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A friend of mine passed along this problem he's having trouble with in a business class, and I have to admit, I'm a bit stumped coming up with the proper mathematical framework for it. Here's the problem:

A company manufactures loudspeakers in plants I and II. Output at plant I is at most 800 units per month, output at plant II is at most 600 units per month. The units are shipped seperately to three warehouses, A, B, and C, whose minimum monthly requirements are 500, 400, and 400 units respectively. Shipping costs for plant I to warehouses A, B, and C are $16, $20, and $22 per unit, repectively, and shipping costs for plant II to each warehouse are $18, $16, and $14, respectively. What shipping schedule will enable the company to meet the warehouses' requirements and at the same time minimize shipping costs?

The attempt at a solution

Basically, I looked at minimizing the cost for the shipping. Since plant I is cheaper to send to warehouse A, and plant II is cheaper to send to B and C, I then looked at which warehouse would minimize costs. So, I had plant II send 400 units to C and 200 units to B. This leaves plant I sending 200 units to B, and 500 units to A.

This looks right to me, but I'm curious, is there a way to set this up with equations to get an exact and correct answer? I tried setting this up with a system of equations like this:

(Plant I's production I'm calling x, y, and z for shipping to the three warehouses, respectively. Plant II's production I'm calling A, B, and C for shipping to the three warehouses)

x + A = 500

y + B = 400

z + C = 400

A + B + C = 600

x + y + z = 700

However, this is six unknowns with only five equations... Also, I'm having to assume that plant II (A + B + C) will be maxed out in it's production. Is there any way to approach this where I don't have to make those sorts of assumptions?

Thanks for your help!

-Max

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# Homework Help: Shipping problem (algebra)

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