Consider the Leontief matrix 0.2 0.5 0.3 C0.4 0.2 0.2, 0.1 0.3 0.3 so that produ

Question

Consider the Leontief matrix 0.2 0.5 0.3 C0.4 0.2 0.2, 0.1 0.3 0.3 so that production r and demand d are related by (1-C)2-d Suppose that the total production is normalized to 1 unit, that is,1 We wish to find the production levels that maximize the total demand (which is d- di d2 ds), with the additional constraint that d2 has to be at least 0.15 units. (a) Formulate this as a linear program: write down the decision variables, objective function, and constraints (b) Find the optimal solution using Ercel.

Explanation / Answer

Answer:

(a) Linear Programming problem: (I-C)x ? 0 and  d2 ? 1.5 implies 0.4x1 +0.8x2 + 0.2x3 ? 1.5

Objective function: Maximise = d1 +d2 +d3 = 1.3x1 + 1.6x2 + 1.2x3

Constraints: di ? 0 , xi ? 0

(b) The optimal solution to this using simplex method is

STEP 1: Z = 1.3 X1 + 1.6 X2 + 1.2 X3 + 0 X4 + 0 X5 + 0 X6 + 0 X7

-0.8 X1 + 0.5 X2 + 0.3 X3 + 1 X4 = 0
-0.4 X1 + 0.8 X2 -0.2 X3 -1 X5 + 1 X7 = 1.5
0.1 X1 + 0.3 X2 -0.7 X3 + 1 X6 = 0

X1, X2, X3, X4, X5, X6, X7 ? 0

STEP 2: Make the simplex tableu

Perform Row operations to get

Thus the solution is unbounded. This same exercise can be performed in excel using LPP solver add-on.

-0.8 X1 + 0.5 X2 + 0.3 X3 + 1 X4 = 0
-0.4 X1 + 0.8 X2 -0.2 X3 -1 X5 + 1 X7 = 1.5
0.1 X1 + 0.3 X2 -0.7 X3 + 1 X6 = 0

X1, X2, X3, X4, X5, X6, X7 ? 0

STEP 2: Make the simplex tableu

Tableau 1 0 0 0 0 0 0 -1 Base Cb P0 P1 P2 P3 P4 P5 P6 P7 P4 0 0 -0.8 0.5 0.3 1 0 0 0 P7 -1 1.5 -0.4 0.8 -0.2 0 -1 0 1 P6 0 0 0.1 0.3 -0.7 0 0 1 0 Z -1.5 0.4 -0.8 0.2 0 1 0 0

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