Question

Nada Dairy is contracted to receive daily 60,000 kg of ripe tomatoes at 7 cents per kg from which it produces canned tomato juice, tomato sauce, and tomato paste. The canned products are packaged in 24-can cases. A can of juice uses 1 kg of fresh tomatoes, a can of sauce uses 1/2 kg, and a can of paste uses 3/4 kg. The company’s daily share of the market is limited to 2000 cases of juice, 5000 cases of sauce, and 6000 cases of paste. The wholesale prices per case of juice, sauce, and paste are SAR 21, SAR 9, and SAR 12, respectively.

a. [2 Marks] Develop a mathematical model that determine the optimum daily production program for the company.

b. [1 Mark] Solve it using software and show the results. c. [2 Mark] If the price per case for juice and paste remains fixed as given in the problem, use sensitivity analysis to determine the unit price range that the company should charge for a case of sauce to keep the optimum product mix unchanged.

Answer 1

(a)

Let 'J', 'S', and 'P' are the cases of Juice, Sauce, and Paste to be produced respectively.

Revenue = 21 * J + 9 * S + 12 * P
Cost = {1 * J + (1/2) * S + (3/4) * P} * 24 * 0.07

Max Z = Total profit = Revenue - Cost = (21 - 24*0.07)*J + (9 - 12*0.07)*S + (12 - 18*0.07)*P

or, Max Z = 19.32*J + 8.16*S + 10.74*P

Subject to,

1 * 24 * J + (1/2) * 24 * S + (3/4) * 24* P <= 60000 or, 24*J + 12*S + 18*P <= 60000

J <= 2000

S <= 5000

P <= 6000

J, S, P >= 0

(b)

Excel model

Solver inputs:

Solution (Answer report)

(c)

Sensitivity report:

The optimality range of profit is [-1, +1.5], the same will hold for selling price because we are not changing the cost of tomatoes.

So, the required range is [9-1, 9+1.5] i.e. [8, 10.5]