29. A company is manufacturing two products, A and B. The manufacturing time required to make them, the profit, and capacity available at each work centre are as follows:

If x1 and x2 represent the number of units of products A and B, respectively, while S1, S2 and S3 represent the slack variables indicating the unused capacity in the three work centres, we can state the problem as follows:

The simplex algorithm, applied to this problem, Yields the following final tableau:

Required:

(a) Identify the values of all the variables and the objective function for the optimal solution to the primal problem.

(b) Formulate the dual for this problem.
(

c) Identify the values of all variables and the objective function for the solution to the dual problem.
(

d) Provide an economic interpretation of the dual variables.

(e) Suppose that the cost of overtime in each of the departments is Rs 15 per hour. Would it be advisable to work either of the departments on an overtime basis? What would be the maximum amount of overtime authorised, if any?
(t) How would your answer in

(e) change if the overtime costs were Rs 8 instead?
(g) Suppose that a price change is under consideration for the product A. This would raise the profit for this product to Rs 100 per unit. Would this change the optimal production plan? What is the maximum amount of change in profit for product A that would not cause a change in the optimum production plan?
(h) Suppose that there would be no change in the price of product A. How far the unit profit of B may vary without changing the optimal production plan?
(i) The company is planning to introduce a new product C with the following requirements per unit:

What profit would be necessary before the company considers the production of this new product?

Transcribed Image Text:

Product A B Total capacity Machining 1 hour Work centre Fabrication 5 hours Assembly Profit per unit (Rs) 3 hours 80 2 hours 4 hours 1 hour 100 720 hours 1,800 hours 900 hours