Heuristic Methods for Locating Emergency Facilities

Transcription

1 Heurstc Methods for Locatng Eergency Facltes L. Caccetta and M. Dzator Western Australan Centre of Excellence n Industral Optsaton, Curtn Unversty of Technology, Kent Street, Bentley WA 602, Australa E-Mal: kedzat@hotal.co Keywords: Heurstcs; Facltes; Eergency; Locaton. EXTENDED ABSTRACT Faclty locaton probles for an portant class of ndustral optzaton probles. These probles typcally nvolve the optal locaton of facltes. For our purposes, a faclty s just a physcal entty that asssts wth the provson of a servce or the producton of a product. Exaples nclude: schools, abulance depots, eergency care centers, frestatons, workstatons, lbrares etc. The objectve ay nvolve factors such as cost, dstance or servce utlzaton. The optzaton probles are coplcated by the need to eet a nuber of specfed constrants. These constrants ay relate to safety, avalable resources, level of servce, te, etc. The optzaton probles are usually grouped nto two categores, naely servce and anufacturng ndustres. In the servce ndustres, the locaton of eergency facltes (abulance, fre staton, eergency centers) affects sgnfcantly on the safety and well-beng of the county. The safety and well-beng of the county depends drectly or ndrectly on the response te of the eergency facltes. The objectve s to locate the faclty where the average response te (te between the recept of a call and the arrval of eergency vehcle) s nzed. The nzaton of the response te easures the perforance of eergency facltes. The perforance of these facltes can be proved by ether ovng the exstng locatons of the eergency facltes or ncreasng the nuber of facltes. However, ncreasng the nuber of facltes s generally lted or possble due to captal constrants. It s, therefore, portant to locate eergency facltes effectvely and effcently. One way to easure the effcency and effectveness of eergency facltes s by evaluatng the average dstance between the custoers and the facltes. When the average dstance decreases, the accessblty of the facltes ncrease and average response tes decrease. Ths s known as the p-edan proble, whch was ntroduces by Hak (964). It s defned as: deterne the locaton of p facltes to nze the average (total) dstance between deands and ther closest faclty. The p-edan proble s coputatonally dffcult to solve by exact ethods because t s NP-hard on general networks (Karv and Hak 979). However, solutons fro the p-edan odel are consdered effcent snce they brng the faclty locatons nto closer proxty of the users. The dffculty of solvng the p-edan proble usng exact ethods has led researchers to consder sub optal solutons generated by heurstc approaches. Heurstcs for solvng the p-edan proble have been dscussed n Daskn (995), Maranzana (964), Tetz and Bart (968) and Densha and Rushton (992). Ths paper dscusses three new heurstc ethods for solvng the p-edan proble. These ethods are otvated by the desre to elnate outlers fro havng strong nfluence over the fnal soluton gven by the heurstcs. These heurstcs wll also prove the delvery of eergency edcal care by properly locatng eergency facltes n an area. In these heurstcs, the faclty locaton proble s forulated as a network optzaton proble as follows. The geographcal regon s parttoned nto a nuber of subregons and a correspondng graph s constructed. Each node of ths graph represents a subregon and each lnk of the graph represents the fact that the correspondng regons share a boundary. Ths gves us a structural odel. Non-structural nforaton s added as weghts on the nodes (reflect expected deand n regon) and the lnks (reflect travel te). Usually the nodes of the network represent possble locatons of facltes. An effcent reducton ethod s then used to address the proble of outlers. Coputatonal results, based on 400 rando unforly generated probles, show that the heurstcs perfor well n ters of qualty of soluton and coputatonal te. Our best heurstc s copared wth the well known exstng p- edan heurstcs. Better solutons are acheved n ost cases. 744

2 . INTRODUCTION The provson and utlzaton of effectve and effcent eergency servces s an portant optzaton proble encountered n all parts of the world. Integer prograng probles and, specfcally, faclty locaton odels have real applcaton n the servce and anufacturng ndustres. Faclty locaton odels are used extensvely n solvng optzaton probles, whch attept to choose the best locaton for facltes such as warehouses, schools, hosptals, abulance statons, fre statons etc. In ths paper, we develop a nuber of new heurstc algorths and test the on sulated data and data fro the lterature. 2. THE P-MEDIAN MODEL AND EMERGENCY FACILITIES The crteron for fndng a good locaton for eergency facltes requres the proveent of the response tes. The response te depends on the dstance between the eergency facltes and the eergency stes. The a s to locate these facltes such that the average (total) dstance traveled by those who vst or use these facltes s nzed. Ths easures the effectveness and effcency of the eergency facltes. It s clear that people tend to travel to the closest faclty regardless of the dstance or te travelled. A good way to acheve ths s by solvng the p- edan proble. The p-edan proble conssts of deternng the locaton of p eergency facltes to nze the weghted dstance between eergency (deand) ponts and ther closest new eergency faclty. A nuber of authors, such as Berln et a (9760, Mrchandan (980), Carson and Batta (990), Serra and Marnov (998), Paluzz (2004), use the p-edan proble soluton to locate eergency facltes. We now present the odel for the p-edan proble. We start wth soe notaton: I = {,...,} s the set of deand locatons, J = {,...,n} s the canddate stes for facltes, d s the shortest dstance between locaton and locaton j, x = f the custoer at locaton s allocated to the faclty at locaton j and 0 otherwse, y j = f a faclty s establshed at locaton j and 0 otherwse, p s the nuber of facltes to be establshed, and a s the populaton at the deand node. The atheatcal forulaton s subject to n Mn ad j J j J = j= X, () x =, I (2) y = p (3) j x y j I, j J (4) y j { 0, }, x { 0, } (5) The objectve () s to nze the total dstance fro custoers or clents to ther nearest faclty. Constrant (2) shows that the deand of each custoer or clent ust be et. Constrant (3) shows the nuber of facltes to be located s p. Constrant (4) shows that custoers ust be suppled fro an open faclty, and constrant (5) restrcts the varables to 0, values. Several extensons have been proposed for the p- edan odel, whch proves ts effcency (Daskn et al., 988). Extensons to the p-edan proble that account for ts stochastc nature have been gven by Ftzsons (973), Weaver and Church (985) and Swoveland et al. (973). 3. SOLUTION METHODS FOR THE P- MEDIAN PROBLEM The p-edan proble s a coputatonally dffcult proble to solve (the proble s NP-hard on general networks). Most soluton ethods are heurstc based because of the large nuber of varables and constrants that arse for a edu szed network. The heurstcs are based on: genetc algorths, sulated annealng, tabu search, node parttonng, node nserton, node substtuton and varous hybrds (Hosage and Goodchld (986), Golden and Skscs (986), Glover (990)). Soe of these heurstcs, together wth Lagrangan relaxaton, whch s one of the ost successful exact ethods, are brefly dscussed below. 3. Lagrangan Relaxaton Lagrangan relaxaton s based on the prncple that reovng constrants fro a proble akes the proble easer to solve. Generally, Lagrangan relaxaton reoves a constrant and solves the revsed proble, whch ntroduces a penalty for volatng the reoved constrant. The soluton procedure for solvng the proble s stated below. The Lagrangan relaxaton for the p-edan s gven as 745

3 L( λ ) = n d + x j x λ j (6) subject to constrants (3)-(5). The expresson r = n 0, λ (7) j d { } s used to nze the objectve functon (6) for the fxed values of the Lagrange ultplers. We then set < f y j = and d λ 0 x = (8) 0 otherwse The lower and upper bounds of the objectve functon are deterned by usng the varables of odfed and unodfed probles respectvely. The next step nvolves the use of subgradent optzaton to update the value of the Lagrange ultplers by usng the equaton below (Daskn 995): + λ = ax 0, λ t x j (9) A (UB L ) t = (0) 2 x j Where A s a constant on the th teraton, t s the stepsze at the th teraton of the Lagrangan procedure, UB s the best (sallest) upper bound on the P-edan objectve functon, L s the value of the objectve functon usng the soluton obtaned fro the relaxed proble and x s the optal value of the allocaton varable at the th teraton. An optal soluton s found f the lower bound s equal to the upper bound. Narula et al. (977) and Galvao (980) and Beasley (993) have successfully appled the subgradent optzaton to solve a nuber of probles. However, for the larger probles tested, the coputatonal te s excessvely large. 3.2 Heurstcs In ths secton, we start our dscusson by observng that t s an easy task to assgn a set of clents to p facltes J wth fxed locatons. We just deterne d { d } * = n,, j J () and assgn custoer to faclty j *. Ths gves us a tool for generatng possble solutons. The procedure s also useful for deternng alternatve solutons through exchange of faclty locatons. We now use the dea above to descrbe three sple heurstcs, whch are copettve wth other ethods Myopc Algorth (MA) The yopc heurstc s a greedy type, whch works n the followng way. Frst, a faclty s located n such a way as to nze the total cost for all custoers. Facltes are then added one by one untl p s reached. For ths heurstc, the locaton that gves the nu cost s selected. The an proble wth ths approach s that once a faclty s selected t stays n all subsequent solutons. Consequently, the fnal soluton attaned ay be far fro optal Neghborhood Search Heurstc (NS) Maranzana (964) proposed ths heurstc, whch s descrbed as follows. We begn wth any set of p faclty nodes. The deand nodes are then dvded nto p subsets and, for each subset, a deand node s allocated to the nearest faclty node. The node gvng the optal for each subset s found, whch results n a new pattern of faclty nodes. Ths process s repeated untl the faclty nodes pattern reans the sae as that n the prevous step Exchange Heurstc (EH) Ths s one of the early heurstcs developed by Tetz and Bart (968) for the p-edan proble. The heurstc starts by choosng an ntal set of p nuber of nodes as the soluton, and then a node, whch s not n the current soluton, s selected to substtute for each of the p nodes n turn. We fnd the objectve value n each case and copare the changes n the objectve functon. The substtuton leadng to the bggest decrease n the objectve functon s selected and s exchanged for a node n the current soluton. Ths exchange of nodes results n a new soluton confguraton and ths process contnues untl there s no further proveent n the objectve value. 4. NEW P-MEDIAN HEURISTICS FOR LOCATING EMERGENCY FACILITIES 4.. Reducton Heurstcs (RH, RH2, RRH) In the prevous secton, the dscusson of soe of the heurstcs (yopc n partcular) for the p- edan proble uses all the values of the dstance atrx wthout any odfcaton to solve the proble of extree values (outlers). In ths secton, we tred to elnate the proble of outlers by usng a reducton technque. Outlers 746

4 can have a strong nfluence over the fnal soluton. We also elnate the uncertanty of choosng a good ntal soluton n the case of the Neghborhood search and Exchange heurstcs by usng a specfc and effcent way of selectng the ntal soluton for the three new heurstcs. We obtaned the ntal soluton set for the heurstcs by frst elnatng the outlers and then su the coluns. We then choose the nodes correspondng to the frst p nodes of the totals arrange n ascendng order. The ntal set s the frst p nodes correspondng to the frst p total, whch s arranged n ascendng order. The a of the heurstcs s to elnate the outlers before usng the data. Ths wll enhance a faclty to be located at nodes that are not far away fro all custoers, so the cost of usng these facltes s nzed. We use the ntal soluton to reduce the dstance atrx by settng the nodes that correspondng to the ntal set for both rows and coluns to zero. Ths s done wth the assupton that custoers at those nodes are not charged to uses the facltes. For RH, the coluns of the resultng dstance atrx are added and the nu value s chosen for substtutng nto the ntal soluton. We fnally choose the set wth the nu objectve value. In the case of RH2, all the nodes not n the ntal soluton are exchanged one-by-one for the nodes n the ntal soluton. We then choose the faclty set wth the nu objectve value as the fnal soluton. However, for both heurstcs, we choose the ntal set as the fnal soluton f there s no proveent n the objectve value after the swappng procedure. Motvated by the perforance of the two new heurstcs (RH and RH2), we extend RH2 and propose a new heurstc, whch we call Repeated Reducton Heurstc (RRH). The process of reducng the atrx s slar to RH2 but, n ths case, the reducton s done repeatedly untl there s no proveent n the fnal soluton. We descrbe the three new reducton heurstcs for the p-edan proble below. 4.2 Reducton Heurstc One (RH) Step : Set the nuber of nodes and facltes to be equal to n and p respectvely. Step 2: Arrange the n values for each colun n ascendng order and delete the last α nuber of values fro each colun. Next, let the resultng nuber of nodes be equal to n (.e. n = n α where α s p for less than twenty nodes, 2p for less than thrty nodes, 3p for less than forty nodes etc. ) Step 3: Su the frst n values for each colun, arrange the values n ascendng order, and choose the frst p nodes as the ntal set. Step 4: Set the coluns and rows correspondng to the ntal set to zero and su the coluns of the resultng dstance atrx. Step 5: Choose the node or nodes correspondng to the nu value and substtute for the nodes n the ntal set. Step 6: Choose the set correspondng to the nu objectve value after the substtuton procedure reaches the fnal soluton. Otherwse, go to step 3 and choose the ntal set as the fnal soluton f that value s lower. 4.3 Reducton Heurstc Two (RH2) For RH2, Steps to 4 s the sae as RH and the reanng steps are outlned below. Step 5: Substtute all the nodes not n the ntal set wth the nodes n the ntal set. Step 6: Choose the set correspondng to the nu value as the fnal soluton. Otherwse, we choose the ntal set as the fnal soluton f that s lower We note that the dfferent swappng procedure lead to an proved fnal soluton as copared wth RH (Secton 5). 4.4 Repeated Reducton Heurstc (RRH) In ths heurstc, we repeatedly use the fnal soluton of RH2 as the ntal set and use step 4 of RH, and steps 5 and 6 of RH2. We contnue ths untl there s no proveent n the fnal soluton. We note that the repeated reducton ncorporated n RRH has ncreased ts perforance as copared wth RH2. The proposed heurstcs are unque n three dfferent ways. Frst, the ethodology s sple and tractable. Second, the elnaton of outlers gves a good ntal soluton. Thrd, the deternaton of swappng a node or nodes and the swappng procedure gves a good fnal soluton. We also note that an proveent procedure can be further ntroduced to reduce the response te. 4.5 Illustratve Exaple

5 We use the data above to llustrate the three new heurstcs. To locate two facltes, we elnate the two greatest values n each colun. Hence, we elnate 67 and 74 n colun, 82 and 87 n colun 2, 5 and 78 n colun 3, 87 and 93 n colun 4 and 97 and 00 n colun 5. Sung the reanng values and arrangng the n ascendng order gves the followng: 2 (55), 3 (64), 4 (7), (82) and 5 (5). We choose nodes 2 and 3 as the ntal soluton for RH, RH2 and RRH. We, therefore, set rows and coluns 2 and 3 of the data to zero and we have the followng table The resultng totals for the non-zero coluns gve node wth the nu value, so, for RH, we substtute nodes 2 and 3 wth node, whch results n the possble soluton sets of {,3} and {,2}. We choose {,2} snce that gves an optal value of 75. In the case of RH2 and RRH, we use all the nodes not n the ntal soluton for substtutng for nodes n the ntal soluton. Ths gves the possble soluton set as follows: {,2}, {,3}, {2,4}, {3,4}, {2,5} and {3,5}. We choose {,2} as the fnal soluton snce t gves an optal value of 75. We contnue the sae process repeatedly for RRH and now use {,2} as ts ntal soluton, whch fnally yeld {,2} as the fnal soluton. We use the sae data to locate three facltes. In ths case, we elnate the three greatest values n each colun and su the values of the reanng coluns. Ths gves the ntal soluton of, 2 and 4. Gong through the sae process, and settng the rows and coluns, 2 and 4 to zero, we have the followng table For RH, node 5 has the nu value, so we substtute node 5 for nodes, 2 and 4. Thus, we have the possble sets of {2,4,5}; {,4,5} and {,2,5}. We choose {,2,5} as the fnal soluton, whch has an optal value of 38. In the case of RH2 and RRH, we use nodes 3 and 5, whch are not n the ntal soluton for substtutng nto nodes, 2 and 4. Ths gves the possble soluton of {2,3,4}, {,3,4}, {,2,3}, {2,4,5}, {,4,5} and {,2,5}. We fnally choose {,2,5} as the fnal soluton, whch has an optal value of 38. For RRH, we agan use {,2,5} as the ntal soluton and contnue the process repeatedly. The fnal soluton s {,2,5}. For the Myopc heurstc, we elnate any extree values, whch gves the followng table When we su all the coluns, node 3 has the nu value of 93. Therefore, one faclty s located at node 3. We note that, for the p-edan proble, a deand s allocated to the nearest faclty. We, therefore, adjust the dstance atrx, whch gves the followng table Node 2 has the nu value of 0 when the coluns of the above atrx are added, so, for two facltes, we have nodes 2 and 3 wth an objectve value of 0. Slarly, we have adjusted the above atrx after the two facltes were located, as shown below Node has the nu value when all the coluns are added, so, for three facltes, we have nodes, 2 and 3 wth an objectve value of 57. We present, n Table, the results of the exaple of the three heurstcs and Myopc Algorth. The three heurstcs gve better results than yopc algorth. Table : Results for RH, RH2, RRH and Myopc P Soluton RH, RH2, RRH Myopc Fac. Obj. Fac. Obj. 2 {,2} 75 {,3} 0 3 {,2,5} 38 {,2,3} COMPUTATIONAL RESULTS The three new heurstcs are pleented n C++ and tested on sets of 20 randoly generated data 748

6 for a [0, 00] atrx wth n rangng fro 0 to 50 n steps of 0 and p rangng fro 2 to 5. The statstc used to easure the qualty of the soluton H O O s gven as 00 where H s the value gven by the pleentaton of the heurstc and O s the optal value deterned by the enueraton ethod. The value of 0% s consdered to be optal. A sall devaton results n a better soluton than a large devaton. Table gves the perforance of the three new heurstcs for locatng 2, 3, 4 and 5 facltes. In Table 2 below, we have the average values for usng ten, twenty, thrty, forty and ffty nodes. Table 2: Average Values for the New Heurstcs Nuber of Average Values (%) Nodes (n) RH RH2 RRH Fro Table 2, the average values for RH ranges fro 2.22% to 4.87%, RH2 ranges fro 0.79% to 2.27% and RRH ranges fro 0.32% to 0.87%. The values of RRH are alost optal, whch s good for locatng eergency facltes and ght gve rse to acceptable response tes. 5. Coparson of the Repeated Reducton Heurstc (RRH) and soe P-Medan Heurstcs Motvated by the perforance of RRH, we copare the heurstc usng data fro the lterature. We copare ths heurstc usng the 55- node network data (Swan 97). The data are gven n Coloe et al. (2003). The data has been used by authors such as Daskn (982, 983), Coloe et al. (2003) and Church and Gerrard (2003) for testng locaton probles. The 55-node data set represents 55 countes n the Washngton D.C (USA) area. Deands for each node were generated n pseudo-rando anner wth ost large deands at the center of the regon and ost sall deands at the outer regon. We copare RRH wth the Myopc algorth (MA), Exchange heurstc (EH) and Neghborhood search (NS) heurstc. We coded the Repeated Reducton Heurstc (RRH) n C++ whle the results of the other heurstcs were obtaned usng the SITATION software (Daskn, 995). The solutons of the heurstcs were copared wth the optal solutons, whch were deterned usng Lagrangan Relaxaton (Daskn, 995). Table 3: Coparson Perforance of RRH and Exstng Heurstc usng 55-node Data Nuber of MA NS EH RRH Facltes (p) H O 00 O Devaton fro Optal Value (%) Nuber of Facltes Myopc Neghborhood Exchange RRH Fgure : Coparson Perforance of Heurstc usng 55-node Data Table 3 and Fgure show the perforance of the new heurstcs and the exstng ones for the 55- node lterature test proble. Fro Table 2 and Fgure, the perforance easured n ters of the nuber of optal solutons gves the rank (fro the best to the worst) of RRH, Exchange heurstc, Neghborhood Search heurstc and Myopc heurstc. The new heurstc RRH perfors better n the locaton of all facltes wth the excepton of the locaton three and sx facltes. 6. CONCLUSION In ths paper, we ntroduced three new heurstc ethods to locate eergency facltes. These heurstcs are based on the p-edan proble and were tested usng about 400 rando data. The perforance of our new heurstcs copared wth the optal soluton and exstng heurstcs s encouragng. The best heurstc aong the three s wthn % of the optal, and, when copared wth other heurstcs, t perfors better. 7. REFERENCES Beasley, J. (993), Lagrangean heurstcs for locaton probles, European Journal of Operatonal Research, 65,

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