Question

A company is producing two products, product A and product B. It takes 2 hours too produce one unit of product A and 1 hour to produce 1 unit of product B. To produce one unit of product B it costs 10 rupees. The total budget available is 100 rupees. It is required that the company produce at least 10 units of product A and product B taken together. However, the company can not produce more than 5 units of product B. It is required to find how many units of A and B should be produced so that the total production time is minimized. Formulate the above problem as a linear programming problem and solve it by the graphical method.

Answer 1

Answer on Question #85669 – Math – Linear Algebra

Question

A company is producing two products, product A and product B. It takes 2 hours to produce one unit of product A and 1 hour to produce 1 unit of product B. To produce one unit of product B it costs 10 rupees. The total budget available is 100 rupees. It is required that the company produces at least 10 units of product A and product B taken together. However, the company cannot produce more than 5 units of product B. It is required to find how many units of A and B should be produced so that the total production time is minimized. Formulate the above problem as a linear programming problem and solve it by the graphical method.

Solution

Let $x$ be the number of units of A should be produced, $y$ be the number of units of B should be produced.

Then the constraints are:

$$\begin{array}{l} 10y \leq 100 \\ x + y \geq 10 \\ y \leq 5 \\ x \geq 0, y \geq 0 \\ \end{array}$$

The objective is: minimize the total production time $2x + y$.

$$\begin{array}{l} 2x + y = x + y + x \geq 10 + x \\ y \leq 5 \\ y + x \geq 10 \Rightarrow x \geq 5 \\ \end{array}$$

A company has to produce 5 units of product A and 5 units of product B so that the total production time is minimized.

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