Hartman Electronics is a small retail electronics store. Among the items they se

Question

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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