A company which manufactures canoes employs 120 employees, each of whom working 30 hours per week. Half of them work in the carpenter department, 20 persons in the plastics department, and the rest of them at the completion department. The company manufactures the simple canoes with net unit profit € 7 and the luxury canoes with corresponding profit € 10. A simple canoe requires 4.5 hours in the carpenter department and two hours in each of the other two departments. The working hours for each luxury canoe are 5, 1 and 4 at the carpenter department, plastics department and completion department respectively. Marketing calculations have shown that not less than 1/3 and not more than 2/3 of the total number of the canoes should be luxurious. How will the company maximize its overall net profit?

Answer:

Available hours in carpentry = 60 employees x 30 hours per employee = 1800 hours
Available hours in plastics = 20 employees x 30 hours per employee = 600 hours
Available hours in completion = 40 employees x 30 hours per employee = 1200 hours

Formulation

Maximize Z = total profit = 7*Simple + 10*Luxury
Subject to,
4.5*Simple + 5.0*Luxury <= 1800 (carpentry hrs. limit)
2.0*Simple + 1.0*Luxury <= 600 (plastics hrs. limit)
2.0*Simple + 4.0*Luxury <= 1200 (completion hrs. limit)

Luxury >= 1/3*(Simple + Luxury) or, -(2/3)*Luxury + (1/3)*Simple <= 0 (min. of Luxury)
Luxury <= 2/3*(Simple + Luxury) or, (1/3)*Luxury - (2/3)*Simple <= 0 (max. of Luxury)

Luxury, Simple >= 0

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