The Warehouse Manager Discussion Paper

Consider a hardware supply warehouse that is contractually obligated to deliver 2,000 units of a specialized fastener to a local manufacturing company each week. Each time the warehouse places an order for these items from its supplier, an ordering and transportation fee of $$30$ is charged to the warehouse. The warehouse pays $$1.00$ for each fastener and charges the local firm $$4.00$ for each fastener. Annual holding cost is 25 percent of inventory value, or $$0.25$ per year. Assume the manufacturing plant operates for 50 weeks a year. (1) [5 pt] How much the warehouse manager need to order when inventory gets to zero to minimize the average cost? (2) [10 pt] If the supplier only allows orders in thousands of units, then how much the warehouse manager need to order? Compared with the optimal average cost in (1), what is the percentage of increase in this case? (3) [10 pt] In an EOQ model, denote $K$ the fixed order cost, $h$ the holding cost per unit per day, $D$ the constant demand per day. Let $Q_{∗}$ be the opitmal order quantity and $C(Q_{∗})$ be the corresponding average cost in terms of time. Prove that for any $b>0$ $C(Q)C(bQ) =21 (b+b1 ) The Warehouse Manager Discussion Paper$

**Step 1/2**

(1) To minimize the average cost, the warehouse manager needs to order the Economic Order Quantity (EOQ). The EOQ formula is given as: EOQ = sqrt((2DS)/H) where D is the weekly demand (2000), S is the ordering and transportation cost ($30), and H is the holding cost per unit ($0.25 per year or 0.25/52 per week). Thus, we have: EOQ = sqrt((2200030)/(0.25/52)) = 1096.07 The warehouse manager should order 1096 units when inventory gets to zero to minimize the average cost. The Warehouse Manager Discussion Paper

**Step 2/2**

(2) If the supplier only allows orders in thousands of units, the warehouse manager should order 1 thousand units (1000 units) as it is the nearest multiple of the EOQ. The total cost for this order will be: Total cost = (D/Q*)S + (Q/2)H + (DP)/Q* where P is the unit cost ($1) and Q* is the EOQ. Substituting the values, we get: Total cost = (2000/1096.07)30 + (1096.07/2)(0.25/52) + (2000*1)/1096.07 = $32.23 The percentage of increase in the total cost compared to the optimal average cost in (1) is: ((32.23 – 28.87)/28.87)*100% = 11.6%

**Final answer**

(3) The average cost in terms of time for the EOQ model is given by: C(Q*) = (DS/Q*)S + (Q/2)*H Let Q = bQ*, where b > 0. Then, the total cost for this order will be: Total cost = (D/Q)S + (Q/2)H + (DP)/Q = (D/bQ)S + (bQ/2)H + (DP)/(bQ*) Substituting Q* = sqrt((2DS)/H) in the above expression, we get: Total cost = (b + sqrt((D/H)(S/P)))/2 * C(Q) Thus, we have: C(bQ*) = (b + sqrt((D/H)(S/P)))/2 * C(Q) This proves the given result. The Warehouse Manager Discussion Paper