For each problem, your responsibility is to formulate the decision environment a
ID: 353457 • 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.
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:
BaseDeluxe
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
Let B1, B2, B3 be the amount of Base DVD ordered in three months
Let D1, D2, D3 be the amount of Delux DVD ordered in three months
The B(i) & D(i) are our decision variable as this will impact the cost (where i = 1,2,3)
Let XB0 & XD0 be the base Inventory for Base & Delux DVDs respectively (It is equal to zero in our case)
Let XB1, XB2, XB3 be the inventory of L left after each month
Let XD1, XD2, XD3 be the inventory of M left after each month
Let DB1, DB2, DB3 be the Demands of Base & DD1, DD2, DD3 be the Demands of Delux for the three months
We know Inventory = Amount Purchased + Previous Inventory - Demand
Thus we get,
XB(i) = B(i) + XB(i-1) - DB(i) where i = 1,2,3
XD(i) = D(i) + XD(i-1) - DD(i) where i = 1,2,3
Let YB1, YB2 & YB3 be the quantity of Base DVD ordered in Higher cost. We know,
YB(i) = XB(i) - 500 where i = 1,2,3
Let YD1, YD2 & YD3 be the quantity of Delux DVD ordered in Higher cost. We know,
YD(i) = XD(i) - 300 where i = 1,2,3
Incremental hike in Price on Base DVD over 500= $5
Incremental hike in Price on Delux DVD over 300= $6
Our Objective is to Minimize the cost fulfilling the orders. We have three types of costs to minimize, Cost from Purchase in Normal Rate, Cost from Purchase in Higher rate & Inventory Cost. Thus our Objective function is-
Min(65*(B1+B2+B3) + 5*(YB1+YB2+YB3) + 2*(XB1+XB2+XB3) + 90*(D1+D2+D3) + 6*(YD1+YD2+YD3) + 2*(XD1+XD2+XD3))
Constraints-
For Demand Fulfillment & Extra Inventory
B(i) + XB(i-1) >= DB(i) where i=1,2,3
D(i) + XD(i-1) >= DD(i) where i = 1,2,3
For Inventory <=100
XB(i) + XD(i) <= 100
As Inventory cant be negative,
XB(i) >= 0 where i = 0,1,2,3
XD(i) >= 0 where i = 0,1,2,3
As Quantity cant be Negative,
B(i), YB(i) >= 0 where i=1,2,3
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