PROBLEM 4: Integer Programming and Its Business Applications

Recall Rolled-Rice from Problem 1 of Case Study #2. The company wants to minimise the total cost of meeting the customers’ demand. The total cost consists of (i) transportation costs, (ii) fixed costs for hangars A-C, (iii) renovation costs for hangars D-E, and (iv) the extension cost for hangar A. Given the following decision variables

 Xij = Number of generators shipped from hangar i to customer j

 Z = Number of places added to hangar A

the integer linear programming (ILP) problem is formulated as follows:

Minimise 300 XA1 + 700 XA2 + … + 900 XE3 + 1000 * (25 YA +50 YB + 45 YC + 30 YD + 25 YE) + 1000 * Z

s.t. XA1 + XA2 + XA3 ≤ 30 YA + Z

XB1 + XB2 + XB3 ≤ 10 YB

XC1 + XC2 + XC3 ≤ 50 YC

XD1 + XD2 + XD3 ≤ 20 YD

XE1 + XE2 + XE3 ≤ 30 YE

Z ≤ 20 YA

XA1 + XB1 + … + XE1 = 20

XA2 + XB2 + …+ XE2 = 60

XA3 + XB3 + … + XE3 = 40

Xij non-negative integers, i=A,B,C,D,E, j=1,2,3

Z non-negative integer

Yi ∈ {0,1}, i=A,B,C,D,E

Based on the given parameters, it is optimal to use hangars A, C, and E and to expand hangar A by 10 units. The minimised cost is €145,000. All three hangars are used to their full capacities.

Question 1:

If hangar A is used, then exactly one of hangars B or D must be used. Use binary variables to modify the ILP formulation above.

Question 2:

Implement the resulting model in Excel and provide the new solution along with the minimised cost.