Question

There are 6 cities (1-6) in Araxi County. The county must determine where to build fire stations. The county wants to build the minimum number of fire stations needed to ensure that at least one fire station is within 15 minutes (driving time) for each city. The times required to drive between the cities in Araxi County are shown in the following table. a. Define the decision variables to formulate an Integer Programming model that will tell Araxi County Fire Department how many fire stations should be built and where they should be located. b. Complete the table below. c. Using the table formulate an integer programming model for the problem. d. Solve the model in Excel using Solver. State the solution.

Answer 1

Answer:

a.

Look at the two attached figures please. The decision variables is the choice to build a fire station in which city.

So there are six decision variables each taking a value of either 0 or 1.

D1 = decision to build fire station in city 1, (1 denotes yes and 0 denotes no)

D2 = decision to build fire station in city 2, (1 denotes yes and 0 denotes no)

D3 = decision to build fire station in city 3, (1 denotes yes and 0 denotes no)

D4 = decision to build fire station in city 4, (1 denotes yes and 0 denotes no)

D5 = decision to build fire station in city 5, (1 denotes yes and 0 denotes no)

D6 = decision to build fire station in city 6, (1 denotes yes and 0 denotes no)

b.

Please look at attachment

C .

The figure 1 denotes the problem formulation in excel and shows all the formulas.

In first step, a matrix is prepared to denote all cities which are within 15 minutes by 1 and others by zero

A array of decision variables for selecting the city is shown. A 1 in cell implies that city is selected and 0 means the city is not selected.

A product of the selection variable and the value in the within 15 minutes matrix is calculated as shown by the formula.

The constraint is to ensure that all cities have at least one fire station mapped within 15 minutes.

d.

The problem is solved in Excel using integer programming and the solution is that City 2 and 4 should be selected for building fire stations and thus total no of fire stations are two with every city within 15 minutes.

Explanation: