1. 1 IE&M Shanghai Jiao Tong University Inventory Control Part 1 Subject to Known Demand By Ming Dong Department of Industrial Engineering & Management Shanghai Jiao Tong University

2. 2 IE&M Shanghai Jiao Tong University Contents Types of Inventories Motivation for Holding Inventories; Characteristics of Inventory Systems; Relevant Costs; The EOQ Model; EOQ Model with Finite Production Rate

3. 3 IE&M Shanghai Jiao Tong University Introduction Definition: Inventory is the stock of any item or resource used in an organization. An inventory system is the set of policies and controls that monitors levels of inventory or determines what levels should be maintained. Generally, inventory is being acquired or produced to meet the need of customers; Dependant demand system - the demand of components and subassemblies (lower levels depend on higher level) -MRP; The fundamental problem of inventory management : When to place order for replenishing the stock ? How much to order?

4. 4 IE&M Shanghai Jiao Tong University Introduction Inventory: plays a key role in the logistical behavior of virtually all manufacturing systems. The classical inventory results: are central to more modern techniques of manufacturing management, such as MRP, JIT, and TBC. The complexity of the resulting model depends on the assumptions about the various parameters of the system - the major distinction is between models for known demand and random demand.

5. 5 IE&M Shanghai Jiao Tong University Introduction The current investment in inventories in USA is enormous; It amounted up to $1.37 trillion in the last quarter of 1999; It accounts for 20-25% of the total annual GNP (general net product); There exists enormous potential for improving the efficiency of economy by scientifically controlling inventories; Breakdown of total investment in inventories

6. 6 IE&M Shanghai Jiao Tong University Types of Inventories A natural classification is based on the value added from manufacturing operations Raw materials: Resources required in the production or processing activity of the firm. Components: Includes parts and subassemblies. Work-in-process (WIP): the inventory either waiting in the system for processing or being processed. The level of WIP is taken as a measure of the efficiency of a production scheduling system. JIT aims at reducing WIP to zero. Finished good: also known as end items or the final products.

7. 7 IE&M Shanghai Jiao Tong University Why Hold Inventories (1) For economies of scale It may be economical to produce a relatively large number of items in each production run and store them for future use. Coping with uncertainties Uncertainty in demand Uncertainty in lead time Uncertainty in supply For speculation Purchase large quantities at current low prices and store them for future use. Cope with labor strike

8. 8 IE&M Shanghai Jiao Tong University Why Hold Inventories (2) Transportation Pipeline inventories is the inventory moving from point to point, e.g., materials moving from suppliers to a plant, from one operation to the next in a plant. Smoothing Producing and storing inventory in anticipation of peak demand helps to alleviate the disruptions caused by changing production rates and workforce level. Logistics To cope with constraints in purchasing, production, or distribution of items, this may cause a system maintain inventory.

9. 9 IE&M Shanghai Jiao Tong University Characteristics of Inventory Systems Demand (patterns and characteristics) Constant versus variable Known versus random Lead Time Ordered from the outside Produced internally Review Continuous: e.g., supermarket Periodic: e.g., warehouse Excess demand demand that cannot be filled immediately from stock backordered or lost. Changing inventory Become obsolete: obsolescence

10. 10 IE&M Shanghai Jiao Tong University Relevant Costs - Holding Cost Holding cost (carrying or inventory cost) The sum of costs that are proportional to the amount of inventory physically on-hand at any point in time Some items of holding costs Cost of providing the physical space to store the items Taxes and insurance Breakage, spoilage, deterioration, and obsolescence Opportunity cost of alternative investment Inventory cost fluctuates with time inventory as a function of time

11. 11 IE&M Shanghai Jiao Tong University Relevant Costs - Holding Cost Inventory as a Function of Time

12. 12 IE&M Shanghai Jiao Tong University Relevant Costs - Order Cost It depend on the amount of inventory that is ordered or produced. Two components The fixed cost K: independent of size of order The variable cost c: incurred on per-unit basis

13. 13 IE&M Shanghai Jiao Tong University Relevant Costs - Order Cost Order Cost Function

14. 14 IE&M Shanghai Jiao Tong University Relevant Costs - Penalty Cost Also know as shortage cost or stock-out cost The cost of not having sufficient stock on-hand to satisfy a demand when it occurs. Two interprets Backorder case: include delay costs may be involved Lost-sale case: include “loss-of-goodwill” cost, a measure of customer satisfaction Two approaches Penalty cost, p, is charged per-unit basis. Each time a demand occurs that cannot be satisfied immediately, a cost p is incurred independent of how long it takes to eventually fill the demand. Charge the penalty cost on a per-unit-time basis.

15. 15 IE&M Shanghai Jiao Tong University EOQ History Introduced in 1913 by Ford W. Harris, “ How Many Parts to Make at Once ” Product types: A, B and C A-B-C-A-B-C: 6 times of setup A-A-B-B-C-C: 3 times of setup A factory producing various products and switching between products causes a costly setup (wages, material and overhead). Therefore, a trade-off between setup cost and production lot size should be determined. Early application of mathematical modeling to Scientific Management

16. 16 IE&M Shanghai Jiao Tong University 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.

17. 17 IE&M Shanghai Jiao Tong University Notation demand rate (units per year) c proportional order cost at c per unit ordered (dollars per unit) K fixed or setup cost to place an order (dollars) hholding 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. Qthe unknown size of the order or lot size

18. 18 IE&M Shanghai Jiao Tong University Inventory vs Time in EOQ Model Q/ 2Q/ 3Q/ 4Q/ Q Inventory Time slope = - Order cycle: T=Q/

19. 19 IE&M Shanghai Jiao Tong University Costs Holding Cost: Setup Costs: K per lot, so The average annual cost:

20. 20 IE&M Shanghai Jiao Tong University 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 hold too much capital in inventory. Question: how many racks should MedEquip order at once?

21. 21 IE&M Shanghai Jiao Tong University MedEquip Example Costs = 1000 racks per year c = $250 K = $500 (estimated from supplier’s pricing) h = i*c + floor space cost = (0.1)($250) + $10 = $35 per unit per year

22. 22 IE&M Shanghai Jiao Tong University Costs in EOQ Model

23. 23 IE&M Shanghai Jiao Tong University Economic Order Quantity MedEquip Solution EOQ Square Root Formula Solution (by taking derivative and setting equal to zero): Since Q”>0 ， G(Q) is convex function of Q

24. 24 IE&M Shanghai Jiao Tong University Another Example Example 2 Pencils are sold at a fairly steady rate of 60 per week; Pencils cost 2 cents each and sell for 15 cents each; Cost $12 to initiate an order, and holding costs are based on annual interest rate of 25%. Determine the optimal number of pencils for the book store to purchase each time and the time between placement of orders Solutions Annual demand rate  =60  52=3,120; The holding cost is the product of the variable cost of the pencil and the annual interest-h=0.02  0.25=0.05

25. 25 IE&M Shanghai Jiao Tong University The EOQ Model-Considering Lead Time Since there exists lead time  (4 moths for Example 2), order should be placed some time ahead of the end of a cycle; Reorder point R-determines when to place order in term of inventory on hand, rather than time. Reorder Point Calculation for Example 2

26. 26 IE&M Shanghai Jiao Tong University The EOQ Model-Considering Lead Time Determine the reorder point when the lead time exceeds a cycle. Example:  Q=25;  =500/yr;   =6 wks;  T=25/500=2.6 wks;   /T=2.31---2.31 cycles are included in LT.  Action: place every order 2.31 cycles in advance. Reorder Point Calculation for Lead Times Exceeding One Cycle Computing R for placing order 2.31 cycles ahead is the same as that 0.31 cycle ahead.

27. 27 IE&M Shanghai Jiao Tong University 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. relax via EOQ Model for Finite Production Rate

28. 28 IE&M Shanghai Jiao Tong University The EOQ Model for Finite Production Rate The EOQ model with finite production rate is a variation of the basic EOQ model Inventory is replenished gradually as the order is produced (which requires the production rate to be greater than the demand rate) Notice that the peak inventory is lower than Q since we are using items as we produce them

29. 29 IE&M Shanghai Jiao Tong University Notation – EOQ Model for Finite Production Rate demand rate (units per year) Pproduction rate (units per year), where P> c unit production cost, not counting setup or inventory costs (dollars per unit) K fixed or setup cost (dollars) hholding 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. Qthe unknown size of the production lot size decision variable

30. 30 IE&M Shanghai Jiao Tong University Inventory vs Time Inventory Time - P- 1. Production run of Q takes Q/P time units (P- )(Q/P) (P- )(Q/P)/2 2. When the inventory reaches 0, production begins until Q products are produced (it takes Q/P time units). During the Q/P time units, the inventory level will increases to (P- )(Q/P) Inventory increase rate Time slope = - slope = P-

31. 31 IE&M Shanghai Jiao Tong University Solution to EOQ Model with Finite Production Rate Annual Cost Function: Solution (by taking first derivative and setting equal to zero): setup holding tends to EOQ as P  otherwise larger than EOQ because replenishment takes longer EOQ model

32. 32 IE&M Shanghai Jiao Tong University Example: Non-Slip Tile Co. Non-Slip Tile Company (NST) has been using production runs of 100,000 tiles, 10 times per year to meet the demand of 1,000,000 tiles annually. The set-up cost is $5000 per run and holding cost is estimated at 10% of the manufacturing cost of $1 per tile. The production capacity of the machine is 500,000 tiles per month. The factory is open 365 days per year.

33. 33 IE&M Shanghai Jiao Tong University Example: Non-Slip Tile Co. (Cont.) This is a “EOQ Model with Finite Production Rate” problem with = 1,000,000 P = 500,000*12 = 6,000,000 h = 0.1 K = 5,000

34. 34 IE&M Shanghai Jiao Tong University Example: Non-Slip Tile Co. (Cont.) Find the Optimal Production Lot Size How many runs should they expect per year? How much will they save annually using EOQ Model with Finite Production Rate?

35. 35 IE&M Shanghai Jiao Tong University Example: Non-Slip Tile Co. (Cont.) Optimal Production Lot Size Number of Production Runs Per Year The number of runs per year = /Q* = 2.89 times per year

36. 36 IE&M Shanghai Jiao Tong University Example: Non-Slip Tile Co. (Cont.) Annual Savings: Setup cost Holding cost TC = 0.04167Q + 5,000,000,000/ Current TC = 0.04167(1,000,000) + 5,000,000,000/(1,000,000) = $54,167 Optimal TC = 0.04167(346,410) + 5,000,000,000/(346,410) = $28,868 Difference = $54,167 - $28,868 = $25,299

37. 37 IE&M Shanghai Jiao Tong University Sensitivity of EOQ Model to Quantity Optimal Unit Cost: Optimal Annual Cost: Multiply G* by and simplify,

38. 38 IE&M Shanghai Jiao Tong University Sensitivity of EOQ Model to Quantity (cont.) Annual Cost from Using Q': Ratio:

39. 39 IE&M Shanghai Jiao Tong University Sensitivity of EOQ Model to Quantity (cont.) Example: If Q' = 2Q*, then the ratio of the actual cost to optimal cost is (1/2)[2 + (1/2)] = 1.25 If Q' = Q*/2, then the ratio of the actual cost to optimal cost is (1/2)[(1/2)+2] = 1.25 A 100% error in lot size results in a 25% error in cost.

40. 40 IE&M Shanghai Jiao Tong University 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: Total cost is relatively insensitive to lot size (so rounding for other reasons, like coordinating shipping, may be attractive).

41. 41 IE&M Shanghai Jiao Tong University Inventory Control Part 2 Inventory Control Subject to Unknown Demand By Ming Dong Department of Industrial Engineering & Management Shanghai Jiao Tong University

42. 42 IE&M Shanghai Jiao Tong University The Wagner-Whitin Model Change is not made without inconvenience, even from worse to better. – Robert Hooker

43. 43 IE&M Shanghai Jiao Tong University EOQ Assumptions 1. Instantaneous production. 2. Immediate delivery. 3. Deterministic demand. 4. Constant demand. 5. Known fixed setup costs. 6. Single product or separable products. WW model relaxes this one

44. 44 IE&M Shanghai Jiao Tong University Dynamic Lot Sizing Notation ta period (e.g., day, week, month); we will consider t = 1, …,T, where T represents the planning horizon. D t demand in period t (in units) c t unit production cost (in dollars per unit), not counting setup or inventory costs in period t A t fixed or setup cost (in dollars) to place an order in period t h t holding cost (in dollars) to carry a unit of inventory from period t to period t +1 Q t the unknown size of the order or lot size in period t decision variables

45. 45 IE&M Shanghai Jiao Tong University Wagner-Whitin Example Data Lot-for-Lot Solution Since production cost c is constant, it can be ignored.

46. 46 IE&M Shanghai Jiao Tong University Wagner-Whitin Example (cont.) Fixed Order Quantity Solution Data

47. 47 IE&M Shanghai Jiao Tong University Wagner-Whitin Property Under an optimal lot-sizing policy (1) either the inventory carried to period t+1 from a previous period will be zero (there is a production in t+1) (2) or the production quantity in period t+1 will be zero (there is no production in t+1) A key observation If we produce items in t (incur a setup cost) for use to satisfy demand in t+1, then it cannot possibly be economical to produce in t+1 (incur another setup cost). Either it is cheaper to produce all of period t+1’s demand in period t, or all of it in t+1; it is never cheaper to produce some in each. Does fixed order quantity solution violate this property? Why?

48. 48 IE&M Shanghai Jiao Tong University Basic Idea of Wagner-Whitin Algorithm By WW Property, either Q t =0 or Q t =D 1 + … +D k for some k. If j k * = last period of production in a k period problem, then we will produce exactly D k + … D T in period j k *. Why? We can then consider periods 1, …, j k *-1 as if they are an independent j k *-1 period problem.

49. 49 IE&M Shanghai Jiao Tong University Wagner-Whitin Example Step 1: Obviously, just satisfy D 1 (note we are neglecting production cost, since it is fixed). Step 2: Two choices, either j 2 * = 1 or j 2 * = 2.

50. 50 IE&M Shanghai Jiao Tong University Wagner-Whitin Example (cont.) Step3: Three choices, j 3 * = 1, 2, 3.

51. 51 IE&M Shanghai Jiao Tong University Wagner-Whitin Example (cont.) Step 4: Four choices, j 4 * = 1, 2, 3, 4.

52. 52 IE&M Shanghai Jiao Tong University Planning Horizon Property In the Example: Given fact: we produce in period 4 for period 4 of a 4 period problem. Question: will we produce in period 3 for period 5 in a 5 period problem? Answer: We would never produce in period 3 for period 5 in a 5 period problem. If j t *=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. (this means that it CANNOT be t-1, t-2 …… )

53. 53 IE&M Shanghai Jiao Tong University Wagner-Whitin Example (cont.) Step 5: Only two choices, j 5 * = 4, 5. Step 6: Three choices, j 6 * = 4, 5, 6. And so on.

54. 54 IE&M Shanghai Jiao Tong University Wagner-Whitin Example Solution 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)

55. 55 IE&M Shanghai Jiao Tong University 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)

56. 56 IE&M Shanghai Jiao Tong University Problems with Wagner-Whitin 1. Fixed setup costs. 2. Deterministic demand and production (no uncertainty) 3. Never produce when there is inventory (WW Property). safety stock (don't let inventory fall to zero) random yields (can't produce for exact no. periods)

57. 57 IE&M Shanghai Jiao Tong University Statistical Reorder Point Models When your pills get down to four, Order more. – Anonymous, from Hadley &Whitin

58. 58 IE&M Shanghai Jiao Tong University EOQ Assumptions 1. Instantaneous production. 2. Immediate delivery. 3. Deterministic demand. 4. Constant demand. 5. Known fixed setup costs. 6. Single product or separable products. WW model relaxes this one newsvendor and (Q,r) relax this one can use constraint approach Chapter 17 extends (Q,r) to multiple product cases lags can be added to EOQ or other models EPL model relaxes this one

59. 59 IE&M Shanghai Jiao Tong University 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

60. 60 IE&M Shanghai Jiao Tong University 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

61. 61 IE&M Shanghai Jiao Tong University Newsvendor Model Notation

62. 62 IE&M Shanghai Jiao Tong University Newsvendor Model Cost Function: Units over = max {Q-X, 0} Units short = max {X-Q, 0} Objective: find the value of Q that minimizes this expected cost.

63. 63 IE&M Shanghai Jiao Tong University Newsvendor Model – Leibnitz’s rule For Y(Q): taking its derivative and setting it to 0. To do this, we need to take the derivative of integrals with limits that are functions of Q. A tool called Leibnitz's rule can do this. Applying this for Y(Q):

64. 64 IE&M Shanghai Jiao Tong University Newsvendor Model (cont.) G(Q*) represents the probability that demand is less than or equal to Q*. G(x) 1 Q* Note: Critical Ratio is that probability stock covers demand

65. 65 IE&M Shanghai Jiao Tong University 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: c s = 15 – 10 = $5 c o = 10 – 8 = $2

66. 66 IE&M Shanghai Jiao Tong University Newsvendor Example – T Shirts (cont.) Solution: Sensitivity: If c o = $10 (i.e., shirts must be discarded) then

67. 67 IE&M Shanghai Jiao Tong University Newsvendor Model with Normal Demand Suppose demand is normally distributed with mean  and standard deviation . Then the critical ratio formula reduces to: Note: Q* increases in both  and  if z is positive (i.e., if ratio is greater than 0.5). z0  (z) is the cumulative distribution function (cdf) of the standard normal distribution. z is the value in the standard normal table.

68. 68 IE&M Shanghai Jiao Tong University 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 c o and c s appropriately represent overage and shortage cost.

69. 69 IE&M Shanghai Jiao Tong University 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?

70. 70 IE&M Shanghai Jiao Tong University Example (cont.) Newsvendor Parameters: c 0 = $0.5 c s = $15 Solution: Every Friday, they should order-up-to 146, that is, if there are x on hand, then order 146-x.

71. 71 IE&M Shanghai Jiao Tong University 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.

72. 72 IE&M Shanghai Jiao Tong University The (Q, r) Approach Decision Variables: Reorder Point: r – affects likelihood of stockout (safety stock). Order Quantity: Q – affects order frequency (cycle inventory). 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.

73. 73 IE&M Shanghai Jiao Tong University Inventory vs Time in (Q,r) Model Q Inventory Time r l Delivery lead time

74. 74 IE&M Shanghai Jiao Tong University The Base-Stock Policy Start with an initial amount of inventory R. Each time a new demand arrives, place a replenishment order with the supplier. An order placed with the supplier is delivered l units of time after it is placed. Because demand is stochastic, we can have multiple orders that have been placed but not delivered yet.

75. 75 IE&M Shanghai Jiao Tong University Base Stock Model What the policy looks like Choose a base-stock level stock level R Each period, order to bring inventory position up to the base-stock level R That order will arrive after a lead time l Costs are incurred based on net inventory Balance holding costs vs. backorder costs

76. 76 IE&M Shanghai Jiao Tong University 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).

77. 77 IE&M Shanghai Jiao Tong University Base Stock Notation Q= 1, order quantity (fixed at one) r= reorder point R= 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)= Prob{demand during lead time l is less than x} h= unit holding cost b= unit backorder cost S(R)= average fill rate (service level) B(R)= average backorder level I(R)= average on-hand inventory level

78. 78 IE&M Shanghai Jiao Tong University Inventory Balance Equations Balance Equation: inventory position = on-hand inventory - backorders + orders Under Base Stock Policy inventory position = R On-hand inventory On-order quantity from suppliers Back-orders from customers Inventory position R

79. 79 IE&M Shanghai Jiao Tong University Inventory Profile for Base Stock System (R=5) l R r # of orders = # of demands

80. 80 IE&M Shanghai Jiao Tong University 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:

81. 81 IE&M Shanghai Jiao Tong University Backorder Level Note: At any point in time, number of orders equals number demands that have occurred during the 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 Expected Backorder Level: simpler version for spreadsheet computing

82. 82 IE&M Shanghai Jiao Tong University Inventory Level Observe: on-hand inventory - backorders = R-X E[X] =  E[backorders] = B(R) from previous slide Result: I(R) = R -  + B(R)

83. 83 IE&M Shanghai Jiao Tong University Base Stock Example l = one month  = 10 units (per month) Assume Poisson demand, so Note: Poisson demand is a good choice when no variability data is available.

84. 84 IE&M Shanghai Jiao Tong University Base Stock Example Calculations Summarize the results in Table

85. 85 IE&M Shanghai Jiao Tong University 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

86. 86 IE&M Shanghai Jiao Tong University “Optimal” Base Stock Levels Objective Function: 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 get Implication: set base stock level so fill rate is b/(h+b). Note: R* increases in b and decreases in h. holding plus backorder cost

87. 87 IE&M Shanghai Jiao Tong University Base Stock Normal Approximation If G is normal( ,  ), then where  (z)=b/(h+b). So R* =  + z  Note: R* increases in  and also increases in  provided z>0. z is the value from the standard normal table.

88. 88 IE&M Shanghai Jiao Tong University “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: 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.

89. 89 IE&M Shanghai Jiao Tong University 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: 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 cycle stocksafety stock

90. 90 IE&M Shanghai Jiao Tong University 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.

91. 91 IE&M Shanghai Jiao Tong University Pooling Example PC’s consist of 6 components (CPU, HD, CD ROM, RAM, removable storage device, keyboard) 3 choices of each component: 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% 36 = 729 different PC’s

92. 92 IE&M Shanghai Jiao Tong University Pooling Example - Stock PC’s 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

93. 93 IE&M Shanghai Jiao Tong University Pooling Example - Stock Components 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! 729 models of PC 3 types of each comp.

94. 94 IE&M Shanghai Jiao Tong University 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.

95. 95 IE&M Shanghai Jiao Tong University 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) As in EOQ, this makes batch production attractive.

96. 96 IE&M Shanghai Jiao Tong University 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.

97. 97 IE&M Shanghai Jiao Tong University (Q,r) Notation

98. 98 IE&M Shanghai Jiao Tong University (Q,r) Notation (cont.) Decision Variables: Performance Measures:

99. 99 IE&M Shanghai Jiao Tong University Inventory and Inventory Position for Q=4, r=4 Inventory Position uniformly distributed between r+1=5 and r+Q=8

100. 100 IE&M Shanghai Jiao Tong University 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)

101. 101 IE&M Shanghai Jiao Tong University Fixed Setup Cost in (Q,r) Model Observation: since the number of orders per year is D/Q,

102. 102 IE&M Shanghai Jiao Tong University 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: Note: this form is easier to use in spreadsheets because it does not involve a sum.

103. 103 IE&M Shanghai Jiao Tong University Service Level Approximations Type I (base stock): Type II: Note: computes number of stockouts per cycle, underestimates S(Q,r) Note: neglects B(r,Q) term, underestimates S(Q,r)

104. 104 IE&M Shanghai Jiao Tong University 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]. (Similar to the fill rate) 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.

105. 105 IE&M Shanghai Jiao Tong University Inventory Costs in (Q,r) Model Approximate Analysis: on average inventory declines from Q+s to s+1 so 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

106. 106 IE&M Shanghai Jiao Tong University Inventory vs Time in (Q,r) Model Inventory Time r s+1=r-  +1 s+Q Expected InventoryActual Inventory Approx I(Q,r) Exact I(Q,r) = Approx I(Q,r) + B(Q,r)

107. 107 IE&M Shanghai Jiao Tong University Expected Inventory Level for Q=4, r=4,  =2

108. 108 IE&M Shanghai Jiao Tong University (Q,r) Model with Backorder Cost Objective Function: 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):

109. 109 IE&M Shanghai Jiao Tong University Results of Approximate Optimization Assumptions: Q,r can be treated as continuous variables G(x) is a continuous cdf Results: if G is normal( ,  ), where  (z)=b/(h+b) Note: this is just the EOQ formula Note: this is just the base stock formula

110. 110 IE&M Shanghai Jiao Tong University (Q,r) Example Stocking Repair Parts: D = 14 units per year c = $150 per unit h= $25 per unit l = 45 days  = (14/365) × 45 = 1.726 units during replenishment lead time A = $10 b = $40 Demand during lead time is Poisson

111. 111 IE&M Shanghai Jiao Tong University Values for Poisson(  ) Distribution 111

112. 112 IE&M Shanghai Jiao Tong University Calculations for Example

113. 113 IE&M Shanghai Jiao Tong University Performance Measures for Example

114. 114 IE&M Shanghai Jiao Tong University 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)

115. 115 IE&M Shanghai Jiao Tong University Varying the Example Change: suppose we order twice as often so F=7 per year, then Q=2 and: which may be too low, so increase r from 2 to 3: This is better. For this policy (Q=2, r=3) 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.

116. 116 IE&M Shanghai Jiao Tong University (Q,r) Model with Stockout Cost Objective Function: 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:

117. 117 IE&M Shanghai Jiao Tong University Results of Approximate Optimization Assumptions: Q,r can be treated as continuous variables G(x) is a continuous cdf Results: if G is normal( ,  ), where  (z)=kD/(kD+hQ) Note: this is just the EOQ formula Note: another version of base stock formula (only z is different)

118. 118 IE&M Shanghai Jiao Tong University Backorder vs. Stockout Model Backorder Model when real concern is about stockout time because B(Q,r) is proportional to time customers 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 solutions Keep track of all performance measures regardless of model B-model will work best for backorders, S-model for stockouts

119. 119 IE&M Shanghai Jiao Tong University 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) D t = demand on day t, a random variable, assumed independent and identically distributed d = E[D t ] = expected daily demand  D = std dev of daily demand (units)

120. 120 IE&M Shanghai Jiao Tong University Including Lead Time Variability in Formulas Standard Deviation of Lead Time Demand: Modified Base Stock Formula (Poisson demand case): Inflation term due to lead time variability Note:  can be used in any base stock or (Q,r) formula as before. In general, it will inflate safety stock. if demand is Poisson

121. 121 IE&M Shanghai Jiao Tong University Single Product (Q,r) Insights Basic Insights: Safety stock provides a buffer against stockouts. Cycle stock increases as replenishment frequency decreases. 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.

122. 122 IE&M Shanghai Jiao Tong University Safety Stock Stock carried to provide a level of protection against costly stockouts due to uncertainty of demand during lead time Stock outs occur when Demand over the lead time is larger than expected service level (1 -  )100%. Service Level

123. 123 IE&M Shanghai Jiao Tong University Computing Inventory Position … Probability {Demand over lead-time < s} = 1-  Service Level 1-   Assumption: Demand over lead-time is normally distributed s Probability distribution of demand over L

124. 124 IE&M Shanghai Jiao Tong University Computing Inventory Position: Normal Distribution  1-  s Probability distribution of demand over L: Mean =  ; Std Dev =  Probability distribution of demand over L: Mean =  ; Std Dev =    1-  z0 1 - .90.95.98.99.999 z1.281.652.052.333.09 From normal table or, in Excel, use: =normsinv (0.90)

125. 125 IE&M Shanghai Jiao Tong University Computation of Variance for Demand over Lead Time: Variability Comes From Two Sources 3: Adding the two terms, we get to our result 1: Suppose only demand d i in day i is variable; lead time is constant at 2: Now, suppose only lead time is variable; daily demand is constant at d i ’s are independentd i ’s are identically distributed

126. 126 IE&M Shanghai Jiao Tong University Safety stock SS More specifically…. Note: If lead time is constant, If demand is constant, Standard deviation of demand over LT Safety factor (std normal table) Mean demand over LT

127. 127 IE&M Shanghai Jiao Tong University Example: Consider inventory management for a certain SKU at Home Depot. Supply lead time is variable (since it depends on order consolidation with other stores) and has a mean of 5 days and std deviation of 2 days. Daily demand for the item is variable with a mean of 30 units and std deviation of 6 units. Find the reorder point for 95% service level. 95% service level  z = 1.64 AVG = 30, STD = 6

128. 128 IE&M Shanghai Jiao Tong University The (s,S) Policy: Fixed Ordering Costs s should be set to cover the lead time demand and together with a safety stock that insures the stock out probability is within the specific limit (When to reorder). S depends on the fixed order cost – EOQ (How much) L R Order placed Order arrives Average demand during lead time Safety Stock s S

129. 129 IE&M Shanghai Jiao Tong University The (s,S) Policy: Fixed Ordering Costs Compute s exactly as in the base-stock model: Compute Q using the EOQ formula, using mean demand D = AVG (be careful about units…): Set S = s + Q Order when: inventory position (IP) drops below s Order how much: bring IP to S

130. 130 IE&M Shanghai Jiao Tong University Example: (s,S) Model Consider previous Home Depot example, however, there are fixed ordering costs, which are estimated at $50. Assume that holding costs are 15% of the product cost ($80) per year. Also, assume that the store is open 360 days a year. (from previous calculations) h = (0.15)*80/360=0.0333; A = K = 50, D = AVG = 30

131. 131 IE&M Shanghai Jiao Tong University Summary of Inventory Models Is demand rate and lead time constant? Use EOQ How much : EOQ formula When : d*L no yes Are there fixed ordering costs? no yes Use (s, S) policy How much : Q = S – s (Q is from EOQ formula) When : IP drops below s (base- stock policy formula) Use base stock (s) policy When : IP drops below s How much : necessary to bring IP back to s