Question

A major retail firm is considering a major change in its distribution system. It is considering relocating its distribution centers such that all retail stores can be replenished within 24 hours of a request for merchandise.

The table below is a matrix of cover coefficients for 10 stores and 10 potential sites for distribution centers. If store i can be replenished within 24 hours from site j, then a_ij=1; otherwise a_ij=0. The cost of installation of the distributions centers vary from zone to zone and are given in the DC cost row. The firm wants to find the optimal location of the distribution centers.  

Store i	Distribution Center Site
	1	2	3	4	5	6	7	8	9	10
1	1	1	0	0	0	0	0	1	1	1
2	1	1	1	0	0	0	0	0	0	0
3	1	1	0	1	1	0	0	1	1	0
4	0	0	1	0	1	1	1	0	1	1
5	0	0	1	1	0	1	1	1	1	0
6	0	0	0	1	1	1	1	0	0	1
7	0	1	0	1	0	1	0	1	1	0
8	1	1	1	0	1	0	0	1	1	1
9	1	0	1	1	1	0	1	0	0	1
10	0	0	1	0	0	1	1	1	1	1
DC COST	193	721	822	156	607	593	246	385	309	867

1-Provide a mathematical formulation of above model

2-Provide a mathematical formulation given a budget restriction of 300.

3-Provide a mathematical formulation given a coverage target more than 85% (firms wants to cover at least 85% of the stores)

3-Solve the 3 models using Lindo/Excel Solver or any optimization software you know

Answer 1

Solution,

1) Let x_i is the binary integer in such a way that x_j = 1 when the site (j = 1,2,...........,10) is selected and x_j = 0 otherwise.

Minimize Z = 193x₁ + 721x₂ + 822x₃ + 156x₄ + 607x₅ + 593x₆ + 246x₇ + 385x₈ + 309x₉ + 867x₁₀

Subject to : for i = 1, 2, ......................, 10.

x_ij = {0,1}

From above table Result is:

	DC
Store	1	2	3	4	5	6	7	8	9	10	Total
1	1	1	0	0	0	0	0	1	1	1	1
2	1	1	1	0	0	0	0	0	0	0	1
3	1	1	0	1	1	0	0	1	1	0	2
4	0	0	1	0	1	1	1	0	1	1	1
5	0	0	1	1	0	1	1	1	1	0	2
6	0	0	1	1	1	1	0	0	0	1	1
7	0	1	0	1	0	1	0	1	1	0	1
8	1	1	1	0	1	0	0	1	1	1	1
9	1	0	1	1	1	0	1	0	0	1	3
10	0	0	1	0	0	1	1	1	1	1	1
Cost	193	721	822	156	607	593	246	385	309	867	595
x_j	1	0	0	1	0	0	1	0	0	0

(2)

Let y_i be the binary integer such that y_i when store - i (i = 1,2,...........,10) is covered and y_i=0 otherwise

Maximize

Subject to : 193x₁ + 721x₂ + 822x₃ + 156x₄ + 607x₅ + 593x₆ + 246x₇ + 385x₈ + 309x₉ + 867x₁₀ 300

x_ij = {0,1}

y_i = {0,1}

Hence the result from table is :

DC
Store	1	2	3	4	5	6	7	8	9	10	Total	y_i	y_i - 10*Total
1	1	1	0	0	0	0	0	1	1	1	1	1	-9
2	1	1	1	0	0	0	0	0	0	0	1	1	-9
3	1	1	0	1	1	0	0	1	1	0	1	1	-9
4	0	0	1	0	1	1	1	0	1	1	0	0	0
5	0	0	1	1	0	1	1	1	1	0	0	0	0
6	0	0	1	1	1	1	0	0	0	1	0	0	0
7	0	1	0	1	0	1	0	1	1	0	0	0	0
8	1	1	1	0	1	0	0	1	1	1	1	1	-9
9	1	0	1	1	1	0	1	0	0	1	1	1	-9
10	0	0	1	0	0	1	1	1	1	1	0	0	0
Cost	193	721	822	156	607	593	246	385	309	867	193	5
x_j	1	0	0	0	0	0	0	0	0	0

(3)

Minimize Z = 193x₁ + 721x₂ + 822x₃ + 156x₄ + 607x₅ + 593x₆ + 246x₇ + 385x₈ + 309x₉ + 867x₁₀

Subject to:

x_ij = {0,1}

y_i = {0,1}

Result

	DC
Store	1	2	3	4	5	6	7	8	9	10	Total	y_i	y_i - 10*Total
1	1	1	0	0	0	0	0	1	1	1	1	1	-9
2	1	1	1	0	0	0	0	0	0	0	0	0	0
3	1	1	0	1	1	0	0	1	1	0	2	1	-19
4	0	0	1	0	1	1	1	0	1	1	1	1	-9
5	0	0	1	1	0	1	1	1	1	0	2	1	-19
6	0	0	1	1	1	1	0	0	0	1	1	1	-9
7	0	1	0	1	0	1	0	1	1	0	2	1	-19
8	1	1	1	0	1	0	0	1	1	1	1	1	-9
9	1	0	1	1	1	0	1	0	0	1	1	1	-9
10	0	0	1	0	0	1	1	1	1	1	1	1	-9
Cost	193	721	822	156	607	593	246	385	309	867	465	9
x_j	0	0	0	1	0	0	0	0	1	0