Problem 2. Linear Programming (50 points) A company is involved in the productio

Question

Problem 2. Linear Programming (50 points) A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic prooessing and oraftsman ime for hand finishing. The table below gives the number of minutes required for each item: Machine time Craftsman time 20 29 tem X 13 tem Y 19 The company has 40 hours of machine time available in the next working week but only 35 hours of craftsman time. Machine time costs £10 per hour worked and craftsman time costs £2 per hour worked. Both machine and craftsman idie times incur no costs The revenue received for each item produced (all production is sold) is £20 for X and £30 for Y.The company has a specific contract to produce 10 items of X per week for a partioutar customer Using the excel file provided here 1. Define the objective function for the firm and the decision variables (10 points) 2. Define the constraints for the problem (20 points) 3. Solve the problem using solver, which is the optimal amount of X and Y that maximize the profit and how much are the profits? (20 points)

Explanation / Answer

13x + 19y <= 2400

20x+29y<=2100

Cost of X = 13*10 +20*2 /60= 170/60

Cost of Y = 19*10+29*2 = 248/60

Maximize : (20-170/60)x + (30-248/60) y such that

13x + 19y <= 2400

20x+29y<=2100

x>=10

This equation can be solved as

x = 10,y = 1900/29

Profit=1865

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