MLP Introduction 2010march

8/9/2019 MLP Introduction 2010march

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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


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



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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]



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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!

MLP Introduction 2010march

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