3. The Heartland Distribution Company is a food warehouse and distributor that h

Question

3. The Heartland Distribution Company is a food warehouse and distributor that has a contract with a grocery store chain in several Midwest and Southeast cities. The company wants to construct new warehouses/distribution centers in some of the cities it services to serve the stores in those cities plus all the other stores in the other cities that don’t have distribution centers. A distribution center can effectively service all stores within a 300-mile radius. The company also wants to limit its fixed annual costs to under $1,200,000. The company wants to build the minimum number of distribution centers possible. The following table shows the cities within 300 miles of every city and the projected fixed annual charge for a distribution center in each city.

(a) Formulate an integer programming model for this problem.
(b) Find the optimal objective value and the optimal solution point by using the computer.
(c) What effect will be if the cost constraint is removed from the original model?

Citv 1. Atlanta 2. Charlotte 3. Cincinnati 4. Cleveland 5. Indianapolis 6. Louisville 7. Nashville 8. Pittsburgh 9. Richmond 10. St. Louis Annual fixed charge (S1ks) 280 260 400 410 290 370 270 330 390 300 Cities within 300 miles 1. 2.7 1. 2.9 3, 4, 5, 6, 7. 10 3, 5, 6,7. 10 1. 3. 5, 6. 7, 10 3, 4, 8,9 2, 8,9 5, 6. 7. 10

Explanation / Answer

Solution :This problem requires a 0–1 integer programming model in which the decision variables are the available cities:
Decision Variable:
Let xi = city i selected for locating distribution center, i = 1,2,…10
Where,
xi = 1, if city i is selected as a hub and
xi = 0, if city i is not selected
Objective function:
The objective of the company is to minimize the number of distribution centers while catering to required cities. The objective function (Z) is written as follows:
Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10
Subject To:
The model constraints establish the set covering requirement (i.e., that each city be within
300 miles of a hub). For example, Atlanta covers itself, Charlotte, and Nashville:
City No.
City
Constraint Equation
1
Atlanta
x1 + x2 + x7 >= 1
2
Charlotte
x1 + x2 + >= 1
3
Cincinnati
x3 + x4 + x5 + x6 + x7 + x8 >= 1
4
Cleveland
x3 + x4 + x5 + x8 >= 1
5
Indianapolis
x3 + x4 + x5 + x6 + x7 + x10 >= 1
6
Louisville
x3 + x5 + x6 + x7 + x10 >= 1
7
Nashville
x1 + x3 + x5 + x6 + x7 + x10 >= 1
8
Pittsburgh
x3 + x4 + x8 + x9 >= 1
9
Richmond
x2 + x8 + x9 >= 1
10
St. Louis
x5 + x6 + x7 + x10 >= 1
Binary integer Constraint
xi = 0 or 1, for all i
Spreadsheet Model and Solver solution:


Formula
Optimal Solution:
Optimal Number of Hubs = 3
Hubs location: Atlanta, Nashville, and Pittsburgh
Total Cost of Locating hubs = $276,000 + $268,000 + $323,000 = $867,000

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