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	South 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.

--create an Excel file to solve the problem and solve the problem – attach the file when you submit answer
--- type out what is the minimum distance and where each group is assigned

-- explain how the distance changes if Group A goes to Wayne, Group B goes to South Sioux City, Group C goes to Winnebago, and Group D goes to Wakefield.

Answer 1

Officials	Wayne	South 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

Formulating LP:

Decision Variables:

Let Xij be the number of times group i travels to game site j

Officials	Wayne	South Sioux City	Winnebago	Wakefield
Group A	X11	X12	X13	X14
Group B	X21	X22	X23	X24
Group C	X31	X32	X33	X34
Group D	X41	X42	X43	X44

Objective Function: Minimize the distance

Z min = 210X11+90X12+180X13+160X14+100X21+70X22+130X23+200X24+175X31+105X32+140X33+170X34+80X41+65X42+105X43+120X44

Constraints: (Each Site needs one group of officials and vice versa)

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

Xij is binary (As one team visits one city, hence the value of one city and group combination will be either 1 or 0)

Xij ≥ 0

Solving in excel solver:

Solver Parameters:

Solution:

Assignment of Groups and Minimum DIstance

Officials	Wayne	South Sioux City	Winnebago	Wakefield
Group A		1
Group B	1
Group C			1
Group D				1

Minimum Distance = 450

If Group A goes to Wayne, Group B goes to South Sioux City, Group C goes to Winnebago, and Group D goes to Wakefield, the minimum distance increases from 450 to 540.