Question

The district manager of basketball officials has to assign each team of officials to one game on Friday night.

There are four sites for games. So each site needs one group of officials. Below are the distances in miles each

group of officials is from each site.

	Game site
Officials	Wayne	Sioux City	Winnebago	Wakefield
Group A	210	90	180	160
Group B	100	70	130	200
Group C	175	105	140	170
Group D	80	65	105	120

Formulate the linear programming problem to minimize distance traveled by the groups of officials in this

assignment problem.

A-Which Group is assigned to each location?

B-what would happen to the distance traveled if instead of the best assignment we had Group A go to

Wayne, Group B to South Sioux City, Group C to Winnebago, and Group D to Wakefield?

Answer 1

Let the assignment of Group A assigned to Wayne be x11, Group B assigned to Wayne be x21 and so on. We get the decision variables as per the table shown below:

Hence, we get the total distance traveled.

Total Distance = 210*x11 + 90*x12 + 180*x13 + 160*x14 + 100*x21 + 70*x22 + 130*x23 + 200*x24 + 175*x31 + 105*x32 + 140*x33 + 170*x34 + 80*x41 + 65*x42 + 105*x43 + 120*x44

We have to minimize this distance

Further,

x11 + x12 + x13 + x14 = 1

x21 + x22 + x23 + x24 = 1

x31 + x32 + x33 + x34 = 1

x41 + x42 + x43 + x44 = 1

x11 + x21 + x31 + x41 = 1

x12 + x22 + x32 + x42 = 1

x13 + x23 + x33 + x43 = 1

x14 + x24 + x34 + x44 = 1

x11, x12, x13, x14, x21, x22, x23, x24, x31, x32, x33, x34, x41, x42, x43, x44 = Binary

We solve the given assignment problem in Excel using Excel solver as shown below:

The above table in the form of formulas along with Excel Solver extract is shown below for better understanding and reference:

As seen from above, the assignment is highlighted in blue:

Group A - South Siox City

Group B - Wayne

Group C - Winnebago

Group D - Wakefield

Total Distance = 450 miles

B. Now if assignment is as below:

Group A - Wayne = 210 miles

Group B - South Siox City = 70 miles

Group C - Winnebago = 140 miles

Group D - Wakefield = 120 miles

Total distance = 210 + 70 +140 + 120 = 540 miles

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