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Factory Location Problem

A manufacturer of circuit board parts has factories in a city at the location coordinates shown below. Each coordinate unit (roughly 100 feet) represents a city block. The yearly demand at each factory is also given.

Factory Demand Coordinate Coordinate
Glen Mills 20,000 20 150
Media 7,000 55 45
Brandywine 5,000 80 90
Springfield 2,000 90 30

Management has decided to build a new factory to supply raw material to these factories. The location of the new factory should be central to the existing factories.

What should be the map coordinates of the new factory? Graph the locations of the four factories and the proposed raw material factory. Is the location of the proposed raw material factory where you expected it to be based on the coordinates of the other factories? Why or why not? What is the main contributing factor leading to the location of the proposed raw material factory?

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Solution Preview

Let (x,y) be the coordinates of the new factory. Therefore using the Pythagorean Theorem, we minimize the squared difference of the coordinates.

The Pythagorean Theorem states that the shortest distance between two points is given by

Shortest distance between two points = √((x1 - x2)2 + (y1 - x2)2)

Minimize:
Distance = √((20 - x)2 + (150 - y)2) + √((55 - x)2 + (45 - y)2) + √((80 - x)2 + (90 - y)2) + √((90 - x)2 + (30 - y)2)

Subject to:
√((20 - x)2 + (150 - y)2) > 0 Glen Mills requirement
√((55 - x)2 + (45 - y)2) > 0 Media requirement
√((80 - x)2 + (90 - y)2) > 0 Brandywine requirement ...

Solution Summary

The expert examines manufacturer factory location coordinates.

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