Answer to Question #310461 in Operations Research for abc

Question #310461

Maximise 1170x1 + 1110x2



Subject to: 9x1 + 5x2 ≥ 500



7x1 + 9x2 ≥ 300



5x1 + 3x2 ≤ 1500



7x1 + 9x2 ≤ 1900



2x1 + 4x2 ≤ 1000



x1, x2 ≥ 0



-Find graphically the feasible region and the optimal solution.


1
Expert's answer
2022-03-15T13:15:14-0400

Given,

"\\text{Maximise}~ 1170x_{1} + 1110x_{2}\\\\\n\\text{subject to:}\\\\\n\\begin{aligned}\n9x_1 + 5x_2 &\u2265 500\\\\\n7x_1 + 9x_2 &\u2265 300\\\\\n5x_1 + 3x_2 &\u2264 1500\\\\\n7x_1 + 9x_2 &\u2264 1900\\\\\n2x_1 + 4x_2 &\u2264 1000\\\\\n x_1, x_2 &\u2265 0\n\\end{aligned}"


To solve it graphically, we consider the constraints as equations and draw straight lines.

"\\begin{aligned}\n9x_1 + 5x_2 &= 500~~\\qquad(1)\\\\\n7x_1 + 9x_2 &= 300~~\\qquad(2)\\\\\n5x_1 + 3x_2 &= 1500\\qquad(3)\\\\\n7x_1 + 9x_2 &= 1900\\qquad(4)\\\\\n2x_1 + 4x_2 &= 1000\\qquad(5)\\\\\n\\end{aligned}"

The graph plotted is shown in the following figure.





The region of feasibility is bounded by the extreme points ABCD. The values of the objective function at the extreme points are given in the following table:


"\\begin{array}{|c|c|}\n\\hline\n\\text{Extreme points}& \\text{Value of ~} z = 1170x_1+1110x_2\\\\\n\\hline\nA(0,100) & 111000\\\\\n&\\\\\nB(500\/9,0) & 65000\\\\\n&\\\\\nC(1900\/7,0) & \\dfrac{2223000}{7}\\\\\n&\\\\\nD(0,1900\/9) & \\dfrac{703000}{3}\\\\\n&\\\\\n\\hline\n\\end{array}"


Hence the maximum values occurs at C(1900/7,0) and the maximum value is "z = \\dfrac{2223000}{7}".


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