Question

A company manufactures two products (A and B) and the profit per unit sold is Rs 3 and Rs 5 respectively. Each product must be assembled on a particular machine, each unit of product A takes 12 minutes of assembly time and each unit of product B takes 25 minutes of assembly time. The company estimates that the machine used for assembly has an effective working week of only 30 hours (due to maintenance/breakdown). Technological constraints mean that for every 5 units of product A produced at least 2 units of product B must be produced. Formulate the problem of how much of each product to produce as a linear program and find the points of intersection for maximization. The company has been offered the chance to hire an extra machine, thereby doubling the effective assembly time available. What is the maximum amount you would be prepared to pay (per week) for the hire of this machine and why?

Answer 1

The problem is formulated as a linear program to determine the optimal number of products A and B to produce, with constraints on assembly time and production ratios. The points of intersection for maximization are where the constraints intersect.

Step-by-step explanation:

To formulate the problem of how much of each product to produce as a linear program, let's denote x as the number of units of product A and y as the number of units of product B. The objective function to maximize is the profit, which is 3x + 5y. The machine has an effective working week of 30 hours, which is 1800 minutes. Therefore, the constraints are 12x + 25y ≤ 1800 (assembly time constraint) and y ≥ 0.4x (technological constraint). The intersection points that need to be evaluated are found by solving the system of equations representing the constraints. As for the maximum amount to pay for hiring an extra machine, it depends on the increased profit that can be achieved with the doubled assembly time, provided no other constraints are limiting factors.

Now, considering the additional information provided, firms can indeed adjust their production methods in response to labor costs, by shifting to more capital-intensive processes which might involve utilizing more machinery and less human labor, thus affecting productivity and labor demand. This can be applied to our scenario where the firm considers the cost/benefit of hiring an additional machine.