LARRY SNYDER DEPT. OF INDUSTRIAL AND SYSTEMS ENGINEERING CENTER FOR VALUE CHAIN RESEARCH LEHIGH UNIVERSITY
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1 Faclty Locaton Models: An Overvew 1 LARRY SNYDER DEPT. OF INDUSTRIAL AND SYSTEMS ENGINEERING CENTER FOR VALUE CHAIN RESEARCH LEHIGH UNIVERSITY EWO SEMINAR SERIES APRIL 21, 2010
2 Outlne Introducton Taxonomy of locaton models IP formulatons for some classcal models Algorthms Extensons Faclty locaton / network desgn software 2
3 Introducton 3
4 Overvew Decde where to locate facltes (factores / warehouses / DCs / retal outlets / etc.) To serve customers In order to acheve some balance between Cost Servce 4
5 Decsons Usually 2 decsons to make: Where to locate? Whch customers are assgned/allocated to whch facltes? Sometmes referred to as locaton allocaton models 5
6 Applcatons of Faclty Locaton Models Wdely appled n publc and prvate sectors: Emergency medcal servces (EMS) / fre statons Arlne hubs Blood banks Hazardous waste dsposal stes Fast-food restaurants Publc swmmng pools Schools Vehcle nspecton statons Bus stops etc. 6
7 Uses for Faclty Locaton Models Also appled to vrtual facltes : Wldlfe reserves Satellte orbts Apparel szes Flexble manufacturng system tool selecton Locaton of bank accounts Poltcal l party platforms Product postonng etc. Sometmes arse as subproblems for other OR problems Vehcle routng 7
8 Taxonomy of Locaton Models 8
9 Topology 9 Contnuous Dscrete Network Locate anywhere on plane Contnuous, non-lnear optmzaton Weber problem Locate at pre-defned ponts Integer programmng We ll consder dscrete problems (Network problems are a specal case) Locate anywhere on network Travel along arcs Integer programmng Hakm property: Optmal to locate at nodes Holds for some (not all) problems
10 Dstance Metrc 2 2 Eucldean: ( x y 10 1 x2) + ( y1 2) Rectlnear / Manhattan: x1 x2 + y1 y2 (travel along streets) Great crcle: accounts for Earth s curvature Hghway / network: shortest path wthn network (e.g., U.S. hghway network) Matrx: dstance between each par gven explctly For sake of generalty, we ll assume matrx dstances Also, dstance = transportaton cost
11 Dstance Obectve Total dstance: total dstance between customers and ther assgned facltes (dstance s usually demand-weghted) 11 dstance to assgned faclty customers Maxmum dstance: maxmum dstance between a customer and ts assgned faclty (dstance s usually unweghted) max customers {dstance to assgned faclty} Coverage: cust. s covered f dstance specfed radus Can appear n obectve functon or constrants
12 Preventng Too Many Facltes Fxed cost: fxed (annual) cost to open/operate p faclty 12 Represents constructon / leasng cost + overhead (lghts, heat, securty, etc.) Independent of volume of demand served by faclty Restrcton on # facltes: requre # of facltes P Restrcton on # facltes: requre # of facltes P n constrants
13 Classcal Models P-medan problem: mnmze demand-weghted dstance s.t. locate P facltes Uncapactated fxed-charge locaton problem (UFLP): 13 mnmze fxed cost + DWD P-center problem: mnmze maxmum dstance s.t. locate P facltes Set coverng locaton problem (SCLP): mnmze # of facltes s.t. cover all customers Maxmum coverng locaton problem (MCLP): maxmze covered demands s.t. locate P facltes
14 Capacty Most models also have a capactated verson Facltes have fxed throughput capacty Capacty s usually an nput But sometmes a decson varable 14 Dscrete choces (50,000 sq ft / 100,000 sq ft / 200,000 sq ft) Contnuous varable (cost s a functon of capacty)
15 IP Formulatons for Some Classcal Models 15
16 Sets I = {customers} J = {potental faclty stes} Parameters Notaton h = annual demand of customer I c = cost to transport one unt from J to I f = fxed (annual) cost to open a faclty at ste J Decson varables x = 1 f faclty J s opened, 0 otherwse y = 1 f faclty J serves customer I, 0 otherwse 16
17 P-Medan Formulaton 17 mn I s.t. J J J y h c y = 1 y x, x = P x {0,1} y {0,1}, Mn demand-weghted dstance (transportaton cost) Satsfy all demands Don t assgn cust to closed faclty Locate P facltes Integralty
18 UFLP Formulaton 18 mn s.t. J J y y x f x = 1 x + {0,1} I J, y { {0,1}, h c y Mn fxed + transportaton cost Satsfy all demands Don t assgn cust to closed faclty Integralty
19 Maxmal Coverng Formulaton 19 max s.t. I J x h z z x = P V x z {0,1} Maxmze covered demand Locate P facltes { 0,1} Integralty Defnton of coverage where V = set of facltes that can cover customer z = 1 f customer s covered, 0 otherwse
20 Algorthms 20
21 Algorthms Most faclty locaton problems are NP-hard But many classcal problems are easy computatonally LP relaxatons are often extremely tght Sometmes nteger solutons for free V t ll t f l th f d t Vrtually every type of algorthm for dscrete optmzaton has been appled to faclty locaton 21
22 Heurstcs: Greedy add/drop Swap Neghborhood search Metaheurstcs Algorthms Genetc algorthms, tabu search, varable neghborhood search, smulated annealng, ant algorthms, bee algorthms, Exact Algorthms: Branch and bound Cuttng gplanes Benders decomposton Column generaton / Dantzg-Wolfe decomposton Lagrangan a g a relaxaton ea ato 22
23 Lagrangan Relaxaton for P-Medan 23 I J y h c mn x y y J, 1 s.t. = RELAX P x y J, = y x, {0,1} {0,1}
24 Lagrangan Subproblem 24 + I J I J y y c h λ 1 mn x y, s.t. + = I I J y c h λ λ ) ( P x y J, = y x, {0,1} {0,1}
25 Faclty- Subproblem Subproblem s separable by Suppose we open ; need to solve Easy solve by nspecton: Would set y = 1 ff h c λ < 0 Beneft of openng s Open P facltes wth smallest β β = I 25 mn{ 0,h c λ } Ths gves lower bound Obtan upper bound from heurstc Update λ and repeat mn I y ( h c st s.t. y {0,1} λ ) y
26 Extensons 26
27 Obnoxous facltes Other Flavors 27 Obnoxous locaton: Maxmze dstance from facltes to customers Dsperson: Maxmze dstance among facltes Compettve locaton Multple players try to capture demand by locatng facltes Mult-obectve models Account for multple stakeholders obectves Hub locaton Flows from facltes to customers but also among facltes Quadratc obectve Dynamc locaton Facltes are located over tme, or move over tme
28 Types of randomness: Uncertanty Demand-sde: Randomness n demands, costs, etc. Supply-sde: Randomness n supply (e.g., dsruptons) 28 Approaches to uncertanty: Stochastc programmng: mn expected cost Robust optmzaton: mnmax cost, mnmax regret, CVaR, etc. Modelng approaches: Scenaro formulatons Interval uncertanty
29 Integrated Models Incorporate tactcal / operatonal costs nto strategc (faclty locaton) decsons Inventory Routng Integrated locaton-nventory model Daskn, Coullard, and Shen (Ann OR, 2002) Obectve functon ncludes two concave terms: Inventory economes of scale (EOQ) Rsk poolng (safety stock) Constrants are same as UFLP Solve va Lagrangan relaxaton Subproblem solved n O( I log I ) tme for each (UFLP: O( I ) tme for each ) 29
30 Network Desgn Mult-echelon faclty locaton models Make open/close decsons for multple ters 30 Geoffron and Graves (MS, 1974) Generalzaton: network desgn problems Usually locate arcs n the network But locatng nodes s equvalent Rch lterature on network desgn e.g., g, Magnant and Wong g( (TS, 1984)
31 Faclty Locaton / Network Desgn Software 31
32 LogcNet / LogcNet Plus Orgnally LogcTools Software Packages 32 Now ILOG Supply Chan Applcatons, part of IBM Consultng SAILS INSIGHT SAP, Oracle modules
33 Key decsons: Capabltes Faclty locatons, capactes, capabltes, volumes Dstrbuton lanes (yes/no, volumes) Make vs. buy Key features: Data mport Output report export GUI, GIS Optmzaton solver Ratng engne What-f scenaros 33
34 New features: Green Rsk management Taxes Seasonalty Cost vs. servce level tradeoffs Recent Developments Thngs most current software can t do (very well): Non-lneartes (e.g., quantty dscounts) Open/close decsons on arcs for larger models Close ntegraton wth nventory modelng Uncertanty Optmzaton under multple scenaros Rsk-averse obectves 34
35 Further Readng Textbooks on faclty locaton Daskn (1995) Drezner (1995) Drezner and Hamacher (2001) Revew artcles / book chapters 35 Dscrete locaton: Current, Daskn, and Schllng (D&H book, 2001) Contnuous locaton: Drezner, et al. (D&H book, 2001) Stochastc locaton: Snyder (IIE Trans, 2006) Locaton wth dsruptons: Snyder, et al. (INFORMS Tutoral, 2006) Locaton-nventory nventory models: Shen (JIMO, 2007)
36 Questons? 36 LARRY. EDU
MILP. LP: max cx ' MILP: some integer. ILP: x integer BLP: x 0,1. x 1. x 2 2 2, c ,
MILP LP: max cx ' s.t. Ax b x 0 MILP: some nteger x max 6x 8x s.t. x x x 7 x, x 0 c A 6 8, 0 b 7 ILP: x nteger BLP: x 0, x 4 x, cx * * 0 4 5 6 x 06 Branch and Bound x 4 0 max 6x 8x s.t. xx x 7 x, x 0 x,
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