Covering Problems. Lawrence V. Snyder. Lehigh University

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1 Coverng Problems Lawrence V. Snyder Lehgh Unversty Forthcomng n Foundatons of Locaton Analyss, H. A. Eselt and V. Maranov (eds.), Sprnger. 1. Introducton The mal-order DVD-rental company Netflx chooses dstrbuton center locatons so that most of ts customers receve ther DVDs wthn one busness day va frst-class U.S. mal. Smlarly, many muncpaltes am to have fre crews reach 911 callers wthn a specfed tme, such as four mnutes. Both of these are examples of the noton of coverage, a concept that s central to several classes of faclty locaton models; t ndcates whether a demand locaton s wthn a pre-specfed radus (measured by dstance, travel tme, cost, or another metrc) of ts assgned faclty. Homeowners are covered f they are wthn four mnutes of the nearest fre staton, and Netflx customers are covered f they are wthn one malng day of a dstrbuton center. Note that n the fre-staton example, muncpaltes typcally want to cover all resdents (whle mnmzng the number of servce statons to open), whereas Netflx wants to cover as many customers as possble (subect to a lmt on the number of warehouses t may operate at any tme, as specfed by ts captal budget). The fre-staton problem s an example of the set coverng locaton problem (SCLP), whle Netflx s problem s an example of the maxmal coverng locaton problem (MCLP). Ths chapter dscusses both problems. The SCLP was frst ntroduced by Hakm (1965) and was later formulated as an nteger programmng problem by Toregas, et al. (1971). The MCLP was ntroduced by Church and ReVelle (1974). Both models, and ther varants, have been appled extensvely to publc-sector faclty locaton problems, such as the locaton of emergency medcal servce (EMS) vehcles (Eaton, et al. 1985), fre statons (Schllng, et al. 1980), bus stops (Gleason 1975), wldlfe reserves (Church, Stoms, and Davs 1996), and emergency

2 ar servces (Flynn and Ratck 1988). They have been appled to a much more lmted extent n the prvate sector (see, e.g., Nozck and Turnqust 2001). The SCLP and MCLP are closely related to the p-center problem, whch ams to locate at most p facltes to mnmze the maxmum dstance, among all customers, between the customer and ts assgned faclty. In the p-center problem, the coverage radus tself consttutes the obectve functon. See the Introducton chapter of ths book for a more thorough dscusson of the relatonshps among these classcal models. Lke most locaton problems, the SCLP and MCLP may be defned as contnuous problems (n whch facltes may be located anywhere on the plane) or as dscrete problems (n whch they may be located only at the nodes of a network). In ths chapter we consder the latter approach. The remander of ths chapter s organzed as follows. In Secton 2, we dscuss classcal papers on the SCLP (n Secton 2.1) and on the MCLP (n Secton 2.2), present the results of computatonal experments, and dscuss more recent varatons. In Secton 3, we dscuss the mpact that these models have had and the bodes of research they have nspred, focusng generalzed notons of coverage. Fnally, we conclude n Secton 4 and suggest some possble future research drectons. 2. Hstorcal Contrbutons We present the classcal models for the SCLP by Hakm (1965) and Toregas, et al. (1971), and for the MCLP by Church and ReVelle (1974), n Sectons 2.1 and 2.2, respectvely. 2.1 The Set Coverng Locaton Problem Hakm (1965) Although the generc (non-locaton) set-coverng problem had been formulated pror to Hakm s (1965) semnal paper on the SCLP, Hakm s work s mportant for, among other thngs, ntroducng the noton of coverage nto faclty locaton models. Hakm s proposed soluton method, whch nvolved the use of

3 Boolean functons, never proved to be effcent enough to warrant ts use n practce; rather, the SCLP s generally solved usng nteger programmng (IP) technques, frst proposed by Toregas, et al. (1971). We dscuss Hakm s model and brefly outlne the Boolean-functon approach n ths secton. In Secton 2.1.2, we present the IP method of Toregas, et al. We consder a graph G = (V, A) and assume that every node n V s both a customer (demand) node and a potental ste for a faclty. (However, one can easly extend the models below to handle the cases n whch some customers may not be facltes, or some facltes are not customers,.e., do not need to be covered. Below, we use terms lke customer and faclty as shorthand for the customer located at node and the faclty located at node. ) Let n = V. The dstance between nodes and s gven by d, and the maxmum allowable dstance between a customer and ts nearest opened faclty the coverage dstance s gven by s. If (, ) A, then d s the length of the arc (, ), and otherwse t s the shortest dstance from to on the graph. (We use the term dstance throughout, but the parameters d and s may ust as well represent travel tmes, costs, or another measure of proxmty.) Therefore, faclty covers customer f d s. We defne V = { V : d s}, that s, V s the set of nodes that cover customer. Note that every V s nonempty, assumng that d = 0 for all. The obectve of the SCLP s to fnd the mnmum-cost (or mnmum-cardnalty) set of locatons such that every node n V s covered by some node n the set. The applcaton that Hakm ctes for the SCLP s that of locatng polcemen along a hghway network so that every ntersecton (vertex of the graph) s wthn one dstance unt of a polceman. Subsequently, the problem has found a much broader range of applcatons, as dscussed earler.

4 We wll assume that facltes may be located only at the nodes of the network, not along the edges. Note that t may be optmal to locate along edges, snce the well known Hakm property whch says that an optmal soluton always exsts n whch facltes are located at the nodes, rather than along the edges, of the network does not apply to the SCLP. (Hakm ntroduced hs famous property n an earler paper (Hakm 1964) n the context of the p-medan problem, not of the SCLP.) A very smple counterexample conssts of two nodes connected by a sngle edge of length 1 and a coverage dstance of 0.5. If facltes are allowed on the edges, the unque optmal soluton conssts of one faclty (located n the mddle of the edge), whereas the optmal nodal soluton conssts of two facltes, one at each node. On the other hand, a problem n whch facltes may be located on edges may be converted to a nodeonly problem by nsertng dummy nodes onto the edges, takng advantage of the fact that there are only a fnte number of possble optmal locatons along edges. See Church and Meadows (1979) for detals. In some applcatons, t s desrable to use a dfferent coverage dstance for each customer for example, f customers have servce agreements that specfy dfferent response tmes. In ths case, the coverage dstance s customer dependent, s, and the set V s gven by V = { V: d s }. The analyss below changes n only mnor ways. The SCLP s closely related to the graph-theoretc vertex cover problem, whose obectve s to fnd a subset of nodes n the graph such that every node n the graph s adacent to some node n the set and such that no strct subset of the set has the same property. Such a set of nodes s called a cover. The optmzaton verson of the vertex cover problem seeks the mnmum-cardnalty cover, and ths problem s a specal case of the SCLP n whch s = 1 and d = 1 for all (, ) A. Indeed, ths specal case s the problem consdered by Hakm (1965), although he s usually credted for ntroducng the more general SCLP, snce he presented the problem explctly n a faclty locaton context. In ths secton we wll

5 assume, followng Hakm, that s = d = 1, though n subsequent sectons we wll allow s and d to be arbtrary. Hakm notes that the assumpton that d = 1 s not overly restrctve, snce f the arc lengths are greater than 1, one could smply ntroduce dummy nodes along the arcs, one unt apart, assumng that the arc lengths are ntegers. Of course, ths modelng trck comes at consderable computatonal expense, especally snce Hakm s method reles on an enumeratve approach whose computatonal complexty ncreases exponentally wth the number of nodes. In the remander of ths secton, we descrbe Hakm s (1965) approach to solvng the SCLP. As noted earler, ths method s not commonly used today and s dscussed here prmarly for ts hstorcal nterest. Recall that V s the set of nodes that cover node ; gven the assumpton of unt arc-lengths and unt coverage dstance, V s smply the set of nodes that are adacent to, plus tself. Let S be a subset of the node set V. For each node, we defne a Boolean (bnary) varable x that equals 1 f S and 0 otherwse. Wth a slght abuse of notaton, we can wrte S = U x, V where x s taken to equal the set {} f x = 1 and the null set otherwse. We also defne X as the sum of the Boolean varables for the nodes n V ; that s, X =. x V Here, represents Boolean summaton, analogous to the or operator, n whch = 1. Then X = 1 f and only f S contans a node that covers node. Fnally, we defne the Boolean functon f, whch takes as nputs the vector of Boolean varables for the nodes and returns a sngle Boolean value: f ( x 1, K, x n ) = X. V

6 Snce node s covered f and only f X = 1, we have the followng theorem: Theorem 1. S contans a coverng of V f and only f f (x 1,,x n ) = 1. The advantage of usng the functon f s that t allows us to use Boolean algebra to construct coverngs of V. Although ths approach stll nvolves enumeratng all coverngs, t allows us to do so wthout enumeratng all subsets of V to dentfy them. In partcular, we wll create a mnmum sum of products,.e., the smallest possble sum of products of x varables that s logcally equvalent to f (x 1,,x n ). Ths method nvolves elmnatng terms that are mpled by others, then usng Boolean algebra to smplfy the resultng formula untl we have an expresson consstng of the sum of smple products of varables such that no product s mpled by (contans) any other. The method s best explaned by use of an example. Example 1. We llustrate the method usng the sample network n Fgure Fgure 1. Sample network. Usng the adacences depcted n the fgure, X 1 = x 1 + x 3, X 2 = x 2 + x 4, and so on. Therefore, f ( x1, K, xn ) = ( x1 + x3 )( x2 + x4 )( x3 + x1 + x2 + x4 + x5 )( x4 + x2 + x3 + x5 )( x5 + x3 + x4 ). By Theorem 1, to fnd all coverngs of the graph, we need to fnd all possble values of {x 1,, x 5 } that make f (x 1,,x n ) = 1,.e., such that all terms n the product above equal 1.

7 To begn, note that the frst term s contaned n the thrd. Snce we need each term to equal 1, the thrd term equals 1 f the frst does; therefore, we can elmnate the thrd term. Smlarly, the fourth term contans the ffth, so we can elmnate the fourth term. The resultng expresson s: f ( x1, K, xn ) = ( x1 + x3 )( x2 + x4 )( x3 + x4 + x5 ). Boolean algebra contans two dstrbutve laws; one says that, for any Boolean varables x, y, and z, x + (yz) = (x + y)(x + z). Applyng ths law to the last two terms, we get f ( x1, K, xn ) = ( x1 + x3 )( x4 + x2 x3 + x2 x5 ). The other Boolean dstrbutve law says that x(y + z) = xy + xz. Applyng ths law to multply the two terms, and repeatedly applyng both Boolean dentty laws (whch say that x + x = x and that xx = x), we get f ( x K n = x x + x x x + x x x + x x + x x + x x x. 1,, x ) Fnally, by the Boolean redundance law (x + xy = x), we can remove the second and last terms: f ( x K n = x x + x x x + x x + x x. 1,, x ) Therefore, the covers for the graph n Fgure 1 are: {1, 4}, {1, 2, 5}, {3, 4}, {2, 3}. All but {1, 2, 5} are mnmum covers Hakm was optmstc that ths enumeratve approach would prove to be practcal: snce the subect of smplfcaton of Boolean functons has been wdely studed and there are effcent dgtal computer programs for such a purpose, the above formulaton s feasble. Twenty-frst-century readers, however, wll recognze that the enumeratve approach s mpractcal for large nstances. Moreover, snce the vertex cover problem s NP-complete (Garey and Johnson 1979), no polynomal-tme exact algorthm for

8 the SCLP exsts. However, more effcent approaches than Hakm s exst; we dscuss a mathematcalprogrammng-based approach n the next secton Toregas, et al. (1971) Toregas, et al. (1971) formulate the SCLP as an nteger programmng (IP) problem and use standard mathematcal programmng methods to solve t. We dscuss ther approach next. The IP has one set of decson varables: x 1, f a faclty s opened at node = 0, otherwse for V. Note that ths varable x has no relaton to the Boolean varables x defned n Secton The IP s formulated as follows: (SCLP) mnmze z = (1) x V subect to x 1 V (2) V x {0,1} V (3) The obectve functon (1) computes the total number of facltes opened. Constrants (2) requre at least one node from the coverage set V to be opened for each node. Constrants (3) are standard ntegralty constrants. Here, we do not assume that s = d = 1 (as we dd n Secton 2.1.1); any values for these parameters may be used n determnng the coverage sets V. Ths formulaton s vrtually dentcal to that of the classcal set-coverng problem; here t s dscussed n the context of locaton theory n partcular. It s well known that the set-coverng problem typcally has a small ntegralty gap; that s, the optmal obectve value of the LP relaxaton (denoted z LP ) s close to that of the IP tself (Bramel and Smch-Lev 1997), and often the LP relaxaton even has all-nteger solutons.

9 In fact, ReVelle (1993) argues that many faclty locaton problems have ths property and dscusses nteger-frendly programmng technques for several classcal problems. However, there do exst nstances of the SCLP whose LP relaxatons do not have all-nteger optmal solutons (otherwse the problem would not be NP-hard). An example s as follows. Example 2. Consder the network depcted n Fgure 2. In ths example, s = 1. An optmal soluton to the LP relaxaton of (SCLP) s gven by x 1 = x 2 = x 3 = 0.5, x 4 = 0, wth an obectve value of z LP = Fgure 2. Network for whch SCLP does not have all-nteger LP relaxaton. Snce the coeffcent of each x s 1 n the obectve functon of (SCLP), t s clear that the obectve functon value s nteger for any soluton to the IP. Snce z LP s a lower bound on z*, the optmal obectve functon value for the IP, and snce z* must be nteger, we can say that z* z LP, where a denotes the smallest nteger greater than or equal to a. Therefore, Toregas, et al. propose addng the followng cut to (SCLP): x J z LP. (4) We denote the resultng problem (SCLP-C). The new cut may elmnate some fractonal solutons, and the LP relaxaton to (SCLP-C) may have an all-nteger soluton as a result.

10 For the example n Fgure 2, (SCLP-C) does ndeed have an nteger soluton: x 1 = x 2 = 1, x 3 = x 4 = 0, for example, wth z* = 2. (It also has optmal fractonal solutons, e.g., x = 0.5 for all, but the smplex method would fnd nteger solutons snce these represent extreme ponts of the feasble regon.) Toregas, et al. therefore propose a two-step soluton procedure for the SCLP: 1. Solve the LP relaxaton of (SCLP). If the optmal soluton s nteger, STOP. 2. Otherwse, solve the LP relaxaton of (SCLP-C) usng the optmal obectve value from step 1 n the rght-hand sde of (4). Even wth constrant (4), the LP relaxaton may not have an nteger soluton. Toregas, et al. report that they found no such nstance n ther computatonal experments, though we found several such nstances n ours; see Secton In fact, Rao (1974) gves two counterexamples; n one, the addton of cut (4) results n a fractonal soluton, and n the other, the addton of cut (4) results n an nteger but nonoptmal soluton. (See also the reply to Rao s note by Toregas, ReVelle, and Swan (1974)). Toregas, et al. also dscuss the relatonshp between the SCLP and a varant of the p-medan problem n whch each customer may only be served by facltes that are wthn a dstance of s. The formulaton s obtaned smply by forcng the assgnment varable to be 0 for faclty customer pars that are more than s unts apart, or, alternately, by ndexng the assgnment varables for each customer over facltes n V, as opposed to all facltes n V. (We omt the formulaton here.) The optmal obectve value of ths p-medan varant changes wth s. For suffcently large s, the obectve functon value s no dfferent from the p-medan wthout dstance constrants; as s decreases, the obectve functon value ncreases as a step functon; and for suffcently small s, the problem s nfeasble. Toregas, et al. argue that the soluton to the SCLP provdes some nformaton about the feasblty of ths problem. In partcular, for a gven value of p, the smallest value of s for whch the p-medan varant s feasble s equal to the smallest value of s for whch the SCLP has an optmal obectve value of p. On the

11 other hand, the soluton to SCLP does not provde any nformaton about the breakponts of the step functon that relates the p-medan obectve to s Experments and Varants In Secton , we dscuss the results of our computatonal experment related to (SCLP). In Secton , we dscuss a technque for reducng the problem sze of the SCLP, and n Secton , we dscuss a varant nvolvng fxed costs Computatonal Experment We performed a computatonal experment to confrm the results reported by Toregas, et al. namely, that the LP gap for (SCLP) s small, and that cut (4) produces nteger solutons. For each value of n = 50, 100, 200, 400, 800, we generated 100 random nstances of the SCLP. Parameters were generated as follows: x- and y-coordnates were drawn from U[0,100] Dstances were calculated usng the Eucldean metrc The coverage dstance s was drawn from U[0,140] (140 maxmum possble dstance between two ponts n grd) For each nstance, we solved the LP relaxaton of (SCLP) usng CPLEX v to obtan z LP. If the optmal soluton to the LP was not nteger, we added cut (4) and solved the LP relaxaton to (SCLP-C) to obtan z LP-C. If the optmal soluton was stll not nteger, we solved (SCLP) as an IP to obtan z*. (If ether of the LP relaxatons resulted n nteger solutons, ther obectve values gve us z*.) The results are gven n Table 1. The columns labeled % Integer lst the percentage of nstances for whch the LP relaxaton produced an nteger optmal soluton. The columns labeled Avg LP Gap and Max LP Gap lst the average and maxmum, respectvely, of the LP gap, measured as (z LP z*) / z* for (SCLP) and (z LP-C z*) / z* for (SCLP-C).

12 Table 1. Performance of LP relaxatons of (SCLP) and (SCLP-C). (SCLP) (SCLP-C) n % Integer Avg LP Max LP % Integer Avg LP Max LP Gap Gap Gap Gap % % % % % % % % % % Total 83.6% % The LP gap for (SCLP) s small and tends to be smaller for smaller values of n. The worst gap we found was 33.5% for a problem wth n = 800. The addton of cut (4) reduces the LP gap substantally (from to , on average), but does not guarantee nteger solutons even wth the cut, 11.2% of nstances had fractonal optmal solutons. Several of these nstances also had nteger optmal solutons, though CPLEX dd not fnd these. In general, CPLEX solved the IP n well under one mnute on a laptop computer, even for the largest problems Row and Column Reducton The sze of (SCLP) can often be reduced substantally by usng row- and column-reducton technques. These methods explot the coverage structure by elmnatng rows and columns that are domnated by others. In partcular: A faclty 1 domnates another faclty 2 f t covers all of the customers that 2 does; that s, f 2 V mples 1 V for all V. In ths case, there s no reason to open faclty 2 snce 1 covers all the same customers and possbly more. Therefore we can set x = 0, or equvalently, elmnate the column correspondng to 2. A customer 1 domnates another customer 2 f every faclty that covers 2 also covers 1 ; that s, f V V. In ths case, f constrant (2) holds for 1 t also holds for 2, and therefore we can 2 1 elmnate the row correspondng to 2. 2

13 Row and column reducton technques are approprate for the SCLP because of the bnary nature of coverage. Most faclty locaton problems wth dstance obectves cannot generally accommodate these technques, except heurstcally, snce under most metrcs t s mpossble for a faclty to domnate another,.e., to be closer to every customer than another faclty s. These technques were proposed by Toregas and ReVelle (1972). See also Daskn (1995) and Eselt and Sandblom (2004) for thorough dscussons and examples of row- and column-reducton technques Faclty Fxed Costs If the facltes each have a dfferent fxed cost f, then the problem becomes to choose facltes to cover all demands at mnmum possble cost. Ths problem can be formulated smply by replacng the obectve functon (1) wth mnmze z = V f x The SCLP as formulated above s a specal case n whch f = 1 for all. The lnear-programmng-based soluton methods descrbed n Secton 2.3 can easly accommodate ths varaton. So can the Booleanfuncton approach: at the fnal step, we smply choose the cover that has the smallest total fxed cost. 2.2 The Maxmal Coverng Locaton Problem Church and ReVelle (1974) Introducton and Formulaton Whereas the SCLP has the form mnmze subect to number of facltes opened cover all demand, the maxmal coverng locaton problem (MCLP) has the nverse form: maxmze subect to demand covered lmt on number of facltes opened.

14 The SCLP treats all demand nodes as equvalent snce the coverage constrant apples equally to all. In the MCLP, n contrast, nodes are weghted by the demand that they generate, and the obectve favors coverage of larger demands over smaller ones. As the number of allowable facltes ncreases, the demand covered naturally ncreases as well. The modeler can plot a tradeoff curve depctng the performance of a range of solutons along these two dmensons; the decson maker can then choose a soluton based on ths tradeoff. Our notaton n ths secton s dentcal to that n Secton 2.1, wth the addton of two new parameters: a s the demand at node per unt tme, and p s the maxmum allowable number of facltes. We also ntroduce a new set of decson varables: y 1, f customer s covered by some faclty = 0, otherwse The MCLP s formulated by Church and ReVelle (1974) as follows: (MCLP) maxmze z = a y (5) subect to V V x y x V = p V (6) x {0,1} V (8) y {0,1} V (9) (7) The obectve functon (5) computes the total demand covered. Constrants (6) prohbt a customer from countng as covered unless some faclty that covers t has been opened. Constrant (7) requres exactly p facltes to be opened. Constrants (8) and (9) are standard ntegralty constrants. (In fact, t suffces to relax constrants (9) to 0 y 1 snce nteger values for the y are optmal f the x are nteger.)

15 Church and ReVelle cte Whte and Case (1973) as formulatng a smlar model to (MCLP) that maxmzes the number of demand nodes covered, rather than the total demand. Case and Whte s model s therefore a specal case of the MCLP n whch a = 1 for all. Church and ReVelle also present an alternate formulaton that uses a new decson varable y = 1 ; that s, y y defned as y 1, f customer s not covered by any faclty = 0, otherwse In the alternate formulaton, constrants (6) are replaced by x + y V 1 V The revsed constrants say that f node s not covered by any faclty ( V x = 0 ), then y must equal 1. The obectve functon (5) can be rewrtten as or equvalently, maxmze a ( 1 y ) = a a y, (10) V V V mnmze a, (11) V y snce the frst term n (10) s a constant. The revsed obectve (11) mnmzes the uncovered demand. The revsed formulaton s then gven by (MCLP2) mnmze z = a y (12) V subect to x + 1 V (13) V x V y = p (14) x {0,1} V (15) y {0,1} V (16) The two formulatons are mathematcally equvalent, as are ther LP relaxatons.

16 The MCLP s NP-hard (Megddo, Zemel, and Hakm 1983). In the next two sectons, we descrbe heurstc and exact approaches to solvng t, all of whch are dscussed by Church and ReVelle (1974) Heurstc Soluton Methods Lke many faclty locaton problems, the MCLP lends tself ncely to greedy heurstcs such as the Greedy Addng (GA) heurstc, whch Church and ReVelle (1974) credt to Church s (1974) Ph.D. dssertaton. The GA heurstc begns wth all facltes closed, then opens p facltes n sequence, choosng at each teraton the faclty that ncreases coverage the most. (For a dscusson of greedy and other heurstcs for faclty locaton problems, see Current, Daskn and Schllng 2002.) Solutons from the GA heurstc are nested n the sense that all of the facltes n the soluton to the p- faclty problem are also opened n the soluton to the (p+1)-faclty problem. Optmal solutons to the MCLP are not, n general, nested n ths way. Therefore, Church and ReVelle also suggest an alternate heurstc, called the Greedy Addng wth Substtuton (GAS) heurstc, whch attempts to rectfy ths problem by allowng an open faclty to be closed and a closed faclty to be opened at each teraton. Lke any heurstc, the GA and GAS are not guaranteed to fnd the optmal soluton. The GAS, however, tends to perform well n practce, and both heurstcs execute very quckly Lnear Programmng Approach Church and ReVelle propose solvng (MCLP2) drectly usng lnear programmng and branch-and-bound. Lke the SCLP, the LP relaxaton of the MCLP often yelds all-nteger solutons: Church and ReVelle report that approxmately 80% of ther test nstances had nteger solutons; we found an even hgher percentage n our computatonal tests (Secton ). Branch-and-bound may be appled to resolve fractonal solutons to the LP relaxaton, but Church and ReVelle also suggest a method that s effectve when solvng the same problem for consecutve values of p.

17 The method takes as nput a fractonal soluton to the p-faclty problem and an nteger soluton to the (p 1)-faclty problem. It s effectve when the (p 1)-faclty soluton covers all but a few nodes. We llustrate the method usng an example. Example 3. Consder an nstance of the MCLP for whch the total demand across all nodes s 100 unts. Suppose we have found an nteger soluton to the 4-faclty problem and that t covers all but two nodes, for a total of 91 demand unts covered. The two uncovered nodes (we ll call them 1 and 2) have demands of 4 and 6, respectvely. Suppose further that the LP relaxaton to the 5-faclty problem s fractonal and covers 98 demand unts. Fnally, suppose that the mnmum a among all nodes s 3. Now, the optmal nteger soluton wth p = 5 cannot cover all of the nodes, snce the LP relaxaton has an obectve value of 98. In fact, the IP soluton may cover at most 97 demand unts, snce at best t leaves the 3-demand node uncovered. We can create an nteger soluton to the p = 5 problem by addng node 2 to the p = 4 soluton. Snce the p = 4 soluton covered 91 demands, not ncludng node 2, ths new soluton covers = 97 demands. Ths soluton must be optmal for p = 5 snce 97 s an upper bound on the obectve value. An optmal soluton for the problem wth p = 6 can now be found by addng node 1 to the p = 5 soluton; the resultng soluton covers all demands. Church and ReVelle refer to ths method as the nspecton method. It can be summarzed as follows. Let z IP (p) be the optmal p-faclty obectve value of (MCLP), that s, the optmal demand covered by p facltes, and let z LP (p) be the optmal p-faclty obectve value of the LP relaxaton of (MCLP). We assume that we know the nteger optmal soluton wth p 1 facltes and that the optmal soluton to the LP relaxaton wth p facltes s not nteger. Let a mn = mn{a : V} and a = V a. We summarze the nspecton method n the followng theorem. (Church and ReVelle llustrate ths method wth an example, rather than statng t formally as a theorem.)

18 Theorem 2. Suppose the followng condtons hold: 1. z LP (p) < a 2. z IP (p 1) + a = a a mn for some node that s not covered n the optmal soluton to the (p 1)- faclty problem. Then an optmal soluton to the p-faclty problem conssts of the optmal soluton to the (p 1)-faclty problem plus node. Church and ReVelle report that, of the 20% of ther test nstances whose LP relaxatons dd not have nteger solutons, half could be solved usng the nspecton method. The other half was solved va branch-and-bound Mandatory Closeness Constrants Church and ReVelle dscuss a varant of the MCLP n whch we requre that all customers be covered wthn a secondary coverage dstance t (t s). For example, we mght want to maxmze the demand covered wthn 50 mles but requre all demands to be covered wthn 100 mles. Ths model, known as the MCLP wth Mandatory Closeness Constrants, can be vewed as a hybrd between the MCLP and the SCLP snce t has a max-coverage obectve plus a hard coverage constrant. Ths problem can be formulated smply by addng the followng constrant to ether formulaton of the MCLP: x 1 V, U where U = { V : d t}. The resultng model can be solved usng lnear programmng and branchand-bound.

19 Suppose we solve the SCLP and fnd that, for a gven nstance, the mnmum number of facltes that covers all demand nodes wth a coverage dstance of t s p*. Generally there are many optmal solutons to ths problem. The MCLP wth mandatory closeness constrants gves us a mechansm for choosng among these, by selectng the soluton that also maxmzes the demands covered wthn some dstance s. In partcular, we solve the MCLP wth mandatory closeness constrants usng p* as the number of facltes to open and t as the secondary coverage dstance Experments and Varants Computatonal Experment We performed a computatonal experment to verfy Church and ReVelle s clam that the MCLP often results n all-nteger solutons. We set n = 50, 100, 200, 400, 800. For each value of n, we generated 100 random nstances and tested three dfferent values of p. The random nstances were generated as descrbed n Secton , wth one addtonal parameter: Demands a were drawn from U[0,100] We solved the LP relaxaton of (SCLP2) usng CPLEX v and, f the soluton was not all nteger, we solved the IP. The results are dsplayed n Table 2. The columns labeled p gves the value of p n (SCLP2). The column labeled Avg LP Gap >0 gves the average ntegralty gap among only those nstances wth a postve ntegralty gap, or f there were no such nstances. All other columns are nterpreted as n Table 1. Table 2. Performance of LP relaxaton of (MCLP2). n p % Integer Avg LP Gap Avg LP Gap > % % % % 98.0% Max LP Gap

20 % % % % % 92.0% 92.0% % 91.0% 89.0% Total 95.3% The LP relaxaton MCLP seems to generate nteger solutons even more frequently than the SCLP does (at least for our test nstances), an average of 95.3% of the tme. When t fals to do so, the ntegralty gap can be qute large, though ths s partly a functon of the mnmzaton obectve, whch may have optmal values near zero and hence any suboptmal soluton may have a large error on a percentage bass. Note that some nstances had fractonal LP solutons but an ntegralty gap of 0, as evdenced by the fact that some rows have % Integer <100% but an average LP gap of 0. For these nstances, an optmal nteger soluton exsts for the LP relaxaton but CPLEX returned a fractonal optmal soluton nstead Tradeoff Curve Fgure 3 dsplays the optmal obectve functon value of (MCLP2) the number of demand unts uncovered as p vares for a partcular random nstance wth n = 100 and s = 15. As expected, the uncovered demand decreases as p ncreases. For p 18, all demands are covered. The convex shape s typcal of tradeoff curves for the MCLP, meanng that addtonal facltes provde decreasng margnal returns n terms of addtonal coverage.

21 Fgure 3. Tradeoff curve: demands uncovered vs. p Lagrangan Relaxaton Approach The MCLP can also be solved usng Lagrangan relaxaton. The key dea s to remove a set of constrants and add a penalty to the obectve functon for volatng the constrants. The resultng problem s easer to solve but may produce solutons that are nfeasble for the MCLP. By adustng the obectve-functon penaltes teratvely, the solutons found approach the optmal soluton for the MCLP. The use of Lagrangan relaxaton for the MCLP was detaled by Galvão and ReVelle (1996), although Daskn, et al. (1989) also report computatonal results from a smlar method wthout provdng detals. See Fsher (1981, 1985) for an excellent overvew of Lagrangan relaxaton. We llustrate the Lagrangan relaxaton method usng formulaton (MCLP), though t can also be appled to (MCLP2). We relax constrants (6) usng Lagrange multplers λ to obtan the followng Lagrangan subproblem:

22 (MCLP-LR) maxmze subect to z = a V y + λ x V V = ( a λ ) y + λ x V V V : V x p V = y (17) (18) x {0,1} V (19) y {0,1} V (20) Ths problem decouples by x and y snce there are no constrants nvolvng both sets of varables. As a result, t can be solved easly. The optmal y-values are gven by y 1, f a λ > 0, = 0, otherwse To fnd the optmal x-values, we set x = 1 for the p facltes wth the largest values of λ. The V : V optmal obectve value of (MCLP-LR) provdes an upper bound on that of (MCLP). Feasble (lower bound) solutons can be found by openng the p facltes that are opened n the upper-bound soluton and settng y = 1 for each customer that s covered by some opened faclty. Lagrange multplers can be updated usng subgradent optmzaton, and branch-and-bound can be used f the Lagrangan procedure fals to yeld a sutably small optmalty gap; see Daskn (1995) for more detals. Daskn, et al. (1989) report that the procedure works qute well, especally when the lower-bound heurstc s supplemented by a substtuton heurstc Budget Constrants We can ncorporate fxed costs nto the model n a smlar manner as we dd for the SCLP n Secton Here, the fxed cost appears n the constrants rather than the obectve functon. In partcular, we replace constrant (7) or (14) wth f V x B,

23 where B s a budget mposed exogenously on the total fxed costs. Ths constrant can be easly handled by the lnear programmng approach n Secton , but t somewhat complcates the Lagrangan approach n Secton snce determnng the optmal x values now requres us to solve the followng knapsack problem: maxmze subect to λ x V V: V f x B V x {0,1} V Although ths problem can be solved qute quckly usng modern codes, t s stll NP-hard, and t may slow the Lagrangan procedure sgnfcantly Relatonshp to p-medan Problem The MCLP can be formulated as a specal case of the p-medan problem (PMP) through a smple transformaton of the dstance matrx. In partcular, we set d 0, f N = 1, otherwse That s, we redefne the dstance metrc so that the dstance from node to node s 0 f covers and 1 otherwse. The PMP s then formulated as usual (see, e.g., Daskn 1995). The optmal soluton wll cover as many demand unts as possble usng p facltes. Any algorthm for the PMP can then be appled to solve the MCLP. 3. Extensons The lterature contans many enhancements to the SCLP and MCLP. In ths secton, we focus n partcular on generalzatons of the noton of coverage. One common crtcsm of the SCLP and MCLP s that they assume that all customers wthn a faclty s coverage radus can be served by the faclty, and served equally. In practce, facltes are not always avalable when needed, especally n the publcsector arena where facltes may represent ambulances, fre crews, etc. One approach to ths ssue s backup coverage, n whch customers are requred or encouraged to be covered by more than one open

24 faclty. Another approach s expected coverage, whch accounts for probablstc nformaton. Moreover, n many cases the coverage beneft changes as the dstance between a customer and ts assgned faclty changes. Ths dependency s captured by the noton of gradual coverage. We brefly dscuss models for backup, expected, and gradual coverage n the next three subsectons. For thorough revews of backup and expected coverage models, see Daskn, Hogan, and ReVelle (1988) or Berman and Krass (2002). 3.1 Backup Coverage Models Both the SCLP and the MCLP have been extended to consder solutons n whch customers are covered by more than one faclty. One may requre backup coverage n order for a customer to count as covered, or one may smply reward solutons for backup coverage Requred Backup Coverage It s smple to formulate a requred-backup verson of ether the SCLP or the MCLP. In the SCLP, we smply modfy constrants (2) to read V x m V, where m s the desred number of tmes that each customer s to be covered. In the MCLP, we can replace constrants (6) wth x my V V. Then y must equal 0 unless at least m facltes that cover customer are open. Ths constrant s lkely to weaken the LP relaxaton of (MCLP), however, as s typcal of such bg-m constrants Rewards for Backup Coverage We focus on models n whch m = 2. Extensons to these models to consder m > 2 are straghtforward but often yeld weaker LP relaxatons, as dscussed above. Let w 1, f customer s covered by two or more facltes = 0, otherwse

25 The models formulated below contan a reward n the obectve functon for each customer that s covered twce. However, the backup-coverage reward s strctly a secondary obectve; n no case should a soluton wth more facltes have a better obectve than one wth fewer facltes, even f t has better backup coverage. Daskn and Stern (1981) propose the followng model for the SCLP wth backup coverage: (SCLP-BC) mnmze z = ( V + 1) x w (21) V subect to x w 1 V V V (22) x {0,1} V (23) w {0,1} V (24) The obectve functon (21) enforces the herarchcal nature of the prmary obectve (mnmzng the number of facltes) and the secondary one (maxmzng twce-covered customers). It does so by multplyng the prmary obectve by a constant large enough that even f the prmary obectve s as small as possble (equal to 1), the secondary obectve can never exceed t. Therefore, the soluton wll never open more facltes than necessary solely to mprove the secondary obectve. Constrants (22) requre each customer to be covered at least once and prohbt w from equalng 1 unless customer s covered at least twce. Another advantage of ths formulaton s that ts solutons avod facltes that are domnated by others n the sense descrbed n Secton As a result, the LP relaxaton to (SCLP-BC) s more lkely to have all-nteger solutons than that of (SCLP) s. (See Daskn and Stern 1981 for ustfcatons of both of these clams.) A smlar herarchcal verson of the MCLP was ntroduced by Storbeck (1982) and reformulated by Daskn, Hogan, and ReVelle (1988). We modfy ther formulaton somewhat n what follows.

26 (MCLP-BC) maxmze z = ( V + 1) a y + w (25) V subect to x y w 0 V x V = p V V (26) (27) x {0,1} V (28) y {0,1} V (29) w {0,1} V (30) The obectve functon (25) maxmzes a sum of the prmary coverage (frst term) and backup coverage (second term); the weght on the frst term ensures that prmary coverage wll never be sacrfced n order to acheve backup coverage. Note that the secondary coverage obectve consders nodes covered, rather than demand unts covered. Ths s requred n order for the weghtng to acheve the desred herarchy. Constrants (26) stpulate that customer may be consdered covered (y = 1) only f at least one faclty n V s open, and may be consdered twce covered (w = 1) only f two such facltes are open. Snce the obectve functon coeffcent for y s greater than that for w, the model wll always set y = 1 before t sets w = 1, thus ensurng the desred coverage herarchy. 3.2 Expected Coverage Models The class of expected coverage models s descended prmarly from the Maxmum Expected Coverng Locaton Problem (MEXCLP) ntroduced by Daskn (1982). Daskn s prmary applcaton s n the stng of emergency medcal servce (EMS) vehcles. The MEXCLP maxmzes the expected coverage of each node, defned usng probablstc nformaton about faclty avalablty, subect to a constrant on the number of facltes. The MEXCLP assumes that the average system-wde probablty that a gven faclty (vehcle) s busy s gven by q. If a customer s covered by k facltes, then the probablty that all those facltes are busy at a gven pont n tme s gven by q k, and the probablty that at least one faclty s avalable s 1 q k. The MEXCLP defnes new varables to keep track of the number of coverng facltes for each customer: let

27 y m 1, f customer s covered by at least m facltes = 0, otherwse for all V and m = 1,, p. Note that f customer s covered by exactly k facltes, then y m = 1 for m = 1,, k and y m = 0 for m = k + 1,, p. Then p m= 1 (1 q ) q m 1 y m = k 1 m= 0 (1 q) q m = 1 q k usng a standard formula for geometrc sums. In other words, the frst summaton n the equaton above expresses the probablty that customer s covered by an avalable faclty n terms of the decson varables y m. Usng ths approach, Daskn formulates the MEXCLP as follows: (MEXCLP) maxmze z = ( 1 q) q p p V m= 1 subect to y m x 0 m= 1 V x = p V m 1 a y m V (31) (32) (33) x {0,1} V (34) y {0,1} V (35) The obectve functon (31) calculates the expected coverage. Constrants (32) allow the total number of y m varables, for fxed, to be no more than the total number of opened facltes that cover. At frst t may seem that the model needs a constrant of the form y V, m = 1, K, p 1 m y, m+1 n order to ensure that y m s set to 1 for the correct values of m; that s, for the k smallest values of m, where k s the number of opened facltes that cover. However, such a constrant s not necessary snce the obectve functon coeffcent s larger for smaller values of m; the model wll automatcally set y m = 1 for the k smallest values of m. Daskn (1983) proposes a heurstc for the MEXCLP based on node exchanges, and several metaheurstcs have been proposed subsequently (e.g., Aytug and Saydam 2002, Raagopalan et al. 2007).

28 The prmary crtcsm that has been leveled at the MEXCLP concerns the assumpton of a unform system-wde avalablty probablty, snce avalablty mght vary based on geographc area, or on the demand assgned to each faclty. ReVelle and Hogan (1989) address ths concern n the Maxmum Avalablty Locaton Problem (MALP), a chance-constraned verson of the MCLP. They formulate two versons of the model, one n whch the avalablty probablty s assumed to be the same throughout the system; the man dfference between ths model and the MEXCLP s that the MALP maxmzes the number of demand unts that are covered wth at least a certan probablty, whereas the MEXCLP ncludes the expected coverage n the obectve. ReVelle and Hogan s second MALP model estmates the busy probablty separately for each customer by assumng that facltes wthn the coverage radus of a gven customer are avalable only to that customer. Obvously ths assumpton s not true but provdes an easy, and farly accurate, estmate of the avalablty probablty. The two models are nearly dentcal once the avalablty probabltes are calculated. Galvão, Chyosh, and Morabto (2005) present a framework that attempts to unfy the MEXCLP and MALP. Batta, Dolan, and Krshnamurthy (1989) embed Larson s (1974, 1975) hypercube queung model nto MEXCLP to compute the avalablty probabltes endogenously. They fnd that ther model dsagrees substantally wth MEXCLP n terms of expected coverage predcted but nevertheless results n smlar sets of facltes chosen. Maranov and ReVelle (1996) formulate a verson of the MEXCLP that endogenously calculates the avalablty property usng a queung model at each faclty. The regon around each customer node s treated as an M/M/s/s queue, where s s the number of servers located wthn the coverage radus. Ther model mplctly assumes that that the call rate n the neghborhood s not too dfferent from that n adacent neghborhoods. The resultng model s structurally smlar to the MALP but uses dfferent (but pre-computable) values for the coverage radus.

29 3.3 Gradual Coverng Models The models dscussed n ths chapter so far all assume that coverage s a bnary concept: ether a customer s covered or t sn t, and the dstance from the customer to the coverng faclty s rrelevant. In practce, though, customers that are located very close to a faclty (e.g., a fre staton) may be served better than those located farther away, even f both customers are wthn the nomnal coverage radus. In ths case, the beneft from coverage s decreasng wth the customer faclty dstance, as llustrated n Fgure 4(a). Moreover, some facltes (e.g., garbage dumps) are most benefcal when they are close (to reduce transportaton costs) but not too close (to reduce odors and truck traffc), as llustrated n Fgure 4(b). (a) (b) Fgure 4. Beneft of coverage vs. dstance: (a) strctly decreasng, (b) non-monotonc. Church and Roberts (1983) ntroduce the Weghted Beneft Maxmal Coverage (WBMC) Model, whch extends the MCLP to accommodate non-bnary coverage benefts. The obectve s to maxmze the sum of all customers coverage benefts (defned as the beneft per unt of demand tmes the demand at that customer) subect to a constrant on the number of facltes located. The formulaton s a relatvely straghtforward modfcaton of (MCLP) and ncludes a coverage varable (y) and a constrant for each customer dstance par. (Each dstance s really a range of dstances, as n Fgure 4.) The number of varables and constrants therefore grows lnearly wth the number of dstance ranges. If the benefts are not monotoncally decreasng wth the dstance, as n Fgure 4(b), then an addtonal set of constrants s requred to ensure that customers are assgned to ther nearest opened facltes, a property that s

30 automatc f benefts are monotoncally decreasng. The resultng formulatons are more complex than (MCLP), but Church and Roberts fnd that they stll retan ther nteger-frendlness : the LP relaxaton s generally very tght, and often all-nteger. 4. Conclusons and Future Research Drectons In ths chapter we have dscussed two classcal models for locatng facltes to ensure coverage of customer nodes. One model, the SCLP, requres every customer to be covered and does so wth the mnmum number of facltes, whle the other, the MCLP, maxmzes the demand covered subect to a lmt on the number of facltes. Both models have garnered consderable attenton n the locaton theory lterature, and both models (and ther extensons) have been wdely appled n practce, especally n publc-sector applcatons such as EMS locaton. The SCLP and MCLP are both reasonably easy to solve, n the sense that modern general-purpose IP solvers such as CPLEX can solve problems wth hundreds or thousands of nodes to optmalty n a few mnutes on a desktop computer. Ths stems, n part, from the fact that the LP relaxatons of both problems tend to be tght, and even yeld nteger optmal solutons for a large percentage of nstances. Therefore, although these problems are NP-hard, they are among the easest problems n that class. On the other hand, many of the extensons of these models are much more computatonally challengng. Daskn s (1982) MEXCLP model, for example, or the queung-based congeston models dscussed by Berman and Krass (2002), have more complex structures than the SCLP or MCLP and therefore cannot be solved usng off-the-shelf solvers, except for small nstances. One mportant drecton for future research, therefore, s the development of effectve, accurate algorthms and heurstcs for extensons of the SCLP and MCLP.

31 Of partcular nterest are stochastc and robust varants of coverage models. Although the lterature on stochastc faclty locaton models s extensve (see Snyder 2006 for a revew), most such models consder cost-based obectves rather than coverage-based ones. (Notable exceptons are the expected-coverage models descrbed n Secton 3.2, and ther varants.) An mportant topc for future study s therefore the ncorporaton of stochastc elements such as demands, travel tmes, server avalabltes, and supply dsruptons nto coverage models. The resultng models are lkely to be sgnfcantly more complex than ther determnstc counterparts, but the stochastc programmng and robust optmzaton lteratures are vast, and many of ther more sophstcated tools have yet to be tapped by the locaton scence communty. The dstncton between cost- and coverage-based models made n the prevous paragraph s an mportant one snce t s often equvalent to the dstncton between prvate- and publc-sector applcatons the former s prmarly concerned wth cost mnmzaton whle the latter s often encouraged or mandated to provde adequate coverage to all demand locatons (ReVelle, Marks, and Lebman 1970). Publc-sector and humantaran applcatons have ganed ncreased attenton n the OR communty n recent years for example, the 2008 INFORMS Annual Meetng featured Dong Good wth OR as a central theme, as dd the February, 2008 ssue of OR/MS Today. The applcaton of coverage models to EMS and other servces has been a publc-or success story for decades, and the renewed nterest provdes an opportunty for exstng and new coverage models to be appled for the publc good. References H. Aytug and C. Saydam. Solvng large-scale maxmum expected coverng locaton problems by genetc algorthms: A comparatve study. European Journal of Operatonal Research, 141(3): , R. Batta, J. M. Dolan, and N. N. Krshnamurthy. The maxmal expected coverng locaton problem: Revsted. Transportaton Scence, 23(4): , 1989.

32 O. Berman and D. Krass. Faclty locaton problems wth stochastc demands and congeston. In Z. Drezner and H. W. Hamacher, edtors, Faclty Locaton: Applcatons and Theory, chapter 11. Sprnger-Verlag, New York, J. Bramel and D. Smch-Lev. On the effectveness of set coverng formulatons for the vehcle routng problem wth tme wndows. Operatons Research, 45(2): , R. Church. Synthess of a Class of Publc Facltes Locaton Models. PhD thess, The Johns Hopkns Unversty, Baltmore, MD, R. Church and C. ReVelle. The maxmal coverng locaton problem. Papers of the Regonal Scence Assocaton, 32: , R. L. Church and B. Meadows. Locaton modellng usng maxmum servce dstance crtera. Geographcal Analyss, 11: , R. L. Church and K. L. Roberts. Generalzed coverage models and publc faclty locaton. Papers n Regonal Scence, 53(1): , R. L. Church, D. M. Stoms, and F. W. Davs. Reserve selecton as a maxmal coverng locaton problem. Bologcal Conservaton, 76(2): , J. Current, M. S. Daskn, and D. Schllng. Dscrete network locaton models. In Z. Drezner and H. W. Hamacher, edtors, Faclty Locaton: Applcatons and Theory, chapter 11. Sprnger- Verlag, New York, M. S. Daskn. Applcaton of an expected coverng model to emergency medcal servce system desgn. Decson Scences, 13: , M. S. Daskn. A maxmum expected coverng locaton model: Formulaton, propertes and heurstc soluton. Transportaton Scence, 17(1):48 70, M. S. Daskn. Network and Dscrete Locaton: Models, Algorthms, and Applcatons. Wley, New York, M. S. Daskn, A. E. Haghan, M. Khanal, and C. Malandrak. Aggregaton effects n maxmum coverng models. Annals of Operatons Research, 18: , M. S. Daskn, K. Hogan, and C. ReVelle. Integraton of multple, excess, backup, and expected coverng models. Envronment and Plannng B, 15(1):15 35, M. S. Daskn and E. H. Stern. A herarchcal obectve set coverng model for emergency medcal servce vehcle deployment. Transportaton Scence, 15: , 1981.