PLEASE start by writing down the objective function, decision variables and constraints clearly on paper .

A plant in a developing country can manufacture four different products in any combination. Each product requires time on each of three machines, as shown in the table below. All figures are in minutes per pound of product. Each machine is available 120 hours per week. Products A,B,C, and D are purely competitive, and any amounts made may be sold at per pound prices of $50,$43,$28, and $28, respectively. Variable labour costs are $30 per hour for machines 1 and 2 , and $60 per hour for machine 3. Material costs are $5 for each pound of products A,B,C and D. 1. Formulate an LP model to help maximize total profit. 2. Find the optimal solution. 3. How many hours are spent working on machine 3 ? 4. What are the units of the shadow prices on the constraints that control machine capacity? 5. What should the selling price of one pound of product D be to get a solution where D is produced? 6. Suppose that any additional time on machine 2 beyond its current capacity must be done on an overtime basis at an additional cost of 50% the regular hourly cost. Workers are willing to work any fraction of time during overtime. Is it economically advantageous to use overtime on Machine 2? Explain your answer and if necessary give the maximum amount of overtime to be used.|