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Partial Capture Location Problems: Facility Location and Design

Partial Capture Location Problems: Facility Location and Design . Dmitry Krass Rotman School of Management, University of Toronto With Robert Aboolian , CSUSM Oded Berman , Rotman. Rotman School of Management Ph.D. Program in Operations Management. Rotman

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Partial Capture Location Problems: Facility Location and Design

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  1. Partial Capture Location Problems:Facility Location and Design Dmitry Krass Rotman School of Management, University of Toronto With Robert Aboolian, CSUSM Oded Berman, Rotman

  2. Rotman School of ManagementPh.D. Program in Operations Management • Rotman • MBA program is among top 50 world-wide (FT, 2013) • #11 International MBA programs (BW, 2012) • Ranked #8 in research (FT, 2013) • Ranked #9 Ph.D. program among all business schools (FT, 2013) • University of Toronto • Ranked #1 in Canada • Ranked #16 in the world by reputation (Times of London, 2012) • RotmanPh.D. Program in OM • 1-2 students per year (25-50 applicants) • All students fully funded ($26K per year) for up to 5 years • Research areas: Supply Chain Management, Queuing, Revenue Management, Location Theory, other OR/OM topics • Our goal is 100% academic placements

  3. Rotman Ph.D. in OM If you have great math skills, are interested in applying them to important managerial problems, want to participate in world-class research, and are interested in a career as a university professor, Contact Dmitry Krass krass@rotman.utoronto.ca program web site: http://www.rotman.utoronto.ca/Degrees/PhD/Academics/MajorAreasofStudy/OperationsManagement.aspx

  4. Overview • Introduction • Facility Location Problems – a quick review • Partial Capture Models: a Unifying framework • Modeling design aspects • Single-Facility Design Problem • Non-linear knapsack approach • Sensitivity analysis • Multi-facility Design and Location Problem • Tangent Line Approximation (TLA) approach to non-linear knapsack-type problems • Iterated TLA method • Conclusions and Future Research

  5. Location Models – Brief Overview • Key interaction: customers and facilities • Application areas • Physical facilities: public, private • Strategic planning • Marketing (perception space), communications (servers, nodes), statistics/data mining (clustering), etc.

  6. Competitive Location Models: basics • Facilities always “compete” for customer demand • “Competitive location models” assume (at a minimum) • Customer choice (not directed) assignments • Not all facilities controlled by the same decision-maker • Goal is to maximize “profit” for a subset of facilities • Facilities outside the subset belong to “competitors”

  7. Modeling Competition • Static models • No reaction from competitor(s); “follower’s model” • “Dynamic” models (“stackelberg games”) • Some form of competitive reaction • Leader’s problem; Leader/follower/leader, etc. • Nash games (simultaneous moves) • Issues: non-existence of equilibria, solution difficulty, limited insights ✓

  8. Location Theory: Key literature • M. Daskin, 1995, “Netowork and Discrete Location Models” - textbook, excellent place to start • Three “state of the art” survey books • P. Mirchandani, R. Francis, 1990, “Discrete Location Theory” • Z. Drezner, 1995, “A Survey Of Applications And Methods” • Z. Drezner, H. Hamachar, 2004, “Location Theory: A survey of Applications and Methods” • New volume in the works… • Also of Interest • S. Nickel, J. Puerto, 2005, “Location Theory: A Unified Approach” – good reference for planar models • V. Marianov, H.A. Eiselt, 2011, “Foundations of Location Analysis” • Vast literature in various OR, OM, IE, Geography, Regional Science journals

  9. Overview • Introduction • Facility Location Problems – a quick review • Partial Capture Models: a Unifying framework • Modeling design aspects • Single-Facility Design Problem • Non-linear knapsack approach • Sensitivity analysis • Multi-facility Design and Location Problem • Tangent Line Approximation (TLA) approach to non-linear knapsack-type problems • Iterated TLA method • Conclusions and Future Research

  10. Goal • Want to model customer choice endogenously • Model should be realistic • Partial capture: good record of applications • Want to capture two key effects • Cannibalization • “Category expansion” • Need to model elastic demand • Need to incorporate facility “attraction” • Need a way to capture design elements • Start with a static model • Complex enough!

  11. Static Location and Design ModelsIncomplete literature review Full-Capture Models (deterministic customer choice) • MAXCAP: Revelle (1986), (…) • Location and Design Models • Plastria (1997), Plastria and Carrizosa (2003) – deterministic customer choice setting on a plane • Eiselt and Laporte (1989) – one facility, constant demand Partial-capture models (“discrete choice models”, “logit models”, “market share games”, etc.) • Spatial Interaction Models • Huff (1962, 1964), Nakanishi and Cooper (1974), (…), Berman and Krass (1998) • Spatial Interaction Models with Elastic Demand • Berman and Krass (2002), Aboolian, Berman, Krass (2006) - TLA • Competitive Facility Location and Design Problem (CFDLP) • Scenario design: Aboolian, Berman, Krass (2007) • Optimal design: Aboolian, Berman, Krass (??)

  12. Facility Location and Design ProblemModel Structure Objective (profit): (Total Captured Demand) - (Total Cost) Customer Demand: % captured by Facility j (customer choice) Di MSij Demand Customer Utility: Ui Utility of facility j for customer i Overall utility uij Travel distance d(i,j) Attractiveness Aj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  13. Model Components: Facility Decisions Location Decisions • Discrete set of potential locations M • Competitive facilities may be present: set C • Must choose subset SM-C, |S|≤m • Binary decision variables xj=1 if location j chosen • Customers located at discrete set of points N • d(i,j) = distance from i to j • Fixed location cost fj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  14. Model Components: Facility Decisions Design Decisions • Attractiveness of facility at location j is given by • Assume design characteristics indexed by k=1,…,K • Typical characteristics: size, signage, #parking spaces, etc • j – attractiveness of “basic” (unimproved) facility at j • yjk – value of “improvement” of the facility with respect to k-th design characteristics • yk{0,1} for qualitative design characteristics • Log-linear form agrees with many marketing models; note concavity Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  15. Model Components: Facility Decisions Design Decisions • Attractiveness of facility at location j is given by • Cost: linear in decision variables Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  16. Model Components: UtilityUtility of facility j for customer i: uij • uij(Aj, d(i,j)) • Non-decreasing in attractiveness Aj • Decreasing in distance d(i,j) Customer Utility: Utility of facility j for customer i uij Travel distance d(i,j) Attractiveness Aj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  17. Model Components: UtilityUtility of a given facility: Functional Form • Log-linear • Used in spatial interaction models • Exponential form is equivalent • Other functional forms can also be used Customer Utility: Utility of facility j for customer i uij Travel distance d(i,j) Attractiveness Aj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  18. Model Components: UtilityOverall Utility Ui • uij(Aj, d(i,j)) • Uiis non-decreasing in uij for all i,j • Used Sum form: Customer Utility: Ui Utility of facility j for customer i Overall utility uij Travel distance d(i,j) Attractiveness Aj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  19. Facility Location and Design ProblemPercent of Realized Customer Demand: Gi - • Gi(Ui) – non-negative, non-decreasing, concave function of total utility; 0≤ Gi(Ui)≤ 1 • D(Ui)= wiGi Customer Demand: % captured by Facility j (customer choice) Di MSij Demand Customer Utility: Ui Utility of facility j for customer i uij Overall utility Travel distance d(i,j) Attractiveness Aj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  20. Model Components:Customer Demand • Gi(Ui) – non-negative, non-decreasing, concave function of total utility • 0≤G(Ui)≤1 represents realized proportion of potential demand from node i • wi- the maximum potential demand at i • Can write • Examples • Exponential demand: • Inelastic demand:

  21. Facility Location and Design ProblemPercent of Realized Customer Demand: Gi - • Spatial Interaction Models: Customer Demand: % captured by Facility j (customer choice) Di MSij Demand Customer Utility: Ui Utility of facility j for customer i uij Overall utility Travel distance d(i,j) Attractiveness Aj Facility Decisions: m xj yjk Number of facilities Locations Design Characteristics

  22. Model ComponentsMarket Share • Spatial Interaction Models: • note that total utility includes competitive facilities • also known as (or equivalent to) “logit”, “discrete choice”, “market-share games”, etc. • Full-capture model: • Total value Vi of customer i if facilities located in set S:

  23. Competitive Facility Location and Design Problem (CDFLP) Maximize total captured demand Design definition (attractiveness) The budgetary constraint Cannot improve unopened facility Select facility set S and design variables y

  24. Unifying Framework • This model unifies • Full and partial capture models • Constant / Elastic demand models • Models with / without design characteristics • General model very hard to solve directly • Non-linear IP; non-linearities in constraints and objective • Solvable cases • Constant demand, constant design (1998, 2002) • Elastic demand, constant design (2006) • Elastic demand, scenario design (2007) • General case: today

  25. Example • Assume a line segment network • No competitive facilities: Ui(C)=0 • Basic attractiveness j = 1 for j=1,2 • Only one design characteristic • yj = 2 or 0 (large or small facility) •  = .9 (large facility is 2.8x more attractive) • Budget allows us to locate two “small” or one “large” facility • Elasticity and distance sensitivity are set at 1 • ==1 distance =1 1 2 w1=1 w2=1

  26. distance =1 1 2 w1=1 w2=1 Illustration:Expansion and Cannibalization Demand Captured • Note that addition of second facility at 2 improved “company” picture, but not necessarily facility 1’s outlook – cannibalization and expansion in action First consider 1 small facility at node 1 Market shares Now add a second small facility at node 2

  27. Market Expansion vs. Cannibalization • Theorem: • Consider facility jX and customer iN • Suppose Gi(U) is concave • Let U.j= X-{j}uik+Ui(C) – utility derived by i from all other facilities • Let Dij(U.j) be the demand from i captured by j viewed as a function of U.j • Then Dij( ) is strictly decreasing in U.j • Implications: • Any improvements by other facilities (better design and/or new facilities by self or competitor) will reduce demand captured at facility j • Cannibalization effect always stronger than market expansion • Consequence of concave demand

  28. distance =1 distance =1 1 1 2 2 w1=1 w1=1 w2=1 w2=1 Corner stores vs. Supermarket Demand Captured • Here, one large facility performs slightly better Option 1: two small facilities Market shares Option 2: one large facility Market shares

  29. One “large” or two “small” facilities?Parametric Analysis – symmetric case Demand Elasticity Distance Sensitivity Design Sensitivity Conclusion: depending on sensitivity parameters, get either “corner store” or “supermarket” solutions

  30. One “large” or two “small” facilities?Competitive case (symmetric) • Assume locations are symmetric, but there are competitive facilities • U1(C) =2, U2(C) =1 (customers at 1 are better served by competition) Note that optimal location for large facility switches between 1 and 2 Why? Shouldn’t 2 be always preferred? Conclusion: depending on sensitivity parameters, get “corner store”, “box store”, or “mall” solutions – very flexible model

  31. CFDLP – Conceptual Solution Approach • Step 1: Solve 1-facility model for specified budget B • Equivalent to finding design characteristics that maximize attractiveness A for the given B • Solvable in closed form (non-linear knapsack) • Single-facility model can be solved by enumerating all potential facility locations • Step 2: Parametric analysis • Analyze A(B) optimal objective as a function of B • Can prove concavity; have quick algorithm for computing A(B) • Step 3: Back to multi-facility case

  32. Step 1: Single-Facility Design Problem(Index j suppressed) • Non-linear concave knapsack problem • Bretthauer and Shetty (EJOR, 2002); Birtran and Hax (MS, 1981) • Optimal solution can be computed in O(K2) time • Three sets: L, U, K-L-U • Characteristics in L “pegged” to LB of 0, • Characteristics in U pegged to the UB • Closed-form solution for all others Optimal Solution:

  33. Step 2: Parametric Analysis to Derive A(B) • For fixed sets L,U, K-L-U, can obtain a closed-form expression of optimal attractiveness as a function of the budget A*(B) • Optimal attractiveness is concave and non-decreasing in B • However, as B changes, so do sets L(B) and U(B) • Can identify (through linear search) a finite set of budgetary breakpoints B1,…BD • For B[Bb, Bb+1], set L(B) and U(B) are invariant and A*(B) is concave, non-decreasing in B • As B crosses a breakpoint, the slope of A*(B)changes • Can prove A*j(B) is concave, continuous and non-decreasing

  34. Parametric Analysis - Example • Theorem: A*(B) function is always concave(the derivative is discontinuous at breakpoints) K=3 B[3.5, 7] B B1=3.5 L= {2,3} U= B1=4 L={3}, U= B1=5 L=, U={1}

  35. CFDLP – Conceptual Solution Approach (cont) • Step 1: Solve 1-facility model for specified budget B • Step 2: Parametric analysis, derive A(B) • Step 3: Back to multi-facility case • All design variables yjk replaced with a single budget variable Bj Still difficult, but much more tractable non-linear IP Has knapsack-type structure Can prove that objective is a concave "superposition" of univariate concave functions

  36. CFDLP – Conceptual Solution Approach (cont) • Step 1: Solve 1-facility model for specified budget B • Step 2: Parametric analysis, derive A(B) • Step 3: Back to multi-facility case: replace design variables with Bj • Step 4: “Iterated TLA” • Utility Uiis separable with respect to A(Bj), concave • Objective function V(Ui) is also concave, composition of a concave function and a sum of univariate concave functions • Can apply a generalization of Tangent Line Approximation (TLA) method developed in Aboolian, Berman, Krass (2006) • Allows us to approximate the non-linear problem with a linear MIP • Approximation accuracy controllable by the user

  37. Tangent Line Approximation (TLA) Approach for a Class of Non-Linear Programs • Theorem (TLA): for any i and ε>0 can construct (in polynomial time) a piece-wise linear function Gεi(u) such that Gi(u) ≤ Gεi(u) and ( Gεi(u) – Gi(u))/Gi(u) ≤ 1-ε • i.e., Gεi(u) is an over-approximator within specified error bound • Moreover, Gεi(u) has the minimal number of linear segments among all piece-wise linear approximators of this precision level • Corollary 1: TLA converts NLP above into an LP whose solution is at most ε away from that of the original model (if original model was non-linear IP, get a linear IP) • For our problem, i(x) is concave in the decision variable, need a second application of TLA: “iterated TLA” • Also results in a single linear IP • Gi( ) is a concave, non-decreasing function, • i(x) – linear functional

  38. Tangent Line Approximation – Main Idea piece-wise linear approximator max relative error

  39. General CDFLP - Algorithm • Step 1: For each potential location derive breakpoints of A(B) • O(|K|2|M|) time • Step 2: Apply TLA approach to get piece-wise linear approximation • Polynomial approximation scheme • Step 3: Solve linear MIP • Size depends on solution accuracy set by the user

  40. Conclusions and Future Research • Very general and flexible framework • Single-facility location and design problem easy • Multi-facility problem tractable • Concave demand problem solvable through “iterated TLA” • Dimensionality grows over the regular TLA, but not too rapidly • Open Problems • Does the same methodology apply to “all or nothing” models • Dynamic competition

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