door 2013, akademgorodok, novosibirsk partial capture location problems: facility location and...

DOOR 2013, Akademgorodok, Novosibirsk

Partial Capture Location Problems:Facility Location and Design

Dmitry KrassRotman School of Management, University of Toronto

With

Robert Aboolian, CSUSMOded Berman, Rotman


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)

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


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

[email protected]

program web site:

http://www.rotman.utoronto.ca/Degrees/PhD/Academics/MajorAreasofStudy/OperationsManagement.aspx

mailto:[email protected]


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


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.


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”


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

Set C of comp. facilities

“We” locate new facilities in set S

Customers re-allocate demand between C and S

✓


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


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



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!


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 (??)


Facility Location and Design ProblemModel Structure

Facility Decisions:m

Number of facilities

xj

Locations

yjk

Design Characteristics

Customer Utility: uijUtility of facility j for customer i

Ui

Overall utility

Travel distance d(i,j) Attractiveness Aj

Customer Demand: Di Demand MSij

% captured byFacility j(customer choice)

Objective (profit): (Total Captured Demand) - (Total Cost)


Model Components: Facility Decisions Location Decisions

Discrete set of potential locations M– Competitive facilities may be present: set C– Must choose subset SM-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



xj

Locations

yjk



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



xj

Locations

yjk


10 where,)1(1

k

K

kjkjj

kyA


Model Components: Facility Decisions Design Decisions Attractiveness of facility at location j is given by

Cost: linear in decision variables



xj

Locations

yjk


10 where,)1(1

k

K

kjkjj

kyA

jkjk yc


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)



xj

Locations

yjk





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



xj

Locations

yjk




0 ,)j)i,(d1( jij Au


Model Components: UtilityOverall Utility Ui

uij(Aj, d(i,j))

Ui is non-decreasing in uij for all i,j Used Sum form:



xj

Locations

yjk



Ui

Overall utility



Facility Location and Design ProblemPercent of Realized Customer Demand: Gi



xj

Locations

yjk



Ui

Overall utility


Customer Demand:Di Demand MSij


- Gi(Ui) – non-negative, non-decreasing, concave function of total utility; 0≤ Gi(Ui)≤ 1

D(Ui)= wiGi


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:

),()( iiii UGwUD

)1()( iiUiii ewUD

iii wUD )(


Facility Location and Design ProblemPercent of Realized Customer Demand: Gi



xj

Locations

yjk



Ui

Overall utility


Customer Demand:Di Demand MSij


- Spatial Interaction Models: i

ijij U

uMS


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:

otherwise0

}{max if1 ikPCkijij

uuMS

i

ijij U

uMS


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


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


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

1 2distance =1

w1=1 w2=1


Illustration:Expansion and Cannibalization

Note that addition of second facility at 2 improved “company” picture, but not necessarily facility 1’s outlook – cannibalization and expansion in action

1 2distance =1

w1=1 w2=1

First consider 1 small facility at node 1

Market shares

Demand Captured

Now add a second small facility at node 2

0

0.3

0.6

0.9

1.2

1.5

1.8

Node 1 Node 2 Total

Dem

and


Market Expansion vs. Cannibalization

Theorem:– Consider facility jX and customer iN

» 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


Corner stores vs. Supermarket

Here, one large facility performs slightly better

1 2distance =1

w1=1 w2=1

Option 1:two small facilities

Market shares

Demand Captured

0

0.3

0.6

0.9

1.2

1.5

1.8

Node 1 Node 2 Total

Dem

and

Option 2:one large facility

Market shares

1 2distance =1

w1=1 w2=1


One “large” or two “small” facilities?Parametric Analysis – symmetric case

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

0 1 2 3

Lambda

To

tal D

em

an

d

2 small facilities

1 Large facility

Conclusion: depending on sensitivity parameters, get either “corner store” or “supermarket” solutions

Demand Elasticity

0

0.2

0.40.6

0.8

1

1.2

1.41.6

1.8

2

0 1 2 3 4

Beta

Tota

l Dem

and

2 small facilities

1 Large facility

Distance Sensitivity

0

0.5

1

1.5

2

2.5

0 1 2 3 4

Theta

To

tal D

em

an

d2 small facilities

1 Large facility

Design Sensitivity


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)

Conclusion: depending on sensitivity parameters, get “corner store”, “box store”, or “mall” solutions – very flexible model

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Lambda

2 small facilities

1 Large facility (L1)

1 Large Facility(L2)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4

Beta

2 small facilities



0

0.5

1

1.5

2

0 1 2 3 4

Theta

2 small facilities



Note that optimal location for large facility switches between 1 and 2

Why? Shouldn’t 2 be always preferred?


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


0

'

..

)1(max

max

1

1

kk

K

kkk

K

kk

yy

fBByc

tS

y k

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

Uky

ULKkc

cycfBLk

y

k

ULKkkk

ULKkk

Ukkkk

k

if

if 1)(

if 0

max

max

*

- 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:


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


Parametric Analysis - Example

Theorem: A*(B) function is always concave (the derivative is discontinuous at breakpoints)

B1=4L={3}, U=

B1=5L=, U={1}

B1=3.5L= {2,3} U=

B

K=3B[3.5, 7]


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 objectiveis a concave "superposition"of univariate concave functions


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 Ui is 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


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 Gε

i(u) such that Gi(u) ≤ Gεi(u) and ( Gε

i(u) – Gi(u))/Gi(u) ≤ 1-ε– i.e., Gε

i(u) is an over-approximator within specified error bound– Moreover, Gε

i(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

m

ii

i

RxbxAtS

xG

~,

~~ ..

))~((max • Gi( ) is a concave, non-decreasing function,

• i(x) – linear functional


Tangent Line Approximation – Main Idea

piece-wise linear approximator

max relative error


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



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

door 2013, akademgorodok, novosibirsk partial capture location problems: facility location and...

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