Integer Allocation Problems of MinMax Type with Quasiconvex Separable Functions

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Integer Allocation Problems of Min-Max Type with
Quasiconvex Separable Functions

• ZEEV ZEITIN
• Delft University of Technology, Netherlands

Operations Research. Vol. 29, no. 1, pp. 207-211.
1981
Presented by Tzu-Cheng Hsieh, OPLab, IM,
NTU 2007/1/23
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Outline

• Introduction
• Optimality Conditions For MinxMaxj Problem
• Solution Algorithm
• Applications

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Outline

• Introduction
• Optimality Conditions For MinxMaxj Problem
• Solution Algorithm
• Applications

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Introduction

• The paper treats the problem of integer resource
  allocations using the min-max criterion.
• The objective function is assumed to be strictly
  quasiconvex(in the integer sense)and separable.
• The optimality conditions are used to obtain a
  solution algorithm which may be simplified in the
  case.

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Outline

• Introduction
• Optimality Conditions For MinxMaxj Problem
• Solution Algorithm
• Applications

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Optimality Conditions For MinxMaxj Problem

• Define
• Theorem
• Proof

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Definition

• A bottleneck integer problems
• Strictly Quasiconvex

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Theorem

• Theorem

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Proof(1/4)

• Necessity

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Proof(2/4)

• Sufficiency(1/3)

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Proof(3/4)

• Sufficiency(2/3)

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Proof(4/4)

• Sufficiency(3/3)

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Outline

• Introduction
• Optimality Conditions For MinxMaxj Problem
• Solution Algorithm
• Applications

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Solution Algorithm

• Solution Algorithm

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Outline

• Introduction
• Optimality Conditions For MinxMaxj Problem
• Solution Algorithm
• Applications

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Application

• Attack and Defense Model
• Optimal Search

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Attack and Defense Model(1/5)

• There are two players pursuing antagonistic
  interests, attack and defense.
• The attacker has available M units and the
  defender N units.
• The collision of attack and defense occurs at n
  targets.
• Let xi(yi) represent the size of the subforces
  attacking(defending) the i-th target.

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Attack and Defense Model(2/5)

• The quantity of subforce Xi penetrating through
  the defense is
• , where qi is the efficiency of the use of
  the defensive means at the i-th target.

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Attack and Defense Model(3/5)

• The total damage inflicted on the complex of n
  targets is
• , where di are the weighting coefficients,
  measuring the relative importance of
  vulnerability of the n targets.

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Attack and Defense Model(4/5)

• Thus, there are two versions of the same problem
• (i) From the attackers point of view

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Attack and Defense Model(5/5)

• (ii) From the defenders point of view

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Optimal Search(1/2)

• Let the search of some object be carried out at n
  areas.
• The efficiency of search at the j-th area is
  fj(xj), which spending the resource xj, if the
  object located at j-th area and 0 otherwise.
• The mathematical expectation of the search
  efficiency is evidently
• , where pj is the probability of the object
  whereabouts being in the j-th area.

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Optimal Search(2/2)

• If are unknown, then an
  optimal strategy for a whereabouts search is
  determined by finding
• , where M is a positive integer, representing
  the budget restriction for the search.

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• The End.
• Thanks for your listening!