2 marksMathematicsConstrained optimisation and linear programmingLinear programming: formulation and graphical solutionModerateabout 2 min

IIT JAM 2022 Question 58: Constrained optimisation and linear programming

Maximize $Z = -0.2x_1 + x_2$ subject to $2x_1 + 5x_2 \le 70$ , $x_1 + x_2 \le 20$ , $x_1, x_2 \ge 0$ . If $x_1 = a$ and $x_2 = b$ is the optimal solution, then $a + b =$ _______ (in integer).

Answer and worked solution

Answer: 14 to 14

The objective rises with $x_2$ and falls with $x_1$ , so the optimum sets $x_1 = 0$ and pushes $x_2$ up. With $x_1 = 0$ , the tighter of $5x_2 \le 70$ (giving $x_2 \le 14$ ) and $x_2 \le 20$ is the first, so $x_2 = 14$ .

Checking the vertices $(0,0), (0,14), (10,10), (20,0)$ , the maximum $Z = 14$ occurs at $(0, 14)$ . Hence $a + b = 0 + 14 = 14$ .

Plain-text version

Maximize Z = -0.2x₁ + x₂ subject to 2x₁ + 5x₂ ≤ 70, x₁ + x₂ ≤ 20, x₁, x₂ ≥ 0. If x₁ = a and x₂ = b is the optimal solution, then a + b = _ (in integer).

Also written as: Maximize Z = -0.2x_1 + x_2 subject to 2x_1 + 5x_2 <= 70, x_1 + x_2 <= 20, x_1, x_2 >= 0. If x_1 = a and x_2 = b is the optimal solution, then a + b = _ (in integer).

Constrained optimisation and linear programming has come up 44 times in 7 exams since 2007.

What this question tests

This question is on Linear programming: formulation and graphical solution, part of Constrained optimisation and linear programming in Mathematics.

Setting up a linear program and finding its optimum at a corner of the feasible region.

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