Feasibility Mapping for Multi-attribute Decision Making

1
Feasibility Mapping for Multi-attribute Decision
Making
Informs/MC34.1

• Liang Zhu, David Kazmer and Yanli Zhao
• Department of Mech. and Ind. Engineering
• University of Mass. Amherst
• November 2000

2
Introduction

• Multi-attribute decision making
• Coupled and competitive multiple objectives
• Implicit preference over multiple attribute
• Fuzzy constraint limits
• Limited decision range in practical applications
• Feasibility mapping
• Visualization of the decision space and the
  performance space (attribute space)
• Efficiency frontier for multi-attributes
• Adjustable constraint limits

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Linear Feasibility Mapping

• Linear formulation
• Objective in the constraint form
• With the minimum acceptable performance level
• Nonnegative decision variables
• Convex feasible space
• Solved by an extensive simplex method
• Explore all adjacent extreme points
• Store the connection graph

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Case with 2 Decision Variables
x2
H5 Primary constraint x2 ? 0
Feasible Space
x1
H4 Primary constraint x1 ? 0
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Pivot Operation
j Min bi/aij bi/aij gt 0
A
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Breadth-First Traversal
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Algorithm Refinement

• Initial extreme point
• Two-phase method
• Identification of an empty feasible space
• Unbounded feasible space
• aij ? 0 when computing Min bi/aij bi/aij gt 0
• Ignore the pivot at aij
• Degeneracy
• bi 0 or tied blocking distance bi/aij
• Alias for extreme point

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Analysis

• Correctness
• All extreme points connected
• Each extreme point reached in the shortest path
• Time
• Dominating pivot operation in O(mn)
• Number of the pivots (extreme points)
• Mean number ? 2n (Berenguer and Smith 1986)
• NP Problem
• More efficient than the exhaustive combinations
• Sensitive to the number of decision variables

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Multi-attribute Space
x2
B

• Duality in linear problems
• Extreme points from x to y
• Decision with multi-attribute space
• Efficiency Frontier ABCD
• Adjust the constraint limits
• Options valuation
• Inverse approach

A
C
Decision Space
D
x1
Cost
A
Attribute Space
B
C
D
Time (s)
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Non-linear Systems

• Simulation
• Sampling within the decision range
• Simple but expensive
• Linear approximation
• Local linearization
• Depending on the initial point
• Linear patch

x2
Feasible space
x1
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Experimental Results

• y1 1-x12/2- x22/2
• y2 1-x12/2- (x2-1)2/8
• 0.1 ? y1, y2 ? 0.95
• 0 ? x1, x2? 1

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Conclusions

• To establish the global feasibility mapping
• An extensive simplex method for the linear model
• Working on the approximation of non-linear
  systems
• To support multi-attribute decision modeling and
  selection
• Rational trade-off on multiple objectives
• Enable the refinement on fuzzy constraints
• Working on options valuation for successive
  decisions