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Inventory Management Operations Management Dr. Ron Tibben-Lembke Purposes of Inventory Meet anticipated demand Demand variability Supply variability Decouple production & distribution permits constant production quantities Take advantage of quantity discounts Hedge against price increases Protect against shortages 2006 13.81 1857 24.0% 446 801 58 1305 9.9 US Inventory, GDP ($B) 14,000 12,000 10,000 8,000 6,000 4,000 2,000 - 84 86 88 90 92 94 96 98 00 02 04 19 19 19 19 19 19 19 19 20 20 20 Business Inventories US GDP US Inventories as % of GDP 25.0% 20.0% 15.0% % of GDP 10.0% 5.0% 0.0% 84 86 88 90 92 94 96 98 00 02 04 19 19 19 19 19 19 19 19 20 20 20 Year Source: CSCMP, Bureau of Economic Analysis Two Questions Two main Inventory Questions: How much to buy? When is it time to buy? Also: Which products to buy? From whom? Types of Inventory Raw Materials Subcomponents Work in progress (WIP) Finished products Defectives Returns Inventory Costs What costs do we experience because we carry inventory? Inventory Costs Costs associated with inventory: Cost of the products Cost of ordering Cost of hanging onto it Cost of having too much / disposal Cost of not having enough (shortage) Shrinkage Costs How much is stolen? 2% for discount, dept. stores, hardware, convenience, sporting goods 3% for toys & hobbies 1.5% for all else Where does the missing stuff go? Employees: 44.5% Shoplifters: 32.7% Administrative / paperwork error: 17.5% Vendor fraud: 5.1% Inventory Holding Costs Category % of Value Housing (building) cost 4% Material handling 3% Labor cost 3% Opportunity/investment 9% Pilferage/scrap/obsolescence 2% Total Holding Cost 21% Inventory Models Fixed order quantity models How much always same, when changes Economic order quantity Production order quantity Quantity discount Fixed order period models How much changes, when always same Economic Order Quantity Assumptions Demand rate is known and constant No order lead time Shortages are not allowed Costs: S - setup cost per order H - holding cost per unit time EOQ Inventory Level Q* Decrease Due to Optimal Constant Demand Order Quantity Time EOQ Inventory Level Q* Instantaneous Optimal Receipt of Optimal Order Order Quantity Quantity Time EOQ Inventory Level Q* Reorder Point (ROP) Time Lead Time EOQ Inventory Level Q* Average Inventory Q/2 Reorder Point (ROP) Time Lead Time Total Costs Average Inventory = Q/2 Annual Holding costs = H * Q/2 # Orders per year = D / Q Annual Ordering Costs = S * D/Q Annual Total Costs = Holding + Ordering Q D TC (Q) H * S * 2 Q How Much to Order? Annual Cost Holding Cost = H * Q/2 Order Quantity How Much to Order? Annual Cost Ordering Cost = S * D/Q Holding Cost = H * Q/2 Order Quantity How Much to Order? Total Cost Annual Cost = Holding + Ordering Order Quantity How Much to Order? Total Cost Annual Cost = Holding + Ordering Optimal Q Order Quantity Optimal Quantity Q D Total Costs = H* S* 2 Q Take derivative H D Set equal with respect to Q = S * 2 0 to zero 2 Q Solve for Q: H DS 2 DS 2 DS 2 Q 2 Q 2 Q H H Adding Lead Time Use same order size 2 DS Q H Order before inventory depleted R = d * L where: d = demand rate (per day) L = lead time (in days) both in same time period (wks, months, etc.) A Question: If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy? Benefits of EOQ Profit function is very shallow Even if conditions don’t hold perfectly, profits are close to optimal Estimated parameters will not throw you off very far Sensitivity Suppose we do not order optimal Q*, but order Q instead. Percentage profit loss given by: TC(Q) 1 Q * Q TC(Q*) 2 Q Q * Should order 100, order 150 (50% over): 0.5*(1.5 + 0.66) =1.08 an 8%cost increase Quantity Discounts How does this all change if price changes depending on order size? Holding cost as function of cost: H=I*C Explicitly consider price: 2DS Q I C Discount Example D = 10,000 S = $20 I = 20% Price Quantity EOQ c = 5.00 Q < 500 633 4.50 501-999 666 3.90 Q >= 1000 716 Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 500 1,000 Order Size Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 500 1,000 Order Size Discount Example Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $299.70 Order 10,000/666 * 20 = $300.00 Mat’l 10,000*4.50 = $45,000.00 45,599.70 Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2=$390.00 Order 10,000/1,000 * 20 = $200.00 Mat’l 10,000*3.90 = $39,000.00 39,590.00 Discount Model 1. Compute EOQ for next cheapest price 2. Is EOQ feasible? (is EOQ in range?) If EOQ is too small, use lowest possible Q to get price. 3. Compute total cost for this quantity 4. Repeat until EOQ is feasible or too big. 5. Select quantity/price with lowest total cost. Inventory Management -- Random Demand Master of the Obvious? If you focus on the things the customers are buying it’s a little easier to stay in stock • James Adamson CEO, Kmart Corp. 3/12/02 Fired Jan, 2003 Random Demand Don’t know how many we will sell Sales will differ by period Average always remains the same Standard deviation remains constant Impact of Random Demand How would our policies change? How would our order quantity change? How would our reorder point change? Mac’s Decision How many papers to buy? Average = 90, st dev = 10 Cost = 0.20, Sales Price = 0.50 Salvage = 0.00 Overage: CO = 0.20 - 0.00 = 0.20 Underage: CU = 0.50 - 0.20 = 0.30 Optimal Policy F(x) = Probability demand <= x Optimal quantity: Cu P(unit sold) Co Cu Mac: F(Q) = 0.3 / (0.2 + 0.3) = 0.6 From standard normal table, z = 0.253 =Normsinv(0.6) = 0.253 Q* = avg + zs = 90+ 2.53*10 = 90 +2.53 = 93 Optimal Policy Model is called “newsboy problem,” newspaper purchasing decision If units are discrete, when in doubt, round up If u units are on hand, order Q - u units Multiple Periods For multiple periods, salvage = cost - holding cost Solve like a regular newsboy Random Demand If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be? Probabilistic Models Safety stock = x m x m From statistics, z s Safety stock Therefore, z = & Safety stock = zs s From normal table z.95 = 1.65 Safety stock = zs= 1.65*10 = 16.5 ROP = m + Safety Stock =350+16.5 = 366.5 ≈ 367 Random Example What should our reorder point be? demand over the lead time is 50 units, with standard deviation of 20 want to satisfy all demand 90% of the time To satisfy 90% of the demand, z = 1.28 Safety stock = zs = 1.28 * 20 = 25.6 R = 50 + 25.6 = 75.6 St Dev Over Lead Time What if we only know the average daily demand, and the standard deviation of daily demand? Lead time = 4 days, daily demand = 10, standard deviation = 5, What should our reorder point be, if z = 3? St Dev Over LT If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40. What is the standard deviation of demand over the lead time? Std. Dev. ≠ 5 * 4 St Dev Over Lead Time Standard deviation of demand = Ls 4 5 10 R = 40 + 3 * 10 = 70 Service Level Criteria Type I: specify probability that you do not run out during the lead time Chance that 100% of customers go home happy Type II: proportion of demands met from stock 100% chance that this many go home happy, on average Service levels easier to estimate Two Types of Service Cycle Demand Stock-Outs 1 180 0 Type I: 2 75 0 8 of 10 periods 3 235 45 80% service 4 140 0 5 180 0 Type II: 6 200 10 1,395 / 1,450 = 7 150 0 96% 8 90 0 9 160 0 10 40 0 Sum 1,450 55 Type I Service a = desired service level We want F(R) = a R=m+s*z Example: a = 0.98, so z = 2.05 if m = 100, and s = 25, then R = 100 + 2.05 * 25 = 151 Type II Service b = desired service level Number of mad cust. = (1- b) EOQ L(z) = EOQ (1- b) / s Example: EOQ = 100, b = 0.98 L(z) = 100 * 0.2 / 25 = 0.8 P. 835: z = 1.02 R = 126 -- A very different answer Inventory Recordkeeping Two ways to order inventory: Keep track of how many delivered, sold Go out and count it every so often If keeping records, still need to double- check Annual physical inventory, or Cycle Counting Cycle Counting Physically counting a sample of total inventory on a regular basis Used often with ABC classification A items counted most often (e.g., daily) Advantages Eliminates annual shut-down for physical inventory count Improves inventory accuracy Allows causes of errors to be identified Fixed-Period Model Answers how much to order Orders placed at fixed intervals Inventory brought up to target amount Amount ordered varies No continuous inventory count Possibility of stockout between intervals Useful when vendors visit routinely Example: P&G rep. calls every 2 weeks Fixed-Period Model: When to Order? Inventory Level Target maximum Period Time Fixed-Period Model: When to Order? Inventory Level Target maximum Period Period Time Fixed-Period Model: When to Order? Inventory Level Target maximum Period Period Time Fixed-Period Model: When to Order? Inventory Level Target maximum Period Period Period Time Fixed-Period Model: When to Order? Inventory Level Target maximum Period Period Period Time Fixed-Period Model: When to Order? Inventory Level Target maximum Period Period Period Time Fixed Order Period Standard deviation of demand over T+L = s T L T Ls T = Review period length (in days) σ = std dev per day Order quantity (12.11) = q d (T L) zs T L I ABC Analysis Divides on-hand inventory into 3 classes A class, B class, C class Basis is usually annual $ volume $ volume = Annual demand x Unit cost Policies based on ABC analysis Develop class A suppliers more Give tighter physical control of A items Forecast A items more carefully Classifying Items as ABC % Annual $ Usage 100 80 60 40 A 20 B C 0 0 50 100 150 % of Inventory Items ABC Classification Solution Stock # Vol. Cost $ Vol. % ABC 206 26,000 $ 36 $936,000 105 200 600 120,000 019 2,000 55 110,000 144 20,000 4 80,000 207 7,000 10 70,000 Total 1,316,000 ABC Classification Solution Stock # Vol. Cost $ Vol. % ABC 206 26,000 $ 36 $936,000 71.1 A 105 200 600 120,000 9.1 A 019 2,000 55 110,000 8.4 B 144 20,000 4 80,000 6.1 B 207 7,000 10 70,000 5.3 C Total 1,316,000 100.0

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inventory management, optimal solution, Operations Management, respect to, Supply Chain Management, Production Planning, cooperative games, valid inequalities, approximation algorithm, joint replenishment

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posted: | 5/21/2010 |

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