1. A firm operates two distribution centers, one in Phoenix and the other is Chi

Question

1. A firm operates two distribution centers, one in Phoenix and the other is Chicago. A customer in College Station, TX needs solar panels produced by the firm. The cost to produce a solar panel in Phoenix is $349, and shipment to College Station is $63. The cost to produce a solar panel in Chicago is $349, and shipment to College Station is $47. Phoenix can supply at most 13 panels, and Chicago up to 29. The customer wants 21 panels. Structure this as an optimization model?

2. How does the problem in #1 change if the cost to produce a panel in each city changes? Specifically, the cost to produce a solar panel in Phoenix is $349 and the cost to produce a solar panel in Chicago is $379. Structure this optimization model.

Explanation / Answer

1. Let 'P' denote the number of solar panel units from Phoenix and 'C' denote the number of units from Chicago.

We need a total of 21 panels. Therefore, P + C =21

Total production cost incurred: 349 * P + 349 * C

Total shipment cost: 63P + 47C

Therefore, overall cost of solar panels (Production + Shipment) = (349+63) * P + (349+47) * C = 412* P + 396 * C

Limit on maximum number of panels that can be supplied from the distribution centers: P <= 13, C <= 29

P, C >=0 (Number of units cannot be a negative number)

All the above information is derived directly from the question.

Our optimization objective function will be to minimize the overall cost, i.e, we need to get the 21 panels to TX at the minimum cost possible.

Hence, the optimization model will be as follows:

Minimize the overall cost: 412* P + 396 * C

subject to the constraints:

P + C = 21

P <= 13

C <= 29

P,C >= 0

2. In the above answer, change the overall cost equation to (349 + 63) * P + (379 + 47) * C = 412*P + 426*C

Thus, the optimziation problem will change as follows (provided all other parameters stay same):

Minimize the overall cost: 412* P + 426 * C

subject to the constraints:

P + C = 21

P <= 13

C <= 29

P,C >= 0

NOTE: It seems that there are too few constraints mentioned in the question. Because of this, the solution to the first optimization problem seems very straightforward. In the first case, cost will be minimum if all panels are ordered from Chicago. In the second question, although cost is lesser in Phoenix, we can order only a mazimum of 13 panels. Remaining have to be from Chicago. Overall cost incurred will be higher in the second scenario.

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