Operations Research : Applications and Algorithms
4th Edition
ISBN: 9780534380588
Author: Wayne L. Winston
Publisher: Brooks Cole
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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
Variable Value Reduced Cost x1 200.000000 0.000000 x2 150.000000 0.000000 x3 0.000000 3.000000 x4 0.000000 0.000000
- Number of iterations=4
- Ranges in which the basis is unchanged:
Variable Current Coefficient Obj Coefficient ranges allowable increase Allowance Decrease x1 9.000000 7.000000 1.000000 x2 8.000000 Infinity 3.000000 x3 5.000000 3.000000 Infinity x4 4.000000 0.500000 Infinity
Row Current RHS Righthand side ranges allowable increase Allowance decrease 2 200.000000 Infinity 0.000000 3 150.000000 0.000000 0.000000 4 350.000000 Infinity 0.000000 5 550.000000 0.000000 400.000000
- x1+x4≤200
- x2+x3≤150
- x1+x2+x3≤350
- 2x1+x2+x3+x4≤550
- x1,x2,x3,x4≥0
- x1+x4≤200
- x2+x3≤150
- x1+x2+x3≤350
- 2x1+x2+x3+x4≤550
| Variable | Value | Reduced Cost |
| x1 | 200.000000 | 0.000000 |
| x2 | 150.000000 | 0.000000 |
| x3 | 0.000000 | 3.000000 |
| x4 | 0.000000 | 0.000000 |
| Variable | Current Coefficient | Obj Coefficient ranges allowable increase | Allowance Decrease |
| x1 | 9.000000 | 7.000000 | 1.000000 |
| x2 | 8.000000 | Infinity | 3.000000 |
| x3 | 5.000000 | 3.000000 | Infinity |
| x4 | 4.000000 | 0.500000 | Infinity |
| Row | Current RHS | Righthand side ranges allowable increase | Allowance decrease |
| 2 | 200.000000 | Infinity | 0.000000 |
| 3 | 150.000000 | 0.000000 | 0.000000 |
| 4 | 350.000000 | Infinity | 0.000000 |
| 5 | 550.000000 | 0.000000 | 400.000000 |
(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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Chapter 5 Solutions
Operations Research : Applications and Algorithms
Ch. 5.1 - Prob. 1PCh. 5.1 - Prob. 2PCh. 5.1 - Prob. 3PCh. 5.1 - Prob. 4PCh. 5.1 - Prob. 5PCh. 5.2 - Prob. 1PCh. 5.2 - Prob. 2PCh. 5.2 - Prob. 3PCh. 5.2 - Prob. 4PCh. 5.2 - Prob. 5P
Ch. 5.2 - Prob. 6PCh. 5.2 - Prob. 7PCh. 5.2 - Prob. 8PCh. 5.3 - Prob. 1PCh. 5.3 - Prob. 3PCh. 5.3 - Prob. 4PCh. 5.3 - Prob. 5PCh. 5.3 - Prob. 6PCh. 5.3 - Prob. 7PCh. 5.3 - Prob. 9PCh. 5.3 - Prob. 10PCh. 5.3 - Prob. 11PCh. 5 - Prob. 1RPCh. 5 - Prob. 2RPCh. 5 - Prob. 3RPCh. 5 - Prob. 4RPCh. 5 - Prob. 6RPCh. 5 - Prob. 7RPCh. 5 - Prob. 8RPCh. 5 - Prob. 9RPCh. 5 - Prob. 10RPCh. 5 - Prob. 11RPCh. 5 - Prob. 12RPCh. 5 - Prob. 13RPCh. 5 - Prob. 14RPCh. 5 - Prob. 15RPCh. 5 - Prob. 16RP
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