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TM 631 Optimization Fall 2006 Dr. Frank Joseph Matejcik 13th Session: Ch. 18 Inventory Theory 12/8/08 Frank Matejcik SD School of Mines & Technology 1 Activities • Review assignments and resources • Exam Preview • No Assignment this week • Chapter 18 alternative Frank Matejcik SD School of Mines & Technology 2 Tentative Schedule Chapters Assigned Chapters Assigned 9/01/2008 Holiday 11/17/2008 9 9.3-3, 9.4-1, 9.5-6 9/08/2008 1, 2 ________ 11/24/2008 9 9.6-1, 9.8-1 9/15/2008 3 3.1-8,3.2-4,3.6-3 12/01/2008 27 27.4-2, 27.5-3, 9/22/2008 4 4.3-6, 4.4-6, 4.7-6 27.6-2, 27.6-3, & 27.8-2 9/29/2008 6 6.3-1, 6.3-5, 12/08/2008 18 None and 6.8-3(abce) 12/15/2008 Final 10/06/2008 Exam 1 10/13/2008 Holiday 10/20/2008 8 8.1-5, 8.1-6, 8.2-6, 8.2-7(ab), 8.2-8 10/27/2008 8 8.4 Answers in Slides 11/03/2008 21 11/10/2008 Exam 2 Frank Matejcik SD School of Mines & Technology 3 Web Resources • Class Web site on the HPCnet system • http://sdmines.sdsmt.edu/sdsmt/directory/co urses/2008fa/tm631M021 • May use D2L, also. It has a password protection • Answers found from the Fall 2006 site http://sdmines.sdsmt.edu/sdsmt/directory/co urses/2006fa/tm631021 Look at the Answer Link and the User guide and password for the answer link. Frank Matejcik SD School of Mines & Technology 4 Exam Study Guide Short • Just problems • Chapter 9 – Shortest Path Network – Minimum Spanning Tree – Maximum Flow • Chapter 27 – Last-Value Forecasting Method – Averaging Forecasting Method – Moving-Average Frank Matejcik SD School of Mines & Technology 5 Exam Study Guide Short • Chapter 27 continued – Exponential Smoothing – Seasonally Adjusted Time Series – Seasonal factors – Exponential Smoothing Method for a Linear Trend – Forecasting Errors MAD, MSE – Linear Regression • Chapter 18 no problems Frank Matejcik SD School of Mines & Technology 6 The EOQ Model To a pessimist, the glass is half empty. to an optimist, it is half full. – Anonymous Frank Matejcik SD School of Mines & Technology 7 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 Frank Matejcik SD School of Mines & Technology 8 MedEquip Example • Small manufacturer of medical diagnostic equipment. • Purchases standard steel “racks” into which 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 doesn’t want to tie up too much precious capital in inventory. Frank Matejcik SD School of Mines & Technology 9 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. Frank Matejcik SD School of Mines & Technology 10 1. EOQ Modeling Assumptions 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. Frank Matejcik SD School of Mines & Technology 11 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 is 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 decision variable Frank Matejcik SD School of Mines & Technology 12 Inventory vs Time in EOQ Inventory Model Q Q/D 2Q/D 3Q/D 4Q/D Time Frank Matejcik SD School of Mines & Technology 13 Costs • Holding Cost: average inventory Q 2 hQ annual holding cost 2 hQ unit holding cost 2D • Setup Costs: A per lot, so unit setup cost A Q • Production Cost: c per unit hQ A Y (Q) c • Cost Function: 2D Q Frank Matejcik SD School of Mines & Technology 14 MedEquip Example Costs • D = 1000 racks per year • c = $250 • A = $500 (estimated from supplier’s pricing) • h = (0.1)($250) + $10 = $35 per unit per year Frank Matejcik SD School of Mines & Technology 15 Costs in EOQ Model 20.00 18.00 16.00 14.00 Cost ($/unit) 12.00 10.00 Y(Q) Q* =169 8.00 6.00 hQ/2D 4.00 2.00 c A/Q 0.00 0 100 200 300 400 500 Order Quantity (Q) Frank Matejcik SD School of Mines & Technology 16 Economic Order Quantity dY (Q) h A 2 0 dQ 2D Q 2 AD Q* EOQ Square Root Formula h 2(500)(1000) Q* 169 MedEquip Solution 35 Frank Matejcik SD School of Mines & Technology 17 EOQ Modeling Assumptions 1. Production is instantaneous – there is no capacity constraint relax via and the entire lot is produced EPL model simultaneously. Frank Matejcik SD School of Mines & Technology 18 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 is consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate. decision variable Q the unknown size of the production lot size Frank Matejcik SD School of Mines & Technology 19 Inventory vs Time in EPL Model Production run of Q takes Q/P time units (P-D)(Q/P) Inventory -D P-D (P-D)(Q/P)/2 Time Frank Matejcik SD School of Mines & Technology 20 Solution to EPL Model • Annual Cost Function: AD h(1 D / P )Q Y (Q ) Dc Q 2 setup holding production • Solution (by taking derivative and setting equal to zero): • tends to EOQ as P 2 AD Q* h(1 P / D) • otherwise larger than EOQ because replenishment takes longer Frank Matejcik SD School of Mines & Technology 21 The Key Insight of EOQ There is a tradeoff between lot size and inventory • Order Frequency: F D Q cQ cD • Inventory Investment: I 2 2F Frank Matejcik SD School of Mines & Technology 22 EOQ Tradeoff Curve 50 45 Inventory Investment 40 35 30 25 20 15 10 5 0 0 20 40 60 80 100 Order/Year Frank Matejcik SD School of Mines & Technology 23 Sensitivity of EOQ Model to Quantity • Optimal Unit Cost: hQ* A We neglect unit cost, c, Y Y (Q ) * * 2 D Q* 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 Frank Matejcik SD School of Mines & Technology 24 Sensitivity of EOQ Model to Quantity (cont.) • Annual Cost from Using Q': hQ AD Y (Q) 2 Q Cost(Q) Y (Q) hQ 2 AD Q 1 Q Q* * • Ratio: Cost(Q* ) Y (Q* ) 2 ADh 2 Q Q • Example: If Q' = 2Q*, then the ratio of the actual to optimal cost is (1/2)[2 + (1/2)] = 1.25 Frank Matejcik SD School of Mines & Technology 25 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 Frank Matejcik SD School of Mines & Technology 26 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 2 1.06 2 Frank Matejcik SD School of Mines & Technology 27 The “Root-Two” Interval 2m T1* 2m 2 T2* 2 m1 divide by multiply by less than less than 2 to get 2 to get to 2m to 2m+1 Frank Matejcik SD School of Mines & Technology 28 Medequip Example • Optimum: Q*=169, so T*=Q*/D =169/1000 years = 62 days hQ * AD 35(169 ) 500 (1000 ) Y (Q*) $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’= Only 0.07%were lucky because we error T’D=(64/365)1000=175. and happened to be hQ' AD 35(175 ) 500 (1000 ) close to a power-of-two. Y (Q ' ) $5,920 But we can’t do worse 2 Q' 2 175 than 6%. Frank Matejcik SD School of Mines & Technology 29 Powers-of-Two Order Intervals Order Interval Week 0 1 2 3 4 5 6 7 8 1 2 0 2 21 4 22 8 23 Frank Matejcik SD School of Mines & Technology 30 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). Frank Matejcik SD School of Mines & Technology 31 The Wagner-Whitin Model Change is not made without inconvenience, even from worse to better. – Robert Hooker Frank Matejcik SD School of Mines & Technology 32 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. Frank Matejcik SD School of Mines & Technology 33 Dynamic Lot Sizing Notation t a 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 decision variables Frank Matejcik SD School of Mines & Technology 34 Wagner-Whitin Example • Data t 1 2 3 4 5 6 7 8 9 10 Dt 20 50 10 50 50 10 20 40 20 30 ct 10 10 10 10 10 10 10 10 10 10 At 100 100 100 100 100 100 100 100 100 100 ht 1 1 1 1 1 1 1 1 1 1 • Lot-for-Lot Solution t 1 2 3 4 5 6 7 8 9 10 Total Dt 20 50 10 50 50 10 20 40 20 30 300 Qt 20 50 10 50 50 10 20 40 20 30 300 It 0 0 0 0 0 0 0 0 0 0 0 Setup cost 100 100 100 100 100 100 100 100 100 100 1000 Holding cost 0 0 0 0 0 0 0 0 0 0 0 Total cost 100 100 100 100 100 100 100 100 100 100 1000 Frank Matejcik SD School of Mines & Technology 35 Wagner-Whitin Example (cont.) • Fixed Order Quantity Solution t 1 2 3 4 5 6 7 8 9 10 Total Dt 20 50 10 50 50 10 20 40 20 30 300 Qt 100 0 0 100 0 0 100 0 0 0 300 It 80 30 20 70 20 10 90 50 30 0 0 Setup cost 100 0 0 100 0 0 100 0 0 0 300 Holding cost 80 30 20 70 20 10 90 50 30 0 400 Total cost 180 30 20 170 20 10 190 50 30 0 700 Frank Matejcik SD School of Mines & Technology 36 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. Frank Matejcik SD School of Mines & Technology 37 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. Frank Matejcik SD School of Mines & Technology 38 Wagner-Whitin Example • Step 1: Obviously, just satisfy D1 (note we are neglecting production cost, since it is fixed). Z * A 100 1 1 j1* 1 • Step 2: Two choices, either j2* = 1 or j2* = 2. A1 h D , produce in 1 Z* min * 1 2 Z1 A2 , produce in 2 2 100 1(50) 150 min 100 100 200 150 j2 1 * Frank Matejcik SD School of Mines & Technology 39 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* A3 , 2 produce in 3 100 1(50) (1 1)10 170 min 100 100 (1)10 210 150 100 250 170 j3 1 * Frank Matejcik SD School of Mines & Technology 40 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 , 1 produce in 2 Z 4 min * * 2 2 3 2 3 4 Z 2 A3 h3 D4 , produce in 3 Z* A4 , 3 produce in 4 100 1(50) (1 1)10 (1 1 1)50 320 100 100 (1)10 (1 1)50 310 min 150 100 (1)50 300 170 100 270 270 j4 4 * Frank Matejcik SD School of Mines & Technology 41 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. Frank Matejcik SD School of Mines & Technology 42 Wagner-Whitin Example (cont.) • Step 5: Only two choices, j5* = 4, 5. Z 3 A4 h4 D5 , produce in 4 * Z min * * Z 4 A5 , 5 produce in 5 170 100 1(50) 320 min 270 100 370 320 j5 4 * • Step 6: Three choices, j6* = 4, 5, 6. And so on. Frank Matejcik SD School of Mines & Technology 43 Wagner-Whitin Example Solution Last Period Planning Horizon (t) with Production 1 2 3 4 5 6 7 8 9 10 1 100 150 170 320 2 200 210 310 3 250 300 4 270 320 340 400 560 5 370 380 420 540 6 420 440 520 7 440 480 520 610 8 500 520 580 9 580 610 10 620 Zt 100 150 170 270 320 340 400 480 520 580 jt 1 1 1 4 4 4 4 7 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 Frank Matejcik SD School of Mines & Technology 44 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) Note: we produce in 7 for an 8 period problem, but this never comes into play in optimal solution. Frank Matejcik SD School of Mines & Technology 45 Wagner-Whitin Example Solution (cont.) t 1 2 3 4 5 6 7 8 9 10 Total Dt 20 50 10 50 50 10 20 40 20 30 300 Qt 80 0 0 130 0 0 0 90 0 0 300 It 60 10 0 80 30 20 0 50 30 0 0 Setup cost 100 0 0 100 0 0 0 100 0 0 300 Holding cost 60 10 0 80 30 20 0 50 30 0 280 Total cost 160 10 0 180 30 20 0 150 30 0 580 Note: we produce in 7 for an 8 period problem, but this never comes into play in optimal solution. Frank Matejcik SD School of Mines & Technology 46 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) Frank Matejcik SD School of Mines & Technology 47 Statistical Reorder Point Models When your pills get down to four, Order more. – Anonymous, from Hadley &Whitin Frank Matejcik SD School of Mines & Technology 48 EOQ Assumptions 1. Instantaneous production. EPL model relaxes this one 2. Immediate delivery. lags can be added to EOQ or other models 3. Deterministic demand. and (Q,r) relax this one newsvendor 4. Constant demand. WW model relaxes this one 5. Known fixed setup costs. can use constraint approach 6. Single product or separable products. Chapter 17 extends (Q,r) to multiple product cases Frank Matejcik SD School of Mines & Technology 49 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 Frank Matejcik SD School of Mines & Technology 50 The Newsvendor Approach • Assumptions: 1. single period 2. random demand with known 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 Frank Matejcik SD School of Mines & Technology 51 Newsvendor Model Notation X demand (in units), a random variable. G ( x) P( X x), cumulative distributi on function of demand (assumed continuous.) d g ( x) G ( x) density function of demand. dx co cost (in dollars) per unit left over after demand is realized. c s cost (in dollars) per unit of shortage. Q production/order quantity (in units); this is the decision variable. Frank Matejcik SD School of Mines & Technology 52 Newsvendor Model • Cost Function: Note: for any given day, we will be either Y ( x) expected overage expectedshortage cost over or short, not both. But in expectation, co E units over c s E units short overage and shortage can both be positive. co maxQ x,0g ( x)dx c s maxx Q,0g ( x)dx 0 0 Q co (Q x) g ( x)dx c s ( x Q) g ( x)dx 0 Q Frank Matejcik SD School of Mines & Technology 53 Newsvendor Model (cont.) • Optimal Solution: taking derivative of Y(Q) with respect to Q, setting equal to zero, and solving yields: c G(Q* ) P X Q* s co c s Critical Ratio is probability stock covers demand Q * co 1 cs G(x) • Notes: Q * cs co c s Q* Frank Matejcik SD School of Mines & Technology 54 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 Frank Matejcik SD School of Mines & Technology 55 Newsvendor Example – T Shirts (cont.) • Solution: Q cs 5 G (Q ) 1 e * 1000 0.714 co cs 2 5 Q* 1,253 • Sensitivity: If co = $10 (i.e., shirts must be discarded) then Q cs 5 G (Q ) 1 e * 1000 0.333 co cs 10 5 Q* 405 Frank Matejcik SD School of Mines & Technology 56 Newsvendor Model with Normal Demand • Suppose demand is normally distributed with mean and standard deviation . Then the critical ratio formula reduces to: Q * 3.00 cs (z) G (Q * ) co c s Q * cs z where ( z ) 0 z co c s 0.00 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157 Note: Q* increases in both Q* z and if z is positive (i.e., if ratio is greater than 0.5). Frank Matejcik SD School of Mines & Technology 57 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 Frank Matejcik SD School of Mines & Technology 58 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? Frank Matejcik SD School of Mines & Technology 59 Example (cont.) • Newsvendor Parameters: c0 = $0.5 cs = $15 15 G (Q * ) 0.9677 0.5 15 Q 100 • Solution: 25 0.9677 Q 100 Every Friday, they should 1.85 order-up-to 146, that is, if 25 Q 100 1.85(25) 146 there are x on hand, then order 146-x. Frank Matejcik SD School of Mines & Technology 60 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. Frank Matejcik SD School of Mines & Technology 61 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). Frank Matejcik SD School of Mines & Technology 62 Inventory vs Time in (Q,r) Model Inventory Q r l Time Frank Matejcik SD School of Mines & Technology 63 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. min fixed setup cost holding cost backordercost Q,r • Objective: Under (1) As in EOQ, this makes batch production attractive. Frank Matejcik SD School of Mines & Technology 64 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. Frank Matejcik SD School of Mines & Technology 65 (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 Frank Matejcik SD School of Mines & Technology 66 (Q,r) Notation (cont.) • Decision Variables: Q order quantity r reorder point s 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 inventorylevel Frank Matejcik SD School of Mines & Technology 67 Inventory and Inventory Position for Q=4, r=4 9 8 Inventory Position uniformly distributed 7 between r+1=5 and 6 r+Q=8 5 Quantity 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 -1 -2 Time Inventory Position Net Inventory Frank Matejcik SD School of Mines & Technology 68 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) Frank Matejcik SD School of Mines & Technology 69 Fixed Setup Cost in (Q,r) Model • Observation: since the number of orders per year is D/Q, D F(Q) Q Frank Matejcik SD School of Mines & Technology 70 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. 1 r Q 1 S (Q, r ) G( x 1) [G(r ) G (r Q 1)] Q x r 1 Q • Result: 1 Note: this form is easier to use S (Q, r ) 1 [ B(r ) B(r Q)] in spreadsheets because it does Q not involve a sum. Frank Matejcik SD School of Mines & Technology 71 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 ) Note: neglects B(r,Q) S (Q, r ) 1 term, underestimates S(Q,r) Q Frank Matejcik SD School of Mines & Technology 72 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]. 1 r Q 1 B(Q, r ) Q x r 1 B( x) [ B(r 1) B(r Q)] Q • Result: 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. Frank Matejcik SD School of Mines & Technology 73 Inventory Costs in (Q,r) Model • Approximate Analysis: on average inventory declines from Q+s to s+1 so (Q s ) ( s 1) Q 1 Q 1 I (Q, r ) s r 2 2 2 • Exact Analysis: this neglects backorders, which add to average inventory since on- Q 1 I (Q, r ) r B (Q, r ) hand inventory can 2 never go below zero. The corrected version turns out to be Frank Matejcik SD School of Mines & Technology 74 Inventory vs Time in (Q,r) Model Expected Inventory Actual Inventory Exact I(Q,r) = s+Q Approx I(Q,r) + B(Q,r) Inventory r Approx I(Q,r) s+1=r-+1 Time Frank Matejcik SD School of Mines & Technology 75 Expected Inventory Level for Q=4, r=4, =2 7 s+Q 6 5 Inventory Level 4 3 s 2 1 0 0 5 10 15 20 25 30 35 Time Frank Matejcik SD School of Mines & Technology 76 (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 Frank Matejcik SD School of Mines & Technology 77 Results of Approximate Optimization • Assumptions: – Q,r can be treated as continuous variables – G(x) is a continuous cdf • Results: 2 AD Q* Note: this is just the EOQ formula h b Note: this is just the G (r*) r * z hb base stock formula if G is normal(,), where (z)=b/(h+b) Frank Matejcik SD School of Mines & Technology 78 (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 Frank Matejcik SD School of Mines & Technology 79 Values for Poisson() Distribution r p(r) G(r) B(r) 0 0.178 0.178 1.726 1 0.307 0.485 0.904 2 0.265 0.750 0.389 3 0.153 0.903 0.140 4 0.066 0.969 0.042 5 0.023 0.991 0.011 6 0.007 0.998 0.003 7 0.002 1.000 0.001 8 0.000 1.000 0.000 9 0.000 1.000 0.000 10 0.000 1.000 0.000 80 Frank Matejcik SD School of Mines & Technology 80 Calculations for Example 2 AD 2(10)(14) Q* 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 Frank Matejcik SD School of Mines & Technology 81 Performance Measures for Example D 14 F (Q*) 3.5 Q* 4 1 1 S(Q * ,r * ) 1 [ B (r*) B (r * Q*)] 1 [ B (2) B (2 4)] Q* Q 1 1 [0.389 0.003] 0.904 4 1 r * Q * 1 B (Q*, r*) Q * x r *1 B ( x) [ B(3) B(4) B(5) B (6)] Q 1 [0.140 0.042 0.011 0.003] 0.049 4 Q * 1 4 1 I (Q*, r*) r * B (Q*, r*) 2 1.726 0.049 2.823 2 2 Frank Matejcik SD School of Mines & Technology 82 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) Frank Matejcik SD School of Mines & Technology 83 • Example Varying theorder twice as often Change: suppose we so F=7 per year, then Q=2 and: 1 1 S (Q, r ) 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: 1 1 S (Q, r ) 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. Frank Matejcik SD School of Mines & Technology 84 (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 Frank Matejcik SD School of Mines & Technology 85 Results of Approximate Optimization • Assumptions: – Q,r can be treated as continuous variables – G(x) is a continuous cdf • Results: 2 AD Q* Note: this is just the EOQ formula h kD Note: another version G(r*) r* z kD hQ of base stock formula if G is normal(,), (only z is different) where (z)=kD/(kD+hQ) Frank Matejcik SD School of Mines & Technology 86 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 Frank Matejcik SD School of Mines & Technology 87 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) Frank Matejcik SD School of Mines & Technology 88 Including Lead Time Variability in Formulas • Standard Deviation of Lead Time Demand: if demand is Poisson D d 2 L d 2 L 2 2 2 Inflation term due to lead time variability R z z d 2 L 2 Note: can be used in any • Modified Base Stock base stock or (Q,r) formula as before. In general, it will Formula (Poisson demand inflate safety stock. case): Frank Matejcik SD School of Mines & Technology 89 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. Frank Matejcik SD School of Mines & Technology 90

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