1. 1 Inventory Management and Control

2. 2AMAZON.com Jeff Bezos, in 1995, started AMAZON.com as a “virtual” retailer – no inventory, no warehouses, no overhead; just a bunch of computers. Growth forced AMAZON.com to excel in inventory management! AMAZON is now a worldwide leader in warehouse management and automation.

3. 3 Order Fulfillment at AMAZON (1 of 2) 1.You order items; computer assigns your order to distribution center [closest facility that has the product(s)] 2.Lights indicate products ordered to workers who retrieve product and reset light. 3.Items placed in crate with items from other orders, and crate is placed on conveyor. Bar code on item is scanned 15 times – virtually eliminating error.

4. 4 Order Fulfillment at AMAZON (2 of 2) 4.Crates arrive at a central point where items are boxed and labeled with new bar code. 5.Gift wrapping done by hand (30 packages per hour) 6.Box is packed, taped, weighed and labeled before leaving warehouse in a truck. 7.Order appears on your doorstep within a week

5. 5 Inventory (Definition of) Inventory is the stock of any item or resource held to meet future demand and can include: raw materials, finished products, component parts, supplies, and work- in-process

6. 6 Inventory Process stage Demand Type Number & Value Other Raw Material WIP Finished Goods Independent Dependent A Items B Items C Items Maintenance Operating Inventory Classifications

7. 7 E(1 ) Inventories by Demand Type B(4) E(2)D(1) C(2) E(3)B(1) A Independent Demand : Demand for the final end-product that are ready to be sold or used. Demand not related to other items; demand created by external customers); eg. Demand for computers Dependent Demand : Derived demand for components of finished products (parts, raw materials, subassemblies) Finished product: eg: Computer Components: eg. parts that make up the computer Independent demand is uncertain Dependent demand is certain

8. 8 Types of Inventories (1 of 2) Raw materials & purchased parts Partially completed goods called work in process Finished-goods inventories (manufacturing firms) or merchandise (retail stores)

9. 9 Types of Inventories (2 of 2) Maintenance and repairs (MRO) inventory, replacement parts, tools, & supplies Goods-in-transit to warehouses or customers (pipeline inventory)

10. 10 The Material Flow Cycle (1 of 2)

11. 11 Run time: Job is at machine and being worked on Setup time: Job is at the work station, and the work station is being "setup." Queue time: Job is where it should be, but is not being processed because other work precedes it. Move time: The time a job spends in transit Wait time: When one process is finished, but the job is waiting to be moved to the next work area. Other: "Just-in-case" inventory. The Material Flow Cycle (2 of 2) Wait Time Move Time Queue Time Setup Time Run Time Input Cycle Time Output

12. 12 Performance Measures Inventory turnover (the ratio of annual cost of goods sold to average inventory investment) Days of inventory on hand (expected number of days of sales that can be supplied from existing inventory)

13. 13 Functions of Inventories (1 of 2) 1.To meet variation in product demand and to protect against stock-outs 2.To “decouple” operations or separate various parts of the production process, ie. to maintain independence of operations 3.To meet unexpected demand & to provide high levels of customer service 3.To smooth production requirements by meeting seasonal or cyclical variations in demand

14. 14 Functions of Inventories (2 of 2) 4. To provide a safeguard for variation in raw material delivery time 5. To provide a stock of goods that will provide a “selection” for customers 6. To take advantage of economic purchase-order size 7. To take advantage of quantity discounts 8. To take advantage of order cycles 9. To hedge against price increases

15. 15 Higher costs –Item cost (if purchased) –Ordering (or setup) cost –Holding (or carrying) cost Difficult to control Hides production problems May decrease flexibility Disadvantages of Inventories

16. 16 Inventory Costs  Holding (or carrying) costs Costs for storage, handling, insurance, etc  Setup (or production change) costs Costs to prepare a machine or process for manufacturing an order, eg. arranging specific equipment setups, etc  Ordering costs (costs of replenishing inventory) Costs of placing an order and receiving goods  Shortage costs Costs incurred when demand exceeds supply

17. 17 Holding (Carrying) Costs Obsolescence Insurance Extra staffing Interest Pilferage Damage Warehousing Etc.

18. 18 Inventory Holding Costs (Approximate Ranges) Category Housing costs (building rent, depreciation, operating cost, taxes, insurance) Material handling costs (equipment, lease or depreciation, power, operating cost) Labor cost from extra handling Investment costs (borrowing costs, taxes, and insurance on inventory) Pilferage, scrap, and obsolescence Overall carrying cost Cost as a % of Inventory Value 6% (3 - 10%) 3% (1 - 3.5%) 3% (3 - 5%) 11% (6 - 24%) 3% (2 - 5%) 26%

19. 19 Ordering Costs Supplies Forms Order processing Clerical support, etc.

20. 20 Setup Costs Clean-up costs Re-tooling costs Adjustment costs, etc.

21. 21 Shortage Costs Backordering cost Cost of lost sales

22. 22 Inventory Control System Defined  An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should be  Management has two basic functions concerning inventory:  Establish a system for tracking items in inventory  Make decisions about:  When to order?  How much to order?

23. 23 Objective of Inventory Control To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds  Level of customer service  Costs of ordering and carrying inventory

24. 24  A system to keep track of inventory  A reliable forecast of demand  Knowledge of lead time and lead time variability  Reasonable estimates of  Holding costs  Ordering costs  Shortage costs  A classification system for inventory items Requirements of an Effective Inventory Management

25. 25 Inventory Counting (Control) Systems Periodic System Physical count of items made at periodic intervals; order is placed for a variable amount after fixed passage of time Perpetual (Continuous) Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item (constant amount is ordered when inventory declines to a predetermined level)

26. 26 Inventory accuracy refers to how well the inventory records agree with physical count. Cycle Counting Physically counting a sample of total inventory on a regular basis Used often with ABC classification –A items counted most often (e.g., daily) Inventory Accuracy and Cycle Counting

27. 27 Cycle Counting Management Cycle counting management –How much accuracy is needed? A items: ± 0.2 percent B items: ± 1 percent C items: ± 5 percent –When should cycle counting be performed? –Who should do it? 12-27

28. 28 Advantages of Cycle Counting Eliminates shutdown and interruption of production necessary for annual physical inventories Eliminates annual inventory adjustments Provides trained personnel to audit the accuracy of inventory Allows the cause of errors to be identified and remedial action to be taken Maintains accurate inventory records

29. 29 Inventory Counting Technologies Universal product code (UPC) –Bar code printed on a label that has information about the item to which it is attached Radio frequency identification (RFID) tags –A technology that uses radio waves to identify objects, such as goods in supply chains 12-29

30. 30 Demand Forecasts and Lead Time Forecasts –Inventories are necessary to satisfy customer demands, so it is important to have a reliable estimates of the amount and timing of demand Lead time –Time interval between ordering and receiving the order Point-of-sale (POS) systems –A system that electronically records actual sales –Such demand information is very useful for enhancing forecasting and inventory management 12-30

31. 31 ABC Classification System

32. 32 ABC Classification System Demand volume and value of items vary Items kept in inventory are not of equal importance in terms of: – dollars invested – profit potential – sales or usage volume – stock-out penalties

33. 33 ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly. A A - very important (10 to 20 percent of the number of items in inventory and about 60 to 70 percent of the annual dollar value B B - mod. important C C - least important (50 to 60 percent of the number of items in inventory but only about 10 to 15 percent of the annual dolar value Annual $ value of items High Low FewMany Number of Items A C B

34. 34  Classify inventory into 3 categories typically on the basis of the dollar value to the firm $ volume = Annual demand x Unit cost  A class, B class, C class Policies based on ABC analysis –Develop class A suppliers more carefully –Give tighter physical control of A items –Forecast A items more carefully ABC Analysis

35. 35 % of Inventory Items Classifying Items as ABC 0 20 40 60 80 100 50100 % Annual $ UsageA B C Class% $ Vol% Items A70-805-15 B1530 C 5-1050-60

36. 36 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE

37. 37 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE TOTAL% OF TOTAL% OF TOTAL PARTVALUEVALUEQUANTITY% CUMMULATIVE 9$30,60035.96.06.0 816,00018.75.011.0 214,00016.44.015.0 15,4006.39.024.0 44,8005.66.030.0 33,9004.610.040.0 63,6004.218.058.0 53,0003.513.071.0 102,4002.812.083.0 71,7002.017.0100.0 $85,400

38. 38 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE TOTAL% OF TOTAL% OF TOTAL PARTVALUEVALUEQUANTITY% CUMULATIVE 9$30,60035.96.06.0 816,00018.75.011.0 214,00016.44.015.0 15,4006.39.024.0 44,8005.66.030.0 33,9004.610.040.0 63,6004.218.058.0 53,0003.513.071.0 102,4002.812.083.0 71,7002.017.0100.0 $85,400 A B C

39. 39 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE TOTAL% OF TOTAL% OF TOTAL PARTVALUEVALUEQUANTITY% CUMMULATIVE 9$30,60035.96.06.0 816,00018.75.011.0 214,00016.44.015.0 15,4006.39.024.0 44,8005.66.030.0 33,9004.610.040.0 63,6004.218.058.0 53,0003.513.071.0 102,4002.812.083.0 71,7002.017.0100.0 $85,400 A B C % OF TOTAL CLASSITEMSVALUEQUANTITY A9, 8, 271.015.0 B1, 4, 316.525.0 C6, 5, 10, 712.560.0

40. 40 ABC Classification 100 100 – 80 80 – 60 60 – 40 40 – 20 20 – 0 0 – |||||| 020406080100 % of Quantity % of Value A B C

41. 41 Inventory Models  Multi-Period Inventory Models  Fixed-Order Quantity Models Event triggered (Example: running out of stock)  Fixed-Time Period Models Time triggered (Example: Monthly sales call by sales representative)  Single-Period Inventory Model  One time purchasing decision (Example: vendor selling t-shirts at a football game)  Seeks to balance the costs of inventory overstock and under stock

42. 42 Multi-Period Inventory Models

43. 43 Multi-Period Inventory Models  Fixed-Order Quantity Models (Types of) The Basic Economic Order Quantity Model Economic Production Order Quantity (Economic Lot Size) Model Economic Order Quantity Model with Quantity Discounts  Fixed Time Period (Fixed Order Interval) Models

44. 44 Fixed Order Quantity Models: Economic Order Quantity Model

45. 45 Economic Order Quantity Model Economic Order Quantity Model The basic EOQ Model is used to find a fixed order quantity that will minimize total annual inventory costs Assumptions: Only one product is involved Demand for the product is known with certainty, is constant and uniform (even) throughout the period Lead time (time from ordering to receipt) is known and constant Price per unit of product is constant (no quantity discounts) Inventory holding cost is based on average inventory

46. 46 Economic Order Quantity Model Ordering or setup costs are constant All demands for the product will be satisfied (no back orders are allowed) No stockouts (shortages) are allowed The order quantity is received all at once. (Instantaneous receipt of material in a single lot) The goal is to calculate the order quantitiy that minimizes total cost

47. 47 Basic Fixed-Order Quantity Model and Reorder Point Behavior R = Reorder point Q = Economic order quantity L = Lead time L L QQQ R Number of units on hand (Inv. Level) 1. You receive an order quantity Q. 2. You start using them up over time.(usage rate) 3. When you reach down to a level of inventory of R, you place your next Q sized order. 4. The cycle then repeats. Place order Receive order Time

48. 48 EOQ Model Reorder Point (ROP) Time Inventory Level Average Inventory (Q/2) Lead Time Order Quantity (Q) Demand rate Order placedOrder received

49. 49 Total Annual Cost Total Annual = Cost Annual Purchase Cost Annual Ordering Cost Annual Holding Cost ++ TC=Total annual cost D =Annual demand C =Cost per unit Q =Order quantity in units S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding (carrying) and storage cost per unit of inventory TC=Total annual cost D =Annual demand C =Cost per unit Q =Order quantity in units S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding (carrying) and storage cost per unit of inventory

50. 50 EOQ Cost Model: How Much to Order? By adding the holding and ordering costs together, we determine the total cost curve, which in turn is used to find the optimal order quantity that minimizes total costs Slope = 0 Total Cost Order Quantity, Q Annual cost ($) Minimum total cost Optimal order Q opt Q opt Carrying Cost = HQHQ22HQHQ222 Ordering Cost = SDSDQQSDSDQQQ The total cost curve is U-Shaped

51. 51 More units must be stored if more are ordered Purchase Order DescriptionQty. Microwave1 Order quantity Purchase Order DescriptionQty. Microwave1000 Order quantity Why Holding Costs Increase?

52. 52 Cost is spread over more units Example: You need 1000 microwave ovens Purchase Order DescriptionQty. Microwave1 Purchase Order DescriptionQty. Microwave1 Purchase Order DescriptionQty. Microwave1 Purchase Order Description Qty. Microwave 1 1 Order (Postage $ 0.33)1000 Orders (Postage $330) Order quantity Purchase Order Description Qty. Microwave1000 Why Ordering Costs Decrease ?

53. 53 Basic Fixed-Order Quantity (EOQ) Model Formula Total Annual = Cost Annual Purchase Cost Annual Ordering Cost Annual Holding Cost ++ TC=Total annual cost D =Annual demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding and storage cost per unit of inventory TC=Total annual cost D =Annual demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding and storage cost per unit of inventory

54. 54 EOQ Cost Model Annual ordering cost = S D Q Annual carrying cost = HQHQ22HQHQ222 Total cost = + S D Q H Q 2 TC = + S DQS DQ H Q 2 = + S DQ2S DQ2 H2H2  TC  Q 0 = + S DQ2S DQ2 H2H2 Q opt = 2SD H Deriving Q opt Proving equality of costs at optimal point = S D Q H Q 2 Q 2 = 2S D H Q opt = 2 S D H Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt. The total cost curve reaches its minimum where the carrying and ordering costs are equal

55. 55 Deriving the EOQ We also need a reorder point to tell us when to place an order How much to order?: When to order?

56. 56 Expected Number of Orders Expected Time Between Orders Working Days / Year Working Days / Year == == = =× N D Q*Q* T N d D ROPdL EOQ Model Equations

57. 57 EOQ Example 1 (1 of 3) Annual Demand = 1,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = $2.50 Lead time = 7 days Cost per unit = $15 Given the information below, what are the EOQ and reorder point?

58. 58 EOQ Example 1(2 of 3) In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

59. 59 EOQ Example 1(3 of 3) TC min = SDQSDQ HQ2HQ2 (10)(1,000) 90 (2,5)(90) 2 TC min = $ 111 + $111 = 222 $ Orders per year =D/Q opt =1000/90 =11 orders/year Order cycle time = 365/(D/Qopt) =365/11 =33.1days + +

60. 60 EOQ Example 2(1 of 2) Annual Demand = 10,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = 10% of cost per unit Lead time = 10 days Cost per unit = $15 Determine the economic order quantity and the reorder point given the following… Determine the economic order quantity and the reorder point given the following…

61. 61 EOQ Example 2(2 of 2) Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

62. 62 EOQ Example 3 H = $0.75 per yardS = $150D = 10,000 yards Q opt = 2 S D H Q opt = 2(150)(10,000) (0.75) Q opt = 2,000 yards TC min = + S D Q H Q 2 TC min = + (150)(10,000) 2,000 (0.75)(2,000) 2 TC min = $750 + $750 = $1,500 Orders per year =D/Q opt =10,000/2,000 =5 orders/year Order cycle time =311 days/(D/Q opt ) =311/5 =62.2 store days

63. 63 When to Reorder with EOQ Ordering ? Reorder Point –When the quantity on hand of an item drops to this amount, the item is reordered ROP = d. L where: d= demand rate (units per period, per day, per week) L= lead time (in the same unts as d) –Determinants of the reorder point 1. the rate of demand 2. the lead time 3. the extent of demand and/or lead time variability 4. the degree of stockout risk acceptable to management

64. 64 Reorder Point Example Demand = 10,000 yards/year Store open 311 days/year Daily demand = 10,000 / 311 = 32.154 yards/day Lead time = L = 10 days R = dL = (32.154)(10) = 321.54 yards

65. 65 Reorder Point: Under Uncertainty 12-65 Demand or lead time uncertainty creates the possibility that demand will be greater than available supply To reduce the likelihood of a stockout, it becomes necessary to carry safety stock Safety Stock Stock that is held in excess of expected demand due to variable demand and/or lead time

66. 66 Safety Stock LT Time Expected demand during lead time Maximum probable demand during lead time ROP Quantity Safety stock Safety stock reduces risk of stockout during lead time

67. 67 Variable Demand with a Reorder Point Reorder point, R Q LT Time LT Inventory level 0

68. 68 Reorder Point with a Safety Stock Reorder point, R Q LT Time LT Inventory level 0 Safety Stock

69. 69 Safety Stock? As the amount of safety stock carried increases, the risk of stockout decreases. –This improves customer service level Service level –The probability that demand will not exceed supply during lead time (probability that inventory available during lead time will meet demand) –Service level = 100% - Stockout risk (probability of stockout) -Higher service level means more safety stock - More safety stock means higher ROP 12-69

70. 70 How Much Safety Stock? The amount of safety stock that is appropriate for a given situation depends upon: 1.The average demand rate and average lead time 2.Demand and lead time variability 3.The desired service level 12-70

71. 71 Reorder Point for Service Level Probability of a stockout R Safety stock dL Expected Demand z  d L The reorder point based on a normal distribution of LT demand

72. 72 Reorder Point ROP Probability of a stockout Service level Expected demand Safety stock 0z Quantity z-scale 12-72 Probability of meeting demand during lead time (Probability of no stockout) = service level The ROP based on a normal distribution of lead time demand z  d √ L

73. 73 Reorder Point With Variable Demand R = dL + z  d L where d=average daily demand L=lead time (same time units as average demand)  d =the standard deviation of daily demand(same time units as average demand) z=number of standard deviations corresponding to the service level probability z  d L=safety stock

74. 74 Reorder Point for Variable Demand (Example) The carpet store wants a reorder point with a 95% service level and a 5% stockout probability d= 30 yards per day L= 10 days  d = 5 yards per day For a 95% service level, z = 1.65 R= dL + z  d L = 30(10) + (1.65)(5)( 10) = 326.1 yards Safety stock= z  d L = (1.65)(5)( 10) = 26.1 yards

75. 75 Reorder Point: Lead Time Uncertainty 12-75

76. 76 Fixed Order Quantity Models: -Noninstantaneous Receipt- Production Order Quantity (Economic Lot Size) Model

77. 77 Economic Production Quantity (EPQ) or Economic Order Quantity or Economic Lot Size Assumptions –Only one product is involved –Annual demand requirements are known –Usage rate is constant –Usage occurs continually, but production occurs periodically –The production rate is constant –Lead time does not vary –There are no quantity discounts 12-77

78. 78  Production done in batches or lots  Capacity to produce a part exceeds that part’s usage or demand rate  Allows partial receipt of material  Other EOQ assumptions apply  Suited for production environment  Material produced, used immediately  Provides production lot size  Lower holding cost than EOQ model  Answers how much to order and when to order Production Order Quantity Model

79. 79 EOQ EPQ: Inventory Profile Q Q*Q* I max Production and usage Production and usage Production and usage Usage only Usage only Cumulative production Amount on hand Time 12-79

80. 80 EOQ POQ Model When To Order Reorder Point (ROP) Time Inventory Level Average Inventory Lead Time Optimal Order Quantity (Q*)

81. 81 POQ Model Inventory Levels Time Inventory Level Production Portion of Cycle Max. Inventory Level Q·(1- u/p) Q* Supply Begins Supply Ends Inventory level with no demand Demand portion of cycle with no supply Average inventory (Q/2)(1- u/p)

82. 82 EPQ – Total Cost 12-82

83. 83 POQ Model Equations Production Order Quantity == - Q H* u p p * 1 2*D*S ( )

84. 84 Production Order Quantity Example (1 of 2) H = $0.75 per yardS = $150D = 10,000 yards u = 10,000/311 = 32.2 yards per dayp = 150 yards per day POQ opt = = = 2,256.8 yards 2 S D H 1 - upup 2(150)(10,000) 0.75 1 - 32.2 150 TC = + 1 - = $1,329 upup S D Q H Q 2 Production run = = = 15.05 days per order QpQp 2,256.8 150

85. 85 Production Quantity Example (2 of 2) H = $0.75 per yardS = $150D = 10,000 yards u= 10,000/311 = 32.2 yards per dayp = 150 yards per day Q opt = = = 2,256.8 yards 2C o D C c 1 - dp 2(150)(10,000) 0.75 1 - 32.2150 TC = + 1 - = $1,329 dp CoDCoDQQCoDCoDQQQ CcQCcQ22CcQCcQ222 Production run = = = 15.05 days per order Qp 2,256.8150

86. 86 Number of production runs = = = 4.43 runs/year DQDQ 10,000 2,256.8 Maximum inventory level =Q (1 - ) = 2,256.8 ( 1 - =1,772 yards upup 32.2 150 Production Quantity Example (2 of 2)

87. 87 Fixed-Order Quantity Models: Economic Order Quantity Model with Quantity Discounts

88. 88 Answers how much to order & when to order Allows quantity discounts –Price reduction offered to customers for placing large orders, ie. Price per unit decreases as order quantity increases –Other EOQ assumptions apply Trade-off is between lower price & increased holding cost Quantity Discount Model TC = + H + PD S D Q Q2 Q2 Where P: Unit Price Total cost with purchasing cost

89. 89 Price-Break Model Formula Based on the same assumptions as the EOQ model, the price-break model has a similar Q opt formula: i = percentage of unit cost attributed to carrying inventory C = cost per unit Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

90. 90 Total Costs with PD Cost EOQ TC with PD TC without PD PD 0 Quantity Adding Purchasing cost doesn’t change EOQ

91. 91 Total Cost with Constant Carrying Costs OC EOQ Quantity Total Cost TC a TC c TC b Decreasing Price CC a,b,c

92. 92 Quantity Discounts 12-92

93. 93 Quantity Discount Schedule Discount Number Discount Quantity Discount (%) Discount Price (P) 10 to 999No discount$5.00 21,000 to 1,9994$4.80 32,000 and over5$4.75

94. 94 Quantity Discount – How Much to Order?

95. 95 Price-Break Example 1 (1 of 3) ORDER SIZEPRICE 0 - 99$10 100 - 1998 ( d 1 ) 200+6 ( d 2 ) For this problem holding cost is given as a constant value, not as a percentage of price, so the optimal order quantity is the same for each of the price ranges. (see the figure)

96. 96 Price Break Example 1 (2 of 3) Q opt Carrying cost Ordering cost Inventory cost ($) Q( d 1 ) = 100 Q( d 2 ) = 200 TC ( d 2 = $6 ) TC ( d 1 = $8 ) TC = ($10 )

97. 97 Price Break Example 1 (3 of 3) Q opt Carrying cost Ordering cost Inventory cost ($) Q( d 1 ) = 100 Q( d 2 ) = 200 TC ( d 2 = $6 ) TC ( d 1 = $8 ) TC = ($10 ) The lowest total cost is at the second price break

98. 98 Price Break Example 2 QUANTITYPRICE 1 - 49$1,400 50 - 891,100 90+900 S =$2,500 S =$2,500 H =$190 per computer D =200 Q opt = = = 72.5 PCs 2SD2SDHH2SD2SDHHH2(2500)(200)190 TC = + + PD = $233,784 SD Q opt H Q opt 2 For Q = 72.5 TC = + + PD = $194,105 SDSDQQSDSDQQQ H Q 2 For Q = 90

99. 99 Price-Break Example 3 (1 of 4) A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units? Order Quantity(units)Price/unit($) 0 to 2,499 $1.20 2,500 to 3,999 1.00 4,000 or more.98

100. 100 Price-Break Example (2 of 4) Annual Demand (D)= 10,000 units Cost to place an order (S)= $4 First, plug data into formula for each price-break value of “C” Carrying cost % of total cost (i)= 2% Cost per unit (C) = $1.20, $1.00, $0.98 Interval from 0 to 2499, the Q opt value is feasible Interval from 2500-3999, the Q opt value is not feasible Interval from 4000 & more, the Q opt value is not feasible Next, determine if the computed Q opt values are feasible or not

101. 101 Price-Break Example 2 (3 of 4) Since the feasible solution occurred in the first price- break, it means that all the other true Q opt values occur at the beginnings of each price-break interval. Why? 0 1826 2500 4000 Order Quantity Total annual costs So the candidates for the price- breaks are 1826, 2500, and 4000 units Because the total annual cost function is a “u” shaped function

102. 102 Price-Break Example 2 (4 of 4) Next, we plug the true Q opt values into the total cost annual cost function to determine the total cost under each price-break TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999)= $10,041 TC(4000&more)= $9,949.20 TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999)= $10,041 TC(4000&more)= $9,949.20 Finally, we select the least costly Q opt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

103. 103 Multi-period Inventory Models: Fixed Time Period (Fixed-Order- Interval) Models

104. 104  Orders are placed at fixed time intervals  Order quantity for next interval? (inventory is brought up to target amount, amount ordered varies)  Risk of stockout between intervals  Reasons for using the FOI model –Supplier’s policy may encourage its use –Grouping orders from the same supplier can produce savings in ordering, packing and shipping costs. –Some circumstances do not lend themselves to continuously monitoring inventory position –Requires only periodic checks of inventory levels (no continous monitoring is required) Fixed-Order-Interval Model

105. 105 Inventory Level in a Fixed Period System Various amounts (Q i ) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to target maximum ppp Q1Q1Q1Q1 Q2Q2Q2Q2 Q3Q3Q3Q3 Q4Q4Q4Q4 Target maximum Time d Inventory

106. 106  Requires a larger safety stock  Increases carrying cost  Costs of periodic reviews Fixed-Interval Disadvantages

107. 107 Fixed-Quantity vs. Fixed-Interval Fixed-Quantity vs. Fixed-Interval Ordering 12-107

108. 108 FOI Model 12-108

109. 109 Fixed-Time Period Model with Safety Stock Formula q = Average demand + Safety stock – Inventory currently on hand

110. 110 Fixed-Time Period Model: Determining the Value of  T+L The standard deviation of a sequence of random events equals the square root of the sum of the variances

111. 111 Order Quantity for a Periodic Inventory System Q = d(T + L) + z  d T + L - I where d= average demand rate T= the fixed time between orders L= lead time  d = standard deviation of demand z  d T + L= safety stock I= inventory level z = the number of standard deviations for a specified service level

112. 112 Fixed-Period Model with Variable Demand (Example 1) d= 6 bottles per day  d = 1.2 bottles T= 60 days L= 5 days I= 8 bottles z= 1.65 (for a 95% service level) Q= d(T + L) + z  d T + L - I = (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8 = 397.96 bottles

113. 113 Fixed-Time Period Model with Variable Demand (Example 2)(1 of 3) Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The standard deviation of daily demand is 4 units. Given the information below, how many units should be ordered?

114. 114 Fixed-Time Period Model with Variable Demand (Example 2)(2 of 3) So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75

115. 115 Fixed-Time Period Model with Variable Demand (Example 2) (3 of 3) So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

116. 116 Miscellaneous Systems: Optional Replenishment System Maximum Inventory Level, M M Actual Inventory Level, I q = M - I I Q = minimum acceptable order quantity If q > Q, order q, otherwise do not order any.

117. 117 Single-Period Inventory Model

118. 118 Single period model: model for ordering of perishables and other items with limited useful lives Shortage cost: generally the unrealized profits per unit C shortage = C s = Revenue per unit – Cost per unit Excess cost: difference between purchase cost and salvage value of items left over at the end of a period C excess = C e = Cost per unit – Salvage value per unit Single Period Model

119. 119 Single-Period Model The goal of the single-period model is to identify the order quantity that will minimize the long-run excess and shortage costs Two categories of problem: –Demand can be characterized by a continuous distribution –Demand can be characterized by a discrete distribution 12-119

120. 120 Continuous stocking levels –Identifies optimal stocking levels –Optimal stocking level balances unit shortage and excess cost Discrete stocking levels –Service levels are discrete rather than continuous –Desired service level is equaled or exceeded Single Period Model

121. 121 Single-Period Model This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cs/Cs+Ce

122. 122 Optimal Stocking Level Service Level So Quantity Ce Cs Balance point Service level = CsCs Cs + Ce Cs = Shortage cost per unit Ce = Excess cost per unit (Optimum Stocking Quantity)

123. 123 Single Period Example 1 Ce = $0.20 per unit Cs = $0.60 per unit Service level = Cs/(Cs+Ce) =.6/(.6+.2) Service level =.75 Service Level = 75% Quantity Ce Cs Stockout risk = 1.00 – 0.75 = 0.25

124. 124 Single Period Model Example 2 Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. How many shirts should we make for the game? C s = $10 and C e = $5; P ≤ $10 / ($10 + $5) =.667 Z.667 =.432 therefore we need 2,400 +.432(350) = 2,551 shirts

125. 125 Last Words Inventories have certain functions. But too much inventory -Tends to hide problems -Costly to maintain So it is desired Reduce lot sizes Reduce safety stocks