For each problem, your responsibility is to formulate the decision environment a
ID: 351129 • Letter: F
Question
For each problem, your responsibility is to formulate the decision environment as a mathematical model. You are not required to solve the problems! Simply write up the appropriate models. This, you should recall, will include complete descriptions of the decision variables, a statement of the objective function, and all relevant constraints. (Note that constraints are not to contain “IF” statements or multiplication or division of variables.) Your models should be written in a form that I can interpret, not one that is meant for Excel. All problems on this part of the exam are worth 15 points each.
Hartman Electronics is a small retail electronics store. Among the items they sell are two different DVD players from a particular manufacturer. There is a base model and a deluxe model. They are planning inventory orders for the next three months; they place orders with the manufacturer on the first of each month. The anticipated demand over the next three months is represented in the following table:
Base Deluxe
Month Model Model
April 400 300
May 700 400
June 375 325
The manufacturer can provide Hartman with up to 500 Base model DVD’s each month at a price of $65 apiece. Anything beyond 500 will require a different production process, so the cost will go up to $70 apiece. The manufacturer can provide Hartman with up to 300 Deluxe model DVD’s each month at $90 apiece. Anything above 300 will cost $96 apiece, for the reasons stated above. Hartman can order more than enough units to cover anticipated demand in one month, and hold the excess units in inventory to help satisfy demand in a subsequent month. Hartman estimates an inventory holding cost of $2.00 per unit, of either model, from one month to the next. Hartman is limited to holding no more than 100 total DVD’s from one month to the next.
Formulate a linear program that will help Hartman to determine the order schedule that will allow him to minimize the cost of obtaining enough DVD’s to satisfy demand.
Explanation / Answer
Linear programming model is following
Let
B1i be the number of Base models produced in month i at cost of $ 65 apiece
B2i, be the number of Base models produced in month i at cost of $ 70 apiece
D1i be the number of Deluxe models produced in month i at cost of $ 90 apiece
D2i, be the number of Deluxe models produced in month i at cost of $ 96 apiece
Vbi be the inventory of Base model in month i
Vdi be the inventory of Deluxe model in month i
Min (B11+B12+B13)*65+(B21+B22+B23)*70+(D11+D12+D13)*90+(D21+D22+D23)*96+(Vb1+Vb2+Vb3+Vd1+Vd2+Vd3)*2
s.t.
B11+B21-Vb1 = 400
B12+B22-Vb2+Vb1 = 700
B13+B23-Vb3+Vb2 = 375
D11+D21-Vd1 = 300
D12+D22-Vd2+Vd1 = 400
D13+D23-Vd3+Vd2 = 325
B11, B12, B13 <= 500
D11, D12, D13 <= 300
Vb1+Vd1 <= 100
Vb2+Vd2 <= 100
Vb3+Vd3 <= 100
B1i, B2i, D1i, D2i, Vbi, Vdi >= 0
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