Partial Capture Location Problems: Facility Location and Design
description
Transcript of Partial Capture Location Problems: Facility Location and Design
![Page 1: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/1.jpg)
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
![Page 2: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/2.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 3: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/3.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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 [email protected] web site:http://www.rotman.utoronto.ca/Degrees/PhD/Academics/
MajorAreasofStudy/OperationsManagement.aspx
![Page 4: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/4.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 5: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/5.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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.
![Page 6: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/6.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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”
![Page 7: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/7.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
✓
![Page 8: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/8.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 9: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/9.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 10: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/10.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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!
![Page 11: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/11.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Static Location and Design ModelsIncomplete literature reviewFull-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 demandPartial-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 (??)
![Page 12: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/12.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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)
![Page 13: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/13.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
Facility Decisions:m
Number of facilities
xj
Locations
yjk
Design Characteristics
![Page 14: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/14.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
Number of facilities
xj
Locations
yjk
Design Characteristics
10 where,)1(1
k
K
kjkjj
kyA
![Page 15: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/15.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Model Components: Facility Decisions Design Decisions Attractiveness of facility at location j is given by
Cost: linear in decision variables
Facility Decisions:m
Number of facilities
xj
Locations
yjk
Design Characteristics
10 where,)1(1
k
K
kjkjj
kyA
jkjk yc
![Page 16: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/16.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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)
Facility Decisions:m
Number of facilities
xj
Locations
yjk
Design Characteristics
Customer Utility: uijUtility of facility j for customer i
Travel distance d(i,j) Attractiveness Aj
![Page 17: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/17.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
Facility Decisions:m
Number of facilities
xj
Locations
yjk
Design Characteristics
Customer Utility: uijUtility of facility j for customer i
Travel distance d(i,j) Attractiveness Aj
0 ,)j)i,(d1( jij Au
![Page 18: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/18.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Model Components: UtilityOverall Utility Ui
uij(Aj, d(i,j)) Ui is non-decreasing in uij for all i,j Used Sum form:
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
![Page 19: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/19.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Facility Location and Design ProblemPercent of Realized Customer Demand: Gi
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)
- Gi(Ui) – non-negative, non-decreasing, concave function of total utility; 0≤ Gi(Ui)≤ 1 D(Ui)= wiGi
![Page 20: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/20.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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 )(
![Page 21: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/21.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Facility Location and Design ProblemPercent of Realized Customer Demand: Gi
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)
- Spatial Interaction Models: i
ijij U
uMS
![Page 22: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/22.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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 ikPCkij
ij
uuMS
i
ijij U
uMS
![Page 23: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/23.jpg)
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
![Page 24: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/24.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 25: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/25.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 26: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/26.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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=1First 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
![Page 27: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/27.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 28: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/28.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Corner stores vs. Supermarket
Here, one large facility performs slightly better
1 2distance =1
w1=1 w2=1Option 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
![Page 29: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/29.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
One “large” or two “small” facilities?Parametric Analysis – symmetric case
0.5
0.7
0.9
1.11.3
1.5
1.7
1.9
0 1 2 3
Lambda
Tota
l Dem
and
2 small facilities
1 Large facility
Conclusion: depending on sensitivity parameters, get either “corner store” or “supermarket” solutions
Demand Elasticity
00.20.40.60.8
11.21.41.61.8
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
Tota
l Dem
and
2 small facilities
1 Large facility
Design Sensitivity
![Page 30: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/30.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
1 Large facility (L1)
1 Large Facility(L2)
0
0.5
1
1.5
2
0 1 2 3 4
Theta
2 small facilities
1 Large facility (L1)
1 Large Facility(L2)
Note that optimal location for large facility switches between 1 and 2
Why? Shouldn’t 2 be always preferred?
![Page 31: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/31.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 32: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/32.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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:
![Page 33: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/33.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 34: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/34.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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]
![Page 35: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/35.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 36: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/36.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 37: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/37.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 38: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/38.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
Tangent Line Approximation – Main Idea
piece-wise linear approximator
max relative error
![Page 39: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/39.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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
![Page 40: Partial Capture Location Problems: Facility Location and Design](https://reader036.fdocuments.in/reader036/viewer/2022062814/56816856550346895dde77da/html5/thumbnails/40.jpg)
DOOR 2013, Akademgorodok, Novosibirsk
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