Question

The city of Squaretown wants to build a recreation center for senior citizens who live in three low-income apartment buildings (A, B, and C). Squaretown is a town with only horizontal and vertical streets and so each city block is a perfect square. Each unit represents one city block. The table below provides the map coordinates of each apartment building and the number of senior citizens in each building:

Apartment building	Map coordinates (x, y)	Number of Senior Citizens
A	(1, 2)	100
B	(3, 8)	300
C	(7, 5)	200

Squaretown believes that the best location for the senior citizens center is the one that minimizes the sum of the products obtained by multiplying each apartment building’s senior citizen population by the building’s distance from the center (as measured by the number of city blocks). For example, if the center were built at map coordinates (4, 5), the sum of products would be

                             100 (|4 – 1|+|5 – 2|) + 300 (|4 – 3| + |5 – 8|) + 200 (|4 – 7| + |5 – 5|) =2400

Formulate a goal programming model that Squaretown can use to determine where to locate the senior citizens center. Give the mathematical (algebraic) model by defining the decision variables, deviation variables, constraints plus objective function. Set up a spreadsheet model and solve it using the Excel Solver to find the optimal solution. What is the optimal location for this center?

Answer 1

Decision variables:
x = x-coordinate of the proposed recreation center
y = y- coordinate of the proposed recreation center
d = Sum of the products of each building from the proposed recreation center
Objective function and Constraints:
Minimize d = 100 (|x – 1|+|y – 2|) + 300 (|x – 3| + |y – 8|) + 200 (|x – 7| + |y – 5|), Subject to x, y are integers

Solver Parameters Set Target Cell $C$21 Equal To: OMax ⓔMin By Changing Cells: Solve Ovalue of: 0 Close SA$21:$8$21 Guess Subject to the Constraints: Options SA$21:$B$21- integer Add Change Delete Reset Al Help

Solution:
x	y	d
3	8	2200
The recreation center should be built at coordinates (3, 8). The distance from the buildings is 2200 blocks.

[Note: If we do not impose constraints that x, y should be integers, the solution would be (x, y) = (3, 7.1), d = 2200.]