# Manufacturing to maximize profit

The ABC corporation makes two products with brand names “Swift” and “Impala”. The netcontribution from each product is Rs. 1 and Rs. 2, respectively, measured in lakhs. There are three workshops that are used to make these products. The first one called Tuning (T) has 50 hours a week available. Each Swift requires 1 hour and each Impala 3 hours in the Tuning workshop. The second workshop is called Finisher (F) and has 40 hours available. Each product requires one hour in this workshop. The third workshop is Assembly (A) and has 120 hours available. Swift requires 3 hours and Impala 2 hours in the Assembly workshop.

(a) Formulate the above problem as a linear program to determine the number of Swift and Impala products to be manufactured to maximize profit. (3 points)

b) Solve this problem graphically. Show the optimal variable values as well as the optimum objective function value on the graph. (3 points)

?(c) Write the dual to your LP formulation from part (a). Using the dual formulation, determine the shadow prices of the constraints in the primal problem? (6 points)

(d) What change do you expect if the T and A workshop hours are simultaneously increased by 10 hours each? (1 point)

(e) Notice that the net profit of the Swift model is half of that of the Impala model. If the profit for Swift is further reduced by Rs. 20,000 (or Rs. 0.2 lakhs), will the company make more Impalas and less Swifts? (2 points)

(f) A new product called Gazelle is being considered for production. It yields a much higher net profit of Rs. 3 lakhs per unit, while requiring 3,4, and 5 hours in the Tuning, Finishing and Assembly workshops, respectively. Should the ABC corporation take up production of the Gazelle model? (1 point)

(g) Marketing says we need to produce at least as many Impalas as Swifts. What constraint needs to be added and what is the new optimal solution? (3 points)

(h) The shadow price of the Tuning constraint has changed to 0.75 in the new model. Why has it changed? (1 point