Fuzzy Programming Models for Analyzing Demand Coverage

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Fuzzy Programming Models for Analyzing Demand
Coverage

• Ioannis Giannikos
• Department of Business Administration
• University of Patras
• EWGLA XVII Meeting
• Elche, 17-19 September 2008

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Demand Covering Models

• Objective
• Select appropriate locations for a number of
  facilities (servers) in order to provide
  sufficient coverage to a given demand set
• Demand
• Usually represented by a discrete set of demand
  points
• Two classes of problems
• Mandatory
• Maximal

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Maximal Covering Models

• First Representative MCLM
• Extensions to cater for
• Definition on a tree
• Congestion
• Primary vs Backup coverage
• For detailed applications see
• Moore and Revelle (1982)
• Current and OKelly (1992)
• Schilling et al (1993)
• Marianov and Serra (2001)

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Common Assumption

• Given a distance (or time) limit S
• If a demand point is within distance (or time) S
  from at least one server, then it is fully
  covered
• Otherwise, it is not covered at all

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Ways to Deal with the Weakness

• Piecewise linear step function (Church Roberts
  1983)
• Capacitated MCLP that considers non-covered DPs
  (Pirkul Schilling 1991)
• Partial coverage (Karasakal Karasakal 2004)

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Generalized Concepts of Coverage

• Quantitative representations
• More complicated concepts
• e.g. it is highly desirable to have a server
  within a distance of 5 km from population centre
  j
• e.g. it is unfair to locate a server at a major
  population centre
• e.g. it is preferable to include locations i and
  k in the solution
• Such concepts typically imply vagueness and
  ambiguity

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Fuzzy Set Theory

• Common way to handle vagueness and ambiguity
• A fuzzy set A is characterized by a membership
  function
• For all e?A
• ?(e) the degree with which e is included in A

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Fuzzy Coverage

• Let I set of candidate locations
• J set of demand points
• For all i ? I and j ? J
• ?(i, j) the degree to which demand point j may
  be covered by a server operating at i

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Types of coverage functions

• Conventional

• Coverage with maximum distance

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Types of coverage functions (cont.)

• Sigmoid coverage

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Additional Objectives

• Maximize population covered
• Maximize backup coverage
• Minimize servers operating cost
• Minimize congestion
• Etc.

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Fuzzy GP Model for Demand Covering (1)

• Three objectives
• Maximize total coverage
• Maximize minimum coverage
• Minimize total distance to servers for demand
  points that are not covered

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Fuzzy GP Model for Demand Covering (2)

• I set of candidate locations
• J set of demand points
• ?(i, j) membership function expressing coverage

w (j) importance of demand point j ? J
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Decision Variables
? coverage level of worst served demand point
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Goals

• (G1) Total coverage maximize
• (G2) Minimum coverage maximize
• (G3) Distance of uncovered points minimize
• where Tk is the aspiration level for goal k and
  the symbol is a fuzzifier representing the
  imprecise fashion in which the goals are stated.

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Constraints


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Formulation of Fuzzy Goals

• Fuzzy sets with linear membership functions
• (AX)i achievement of goal i
• DRi maximum admissible violation from Ti

for i 1, 2
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Formulation of Fuzzy Goals/2

• Fuzzy sets with linear membership functions
• (AX)i achievement of goal i
• DRi maximum admissible violation from Ti

for i 3 (minimization)
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Solution Approaches

• Model (M1)
• Max l
• Subject to

(8)
(9)
(10)
for i 1,,3
(11)
for i 1,,3
(12)
??0, di?0
constraints (1) (7) of FGP
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Solution Approaches

• Model (M2)
• Max b1?1b2 ?2 b3?3
• Subject to constraints (8) to (10) of model
  (M1)

for i 1,,3
for i 1,,3
?i?0, di?0
constraints (1) (7) of FGP
Model (M3) Max ?1 ?2 ?3 Subject to ?1?
?2 ?2? ?3 constraints of model (M2)
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Computational Experiments

• Randomly generated problems (100 demand points,
  100 candidate locations)
• Case study of three municipalities of the greater
  Athens area
• 53 demand points
• 59 demand points
• 65 demand points
• Solution with Premium Solver V8, Express-MP
  Solver engine
• Problem with 10107 variables and around 8000
  constraints solvable in 158 sec

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Main Results

• Experimentation with
• S coverage distance
• T maximum allowable distance
• Number of servers (K)
• Priorities of the goals
• For each experiment
• Payoff table was calculated
• Models (M1), (M2) and (M3) were tested
• In all cases results were improved with respect
  to conventional models

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Main Results/Example

• Municipality of Athens
• S1200
• T200
• K6
• Final solution by (M1)
• Obj145.80 (opt46)
• Obj20.7 (opt1)
• Obj326.86 (opt24)
• Final solution by (M3)
• Obj144.84 (opt46)
• Obj20.84 (opt1)
• Obj326.96 (opt24)