stock-keeping unit (SKU). An SKU is a specific item at a particular geographic location.

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1 Chapter 12: Inventory Management Have you ever been in a rush to get through the grocery checkout to be stuck in line behind a person buying numerous varieties of the same item - 20 cans of pet food, with each can being a different flavour? The cashier scans each item, and you wander why not scan one and enter a quantity of 20. From inventory 18 management perspective, it is critical that the cashier scan each individual can. Why? The information is used to update the inventory record of the store to determine how much and when a replenishment order should be place. Learning objectives Conduct an ABC analysis Explain and use cycle counting Explain and use the EOQ model for independent inventory demand Compute a reorder point and safety stock Apply the production order quantity model Explain and use the quantity discount model Understand service levels and probabilistic inventory models The Newsvendor model 18 Inventory management and control are done at the level of the individual item or stock-keeping unit (SKU). An SKU is a specific item at a particular geographic location. 244 / 539 Inventory Management The objective of inventory management is to: provide the desired level of customer services 19 allow cost-efficient operations minimize the inventory investment Importance of inventory One of the most expensive assets of many companies representing as much as 50% of total invested capital Operations managers must balance inventory investment and customer service 19 Customer service is a company s ability to satisfy the needs of its customers. That is, whether or not the product is available for the customer when the customer want it. 245 / 539

2 Functions of Inventory Management To decouple or separate various parts of the production process Anticipation inventory: items built in anticipation of future demand. To decouple the firm from fluctuations in demand and provide a stock of goods that will provide a selection for customers Fluctuation inventory: Protect against unexpected demand variations. To take advantage of quantity discounts: Economies of scale Lot-size inventory: results from the actual quantity purchased. Allows for lower unit costs. To hedge against inflation Speculative inventory: Extra inventory built up or purchased to protect against some future event. 246 / 539 Types of Inventory Raw material: refers to items purchased but not processed Work-in-process: refers to items undergone some change but not completed Maintenance/repair/operating (MRO): refers to items necessary to keep machinery and processes productive Finished goods: refers to completed product awaiting shipment 247 / 539

3 Inventory Management System Two ingredients of inventory management system: How inventory items can be classified (ABC analysis). A method for determining level of control and frequency of review of inventory. How accurate inventory records can be maintained (Record accuracy) 248 / 539 ABC Analysis ABC analysis is a method of dividing on-hand inventory into three classes based on annual dollar volume. ABC is used to determine level of control and frequency of review of inventory items. Class A - high annual dollar volume (70% -80%) Class B - medium annual dollar volume (15% - 25%) Class C - low annual dollar volume ( 5%) ABC analysis is an inventory application of what is known as Pareto Principle. The Pareto principle states that there are a critical few and trivial many. The Pareto principle suggests that about 20% of the inventory items will account for about 80% of the inventory value. The idea is to establish policies that focus on the few critical parts and not the many trivial ones. It is not realistic to monitor inexpensive items with the same intensity as very expensive items. 249 / 539

4 ABC Analysis Procedure for an ABC inventory Analysis: 1. Calculate the annual dollar usage for each item. 2. List the items in descending order based on annual dollar usage. 3. Calculate the cumulative annual dollar volume percentage. 4. Classify the items into groups. 250 / 539 ABC Analysis 251 / 539

5 ABC Analysis 252 / 539 Figure It Out: ABC Analysis Perform an ABC analysis on the following set of products. Item Annual Demand Unit Cost A $9 B $90 C $6 D $150 E $2,000 F $120 G $90 H $ / 539

6 Solution: ABC Analysis The table below details the contribution of each of the eight products. Item Annual Unit Dollar % Dollar Cumulative Demand Cost volume volume $ vol % G $90 $90,000 30% 30% E $2,000 $70,000 23% 53% D $150 $60,000 20% 72% F $120 $30,000 10% 82% C $6 $27,000 9% 91% A $9 $10,800 4% 95% B $90 $9,000 3% 98% H $75 $7,500 2% 100% Total= $304,300 Item G473 is clearly an Aitem, and items A211, B390, and H921 are all Citems. Other classifications are somewhat subjective, but one choice is to label E707 and D100 as Aitems, and F660 and C003 as Bitems. 254 / 539 Solution: ABC Analysis 255 / 539

7 ABC Analysis Policies that may be based on ABC analysis include the following: 1. More emphasis on supplier development for A items. 2. Tighter physical inventory control for A items; A items belong to more secure areas; inventory records for A items should be verified more frequently. 3. More care in forecasting A items. 256 / 539 Inventory Record Accuracy For effective inventory use, the inventory record must accurately reflect the quantity of materials available. Accurate records are critical in production and inventory systems. Allows organization to focus on what is needed. Necessary to make precise decisions about ordering, scheduling, and shipping. Incoming and outgoing record keeping must be accurate. Stockrooms should be secure. Inaccurate inventory record can result in: lost sales (e.g., finished good not available at time of sale), disrupted operations (not enough of a component or raw material to complete a job), poor customer service (late deliveries to customers), lower productivity (additional setups to complete a job), poor material planning (the inventory records are critical in determining MRP quantities), and expediting (trying to obtain necessary items in less than normal lead time). 257 / 539

8 Cycle Counting Cycle counting is a regular physical count of the items in inventory on a cyclic schedule. Items are counted and records updated on a periodic basis Often used with ABC analysis to determine cycle Cycle counting : 1. eliminates shutdowns and interruptions 2. eliminates annual inventory adjustment 3. allows trained personnel audit inventory accuracy 4. allows causes of errors to be identified and corrected 5. allows to maintain accurate inventory records 258 / 539 Figure It Out: Cycle Counting Imagine a company with 5,000 items in inventory, 500 A items, 1,750 B items, 2,750 C items. The company s policy is to count A items every month (20 working days), B items every quarter (60 days), and C items every six months (120 days). How many items are counted every day? Solution: Seventy-seven items are counted every day. The daily audit of 77 item is much more efficient than conducting a massive inventory count once a year. 259 / 539

9 Inventory Costs Holding costs the variable costs of holding or carrying inventory over time. It also includes obsolescence and cost related to storage, such as insurance, extra staffing, and interest payments. Ordering costs the costs of placing an order and receiving goods; cost of supplies, form, order processing, purchasing, clerical support, etc. Setup costs cost to prepare a machine or process for manufacturing an order. 260 / 539 Inventory Model for Independent Demand The independent inventory model assumes that demand for an item is independent of the demand for other items. The independent demand inventory models are used to address two important inventory related questions: when to order and how much to order. The following are mathematical models that determine order quantity (i.e., how much to order) and minimize inventory cost: Basic economic order quantity (EOQ) Production order quantity (POQ) Quantity discount model (QDM) The objective of most inventory model is to minimize total costs: setup( or ordering) costs and holding (or carrying) cost. 261 / 539

10 The Basic Economic Order Quantity (EOQ) Model The EOQ model is one of the most commonly used inventory control techniques. The EOQ model is based on the following assumptions: Demand for an item is known, constant, and independent of decisions for other items. Lead time 20 is known and constant. Receipt of inventory is instantaneous and complete. Quantity discounts are not possible. The only costs are the costs of setting up and cost of holding inventory over time. Stockouts 21 can be completely avoided if the orders are placed at the right time. 20 Lead time is the time between ordering and receiving the order. That is, the time it will take for the order to be delivered. 21 A stockout is the inability to satisfy the demand for an item. 262 / 539 Inventory Usage over Time 263 / 539

11 The Objective of EOQ is to Minimize Total Cost Total cost = Setup Cost + Holding Cost 264 / 539 The EOQ Model: Holding or Setup Cost Let 265 / 539

12 The EOQ Model: Holding Cost Let 266 / 539 The EOQ Model For a given order quantity, Annual holding costs = Q H 2 Annual setup costs = D S Q The total annual inventory cost: (26) (27) TC(Q) = D Q S + Q 2 H (28) 267 / 539

13 The EOQ Model Optimal order quantity is found when annual setup cost equals annual holding cost Solving for Q: At the EOQ,Q, Q = D Q S = Q 2 H (29) 2DS H = EOQ (30) TC(Q )= 2 D S H. (31) At the EOQ, the cost per unit can be computed as: ATC (Q )= TC(Q ) D = 2 S H. (32) D 268 / 539 Figure It Out: AnEOQModel A product has demand of 4000 units per year. Ordering cost is $20 and holding cost is $4 per unit per year. The EOQ model is appropriate. Question 1: What is the cost-minimizing level of inventory? A) 400 B) 800 C) 200 D) Zero; this is a class C item. E) Cannot be determined because unit price is not known. Question 2: What is the total inventory cost? A) $400 B) $800 C) $200 D) Zero; this is a class C item. E) Cannot be determined because unit price is not known. 269 / 539

14 Solution: An EOQ Model We know: D=4000 per year; S=$20 ; H=$4 per unit per year. Want to know: EOQ = Q and total inventory cost 2DS EOQ = Q = H = 4 The total cost of inventory = 200units TC = D Q S + Q 2 H = = $400 + $400 = $800 2 What happens to annual inventory costs when a non-eoq order quantity is used, say 100 unit or 300 units? TC = D Q S + Q 2 H = = $800 + $200 = $ TC = D Q S + Q 2 H = = $200 + $800 = $ The difference between the EOQ policy and any other policy is a penalty cost incurred by the company for not using the EOQ policy. 270 / 539 Figure It Out: The Expected Total Number of Orders A product has demand of 4000 units per year. Ordering cost is $20 and holding cost is $4 per unit per year. The EOQ model is appropriate. What is the expected total number of orders per year? A) 20 B) 200 C) 50 D) 40 E) 365 Solution: The expected total number of orders: N = Demand Order Quantity = D Q N = D Q = =20 orders per year 271 / 539

15 Figure It Out: The Expected time between orders A product has demand of 4000 units per year. Ordering cost is $20 and holding cost is $4 per unit per year. The EOQ model is appropriate. What is the expected time between orders? Assume 250 working days. A) 4 B) 12.5 C) 25 D) 40 E) 50 Solution: The expected time between orders T = The number of working days per year N = 250 N T = 250 N = =12.5 days 272 / 539 Figure It Out: Total Annual Inventory Cost: Revisited A product has demand of 4000 units per year. Ordering cost is $20 and holding cost is $4 per unit per year. The EOQ model is appropriate. What is the total annual inventory cost and per unit inventory cost? A) $4000, $1.00 B) $600, $0.15 C) $200, $0.05 D) $800, $0.20 E) $500, $0.125 Solution: The expected time between orders Total Annual Cost= Annual Setup Cost + Annual Holding Cost TC = SC + HC = D Q S + Q 2 H TC(200) = D Q S + Q 2 H = =$800 ATC (200) = $800/4000 = $ / 539

16 The EOQ Model: Robust Note that the EOQ model is robust works even if all parameters and assumptions are not met the total cost curve is relatively flat intheareaoftheeoq For our earlier example, imagine that the management overestimated the annual demand by 25%. Ordering cost is $20 and holding cost is $4 per unit per year. The EOQ model is appropriate. What is the total annual inventory cost? D = 4000 x (1-0.25) = 3000, then 2DS Q = H = = units per year 4 and TC = D Q S + Q 2 H = = $ The total cost decreases by 13.40% only. 274 / 539 Sensitivity Analysis of the EOQ Examining the sensitivity of the EOQ model to small changes, errors, or uncertainties, in cost can yield valuable insights into the management of inventories. Sensitivity analysis is a technique for systematically changing crucial parameters to determine the implications of those changes. The following table shows the effect on EOQ of changes in the parameters of the formula. Parameters EOQ Parameters Change EOQ Change Demand (D) Order/setup costs (S) Holding costs (H) 2DS 2DS H H 2DS H Let s continue using our previous example where EOQ = 200, and examine the effects on EOQ of doubling D, S, and H individually Parameters EOQ Parameters Change EOQ Change Demand, D Setup cost, S Holding cost, H 2DS 2DS H D: Q : H S: Q : DS H H: 4 8 Q : / 539

17 Reorder Point (ROP): When to Order? A reorder point (ROP) is the inventory level (point) at which action is taken (an order placed) to replenish the stocked item. EOQ answers the how much question The reorder point (ROP) tells when to order When we have no demand uncertainty, we set the reorder point (ROP)toequaltoaverage demand during lead time, thatis ROP = d L where d= daily demand in units, d=d (number of working days per year); L = Lead time for a new order in days; and D = annual demand. 276 / 539 Figure It Out: Reorder Point (ROP) Imagine that the lead time for the product in our previous example with annual demand (D) of 4000 units is 4 days. Assuming 250 working days, what would be an appropriate re-order point? Solution: d = D/250 = 4000/250 = 16 ROP = d L =16 4=64 units The reorder point is 64 units. Since we know demand and lead time for certainty, the replenishment order arrives just as the on-hand inventory is depleted. The demand during lead time is 64 units. 277 / 539

18 Reorder Point Curve 278 / 539 Figure It Out: Reorder Point A person takes two special tablets per day, which are delivered to his home seven days after an order is called in. At what point should the person reorder? Usages, d = 2 tablets per day Lead time, L = 7 days ROP= usage Lead time = d L = 2 tablets per day 7 days = 14 tablets Thus, the person should reorder when 14 tablets are left. In this example, the demand during lead time is 14 tablets. 279 / 539

19 Figure It Out: EOQ and Reorder Point An inventory decision rule states when the inventory level goes down to 14 gearboxes, 100 gearboxes will be ordered. Which of the following statements is true? 22 A One hundred is the reorder point, and 14 is the order quantity. B Fourteen is the reorder point, and 100 is the order quantity. C The number 100 is a function of demand during lead time. D Fourteen is the safety stock, and 100 is the reorder point. E None of the above is true. 22 ANS: B 280 / 539 Production Order Quantity Model An EOQ Model with non-instantaneous Receipt Used when inventory builds up over a period of time after an order is placed Used when units are produced and sold simultaneously 281 / 539

20 Production Order Quantity Model The demand rate cannot exceed the production rate, since we are still assuming that no shortages are possible, and, if d = p, there is no order size, since items are used as fast as they are produced. For this model the production rate must exceed the demand rate, or p > d. 282 / 539 Production Order Quantity Model 283 / 539

21 Production Order Quantity Model 284 / 539 Figure It Out: Production Order Quantity Model A manufacturing company produces a product for which the annual demand is 10,000 units. Production averages 100 per day, while demand is 40 per day. Holding costs are $2.00 per unit per year; set-up costs $200 per order. 1. If they wish to produce this product in economic batches, what size batch should be used? 2. What is the maximum inventory level? 3. How many order cycles are there per year? 4. How much does management of this good in inventory cost the firm each year? Note that this problem requires economic order quantity model with a noninstantaneous delivery. 285 / 539

22 Solution: Production Order Quantity 1. order size Qp = 2DS H(1 d/p) = = (1 40/100) or maximum inventory = Q p (1 d/p) = (1 40/100) = or Number of order cycles per year N = D Qp = =5.48 cycles per year Total Inventory Costs? = D Q S + HQ(1 (d/p)) 2 = (1 (40/100)) 2 =$2, / 539 Figure It Out: Production Order Quantity Model A production order quantity problem has daily demand rate = 10 and daily production rate = 50. The production order quantity for this problem is approximately 612 units. The average inventory for this problem is approximately A 61 B 245 C 300 D 306 E 490 Solution: Maximum inventory: = Q p (1 (d/p)) = 612 (1 (10/50)) = Average inventory: = 490/2 = / 539

23 EOQ and Quantity Discounts When procuring inventory in a logistics or retailing setting, we frequently are given the opportunities to benefit from quantity discounts: We might be offered a discount for ordering a full truckload of supply. We might receive a free unit every five units we order (just as in customer retailing settings of buy one, get one free ). We might receive a discount for all units ordered over 100 units. We might receive a discount for the entire order if the order volume exceeds 500 units (or say $20,000). 288 / 539 Quantity Discount Models Quantity discounts are price reductions for large orders offered to customers to induce them to buy in large quantities. Reduced prices are often available when larger quantities are purchased. The trade-off is between reduced product cost and increased holding cost. TC = Annual holding cost + Annual ordering cost + Annual purchasing cost TC = Q 2 H + D Q S + P D where P = unit price. Adding the annual purchase cost does not change the EOQ if there is no quantity discount. The rationale for not including purchasing cost is that under the assumption of no quantity discounts, annual purchasing cost is not affected by order quantity. 289 / 539

24 Steps in Analyzing a Quantity Discount 1. For each discount category, calculate Q*. 2. If Q* for a discount does not qualify, choose the smallest possible order size to get the discount. 3. Compute the total cost for each Q* or adjusted value from Step 2 4. Select the Q* that gives the lowest total cost. The quantity that yields the lowest total cost is optimal. If the order quantity we obtain from EOQ model is sufficiently large to obtain the largest discount (the lowest per unit procurement cost), then the discount has no impact on our order size. We go ahead and order the economic order quantity. The more interesting case occurs when the EOQ is less than the discount threshold. Then we must decide if we wish to order more than the economic order quantity to take advantage of the discount offered to us. 290 / 539 Figure It Out: Quantity Discount Models Let s revisit our previous example on EOQ: A product has demand of 4000 units per year. Ordering cost is $20 and holding cost is $4 per unit per year. The purchase price of the product is $40 per unit. Suppose we are offered a discount of 0.1% off the entire order if the order exceeds 500 units. Recall that our EOQ was only 200 units. Thus, the question is should we increase the order size to 500 units in order to get a 0.1% discount, yet incur higher inventory costs, or should we simply order 200 units? We surely will not order more that 500 units; any larger order does not generate additional purchase cost savings but does increase inventory costs. So, we have two choices: either stick with the EOQ or increase our order to 500 units. 291 / 539

25 Solution: Quantity Discount Models 1. If we order Q = 200 units, the total cost is: TC = D Q S + Q 2 H + P D = = $400 + $400 + $160, 000 = $160, If we increase the order quantity to 500 units, the total cost is: TC = D Q S + Q H +( ) P D 2 = = $160+$1000+$159, 840 = $ Given the cost is higher in the case of increased order quantity (i.e., 500 units), we will not take advantage of the quantity discount. 3. Should we increase the order size to 500 units in order to get a 1% discount? TC = D Q S + Q H +(1 0.01) P D 2 = = $160+$1000+$158, 400 = $159, Given the cost is lower in the case of increased order quantity (i.e., 500 units), we will take advantage of the quantity discount. 292 / 539 Figure It Out: Quantity Discount Models A factory uses 4,000 light bulbs a year. Light bulbs are priced as follows: 1 to 499, 90 cents each; 500 to 999, 85 cents each; and 1,000 or more, 80 cents. It costs approximately $30 to prepare a purchase order and receive it, and holding cost is 40% of purchase price per unit on annual basis. Determine the optimal order quantity and the total annual cost. D = 4,000 light bulbs per year; S = $30 ; H = 0.4P; P=unit price Range Discount Unit Price, R holding cost 1 to 499 No discount $ (0.90)= to cents off $ (0.85)=0.34 1,000 or more 10 cents off $ (0.80)=0.32 Find the EOQ for each price. 293 / 539

26 Solution: Quantity Discount Models For $0.80 unit price (1,000 and more) 2DS EOQ 0.80 = H = = 866light bulbs 1, Because an order size of 866 light bulbs will cost $0.85 rather than $0.80 each, 866 is not feasible (does not qualify) for an $0.80 per light bulb. Next, try $0.85 per unit. Then, we choose the smallest possible order size to get the discount. In this case, the smallest possible order quantity would be 1,000 units. For $0.85 unit price (500 to 999) 2DS EOQ 0.85 = H = = 840light bulbs 0.34 This is feasible. It falls in the $0.85 per unit price quantity range (500 to 999). For $0.90 unit price (1 to 499) 2DS EOQ 0.90 = H = = 816light bulbs / 539 Solution: Quantity Discount Models: Total Costs Now calculate the total cost for each order quantity, and select order quantity with the lowest total cost. TC 840 = (0.34) = $3, 686 TC 1000 = (0.32) TC 816 = (0.36) = $3, = $3, 894 The order quantity of 1,000 units will minimize the total cost. How about order quantity 500? TC 500 = (0.34) = $3, / 539

27 Probabilistic Models and Safety Stock All the inventory models discussed so far make the assumption that demand for a product is constant and certain. We now relax this assumption using probabilistic model. When variability in demand or lead time is present, it creates the possibility that actual demand during lead time may exceed expected demand. Thus, it is necessary to carry additional inventory, called safety stock, to reduce the risk of running out of inventory (a stockout) during lead time. The reorder point then increases by the amount of the safety stock: ROP = Expected demand during lead time + safety stock = dxl + ss (33) where ss = safety stock d = demand per period L=leadtime 296 / 539 Definitions: Probabilistic Models and Safety Stock Safety stock: stock that is held in excess of expected demand due to variable demand rate and/or variable lead time. Stockout occurs when demand arrives and there is no inventory available to satisfy that demand immediately. We are in stock in a period if all demand was satisfied in that period. Service level: the probability that demand will not exceed inventory/supply during lead time. A service level of 95% implies a probability of 0.95 that demand will not exceed inventory during lead time. The risk of a stockout is the complement of service level; a service level of 95% implies a stockout of 5%. The probability of stock out is the probability a stockout occurs. Service level = 100 % - stockout level. 297 / 539

28 Probabilistic Models and Safety Stock The amount of safety stock depends on: 1. Demand and lead time variabilities 2. The desired service level Selection of a service level may 1. reflect stockout costs (e.g., lost sales, customer dissatisfaction), or 2. it might simply be a policy variable (e.g., the manager wants to achieve a specified service level for certain groups of products). 298 / 539 Probabilistic Models and Safety Stock The reorder point: where ROP = dxl + ss = dxl + zσ dlt (34) z = safety factor; the number of standard deviations above the expected demand. z is related to the service level, the value of z can be directly determined from the(standard) Normal probability table(appendix I) σ dlt = standard deviation of demand during a lead time. ss = zσ dlt 299 / 539

29 Figure It Out: Probabilistic Models and Safety Stock The manager of a hardware store has determined, from historical records, that demand for a type and size of bagged cement during its lead time could be described by a normal distribution that has a mean of 50 bags and a standard deviation of 5 bags. Answer the following questions, assuming that the manager is willing to accept a stockout risk of no more that 3 % during a lead time. Holding costs are $2.00 per unit per year. 1. What value of z is appropriate? 2. How many safety stock should be held? 3. What reorder point should be used? 4. What is the annual holding cost of maintaining the level of safety stock? 300 / 539 Figure It Out: Visually 301 / 539

30 Getting Z-value from Probability Question: What Z value would be appropriate for a 3% risk or stockout or 97% service level? Answer: Z= / 539 Solution: Probabilistic Models and Safety Stock Given: Expected demand during a lead time= d L = 50 bags; σ dlt = 5 bags; stockout risk =3%. 1. What value of z is appropriate? From Appendix I, using a service level of = 0.97, we obtain a value of z= How many safety stock should be held? ss = zσ dlt =1.88 5=9.40bags 3. What reorder point should be used ROP = dxl + zσ dlt = =59.40bags 4. Annual holding cost of maintaining the level of safety stock ROP =9.4 $2 = $18.8peryear 303 / 539

31 Solution: Probabilistic Models and Safety Stock 304 / 539 Figure It Out The demand for a product during lead time is normally distributed with an average of 20 units per week and standard deviation of (σ dlt )is 6 units per week. What safety stock should be held to achieve a 90% service level? What is the ROP? Solution: We first must find z,the number of standard deviations to the right of average demand during lead time,such that 90% of the normal distribution curve will be to the left of that point. This is done by looking up the number in the normal distribution table (appendix A). The closest number in the table is , which corresponds to 1.28 (1.2 in the row heading plus 0.08 in the column heading; adding this together 1.28). Safety stock = zσ dlt = 1.28 x 6 = 7.68 or 8 ROP = d x L + zσ dlt =20+8= / 539

32 Other Probabilistic Models When data on demand during lead time are not readily available, the above formula cannot be used. Data are generally available on daily or weekly or monthly demand and on the length of lead times. Using these data, a manager can determine whether demand and lead time are variable, and the related standard deviations. For those situations, the following models (formula) can be used: 1. Model 1: When demand is variable and lead time is constant ROP = d L + z L σ d (35) 2. Model 2: When lead time is variable and demand is constant ROP = d L + z d σ L (36) 3. Model 3: When both demand and lead time are variable ROP = d L + z L σd 2 + d 2 σl 2 (37) 306 / 539 Model 1: Variable Demand, d 24 When demand is variable and lead time is constant ROP = d L + z L σ d where d = average daily or weekly or monthly demand; σ d = standard deviation of daily or weekly or monthly demand; L = lead time in days or weeks or months. Figure It Out A restaurant uses an average of 50 jars of a special sauce each week. Weekly usage of sauce has standard deviation of three jars. The manager is willing to accept no more that 10% risk of stock-out during a lead time, which is two weeks. Assume the distribution of usage is normal. 1. Determine the value of z. 2. How many safety stock should be held? 3. Determine the ROP. 24 Suppose that demands are identically and independently distributed. mean demand during lead time = E( L i d i )=L d and σ dlt = var( L i d i )=L σd / 539

33 Solution: Model 1 Given: d = 50 jars per week; L=2 weeks;σ d =3 jars per week acceptable stockout risk =0.10; 1. Determine the value of z. From Appendix I using service level of 0.90, we obtain z= Safety stock ss = zσ dlt = z L σ d = =5jars 3. Determine the ROP. ROP = d L + z L σ d = =105jars 308 / 539 Model 2: Variable Lead Time, L When lead time is variable and demand is constant ROP = d L + z d σ L Where d = daily or weekly or monthly demand; σ L = standard deviation of lead time in days or weeks or months; L = average lead time in days or weeks or months. Figure It Out: A motel replaces broken glasses at a rate of 25 per day and constant. Glasses are ordered from a distant supplier. Lead time is normally distributed with an average of 10 days and a standard deviation of 2 days. How many safety stock should be held? What is the standard deviation of demand during lead time? What ROP should be used to achieve a lead time service level of 95%? 309 / 539

34 Solution: Model 2- Variable Lead Time, L Given L = 10; d = 25; σ L =2 Using a standard normal table (see Appendix I of your textbook), we find a z-value of 1.65 standard deviations from the mean for 95% service level. How many safety stock should be held? ss = zσ dlt = z d σ L = = What is the standard deviation of demand during lead time? What ROP should be used to achieve a lead time service level of 95%? ROP = d L + z d σ L / 539 Model 3: Variable Demand and Lead Time When both demand and lead time are variable ROP = d L + z L σd 2 + d 2 σl 2 where d = average daily or weekly or monthly demand; σ L = standard deviationofleadtimeindaysorweeksormonths; L = average lead time in days or weeks or months. Figure It Out: A motel replaces broken glasses at a rate of 25 per day. In the past, this quantity has tended to vary normally and has a standard deviation os three glasses per day. Glasses are ordered from a distant supplier. Lead time is normally distributed with an average of 10 days and a standard deviation of 2 days. How many safety stock should be held? What ROP should be used to achieve a lead time service level of 95% 311 / 539

35 Solution: Model 3 Given: d = 25 glasses per day; L=10 days; σ d =3 glasses per day; σ L =2 days; service level =0.95,so z=1.65 (Appendix I) Solution: How many safety stock should be held? ss = zσ dlt = z L σd 2 + d 2 σl 2 = What ROP should be used to achieve a lead time service level of 95% ROP = d L + z L σd 2 + d 2 σl 2 = glasses 312 / 539 Try It Out: ROP and Safety Stock 25 Demand for dishwasher water pumps is distributed normally with a mean of 8 per day and standard deviation of demand is 3 per day. The order lead time is four days. The service level is 95% (z=1.65). What should the safety stock and reorder point be, respectively? A about 10 and 18 B about 3 and 24 C about 12 and 32 D about and 32 and 38 E about 10 and ANS: E 313 / 539

36 The Single Period Inventory Model The single period model (sometimes referred to as the newsvender problem) is used for ordering perishables (e.g., fresh fruits and vegetables, baked goods, seafood, cut flowers) and other items that have limited useful life (e.g., newspapers, magazines, spare parts for specialized equipments). The newsvendor model considers a setting in which you have only one production or procurement opportunity. Because that opportunity occurs well in advance of a single selling season, you receive your entire order just before the selling starts. Stochastic demand occurs during the selling season. If demand exceeds your order quantity, then you sell your entire order. But if demand is less than your order quantity, then you have leftover inventory at the end of the season. In summary, the newsvendor model represents a situation in which a decision maker must make a single bet (e.g., the order quantity) before some random event occurs (e.g., demand). 314 / 539 The Single Period Inventory Model Analysis of a single period model generally focusses on two costs: Cost of underage (Shortage): The cost of ordering one less unit than what you would have ordered had you known demand. In other words, you had lost sales (i.e., you under ordered). Cs is the increase in profit you would have enjoyed had you ordered one more unit. may include a charge for loss of customer goodwill as well as the opportunity cost of lost sale (i.e., unrealized profit per unit). C s = Cost of shortage = Sales price per unit - Cost per unit Cost of overage (excess): the cost of ordering one more unit than what you would have ordered had you known demand. - In other words, you had leftover inventory (i.e., you over ordered). Co is the increase in profit you would have enjoyed had you ordered one less unit. relates to items left over at the end of the period. Excess cost is the difference between purchase cost and salvage value. C o = Cost of overage = Cost per unit - Salvage value Therefore, the newsvendor model involves trade-off between doing too much - cost of overage and doing too little - cost of shortage. 315 / 539

37 The Single Period Inventory Model Which of the following items is most likely managed using a single-period order model? A Christmas trees B canned food at the grocery store C automobiles at a dealership D metal for a manufacturing process E gas sold to a gas station Which of the following items is less likely managed using a single-period order model? A Imported wine B T-shirts for a sporting event C Fashion items D Overbooking of airline flights E Fresh pies 316 / 539 Balancing the risk and benefit of ordering a unit Ordering one more unit increases the chance of overage. Expected loss on the S th unit is = C o xf (S), (38) where F (S) =Prob(Demand S), S=stocking level The benefit of ordering one more unit is the reduction in the chance of underage. Expected benefit on the Q th unit is = C s x(1 F (S))), (39) 317 / 539

38 Expected Profit Maximizing Order Quantity To minimize the expected total cost of underage and overage, order S units so that the expected marginal cost with the S th unit equals the expected marginal benefit with the S th unit: Rearrange terms in the above equation C o xf (S) =C s x(1 F (S))) (40) F (S) = The ratio C s /(C s + C o ) is called the critical ratio. C s C s + C o (41) Hence, to minimize the expected total cost of underage and overage, choose stocking level, S, such that we do not have lost sales (i.e., demand is S or lower) with a probability that equals the critical ratio 318 / 539 Objective- The Single Period Model The objective of the single period model is to find an order quantity,or stocking level, that will minimize the long-run total shortage and overage cost. The service level for single period model is given by: Service level = C s C s + C o (42) Then the optimal stocking level, S o, can be determined from the demand distribution. For normal distribution, optimal stocking is given as: S o = μ + zσ (43) where μ is the mean (expected) demand; σ is the standard deviation. 319 / 539

39 Figure It Out: The Single Period Model Muffins are delivered daily to Starbucks at Science Complex. Demand is normally distributed with a mean of 40 muffins per day and a standard deviation of 5 muffins per day. Starbucks pays 50 cents per muffin and charges $2.00 for it. Unsold muffins have no salvage value and cannot be carried over into the next day due to spoilage. Find the optimal stocking level and the risk of stock-out for that quantity. C s = = 1.5 C o = = 0.5 Service level = =0.75 This indicates that 75 percent of the area under the normal curve must be to the left of the stocking level. Appendix I shows that a value of z between 0.67 and 0.68, say, 0.675, will satisfy this; μ=40, σ = 5. Thus, S o = muffins. 320 / 539 Figure It Out: The Single Period Model (Q12.37 Textbook). Henrique Correa s bakery prepares all its cakes between 4:00 A.M.and 6:00 A.M. so they will be fresh when customers arrive. Day-old cakes are virtually always sold, but at a 50% discount off the regular $10 price. The cost of baking a cake is $6, and demand is estimated to be normally distributed, with a mean of 25 and a standard deviation of 4. What is the optimal stocking level? C s = = 4.00 C o = =1.00 Service level = =0.80 (44) This indicates that 80 percent of the area under the normal curve must be to the left of the stocking level. Appendix I shows that a value of z is 0.84, will satisfy this; μ=25, σ = 4. Thus, S o = / 539

40 Figure It Out: The Single Period Model Muffins are delivered daily to Starbucks at Science Complex. Demand is uniformly distributed between 30 and 50 muffins per day. Starbucks pays 50 cents per muffin and charges $2.00 for it. Unsold muffins have no salvage value and cannot be carried over into the next day due to spoilage. Find the optimal stocking level and the risk of stock-out for that quantity. C s = = 1.5 C o = = 0.5 Service level = =0.75 Thus, the optimal stocking level must satisfy demand 0.75 percent of the time. For uniform distribution, this will be at a point equal to the minimum demand plus 75 percent of the difference between maximum and minimum demands. That is S o = minimum demand (maximum - minimum) S o = (50-30)=45 muffins 322 / 539 Figure It Out: EOQ and ROP Daily demand for a certain product is normally distributed with a mean of 60 and standard deviation of 7. The source of supply is reliable and maintains a constant lead time of six days. The cost of placing the order is $10 and annual holding costs are $0.50 per unit. There are no stock-out costs, and unfilled orders are filled as soon as the order arrives. Assume sales occur over the entire 365 days of the year. Find the order quantity and reorder point to satisfy a 95 percent probability of not stocking out during the lead time. Given: d= 60 units; σ d = 7; D = = 21, 900; S = $10; H =$0.50; L =6. Answer: The policy for this example is to place an order, EOQ, for 936 units is whenever the number of units remaining in inventory drops, ROP, to / 539

41 Figure It Out: Newsvendor A convenience store selling papers in the sales stand had collected data over a few months and had found that on average each Monday 90 papers were sold with a standard deviation of 10 papers. Suppose that the store pays $0.20 for each paper and sells the papers for $0.50. Unsold papers have no salvage value, $0.00, and cannot be carried over into the next day, and have to be thrown into the recycling bin. Solution: The cost of shortage, Cs, is: $ $0.20 = $0.30 per unit The cost of overage, Co, is: $ $0.00 = $0.20 per unit The probability therefore is 0.3/( ) = 0.6. Now, we need to find the point on our demand distribution that corresponds to the cumulative probability of 0.6. Using the NORMSINV function in excel (or Appendix I of the textbook) to get the number of standard deviations (commonly referred to as the Z -score) of extra newspapers to carry, we get 0.253, which means that we should stock. =zxσ = 0.253x10 = 2.53 or 3 extra papers. The total number of papers for the stand each Monday morning, therefore, should be 93 papers. = μ + z σ = / 539 Figure It Out: Hotel A hotel near a stadium always fills up on the evening before football games. History has shown that when the hotel is fully booked, the number of last-minute cancellations has a mean of 5 and standard deviation of 3. The average room rate is $80. When the hotel is overbooked, the policy is to find a room in a nearby hotel and to pay for the room for the customer. This usually costs the hotel approximately $200 since rooms booked on such late notice are expensive. How many rooms should the hotel overbook? Solution: The cost of underestimating, Cs, the number of cancellations is $80 and the cost of overestimating cancellations, Co, is $200. Service level = 80/(80+200)= Using NORMSINV(0.2857) from Excel (or Appendix I) gives a Z-score of is The negative value indicates that we should overbook by a value less than the average of 5. The actual value should be, So, So = μ + z σ = The hotel should overbook three reservations on the evening prior to a football game. 325 / 539

42 Questions 1. The assumptions of the production order quantity model are met in a situation where annual demand is 3650 units, setup cost is $50, holding cost is $12 per unit per year, the daily demand rate is 10 and the daily production rate is 100. The production order quantity for this problem is approximately. 2. A specific product has demand during lead time of 100 units, with a standard deviation of 25 units. What safety stock (approximately) provides a 95% service level? 3. If daily demand is normally distributed with a mean of 15 and standard deviation of 5, and lead time is constant at 4 days, 90 percent service level will require safety stock of approximately. 4. If daily demand is constant at 10 units per day, and lead time averages 12 days with a standard deviation of 3 days, 95 percent service requires a safety stock of approximately. 5. In a safety stock problem where both demand and lead time are variable, demand averages 150 units per day with a daily standard deviation of 16, and lead time averages 5 days with a standard deviation of 1 day. The standard deviation of demand during lead time is approximately. 326 / 539 Questions 6. A local club is selling Christmas trees and is deciding how many to stock for the month of December. If demand is normally distributed with a mean of 100 and standard deviation of 20, trees have no salvage value at the end of the month, trees cost $20, and trees sell for $50 what is the service level? 7. A bakery wants to determine how many trays of doughnuts it should prepare each day. Demand is normal with a mean of 5 trays and standard deviation of 1 tray. If the owner wants a service level of at least 95% how many trays should he prepare (round to the nearest whole tray)? Assume doughnuts have no salvage value after the day is complete. 8. A Bike Shop stocks a high volume item that has a normally distributed demand during the reorder period. The average daily demand is 70 units, the lead time is 4 days, and the standard deviation of demand during the reorder period is 15. How much safety stock provides a 95% service level to the shop owner? What should the reorder point be? 327 / 539

43 Questions Average daily demand for a product is normally distributed with a mean of 20 units and a standard deviation of 3 units. Lead time is fixed at 25 days. What reorder point provides for a service level of 95 percent? 10. Demand for a product is approximately normal, averaging 5 units per day with a standard deviation of 1 unit per day. Lead time for this product is approximately normal, averaging 10 days with a standard deviation of 3 days. What reorder point provides a service level of 90 percent? 11. An organization has had a policy of ordering 70 units at a time. Their annual demand is 340 units, and the item has an annual carrying cost of $2. The assumptions of the EOQ are thought to apply. For what value of ordering cost would this order size be optimal? 12. Consider a product with a daily demand of 400 units, a setup cost per production run of $100, a monthly holding cost per unit of $2.00, and an annual production rate of 292,000 units. The firm operates and experiences demand 365 days per year. Suppose that management mistakenly used the basic EOQ model to calculate the batch size instead of using the POQ model. How much money per year has that mistake cost the company? 26 1) 184 2) 41 3) 31 4) 49 5) 154 6) 60 7) 7 8) 25 and 305 9) ) ) ) $1, / 539