# Applied Linear Programming: For the Socioeconomic and by Michael R. Greenberg

By Michael R. Greenberg

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**Extra resources for Applied Linear Programming: For the Socioeconomic and Environmental Sciences**

**Example text**

The lowest b/a ratio becomes the pivot element. In the sample problem, vector 1 will be inserted into the problem. BV v 3 K4 V5 b vl 10 12 15 1 2 3 _ For V3, b3/a5l = 10/1 = 10 For V4,bJaAl = 12/2 = 6 For K 5 ,/? 5 /a 51 = 15/3 = 5 The b/a ratio of the intersection of row vector V5 and column vector Vu element asl is 5; it is the smallest ratio. The mechanical procedure has a specific geometric translation. The lowest b/a ratio indicates the most impinging constraint in the column. While you should use the most impinging constraint, do not use a b/a ratio which is negative, or one for which the divisor or the numerator is 0.

POSTOPTIMAL ANALYSIS A linear programming model that has been carefully designed and tested for many months should represent a mine of information. An optimal solution is the most obvious output. But the optimal solution only scratches the surface of the information that can readily be mined. Three other packages of information may be developed: (1) the sensitivity of the optimal solution to changes in the activities, the constraints, and the matrix coefficients, (2) the optimal solution to the problem when one or more of the parameters (objective function weights, constraints, matrix coefficients) is allowed to systematically vary, and (3) spot tests of suboptimal solutions and specific parameters.

BV b Cj 1 3 4 5 7 10 12 15 4 0 0 0 -1 1 2 3 [1] Zj- Z = -4 Cj 3 4 5 6 7 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 1 2 2 3 1 1 - 1 0 -1 0 1 0 0 The initial value of the problem is —4. It can be improved because column vectors 1 and 2 have negative Zj — Cj values. The artificial vector 7 may be eliminated by introducing vectors 1 or 2 into the basis. We will pivot on αΊί. The pivot yields the following matrix: 7 BV b C: 1 3 4 5 1 6 4 3 4 0 0 0 0 0 0 0 1 — C Z =0 2 5 6 7 1 1 0 0 1 0 1 0 -2 0 0 1 1 0 0 0 1 2 3 -1 -1 -2 -3 1 0 1 0 0 3 4 0 0 0 Vector 7 has been eliminated and the first phase has been completed.