# Answered! A local auto mechanic is developing a schedule for purchasing and storing part X for the next four quarters. Over the…

A local auto mechanic is developing a schedule for purchasing and storing part X for the next four quarters. Over the next four quarters she has to have at least 200, 300, 450, and 200 units respectively on hand to accommodate expected requests for repairs in each quarter. The price she pays per unit starts at \$100 in the first quarter and increases by \$3 per quarter thereafter. Cost of storage per unit from one quarter to the next is \$4. Because of supply limitations she cannot purchase more than 350 units per quarter. Her maximum storage capacity is 100 units from any one quarter to the next. At the end of the fourth quarter she does not want to have anything in inventory and expects to develop a new plan based on the updated forecasts of demand. Develop an LP model to determine the optimal schedule for purchasing and storage of part X. (Do not solve the model or try to figure out the actual schedule, but rather develop an LP model which can be easily modified and resolved if one of the parameters changes.)

The LP model for aggregate planning is as follows

Decision variables:

Pi = Quantity purchased in quarter i

Si = Inventory at the end of quarter i

Objective:

Min 100P1 + 103P2 + 106P3 + 109P4 + 4*(S1+S2+S3+S4)

s.t.

P1 – S1 = 200

S1 + P2 – S2 = 300

S2 + P3 – S3 = 450

S3 + P4 – S4 = 200

S4 = 0

S1, S2, S3, S4 <= 100

P1, P2, P3, P4 <= 350

Si, Pi >= 0