Inventories are important to all types of

12 Inventory Management PowerPoint Slides by Jeff Heyl For Operations Management, 9e by Krajewski/Ritzman/Malhotra 2010 Pearson Education 12 1 Inventory Management Inventories are important to all types of organizations They have to be counted, paid for, used in operations, used to satisfy customers, and managed Too much inventory reduces profitability Too little inventory damages customer confidence Inventory trade-offs 12 2 1

ABC Analysis Stock-keeping units (SKU) Identify the classes so management can control inventory levels A Pareto chart Cycle counting 12 3 ABC Analysis Perc centage of dollar value 100 90 Class A 80 70 60 50 40 30 20 10 Class B Class C 0 10 20 30 40 50 60 70 80 90 100 Percentage of SKUs Figure 12.1 Typical Chart Using ABC Analysis 12 4 2

Economic Order Quantity The lot size, Q, that minimizes total annual inventory holding and ordering costs Five assumptions 1. Demand rate is constant and known with certainty 2. No constraints are placed on the size of each lot 3. The only two relevant costs are the inventory holding cost and the fixed cost per lot for ordering or setup 4. Decisions for one item can be made independently of decisions for other items 5. The lead time is constant and known with certainty 12 5 Economic Order Quantity Don t use the EOQ Make-to-order strategy Order size is constrained Modify the EOQ Quantity discounts Replenishment not instantaneous Use the EOQ Make-to-stock Carrying and setup costs are known and relatively stable 12 6 3

Calculating EOQ hand inventory (units s) On- Q Q 2 Receive order Inventory depletion (demand rate) Average cycle inventory Figure 12.2 Cycle-Inventory Levels 1 cycle Time 12 7 Calculating EOQ Annual holding cost Annual holding cost = (Average cycle inventory) (Unit holding cost) Annual ordering cost Annual ordering cost = (Number of orders/year) (Ordering or setup costs) Total annual cycle-inventory y cost Total costs = Annual holding cost + Annual ordering or setup cost 12 8 4

Calculating EOQ Annual cost (dollars) Total cost Holding cost Ordering cost Lot Size (Q) Figure 12.3 Graphs of Annual Holding, Ordering, and Total Costs 12 9 Calculating EOQ Total annual cycle-inventory cost Q D C = (H) + (S) 2 Q where C = total annual cycle-inventory cost Q = lot size H = holding cost per unit per year D = annual demand S = ordering or setup costs per lot 12 10 5

The Cost of a Lot-Sizing Policy EXAMPLE 12.1 A museum of natural history opened a gift shop which operates 52 weeks per year. Managing inventories has become a problem. Top-selling SKU is a bird feeder. Sales are 18 units per week, the supplier charges $60 per unit. Ordering cost is $45. Annual holding cost is 25 percent of a feeder s value. Management chose a 390-unit lot size. What is the annual cycle-inventory cost of the current policy of using a 390-unit lot size? Would a lot size of 468 be better? 12 11 The Cost of a Lot-Sizing Policy We begin by computing the annual demand and holding cost as D = (18 units/week)(52 weeks/year) = 936 units H = 0.25($60/unit) = $15 The total annual cycle-inventory cost for the current policy is Q D 390 936 C = 2 (H) + Q (S) = ($15) + ($45) 2 390 = $2,925 + $108 = $3,033 The total annual cycle-inventory cost for the alternative lot size is 468 936 C = ($15) + ($45) = $3,510 + $90 = $3,600 2 468 12 12 6

The Cost of a Lot-Sizing Policy Current cost Figure 12.4 Total Annual Cycle-Inventory Cost Function for the Bird Feeder Annual cost (dollars) Lowest cost 3000 2000 1000 Total cost Q D = 2 (H) + (S) Q Holding cost = 0 50 100 150 200 250 300 350 400 Best Q (EOQ) Lot Size (Q) Ordering cost = Current Q Q 2 D Q (H) (S) 12 13 Calculating EOQ The EOQ formula: EOQ = 2DS H Time between orders TBO EOQ = EOQ D (12 months/year) 12 14 7

Finding the EOQ, Total Cost, TBO EXAMPLE 12.2 For the bird feeders in Example 12.1, calculate the EOQ and its total annual cycle-inventory cost. How frequently will orders be placed if the EOQ is used? Using the formulas for EOQ and annual cost, we get EOQ = 2DS = H 2(936)(45) 15 = 74.94 or 75 units 12 15 Finding the EOQ, Total Cost, TBO Figure 12.5 shows that the total annual cost is much less than the $3,033 cost of the current policy of placing 390-unit orders. Figure 12.5 Total Annual Cycle-Inventory Costs Based on EOQ Using Tutor 12.2 12 16 8

Finding the EOQ, Total Cost, TBO When the EOQ is used, the TBO can be expressed in various ways for the same time period. TBO EOQ = EOQ D 75 = = 0.080 year 936 TBO EOQ = EOQ D (12 months/year) 75 = (12) = 0.96 month 936 EOQ TBO EOQ = D (52 weeks/year) 75 = (52) = 4.17 weeks 936 TBO EOQ = EOQ D (365 days/year) 75 = (365) = 29.25 days 936 12 17 Application 12.1 Suppose that you are reviewing the inventory policies on an $80 item stocked at a hardware store. The current policy is to replenish inventory by ordering in lots of 360 units. Additional information is: D = 60 units per week, or 3,120 units per year S = $30 per order H = 25% of selling price, or $20 per unit per year What is the EOQ? 2DS EOQ = = H 2(3,120)(30) 20 = 97 units 12 18 9

Application 12.1 What is the total annual cost of the current policy (Q = 360), and how does it compare with the cost with using the EOQ? Current Policy Q = 360 units C = (360/2)(20) + (3,120/360)(30) C = 3,600 + 260 C = $3,860 EOQ Policy Q = 97 units C = (97/2)(20) + (3,120/97)(30) C = 970 + 965 C = $1,935 12 19 Application 12.1 What is the time between orders (TBO) for the current policy and the EOQ policy, expressed in weeks? TBO 360 = 360 3,120 (52 weeks per year) = 6 weeks 97 TBO EOQ = (52 weeks per year) = 1.6 weeks 3,120 12 20 10

Managerial Insights TABLE 12.1 SENSITIVITY ANALYSIS OF THE EOQ Parameter EOQ Parameter Change EOQ Change Comments 2DS Demand H Order/Setup 2DS Costs H Increase in lot size is in proportion to the square root of D. Weeks of supply decreases and inventory turnover increases because the lot size decreases. Holding 2DS Larger lots are justified when holding H costs decrease. Costs 12 21 Inventory Control Systems Two important questions: How much? When? Nature of demand Independent demand Dependent demand 12 22 11

Inventory Control Systems Continuous review (Q) system Reorder point system (ROP) and fixed order quantity system For independent demand items Tracks inventory position (IP) Includes scheduled receipts (SR), on-hand inventory (OH), and back orders (BO) Inventory position = On-hand inventory + Scheduled receipts Backorders IP = OH + SR BO 12 23 Selecting the Reorder Point Order received IP Order received IP Order received IP Order received On-hand inventory R OH Q OH Q OH Q Order placed Order placed Order placed TBO L TBO L TBO L Time Figure 12.6 Q System When Demand and Lead Time Are Constant and Certain 12 24 12

Application 12.2 The on-hand inventory is only 10 units, and the reorder point R is 100. There are no backorders and one open order for 200 units. Should a new order be placed? IP = OH + SR BO = 10 + 200 0 = 210 R = 100 Decision: Place no new order 12 25 Placing a New Order EXAMPLE 12.3 Demand for chicken soup at a supermarket is always 25 cases a day and the lead time is always 4 days. The shelves were just restocked with chicken soup, leaving an on-hand inventory of only 10 cases. No backorders currently exist, but there is one open order in the pipeline for 200 cases. What is the inventory position? Should a new order be placed? R = Total demand during lead time = (25)(4) = 100 cases IP = OH + SR BO = 10 + 200 0 = 210 cases 12 26 13

Continuous Review Systems Selecting the reorder point with variable demand and constant lead time Reorder point = Average demand during lead time + Safety stock = dl + safety stock where d = average demand d per week (or day or months) L = constant lead time in weeks (or days or months) 12 27 Continuous Review Systems On-hand inventory R Order received Order placed Q IP Order received Order placed Q IP Od Order received Order placed Q IP Order received 0 L 1 L 2 L 3 TBO 1 TBO 2 TBO 3 Time Figure 12.7 Q System When Demand Is Uncertain 12 28 14

Reorder Point 1. Choose an appropriate service-level policy Select service level or cycle service level Protection interval 2. Determine the demand during lead time probability distribution 3. Determine the safety stock and reorder point levels 12 29 Demand During Lead Time Specify mean and standard deviation Standard deviation of demand during lead time σ dlt = σ d2 L = σ d L Safety stock and reorder point Safety stock = zσ dlt where z = number of standard deviations needed to achieve the cycle-service level σ dlt = stand deviation of demand during lead time Reorder point = R = dl + safety stock 12 30 15

Demand During Lead Time σ t = 15 75 Demand for week 1 σ t = 15 + + 75 Demand for week 2 σ t = 25.98 σ t = 15 75 Demand for week 3 = 225 Demand for 3-week lead time Figure 12.8 Development of Demand Distribution for the Lead Time 12 31 Demand During Lead Time Cycle-service level = 85% Average demand during lead time R Probability of stockout (1.0 0.85 = 0.15) zσ dlt Figure 12.9 Finding Safety Stock with a Normal Probability Distribution for an 85 Percent Cycle-Service Level 12 32 16

Reorder Point for Variable Demand EXAMPLE 12.4 Let us return to the bird feeder in Example 12.2. The EOQ is 75 units. Suppose that the average demand is 18 units per week with a standard deviation of 5 units. The lead time is constant at two weeks. Determine the safety stock and reorder point if management wants a 90 percent cycleservice level. 12 33 Reorder Point for Variable Demand In this case, σ d = 5, d = 18 units, and L = 2 weeks, so σ dlt = σ d L = 5 2 = 7.07. Consult the body of the table in the Normal Distribution appendix for 0.9000, which corresponds to a 90 percent cycle-service level. The closest number is 0.8997, which corresponds to 1.2 in the row heading and 0.08 in the column heading. Adding these values gives a z value of 1.28. With this information, we calculate the safety stock and reorder point as follows: Safety stock = zσ dlt = 1.28(7.07) = 9.05 or 9 units Reorder point = dl + Safety stock = 2(18) + 9 = 45 units 12 34 17

Application 12.3 Suppose that the demand during lead time is normally distributed with an average of 85 and σ dlt = 40. Find the safety stock, and reorder point R, for a 95 percent cycle-service level. Safety stock = zσ dlt = 1.645(40) = 65.8 or 66 units R = Average demand during lead time + Safety stock R = 85 + 66 = 151 units Find the safety stock, and reorder point R, for an 85 percent cycle-service level. Safety stock = zσ dlt = 1.04(40) = 41.6 or 42 units R = Average demand during lead time + Safety stock R = 85 + 42 = 127 units 12 35 Reorder Point for Variable Demand and Lead Time Often the case that both are variable The equations are more complicated Safety stock = zσ dlt R = (Average weekly demand Average lead time) + Safety stock = dl + Safety stock where d = Average weekly (or daily or monthly) demand d L = Average lead time σ d = Standard deviation of weekly (or daily or monthly) demand σ LT = Standard deviation of the lead time σ dlt = Lσ d2 + d 2 σ 2 LT 12 36 18

Reorder Point EXAMPLE 12.5 The Office Supply Shop estimates that the average age demand d for a popular ball-point pen is 12,000 pens per week with a standard deviation of 3,000 pens. The current inventory policy calls for replenishment orders of 156,000 pens. The average lead time from the distributor is 5 weeks, with a standard deviation of 2 weeks. If management wants a 95 percent cycleservice level, what should the reorder point be? 12 37 Reorder Point We have d = 12,000 pens, σ d = 3,000 pens, L = 5 weeks, and σ LT = 2 weeks σ dlt = Lσ d2 + d 2 σ LT2 = (5)(3,000) 2 + (12,000) 2 (2) 2 = 24,919.87 pens From the Normal Distribution appendix for 0.9500, the appropriate z value = 1.65. We calculate the safety stock and reorder point as follows: Safety stock = zσ dlt = (1.65)(24,919.87) = 41,117.79 or 41,118 pens Reorder point = dl + Safety stock = (12,000)(5) + 41.118 = 101,118 pens 12 38 19

Application 12.4 Grey Wolf lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all of the room service items, including a special pint-scented bar soap. The daily demand for the soap is 275 bars, with a standard deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year. What should the reorder point be for the bar of soap if management wants to have a 99 percent cycle-service? 12 39 Application 12.4 d = 275 bars L =5 days σ d = 30 bars σ LT =1 day σ dlt = Lσ d2 + d 2 σ 2 LT = 283.06 bars From the Normal Distribution appendix for 0.9900, z = 2.33. We calculate the safety stock and reorder point as follows; Safety stock = zσ dlt = (2.33)(283.06) = 659.53 or 660 bars Reorder point + safety stock = dl + safety stock = (275)(5) + 660 = 2,035 bars 12 40 20

Continuous Review Systems Two-Bin system Visual system An empty first bin signals the need to place an order Calculating total systems costs Total cost = Annual cycle inventory holding cost + Annual ordering cost + Annual safety stock holding cost Q D C = (H) + (S) + (H) (Safety stock) 2 Q 12 41 Application 12.5 The Discount Appliance Store uses a continuous review system (Q system). One of the company s items has the following characteristics: Demand = 10 units/wk (assume 52 weeks per year) Ordering and setup cost (S) = $45/order Holding cost (H) = $12/unit/year Lead time (L) = 3 weeks (constant) Standard deviation in weekly demand = 8 units Cycle-service level = 70% 12 42 21

Application 12.5 What is the EOQ for this item? D = 10/wk 52 wks/yr = 520 units 2DS EOQ = = H 2(520)(45) 12 = 62 units What is the desired safety stock? σ dlt = σ d L = 8 3 = 14 units Safety stock = zσ dlt = 0.525(14) = 8 units 12 43 Application 12.5 What is the desired reorder point R? R = Average demand during lead time + Safety stock R = 3(10) + 8 = 38 units What is the total annual cost? 62 520 C = 2 ($12) + 62 ($45) + 8($12) = $845.4242 12 44 22

Application 12.5 Suppose that the current policy is Q = 80 and R = 150. What will be the changes in average cycle inventory and safety stock if your EOQ and R values are implemented? Reducing Q from 80 to 62 Cycle inventory reduction = 40 31 = 9 units Safety stock reduction = 120 8 = 112 units Reducing R from 150 to 38 12 45 Periodic Review System (P) Fixed interval reorder system or periodic reorder system Four of the original EOQ assumptions maintained No constraints are placed on lot size Holding and ordering costs Independent demand Lead times are certain Order is placed to bring the inventory position up to the target inventory level, T, when the predetermined time, P, has elapsed 12 46 23

Periodic Review System (P) On-hand inventory T IP 1 IP 3 IP 2 IP IP IP Order Order Order received received received Q Q 3 1 OH Q 2 OH Order placed Order placed L L L P P Protection interval Time Figure 12.10 P System When Demand Is Uncertain 12 47 How Much to Order in a P System EXAMPLE 12.6 A distribution center has a backorder for five 36-inch color TV sets. No inventory is currently on hand, and now is the time to review. How many should be reordered if T = 400 and no receipts are scheduled? IP = OH + SR BO = 0 + 0 5 = 5 sets T IP = 400 ( 5) = 405 sets That is, 405 sets must be ordered to bring the inventory position up to T sets. 12 48 24

Application 12.6 The on-hand inventory is 10 units, and T is 400. There are no back orders, but one scheduled receipt of 200 units. Now is the time to review. How much should be reordered? IP = OH + SR BO = 10 + 200 0 = 210 T IP = 400 210 = 190 The decision is to order 190 units 12 49 Periodic Review System Selecting the time between reviews, choosing P and T Selecting T when demand is variable and lead time is constant IP covers demand over a protection interval of P + L The average demand during the protection interval is d(p + L), or T = d(p + L) + safety stock for protection interval Safety stock = zσ P + L, where σ P + L = d P L 12 50 25

Calculating P and T EXAMPLE 12.7 Again, let us return to the bird feeder example. Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system developed in Example 12.4 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent. What is the equivalent P system? Answers are to be rounded to the nearest integer. 12 51 Calculating P and T We first define D and then P. Here, P is the time between reviews, expressed in weeks because the data are expressed as demand per week: D = (18 units/week)(52 weeks/year) = 936 units EOQ P = (52) = D 75 936 (52) = 4.2 or 4 weeks With d = 18 units per week, an alternative approach is to calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4 weeks. Either way, we would review the bird feeder inventory every 4 weeks. 12 52 26

Calculating P and T We now find the standard deviation of demand over the protection interval (P + L) = 6: 12.25units P L d P L calculating T, we also need 65Before a z value. For a 90 percent cycle-service level z = 1.28. The safety stock becomes Safety stock = zσ P + L = 1.28(12.25) = 15.68 or 16 units We now solve for T: T = Average demand during the protection interval + Safety stock = d(p + L) + safety stock = (18 units/week)(6 weeks) + 16 units = 124 units 12 53 Periodic Review System Use simulation when both demand and lead time are variable Suitable to single-bin systems Total costs for the P system are the sum of the same three cost elements as in the Q system Order quantity and safety stock are calculated differently dp D C = 2 (H) + dp (S) + Hzσ P + L 12 54 27

nits28-jun-10 Application 12.7 Return to Discount Appliance Store (Application 12.4), but now use the P system for the item. Previous information Demand = 10 units/wk (assume 52 weeks per year) = 520 EOQ = 62 units (with reorder point system) Lead time (L) = 3 weeks Standard deviation in weekly demand = 8 units z = 0.525 (for cycle-service level of 70%) Reorder interval P, if you make the average lot size using the Periodic Review System approximate the EOQ. 12 55 Application 12.7 Reorder interval P, if you make the average lot size using the Periodic Review System approximate the EOQ. Safety stock P = (EOQ/D)(52) = (62/529)(52) = 6.2 or 6 weeks Safety stock = d P L 0.51o13u258632.6 rtarget inventory T = d(p + L) + safety stock for protection interval T = 10(6 + 3) + 13 = 103 units 12 56 28

Application 12.7 Total cost dp D C = 2 (H) + dp (S) + Hzσ P + L 10(6) 520 = ($12) + ($45) + (13)($12) = $906.00 2 10(6) 12 57 Comparative Advantages Primary advantages of P systems Convenient Orders can be combined Only need to know IP when review is made Primary advantages of Q systems Review frequency may be individualized Fixed lot sizes can result in quantity discounts Lower safety stocks 12 58 29

Hybrid systems Optional replenishment systems Optimal review, min-max, or (s,s) system, like the P system Reviews IP at fixed time intervals and places a variablesized order to cover expected needs Ensures that a reasonable large order is placed Base-stock system Replenishment order is issued each time a withdrawal is made Order quantities vary to keep the inventory position at R Minimizes cycle inventory, but increases ordering costs Appropriate for expensive items 12 59 Solved Problem 1 Booker s Book Bindery divides SKUs into three classes, according to their dollar usage. Calculate the usage values of the following SKUs and determine which is most likely to be classified as class A. SKU Number Description Quantity Used per Year Unit Value ($) 1 Boxes 500 3.00 2 Cardboard 18,000 0.02 (square feet) 3 Cover stock 10,000 0.75 4 Glue (gallons) 75 40.00 5 Inside covers 20,000 0.05 6 Reinforcing tape 3,000 0.15 (meters) 7 Signatures 150,000 0.45 12 60 30

Solved Problem 1 The annual dollar usage for each item is determined by multiplying the annual usage quantity by the value per unit. As shown in Figure 12.11, the SKUs are then sorted by annual dollar usage, in declining order. Finally, A B and B C class lines are drawn roughly, according to the guidelines presented in the text. Here, class A includes only one SKU (signatures), which represents only 1/7, or 14 percent, of the SKUs but accounts for 83 percent of annual dollar usage. Class B includes the next two SKUs, which taken together represent 28 percent of the SKUs and account for 13 percent of annual dollar usage. The final four SKUs, class C, represent over half the number of SKUs but only 4 percent of total annual dollar usage. 12 61 Solved Problem 1 Quantity SKU Unit Value Annual Dollar Description Used per Number ($) Usage ($) Year 1 Boxes 500 3.00 = 1,500 2 Cardboard 18,000 0.02 = 360 (square feet) 3 Cover stock 10,000 0.75 = 7,500 4 Glue (gallons) 75 40.00 = 3,000 5 Inside covers 20,000 0.05 = 1,000 6 Reinforcing tape 3,000 0.15 = 450 (meters) 7 Signatures 150,000 0.45 = 67,500 Total 81,310 12 62 31

Solved Problem 1 Percentage of Dol llar Value 100 Class B Class C 90 Class A 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100 Percentage of SKUs Figure 12.11 Annual Dollar Usage for Class A, B, and C SKUs Using Tutor 12.2 12 63 Solved Problem 2 Nelson s Hardware Store stocks a 19.2 volt cordless drill that is a popular seller. Annual demand is 5,000 units, the ordering cost is $15, and the inventory holding cost is $4/unit/year. a. What is the economic order quantity? b. What is the total annual cost for this inventory item? a. The order quantity is 2DS 2(5,000)($15) EOQ = = H $4 = 37,500 = 193.65 or 194 drills b. The total annual cost is Q D 194 5,000 C = 2 (H) + (S) = ($4) + ($15) = $774.60 Q 2 194 12 64 32

Solved Problem 3 A regional distributor purchases discontinued appliances from various suppliers and then sells them on demand to retailers in the region. The distributor operates 5 days per week, 52 weeks per year. Only when it is open for business can orders be received. Management wants to reevaluate its current inventory policy, which calls for order quantities of 440 counter-top mixers. The following data are estimated for the mixer: Average daily demand (d) = 100 mixers Standard deviation of daily demand (σ d ) = 30 mixers Lead time (L) = 3 days Holding cost (H) = $9.40/unit/year Ordering cost (S) = $35/order Cycle-service level = 92 percent The distributor uses a continuous review (Q) system 12 65 Solved Problem 3 a. What order quantity Q, and reorder point, R, should be used? b. What is the total annual cost of the system? c. If on-hand inventory is 40 units, one open order for 440 mixers is pending, and no backorders exist, should a new order be placed? 12 66 33

Solved Problem 3 a. Annual demand is D = (5 days/week)(52 weeks/year)(100 mixers/day) = 26,000 mixers/year The order quantity is 2DS 2(26,000)($35) EOQ = = H $9.40 = 193,167 = 440.02 or 440 mixers 12 67 Solved Problem 3 The standard deviation of the demand during lead time distribution is σ dlt = σ d L = 30 3 = 51.96 A 92 percent cycle-service level corresponds to z = 1.41 Safety stock = zσ dlt = 1.41(51.96 mixers) = 73.26 or 73 mixers Average demand during lead time = dl = 100(3) = 300 mixers Reorder point (R) = Average demand during lead time + Safety stock = 300 mixers + 73 mixers = 373 mixers With a continuous review system, Q = 440 and R = 373 12 68 34

Solved Problem 3 b. The total annual cost for the Q systems is Q D C = 2 (H) + Q (S) +(H)(Safety stock) 440 26,000 C = ($9.40) + ($35) + ($9.40)(73) = $4,822.38 2 440 c. Inventory position = On-hand inventory + Scheduled receipts Backorders IP = OH + SR BO = 40 + 440 0 = 480 mixers Because IP (480) exceeds R (373), do not place a new order 12 69 Solved Problem 4 Suppose that a periodic review (P) system is used at the distributor in Solved Problem 3, but otherwise the data are the same. a. Calculate the P (in workdays, rounded to the nearest day) that gives approximately the same number of orders per year as the EOQ. b. What is the target inventory level, T? Compare the P system to the Q system in Solved Problem 3. c. What is the total annual cost of the P system? d. It is time to review the item. On-hand inventory is 40 mixers; receipt of 440 mixers is scheduled, and no backorders exist. How much should be reordered? 12 70 35

Solved Problem 4 a. The time between orders is EOQ 440 P = (260 days/year) = (260) = 4.4 or 4 days D 26,000 b. Figure 12.12 shows that T = 812 and safety stock = (1.41)(79.37) = 111.91 or about 112 mixers. The corresponding Q system for the counter-top mixer requires less safety stock. Figure 12.12 OM Explorer Solver for Inventory Systems 12 71 Solved Problem 4 c. The total annual cost of the P system is dp D C = 2 (H) + dp (S) + (H)(Safety stock) 100(4) 26,000 C = ($9.40) + ($35) + ($9.40)(1.41)(79.37) 2 100(4) = $5,207.80 d. Inventory position is the amount on hand plus scheduled receipts minus backorders, or IP = OH + SR BO = 40 + 440 0 = 480 mixers The order quantity is the target inventory level minus the inventory position, or Q = T IP = 812 mixers 480 mixers = 332 mixers An order for 332 mixers should be placed. 12 72 36

Solved Problem 5 Grey Wolf Lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all room service items, including a special pine-scented bar soap. The daily demand d for the soap is 275 bars, with a standard d deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year. a. What is the economic order quantity for the bar of soap? b. What should the reorder point be for the bar of soap if management wants to have a 99 percent cycle-service level? c. What is the total t annual cost for the bar of soap, assuming a Q system will be used? 12 73 Solved Problem 5 a. We have D = (275)(365) = 100,375 bars of soap; S = $10; and H = $0.30. The EOQ for the bar of soap is 2DS EOQ = = H 2(100,375)($10) $0.30 = 6,691,666.7 = 2,586.83 or 2,587 bars 12 74 37

Solved Problem 5 b. We have d = 275 bars/day, σ d = 30 bars, L = 5 days, and σ LT = 1 day. σ dlt = Lσ d2 + d 2 σ LT 2 = (5)(30) 2 + (275) 2 (1) 2 = 283.06 bars Consult the body of the Normal Distribution appendix for 0.9900. The closest value is 0.9901, which corresponds to a z value of 2.33. We calculate the safety stock and reorder point as follows: Safety stock = zσ dlt = (2.33)(283.06) = 659.53 or 660 bars Reorder point = dl + Safety stock = (275)(5) + 660 = 2,035 bars 12 75 Solved Problem 5 c. The total annual cost for the Q system is Q D C = 2 (H) + Q (S) + (H)(Safety stock) 2,587 100,375 C = ($0.30) + ($10) + ($0.30)(660) = $974.05 2 2,587 12 76 38

Solved Problem 6 Zeke s Hardware Store sells furnace filters. The cost to place an order to the distributor is $25 and the annual cost to hold a filter in stock is $2. The average demand per week for the filters is 32 units, and the store operates 50 weeks per year. The weekly demand for filters has the probability distribution shown on the left below. The delivery lead time from the distributor is uncertain and has the probability distribution shown on the right below. Suppose Zeke wants to use a P system with P = 6 weeks and a cycle-service level of 90 percent. What is the appropriate value for T and the associated annual cost of the system? 12 77 Solved Problem 6 Demand Probability Lead Time (wks) Probability 24 0.15 28 0.20 32 0.30 36 0.20 40 0.15 1 0.05 2 0.25 3 0.40 4 0.25 5 0.05 12 78 39

Solved Problem 6 Figure 12.13 contains output from the Demand During the Protection Interval Simulator from OM Explorer. Figure 12.13 OM Explorer Solver for Demand during the Protection Interval 12 79 Solved Problem 6 Given the desired cycle-service level of 90 percent, the appropriate T value is 322 units. The simulation estimated the average demand during the protection interval to be 289 units, consequently the safety stock is 322 289 = 33 units. The annual cost of this P system is 6(32) 50(32) C = ($2) + ($25) + (33)($2) 2 6(32) = $192.00 + $208.33 + $66.00 = $466.33 12 80 40

Solved Problem 7 Consider Zeke s inventory in Solved Problem 6. Suppose that he wants to use a continuous review (Q) system for the filters, with an order quantity of 200 and a reorder point of 140. Initial inventory is 170 units. If the stockout t cost is $5 per unit, and all of the other data in Solved Problem 6 are the same, what is the expected cost per week of using the Q system? Figure 12.14 shows output from the Q System Simulator in OM Explorer. Only weeks 1 through 13 and weeks 41 through 50 are shown in the figure. The average total cost per week is $305.62. Notice that no stockouts occurred in this simulation. These results are dependent on Zeke s choices for the reorder point and lot size. It is possible that stockouts would occur if the simulation were run for more than 50 weeks. 12 81 Solved Problem 7 Figure 12.14 OM Explorer Q System Simulator 12 82 41

12 83 42