mlp introduction 2010march
TRANSCRIPT
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Multiple Level Programming:A Review
Multiple Level Programming:A Review
Hsu-Shih Shih, Ph.D.
Graduate Institute of Management SciencesTamkang University, Tamsui
Taipei, Taiwan, ROC
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Personal BackgroundPersonal BackgroundPh.D., Industrial Engineering, Kansas State
Univ.Visiting Professor, Univ. of Pittsburgh
Major InterestsDecision analysis
Decision support
Operations research
Soft computing
Website
http://163.13.193.161
http://163.13.193.161/http://163.13.193.161/ -
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Brief ContentsBrief Contents
Introduction
Definition
Characteristics
Applications
TechniquesFuture Research
Questions and Comments
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IntroductionIntroductionWhat is multiple level programming?
Decentralized planning in organizations
Where are its applications?
Many areas with conflict resolution
Whats techniques deal with the
problems?Traditional and non-traditional techniques
Future Research
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DefinitionDefinition
Multiple Level Programming (MLP)
To solve decentralized planning problems
with multiple executors in a hierarchical
organization
Explicitly assigns each agent a unique
objective and set of decision variables aswell as a set of common constraints that
affects all agents
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Hierarchical StructureHierarchical Structure
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MLP FormulationMLP Formulation
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CharacteristicsCharacteristics Common Characteristics of MLP
1) Interactive decision-making units exit within apredominantly hierarchical structure
2) Execution of decisions is sequential, from top
level to bottom level3) Each unit independently maximizes its own net
benefits, but is affected by actions of other units
through externalities4) The external effect on a decision-makers
problem can be reflected in both his objective
function and his set of feasible decision space
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Bi-level Programming a simple caseBi-level Programming a simple caseProblem formulation
Max f1 (x1, x2) = c11T x1 + c12
T x2 (upper level)x1
where x2 solves,
Max f2 (x1, x2) = c21T x1 + c22
T x2 (lower level)x2
s.t.
(x1, x2) X={(x1, x2)| A1 x1+A2 x2 b, and x1, x2 0}where c11, c12, c21, c22, and bare vectors, A1 and
A2 are matrices, and X represents the constraint region.
Wen and Hsu 1991
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Bi-level ProgrammingBi-level ProgrammingA Special Case of Two-person, Non-
zero Sum Non-cooperative Game
A general Stackelbergs (leader-follower)
duopoly modelNested Optimization Problem
NP-hard complexity
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Applications (I)Applications (I) Agricultural model
Agricultural policy- Nile Valley case (Parraga 1981) Milk industry (Candler and Norton 1977)
Mexican agriculture model (Candler and Norton 1977)
Water supply model (Candler et al. 1981)
Government policy Distribution of government resources (Kyland 1975)
Environmental regulation (Kolstad 1982)
Finance model Bank asset portfolio (Parraga 1981)
Commission rate setting (Wen and Jiang 1988)
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Applications (II)Applications (II) Economic systems
Distribution center problem (Fortuny and McCarl 1981) Principle-agent model (Arrow 1986)
Price ceilings in the oil industry (DeSilva 1978)
Welfare Allocation model of strategic weapons (Bracken et al.1977)
Transportation Highway network system (LeBlance and Boyce 1986)
Others Network flows (Shih and Lee 1999, Shih 2005)
Supply chain (Viswanarthan et al. 2001)
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Techniques (I)Techniques (I) Extreme-point Search
Kth-best algorithm Grid-search algorithm
Fuzzy approach (Shih 1995, Shih et al. 1996)
Interactive approach (Shih 2002)
Transformation Approach Complement pivot
Branch-and-bound
Penalty function
Interior Point Primal-dual algorithm
Lee and Shih 2001, Shih et al. 2004
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Techniques (II)Techniques (II) Decent and Heuristics
Descent method
Branch-and-bound
Cutting plane
Dynamic programming (Shih and Lee 2001, Shih 2005)
Intelligent Computation Tabu search
Simulated annealing
Genetic algorithm
Artificial neural network (Shih et al. 2004)
Lee and Shih 2001, Shih et al. 2004
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Categories of TechniquesCategories of Techniques
Lee and Shih 2001,
Shih et al. 2004
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Example 1. A Trade-off Problem
between Exports and Imports
Example 1. A Trade-off Problem
between Exports and ImportsProblem formulation
Maxf1 = 2
x1 -
x2 (effect on the export trade - 1st objective )
x1
where x2 solves,
Max f2 = x1 + 2 x2 (profits on the product - 2nd objective )
s.t.
3 x1 - 5 x2 15 ( capacity )
3 x1 - x2 21 ( management )
3 x1 + x2 27 ( space )
3 x1 + 4 x2 45 ( material )
x1 + 3 x2 30 ( labor hours )
x1 , x2 0 ( non-negative )
Shih 1995, Shih et al. 1996
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Kth-best Algorithm Extreme-pointKth-best Algorithm Extreme-point Solving procedure
Step 1. Solve the upper-level problemi=1, x[1]*= (7.5,1.5) at vertex B
Step 2. Solve the lower-level problem with x1 = 7.5
Solution x+= (7.5,4.5) between vertex D and vertex C
x+ x[1]*, go to Step 3.
Step 3. Consider the neighboring set of x[1]* (vertex A andvertex C)
Step 4. Update labeli
=i
+1=2, and choose x[2]* = (8,3)(vertex C). Go to Step 2.
Step 2. Let x1 = 8 to the lower level problem
Solution x+= (8,3). Since x+= x[2]*, the procedure is
terminated. x[2]* is the optimum
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Decision (Variable) SpaceDecision (Variable) Space
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Objective (Function) SpaceObjective (Function) Space
K h K h T k C di i
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Karush-Kuhn-Tucker Conditions
Transformation approach
Karush-Kuhn-Tucker Conditions
Transformation approach
Lee and Shih 2001
Problem formulation
Maxf1 = 2
x1 -
x2
x1, x2s.t.
(x1 , x2 ) X
w1 ( 3 x1 + 5 x2 + 15) = 0
w2 ( 3 x1 + x2 + 21) = 0
w3 ( 3 x1 x2 + 27) = 0
w4 ( 3 x1 4 x2 + 45) = 0w5 ( x1 3 x2 + 30) = 0
5 w1 w2 + w3 + 4 w4 + 3 w5 = 2
w1 ,
w2 ,
w3 ,
w4 ,
w5 ,
x1 ,
x2 0
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Separation ProcedureSeparation Procedure
Lee and Shih 2001
Problem formulation
The constraint set
wT (A1 x1 + A2 x2 b ) = 0, where w is a dual vector.
The transformed two terms
w (1 ) M , andA
1
x1
+ A2
x2
b Mwhere {0, 1} and M is a large positive constant
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Concept of Fuzzy ApproachConcept of Fuzzy Approach
Shih 1995, Shih et al. 1996
Fuzzy Membership Functions (Zadeh
1965) Tolerance of decisions
Achievement of goal
Fuzzy Multi-objective Decision Making(Zimmermann 1985)
Information aggregation
Supervised search procedure
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Fuzzy ApproachFuzzy ApproachProblem formulation
Max f2 = 2 x1 - x2
s.t.
(x1 , x2 ) X
f1( f1(x))
x1( x1)
[0, 1] and [0, 1]
x1 , x2 0
Shih 1995, Shih et al. 1996
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Fuzzy DecisionFuzzy DecisionProblem formulation
Max {, , }
s.t.
(x1 , x2 ) X
f1( f1(x)) = (f1 0) / (13.5 0)
x1(x1) = (x1 4.5) / (7.5 4.5)
x1(x1) = (8 x1 ) / (8 7.5)
f2( f2(x)) = (f2 10.5) / (21 10.5)
x1 , x2 0
, , [0, 1]
Shih 1995, Shih et al. 1996
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Shih 2002
Interactive ApproachInteractive Approach
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Advantages of Fuzzy ApproachAdvantages of Fuzzy Approach
Advantages
Approximation of the natural of Large
MLPPs
Not increase the computational complexity
Ease to extend to multiple levels
DMs involve the processEfficient (Pareto) solution
Nested Optimization Sequential Optimization
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Extension to Vague InformationExtension to Vague Information
Vague/Imprecise data Possibilistic Distribution
Shih 2002
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Dynamic Aspect of MLPDynamic Aspect of MLP
Dynamic environment Multi-stage MLP
(discrete space)
Shih 2005
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Neural Network ApproachNeural Network ApproachUse of dynamic behavior of artificial
neural networks with parallel processing
Based on Hopfield and Tank (1985)-
recurrent networkTransforming to the energy function
without constraintsOptimum solution with a steady state
Shih et al. 2004
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Neural Network ApproachNeural Network Approach
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Future ResearchFuture ResearchUse of hybrid algorithms for uncertainty
Solutions of multi-subunits
Extend to n-level problems
Conditions of existing Pareto-optimal
Applications of real-world problems
(nonlinear coefficients)
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ReferenceReference
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Questions & CommentsQuestions & Comments
Thank you!