Chapter 7Linear Programming

I. Simplex Method

Ding-Zhu Du

LP examples

• A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in the table. Union rules state that each full-time employee must work five consecutive days and then receive two days off. The post office wants to meet its daily requirements using only full-time employees. Formulate an LP that the post office can use to minimize the number of full-time employees that must be hired.

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Feasible domain

Optimal occurs at a vertex!!!

.0,9,0

6032 s.t.

54 min

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Slack Form

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s.t.

min

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Basic Feasible Solution

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Optimality Condition

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Nondegeneracy Assumption

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Remark

solution. basic feasible oneexactly with associated is

basis feasibleevery condition,acy nondegenerUnder

Necessary

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Simplex Method

method.

simplex called g,propromminlinear thesolve

tomethod a givesy necessarit of proof The

Simplex Table

1 0 0 2 1 4 36

0 1 0 5 2 2 24

0 0 1 3 1 1 30

0 0 0 2 1 3

0,,,,,

3624

24522

303 s.t.

23 min

654321

6321

5321

4321

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2 1 3

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1 1/2 21/ 0 3 24

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}2{})5{\}6,5,4({ 1

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1 1/2 21/ 0 3 24

1/2 5/2 1 1 12

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1/2 1/2 0 2 12

}6,2,4{ 1

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1/3 1/6 61/ 0 1 8

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Puzzle 1

? basis feasible

1st or thesolution feasible basic1st thefind wedo How

Artificial Feasible Basis

m

m

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2

1

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where

0 ,0

s.t.

min

Puzzle 2

LP? solve

tohow hold,not does assumptionacy nondegenerWhen

lexicographical ordering

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Method

positive.hically lexicograp is row topthe

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columns. first at the placed is basis feasible initial the

such that columns of ordering therearrange Initially,

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Method(cont’)

pivoting.

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Method (cont’)

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each positive,hically lexicograp are row top theother than rows all Since

1 AAc - c

I

II

Theorem

optimal. is with associatedsolution feasible

basic then the0, satisfies basis feasible a if Moreove,

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solution, optimalan has gprogramminlinear theIf

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Puzzle 3Method?Simplex of timerunning theisWhat

Complexity of LP

time.-polynomialin computed becn

point extreme optimalan solution, optimalan Given

time.)(in runs MethodPoint Interior

time.)(in runs Method Ellepsoid

time.lexponentiain runs MethodSimplex

5.3

6

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Thanks, End