Question

Your express package delivery company is drawing up new zones for the location of drop boxes for customers. The city has been divided into six zones. You have targeted seven possible locations for drop boxes. The list of which zones the drop boxes could serve is shown below

Possible Drop Box Location	Can serve Zones
Location 1	2, 5, 6
Location 2	1, 3, 6
Location 3	2, 4, 5
Location 4	1, 2, 6
Location 5	3, 4, 5
Location 6	1, 3, 5
Location 7	2, 3, 4

Develop a model to find out the locations to use that provides the smallest number of locations yet make sure that each zone is covered by at least two boxes. Define variables if needed

Answer 1

Linear Programming Model:

Decision Variables:

Xj = Location i,

Value of Xj = 1 or 0 (Yes or No)

Objective Function:

Constrain:

		Location
		1	2	3	4	5	6	7
Zone	1		1		1		1
2	1		1	1			1
3		1			1	1	1
4			1		1		1
5	1		1		1	1
6	1	1		1

Every Zone should have at least two location boxes

Let's Define the combination of Zone and Location as Yij as binary variable.

For example Y11 = 0 it means location 1 box can't serve zone 1

While Y12 = 0 it means Location 2 box can serve Zone 1

We can formulate this in excel as below:

Solution:

No. of Location = 4

Drop Box Location = 2,3,4,5

		Location
		1	2	3	4	5	6	7
	Decision Variable	0	1	1	1	1	0	0
Zone	1		1		1		1		2	>=	2
2	1		1	1			1	2	>=	2
3		1			1	1	1	2	>=	2
4			1		1		1	2	>=	2
5	1		1		1	1		2	>=	2
6	1	1		1				2	>=	2
	No. of location	4

Excel Formula:

Solver Equation: