z z z z z z z z z z z z z 4 5 Then the complete model s: mn Z(z, y) = z z z z z z z z z z z z z z z z z z5 4 5 y j = 2 z 1 + y j 1 for = 1, 2,..., 5 z k + j:d j =D 1 y j z k 1 for = 1, 2,..., 5; k = 1, 2,..., K j:d j =D k z K = 0 for = 1, 2,..., 5 z k 0 for = 1, 2,..., 5; k = 1, 2,..., K y j 0, 1} for j = 1, 2,..., 5 4 A pseudo-boolean representaton of the Elloum model Havng dscovered that the pseudo-boolean formulaton developed by AlBdaw et al. s functonally equvalent to the formulaton presented by Cornuejols et al., we are motvated to present a pseudo-boolean formulaton of the model presented by Elloum. In the paper ntroducng Elloum s model, EF ran sgnfcantly faster than both CF and RF solvng a 12

13 varety of benchmark p-medan problems from ORLIB (Beasley 1990) and rw (Resende and Werneck 2004). Ths leads us to beleve that a pseudo-boolean formulaton of EF wll run faster than CF s pseudo-boolean counterpart, AF. We set about convertng the pseudo-boolean AF nto a pseudo-boolean formulaton followng the same set of constrants. We begn wth the AlBdaw formulaton and make the substtuton x j = (1 y j ). Remember that y j n AF s ȳ j n all other formulatons, so x j wll equal y j outsde of AF. Makng ths substtuton, the orgnal AF becomes The AlBdaw formulaton (AF) } n m k 1 mn B(y) = δ 1 + δ k y πr n y = n p k=2 S k y z k S k 1 for each S k correspondng to each z k y 0, 1} for j = 1, 2,..., n z k 0 for each k constructed. r=1 The altered AlBdaw formulaton } n m k 1 mn B(y) = δ 1 + δ k (1 x πr ) k=2 r=1 (33) n x j = p (34) j S k x j + z k 1 for each S k correspondng to each z k (35) x j 0, 1} for j = 1, 2,..., m (36) z k 0 for each k constructed. (37) Calculatons Constrants (34), (36), and (37), as well as the objectve functon (33) are dentcal to the calculatons performed n substtutng ȳ j for y j n CF, for (21), (24), (25), and (20) respectvely. Constrant (35) s smple but not necessarly apparent. j S k y j z k S k 1 = j S k (1 x j ) z k S k 1 13

14 = S k j S k x j z k S k 1 = j S k x j + z k 1 At ths pont, the altered AF s begnnng to resemble EF. All that remans s to convert constrant (35): j S k x j + z k 1 for each S k correspondng to each z k We recall the defntons of S k and D k. In AF, for a mult-part monomal y a y b y c... = z k, S k s the set of a, b, c,... correspondng to the ndces of the varables makng up z k. In applcaton, S k represents the t closest stes to a ste j where t = S k. D k s the ordered set of dstances of the k closest stes to a ste. In order to make constrant (35) consstent wth the notaton of EF, we change our notaton from S k to D k. We want to preserve the contents of the set, so j : j S k } = j : d j D k }. But n dong ths, we also must change z k to z k. We now can reformulate constrant (35) as x j + z k 1 for = 1, 2,..., n j:d j D k Ths essentally gves us the formulaton of Cornuejols et al. The pseudo-boolean Cornuejols model (PBC) } n m k 1 mn Z(y) = δ 1 + δ k (1 x πr ) (38) k=2 r=1 m x j = p (39) j:d j D k x j + z k 1 for = 1, 2,..., n (40) x j 0, 1} for j = 1, 2,..., m (41) z k 0 for each k constructed. (42) By splttng constrant (35) nto two new constrants, we now have a pseudo-boolean formulaton of Elloum s model: The pseudo-boolean Elloum model (PBE) } n m k 1 mn Z(y) = δ 1 + δ k (1 x πr ) k=2 r=1 (43) 14

15 m x j = p (44) z 1 + x j 1 for = 1, 2,..., n (45) z k + j:d j =D 1 x j z k 1 for = 1, 2,..., n (46) j:d j =D k x j 0, 1} for j = 1, 2,..., m (47) z k 0 for each k constructed. (48) Ths PBE model essentally uses the pseudo-boolean objectve functon of AF and the constrants of EF or CF. We may also apply our choce of reducton technques from the two formulatons: whle they ultmately end n the same result, one reducton may be preferred over the other n a gven stuaton. For example, dependng on the modelng language, the AF reductons may be easer because they merely nvolve polynomal algebra, whle the EF reductons rely on set content comparson. Thus the prmary beneft to hybrdzng these two models s the retenton of the pseudo-boolean objectve functon from AF and the flexblty of reducton optons from both models. Though the sets of constrants are nearly dentcal between AF and EF, usng the constrants of EF provdes one small advantage compared to AF: constrants (45) and (46) n the PBE are more nteger-frendly than constrant (11). Compare wth the AF constrant z 1 + j:d j D x 1 j 1 for = 1, 2,..., n (45) z k + j:d j D x k j z k 1 for = 1, 2,..., n (46) S k y z k S k 1 for each S k correspondng to each z k (11) Revelle (1993) dscusses how the formulaton of a model affects the nature of ts soluton, partcularly whether the soluton s nteger or fractonal. He concludes that, generally, a soluton s more lkely to be nteger f the coeffcents of the varables n the constrant set are -1, 0, or 1. He refers to ths property as beng nteger-frendly In the PBE constrants (45) and (46), we have each of our varables and our bound takng values of 0 or 1. In the AF constrant (11), we have the varables takng values of 0 or 1, but the boundng constrant can take a value up to n 1 n sze. Whle there may be some unexplored repercusson of havng an unbounded greater-than nequalty, what wth the lesser-than nequalty bounded below by zero, Revelle s noton of nteger-frendlness makes the PBE or CBE formulaton of the constrants preferable for now. 5 Computatonal Comparson For the computatonal aspect of ths paper, we developed models for the AF and PBC models. We assumed that the PBE model would be faster than the PBC model, so f we can show that the PBC model s faster than the AF model, we assume that the PBE model s faster than the AF model as well. Comparng the PBC and AF models s tenuous, as they are nearly dentcal wth the excepton of the PBC model s nteger frendly set of constrants. 15

16 As such, we expect only margnal advantages n run tme. To best solate the effect that the constrant formulaton has on model speed, we refran from performng other reductons on these models. Thus the runtmes provded and varables and constrants elmnated are not lower bounds: the results of the PBC model are ntended solely to be taken n context of comparson, not as a benchmark. Usng Code::Blocks C++ as our modelng language, we have developed algorthms to perform the reductons approprate to each formulaton and to formulate the AlBdaw and hybrd models. We run our models aganst several problem datasets from the Beasley OR lbrary, varyng n ther szes and values of p. Preprocessng occurred on a portable PC wth a 2.00 GHz Intel Core 2 Duo processor and 3.00 GB of RAM. The models were run on a non-portable PC wth a 2.13 GHz Intel Core 2 Duo processor and 2.00 GB of RAM. The results are solved n Xpress IVE (FICO), courtesy of the UCSB Department of Geography. We then compare the reducton effcency, preprocessng tmes, and solvng tmes of each model. 5.1 Dscusson of Models In ths secton, we focus exclusvely on AlBdaw et al. s pseudo-boolean mode and the pseudo-boolean Cornuejols model developed n ths paper. The classc model developed by Revelle has been thoroughly tested already, and n ther respectve papers, AlBdaw et al. and Elloum both show ther respectve models to be faster n regard to certan benchmark problems. Smlarly, Elloum shows her model to be more effcent than that of Cornuejols et al. For these reasons, we choose to focus on AF and PBC, as we assume that the PBE s faster than the PBC model. In runnng these models, we seek to nvestgate the effects of the PBE model s nteger frendly constrants, as opposed to the non-nteger frendly constrants of the AF model. For ths reason, we refran from ncludng any of the aforementoned reducton methods n order to solate the effects of the dfferent constrant formulatons on the model run tme. 5.2 Computatonal Results The Beasley lbrary from whch we have drawn these problems has n=m; that s, the problem treats each node as a potental faclty ste and a potental clent. Thus, the problem sze, number of demand stes, and number of faclty stes are all the same for the pmed problems. Smlarly, notce that the number of entres n the cost matrx s equal to n 2. The percent varable reducton s gven by (n 2 remanng varables)/n 2. Notce that snce the AF and PBE models use the same objectve functon, the number of remanng varables and varable reducton are the same for each model. The preprocessng and run tmes are rounded to the nearest hundredth of a second. In the AF model, we see that preprocessng tme ncreases exponentally as n ncreases. The run tmes for ths model fluctuate broadly, wth lttle reason mmedately apparent. We attrbute t to dfferences n the complexty of the problem datasets, as the problems from 16

17 AF AF PBC PBC Sze Var. Var. Preproc. Run tme Preproc. Run tme (n) p after red. (%) (sec) (sec) (sec) (sec) pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed pmed Table 2: Reductons of monomals of the objectve functon as well as preprocessng and run tmes of the AF and PBC models for benchmark problems 17

18 the Beasley lbrary are all dstnct. That s, they do not use the same dataset for each n and vary p, but rather use a dfferent dataset for each nstance. In the PBE model,the preprocessng tmes are comparable to the AF model. There are a few dscrepances ether way, as wth pmed21 and pmed22, but we attrbute these to anomales of the computer. Lke the AF model, the run tme for the PBE model does not correlate smoothly wth the sze of the dataset. Agan, we attrbute these dscrepances to the rregularty of the Beasley datasets. Comparng the run tme results to the AF model, we see that they are just slghtly faster for n up to 500. Whle we mght normally dsmss ths as a computatonal anamoly, ts consstency makes an argument for t. In partcular, we see that n all but three of the nstances, the PBC model runs faster than the AF model. In some cases, such as pmed13 and pmed17, PBC runs over three tmes as fast as AF. Averagng the dfference n tmes for each of these smulatons, PBC was over 5 seconds faster than AF. Takng the average wth dfferences greater than three seconds removed, we see that PBC ran.48 seconds faster than AF. As we had predcted ntally, ths ndcates that the nteger-frendly constrant (45) of the PBC model has a slght advantage over the normal constrant (11) of the AF model, at least n the solvng software Xpress IVE. On the whole, the PBE model performed better than the AF model n both preprocessng and run tme. The advantage n preprocessng s unexpected and unattrbuted. The advantage n run tme s margnal, but present. The models could be run on larger values of n to see f the advantage held by the PBC model wdens as n ncreases. 6 Concluson Ths paper begns wth an overvew of the p-medan problem, ts hstory, and the classc formulaton by Revelle and Swan (1970). It then gves a bref explanaton of the development of the respectve models developed by AlBdaw et al. and Elloum, presents the models, explans the reducton technques nherent n each model, and provdes a worked example of a sample problem for each. We then set about convertng Elloum s model nto a pseudo-boolean form. In ths converson process, we determne that the model developed by Cornuejols et al. (1980), a predecessor of Elloum s model, s functonally equvalent to AlBdaw et al. s model. Upon a partcular varable substtuton and gnorng extraneous constrants, we can see that the two models are equvalent. Ths motvated us to develop a pseudo-boolean formulaton of Elloum s model. In Elloum s paper featurng her model, that model had proved more effcent than the model developed by Cornuejols et al. Ths led to the actual development of models for the AlBdaw et al. and pseudo-boolean Cornuejols formulatons, operatng under the assumpton that the pseudo-boolean Elloum model would outperform the PBC model, and hence the AF model. In formulatng the hybrd model, we essentally ended wth the objectve functon from the AlBdaw et al. model and the constrants of the Cornuejols model Ths allowed us some modcum of flexblty n choosng whch reducton technque to choose: the algebrac combnaton of the objectve functon, or the neghborhood comparson of the constrants. In our computatonal analyss, we compared the AF and the PBC models usng benchmark problems of varous szes from the Beasley OR lbrary. Snce both models use dentcal objectve functons, the varable reducton was the same. And snce the constrants are 18

19 functonally dentcal, the constrant reductons were also close to dentcal. Our man queston n comparng the two models was whether the nteger-frendly constrants of the PBC model would provde a sgnfcant computatonal advantage over the non-nteger frendly constrants of the AF model. The PBC model already has the advantage of flexblty n ts constrants; that t may adopt those of ether the AlBdaw or the Cornuejols model, and lkewse for ts objectve functon. For ths comparson, we chose to adapt the constrants of the Cornuejols model to the objectve functon of the AlBdaw model n order to provde a pseudo-boolean formulaton of the Cornuejols model. The results of the computaton comparson were mldly conclusve. Whle the PBC model had a slghtly faster preprocessng speed and run tme, the preprocessng speed s unexpected and unattrbuted, as the two models have comparable preprocessng algorthms. Addtonally, the run tme advantage held by the PBC was only moderate, and over a lmted samplng of sets. Its consstency, however, s encouragng. For now, we conclude that the PBC model s nteger frendly constrant may have a run tme advantage over the AF model, and propose further tests to verfy t. When runnng these computatonal models, we began to face memory and tme ssues as the problem szes ncreased, whch s why our largest problem has only n=500 stes. Dfferent solvng software on dfferent computers would lkely gve dfferent results. Addtonally, the code used n preprocessng was less effcent than t could have been: snce the two algorthms were near dentcal, the correctness of the result was more mportant than the speed of preprocessng. Addtonal research mght nvestgate the effcacy of the two reducton methods that were mentoned n ths paper but not ncluded n the computatonal comparson. Whle ther end result s essentally the same, the reducton algorthms mght work better n dfferent condtons. If so, t would be useful to determne what these condtons are. Addtonally, as suggested by AlBdaw et al., a potentally worthwhle course of study would be to nvestgate the effcacy of the pseudo-boolean formulaton of the p-medan problem. Fnally, t should reman productve to apply dfferent formulaton methods to problems n related felds. As Elloum dd wth the smple plant locaton problem that Cornuejols et al. had orgnally developed, takng an exstng formulaton n a related feld and adaptng t to a current problem can lead to new applcatons. Acknowledgements I would especally lke to thank Professor Rchard Church for the tremendous generosty he has shown wth hs tme. Hs gudance and nsght have been nvaluable. 19

20 References [1] AlBdaw, B.F., Ghosh, D., and Goldengorn, B. (2011). Data aggregaton for p-medan problems. Journal of Combnatoral Optmzaton 21, [2] AlBdaw, B.F. and Goldengorn, B. (2009). Equvalent nstances of the smple plant locaton problem. Journal of Computaton and Appled Mathematcs 57, [3] Avella, P., Sassano, A., and Vasl ev, I. (2007). Computatonal study of large-scale p-medan problems. Mathematcal Programmng A [4] Boros, E. and Hammer, P.L. (2002). Pseudo-Boolean optmalzaton. Dscrete Appedl Mathematcs 123, [5] Church, R.L. (2003). COBRA: A new formulaton of the classc p-medan locaton problem. Annals of Operatons Research 122, [6] Cornuejols, G., Nemhauser, G.L., and Wolsey, L.A. (1980). A canoncal representaton of smple plant locaton problems and ts applcatons. Sam Journal on Matrx Analyss and Applcatons 1, [7] Hakm, S.L. (1964). Optmum locatons of swtchng centers and the absolute centers and medans of a graph. Operatons Research 12, [8] Hakm, S.L. (1965). Optmum dstrbuton of swtchng centers n a communcaton network and some related graph theoretc problems. Operatons Research 13, [9] Morrs, J.G. (1978). On the extent to whch certan fxed-charge depot locaton problems can be solved by LP. The Journal of the Operatonal Research Socety 29, [10] Revelle, C. S. (1993). Faclty stng and nteger-frendly programmng. European Journal of Operatonal Research 65, [11] ReVelle, C. S. and Swan, R. W. (1970). Central facltes locaton. Geographcal Analyss 2: [12] Rosng, K.E. and Revelle, C.S. (1997). Heurstc concentraton: two-stage soluton concentraton. European Journal of Operatonal Research 97, [13] Rosng, K.E., Revelle, C.S., and Rosng-Vogelaar, H. (1979). The p-medan and ts lnear programmng relaxaton: an approach to large problems. The Journal of the Operatonal Research Socety 30, [14] Weber, A. (1909). Über den Standort der Industren, Eerster Tel: Rene Theore des Standortes. Mohr, Tübngen 20