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DOOR 2013, Akademgorodok, Novosibirsk Partial Capture Location Problems: Facility Location and Design Dmitry Krass Rotman School of Management, University of Toronto With Robert Aboolian, CSUSM Oded Berman, Rotman

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DOOR 2013, Akademgorodok, Novosibirsk Short Biography Born Akademgorodok, 1963Akademgorodok, 1963 Школа #130, Терешковой 44Терешковой 44 Весенний проезд 17Весенний проезд 17…. US, Canada, etc., etc.US, Canada, etc., etc. Rotman School of Management, Univ. of Toronto ?? ?? Co-ordinator, Ph.D. program in O.M.Co-ordinator, Ph.D. program in O.M.

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DOOR 2013, Akademgorodok, Novosibirsk Rotman School of Management Ph.D. Program in Operations Management u 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) u University of Toronto –Ranked #1 in Canada –Ranked #16 in the world by reputation (Times of London, 2012) u Rotman Ph.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

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DOOR 2013, Akademgorodok, Novosibirsk Rotman Ph.D. in OM

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DOOR 2013, Akademgorodok, Novosibirsk Overview u Introduction u Facility Location Problems – a quick review u Partial Capture Models: a Unifying framework –Modeling design aspects u Single-Facility Design Problem –Non-linear knapsack approach –Sensitivity analysis u Multi-facility Design and Location Problem –Tangent Line Approximation (TLA) approach to non-linear knapsack-type problems –Iterated TLA method u Conclusions and Future Research

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DOOR 2013, Akademgorodok, Novosibirsk Location Models – Brief Overview u Key interaction: customers and facilities u Application areas –Physical facilities: public, private –Strategic planning –Marketing (perception space), communications (servers, nodes), statistics/data mining (clustering), etc.

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DOOR 2013, Akademgorodok, Novosibirsk Location Models: Solution Space Demand Space Where does customer demand originate? Discrete set N Nodes of network, points on plane Other Uniformly distributed over a certain region, etc. Location Space Where can facilities be located Discrete Set M (subset of N) => Discrete FLM Region of R 2 => Planar FLM Along links or at nodes of a network => Network FLM ✓ ✓

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DOOR 2013, Akademgorodok, Novosibirsk Customers-Facility Interaction Customer Demand Fixed? Elastic with respect to ….? (travel distance, facility attraction, price, service, etc.) Types of Facilities Identical? Different how? (size, design, access) Fixed or unlimited capacity? Customer-Facility Allocation Directed Assignment? Customer choice? Customer decision rules? Features Full vs. Partial Capture Deterministic vs. Stochastic Models Reliable vs. Unreliable Facilities / Links Objectives Median / Center/ Cover

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DOOR 2013, Akademgorodok, Novosibirsk Competitive Location Models: basics u Facilities always “compete” for customer demand u “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”

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DOOR 2013, Akademgorodok, Novosibirsk Modeling Competition u Static models –No reaction from competitor(s); “follower’s model” u “Dynamic” models (“stackelberg games”) –Some form of competitive reaction –Leader’s problem; Leader/follower/leader, etc. u Nash games (simultaneous moves) u Issues: non-existence of equilibria, solution difficulty, limited insights Set C of comp. facilities “We” locate new facilities in set S Customers re- allocate demand between C and S ✓

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DOOR 2013, Akademgorodok, Novosibirsk Location Theory: Key literature u M. Daskin, 1995, “Netowork and Discrete Location Models” - textbook, excellent place to start u 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… u 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” u Vast literature in various OR, OM, IE, Geography, Regional Science journals

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DOOR 2013, Akademgorodok, Novosibirsk Overview u Introduction u Facility Location Problems – a quick review u Partial Capture Models: a Unifying framework –Modeling design aspects u Single-Facility Design Problem –Non-linear knapsack approach –Sensitivity analysis u Multi-facility Design and Location Problem –Tangent Line Approximation (TLA) approach to non-linear knapsack-type problems –Iterated TLA method u Conclusions and Future Research

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DOOR 2013, Akademgorodok, Novosibirsk Goal u Want to model customer choice endogenously u Model should be realistic –Partial capture: good record of applications u Want to capture two key effects –Cannibalization –“Category expansion” »Need to model elastic demand u Need to incorporate facility “attraction” –Need a way to capture design elements u Start with a static model –Complex enough!

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DOOR 2013, Akademgorodok, Novosibirsk Static Location and Design Models Incomplete literature review Full-Capture Models (deterministic customer choice) u MAXCAP: Revelle (1986), (…) u 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.) u Spatial Interaction Models –Huff (1962, 1964), Nakanishi and Cooper (1974), (…), Berman and Krass (1998) u Spatial Interaction Models with Elastic Demand –Berman and Krass (2002), Aboolian, Berman, Krass (2006) - TLA u Competitive Facility Location and Design Problem (CFDLP) –Scenario design: Aboolian, Berman, Krass (2007) –Optimal design: Aboolian, Berman, Krass (??)

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DOOR 2013, Akademgorodok, Novosibirsk Facility Location and Design Problem Model Structure Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics Customer Utility: u ij Utility of facility j for customer i UiUiUiUi Overall utility Travel distance d(i,j) Attractiveness A j Customer Demand: DiDiDiDi Demand MS ij % captured by Facility j (customer choice) Objective (profit): (Total Captured Demand) (Total Captured Demand) - (Total Cost) (Total Cost)

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Facility Decisions Location Decisions u Discrete set of potential locations M –Competitive facilities may be present: set C –Must choose subset S M-C, |S|≤m –Binary decision variables x j =1 if location j chosen u Customers located at discrete set of points N –d(i,j) = distance from i to j u Fixed location cost f j Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Facility Decisions Design Decisions u 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 –y jk – value of “improvement” of the facility with respect to k-th design characteristics »y k {0,1} for qualitative design characteristics –Log-linear form agrees with many marketing models; note concavity Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Facility Decisions Design Decisions u Attractiveness of facility at location j is given by u Cost: linear in decision variables Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Utility Utility of facility j for customer i: u ij u u ij (A j, d(i,j)) –Non-decreasing in attractiveness A j –Decreasing in distance d(i,j) Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics Customer Utility: u ij Utility of facility j for customer i Travel distance d(i,j) Attractiveness A j

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Utility Utility of a given facility: Functional Form u Log-linear –Used in spatial interaction models –Exponential form is equivalent u Other functional forms can also be used Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics Customer Utility: u ij Utility of facility j for customer i Travel distance d(i,j) Attractiveness A j

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Utility Overall Utility U i u u ij (A j, d(i,j)) u U i is non-decreasing in u ij for all i,j u Used Sum form: Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics Customer Utility: u ij Utility of facility j for customer i UiUiUiUi Overall utility Travel distance d(i,j) Attractiveness A j

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DOOR 2013, Akademgorodok, Novosibirsk Facility Location and Design Problem Percent of Realized Customer Demand: G i Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics Customer Utility: u ij Utility of facility j for customer i UiUiUiUi Overall utility Travel distance d(i,j) Attractiveness A j Customer Demand: DiDiDiDi Demand MS ij % captured by Facility j (customer choice) - u G i (U i ) – non-negative, non-decreasing, concave function of total utility; 0≤ G i (U i )≤ 1 u D(U i )= w i G i

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DOOR 2013, Akademgorodok, Novosibirsk Model Components: Customer Demand u G i (U i ) – non-negative, non-decreasing, concave function of total utility –0≤G(Ui)≤1 represents realized proportion of potential demand from node i u w i - the maximum potential demand at i u Can write u Examples – Exponential demand: –Inelastic demand:

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DOOR 2013, Akademgorodok, Novosibirsk Facility Location and Design Problem Percent of Realized Customer Demand: G i Facility Decisions: m Number of facilities xjxjxjxj Locations y jk Design Characteristics Customer Utility: u ij Utility of facility j for customer i UiUiUiUi Overall utility Travel distance d(i,j) Attractiveness A j Customer Demand: DiDiDiDi Demand MS ij % captured by Facility j (customer choice) - u Spatial Interaction Models:

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DOOR 2013, Akademgorodok, Novosibirsk Model Components Market Share u Spatial Interaction Models: – note that total utility includes competitive facilities –also known as (or equivalent to) “logit”, “discrete choice”, “market-share games”, etc. u Full-capture model: u Total value V i of customer i if facilities located in set S:

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DOOR 2013, Akademgorodok, Novosibirsk Competitive Facility Location and Design Problem (CDFLP) Maximize total captured demand The budgetary constraint Cannot improve unopened facility Design definition (attractiveness) Select facility set S and design variables y

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DOOR 2013, Akademgorodok, Novosibirsk Unifying Framework u This model unifies –Full and partial capture models –Constant / Elastic demand models –Models with / without design characteristics u General model very hard to solve directly –Non-linear IP; non-linearities in constraints and objective u Solvable cases –Constant demand, constant design (1998, 2002) –Elastic demand, constant design (2006) –Elastic demand, scenario design (2007) –General case: today

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DOOR 2013, Akademgorodok, Novosibirsk Example u Assume a line segment network u No competitive facilities: U i (C)=0 u Basic attractiveness j = 1 for j=1,2 u Only one design characteristic –y j = 2 or 0 (large or small facility) – =.9 (large facility is 2.8x more attractive) u Budget allows us to locate two “small” or one “large” facility u Elasticity and distance sensitivity are set at 1 – = =1 12 distance =1 w 1 =1 w 2 =1

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DOOR 2013, Akademgorodok, Novosibirsk Illustration: Expansion and Cannibalization u Note that addition of second facility at 2 improved “company” picture, but not necessarily facility 1’s outlook – cannibalization and expansion in action 12 distance =1 w 1 =1 w 2 =1 First consider 1 small facility at node 1 Market shares Demand Captured Now add a second small facility at node 2

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DOOR 2013, Akademgorodok, Novosibirsk Market Expansion vs. Cannibalization u Theorem: –Consider facility j X and customer i N »Suppose G i (U) is concave »Let U.j = X-{j} u ik +U i (C) – utility derived by i from all other facilities »Let D ij (U.j ) be the demand from i captured by j viewed as a function of U.j –Then D ij ( ) is strictly decreasing in U.j u 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

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DOOR 2013, Akademgorodok, Novosibirsk Corner stores vs. Supermarket u Here, one large facility performs slightly better 12 distance =1 w 1 =1 w 2 =1 Option 1: two small facilities Market shares Demand Captured Option 2: one large facility Market shares 12 distance =1 w 1 =1 w 2 =1

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DOOR 2013, Akademgorodok, Novosibirsk One “large” or two “small” facilities? Parametric Analysis – symmetric case Conclusion: depending on sensitivity parameters, get either “corner store” or “supermarket” solutions Demand Elasticity Distance Sensitivity Design Sensitivity

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DOOR 2013, Akademgorodok, Novosibirsk One “large” or two “small” facilities? Competitive case (symmetric) u Assume locations are symmetric, but there are competitive facilities –U 1 (C) =2, U 2 (C) =1 (customers at 1 are better served by competition) Conclusion: depending on sensitivity parameters, get “corner store”, “box store”, or “mall” solutions – very flexible model Note that optimal location for large facility switches between 1 and 2 Why? Shouldn’t 2 be always preferred?

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DOOR 2013, Akademgorodok, Novosibirsk CFDLP – Conceptual Solution Approach u 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 u Step 2: Parametric analysis –Analyze A(B) optimal objective as a function of B »Can prove concavity; have quick algorithm for computing A(B) u Step 3: Back to multi-facility case

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DOOR 2013, Akademgorodok, Novosibirsk Step 1: Single-Facility Design Problem (Index j suppressed) u Non-linear concave knapsack problem –Bretthauer and Shetty (EJOR, 2002); Birtran and Hax (MS, 1981) u Optimal solution can be computed in O(K 2 ) 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:

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DOOR 2013, Akademgorodok, Novosibirsk Step 2: Parametric Analysis to Derive A(B) u 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 u However, as B changes, so do sets L(B) and U(B) u Can identify (through linear search) a finite set of budgetary breakpoints B 1,…B D –For B [B b, B b+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

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DOOR 2013, Akademgorodok, Novosibirsk Parametric Analysis - Example u Theorem: A*(B) function is always concave (the derivative is discontinuous at breakpoints) B1=4 L={3}, U= B1=5 L= , U={1} B1=3.5 L= {2,3} U= B K=3 B [3.5, 7]

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DOOR 2013, Akademgorodok, Novosibirsk CFDLP – Conceptual Solution Approach (cont) u Step 1: Solve 1-facility model for specified budget B u Step 2: Parametric analysis, derive A(B) u Step 3: Back to multi-facility case –All design variables y jk replaced with a single budget variable B j 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

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DOOR 2013, Akademgorodok, Novosibirsk CFDLP – Conceptual Solution Approach (cont) u Step 1: Solve 1-facility model for specified budget B u Step 2: Parametric analysis, derive A(B) u Step 3: Back to multi-facility case: replace design variables with B j u Step 4: “Iterated TLA” –Utility U i is separable with respect to A(B j ), concave –Objective function V(U i ) 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

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DOOR 2013, Akademgorodok, Novosibirsk Tangent Line Approximation (TLA) Approach for a Class of Non-Linear Programs u Theorem (TLA): for any i and ε>0 can construct (in polynomial time) a piece-wise linear function G ε i (u) such that G i (u) ≤ G ε i (u) and ( G ε i (u) – G i (u))/G i (u) ≤ 1-ε –i.e., G ε i (u) is an over-approximator within specified error bound –Moreover, G ε i (u) has the minimal number of linear segments among all piece-wise linear approximators of this precision level u 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) u For our problem, i(x) is concave in the decision variable, need a second application of TLA: “iterated TLA” u Also results in a single linear IP Gi( ) is a concave, non- decreasing function, i (x) – linear functional

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DOOR 2013, Akademgorodok, Novosibirsk Tangent Line Approximation – Main Idea piece-wise linear approximator max relative error

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DOOR 2013, Akademgorodok, Novosibirsk General CDFLP - Algorithm u Step 1: For each potential location derive breakpoints of A(B) –O(|K| 2 |M|) time u Step 2: Apply TLA approach to get piece-wise linear approximation –Polynomial approximation scheme u Step 3: Solve linear MIP –Size depends on solution accuracy set by the user

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DOOR 2013, Akademgorodok, Novosibirsk Conclusions and Future Research u Very general and flexible framework u Single-facility location and design problem easy u Multi-facility problem tractable –Concave demand problem solvable through “iterated TLA” –Dimensionality grows over the regular TLA, but not too rapidly u Open Problems –Does the same methodology apply to “all or nothing” models –Dynamic competition

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