Question 1 (Compulsory) (25 marks)

A company makes three types of precast concrete products: A, B, and C. Each product A is anticipated to generate 14,000 dollars of profit, each product B is anticipated to generate 14,500 dollars of profit, and each product C is anticipated to generate 15,000 dollars of profit. The company tries to optimize its production plan during a production period to maximize the profit. However, the company is subject to the following constraints.

Each product A needs 10 square meters to produce, each product B needs 20 square meters to produce, and each product C needs 25 square meters to produce. The total production area is 800 square meters.

On average, each product A needs to employ 1 worker, each product B needs to employ 1 worker, and 2 product C need to employ 3 workers. A total of 30 workers are available.

The company has signed a long-term supply contract with a construction company. According to the contract, the company needs to supply a minimal of 6 Product C to the contractor.

All the other resources are assumed to be unlimited.

You are asked to:
(a) Formulate the problem using linear programming method. (4 marks)
(b) Using the simplex method, determine the quantities of the three products to be produced in order to maximize the profit (A total of 12 marks; 3 marks will be deducted from each mistake until zero)
(c) Identify the ranges of optimality for the objective function coefficients of product A and B. (6 marks)
(d) Identify the shadow price for the number of workers and explain the meaning of this shadow price. (3 marks)
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