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Distribution Inventory Systems Dr. Everette S. Gardner, Jr.

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Inventory2 Competing interests in inventory management Controller: Inventory investment Marketing manager: Operations manager: Customer service Stock replenishment workload

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Inventory3 Average inventory behavior with stable demand Stock Day 12 16 0 4 820 24 28 On hand Avg. inv. ROP 0 1 2 3 4 5 6 7 8 9 10 11 12

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Inventory4 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

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Inventory5 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

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Inventory6 The economic order quantity (cont.) Holding costs $ Total costs Ordering costs EOQ

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Inventory7 Economic order quantity (cont.) EOQ in units of stock Q U = ((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 EOQ.xls

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Inventory8 Economic order quantity (cont.) Example annual demand= 100 units cost per order= $10 holding cost per unit= $5 unit price= $25 holding rate=.20 Q U = ((2 * 100 * 10) / 5) 1/2 = 20 units Q$ = ((2 * 25 * 100 * 10) /.20) 1/2 = $500

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Inventory9 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.

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Inventory10 EOQ in perspective (cont.) Assumption: Reality: Total ordering costs Nbr. of orders In reality, holding costs depend on executive judgments on the cost of capital.

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Inventory11 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

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Inventory12 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

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Inventory13 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

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Inventory14 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

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Inventory15 Tradeoffs between investment and workload Avg. invest. =Workload = Korder qty./2 demand/order qty. 1$15.8131.62 orders 2 31.6215.8 3 47.4310.5 4 63.24 7.9 5 79.04 6.3 6 94.86 5.3 6.32100.00 5.0 8126.48 4.0 10158.10 3.5 20316.20 1.6

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Inventory16 Tradeoffs between investment and workload (cont.) $ 300 250 The optimal policy curve 200 150 100 50 0 5 10 15 20 25 30 Workload Avg. investment

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Inventory17 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

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Inventory18 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

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Inventory19 Multi-item example 5,000 line-item inventory Σ (demand in dollars) 1/2 = $250,000 K K/2 Investment 1/KWorkload 10.5 $125,0001.0250,000 orders 21.0 250,0000.5125,000 52.5 625,0000.2 50,000 105.0 1,250,0000.1 25,000

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Inventory20 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

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Inventory21 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)

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Inventory22 Average inventory behavior with uncertain demand Stock Day 12 16 0 4 820 24 28 On hand Avg. inv. ROP SS 0 1 2 3 4 5 6 7 8 9 10 11 12

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Inventory23 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

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Inventory24 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

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Inventory25 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.

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Inventory26 Safety factor and probability of shortage z P(z) z P(z) 0.000.500002.300.01072 0.100.460172.400.00820 0.200.420742.500.00621 0.300.382092.600.00466 0.400.344582.700.00347 0.500.308542.800.00256 0.600.274252.900.00187 0.700.241963.000.00135 0.800.211863.100.00097 0.900.184063.200.00069 1.000.158663.300.00048 1.100.135673.400.00034 1.200.115073.500.00023 1.300.096803.600.00016 1.400.080763.700.00011 1.500.066813.800.00007 1.600.054803.900.00005 1.700.044574.000.00003 1.800.035934.100.00002 1.900.028724.200.00001 2.000.022754.300.00001 2.100.017864.400.00001 2.200.01390 4.500.00000

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Inventory27 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 ROP.xls

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Inventory28 Probability of shortage on one order cycle (cont.) From table, P(Z) =.06681.50.43319 Z 0 1.5 X 100 130

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Inventory29 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 ofNumber of SO = shortage on one *annual order cycleorder cycles

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Inventory30 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

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Inventory31 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

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Inventory32 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

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Inventory33 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

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Inventory34 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?

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Inventory35 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.

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Inventory36 Inventory tradeoff curves (cont.) $“Isoservice curve” -- each point yields the same dollars backordered Safety stock Optimal policy or cycle stock curve Workload Investment

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Inventory37 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

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Inventory38 U.S. Navy application of tradeoff curves (cont.) Investment allocation strategies OldNew Safety stock1.5 months1.0 months Cycle stock1.0 months1.5 months Total2.52.5 Results of new allocation Reordering workload cut from 840,000 to 670,000 per year $2 million annual savings in manpower

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Inventory39 Workload/service tradeoffs Customer service New policy Previous policy 90% 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) (Total investment fixed at 2.5 months)

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Inventory40 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

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Inventory41 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 SalesAverageInvestment SalesGrowthQ $ InvestmentGrowth 100 ---1010/2 = 5 --- 200100%1414/2 = 7 40%

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Inventory42 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

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Inventory43 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%

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Inventory44 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.

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Inventory45 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.

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Inventory46 Comparison of shortage values Inventory statistics 5,790 line items $45 million annual sales Shortage values at investment constraint of $5 million Shortage valueShortageDollars minimized occurrencesbackordered shortage occurrences1,120$3.46 million dollars backordered2,812 1.48

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MBA 8452 Systems and Operations Management

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