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Exercise 1:

Your company produces three types of products in a two-stage production process (two departments, each department represents one stage). The following optimization model has been formulated in order to determine the maximum profit margin production program. xi denotes the volume of product i to be produced.

(1)

s.t.: Maximize 3 x1 + 5 x2 + 6 x3

(2)
10 x1 + 12 x2 + 15 x3 ≤ 22,000
(capacity of department 1)
(3) 13 x1 + 11 x2 + 17 x3 ≤ 26,000 (capacity of department 2)

(4)
x1 ≤ 1,000
(maximum sales volume of product 1)
(5) x2 ≤ 1,500 (maximum sales volume of product 2)
(6) x3 ≤ 800 (maximum sales volume of product 3)

(7)
x1, x2, x3 ≥ 0
(non-negativity constraints)


Refer to the formula numbers in the model above when describing your amendments.
Be careful again to formulate linear or mixed integer / binary models. Avoid other non- linearities. Define all additional symbols used. No solution is required, just the model amendments!

a) Amend the model for the case that contract manufacture (outside the company) is available for department 2. The capacity can be increased unlimited. For each capacity unit used there is a cost of 2 € incurred.
b) Amend the model for the case that a staff member of a temporary employment company could increase the capacity of department 1 by a maximum of 1,500 capacity units. For this staff member a fixed cost of 500 € and a piece-rate of 0.4
€/capacity unit is incurred.
c) Amend the model for the case that market conditions changed such that sale of products 2 and 3 is mutually exclusive.

Exercise 2:

Formulate a mathematical model for the “Third Example” within the paper “Caution: Common Sense Planning Methods Can Be Hazardous to Your Corporate Health”. No solution is required.
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