Inventory Review Problems page PAGE 1 EOQ 1. A specialty coffeehouse sells Colombian coffee at a fairly steady rate of 280 pounds annually. The beans are purchased from a local supplier for $2.40 per pound. The coffee house estimates that it costs $45 in paperwork and labor to place an order for the coffee, and holding costs are based on a 20 percent annual interest rate. The optimal order quantity is SQRT(2 x 280 x 45 / .48) = 229 units The time between placements of orders is 229 / 280 = .818 years The annual cost of holding is $0.48 x 229/2 = $55 The annual cost of ordering (not including the purchase cost) is $45 x 280 / 229 = $55 2. A large automobile repair shop installs about 1,250 mufflers per year, 18 percent of which are for imported cars. (Assume a very steady demand rate.) All of the imported-car mufflers are purchased from a single local supplier at a cost of $18.50 each. The shop uses a holding cost based on a 25 percent cost of capital. The fixed cost for placing an order is estimated to be $28. Determine the optimal number of imported-car mufflers the shop should purchase each time an order is placed, and the time between placements of orders. D = .18 x 1250 =225, H = .25 x 18.5 = $4.625, S = $28 Q = SQRT(2 x 225 x 28 / 4.625) = 52 units Time between Orders = 52/225 = .231 years Newsvendor 1. The Year-End Calendar Company must decide how many ?Boilermaker Special Calendars? to produce this year. Since the calendars are to be manufactured well in advance of the selling season (and only one manufacturing run is practical), the batch size must be selected before any calendars are actually sold. Based on historical data from past years, Year-End has estimated the probability of demand to be: DEMAND Prob. Cum Prob. 1,000 .05 .05 2,000 .10 .15 3,000 .20 .35 4,000 .30 .65 5,000 .20 .85 6,000 .10 .95 7,000 .05 1.00 Further, Year-End estimates that each unsold calendar it produces costs them $1 (in wasted production costs plus disposal costs) and if demand exceeds supply, each unfilled demand costs them $9 (in lost profit since they sell the calendars for $10). HOW MANY CALENDARS SHOULD THEY PRODUCE? CUS /( COS + CUS) = 9/(1+9) = .9 Need to produce 6,000 to have a service level of at least .9 2: Same situation as above except now suppose that demand is normal with mean of 4,000 and standard deviation of 1,000. a critical ratio of .9 translates into a z value of 1.28, so Stocking Quantity = 4,000 + 1.28*1000 = 5,280 3. A factory makes two products. It has capacity to produce a combined total of 100 units per week, in any combination. Product 1 costs $20 per unit to make, and sells for $30. Demand for this product exceeds the factory?s capacity. Product 2 costs $10 per unit to make, and sells for $100, but demand is uncertain. Product 2 is produced in batches once a week. Any units that are not sold by the end of the week must be thrown out: they have no salvage value. The probability distribution of demand for Product 2 is shown below. How many units, if any, should be produced? Demand Probability 5 0.10 6 0.20 7 0.25 8 0.20 9 0.20 10 0.05 The newsvendor decision is ?How many units of Product 2 to produce?? COS = cost to produce product 2 + lost profit from 1 less product 1 = $10 +$10 = $20 CUS = lost contribution from product 2 ? contribution from product 1 = $90 - $10 = $80 Target cycle fill rate is .8, so we should produce 9 units of Product 2, since ? Prob(demand ( 9) >.8 but Prob(demand ( 8) < .8 Continuous Review Your firm uses a continuous review system, where inventory position of an item is updated after each transaction. The firm operates 365 days per year. One of the items has the following characteristics: Average Demand (D) = 100 units/day Ordering cost (S) = $64/order Holding Cost to hold one unit for 1 year (H) = $7.30 Replenishment lead-time (L) = 4 days Standard deviation of daily demand = 50 units Target cycle fill rate = 95% Shortage cost per unit short = $16 How many should be ordered each time? SQRT(2 x 36500 x 64 / 7.3) = 800 How many orders per year (on average)? 36,500 / 800 = 45.625 What should the Reorder-point be? R = 4 x 100 + SQRT(4) x 50 x 1.64 = 400 + 164 = 564 Amount of Safety Stock? 164 Average Inventory? 800 / 2 + 164 = 564 (This is a just a coincidence!) Expected Shortages per order cycle? E(Z) for 95% is .021 SQRT(4) x 50 x .021 = 2.1 Expected Shortages per year? 2.1 x 45.625 = 95.8 Expected Unit Fill Rate? (36500 ? 95.8) / 36500 = .997 --------------------------------------------------------------------------- If we use newsvendor model to find best cycle fill rate (instead of 95%) Target cycle fill rate = CUS / (CUS + COS ) = CUS = $16 COS: extra holding cost per unit of any units still in inventory when the next batch of Q arrives. = holding cost of one unit for 1 order cycle. = H * Q/D COS = $7.30 x 800 / 36500 = .16 Target cycle fill rate = CUS / (CUS + COS ) = .99 (z-value of 2.33) ROP = 400 x SQRT(4) x 50 x 2.33 = 633 Periodic Review 1. A retailer orders Imperial Yo-Yo?s from its supplier every other Monday. It takes the supplier 2 days to get the order to the retailer. The retailer?s daily customer demand for these Yo-Yo?s is normally distributed with mean = 10 and standard deviation = 4. The retailer believes that allowing a 5% chance of running out just before each new order arrives is the appropriate target cycle fill rate. Today an order is to be placed and the retailer currently has 30 Imperials in stock. How many should the retailer order today? T+L = 16 days Demand during 16 days is normal with mean of 10 x 16 = 160 standard deviation of SQRT(16) x 4 = 16 Target Cycle Fill Rate = 95% Translates into z-value = 1.64 Base Stock (B) = 160 + 16 x 1.64 = 187 (after rounding up) Amount of Order Placed = 187 ? 30 = 157 General Formula is: B = (T+L) x Mean + SQRT(T+L) x Stand.Dev. x z-value Follow-up: What is The amount of safety stock? 27 Average Inventory when an order arrives? 27 Average Order quantity? 14 x 10 =140 (if there are no lost sales) Expected number of Order Cycles per year that end in shortages? .05 x 26 = 1.3 Average number of units short per order cycle? (E(Z) is .021) SQRT(16) x 4 x .021 = .336 Expected number of units short per year? .336 x 26 = 8.7 Expected unit fill rate? (140 - .336) / 140 = .9976 Suppose that instead of being given a Target Cycle Fill Rate, we know that the cost per unit backordered is $5 (estimated loss of future profit) and that the holding cost per unit per day is 1¢. Thus CUS = $5, COS = .14 (holding cost of .01 per day for 14 days), so The Target Cycle Fill Rate = 5/5.14 = .97 The corresponding z-value is 1.88 Safety Stock is 30.08 Base Stock, B = 191 (after roundup) 2. The daily demand for a widget at Clem?s Discount Store is normally distributed with a mean of 100 units and a standard deviation of 25 units. Clem uses a review period of seven days, and the lead-time is five days. At the beginning of this review period he has 700 widgets in inventory. Clem would like to maintain a 95 percent cycle fill rate. How many units should he order? T + L = 12 days Mean demand over 12 days = 1200 Standard deviation of demand over 12 days = SQRT(12) x 25 = 86.6 Z-value corresponding to 95% cycle fill rate = 1.64 Desired safety stock = 1.64 x 86.6 = 142 Base Stock, B = 1200 + 142 = 1342 Units to order this time = 1342 ? 700 = 642 Average Order Size = 7 x 100 = 700 Expected units short per review period = 86.6 x .021 = 1.8 Expected units short per year (assume 52 cycles per year) = 52 x 1.8 = 94.6 Expected unit fill rate = (36500 ? 94.6) / 36500 = .9974