Sample_quiz2_solutions

# Q 0 29 29 29 29 29 2 8 600 q 1 29 29 29 29 29 2 8 600

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Q (0) = ( 29 ( 29 ( 29 ( 29 ( 29 2 8 600 400 0.2 0.3 = Q (1) = ( 29 ( 29 ( 29 ( 29 ( 29 2 8 600 406 0.2 0.29 = Q (2) = ( 29 ( 29 ( 29 ( 29 ( 29 2 8 600 414 0.2 0.28 = The only realizable EOQ value is Q (0) = 400. We need to check cost at Q (0) = 400, and at the breakpoints of 500 and 1000. G 0 (400) = (600)(0.3) + ( 29 ( 29 ( 29 ( 29 ( 29 2 8 600 0.2 0.3 = \$204 G 1 (500) = (500)(0.29) + (600)(8)/500 + (0.2)(0.29)(500)/2 = \$198.10 G 2 (1000) = (1000)(0.28) + (600)(8)/1000 + (0.2)(0.28)(1000)/2 = \$200.80 So the optimal order quantity is 500. 3

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(b) It costs the weighty trash bag company \$24 each time Publix places an order, it costs them 20 cents to produce a bag, and they also use a holding cost rate of 20%. The weighty trash bag company produces at a steady rate of 600 per year (the same rate as demand) and each time an order is placed by Publix, it is immediately transported to Publix (with negligible lead time; that is, the bag producer builds up inventory at a steady rate until the exact order/delivery quantity is on hand, at which point it is immediately shipped to Publix, and this process repeats itself infinitely many times). What is the Publix order quantity that would maximize the profit of the weighty trash bag company? Q * = ( 29 ( 29 ( 29 ( 29 ( 29 2 24 600 0.2 0.2 = 848.5 (c) What order quantity would minimize total supply chain costs? (assume that the 20 cent production cost includes the per unit transportation cost to Publix.) The optimal order quantity for the system must be between 500 and 848.53, which implies a selling price of 0.29. Let us consider the system costs at this price. Setup Costs = supplier + retailer setup costs = (24 + 8)(600)/Q Holding Costs = supplier + retailer holding costs = (0.2)(0.2 + 0.29)Q/2 Q * = ( 29 ( 29 ( 29 ( 29 ( 29 2 24 8 600 0.2 0.2 0.29 + + = 626 4
Problem 4 Consider a simple two-stage supply chain where a manufacturer directly supplies a retailer with a single product. Assume that the manufacturer is the only supplier for the retailer and the retailer is the manufacturer’s only customer for the product. The retailer observes customer demand for the product at a steady rate of D units per year and no shortages are allowed. The manufacturer’s production rate is exactly matched to the demand rate. Each time the retailer places an order to the manufacturer it costs \$ S in administrative and processing costs. The retailer incurs a cost of \$ H for holding a unit in inventory for one year. When the retailer places an order, the manufacturer makes a shipment, which costs the manufacturer \$ β S in shipping costs (a fixed cost per shipment; assume a constant or negligible lead time). The manufacturer incurs a cost of \$ α H for holding a unit in inventory for one year. Each time the retailer places an order it is for the same quantity, Q . (a) Draw the graph of the inventory level over time for both the retailer and manufacturer below. (b) In terms of the parameters given, what is the retailer’s optimal order quantity? Q R * = sqrt(2 SD / H ) 5 time Retailer’s Inventory level time Manufacturer’s Inventory level

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(c) In terms of the parameters given, what is the optimal quantity for the manufacturer to ship?
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