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Distribution Inventory Systems Dr. Everette S. Gardner, Jr. Competing interests in inventory management Controller: Inventory investment Marketing manager: Operations manager: Customer service Stock replenishment workload Inventory 2 Average inventory behavior with stable demand 12 11 10 On hand 9 8 Stock 7 6 5 Avg. inv. 4 3 2 ROP 1 0 0 4 8 12 16 20 24 28 Day Inventory 3 Average inventory behavior with stable demand (cont.) Demand = 1 unit per day Leadtime = 2 days Leadtime demand (LTD) = 2 units Reorder point (ROP) = LTD = 2 units Order quantity (Q) = 10 units Maximum inventory = Q = 10 units Avg. investment = Q/2 = 5 units Inventory 4 The economic order quantity Objective Minimize total variable costs (TVC) Ordering costs (OC) OC = Cost per order x Nbr. of orders Nbr. of orders = Demand/Order qty. Holding costs (HC) HC = Holding cost per unit per year x Avg. inv. balance Avg. inv. balance = Order qty./2 Total variable costs TVC = OC + HC Inventory 5 The economic order quantity (cont.) $ Total costs Ordering costs EOQ Inventory 6 Economic order quantity (cont.) EOQ in units of stock QU = ((2 * demand in units * cost per order) / holding cost per unit per year)1/2 EOQ in dollars of stock Q$ = ((2 * demand in dollars * cost per order) / holding rate)1/2 where the holding rate is a fraction of inventory value Inventory EOQ.xls 7 Economic order quantity (cont.) Example annual demand = 100 units cost per order = $10 holding cost per unit = $5 unit price = $25 holding rate = .20 QU = ((2 * 100 * 10) / 5)1/2 = 20 units Q$ = ((2 * 25 * 100 * 10) / .20)1/2 = $500 Inventory 8 EOQ in perspective • Ordering and holding costs should be marginal (out of pocket) costs. • Accounting systems generate average costs. • In reality, ordering costs are semifixed. Inventory 9 EOQ in perspective (cont.) Assumption: Reality: Total ordering costs Nbr. of orders Nbr. of orders • In reality, holding costs depend on executive judgments on the cost of capital. Inventory 10 Using the EOQ when costs are unknown Costs can be taken out of EOQ formulas and used as policy variables to achieve management goals for workload and average cycle stock investment. Formula for EOQ in dollars Q$ = ((2 * demand in dollars * cost per order) / holding rate)1/2 Inventory 11 Using the EOQ when costs are unknown (cont.) Remove all constants and unknowns K = ((2 * cost per order) / holding rate)1/2 K is called the EOQ constant Simplified EOQ formula Q$ = K (demand in $)1/2 Inventory 12 Calculations with the EOQ cost constant Let unit price = $10, annual demand = 100 units Low investment, high workload policy K=1 Q = K (demand in $)1/2 = 1 ($10 * 100)1/2 = $31.62 Avg. investment = order qty./2 = $31.62/2 = $15.81 Workload = demand/order qty. = $1000/$31.62 = 31.62 orders Inventory 13 Calculations with the EOQ cost constant (cont.) High investment, low workload policy K=6 Q = K (demand in $)1/2 = 6 ($10 * 100)1/2 = $189.74 Avg. investment = order qty./2 = $189.74 / 2 = $94.87 Workload = demand/order qty. = $1000 / $189.74 = 5.3 orders Inventory 14 Tradeoffs between investment and workload Avg. invest. = Workload = K order qty./2 demand/order qty. 1 $15.81 31.62 orders 2 31.62 15.8 3 47.43 10.5 4 63.24 7.9 5 79.04 6.3 6 94.86 5.3 6.32 100.00 5.0 8 126.48 4.0 10 158.10 3.5 20 316.20 1.6 Inventory 15 Tradeoffs between investment and workload (cont.) $ 300 250 The optimal Avg. investment policy curve 200 150 100 50 0 5 10 15 20 25 30 Workload Inventory 16 Optimal policies for multi-item inventories Given only the sum of square roots of demand in $, you can compute aggregate workload and investment. Read Σ as “the sum of”: Investment formula Single-item Multi-item Q$ = K (demand in $)1/2 Σ Q$ = Σ K (demand in $)1/2 Σ Q$ = K Σ (demand in $)1/2 Q$ / 2 = (K/2) * (demand in $)1/2 Σ Q$ / 2 = K/2 Σ (demand in $)1/2 Inventory 17 Optimal policies for multi-item inventories (cont.) Workload (F) formulas Single-item Multi-item F = (demand in $) / Q$ Σ F = Σ (demand in $) / Q$ F = (1/K) * (demand in $)1/2 Σ F = 1/K Σ (demand in $)1/2 Inventory 18 Multi-item example 5,000 line-item inventory Σ (demand in dollars)1/2 = $250,000 K K/2 Investment 1/K Workload 1 0.5 $125,000 1.0 250,000 orders 2 1.0 250,000 0.5 125,000 5 2.5 625,000 0.2 50,000 10 5.0 1,250,000 0.1 25,000 Inventory 19 Multi-item example (cont.) For K = 5: avg. investment = Σ Q$/2 Σ Q$/2 = (K/2) Σ (demand in $)1/2 = 2.5 * 250,000 = 625,000 workload = 1/K Σ (demand in $)1/2 = 0.2 * $250,000 = 50,000 orders Inventory 20 Achieving management goals for investment Goal = Σ Q$/2 Goal = (K/2) * Σ (demand in $)1/2 Solving for K yields: K = (2 * goal) / Σ (demand in $)1/2 This value of K meets the investment goal exactly. The workload for that K is: Σ F = (1/K) * Σ (demand in $)1/2 = (Σ (demand in $)1/2)2 / (2 * goal) Inventory 21 Average inventory behavior with uncertain demand 12 On hand 11 10 9 8 Stock 7 Avg. inv. 6 5 4 ROP 3 2 SS 1 0 0 4 8 12 16 20 24 28 Day Inventory 22 Average inventory behavior with uncertain demand (cont.) Demand = 1 unit per day Leadtime = 2 days Leadtime demand (LTD) = 2 units Safety stock (SS) = 2 units Reorder point (ROP) = LTD + SS = 4 units Order quantity (Q) = 10 units Maximum inventory = Q + SS = 12 units Avg. investment = Q/2 + SS = 7 units Inventory 23 Reorder point with uncertain demand Assumption Length of leadtime is constant Concepts Reorder point = mean demand + safety during leadtime stock standard deviation Safety stock = safety factor * of forecast errors during leadtime Inventory 24 Reorder point with uncertain demand (cont.) The standard deviation is a measure of variability of the forecast errors. With a perfect forecast, the standard deviation is zero. As forecast accuracy gets worse, the standard deviation gets larger. The larger the safety factor, the smaller the risk of running out of stock. Inventory 25 Safety factor and probability of shortage z P(z) z P(z) 0.00 0.50000 2.30 0.01072 0.10 0.46017 2.40 0.00820 0.20 0.42074 2.50 0.00621 0.30 0.38209 2.60 0.00466 0.40 0.34458 2.70 0.00347 0.50 0.30854 2.80 0.00256 0.60 0.27425 2.90 0.00187 0.70 0.24196 3.00 0.00135 0.80 0.21186 3.10 0.00097 0.90 0.18406 3.20 0.00069 1.00 0.15866 3.30 0.00048 1.10 0.13567 3.40 0.00034 1.20 0.11507 3.50 0.00023 1.30 0.09680 3.60 0.00016 1.40 0.08076 3.70 0.00011 1.50 0.06681 3.80 0.00007 1.60 0.05480 3.90 0.00005 1.70 0.04457 4.00 0.00003 1.80 0.03593 4.10 0.00002 1.90 0.02872 4.20 0.00001 2.00 0.02275 4.30 0.00001 2.10 0.01786 4.40 0.00001 2.20 0.01390 4.50 0.00000 Inventory 26 Probability of shortage on one order cycle P(Z) = Probability demand will exceed Z standard deviations of safety stock on one order cycle Example: Mean demand during LT = 100 units Std. dev. = 20 units Safety factor (Z) = 1.5 Reorder point = mean demand + Z (std. dev.) during LT = 100 + 1.5 (20) = 130 units From table, P(Z) = .06681 Inventory ROP.xls 27 Probability of shortage on one order cycle (cont.) From table, P(Z) = .06681 .50 .43319 Z 0 1.5 X 100 130 Inventory 28 Number of annual shortage occurrences (SO) The probability of shortage on one order cycle is misleading since the ordering rate can vary widely across the inventory. A better measure of customer service is the number of annual shortage occurrences. Probability of Number of SO = shortage on one * annual order cycle order cycles Inventory 29 Number of annual shortage occurrences (SO) (cont.) Example: Safety factor = 1.5 Probability = .06681 Annual demand = 1000 units Order quantity = 50 units Number of annual order cycles = 1000/50 = 20 SO = .06681 * 20 = 1.34 Inventory 30 Units or dollars backordered as a service measure E(Z) = Expected units backordered for a distribution with mean = 0 and standard deviation = 1 on one order cycle E(Z)σ = Expected units backordered for a distribution with mean = X and standard deviation = σ on one order cycle Inventory 31 Units or dollars backordered as a service measure (cont.) Example: Annual demand = 1000 units Order quantity = 50 units X = 25 units σ=5 Z = 1.2 From table, E(Z) = .05610 Reorder point = 25 + 5 (1.2) = 31 Units short per cycle = .05610 (5) = .2805 Annual order cycles = 1000/50 = 20 Units short per year = 20 (.2805) = 5.61 Inventory 32 Quiz #1: Computing customer service measures Given: Annual demand = $2,000 Order quantity = $100 Mean demand during leadtime = $80 Standard deviation = $60 Suppose we want the probability of shortage on one order cycle to be .09680. Compute the following: Safety stock Reorder point Number of annual shortage occurrences Dollars backordered during one order cycle Dollars backordered per year Average cycle stock investment Average inventory investment Inventory 33 Quiz #2: Computing customer service measures For the same data as the previous problem, what reorder point will yield: a. 3 shortage occurrences per year? b. 5% of annual sales backordered? Inventory 34 Inventory tradeoff curves A variety of different workload and investment combinations yield exactly the same customer service. To develop a tradeoff curve for dollars backordered: 1. Compute and plot the optimal policy curve showing tradeoffs between cycle stock investment and workload. 2. Select a percentage goal for dollars backordered. 3. For each workload, compute the safety stock needed to meet the goal. 4. Add cycle stock to safety stock to obtain total investment. Inventory 35 Inventory tradeoff curves (cont.) $ “Isoservice curve” -- each point yields the same dollars backordered Investment Safety stock Optimal policy or cycle stock curve Workload Inventory 36 U.S. Navy application of tradeoff curves Inventory system 8 Naval supply centers Average inventory statistics at each center 80,000 line items $25 million investment Budget constraint Average investment limited to 2.5 months of stock Inventory 37 U.S. Navy application of tradeoff curves (cont.) Investment allocation strategies Old New Safety stock 1.5 months 1.0 months Cycle stock 1.0 months 1.5 months Total 2.5 2.5 Results of new allocation Reordering workload cut from 840,000 to 670,000 per year $2 million annual savings in manpower Inventory 38 Workload/service tradeoffs (Total investment fixed at 2.5 months) 90% New policy Previous policy Customer service 85% 80% 75% 0.0 0.5 1.0 1.5 2.0 Safety stock (months) 112 121 184 240 289 Workload (thousands of orders) Inventory 39 Strategic problems in managing distribution inventories 1. Controlling inventory growth as sales increase 2. Controlling inventory growth as new locations are added 3. Push vs. pull decision rules 4. Continuous review of stock balances vs. periodic review 5. Choosing a customer service measure Inventory 40 Inventory growth Inventories should grow at slower rate than sales. Why? Order quantities are proportional to the square root of sales. Example: One inventory item K=1 Q$ = K (demand in $)1/2 Sales Average Investment Sales Growth Q$ Investment Growth 100 --- 10 10/2 = 5 --- 200 100% 14 14/2 = 7 40% Inventory 41 Effects of adding inventory locations Inventories must increase as new locations are added. One reason is that forecasting is easier when customer demands are consolidated. Thus forecast errors are smaller and less safety stock is required. Another reason stems from the EOQ: One inventory item Sales of $100 K=1 With one location: Q$ = K (demand in $)1/2 Q$ = 1 (100)1/2 = 10 Average investment = 10/2 = 5 Inventory 42 Effects of adding inventory locations (cont.) With two locations: Location 1: Q$ = 1 (50)1/2 = 7.07 Location 2: Q$ = 1 (50)1/2 = 7.07 Average investment = (7.07 + 7.07) / 2 = 7.07 Investment increase = 7.05 – 5 = 2.05 Percentage increase = 2.05 / 5 = 41% Inventory 43 Continuous review vs. periodic review systems Continuous review Check stock balance after each transaction If stock on hand below reorder point, place new order in a fixed amount Periodic review Check stock balance on a periodic schedule If stock on hand below reorder point, place new order: in a fixed amount, or in a variable amount (maximum level – on hand) Investment requirements Periodic review always requires more investment than continuous review to meet any customer service goal. Inventory 44 Push vs. pull control systems Push or centralized system Central authority forecasts demand, sets stock levels, and pushes stock to each location. Pull or decentralized system Each location forecasts its own demand and sets its own stock levels. Investment requirements A pull system always requires more investment than a push system to meet any customer service goal. Inventory 45 Comparison of shortage values Inventory statistics 5,790 line items $45 million annual sales Shortage values at investment constraint of $5 million Shortage value Shortage Dollars minimized occurrences backordered shortage occurrences 1,120 $3.46 million dollars backordered 2,812 1.48 Inventory 46

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inventory systems, inventory system, supply chain, inventory control, deterministic models, demand systems, Principles of Inventory and Materials Management, lead time, Richard J. Tersine, International Journal of Production Economics

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posted: | 2/27/2010 |

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