Distribution Inventory Systems Dr Everette S Gardner Jr

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

Competing interests in inventory management Controller: Inventory investment Marketing manager: Customer service Operations manager: Stock replenishment workload Inventory 2

Average inventory behavior with stable demand 12 11 On hand 10 Stock 9 8 7 6 Avg. inv. 5 4 3 2 ROP 1 0 0 4 8 12 16 Day Inventory 20 24 28 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 = Demand/Order qty. Holding costs (HC) HC = Holding cost per unit per year x Avg. inv. balance = Order qty. /2 Total variable costs TVC = OC + HC Inventory 5

The economic order quantity (cont. ) $ Total costs s g in d l o t cos H 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 cost per order holding cost per unit price holding rate = = = 100 units $10 $5 $25. 20 QU = ((2 * 100 * 10) / 5)1/2 = 20 units Q$ = ((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 • 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 K 1 2 3 4 5 6 6. 32 8 10 20 Avg. invest. = order qty. /2 $15. 81 31. 62 47. 43 63. 24 79. 04 94. 86 100. 00 126. 48 158. 10 316. 20 Inventory Workload = demand/order qty. 31. 62 orders 15. 8 10. 5 7. 9 6. 3 5. 0 4. 0 3. 5 1. 6 15

Tradeoffs between investment and workload (cont. ) $ Avg. investment 300 The optimal policy curve 250 200 150 100 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 $)1/2 Σ Q$ / 2 = K/2 Σ (demand in $)1/2 Q$ / 2 = (K/2) * (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 1 2 5 10 K/2 0. 5 1. 0 2. 5 5. 0 Investment $125, 000 250, 000 625, 000 1, 250, 000 1/K 1. 0 0. 5 0. 2 0. 1 Inventory Workload 250, 000 orders 125, 000 50, 000 25, 000 19

Multi-item example (cont. ) For K = 5: avg. investment = Σ Q$/2 workload = (K/2) Σ (demand in $)1/2 = 2. 5 * 250, 000 = 625, 000 = 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 On hand 12 11 10 Stock 9 8 Avg. inv. 7 6 5 4 ROP 3 2 SS 1 0 0 4 8 12 16 Day Inventory 20 24 28 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 during leadtime + safety 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 0. 00 0. 10 0. 20 0. 30 0. 40 0. 50 0. 60 0. 70 0. 80 0. 90 1. 00 1. 10 1. 20 1. 30 1. 40 1. 50 1. 60 1. 70 1. 80 1. 90 2. 00 2. 10 2. 20 P(z) 0. 50000 0. 46017 0. 42074 0. 38209 0. 34458 0. 30854 0. 27425 0. 24196 0. 21186 0. 18406 0. 15866 0. 13567 0. 11507 0. 09680 0. 08076 0. 06681 0. 05480 0. 04457 0. 03593 0. 02872 0. 02275 0. 01786 0. 01390 z 2. 30 2. 40 2. 50 2. 60 2. 70 2. 80 2. 90 3. 00 3. 10 3. 20 3. 30 3. 40 3. 50 3. 60 3. 70 3. 80 3. 90 4. 00 4. 10 4. 20 4. 30 4. 40 4. 50 P(z) 0. 01072 0. 00820 0. 00621 0. 00466 0. 00347 0. 00256 0. 00187 0. 00135 0. 00097 0. 00069 0. 00048 0. 00034 0. 00023 0. 00016 0. 00011 0. 00007 0. 00005 0. 00003 0. 00002 0. 00001 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. 5 100 130 X 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. SO = Probability of shortage on one order cycle * Inventory Number of annual order cycles 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) = E(Z)σ = Expected units backordered for a distribution with mean = 0 and standard deviation = 1 on one order cycle 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 Safety stock months Cycle stock months Total Old 1. 5 months New 1. 0 months 1. 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% Customer service New policy Previous policy 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 $)1/2 Sales Growth Q$ Average Investment 100 200 --100% 10 14 10/2 = 5 14/2 = 7 Inventory Investment Growth --40% 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 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 minimized Shortage occurrences Dollars backordered shortage occurrences dollars backordered 1, 120 2, 812 $3. 46 million 1. 48 Inventory 46