Question

please show all steps clearly

1. A company manufactures three types of items: A, B, and C. Each type of item produced incurs a variable cost and a raw material cost. Table 1 gives associated costs together with the selling price of each product. There are two types of resources (labor) used for producing each item, namely machinery time and craftsman time (Table 2). Each week, the company can obtain all the needed raw material but only 140 craftsman hours and 100 machinery hours. The company has a specific contract to produce 12 items of B for a specific customer. Item C has a weekly demand of 10 while items A and B have unlimited weekly demands. The company wants to maximize weekly profit (revenues · costs). Formulate a Linear Programming (LP) model to maximize the company's weekly profit. (Use decision variables xi = number of items i produced each week, where i = A, B, C in your model formulation.) Table 1: Costs and Selling price($) A B с Variable cost 5 8 10 Raw material cost 9 12 18 Selling price 21 30 42 Table 2: labor times (hr) A B с Machinery time 1 1.5 2 Craftsman time 1.5 2 2.5

Answer 1

Let x1, x2 and x3 be no of items produced of A,B and C respectively

Profit per unit for A = Selling price – RM cost – Variable cost = 21 – 9 – 5 = 7

Profit per unit for B = 30-12-8 = 10

Profit per unit for C = 42-18-10 = 14

So total profit z = 7x1 + 10x2 + 14x3

Further, machinery time required by A, B and C are 1, 1.5 and 2 hrs respectively

So x1 + 1.5x2 + 2x3 <= 100 (since 100 machinery hours are available)

Also, craftsmen time required by A, B and C are 1.5, 2 and 2.5 hrs respectively

So 1.5x1 + 2x2 + 2.5x3 <= 140 (since 140 craftsman hours are available)

Further, company has specific contract to produce 12 items of B and B also has unlimited demand

i.e. x2 >= 12

Item C has a weekly demand of 10

i.e x3 <= 10

Item A has unlimited weekly demand

i.e. x1 >=0

Also, x1, x2, x3 >= 0

Hence the problem is formulated below:

Maximize 7x1 + 10x2 + 14x3

Subject to

x1 + 1.5x2 + 2x3 <= 100

1.5x1 + 2x2 + 2.5x3 <= 140

x2 >= 12

x3 <= 10

x1, x2, x3 >= 0