A retailer has decided to carry a new product, for which the demand will be dete
ID: 442159 • Letter: A
Question
A retailer has decided to carry a new product, for which the demand will be deterministic and stable at a rate of ? units per day. The product's distributor charges c per unit. The retailer expects to incur a holding cost of h per unit per day. The retailer's policy does not allow shortages. The distributor offered the retailer the following plan: Whenever the retailer places an order, the distributor will send the order in two shipments. (Assume that the lead times for the shipments are negligible.) The size of the first shipment will be two thirds of the order. The remainder of the order will be sent in the second shipment, which will be timed so that the retailer receives the second shipment just before it runs out of inventory. The distributor will charge the retailer K for the two shipments combined. Let G(Q) denote the retailer's total daily cost (the sum of shipment, holding and purchase costs per day) when the order size is Q. Derive an expression for G(Q). Derive the optimal order quantity for this case. Please make sure to verify the quantity minimizes G(Q) which you derived in part a
Explanation / Answer
Unit per day is posted as “?” so assuming it to be “D”
Unit demanded per day= D (Assuming as “?” is only visible)
Let purchase cot per unit be P
Purchase cost= purchase unit price × demand quantity=P*D
Holding cost= average quantity in stock is Q/2=h × Q/2
Shipment cost=each order has a fixed cost K, and we need to order D/Q times daily=D/Q
G(Q)=Purchase cost+ Holding cost + Shipment cost
G(Q)=D*P+Q*h/2+D*K/Q
For minimum point dG(Q)/dQ=0
d(D*P+Q*h/2+D*K/Q)/dQ=0
-D*K/(Q^2)+h/2=0
Solving for Q gives the optimal order quantity
Q^2=2*D*K/h
Optimal Order Quantity=(2*D*K/h)^(1/2)
Where D= Daily Qunatity Demanded
K=Shipment Cost
h=Holding cost
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