In your new position as production manager, you need to plan production for two different products X and Y. You know that your firm earns a profit of 1.5 Euros on product X and 2 Euros on product Y.

To produce the different products you have to book two workers with different expertise from the workers pool - an electrician and a mechanic. Product X needs 1.2 days of electrician's time and 0.5 days of a mechanic's time. Product Y needs 1 day of an electrician's time and 1.6 days of a mechanic's time.

You were able to reserve 7.6 days of capacity from electricians and 6.5 days of mechanics.

Part 1: Linear Programming

Suppose you ignore that you can not produce partial products. Formulate a linear program in which the decision variables for the two products X and Y are continuous (notinteger) to answer the following questions.

Q1)In the solution of the linear program, how many units of product X maximizes profit?

Q2) In the solution of the linear program, how many units of product Y maximizes profit?

Q3) What is your optimal profit?

Looking at the results in Part 1, you notice that you will never produce partial products. You wonder if it is profit maximizing if you would simply round down the result in Part 1 and produce X=3 and Y=2. You recall the Sandbox from your SC0x course.

Put the numbers of this problem into the Sandbox and move the slider below the figure to gain intuition if it is optimal to round down. The optimal discrete solution will light up as a green dot.

Questions you could ask yourself while working in the Sandbox:

• Consider the result in Part 1 what integer values could you use to look for the optimum?
• How does discrete optimization change the number of potential solutions?
• Is there a chance that you can profit from only having discrete solutions available?
• How manyproductof X and Y should you produce if your profit for product X and product Y are both 2?Isthis solutionsunique?

Part 2: Discrete Optimization (cont.)

Q1)What is the profit maximizing number of product X if you only allow entire products to be produced.

Q2) What is the profit maximizing number of product Y if you only allow entire products to be produced.

Q3) Calculate the percentage profit drop compared to the result in Part 1: