SIM Machinery is a local precision engineering manufacturer that specialised in manufacturing complex special process components for Oil & Gas and Aerospace Industries. SIM Machinery has 4 main product family: Gas Lift Valves, Dummy Valves, Orifice Valves and Safety Valve. All four products require performance test and reliability test before they can be shipped to customers. A total of 2,880 minutes of performance test time and 1,680 minutes of reliability testing time are available next week. SIM Machinery planner has provided the possible market demand (maximum and minimum sales) for each product type. Also, total gas lift valve produced should not exceed 30% of the total production for all four product family, while total production for Dummy and Safety valve to should account for at least 25% of the total production. Table 1: Profile of Valve Type Gas Lift Valve Dummy Valve Orifice Valve Safety Valve Performance Test Time (min) 45 20 30 25 Reliability Test Time (min) 20 15 25 10 Profit ($)/unit $370.00 $175.00 $290.00 $300.00 Maximum sales 18 45 40 30 Minimum sales 7 16 20 15 (a) Develop a linear programming problem to determine how many pieces of each product family the company should make next week to maximize the company's profit. (