OPERATIONS RESEARCH >INTERNATIONAL EDITI
OPERATIONS RESEARCH >INTERNATIONAL EDITI
4th Edition
ISBN: 9780534423629
Author: WINSTON
Publisher: CENGAGE L
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Chapter 4, Problem 3RP

Explanation of Solution

Optimal solution:

Consider the following linear programing problem:

min z= 5x1x2

Subject to the constraints:

  2x1+x2= 6  x1+x24  x1,x2 0

Use Max z = -Min(-z) to reduce the given problem to a minimization problem as: -(min z'= -5x1+x2)

Subject to the constraints:

2x1+x2= 6  x1+x24  x1+2x24 x1,x2 0

Add slack variables s1,s2 and artificial variable a1 to get:

-(min w′ = -5x1+x2+a1)

Subject to the constraints:

2x1+x2+a1= 6x1+x2+s1= 4x1+2nx2+s2=5x1,x2,s1,s2,a10

Two Phase Method:

Phase I linear programming problem is,

  min w'= a1

Subject to the constraints:

2x1+x2+a1= 6  x1+x2+s1= 4  x1+2x2+s2=5  x1,x2,s1,s2,a10

The initial simplex table is given below:

 w′x1x2a1s1s2rhsbasic variable
R0100-1000w′=0
R10211006a1=6
R20110104s1=4
R30120015s2=5
  • Since, the basic variable a1 value in R0 is non-zero, therefore, do the transformations

R0R0+R1  to get:

 w′x1x2a1s1s2rhsbasic variable
R0100-1000w′=0
R10211006a1=6
R20110104s1=4
R30120015s2=5

Since the highest positive entry 2 in R0 corresponds to x1, x1 enters the basis.

 w′x1x2a1s1s2rhsratio
R01210006-
R102110063*
R201101044
R312000155

Apply the simplex method further:

 w′x1x2a1s1s2rhsbasic variable
R0100-1000w′=0
R1011212006x1 = 3
R2001212101s1=1
R3003212012s2=2
  • Optimally reached for phase 1. Proceed to phase 2 with the actual objective function

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Chapter 4 Solutions

OPERATIONS RESEARCH >INTERNATIONAL EDITI

Ch. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 3PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.7 - Prob. 1PCh. 4.7 - Prob. 2PCh. 4.7 - Prob. 3PCh. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - Prob. 7PCh. 4.7 - Prob. 8PCh. 4.7 - Prob. 9PCh. 4.8 - Prob. 1PCh. 4.8 - Prob. 2PCh. 4.8 - Prob. 3PCh. 4.8 - Prob. 4PCh. 4.8 - Prob. 5PCh. 4.8 - Prob. 6PCh. 4.10 - Prob. 1PCh. 4.10 - Prob. 2PCh. 4.10 - Prob. 3PCh. 4.10 - Prob. 4PCh. 4.10 - Prob. 5PCh. 4.11 - Prob. 1PCh. 4.11 - Prob. 2PCh. 4.11 - Prob. 3PCh. 4.11 - Prob. 4PCh. 4.11 - Prob. 5PCh. 4.11 - Prob. 6PCh. 4.12 - Prob. 1PCh. 4.12 - Prob. 2PCh. 4.12 - Prob. 3PCh. 4.12 - Prob. 4PCh. 4.12 - Prob. 5PCh. 4.12 - Prob. 6PCh. 4.13 - Prob. 2PCh. 4.14 - Prob. 1PCh. 4.14 - Prob. 2PCh. 4.14 - Prob. 3PCh. 4.14 - Prob. 4PCh. 4.14 - Prob. 5PCh. 4.14 - Prob. 6PCh. 4.14 - Prob. 7PCh. 4.16 - Prob. 1PCh. 4.16 - Prob. 2PCh. 4.16 - Prob. 3PCh. 4.16 - Prob. 5PCh. 4.16 - Prob. 7PCh. 4.16 - Prob. 8PCh. 4.16 - Prob. 9PCh. 4.16 - Prob. 10PCh. 4.16 - Prob. 11PCh. 4.16 - Prob. 12PCh. 4.16 - Prob. 13PCh. 4.16 - Prob. 14PCh. 4.17 - Prob. 1PCh. 4.17 - Prob. 2PCh. 4.17 - Prob. 3PCh. 4.17 - Prob. 4PCh. 4.17 - Prob. 5PCh. 4.17 - Prob. 7PCh. 4.17 - Prob. 8PCh. 4 - Prob. 1RPCh. 4 - Prob. 2RPCh. 4 - Prob. 3RPCh. 4 - Prob. 4RPCh. 4 - Prob. 5RPCh. 4 - Prob. 6RPCh. 4 - Prob. 7RPCh. 4 - Prob. 8RPCh. 4 - Prob. 9RPCh. 4 - Prob. 10RPCh. 4 - Prob. 12RPCh. 4 - Prob. 13RPCh. 4 - Prob. 14RPCh. 4 - Prob. 16RPCh. 4 - Prob. 17RPCh. 4 - Prob. 18RPCh. 4 - Prob. 19RPCh. 4 - Prob. 20RPCh. 4 - Prob. 21RPCh. 4 - Prob. 22RPCh. 4 - Prob. 23RPCh. 4 - Prob. 24RPCh. 4 - Prob. 26RPCh. 4 - Prob. 27RPCh. 4 - Prob. 28RP
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