# The EOQ Model - Dokuz Eylül University The EOQ Model To a pessimist, the glass is half empty. to an optimist, it is half full. Anonymous Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 1 EOQ History Introduced in 1913 by Ford W. Harris, How Many Parts to Make at Once Interest on capital tied up in wages, material and overhead sets a maximum limit to the quantity of parts which can be profitably manufactured at one time; set-up costs on the job fix the minimum. Experience has shown one manager a way to determine the economical size of lots. Early application of mathematical modeling to Scientific Management Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 2 MedEquip Example

Small manufacturer of medical diagnostic equipment. Purchases standard steel racks into which electronic components are mounted. Metal working shop can produce (and sell) racks more cheaply if they are produced in batches due to wasted time setting up shop. MedEquip doesnt want to tie up too much precious capital in inventory. Question: how many racks should MedEquip order at once? Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 3 EOQ Modeling Assumptions 1. Production is instantaneous there is no capacity constraint and the entire lot is produced simultaneously.

2. Delivery is immediate there is no time lag between production and availability to satisfy demand. 3. Demand is deterministic there is no uncertainty about the quantity or timing of demand. 4. Demand is constant over time in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5. A production run incurs a fixed setup cost regardless of the size of the lot or the status of the factory, the setup cost is constant. 6. Products can be analyzed singly either there is only a single product or conditions exist that ensure separability of products. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 4 Notation D demand rate (units per year) c unit production cost, not counting setup or inventory costs (dollars per unit) A fixed or setup cost to place an order (dollars) h holding cost (dollars per year); if the holding cost consists entirely of

interest on money tied up in inventory, then h = ic where i is an annual interest rate. Q the unknown size of the order or lot size Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com decision variable 5 Inventory Inventory vs Time in EOQ Model Q Q/D 2Q/D 3Q/D 4Q/D Time Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

6 Costs Holding Cost: Q 2 hQ annual holding cost 2 hQ unit holding cost 2D average inventory Setup Costs: A per lot, so unit setup cost A Q Production Cost: c per unit Cost Function: Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Y (Q ) hQ A c

2D Q http://www.factory-physics.com 7 MedEquip Example Costs D = 1000 racks per year c = \$250 A = \$500 (estimated from suppliers pricing) h = (0.1)(\$250) + \$10 = \$35 per unit per year i = 10% Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 8 Costs in EOQ Model 20.00 18.00 16.00 Cost (\$/unit) 14.00 12.00 Y(Q)

10.00 Q* =169 8.00 hQ/2D 6.00 4.00 2.00 A/Q c 0.00 0 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 100 200 300 400

Order Quantity (Q) http://www.factory-physics.com 500 9 Economic Order Quantity dY (Q) h A 2 0 dQ 2D Q Q* 2 AD h Q* 2(500)(1000) 169 35 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 EOQ Square Root Formula

http://www.factory-physics.com MedEquip Solution 10 EOQ Modeling Assumptions 1. Production is instantaneous there is no capacity constraint and the entire lot is produced simultaneously. relax via EPL model 2. Delivery is immediate there is no time lag between production and availability to satisfy demand. 3. Demand is deterministic there is no uncertainty about the quantity or timing of demand. 4. Demand is constant over time in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5. A production run incurs a fixed setup cost regardless of the size of the lot or the status of the factory, the setup cost is constant. 6. Products can be analyzed singly either there is only a single product or conditions exist that ensure separability of products. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

11 Notation EPL Model D demand rate (units per year) P production rate (units per year), where P>D c unit production cost, not counting setup or inventory costs (dollars per unit) A fixed or setup cost to place an order (dollars) h holding cost (dollars per year); if the holding cost consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate. Q the unknown size of the production lot size Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com decision variable 12 Inventory vs Time in EPL Model Production run of Q takes Q/P time units Inventory

(P-D)(Q/P) -D P-D (P-D)(Q/P)/2 Time Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 13 Solution to EPL Model Annual Cost Function: Y (Q ) AD h (1 D / P )Q Dc Q 2 setup holding production

Solution (by taking derivative and setting equal to zero): tends to EOQ as P Q* 2 AD h(1 D / P ) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 otherwise larger than EOQ because replenishment takes longer http://www.factory-physics.com 14 The Key Insight of EOQ There is a tradeoff between lot size and inventory Order Frequency: Inventory Investment: Wallace J. Hopp, Mark L. Spearman, 1996, 2000 F D Q

cQ cD I 2 2F http://www.factory-physics.com 15 EOQ Tradeoff Curve Inventory Investment 50 45 40 35 30 25 20 15 10 5 0 0 20

40 60 80 100 Order/Year Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 16 Sensitivity of EOQ Model to Quantity Optimal Unit Cost: hQ* A Y Y (Q ) 2 D Q* * * We neglect unit cost, c,

since it does not affect Q* h 2 AD h A 2D 2 AD h 2A 2 AD h Optimal Annual Cost: Multiply Y* by D and simplify, Annual Cost 2 ADh Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 17 Sensitivity of EOQ Model to Quantity (cont.) Annual Cost from Using Q': Y (Q) hQ AD 2 Q Ratio: Cost (Q) Y (Q) hQ 2 AD Q 1 Q Q*

* Cost (Q * ) Y (Q * ) 2Q Q 2 ADh Example: If Q' = 2Q*, then the ratio of the actual to optimal cost is (1/2)[2 + (1/2)] = 1.25 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 18 Sensitivity of EOQ Model to Order Interval Order Interval: Let T represent time (in years) between orders (production runs) Q T D Optimal Order Interval: 2 AD Q* h 2A *

T D D hD Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 19 Sensitivity of EOQ Model to Order Interval (cont.) Ratio of Actual to Optimal Costs: If we use T' instead of T* annual cost under T 1 T T * annual cost under T * 2 T * T Powers-of-Two Order Intervals: The optimal order interval, T* must lie within a multiplicative factor of 2 of a power-of-two. Hence, the maximum error from using the best power-of-two is 1 1 2 1.06 2 2 Wallace J. Hopp, Mark L. Spearman, 1996, 2000

http://www.factory-physics.com 20 The Root-Two Interval 2m T1* divide by less than 2 to get to 2m Wallace J. Hopp, Mark L. Spearman, 1996, 2000 2m 2 T2* 2 m 1 multiply by less than 2 to get to 2m+1 http://www.factory-physics.com

21 Medequip Example Optimum: Q*=169, so T*=Q*/D =169/1000 years = 62 days Y (Q*) hQ * AD 35(169) 500(1000) \$5,916 2 Q* 2 169 Round to Nearest Power-of-Two: 62 is between 32 and 64, but since 322=45.25, it is closest to 64. So, round to T=64 days or Q= TD=(64/365)1000=175. Y (Q ' ) hQ ' AD 35(175) 500(1000) \$5,920 2 Q' 2 175

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com Only 0.07% error because we were lucky and happened to be close to a powerof-two. But we cant do worse than 6%. 22 Powers-of-Two Order Intervals Order Interval 1 2 0 Week 0 1 2 3 4 5

6 7 8 2 21 4 2 2 8 23 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 23 EOQ Takeaways Batching causes inventory (i.e., larger lot sizes translate into more stock). Under specific modeling assumptions the lot size that optimally balances holding and setup costs is given by the square root formula: 2 AD Q*

h Total cost is relatively insensitive to lot size (so rounding for other reasons, like coordinating shipping, may be attractive). Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 24 The Wagner-Whitin Model Change is not made without inconvenience, even from worse to better. Robert Hooker Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 25 EOQ Assumptions 1. Instantaneous production. 2. Immediate delivery. 3. Deterministic demand.

4. Constant demand. WW model relaxes this one 5. Known fixed setup costs. 6. Single product or separable products. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 26 Dynamic Lot Sizing Notation t a time period (e.g., day, week, month); we will consider t = 1, ,T, where T represents the planning horizon. Dt demand in period t (in units) ct unit production cost (in dollars per unit), not counting setup or inventory costs in period t At fixed or setup cost (in dollars) to place an order in period t ht holding cost (in dollars) to carry a unit of inventory from period t to period t +1 Qt the unknown size of the order or lot size in period t Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

decision variables 27 Wagner-Whitin Example Data t Dt ct At ht 1 2 3 4 5 6 7 8 9 10 20 50 10 50 50 10 20 40 20 30 10 10 10 10 10 10 10 10 10 10 100 100 100 100 100 100 100 100 100 100 1 1 1 1

1 1 1 1 1 1 Lot-for-Lot Solution t Dt Qt It Setup cost Holding cost Total cost 1 20 20 0 100 0 100 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 2 50 50 0

100 0 100 3 10 10 0 100 0 100 4 50 50 0 100 0 100 5 50 50 0 100 0 100 6 10

10 0 100 0 100 7 20 20 0 100 0 100 http://www.factory-physics.com 8 40 40 0 100 0 100 9 20 20 0 100 0

100 10 30 30 0 100 0 100 Total 300 300 0 1000 0 1000 28 Wagner-Whitin Example (cont.) Fixed Order Quantity Solution t Dt Qt It Setup cost Holding cost Total cost 1

20 100 80 100 80 180 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 2 50 0 30 0 30 30 3 4 5 10 50 50 0 100 0 20 70 20 0 100 0 20 70 20 20 170 20 6 7 8

10 20 40 0 100 0 10 90 50 0 100 0 10 90 50 10 190 50 http://www.factory-physics.com 9 20 0 30 0 30 30 10 Total 30 300 0 300 0 0 0 300 0 400 0 700 29 Wagner-Whitin Property Under an optimal lot-sizing policy either the inventory carried to

period t+1 from a previous period will be zero or the production quantity in period t+1 will be zero. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 30 Basic Idea of Wagner-Whitin Algorithm By WW Property I, either Qt=0 or Qt=D1++Dk for some k. If jk* = last period of production in a k period problem then we will produce exactly Dk+...+DT in period jk*. We can then consider periods 1, , jk*-1 as if they are an independent jk*1 period problem. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 31 Wagner-Whitin Example Step 1: Obviously, just satisfy D1 (note we are neglecting production cost, since it is fixed). Z1* A1 100 j1* 1 Step 2: Two choices, either j2* = 1 or j2* = 2. A h D , produce in 1

Z 2* min *1 1 2 Z1 A2 , produce in 2 100 1(50) 150 min 100 100 200 150 j2* 1 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 32 Wagner-Whitin Example (cont.) Step3: Three choices, j3* = 1, 2, 3. A1 h1 D2 (h1 h2 ) D3 , produce in 1 Z 3* min Z1* A2 h2 D3 , produce in 2 Z*2 A3 , produce in 3 100 1(50) (1 1)10 170 min 100 100 (1)10 210 150 100 250 170 j3* 1

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 33 Wagner-Whitin Example (cont.) Step 4: Four choices, j4* = 1, 2, 3, 4. A1 h1 D2 (h1 h2 ) D3 (h1 h2 h3 ) D4 , produce in 1 Z* A h D ( h h ) D , produce in 2 1 2 2 3 2 3 4 * Z 4 min * produce in 3 Z 2 A3 h3 D4 , Z*3 A4 , produce in 4 100 1(50) (1 1)10 (1 1 1)50 320 100 100 (1)10 (1 1)50 310 min 300 150 100 (1)50

170 100 270 270 j4* 4 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 34 Planning Horizon Property If jt*=t, then the last period in which production occurs in an optimal t+1 period policy must be in the set t, t+1,t+1. In the Example: We produce in period 4 for period 4 of a 4 period problem. We would never produce in period 3 for period 5 in a 5 period problem. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 35 Wagner-Whitin Example (cont.) Step 5: Only two choices, j5* = 4, 5. Z 3* A4 h4 D5 , produce in 4 Z min * produce in 5 Z 4 A5 ,

170 100 1(50) 320 min 370 270 100 320 * 5 j5* 4 Step 6: Three choices, j6* = 4, 5, 6. And so on. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 36 Wagner-Whitin Example Solution Last Period with Production 1 2 3 4 5 6 7 8

9 10 Zt jt Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Planning Horizon (t) 1 2 3 4 5 6 7 8 100 150 170 320 200 210 310 250 300 270 320 340 400 560 370 380 420 540 420 440 520 440 480 500 100 150 170 270 320 340 400 480 1 1

1 4 4 4 4 7 9 10 520 520 580 520 610 580 610 620 580 7 or 8 8

Produce in period 1 Produce in period 4 Produce in period 8 for 1, 2, 3 (20 + 50 + for 4, 5, 6, 7 (50 + 50 + for 8, 9, 10 (40 + 20 + 10 = 80 units) 10 + 20 = 130 units) 30 = 90 units 37 http://www.factory-physics.com Wagner-Whitin Example Solution (cont.) Optimal Policy: Produce in period 8 for 8, 9, 10 (40 + 20 + 30 = 90 units) Produce in period 4 for 4, 5, 6, 7 (50 + 50 + 10 + 20 = 130 units) Produce in period 1 for 1, 2, 3 (20 + 50 + 10 = 80 units) t Dt Qt It Setup cost Holding cost Total cost 1 20 80 60 100 60 160

2 50 0 10 0 10 10 3 4 5 10 50 50 0 130 0 0 80 30 0 100 0 0 80 30 0 180 30 6 10 0 20 0 20 20 7 8 9

20 40 20 0 90 0 0 50 30 0 100 0 0 50 30 0 150 30 10 Total 30 300 0 300 0 0 0 300 0 280 0 580 Note: we produce in 7 for an 8 period problem, but this never comes into play in optimal solution. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 38 Problems with Wagner-Whitin 1. Fixed setup costs. 2. Deterministic demand and production (no uncertainty) 3. Never produce when there is inventory (WW Property I). safety stock (don't let inventory fall to zero) random yields (can't produce for exact no. periods)

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 39 Statistical Reorder Point Models When your pills get down to four, Order more. Anonymous, from Hadley &Whitin Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 40 EOQ Assumptions 1. Instantaneous production. EPL model relaxes this one lags can be added to EOQ or other models 2. Immediate delivery. 3. Deterministic demand. newsvendor and (Q,r) relax this one

WW model relaxes this one 4. Constant demand. 5. Known fixed setup costs. can use constraint approach 6. Single product or separable products. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Chapter 17 extends (Q,r) to multiple product cases http://www.factory-physics.com 41 Modeling Philosophies for Handling Uncertainty 1. Use deterministic model adjust solution - EOQ to compute order quantity, then add safety stock - deterministic scheduling algorithm, then add safety lead time 2. Use stochastic model - news vendor model - base stock and (Q,r) models - variance constrained investment models Wallace J. Hopp, Mark L. Spearman, 1996, 2000

http://www.factory-physics.com 42 The Newsvendor Approach Assumptions: 1. single period 2. random demand with known probability distribution 3. linear overage/shortage costs 4. minimum expected cost criterion Examples: newspapers or other items with rapid obsolescence Christmas trees or other seasonal items capacity for short-life products Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 43 Newsvendor Model Notation X demand (in units), a random variable. G ( x) P( X x), cumulative distribution function of demand (assumed continuous.) g ( x) d

G ( x) density function of demand. dx co cost (in dollars) per unit left over after demand is realized. cs cost (in dollars) per unit of shortage. Q production/order quantity (in units); this is the decision variable. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 44 Newsvendor Model Cost Function: Y ( x) expected overage expected shortage cost co E units over c s E units short Note: for any given day, we will be either over or short, not both. But in expectation, overage and shortage can both be positive. co max Q x,0 g ( x)dx c s max x Q,0 g ( x)dx

0 co 0 Q (Q x) g ( x)dx c ( x Q) g ( x)dx 0 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 s Q http://www.factory-physics.com 45 Newsvendor Model (cont.) Optimal Solution: taking derivative of Y(Q) with respect to Q, setting equal to zero, and solving yields:

G (Q * ) P X Q * cs co c s Critical Ratio is probability stock covers demand Notes: Q* co Q * cs 1 G(x) cs co c s Q* Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 46

Newsvendor Example T Shirts Scenario: Demand for T-shirts is exponential with mean 1000 (i.e., G(x) = P(X x) = 1- e-x/1000). (Note - this is an odd demand distribution; Poisson or Normal would probably be better modeling choices.) Cost of shirts is \$10. Selling price is \$15. Unsold shirts can be sold off at \$8. Model Parameters: cs = 15 10 = \$5 co = 10 8 = \$2 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 47 Newsvendor Example T Shirts (cont.) Solution: * G (Q ) 1 e Q 1000

cs 5 0.714 co cs 2 5 Q* 1,253 Sensitivity: If co = \$10 (i.e., shirts must be discarded) then * G (Q ) 1 e Q 1000 cs 5 0.333 co cs 10 5 Q* 405

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 48 Newsvendor Model with Normal Demand Suppose demand is normally distributed with mean and standard deviation . Then the critical ratio formula reduces to: 3.00 c Q* G (Q * ) s co c s c Q* z where ( z ) s co c s Q* z Wallace J. Hopp, Mark L. Spearman, 1996, 2000 (z)z) 0.00

1 7 13 19 25 31 37 43 49 55 61 Note: Q* increases in both and if z is positive (i.e., if ratio is greater than 0.5). http://www.factory-physics.com 67 73

79 85 0 91 97 103 109 115 121 127 133 139 145 151 157 z 49 Multiple Period Problems Difficulty: Technically, Newsvendor model is for a single period. Extensions: But Newsvendor model can be applied to multiple period situations, provided: demand during each period is iid, distributed according to G(x) there is no setup cost associated with placing an order stockouts are either lost or backordered Key: make sure co and cs appropriately represent overage and shortage cost. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

50 Example Scenario: GAP orders a particular clothing item every Friday mean weekly demand is 100, std dev is 25 wholesale cost is \$10, retail is \$25 holding cost has been set at \$0.5 per week (to reflect obsolescence, damage, etc.) Problem: how should they set order amounts? Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 51 Example (cont.) Newsvendor Parameters: c0 = \$0.5 cs = \$15 Solution:

15 0.9677 0.5 15 Q 100 0.9677 25 Q 100 1.85 25 Q 100 1.85(25) 146 G (Q * ) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Every Friday, they should order-up-to 146, that is, if there are x on hand, then order 146-x. http://www.factory-physics.com 52 Newsvendor Takeaways Inventory is a hedge against demand uncertainty.

Amount of protection depends on overage and shortage costs, as well as distribution of demand. If shortage cost exceeds overage cost, optimal order quantity generally increases in both the mean and standard deviation of demand. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 53 The (Q,r) Approach Assumptions: 1. Continuous review of inventory. 2. Demands occur one at a time. 3. Unfilled demand is backordered. 4. Replenishment lead times are fixed and known. Decision Variables: Reorder Point: r affects likelihood of stockout (safety stock). Order Quantity: Q affects order frequency (cycle inventory). Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

54 Inventory Inventory vs Time in (Q,r) Model Q r l Time Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 55 Base Stock Model Assumptions 1. There is no fixed cost associated with placing an order. 2. There is no constraint on the number of orders that can be placed per year. That is, we can replenish one at a time (Q=1). Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

56 Base Stock Notation Q r R = 1, order quantity (fixed at one) = reorder point = r +1, base stock level l = delivery lead time = mean demand during l = std dev of demand during l p(x) = Prob{demand during lead time l equals x} G(x) h

b S(R) B(R) I(R) = = = = = = Prob{demand during lead time l is less than x} unit holding cost unit backorder cost average fill rate (service level) average backorder level average on-hand inventory level Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 57 Inventory Balance Equations Balance Equation: inventory position = on-hand inventory - backorders + orders Under Base Stock Policy

inventory position = R Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 58 Inventory Profile for Base Stock System (R=5) 7 6 R 5 r 4 l On Hand Inventory Backorders Orders Inventory Position 3

2 1 0 0 5 10 15 20 25 30 Time Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 35 59 Service Level (Fill Rate) Let:

X = (random) demand during lead time l so E[X] = . Consider a specific replenishment order. Since inventory position is always R, the only way this item can stock out is if X R. Expected Service Level: G ( R), if G is continuous S ( R ) P ( X R ) G ( R 1) G (r ), if G is discrete Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 60 Backorder Level Note: At any point in time, number of orders equals number demands during last l time units (X) so from our previous balance equation: R = on-hand inventory - backorders + orders on-hand inventory - backorders = R - X Note: on-hand inventory and backorders are never positive at the same time, so if X=x, then if x R 0, backorders x R, if x R simpler version for Expected Backorder Level: spreadsheet computing

B( R) ( x R) p( x) p( R) ( R)[1 G( R)] x R Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 61 Inventory Level Observe: on-hand inventory - backorders = R-X E[X] = from data E[backorders] = B(R) from previous slide Result: I(R) = R - + B(R) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 62 Base Stock Example l = one month

= 10 units (per month) Assume Poisson demand, so 10 k e 10 G ( x) p (k ) k! k 0 k 0 x Wallace J. Hopp, Mark L. Spearman, 1996, 2000 x http://www.factory-physics.com Note: Poisson demand is a good choice when no variability data is available. 63 Base Stock Example Calculations R 0 1 2 3 4

5 6 7 8 9 10 11 p(R) 0.000 0.000 0.002 0.008 0.019 0.038 0.063 0.090 0.113 0.125 0.125 0.114 G(R) 0.000 0.000 0.003 0.010 0.029 0.067 0.130

0.220 0.333 0.458 0.583 0.697 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 B(R) 10.000 9.000 8.001 7.003 6.014 5.043 4.110 3.240 2.460 1.793 1.251 0.834 R 12 13 14 15 16 17 18

19 20 21 22 23 p(R) 0.095 0.073 0.052 0.035 0.022 0.013 0.007 0.004 0.002 0.001 0.000 0.000 http://www.factory-physics.com G(R) 0.792 0.864 0.917 0.951 0.973 0.986 0.993

0.997 0.998 0.999 0.999 1.000 B(R) 0.531 0.322 0.187 0.103 0.055 0.028 0.013 0.006 0.003 0.001 0.000 0.000 64 Base Stock Example Results Service Level: For fill rate of 90%, we must set R-1= r =14, so R=15 and safety stock s = r- = 4. Resulting service is 91.7%. Backorder Level: B(R) = B(15) = 0.103 Inventory Level: I(R) = R - + B(R) = 15 - 10 + 0.103 = 5.103

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 65 Optimal Base Stock Levels Objective Function: holding plus backorder cost Y(R) = hI(R) + bB(R) = h(R-+B(R)) + bB(R) = h(R- ) + (h+b)B(R) Solution: if we assume G is continuous, we can use calculus to get b G( R* ) h b Implication: set base stock level so fill rate is b/(h+b). Note: R* increases in b and decreases in h. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 66 Base Stock Normal Approximation

If G is normal(,), then b G ( R*) h b R* z where (z)=b/(h+b). So R* = + z Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Note: R* increases in and also increases in provided z>0. http://www.factory-physics.com 67 Optimal Base Stock Example Data: Approximate Poisson with mean 10 by normal with mean 10 units/month and standard deviation 10 = 3.16 units/month. Set h=\$15, b=\$25.

Calculations: b 25 0.625 h b 15 25 since (0.32) = 0.625, z=0.32 and hence R* = + z = 10 + 0.32(3.16) = 11.01 11 Observation: from previous table fill rate is G(10) = 0.583, so maybe backorder cost is too low. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 68 Inventory Pooling Situation: n different parts with lead time demand normal(, ) z=2 for all parts (i.e., fill rate is around 97.5%) Specialized Inventory: cycle stock safety stock base stock level for each item = + 2

total safety stock = 2n Pooled Inventory: suppose parts are substitutes for one another lead time demand is normal (n ,n ) base stock level (for same service) = n +2 n ratio of safety stock to specialized safety stock = 1/ n Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 69 Effect of Pooling on Safety Stock Conclusion: cycle stock is not affected by pooling, but safety stock falls dramatically. So, for systems with high safety stock, pooling (through product design, late customization, etc.) can be an attractive strategy. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 70 Pooling Example

PCs consist of 6 components (CPU, HD, CD ROM, RAM, removable storage device, keyboard) 3 choices of each component: 36 = 729 different PCs Each component costs \$150 (\$900 material cost per PC) Demand for all models is Poisson distributed with mean 100 per year Replenishment lead time is 3 months (0.25 years) Use base stock policy with fill rate of 99% Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

71 Pooling Example - Stock PCs Base Stock Level for Each PC: = 100 0.25 = 25, so using Poisson formulas, G(R-1) 0.99 R = 38 units On-Hand Inventory for Each PC: I(R) = R - + B(R) = 38 - 25 + 0.0138 = 13.0138 units Total On-Hand Inventory : 13.0138 729 \$900 = \$8,538,358 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 72 Pooling Example - Stock Components 729 models of PC 3 types of each comp. Necessary Service for Each Component: S = (0.99)1/6 = 0.9983 Base Stock Level for Components: = (100 729/3)0.25 = 6075, so G(R-1) 0.9983

R = 6306 On-Hand Inventory Level for Each Component: I(R) = R - + B(R) = 6306-6075+0.0363 = 231.0363 units Total On-Hand Inventory : 231.0363 18 \$150 = \$623,798 93% reduction! Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 73 Base Stock Insights 1. Reorder points control prob of stockouts by establishing safety stock. 2. To achieve a given fill rate, the required base stock level (and hence safety stock) is an increasing function of mean and (provided backorder cost exceeds shortage cost) std dev of demand during replenishment lead time. 3. The optimal fill rate is an increasing in the backorder cost and a decreasing in the holding cost. We can use either a service constraint

or a backorder cost to determine the appropriate base stock level. 4. Base stock levels in multi-stage production systems are very similar to kanban systems and therefore the above insights apply. 5. Base stock model allows us to quantify benefits of inventory pooling. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 74 The Single Product (Q,r) Model Motivation: Either 1. Fixed cost associated with replenishment orders and cost per backorder. 2. Constraint on number of replenishment orders per year and service constraint. Objective: Under (1) min fixed setup cost holding cost backorder or stockout cost Q,r As in EOQ, this makes batch production attractive.

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 75 Summary of (Q,r) Model Assumptions 1. One-at-a-time demands. 2. Demand is uncertain, but stationary over time and distribution is known. 3. Continuous review of inventory level. 4. Fixed replenishment lead time. 5. Constant replenishment batch sizes. 6. Stockouts are backordered. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 76 (Q,r) Notation D expected demand per year replenishment lead time (assumed constant) X (random) demand during replenishment lead time E[ X ] expected demand during replenishment lead time standard deviation of demand during replenishment lead time p(x) P ( X x) pmf of demand during lead time

G ( x) P ( X x ) cdf of demand during lead time A fixed cost per order c unit cost of an item h annual unit holding cost k cost per stockout b annual unit backorder cost Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 77 (Q,r) Notation (cont.) Decision Variables: Q r s order quantity reorder point r safety stock implied by r Performance Measures: F (Q) average order frequency S (Q, r ) average service level (fill rate) B(Q, r ) average backorder level I (Q, r ) average inventory level Wallace J. Hopp, Mark L. Spearman, 1996, 2000

http://www.factory-physics.com 78 Inventory and Inventory Position for Q=4, r=4 9 Inventory Position uniformly distributed between r+1=5 and r+Q=8 8 7 6 Quantity 5 4 3 2 1 0 -1 0 2

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

-2 Time Inventory Position Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Net Inventory http://www.factory-physics.com 79 Costs in (Q,r) Model Fixed Setup Cost: AF(Q) Stockout Cost: kD(1-S(Q,r)), where k is cost per stockout Backorder Cost: bB(Q,r) Inventory Carrying Costs: cI(Q,r) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 80 Fixed Setup Cost in (Q,r) Model Observation: since the number of orders per year is D/Q, F(Q) Wallace J. Hopp, Mark L. Spearman, 1996, 2000

D Q http://www.factory-physics.com 81 Stockout Cost in (Q,r) Model Key Observation: inventory position is uniformly distributed between r+1 and r+Q. So, service in (Q,r) model is weighted sum of service in base stock model. Result: 1 r Q 1 S (Q, r ) G ( x 1) [G (r ) G (r Q 1)] Q x r 1 Q Note: this form is easier to use 1 S (Q, r ) 1 [ B (r ) B(r Q)] in spreadsheets because it does Q not involve a sum. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

82 Service Level Approximations Type I (base stock): S (Q, r ) G (r ) Note: computes number of stockouts per cycle, underestimates S(Q,r) Type II: B(r ) S (Q, r ) 1 Q Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Note: neglects B(r,Q) term, underestimates S(Q,r) http://www.factory-physics.com 83 Backorder Costs in (Q,r) Model Key Observation: B(Q,r) can also be computed by averaging base stock backorder level function over the range [r+1,r+Q]. Result: 1 r Q

1 B (Q, r ) B ( x) [ B (r 1) B (r Q)] Q x r 1 Q Notes: 1. B(Q,r) B(r) is a base stock approximation for backorder level. 2. If we can compute B(x) (base stock backorder level function), then we can compute stockout and backorder costs in (Q,r) model. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 84 Inventory Costs in (Q,r) Model Approximate Analysis: on average inventory declines from Q+s to s+1 so I (Q, r ) (Q s ) ( s 1) Q 1 Q 1 s r 2 2 2 Exact Analysis: this neglects backorders, which add to average inventory since on-hand inventory can never go below zero. The

corrected version turns out to be I (Q, r ) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Q 1 r B (Q, r ) 2 http://www.factory-physics.com 85 Inventory vs Time in (Q,r) Model Expected Inventory Actual Inventory Inventory s+Q Exact I(Q,r) = Approx I(Q,r) + B(Q,r) Approx I(Q,r) r s+1=r-+1

Time Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 86 Expected Inventory Level for Q=4, r=4, =2 7 Inventory Level s+Q 6 5 4 3 s 2 1 0 0 5 10 15

20 25 30 35 Time Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 87 (Q,r) Model with Backorder Cost Objective Function: D Y (Q, r ) A bB(Q, r ) hI (Q, r ) Q Approximation: B(Q,r) makes optimization complicated because it depends on both Q and r. To simplify, approximate with base stock backorder formula, B(r): ~ D Q 1 Y (Q, r ) Y (Q, r ) A bB(r ) h( r B (r )) Q

2 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 88 Results of Approximate Optimization Assumptions: Q,r can be treated as continuous variables G(x) is a continuous cdf Results: 2 AD h b G (r*) h b Q* Note: this is just the EOQ formula r* z Note: this is just the base stock formula

if G is normal(,), where (z)=b/(h+b) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 89 (Q,r) Example Stocking Repair Parts: D = 14 units per year c = \$150 per unit h = 0.1 150 + 10 = \$25 per unit l = 45 days = (14 45)/365 = 1.726 units during replenishment lead time A = \$10 b = \$40 Demand during lead time is Poisson Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 90 Values for Poisson() Distribution r p(r)

G(r) B(r) 0 1 2 3 4 5 6 7 8 9 10 0.178 0.307 0.265 0.153 0.066 0.023 0.007 0.002 0.000 0.000 0.000 0.178 0.485

0.750 0.903 0.969 0.991 0.998 1.000 1.000 1.000 1.000 1.726 0.904 0.389 0.140 0.042 0.011 0.003 0.001 0.000 0.000 0.000 91 Calculations for Example Q* 2 AD 2(10)(14)

4.3 4 h 15 b 40 0.615 h b 25 40 (0.29) 0.615, so z 0.29 r* z 1.726 0.29(1.314) 2.107 2 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 92 Performance Measures for Example F (Q*) D 14 3.5 Q* 4 1 1 [ B (r*) B (r * Q*)] 1 [ B (2) B (2 4)] Q*

Q 1 1 [0.389 0.003] 0.904 4 S(Q * ,r * ) 1 1 r * Q * 1 B (Q*, r*) B ( x ) [ B (3) B (4) B (5) B (6)] Q * x r *1 Q 1 [0.140 0.042 0.011 0.003] 0.049 4 I (Q*, r*) Q * 1 4 1 r * B (Q*, r*) 2 1.726 0.049 2.823 2 2

Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 93 Observations on Example Orders placed at rate of 3.5 per year Fill rate fairly high (90.4%) Very few outstanding backorders (0.049 on average) Average on-hand inventory just below 3 (2.823) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 94

Varying the Example Change: suppose we order twice as often so F=7 per year, then Q=2 and: S (Q, r ) 1 1 1 [ B(r ) B(r Q)] 1 [0.389 0.042] 0.826 Q 2 which may be too low, so increase r from 2 to 3: S (Q, r ) 1 1 1 [ B(r ) B(r Q)] 1 [0.140 0.011] 0.936 Q 2 This is better. For this policy (Q=2, r=4) we can compute B(2,3)=0.026, I(Q,r)=2.80. Conclusion: this has higher service and lower inventory than the original policy (Q=4, r=2). But the cost of achieving this is an extra 3.5 replenishment orders per year. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com

95 (Q,r) Model with Stockout Cost Objective Function: D Y (Q, r ) A kD(1 S (Q, r )) hI (Q, r ) Q Approximation: Assume we can still use EOQ to compute Q* but replace S(Q,r) by Type II approximation and B(Q,r) by base stock approximation: ~ D B(r ) Q 1 Y (Q, r ) Y (Q, r ) A kD h( r B (r )) Q Q 2 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 96 Results of Approximate Optimization Assumptions:

Q,r can be treated as continuous variables G(x) is a continuous cdf Results: 2 AD Q* h kD G (r*) kD hQ Note: this is just the EOQ formula r* z if G is normal(,), where (z)=kD/(kD+hQ) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com Note: another version of base stock formula (only z is different) 97 Backorder vs. Stockout Model

Backorder Model when real concern is about stockout time because B(Q,r) is proportional to time orders wait for backorders useful in multi-level systems Stockout Model when concern is about fill rate better approximation of lost sales situations (e.g., retail) Note: We can use either model to generate frontier of solutions Keep track of all performance measures regardless of model B-model will work best for backorders, S-model for stockouts Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 98 Lead Time Variability Problem: replenishment lead times may be variable, which increases variability of lead time demand. Notation: L = replenishment lead time (days), a random variable l = E[L] = expected replenishment lead time (days) L= std dev of replenishment lead time (days)

Dt = demand on day t, a random variable, assumed independent and identically distributed d = E[Dt] = expected daily demand D= std dev of daily demand (units) Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 99 Including Lead Time Variability in Formulas Standard Deviation of Lead Time Demand: if demand is Poisson D2 d 2 L2 d 2 L2 Inflation term due to lead time variability Modified Base Stock Formula (Poisson demand case): R z z d 2 L2 Wallace J. Hopp, Mark L. Spearman, 1996, 2000 Note: can be used in any base stock or (Q,r) formula as before. In general, it will inflate safety stock. http://www.factory-physics.com

100 Single Product (Q,r) Insights Basic Insights: Safety stock provides a buffer against stockouts. Cycle stock is an alternative to setups/orders. Other Insights: 1. Increasing D tends to increase optimal order quantity Q. 2. Increasing tends to increase the optimal reorder point. (Note: either increasing D or l increases .) 3. Increasing the variability of the demand process tends to increase the optimal reorder point (provided z > 0). 4. Increasing the holding cost tends to decrease the optimal order quantity and reorder point. Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 101