OPERATIONS RESEARCH >INTERNATIONAL EDITI
OPERATIONS RESEARCH >INTERNATIONAL EDITI
4th Edition
ISBN: 9780534423629
Author: WINSTON
Publisher: CENGAGE L
Question
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Chapter 5.2, Problem 8P

(a)

Program Plan Intro

  • Let us consider on the following Linear programming;
  • max z=9x1+8x2+5x3+4x4
  • Such that
    • x1+x4≤200
    • x2+x3≤150
    • x1+x2+x3≤350
    • 2x1+x2+x3+x4≤550
    • x1,x2,x3,x4≥0
  • The LINDO output for this Linear Programming is given below:
  • Max 9x1+8x2+5x3+4x4
  • Subject to constraints:
    • x1+x4≤200
    • x2+x3≤150
    • x1+x2+x3≤350
    • 2x1+x2+x3+x4≤550
  • End
  • LP optimum found at step 4
  • Objective function value: 3000.000
VariableValueReduced Cost
x1200.0000000.000000
x2150.0000000.000000
x30.0000003.000000
x40.0000000.000000
  • Number of iterations=4
  • Ranges in which the basis is unchanged:
VariableCurrent CoefficientObj Coefficient ranges allowable increaseAllowance Decrease
x19.0000007.0000001.000000
x28.000000Infinity3.000000
x35.0000003.000000Infinity
x44.0000000.500000Infinity
RowCurrent RHSRighthand side ranges allowable increaseAllowance decrease
2200.000000Infinity0.000000
3150.0000000.0000000.000000
4350.000000Infinity0.000000
5550.0000000.000000400.000000

(b)

Explanation of Solution

  • Here, three oddities that may occur when the optimal solution found by LINDO is degenerate.
  • Oddity 1: In the ranges in which the basis is unchanged, at least one constraint will have a 0. Allowable increase or Allowable decrease.
  • This means that for at least one constraint, the dual price can tell us about the new z-value for either an increase or decrease in the right-hand side, but not both.
  • To understand Oddity 1, consider the second constraint. Its allowable increase is 0.
  • This means that the second constraint’s dual price of 3 cannot be used to determine a new z-value resulting from any increase in the first constraint’s right-hand side.
  • Oddity 2: For a non-basic variable to become positive, its objective function coefficient may have to be improved by more than it reduced cost...

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