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Overview

GLOBAL COMPANY PROFILE: Amazon.com ◆ The Importance of Inventory 490 ◆ Managing Inventory 491 ◆ Inventory Models 495 ◆ Inventory Models for Independent Demand 496



◆ ◆

Probabilistic Models and Safety Stock 508 Single-Period Model 513 Fixed-Period (P ) Systems 514

Alaska Airlines

CHAPTER OUTLINE

C H A P T E R

12

Inventory Management

10 OM STRATEGY DECISIONS

• • • • • • •

Design of Goods and Services Managing Quality Process Strategy Location Strategies Layout Strategies Human Resources Supply-Chain Management

• Inventory Management j Independent Demand (Ch. 12) j Dependent Demand (Ch. 14) j Lean Operations (Ch. 16)

• Scheduling • Maintenance 487

C H A P T E R

1 2

Inventory Management Provides Competitive Advantage at Amazon.com

GLOBAL COMPANY PROFILE Amazon.com

W

hen Jeff Bezos opened his revolutionary business in 1995, Amazon.com was intended to be a “virtual” retailer—no inventory, no warehouses, no overhead—just a bunch of computers taking orders for books and authorizing others to fill them. Things clearly

didn’t work out that way. Now, Amazon stocks millions of items of inventory, amid hundreds of

1. You order three items, and a computer in Seattle takes charge. A computer assigns your order—a book, a game, and a digital camera—to one of Amazon’s massive U.S. distribution centers. 2. The “flow meister” at the distribution center receives your order. She determines which workers go where to fill your order.

Bernhard Classen/Alamy

Marilyn Newton/Reno Gazette-Journal

thousands of bins on shelves in over 150 warehouses around the world. Additionally, Amazon’s

3. Amazon’s current system doubles the picking speed of manual operators and drops the error rate to nearly zero.

Ben Cawthra/Sipa USA/Newscom

4. Your items are put into crates on moving belts. Each item goes into a large yellow crate that contains many customers’ orders. When full, the crates ride a series of conveyor belts that wind more than 10 miles through the plant at a constant speed of 2.9 feet per second. The bar code on each item is scanned 15 times, by machines and by many of the 600 workers. The goal is to reduce errors to zero—returns are very expensive.

488

5. All three items converge in a chute and then inside a box. All the crates arrive at a central point where bar codes are matched with order numbers to determine who gets what. Your three items end up in a 3-foot-wide chute— one of several thousand—and are placed into a corrugated box with a new bar code that identifies your order. Picking is sequenced to reduce operator travel. 6. Any gifts you’ve chosen are wrapped by hand. Amazon trains an elite group of gift wrappers, each of whom processes 30 packages an hour.

software is so good that Amazon sells its order taking, REX/Newscom

processing, and billing expertise to others. It is estimated that 200 million items are now available via the Amazon Web site. Bezos expects the customer experience at Amazon to

and management. The time to receive, process, and position

be one that yields the lowest price, the fastest delivery, and

the stock in storage and to then accurately “pull” and pack-

an error-free order fulfillment process so no other contact with

age an order requires a labor investment of less than 3 min-

Amazon is necessary. Exchanges and returns are very expensive.

utes. And 70% of these orders are multiproduct orders. This

Managing this massive inventory precisely is the key for Amazon to be the world-class leader in warehouse automation

underlines the high benchmark that Amazon has achieved. This is world-class performance. When you place an order with Amazon .com, you are doing business with a company that obtains competitive advantage through inventory management. This Global Company Profile shows how

Adrian Sherratt/Alamy

Amazon does it.

7. The box is packed, taped, weighed, and labeled before leaving the warehouse in a truck. A typical plant is designed to ship as many as 200,000 pieces a day. About 60% of orders are shipped via the U.S. Postal Service; nearly everything else goes through United Parcel Service. 8. Your order arrives at your doorstep. In 1 or 2 days, your order is delivered.

489

LE A RNING OBJECTIVES

LO LO LO LO LO LO LO

12.1 12.2 12.3 12.4 12.5 12.6 12.7

Conduct an ABC analysis 492 Explain and use cycle counting 493 Explain and use the EOQ model for independent inventory demand 496 Compute a reorder point and explain safety stock 502 Apply the production order quantity model 503 Explain and use the quantity discount model 505 Understand service levels and probabilistic inventory models 511

The Importance of Inventory As Amazon.com well knows, inventory is one of the most expensive assets of many companies, representing as much as 50% of total invested capital. Operations managers around the globe have long recognized that good inventory management is crucial. On the one hand, a firm can reduce costs by reducing inventory. On the other hand, production may stop and customers become dissatisfied when an item is out of stock. The objective of inventory management is to strike a balance between inventory investment and customer service. You can never achieve a low-cost strategy without good inventory management. All organizations have some type of inventory planning and control system. A bank has methods to control its inventory of cash. A hospital has methods to control blood supplies and pharmaceuticals. Government agencies, schools, and, of course, virtually every manufacturing and production organization are concerned with inventory planning and control. In cases involving physical products, the organization must determine whether to produce goods or to purchase them. Once this decision has been made, the next step is to forecast demand, as discussed in Chapter 4. Then operations managers determine the inventory necessary to service that demand. In this chapter, we discuss the functions, types, and management of inventory. We then address two basic inventory issues: how much to order and when to order.

Functions of Inventory VIDEO 12.1

Managing Inventory at Frito-Lay

Inventory can serve several functions that add flexibility to a firm’s operations. The four functions of inventory are: 1. To provide a selection of goods for anticipated customer demand and to separate the firm from fluctuations in that demand. Such inventories are typical in retail establishments. 2. To decouple various parts of the production process. For example, if a firm’s supplies fluctuate, extra inventory may be necessary to decouple the production process from suppliers. 3. To take advantage of quantity discounts, because purchases in larger quantities may reduce the cost of goods or their delivery. 4. To hedge against inflation and upward price changes.

Types of Inventory Raw material inventory Materials that are usually purchased but have yet to enter the manufacturing process.

Work-in-process (WIP) inventory Products or components that are no longer raw materials but have yet to become finished products.

Maintenance/repair/operating (MRO) inventory Maintenance, repair, and operating materials.

490

To accommodate the functions of inventory, firms maintain four types of inventories: (1) raw material inventory, (2) work-in-process inventory, (3) maintenance/repair/operating supply (MRO) inventory, and (4) finished-goods inventory. Raw material inventory has been purchased but not processed. This inventory can be used to decouple (i.e., separate) suppliers from the production process. However, the preferred approach is to eliminate supplier variability in quality, quantity, or delivery time so that separation is not needed. Work-in-process (WIP) inventory is components or raw material that have undergone some change but are not completed. WIP exists because of the time it takes for a product to be made (called cycle time). Reducing cycle time reduces inventory. Often this task is not difficult: during most of the time a product is “being made,” it is in fact sitting idle. As Figure 12.1 shows, actual work time, or “run” time, is a small portion of the material flow time, perhaps as low as 5%. MROs are inventories devoted to maintenance/repair/operating supplies necessary to keep machinery and processes productive. They exist because the need and timing for maintenance and

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491

Cycle time 95% Input

Figure

Wait for inspection

Wait to be moved

Move time

5% Wait in queue for operator

Setup time

Run time

Output

12.1

The Material Flow Cycle Most of the time that work is in-process (95% of the cycle time) is not productive time.

repair of some equipment are unknown. Although the demand for MRO inventory is often a function of maintenance schedules, other unscheduled MRO demands must be anticipated. Finished-goods inventory is completed product awaiting shipment. Finished goods may be inventoried because future customer demands are unknown.

Finished-goods inventory An end item ready to be sold, but still an asset on the company’s books.

Managing Inventory Operations managers establish systems for managing inventory. In this section, we briefly examine two ingredients of such systems: (1) how inventory items can be classified (called ABC analysis) and (2) how accurate inventory records can be maintained. We will then look at inventory control in the service sector.

ABC Analysis

Percentage of annual dollar usage

ABC analysis divides on-hand inventory into three classifications on the basis of annual dollar volume. ABC analysis is an inventory application of what is known as the Pareto principle (named after Vilfredo Pareto, a 19th-century Italian economist). The Pareto principle states that there are a “critical few and trivial many.” The idea is to establish inventory policies that focus resources on the few critical inventory parts and not the many trivial ones. It is not realistic to monitor inexpensive items with the same intensity as very expensive items. To determine annual dollar volume for ABC analysis, we measure the annual demand of each inventory item times the cost per unit. Class A items are those on which the annual dollar volume is high. Although such items may represent only about 15% of the total inventory items, they represent 70% to 80% of the total dollar usage. Class B items are those inventory items of medium annual dollar volume. These items may represent about 30% of inventory items and 15% to 25% of the total value. Those with low annual dollar volume are Class C, which may represent only 5% of the annual dollar volume but about 55% of the total inventory items. Graphically, the inventory of many organizations would appear as presented in Figure 12.2.

80 70 60 50 40 30 20 10 0

ABC analysis A method for dividing on-hand inventory into three classifications based on annual dollar volume.

STUDENT TIP A, B, and C categories need not be exact. The idea is to recognize that levels of control should match the risk.

Figure

A items

12.2

Graphic Representation of ABC Analysis

B items C items 10

20

30 40 50 60 70 80 Percentage of inventory items

90

100

492 P A R T 3 | M AN AG I N G O P ERAT I O N S An example of the use of ABC analysis is shown in Example 1.

Example 1

ABC ANALYSIS FOR A CHIP MANUFACTURER Silicon Chips, Inc., maker of superfast DRAM chips, wants to categorize its 10 major inventory items using ABC analysis. APPROACH c ABC analysis organizes the items on an annual dollar-volume basis. Shown below (in columns 1–4) are the 10 items (identified by stock numbers), their annual demands, and unit costs. SOLUTION c Annual dollar volume is computed in column 5, along with the percentage of the total represented by each item in column 6. Column 7 groups the 10 items into A, B, and C categories. ABC Calculation (3)

ITEM STOCK NUMBER

PERCENTAGE OF NUMBER OF ITEMS STOCKED

ANNUAL VOLUME (UNITS)

#10286 J #11526 #12760 #10867 6 #10500 #12572 #14075 #01036 #01307 #10572

3

(4)

(5)

(6)

(7)

UNIT COST

ANNUAL DOLLAR VOLUME

PERCENTAGE OF ANNUAL DOLLAR VOLUME

CLASS

5

20%

1,000 500

$ 90.00 154.00

$ 90,000 77,000

38.8%J 33.2%

72%

A A

30%

1,550 350 1,000

17.00 42.86 12.50

26,350 15,001 12,500

11.3% 6.4%6 5.4%

23%

B B B

50%

600 2,000 100 1,200 250

14.17 .60 8.50 .42 .60

8,502 1,200 850 504 150

3.7% .5% .4% .2% .1%

5%

C C C C C

$232,057

100.0%

8,550

(')'*

ABC analysis

(2)

(')'*

LO 12.1 Conduct an

(1)

INSIGHT c The breakdown into A, B, and C categories is not hard and fast. The objective is to try to separate the “important” from the “unimportant.” LEARNING EXERCISE c The unit cost for Item #10286 has increased from $90.00 to $120.00. How does this impact the ABC analysis? [Answer: The total annual dollar volume increases by $30,000, to $262,057, and the two A items now comprise 75% of that amount.] RELATED PROBLEMS c

12.1, 12.2, 12.3 (12.5–12.6 are available in MyOMLab)

EXCEL OM Data File Ch12Ex1.xls can be found in MyOMLab.

Criteria other than annual dollar volume can determine item classification. For instance, high shortage or holding cost, anticipated engineering changes, delivery problems, or quality problems may dictate upgrading items to a higher classification. The advantage of dividing inventory items into classes allows policies and controls to be established for each class. Policies that may be based on ABC analysis include the following: 1. Purchasing resources expended on supplier development should be much higher for individual A items than for C items. 2. A items, as opposed to B and C items, should have tighter physical inventory control; perhaps they belong in a more secure area, and perhaps the accuracy of inventory records for A items should be verified more frequently. 3. Forecasting A items may warrant more care than forecasting other items. Better forecasting, physical control, supplier reliability, and an ultimate reduction in inventory can all result from classification systems such as ABC analysis.

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493

Inventory Accuracy at Milton Bradley

Milton Bradley, a division of Hasbro, Inc., has been manufacturing toys for 150 years. Founded by Milton Bradley in 1860, the company started by making a lithograph of Abraham Lincoln. Using his printing skills, Bradley developed games, including The Game of Life, Chutes and Ladders, Candy Land, Scrabble, and Lite Brite. Today, the company produces hundreds of games, requiring billions of plastic parts. Once Milton Bradley has determined the optimal quantities for each production run, it must make them and assemble them as a part of the proper game. Some games require literally hundreds of plastic parts, including spinners, hotels, people, animals, cars, and so on. According to Gary Brennan, director of manufacturing, getting the right number of pieces to the right toys and production lines is the most important issue for the credibility of the company. Some orders can require 20,000 or more perfectly assembled games delivered to their warehouses in a matter of days. Games with the incorrect number of parts and pieces can result in some very unhappy customers. It is also time-consuming and expensive for Milton Bradley to

supply the extra parts or to have toys or games returned. When shortages are found during the assembly stage, the entire production run is stopped until the problem is corrected. Counting parts by hand or machine is not always accurate. As a result, Milton Bradley now weighs pieces and completed games to determine if the correct number of parts have been included. If the weight is not exact, there is a problem that is resolved before shipment. Using highly accurate digital scales, Milton Bradley is now able to get the right parts in the right game at the right time. Without this simple innovation, the company’s most sophisticated production schedule would be meaningless. Anthony Labbe/Photofulcrum.com

OM in Action

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Sources: Forbes (February 7, 2011); and The Wall Street Journal (April 15, 1999).

Record accuracy is a prerequisite to inventory management, production scheduling, and, ultimately, sales. Accuracy can be maintained by either periodic or perpetual systems. Periodic systems require regular (periodic) checks of inventory to determine quantity on hand. Some small retailers and facilities with vendor-managed inventory (the vendor checks quantity on hand and resupplies as necessary) use these systems. However, the downside is lack of control between reviews and the necessity of carrying extra inventory to protect against shortages. A variation of the periodic system is a two-bin system. In practice, a store manager sets up two containers (each with adequate inventory to cover demand during the time required to receive another order) and places an order when the first container is empty. Alternatively, perpetual inventory tracks both receipts and subtractions from inventory on a continuing basis. Receipts are usually noted in the receiving department in some semiautomated way, such as via a bar-code reader, and disbursements are noted as items leave the stockroom or, in retailing establishments, at the point-of-sale (POS) cash register. Regardless of the inventory system, record accuracy requires good incoming and outgoing record keeping as well as good security. Stockrooms will have limited access, good housekeeping, and storage areas that hold fixed amounts of inventory. In both manufacturing and retail facilities, bins, shelf space, and individual items must be stored and labeled accurately. Meaningful decisions about ordering, scheduling, and shipping, are made only when the firm knows what it has on hand. (See the OM in Action box, “Inventory Accuracy at Milton Bradley.”)

Omnicell

Record Accuracy

In this hospital, these vertically rotating storage carousels provide rapid access to hundreds of critical items and at the same time save floor space. This Omnicell inventory management carousel is also secure and has the added advantage of printing bar code labels.

Cycle Counting Even though an organization may have made substantial efforts to record inventory accurately, these records must be verified through a continuing audit. Such audits are known as cycle counting. Historically, many firms performed annual physical inventories. This practice often meant shutting down the facility and having inexperienced people count parts and material. Inventory records should instead be verified via cycle counting. Cycle counting uses inventory classifications developed through ABC analysis. With cycle counting procedures, items are counted, records are verified, and inaccuracies are periodically documented. The cause of inaccuracies is then traced and appropriate remedial action taken to ensure integrity of the inventory system. A items will be counted frequently, perhaps once a month; B items will be counted less frequently, perhaps once a quarter; and C items will be counted perhaps once every 6 months. Example 2 illustrates how to compute the number of items of each classification to be counted each day.

Cycle counting A continuing reconciliation of inventory with inventory records.

LO 12.2 Explain and use cycle counting

494 P A R T 3 | M AN AG I N G O P ERAT I O N S

Example 2

CYCLE COUNTING AT COLE’S TRUCKS, INC. Cole’s Trucks, Inc., a builder of high-quality refuse trucks, has about 5,000 items in its inventory. It wants to determine how many items to cycle count each day. APPROACH c After hiring Matt Clark, a bright young OM student, for the summer, the firm determined that it has 500 A items, 1,750 B items, and 2,750 C items. Company policy is to count all A items every month (every 20 working days), all B items every quarter (every 60 working days), and all C items every 6 months (every 120 working days). The firm then allocates some items to be counted each day. SOLUTION c ITEM CLASS

QUANTITY

CYCLE-COUNTING POLICY

NUMBER OF ITEMS COUNTED PER DAY

500Y20 5 25Yday

A

500

Each month (20 working days)

B

1,750

Each quarter (60 working days)

C

2,750

Every 6 months (120 working days)

1,750Y60 5 29Yday 2,750Y120 5 23Yday 77Yday

Each day, 77 items are counted. INSIGHT c This daily audit of 77 items is much more efficient and accurate than conducting a massive inventory count once a year. LEARNING EXERCISE c Cole’s reclassifies some B and C items so there are now 1,500 B items and 3,000 C items. How does this change the cycle count? [Answer: B and C both change to 25 items each per day, for a total of 75 items per day.] RELATED PROBLEM c

12.4

In Example 2, the particular items to be cycle counted can be sequentially or randomly selected each day. Another option is to cycle count items when they are reordered. Cycle counting also has the following advantages: Shrinkage Retail inventory that is unaccounted for between receipt and sale.

Pilferage A small amount of theft.

1. Eliminates the shutdown and interruption of production necessary for annual physical inventories. 2. Eliminates annual inventory adjustments. 3. Trained personnel audit the accuracy of inventory. 4. Allows the cause of the errors to be identified and remedial action to be taken. 5. Maintains accurate inventory records.

Robin Nelson/ZUMA Press, Inc./Alamy

Control of Service Inventories

Pharmaceutical distributor McKesson Corp., which is one of Arnold Palmer Hospital’s main suppliers of surgical materials, makes heavy use of bar-code readers to automate inventory control. The device on the warehouse worker’s arm combines a scanner, a computer, and a two-way radio to check orders. With rapid and accurate data, items are easily verified, improving inventory and shipment accuracy.

Although we may think of the service sector of our economy as not having inventory, that is seldom the case. Extensive inventory is held in wholesale and retail businesses, making inventory management crucial. In the food-service business, control of inventory is often the difference between success and failure. Moreover, inventory that is in transit or idle in a warehouse is lost value. Similarly, inventory damaged or stolen prior to sale is a loss. In retailing, inventory that is unaccounted for between receipt and time of sale is known as shrinkage. Shrinkage occurs from damage and theft as well as from sloppy paperwork. Inventory theft is also known as pilferage. Retail inventory loss of 1% of sales is considered good, with losses in many stores exceeding 3%. Because the impact on profitability is substantial, inventory accuracy and control are critical. Applicable techniques include the following: 1. Good personnel selection, training, and discipline: These are never easy but very necessary in food-service, wholesale, and retail operations, where employees have access to directly consumable merchandise.

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495

Retail’s Last 10 Yards

Retail managers commit huge resources to inventory and its management. Even with retail inventory representing 36% of total assets, nearly 1 of 6 items a retail store thinks it has available to its customers is not! Amazingly, close to two-thirds of inventory records are wrong. Failure to have product available is due to poor ordering, poor stocking, mislabeling, merchandise exchange errors, and merchandise being in the wrong location. Despite major investments in bar coding, RFID, and IT, the last 10 yards of retail inventory management is a disaster. The huge number and variety of stock keeping units (SKUs) at the retail level adds complexity to inventory management. Does the customer really need 32 different offerings of Crest toothpaste or 26 offerings of Colgate? The proliferation of SKUs increases confusion, store size, purchasing, inventory, and stocking costs, as well as subsequent markdown costs. With

so many SKUs, stores have little space to stock and display a full case of many products, leading to labeling and “broken case” issues in the back room. Supervalu, the nation’s 4th largest food retailer, is reducing the number of SKUs by 25% as one way to cut costs and add focus to its own store-branded items. Reducing the variation in delivery lead time, improving forecasting accuracy, and cutting the huge variety of SKUs may all help. But reducing the number of SKUs may not improve customer service. Training and educating employees about the importance of inventory management may be a better way to improve the last 10 yards. Sources: The Wall Street Journal (January 13, 2010); Management Science (February 2005); and California Management Review (Spring 2001).

2. Tight control of incoming shipments: This task is being addressed by many firms through the use of Universal Product Code (or bar code) and radio frequency ID (RFID) systems that read every incoming shipment and automatically check tallies against purchase orders. When properly designed, these systems—where each stock keeping unit (SKU; pronounced “skew”) has its own identifier—can be very hard to defeat. 3. Effective control of all goods leaving the facility: This job is accomplished with bar codes, RFID tags, or magnetic strips on merchandise, and via direct observation. Direct observation can be personnel stationed at exits (as at Costco and Sam’s Club wholesale stores) and in potentially high-loss areas or can take the form of one-way mirrors and video surveillance.

A handheld reader can scan RFID tags, aiding control of both incoming and outgoing shipments.

Successful retail operations require very good store-level control with accurate inventory in its proper location. Major retailers lose 10% to 25% of overall profits due to poor or inaccurate inventory records.1 (See the OM in Action box, “Retail’s Last 10 Yards.”)

Inventory Models We now examine a variety of inventory models and the costs associated with them.

VIDEO 12.2

Inventory Control at Wheeled Coach Ambulance

Independent vs. Dependent Demand Inventory control models assume that demand for an item is either independent of or dependent on the demand for other items. For example, the demand for refrigerators is independent of the demand for toaster ovens. However, the demand for toaster oven components is dependent on the requirements of toaster ovens. This chapter focuses on managing inventory where demand is independent. Chapter 14 presents dependent demand management.

Holding, Ordering, and Setup Costs Holding costs are the costs associated with holding or “carrying” inventory over time. Therefore,

Holding cost

holding costs also include obsolescence and costs related to storage, such as insurance, extra staffing, and interest payments. Table 12.1 shows the kinds of costs that need to be evaluated to determine holding costs. Many firms fail to include all the inventory holding costs. Consequently, inventory holding costs are often understated. Ordering cost includes costs of supplies, forms, order processing, purchasing, clerical support, and so forth. When orders are being manufactured, ordering costs also exist, but they are a part

The cost to keep or carry inventory in stock.

Ordering cost The cost of the ordering process.

496 P A R T 3 | M AN AG I N G O P ERAT I O N S TABLE 12.1

Determining Inventory Holding Costs

CATEGORY

COST (AND RANGE) AS A PERCENTAGE OF INVENTORY VALUE

Housing costs (building rent or depreciation, operating cost, taxes, insurance) STUDENT TIP An overall inventory carrying cost of less than 15% is very unlikely, but this cost can exceed 40%, especially in high-tech and fashion industries.

6% (3–10%)

Material-handling costs (equipment lease or depreciation, power, operating cost)

3% (1–3.5%)

Labor cost (receiving, warehousing, security)

3% (3–5%)

Investment costs (borrowing costs, taxes, and insurance on inventory) Pilferage, scrap, and obsolescence (much higher in industries undergoing rapid change like tablets and smart phones) Overall carrying cost

11% (6–24%) 3% (2–5%) 26%

Note: All numbers are approximate, as they vary substantially depending on the nature of the business, location, and current interest rates.

Setup cost The cost to prepare a machine or process for production.

Setup time The time required to prepare a machine or process for production.

of what is called setup costs. Setup cost is the cost to prepare a machine or process for manufacturing an order. This includes time and labor to clean and change tools or holders. Operations managers can lower ordering costs by reducing setup costs and by using such efficient procedures as electronic ordering and payment. In manufacturing environments, setup cost is highly correlated with setup time. Setups usually require a substantial amount of work even before a setup is actually performed at the work center. With proper planning, much of the preparation required by a setup can be done prior to shutting down the machine or process. Setup times can thus be reduced substantially. Machines and processes that traditionally have taken hours to set up are now being set up in less than a minute by the more imaginative world-class manufacturers. Reducing setup times is an excellent way to reduce inventory investment and to improve productivity.

Inventory Models for Independent Demand In this section, we introduce three inventory models that address two important questions: when to order and how much to order . These independent demand models are: 1. Basic economic order quantity (EOQ) model 2. Production order quantity model 3. Quantity discount model

The Basic Economic Order Quantity (EOQ) Model Economic order quantity (EOQ) model An inventory-control technique that minimizes the total of ordering and holding costs.

LO 12.3 Explain and use the EOQ model for independent inventory demand

The economic order quantity (EOQ) model is one of the most commonly used inventory-control techniques. This technique is relatively easy to use but is based on several assumptions: 1. Demand for an item is known, reasonably constant, and independent of decisions for other items. 2. Lead time—that is, the time between placement and receipt of the order—is known and consistent. 3. Receipt of inventory is instantaneous and complete. In other words, the inventory from an order arrives in one batch at one time. 4. Quantity discounts are not possible. 5. The only variable costs are the cost of setting up or placing an order (setup or ordering cost) and the cost of holding or storing inventory over time (holding or carrying cost). These costs were discussed in the previous section. 6. Stockouts (shortages) can be completely avoided if orders are placed at the right time. With these assumptions, the graph of inventory usage over time has a sawtooth shape, as in Figure 12.3. In Figure 12.3, Q represents the amount that is ordered. If this amount is 500 dresses, all 500 dresses arrive at one time (when an order is received). Thus, the inventory

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Inventory level

Total order received

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Inventory Usage over Time

Average inventory on hand Q — 2

Order quantity = Q (maximum inventory level)

12.3

Figure

Usage rate

497

( )

STUDENT TIP If the maximum we can ever have is Q (say, 500 units) and the minimum is zero, then if inventory is used (or sold) at a fairly steady rate, the average 5 (Q 1 0)Y 2 5 QY 2.

Minimum inventory 0 Time

level jumps from 0 to 500 dresses. In general, an inventory level increases from 0 to Q units when an order arrives. Because demand is constant over time, inventory drops at a uniform rate over time. (Refer to the sloped lines in Figure 12.3.) Each time the inventory is received, the inventory level again jumps to Q units (represented by the vertical lines). This process continues indefinitely over time.

Minimizing Costs The objective of most inventory models is to minimize total costs. With the assumptions just given, significant costs are setup (or ordering) cost and holding (or carrying) cost. All other costs, such as the cost of the inventory itself, are constant. Thus, if we minimize the sum of setup and holding costs, we will also be minimizing total costs. To help you visualize this, in Figure 12.4 we graph total costs as a function of the order quantity, Q. The optimal order size, Q*, will be the quantity that minimizes the total costs. As the quantity ordered increases, the total number of orders placed per year will decrease. Thus, as the quantity ordered increases, the annual setup or ordering cost will decrease [Figure 12.4 (a)]. But as the order quantity increases, the holding cost will increase due to the larger average inventories that are maintained [Figure 12.4(b)]. As we can see in Figure 12.4(c), a reduction in either holding or setup cost will reduce the total cost curve. A reduction in the setup cost curve also reduces the optimal order quantity (lot size). In addition, smaller lot sizes have a positive impact on quality and production flexibility. At Toshiba, the $77 billion Japanese conglomerate, workers can make as few as 10 laptop computers before changing models. This lot-size flexibility has allowed Toshiba to move toward a “build-to-order” mass customization system, an important ability in an industry that has product life cycles measured in months, not years. You should note that in Figure 12.4(c), the optimal order quantity occurs at the point where the ordering-cost curve and the carrying-cost curve intersect. This was not by chance. With the EOQ model, the optimal order quantity will occur at a point where the total setup cost is equal

STUDENT TIP Figure 12.4 is the heart of EOQ inventory modeling. We want to find the smallest total cost (top curve), which is the sum of the two curves below it.

Order quantity (a) Annual setup (order) cost

Minimum total cost Holding cost

Order quantity (b) Annual holding cost

Annual cost

Setup (order) cost

Annual cost

Annual cost

Total cost for holding and setup (order)

Holding cost Setup (order) cost

Optimal order quantity (Q *)

Order quantity

(c) Total costs

Figure

12.4

Costs as a Function of Order Quantity

498 P A R T 3 | M AN AG I N G O P ERAT I O N S to the total holding cost.2 We use this fact to develop equations that solve directly for Q*. The necessary steps are: 1. 2. 3. 4.

Develop an expression for setup or ordering cost. Develop an expression for holding cost. Set setup (order) cost equal to holding cost. Solve the equation for the optimal order quantity.

Using the following variables, we can determine setup and holding costs and solve for Q*: Q 5 Number of units per order Q* 5 Optimum number of units per order (EOQ) D 5 Annual demand in units for the inventory item S 5 Setup or ordering cost for each order H 5 Holding or carrying cost per unit per year 1. Annual setup cost 5 (Number of orders placed per year) 3 (Setup or order cost per order) = ¢

= ¢

Annual demand ≤ (Setup or order cost per order) Number of units in each order

D D ≤ ( S) = S Q Q

2. Annual holding cost 5 (Average inventory level) 3 (Holding cost per unit per year) = ¢

= ¢

Order quantity ≤ (Holding cost per unit per year) 2 Q Q ≤(H) = H 2 2

3. Optimal order quantity is found when annual setup (order) cost equals annual holding cost, namely: Q D S = H 2 Q 4. To solve for Q *, simply cross-multiply terms and isolate Q on the left of the equal sign: 2DS = Q 2H 2DS H 2DS Q* = A H Q2 =

(12-1)

Now that we have derived the equation for the optimal order quantity, Q*, it is possible to solve inventory problems directly, as in Example 3.

Example 3

FINDING THE OPTIMAL ORDER SIZE AT SHARP, INC. Sharp, Inc., a company that markets painless hypodermic needles to hospitals, would like to reduce its inventory cost by determining the optimal number of hypodermic needles to obtain per order. APPROACH c The annual demand is 1,000 units; the setup or ordering cost is $10 per order; and the holding cost per unit per year is $.50.

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SOLUTION c

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499

Using these figures, we can calculate the optimal number of units per order: Q* = A Q* = A

2DS H 2(1,000)(10) = 240,000 = 200 units 0.50

Sharp, Inc., now knows how many needles to order per order. The firm also has a basis for determining ordering and holding costs for this item, as well as the number of orders to be processed by the receiving and inventory departments.

INSIGHT c

LEARNING EXERCISE c

If D increases to 1,200 units, what is the new Q*? [Answer: Q* 5 219 units.]

RELATED PROBLEMS c 12.7, 12.8, 12.9, 12.10, 12.11, 12.14, 12.15, 12.17, 12.29 (12.31, 12.32, 12.33a, 12.35a are available in MyOMLab) EXCEL OM Data File Ch12Ex3.xls can be found in MyOMLab. ACTIVE MODEL 12.1 This example is further illustrated in Active Model 12.1 in MyOMLab.

We can also determine the expected number of orders placed during the year (N) and the expected time between orders (T), as follows: Demand D = Expected number of orders = N = (12-2) Q* Order quantity Expected time between orders = T =

Number of working days per year N

(12-3)

Example 4 illustrates this concept.

Example 4

COMPUTING NUMBER OF ORDERS AND TIME BETWEEN ORDERS AT SHARP, INC. Sharp, Inc. (in Example 3) has a 250-day working year and wants to find the number of orders (N) and the expected time between orders (T). APPROACH c SOLUTION c

Using Equations (12-2) and (12-3), Sharp enters the data given in Example 3. Demand Order quantity 1,000 = 5 orders per year = 200 Number of working days per year T = Expected number of orders 250 working days per year = 50 days between orders = 5 orders

N =

The company now knows not only how many needles to order per order but that the time between orders is 50 days and that there are five orders per year.

INSIGHT c

LEARNING EXERCISE c If D 5 1,200 units instead of 1,000, find N and T. [Answer: N > 5.48, T 5 45.62.] RELATED PROBLEMS c

12.14, 12.15, 12.17 (12.35c,d are available in MyOMLab)

As mentioned earlier in this section, the total annual variable inventory cost is the sum of setup and holding costs: Total annual cost = Setup (order) cost + Holding cost In terms of the variables in the model, we can express the total cost TC as: Q D TC = S + H 2 Q

(12-4)

(12-5)

500 P A R T 3 | M AN AG I N G O P ERAT I O N S Example 5 shows how to use this formula.

Example 5

COMPUTING COMBINED COST OF ORDERING AND HOLDING Sharp, Inc. (from Examples 3 and 4) wants to determine the combined annual ordering and holding costs. APPROACH c

Apply Equation (12-5), using the data in Example 3.

SOLUTION c

Q D S + H 2 Q 1,000 200 (+10) + ( +.50) = 200 2 = (5) (+ 10) + (100) ( + .50)

TC =

= + 50 + + 50 = +100 INSIGHT c

These are the annual setup and holding costs. The $100 total does not include the actual cost of goods. Notice that in the EOQ model, holding costs always equal setup (order) costs.

LEARNING EXERCISE c

Find the total annual cost if D 5 1,200 units in Example 3. [Answer: $109.54.]

RELATED PROBLEMS c 12.11, 12.14, 12.15, 12.16 (12.33b,c; 12.35e; 12.36a,b are available in MyOMLab)

Inventory costs may also be expressed to include the actual cost of the material purchased. If we assume that the annual demand and the price per hypodermic needle are known values (e.g., 1,000 hypodermics per year at P 5 $10) and total annual cost should include purchase cost, then Equation (12-5) becomes: TC =

Q D S + H + PD 2 Q

Because material cost does not depend on the particular order policy, we still incur an annual material cost of D * P = (1,000)(+10) = +10,000. (Later in this chapter we will discuss the case in which this may not be true—namely, when a quantity discount is available.)3 Robust Giving satisfactory answers even with substantial variation in the parameters.

Example 6

A benefit of the EOQ model is that it is robust. By robust we mean that it gives satisfactory answers even with substantial variation in its parameters. As we have observed, determining accurate ordering costs and holding costs for inventory is often difficult. Consequently, a robust model is advantageous. The total cost of the EOQ changes little in the neighborhood of the minimum. The curve is very shallow. This means that variations in setup costs, holding costs, demand, or even EOQ make relatively modest differences in total cost. Example 6 shows the robustness of EOQ. Robust Model

EOQ IS A ROBUST MODEL Management in the Sharp, Inc., examples underestimates total annual demand by 50% (say demand is actually 1,500 needles rather than 1,000 needles) while using the same Q. How will the annual inventory cost be impacted? APPROACH c We will solve for annual costs twice. First, we will apply the wrong EOQ; then we will recompute costs with the correct EOQ.

If demand in Example 5 is actually 1,500 needles rather than 1,000, but management uses an order quantity of Q 5 200 (when it should be Q 5 244.9 based on D 5 1,500), the sum of holding and ordering cost increases to $125:

SOLUTION c

Q D S + H Q 2 1,500 200 (+10) + (+.50) = 200 2 = + 75 + + 50 = +125

Annual cost =

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501

However, had we known that the demand was for 1,500 with an EOQ of 244.9 units, we would have spent $122.47, as shown: 244.9 1,500 ( +.50) (+10) + 2 244.9 = 6.125(+ 10) + 122.45( + .50)

Annual cost =

= + 61.25 + + 61.22 = +122.47 Note that the expenditure of $125.00, made with an estimate of demand that was substantially wrong, is only 2% ($2.52Y$122.47) higher than we would have paid had we known the actual demand and ordered accordingly. Note also that were it not due to rounding, the annual holding costs and ordering costs would be exactly equal. INSIGHT c

LEARNING EXERCISE c Demand at Sharp remains at 1,000, H is still $.50, and we order 200 needles at a time (as in Example 5). But if the true order cost 5 S 5 $15 (rather than $10), what is the annual cost? [Answer: Annual order cost increases to $75, and annual holding cost stays at $50. So the total cost 5 $125.] RELATED PROBLEMS c

12.10b, 12.16 (12.36a,b are available in MyOMLab)

We may conclude that the EOQ is indeed robust and that significant errors do not cost us very much. This attribute of the EOQ model is most convenient because our ability to accurately determine demand, holding cost, and ordering cost is limited.

Reorder Points Now that we have decided how much to order, we will look at the second inventory question, when to order. Simple inventory models assume that receipt of an order is instantaneous. In other words, they assume (1) that a firm will place an order when the inventory level for that particular item reaches zero and (2) that it will receive the ordered items immediately. However, the time between placement and receipt of an order, called lead time, or delivery time, can be as short as a few hours or as long as months. Thus, the whento-order decision is usually expressed in terms of a reorder point (ROP)—the inventory level at which an order should be placed (see Figure 12.5 ). The reorder point (ROP) is given as:

In purchasing systems, the time between placing an order and receiving it; in production systems, the wait, move, queue, setup, and run times for each component produced.

Reorder point (ROP) The inventory level (point) at which action is taken to replenish the stocked item.

Safety stock (ss) Extra stock to allow for uneven demand; a buffer.

Q*

Lead time for a new order in days (12-6)

This equation for ROP assumes that demand during lead time and lead time itself are constant . When this is not the case, extra stock, often called safety stock (ss), should be added. The reorder point with safety stock then becomes:

Inventory level (units)

ROP = Demand per day * ROP = d * L

Lead time

Resupply takes place as order arrives Slope = units/day = d

ROP (units)

ROP = Expected demand during lead time + Safety stock 0

The demand per day, d, is found by dividing the annual demand, D, by the number of working days in a year:

Lead time = L

Figure

D d = Number of working days in a year

Time (days)

12.5

The Reorder Point (ROP) Q* is the optimum order quantity, and lead time represents the time between placing and receiving an order.

502 P A R T 3 | M AN AG I N G O P ERAT I O N S Computing the reorder point is demonstrated in Example 7.

Example 7

COMPUTING REORDER POINTS (ROP) FOR IPHONES WITH AND WITHOUT SAFETY STOCK An Apple store has a demand (D) for 8,000 iPhones per year. The firm operates a 250-day working year. On average, delivery of an order takes 3 working days, but has been known to take as long as 4 days. The store wants to calculate the reorder point without a safety stock and then with a one-day safety stock. APPROACH c

First compute the daily demand and then apply Equation (12-6) for the ROP. Then compute the ROP with safety stock.

SOLUTION c

LO 12.4 Compute a

d =

reorder point and explain safety stock

8,000 D = = 32 units Number of working days in a year 250

ROP = Reorder point = d * L = 32 units per day * 3 days = 96 units ROP with safety stock adds 1 day’s demand (32 units) to the ROP (for 128 units). INSIGHT c

When iPhone inventory stock drops to 96 units, an order should be placed. If the safety stock for a possible one-day delay in delivery is added, the ROP is 128 (5 96 1 32). LEARNING EXERCISE c If there are only 200 working days per year, what is the correct ROP, without safety stock and with safety stock? [Answer: 120 iPhones without safety stock and 160 with safety stock.] RELATED PROBLEMS c

12.11d, 12.12, 12.13, 12.15f (12.33d, 12.34, 12.35f, 12.36c are available

in MyOMLab)

When demand is not constant or variability exists in the supply chain, safety stock can be critical. We discuss safety stock in more detail later in this chapter.

Production Order Quantity Model

Production order quantity model An economic order quantity technique applied to production orders.

12.6

Change in Inventory Levels over Time for the Production Model

Inventory level

Figure

In the previous inventory model, we assumed that the entire inventory order was received at one time. There are times, however, when the firm may receive its inventory over a period of time. Such cases require a different model, one that does not require the instantaneous-receipt assumption. This model is applicable under two situations: (1) when inventory continuously flows or builds up over a period of time after an order has been placed or (2) when units are produced and sold simultaneously. Under these circumstances, we take into account daily production (or inventory-flow) rate and daily demand rate. Figure 12.6 shows inventory levels as a function of time (and inventory dropping to zero between orders). Because this model is especially suitable for the production environment, it is commonly called the production order quantity model. It is useful when inventory continuously builds up over time, and traditional economic order quantity assumptions are valid. We derive this model by setting ordering or setup costs equal to holding costs and solving for optimal order size, Q*. Using the following symbols, we can determine the expression for annual inventory holding cost for the production order quantity model:

Part of inventory cycle during which production (and usage) takes place Demand part of cycle with no production (only usage takes place)

Maximum inventory

t

Time

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Q = Number of units per order H = Holding cost per unit per year p = Daily production rate d = Daily demand rate, or usage rate t = Length of the production run in days 1. ¢

Annual inventory Holding cost ≤ = (Average inventory level) * ¢ ≤ holding cost per unit per year

LO 12.5 Apply the production order quantity model

2. (Average inventory level) = (Maximum inventory level)>2 3. ¢

Total used during Total production during Maximum ≤ ≤ - ¢ ≤ = ¢ the production run the production run inventory level = pt - dt

However, Q = total produced = pt, and thus t = Q > p. Therefore: Maximum inventory level = p ¢

STUDENT TIP Note in Figure 12.6 that inventory buildup is not instantaneous but gradual. So the formula reduces the average inventory and thus the holding cost by the ratio of that buildup.

Q Q d ≤ - d¢ ≤ = Q - Q p p p

= Q ¢1 -

d ≤ p

4. Annual inventory holding cost (or simply holding cost) 5 Maximum inventory level Q d (H ) = C 1 - ¢ ≤ S H p 2 2 Using this expression for holding cost and the expression for setup cost developed in the basic EOQ model, we solve for the optimal number of pieces per order by equating setup cost and holding cost: Holding cost = 12 HQ [1 - (d > p)]

Setup cost = (D > Q)S

Set ordering cost equal to holding cost to obtain Q*p : D 1 S = 2 HQ[1 - (d > p)] Q 2DS Q2 = H[1 - (d > p)]

Dmitry Kalinovsky/Shutterstock

Q* p =

2DS A H[1 - (d > p)]

(12-7)

Each order may require a change in the way a machine or process is set up. Reducing setup time usually means a reduction in setup cost, and reductions in setup costs make smaller batches (lots) economical to produce. Increasingly, setup (and operation) is performed by computer-controlled machines, such as this one, operating from previously written programs.

504 P A R T 3 | M AN AG I N G O P ERAT I O N S In Example 8, we use the above equation, Q*p, to solve for the optimum order or production quantity when inventory is consumed as it is produced.

Example 8

A PRODUCTION ORDER QUANTITY MODEL Nathan Manufacturing, Inc., makes and sells specialty hubcaps for the retail automobile aftermarket. Nathan’s forecast for its wire-wheel hubcap is 1,000 units next year, with an average daily demand of 4units. However, the production process is most efficient at 8 units per day. So the company produces 8per day but uses only 4 per day. The company wants to solve for the optimum number of units per order. (Note: This plant schedules production of this hubcap only as needed, during the 250 days per year the shop operates.) APPROACH c

Gather the cost data and apply Equation (12-7): Annual demand = D = 1,000 units Setup costs = S = +10 Holding cost = H = +0.50 per unit per year Daily production rate = p = 8 units daily Daily demand rate = d = 4 units daily

SOLUTION c

Qp* =

2DS A H 31 - ( d > p)4

2(1,000)(10) Q*p = A 0.503 1 - (4 > 8)4 =

20,000 = 280,000 = 282.8 hubcaps, or 283 hubcaps A 0.50(1>2)

INSIGHT c The difference between the production order quantity model and the basic EOQ model is that the effective annual holding cost per unit is reduced in the production order quantity model because the entire order does not arrive at once.

If Nathan can increase its daily production rate from 8 to 10, how does Qp *

LEARNING EXERCISE c

change? [Answer: Qp* 5 258.] 12.18, 12.19, 12.20, 12.30 (12.37 is available in MyOMLab)

RELATED PROBLEMS c

EXCEL OM Data File Ch12Ex8.xls can be found in MyOMLab. ACTIVE MODEL 12.2 This example is further illustrated in Active Model 12.2 in MyOMLab.

You may want to compare this solution with the answer in Example 3 , which had identical D, S, and H values. Eliminating the instantaneous-receipt assumption, where p 5 8 and d 5 4, resulted in an increase in Q* from 200 in Example 3 to 283 in Example 8 . This increase in Q* occurred because holding cost dropped from $.50 to [$.50 3 (1 2 d Yp)], making a larger order quantity optimal. Also note that: d = 4 =

1,000 D = Number of days the plant is in operation 250

We can also calculate Qp*when annual data are available. When annual data are used, we can express Q*p as: 2DS

Q*p = H

H¢ 1 -

Annual demand rate ≤ Annual production rate

(12-8)

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Quantity Discount Models Quantity discounts appear everywhere—you cannot go into a grocery store without seeing them on nearly every shelf. In fact, researchers have found that most companies either offer or receive quantity discounts for at least some of the products that they sell or purchase. A quantity discount is simply a reduced price ( P) for an item when it is purchased in larger quantities. A typical quantity discount schedule appears in Table 12.2. As can be seen in the table, the normal price of the item is $100. When 120 to 1,499 units are ordered at one time, the price per unit drops to $98; when the quantity ordered at one time is 1,500 units or more, the price is $96 per unit. The 120 quantity and the 1,500 quantity are called price-break quantities because they represent the first order amount that would lead to a new lower price. As always, management must decide when and how much to order. However, given these quantity discounts, how does the operations manager make these decisions? As with other inventory models, the objective is to minimize total cost. Because the unit cost for the second discount in Table 12.2 is the lowest, you may be tempted to order 1,500 units. Placing an order for that quantity, however, even with the greatest discount price, may not minimize total inventory cost. This is because holding cost increases. Thus, the major tradeoff when considering quantity discounts is between reduced product cost and increased holding cost. When we include the cost of the product, the equation for the total annual inventory cost can be calculated as follows: Total annual cost 5 Annual setup (ordering) cost 1 Annual holding cost 1 Annual product cost, or Q D TC = S + IP + PD 2 Q

(12-9)

where Q 5 Quantity ordered D 5 Annual demand in units S 5 Setup or ordering cost per order P 5 Price per unit I 5 Holding cost per unit per year expressed as a percent of price P Note that holding cost is IP instead of H as seen in the regular EOQ model. Because the price of the item is a factor in annual holding cost, we do not assume that the holding cost is a constant when the price per unit changes for each quantity discount. Thus, it is common to express the holding cost as a percent (I) of unit price (P) when evaluating costs of quantity discount schedules. The EOQ formula (12-1) is modified for the quantity discount problem as follows: Q* =

2DS A IP

(12-10)

The solution procedure uses the concept of a feasible EOQ. An EOQ is feasible if it lies in the quantity range that leads to the same price P used to compute it in Equation (12-10). For example, suppose that D 5 5,200, S 5 $200, and I 5 28%. Using Table 12.2 and Equation (12-10), the EOQ for the $96 price equals 22(5,200)(200)>[(.28)(96)] = 278 units. Because 278 , 1,500 (the price-break quantity needed to receive the $96 price), the EOQ for the $96 price is not feasible. On the other hand, the EOQ for the $98 price equals 275 units. This amount is feasible because if 275 units were actually ordered, the firm would indeed receive the $98 purchase price. TABLE 12.2

A Quantity Discount Schedule

PRICE RANGE

QUANTITY ORDERED

PRICE PER UNIT P

Initial price

1–119

$100

Discount price 1

120–1,499

$ 98

Discount price 2

1,500 and over

$ 96

Quantity discount A reduced price for items purchased in large quantities.

LO 12.6 Explain and use the quantity discount model

506 P A R T 3 | M AN AG I N G O P ERAT I O N S Figure

12.7

EOQs and Possible Best Order Quantities for the Quantity Discount Problem with Three Prices in Table 12.2

Initial Price

Discount Price 1

Discount Price 2 TC for No Discount

540,000 TC for Discount 1 Annual Total Cost

The solid black curves represent the realized total annual setup plus holding plus purchasing cost at the applicable order quantities. The black curve drops to the total cost curve for the next discount level when each price-break quantity is reached.

550,000

530,000 Not Feasible

TC for Discount 2

520,053 517,155 Feasible 510,000 Possible Order Quantities

Not Feasible 500,000

120

1,500 Order Quantity

Now we have to determine the quantity that will minimize the total annual inventory cost. Because there are a few discounts, this process involves two steps. In Step 1, we identify all possible order quantities that could be the best solution. In Step 2, we calculate the total cost of all possible best order quantities, and the least expensive order quantity is selected. Solution Procedure STEP 1: Starting with the lowest possible purchase price in a quantity discount schedule and

working toward the highest price, keep calculating Q* from Equation (12-10) until the rst feasible EOQ is found. The rst feasible EOQ is a possible best order quantity, along with all price-break quantities for all lower prices. STEP 2: Calculate the total annual cost TC using Equation (12-9) for each of the possible best order quantities determined in Step 1. Select the quantity that has the lowest total cost. Note that no quantities need to be considered for any prices greater than the first feasible EOQ found in Step 1. This occurs because if an EOQ for a given price is feasible, then the EOQ for any higher price cannot lead to a lower cost (TC is guaranteed to be higher). Figure 12.7 provides a graphical illustration of Step 1 using the three price ranges from Table 12.2. In that example, the EOQ for the lowest price is infeasible, but the EOQ for the second-lowest price is feasible. So the EOQ for the second-lowest price, along with the price-break quantity for the lowest price, are the possible best order quantities. Finally, the highest price (no discount) can be ignored because a feasible EOQ has already been found for a lower price. Example 9 illustrates how the full solution procedure can be applied.

Example 9

QUANTITY DISCOUNT MODEL Chris Beehner Electronics stocks toy remote control flying drones. Recently, the store has been offered a quantity discount schedule for these drones. This quantity schedule was shown in Table 12.2. Furthermore, setup cost is $200 per order, annual demand is 5,200 units, and annual inventory carrying charge as a percent of cost, I, is 28%. What order quantity will minimize the total inventory cost?

CHA P T E R 1 2

APPROACH c SOLUTION c

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We will follow the two steps just outlined for the quantity discount model. First we calculate the Q* for the lowest possible price of $96, as we did earlier: Q* $96 =

A

2(5,200)($200) = 278 flying drones per order (.28)($96)

Because 278 , 1,500, this EOQ is infeasible for the $96 price. So now we calculate Q* for the next-higher price of $98: Q* $98 =

A

2(5,200)($200) = 275 flying drones per order (.28)($98)

Because 275 is between 120 and 1,499 units, this EOQ is feasible for the $98 price. Thus, the possible best order quantities are 275 (the first feasible EOQ) and 1,500 (the price-break quantity for the lower price of $96). We need not bother to compute Q* for the initial price of $100 because we found a feasible EOQ for a lower price. Step 2 uses Equation (12-9) to compute the total cost for each of the possible best order quantities. This step is taken with the aid of Table 12.3.

TABLE 12.3

ORDER QUANTITY

Total Cost Computations for Chris Beehner Electronics

UNIT PRICE

ANNUAL ORDERING COST

ANNUAL HOLDING COST

ANNUAL PRODUCT COST

TOTAL ANNUAL COST

275

$98

$3,782

$ 3,773

$509,600

$517,155

1,500

$96

$ 693

$20,160

$499,200

$520,053

Because the total annual cost for 275 units is lower, 275 units should be ordered. The costs for this example are shown in Figure 12.7. INSIGHT c Even though Beehner Electronics could save more than $10,000 in annual product costs, ordering 1,500 units (28.8% of annual demand) at a time would generate even more than that in increased holding costs. So in this example it is not in the store’s best interest to order enough to attain the lowest possible purchase price per unit. On the other hand, if the price-break quantity for the $96 had been 1,000 units rather than 1,500 units, then total annual costs would have been $513,680, which would have been cheaper than ordering 275 units at $98.

Resolve the problem with D 5 2,000, S 5 $5, I 5 50%, discount price 1 5 $99, and discount price 2 5 $98. [Answer: only 20 units should be ordered each time, which is the EOQ at the $100 price.] LEARNING EXERCISE c

RELATED PROBLEMS c

12.21–12.28 (12.38–12.40 are available in MyOMLab)

EXCEL OM Data file Ch12Ex9.xls can be found in MyOMLab.

In this section we have studied the most popular form of single-purchase quantity discount called the all-units discount. In practice, quantity discounts appear in a variety of forms. For example, i ncremental quantity discounts apply only to those units purchased beyond the pricebreak quantities rather than to all units. Fixed fees, such as a fixed shipping and processing cost for a catalog order or a $5,000 tooling setup cost for any order placed with a manufacturer, encourage buyers to purchase more units at a time. Some discounts are aggregated over items or time. Item aggregation bases price breaks on total units or dollars purchased. Time aggregation applies to total items or dollars spent over a specific time period such as one year. Truckload discounts, buy-one-get-one-free offers, and one-time-only sales also represent types of quantity discounts in that they provide price incentives for buyers to purchase more units at one time. Most purchasing managers deal with some form of quantity discounts on a regular basis.

508 P A R T 3 | M AN AG I N G O P ERAT I O N S

Probabilistic Models and Safety Stock Probabilistic model A statistical model applicable when product demand or any other variable is not known but can be specified by means of a probability distribution.

Service level The probability that demand will not be greater than supply during lead time. It is the complement of the probability of a stockout.

All the inventory models we have discussed so far make the assumption that demand for a product is constant and certain. We now relax this assumption. The following inventory models apply when product demand is not known but can be specified by means of a probability distribution. These types of models are called probabilistic models. Probabilistic models are a real-world adjustment because demand and lead time won’t always be known and constant. An important concern of management is maintaining an adequate service level in the face of uncertain demand. The service level is the complement of the probability of a stockout. For instance, if the probability of a stockout is 0.05, then the service level is .95. Uncertain demand raises the possibility of a stockout. One method of reducing stockouts is to hold extra units in inventory. As we noted earlier such inventory is referred to as safety stock. Safety stock involves adding a number of units as a buffer to the reorder point. As you recall: Reorder point = ROP = d * L where d 5 Daily demand L 5 Order lead time, or number of working days it takes to deliver an order The inclusion of safety stock (ss) changed the expression to: ROP = d * L + ss

(12-11)

The amount of safety stock maintained depends on the cost of incurring a stockout and the cost of holding the extra inventory. Annual stockout cost is computed as follows: Annual stockout costs = The sum of the units short for each demand level * The probability of that demand level * The stockout cost>unit * The number of orders per year

(12-12)

Example 10 illustrates this concept.

Example 10

DETERMINING SAFETY STOCK WITH PROBABILISTIC DEMAND AND CONSTANT LEAD TIME David Rivera Optical has determined that its reorder point for eyeglass frames is 50 (d * L) units. Its carrying cost per frame per year is $5, and stockout (or lost sale) cost is $40 per frame. The store has experienced the following probability distribution for inventory demand during the lead time (reorder period). The optimum number of orders per year is six. NUMBER OF UNITS

PROBABILITY

30

.2

40

.2

ROP S 50

.3

60

.2

70

.1 1.0

How much safety stock should David Rivera keep on hand? APPROACH c The objective is to find the amount of safety stock that minimizes the sum of the additional inventory holding costs and stockout costs. The annual holding cost is simply the holding cost per unit multiplied by the units added to the ROP. For example, a safety stock of 20 frames, which implies that the new ROP, with safety stock, is 70 (5 50 1 20), raises the annual carrying cost by $5(20) 5 $100. However, computing annual stockout cost is more interesting. For any level of safety stock, stockout cost is the expected cost of stocking out. We can compute it, as in Equation (12-12), by multiplying the number of frames short (Demand – ROP) by the probability of demand at that level, by the stockout cost, by the number of times per year the stockout can occur (which in our case is the number of orders 4 per year). Then we add stockout costs for each possible stockout level for a given ROP.

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SOLUTION c We begin by looking at zero safety stock. For this safety stock, a shortage of 10 frames will occur if demand is 60, and a shortage of 20 frames will occur if the demand is 70. Thus the stockout costs for zero safety stock are:

(10 frames short) (.2) (+40 per stockout) (6 possible stockouts per year) + (20 frames short) (.1) (+40)(6) = +960 The following table summarizes the total costs for each of the three alternatives: SAFETY STOCK

ADDITIONAL HOLDING COST

20

(20) ($5) 5 $100

10

(10) ($5) 5 $ 50

0

$

TOTAL COST

STOCKOUT COST

0

$100

5 $240

$

$290

(10) (.2) ($40) (6) 1 (20) (.1) ($40) (6) 5 $960

$960

(10) (.1) ($40) (6)

0

The safety stock with the lowest total cost is 20 frames. Therefore, this safety stock changes the reorder point to 50 1 20 5 70 frames. INSIGHT c

The optical company now knows that a safety stock of 20 frames will be the most economi-

cal decision. LEARNING EXERCISE c David Rivera’s holding cost per frame is now estimated to be $20, while the stockout cost is $30 per frame. Does the reorder point change? [Answer: Safety stock 5 10 now, with a total cost of $380, which is the lowest of the three. ROP 5 60 frames.] RELATED PROBLEMS c

12.43, 12.44, 12.45

When it is difficult or impossible to determine the cost of being out of stock, a manager may decide to follow a policy of keeping enough safety stock on hand to meet a prescribed customer service level. For instance, Figure 12.8 shows the use of safety stock when demand (for hospital resuscitation kits) is probabilistic. We see that the safety stock in Figure 12.8 is 16.5 units, and the reorder point is also increased by 16.5. The manager may want to define the service level as meeting 95% of the demand (or, conversely, having stockouts only 5% of the time). Assuming that demand during lead time (the reorder period) follows a normal curve, only the mean and standard deviation are needed to define the inventory requirements for any given service level. Sales data are usually adequate for computing the mean and standard deviation. Example 11 uses a normal curve with a known mean (m) and standard deviation (s) to determine the reorder point and safety stock necessary Figure

12.8

Probabilistic Demand for a Hospital Item Expected number of kits needed during lead time is 350, but for a 95% service level, the reorder point should be raised to 366.5.

Minimum demand during lead time Inventory level

Maximum demand satisfied during lead time Mean demand during lead time ROP = 350 + safety stock of 16.5 = 366.5

ROP (reorder point)

Normal distribution probability of demand during lead time Expected demand during lead time (350 kits)

Safety stock

16.5 units

0 Lead time Place Receive order order

Time Risk of stockout

510 P A R T 3 | M AN AG I N G O P ERAT I O N S for a 95% service level. We use the following formula: ROP = Expected demand during lead time + Zs dLT where

Example 11

(12-13)

Z 5 Number of standard deviations sdLT 5 Standard deviation of demand during lead time

SAFETY STOCK WITH PROBABILISTIC DEMAND Memphis Regional Hospital stocks a “code blue” resuscitation kit that has a normally distributed demand during the reorder period. The mean (average) demand during the reorder period is 350 kits, and the standard deviation is 10 kits. The hospital administrator wants to follow a policy that results in stockouts only 5% of the time. (a) What is the appropriate value of Z? (b) How much safety stock should the hospital maintain? (c) What reorder point should be used? APPROACH c The hospital determines how much inventory is needed to meet the demand 95% of the time. The figure in this example may help you visualize the approach. The data are as follows:

m = Mean demand = 350 kits sdLT = Standard deviation of demand during lead time = 10 kits Z = Number of standard normal deviations

No stockout in 95% of the periods in the long run

Risk of a stockout (5% of area of normal curve)

STUDENT TIP Recall that the service level is 1 minus the risk of a stockout.

Mean ROP = ? kits Quantity demand 350 Safety stock 0 z Number of standard deviations

SOLUTION c

a) We use the properties of a standardized normal curve to get a Z-value for an area under the normal curve of .95 (or 1 - .05). Using a normal table (see Appendix I) or the Excel formula 5NORMSINV(.95), we find a Z-value of 1.645 standard deviations from the mean. b) Because: and: then:

Safety stock = x - m x - m Z = sdLT Safety stock = ZsdLT

(12-14)

Solving for safety stock, as in Equation (12-14), gives: Safety stock = 1.645(10) = 16.5 kits This is the situation illustrated in Figure 12.8. c) The reorder point is: ROP = Expected demand during lead time + Safety stock = 350 kits + 16.5 kits of safety stock = 366.5, or 367 kits INSIGHT c

The cost of the inventory policy increases dramatically (exponentially) with an increase in

service levels. What policy results in stockouts 10% of the time? [Answer: Z 5 1.28; safety stock 5 12.8; ROP 5 363 kits.]

LEARNING EXERCISE c

RELATED PROBLEMS c

12.41, 12.42, 12.49 (12.50 is available in MyOMLab)

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Other Probabilistic Models Equations (12-13) and (12-14) assume that both an estimate of expected demand during lead times and its standard deviation are available. When data on lead time demand are not available, the preceding formulas cannot be applied. However, three other models are available. We need to determine which model to use for three situations: 1. Demand is variable and lead time is constant 2. Lead time is variable and demand is constant 3. Both demand and lead time are variable

LO 12.7 Understand

All three models assume that demand and lead time are independent variables. Note that our examples use days, but weeks can also be used. Let us examine these three situations separately, because a different formula for the ROP is needed for each. Demand Is Variable and Lead Time Is Constant

service levels and probabilistic inventory models

(See Example 12.) When only

the demand is variable, then:5 ROP = (Average daily demand * Lead time in days) + ZsdLT

(12-15)

where s dLT 5 Standard deviation of demand during lead time 5 sd 2Lead time and sd 5 Standard deviation of demand per day

Example 12

ROP FOR VARIABLE DEMAND AND CONSTANT LEAD TIME The average daily demand for Lenovo laptop computers at a Circuit Town store is 15, with a standard deviation of 5 units. The lead time is constant at 2 days. Find the reorder point if management wants a 90% service level (i.e., risk stockouts only 10% of the time). How much of this is safety stock? APPROACH c

Apply Equation (12-15) to the following data:

Average daily demand (normally distributed) 5 15 Lead time in days (constant) 5 2 Standard deviation of daily demand 5 s d 5 5 Service level 5 90% From the normal table (Appendix I) or the Excel formula =NORMSINV(.90), we derive a Z-value for 90% of 1.28. Then:

SOLUTION c

ROP = (15 units * 2 days) + Zsd 2Lead time = 30 + 1.28(5) (22 ) = 30 + 1.28(5) (1.41) = 30 + 9.02 = 39.02 _ 39 Thus, safety stock is about 9 Lenovo computers. INSIGHT c The value of Z depends on the manager’s stockout risk level. The smaller the risk, the higher the Z. LEARNING EXERCISE c If the Circuit Town manager wants a 95% service level, what is the new ROP? [Answer: ROP 5 41.63, or 42.] RELATED PROBLEM c

12.46

Lead Time Is Variable and Demand Is Constant

When the demand is constant

and only the lead time is variable, then: ROP = (Daily demand * Average lead time in days) + Z * Daily demand * sLT (12-16) where s LT 5 Standard deviation of lead time in days

512 P A R T 3 | M AN AG I N G O P ERAT I O N S

Example 13

ROP FOR CONSTANT DEMAND AND VARIABLE LEAD TIME The Circuit Town store in Example 12 sells about 10 digital cameras a day (almost a constant quantity). Lead time for camera delivery is normally distributed with a mean time of 6 days and a standard deviation of 1 day. A 98% service level is set. Find the ROP. APPROACH c

Apply Equation (12-16) to the following data:

Daily demand 5 10 Average lead time 5 6 days Standard deviation of lead time 5 sLT 5 1 day Service level 5 98%, so Z (from Appendix I or the Excel formula =NORMSINV(.98)) 5 2.055 SOLUTION c

From the equation we get: ROP = (10 units * 6 days) + 2.055(10 units) (1) = 60 + 20.55 = 80.55

The reorder point is about 81 cameras. INSIGHT c

Note how the very high service level of 98% drives the ROP up.

If a 90% service level is applied, what does the ROP drop to? [Answer: ROP 5 60 1 (1.28) (10) (1) 5 60 1 12.8 5 72.8 because the Z-value is only 1.28.] LEARNING EXERCISE c RELATED PROBLEM c

12.47

When both the demand and lead time are variable, the formula for reorder point becomes more complex: 6

Both Demand and Lead Time Are Variable

ROP = (Average daily demand * Average lead time in days) + ZsdLT

(12-17)

where sd = Standard deviation of demand per day sLT = Standard deviation of lead time in days 2 and sdLT = 2(Average lead time * sd2) + (Average daily demand)2 sLT

Example 14

ROP FOR VARIABLE DEMAND AND VARIABLE LEAD TIME The Circuit Town store’s most popular item is six-packs of 9-volt batteries. About 150 packs are sold per day, following a normal distribution with a standard deviation of 16 packs. Batteries are ordered from an out-of-state distributor; lead time is normally distributed with an average of 5 days and a standard deviation of 1 day. To maintain a 95% service level, what ROP is appropriate? APPROACH c

Determine a quantity at which to reorder by applying Equation (12-17) to the following

data: Average daily demand 5 150 packs Standard deviation of demand 5 sd 5 16 packs Average lead time 5 5 days Standard deviation of lead time 5 sLT 5 1 day Service level 5 95%, so Z 51.645 (from Appendix I or the Excel formula =NORMSINV(.95)) SOLUTION c

From the equation we compute: ROP = (150 packs * 5 days) + 1.645 sdLT

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513

) + (150 2 * 12)

= 2(5 days * 16 = 2(5 * 256) + (22,500 * 1) = 21,280 + 22,500 = 223,780 _ 154 So ROP 5 (150 3 5) 1 1.645(154) > 750 1 253 5 1,003 packs When both demand and lead time are variable, the formula looks quite complex. But it is just the result of squaring the standard deviations in Equations (12-15) and (12-16) to get their variances, then summing them, and finally taking the square root.

INSIGHT c

LEARNING EXERCISE c

For an 80% service level, what is the ROP? [Answer: Z 5 .84 and ROP

5 879 packs.] RELATED PROBLEM c

12.48

Single-Period Model Single-period inventory model A system for ordering items that have little or no value at the end of a sales period (perishables).

A single-period inventory model describes a situation in which one order is placed for a product. At the end of the sales period, any remaining product has little or no value. This is a typical problem for Christmas trees, seasonal goods, bakery goods, newspapers, and magazines. (Indeed, this inventory issue is often called the “newsstand problem.”) In other words, even though items at a newsstand are ordered weekly or daily, they cannot be held over and used as inventory in the next sales period. So our decision is how much to order at the beginning of the period. Because the exact demand for such seasonal products is never known, we consider a probability distribution related to demand. If the normal distribution is assumed, and we stocked and sold an average (mean) of 100 Christmas trees each season, then there is a 50% chance we would stock out and a 50% chance we would have trees left over. To determine the optimal stocking policy for trees before the season begins, we also need to know the standard deviation and consider these two marginal costs: Cs = Cost of shortage (we underestimated) = Sales price per unit - Cost per unit Co = Cost of overage (we overestimated) = Cost per unit - Salvage value per unit (if there is any) The service level, that is, the probability of not stocking out, is set at: Cs Service level = (12-18) C s + Co Therefore, we should consider increasing our order quantity until the service level is equal to or more than the ratio of [C sY (Cs 1 Co )]. This model, illustrated in Example 15, is used in many service industries, from hotels to airlines to bakeries to clothing retailers.

Example 15

SINGLE-PERIOD INVENTORY DECISION Chris Ellis’s newsstand, just outside the Smithsonian subway station in Washington, DC, usually sells 120 copies of the Washington Post each day. Chris believes the sale of the Post is normally distributed, with a standard deviation of 15 papers. He pays 70 cents for each paper, which sells for $1.25. The Post gives him a 30-cent credit for each unsold paper. He wants to determine how many papers he should order each day and the stockout risk for that quantity. APPROACH c

Chris’s data are as follows: Cs = cost of shortage = + 1.25 - + .70 = +.55 Co = cost of overage = + .70 - + .30 (salvage value) = +.40

Chris will apply Equation (12-18) and the normal table, using m 5 120 and s 5 15.

514 P A R T 3 | M AN AG I N G O P ERAT I O N S SOLUTION c

a) Service level 5

Cs

=

Cs + C o

.55 .55 = = .579 .55 + .40 .95

b) Chris needs to find the Z score for his normal distribution that yields a probability of .579.

u = 15 copies Service level 57.9%

o = 120 Optimal stocking level

So 57.9% of the area under the normal curve must be to the left of the optimal stocking level. c) Using Appendix I or the Excel formula =NORMSINV(.578), for an area of .578, the Z value > .195. Then, the optimal stocking level = 120 copies + (.195) (s) = 120 + (.195) (15) = 120 + 3 = 123 papers The stockout risk if Chris orders 123 copies of the Post each day is 1 2 Service level 5 1 2 .578 5 .422 5 42.2%. INSIGHT c

If the service level is ever under .50, Chris should order fewer than 120 copies per day.

LEARNING EXERCISE c

How does Chris’s decision change if the Post changes its policy and offers no credit for unsold papers, a policy many publishers are adopting?

[Answer: Service level 5 .44, Z 5 2.15. Therefore, stock 120 1 (2.15)(15) 5 117.75, or 118 papers.] RELATED PROBLEMS c

An ordering system with the same order amount each time.

Perpetual inventory system A system that keeps track of each withdrawal or addition to inventory continuously, so records are always current.

Fixed-period (P ) system A system in which inventory orders are made at regular time intervals.

Figure

12.9

Inventory Level in a Fixed-Period (P ) System Various amounts (Q1 , Q2 , Q 3, etc.) are ordered at regular time intervals (P ) based on the quantity necessary to bring inventory up to the target quantity (T ).

Fixed-Period (P ) Systems The inventory models that we have considered so far are fixed-quantity, or Q, systems. That is, the same fixed amount is added to inventory every time an order for an item is placed. We saw that orders are event triggered. When inventory decreases to the reorder point (ROP), a new order for Q units is placed. To use the fixed-quantity model, inventory must be continuously monitored. 7 This requires a perpetual inventory system. Every time an item is added to or withdrawn from inventory, records must be updated to determine whether the ROP has been reached. In a fixed-period system (also called a periodic review, or P system), on the other hand, inventory is ordered at the end of a given period. Then, and only then, is on-hand inventory counted. Only the amount necessary to bring total inventory up to a prespecified target level (T) is ordered. Figure 12.9 illustrates this concept. Fixed-period systems have several of Target quantity (T ) the same assumptions as the basic EOQ fixed-quantity system: On-hand inventory

Fixed-quantity (Q ) system

12.51, 12.52, 12.53

Q4

Q2 Q1

Q3



P ◆ ◆

P

P Time

The only relevant costs are the ordering and holding costs. Lead times are known and constant. Items are independent of one another.

The downward-sloped lines in Figure 12.9 again represent on-hand inventory levels. But now, when the time between orders (P) passes, we place an order to raise inventory up to the target quantity (T).

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The amount ordered during the first period may be Q 1, the second period Q2, and so on. The Qi value is the difference between current on-hand inventory and the target inventory level. The advantage of the fixed-period system is that there is no physical count of inventory items after an item is withdrawn—this occurs only when the time for the next review comes up. This procedure is also convenient administratively. A fixed-period (P) system is appropriate when vendors make routine (i.e., at fixed-time interval) visits to customers to take fresh orders or when purchasers want to combine orders to save ordering and transportation costs (therefore, they will have the same review period for similar inventory items). For example, a vending machine company may come to refill its machines every Tuesday. This is also the case at Anheuser-Busch, whose sales reps may visit a store every 5 days. The disadvantage of the P system is that because there is no tally of inventory during the review period, there is the possibility of a stockout during this time. This scenario is possible if a large order draws the inventory level down to zero right after an order is placed. Therefore, a higher level of safety stock (as compared to a fixed-quantity system) needs to be maintained to provide protection against stockout during both the time between reviews and the lead time.

515

STUDENT TIP A fixed-period model potentially orders a different quantity each time.

Summary Inventory represents a major investment for many firms. This investment is often larger than it should be because firms find it easier to have “just-in-case” inventory rather than “just-in-time” inventory. Inventories are of four types: 1. Raw material and purchased components 2. Work-in-process 3. Maintenance, repair, and operating (MRO)

4. Finished goods

In this chapter, we discussed independent inventory, ABC analysis, record accuracy, cycle counting, and inventory models used to control independent demands. The EOQ model, production order quantity model, and quantity discount model can all be solved using Excel, Excel OM, or POM for Windows software.

Key Terms Holding cost (p. 495) Ordering cost (p. 495) Setup cost (p. 496) Setup time (p. 496) Economic order quantity (EOQ) model (p. 496) Robust (p. 500) Lead time (p. 501) Reorder point (ROP) (p. 501)

Ethical Dilemma Wayne Hills Hospital in tiny Wayne, Nebraska, faces a problem common to large, urban hospitals as well as to small, remote ones like itself. That problem is deciding how much of each type of whole blood to keep in stock. Because blood is expensive and has a limited shelf life (up to 5 weeks under 1–6°C refrigeration), Wayne Hills naturally wants to keep its stock as low as possible. Unfortunately, past disasters such as a major tornado and a train wreck demonstrated that lives would be lost when not enough blood was available to handle massive needs. The hospital

Safety stock (ss) (p. 501) Production order quantity model (p. 502) Quantity discount (p. 505) Probabilistic model (p. 508) Service level (p. 508) Single-period inventory model (p. 513) Fixed-quantity (Q) system (p. 514) Perpetual inventory system (p. 514) Fixed-period (P) system (p. 514)

administrator wants to set an 85% service level based on demand over the past decade. Discuss the implications of this decision. What is the hospital’s responsibility with regard to stocking lifesaving medicines with short shelf lives? How would you set the inventory level for a commodity such as blood? Ginasanders/Fotolia

Raw material inventory (p. 490) Work-in-process (WIP) inventory (p. 490) Maintenance/repair/operating (MRO) inventory (p. 490) Finished-goods inventory (p. 491) ABC analysis (p. 491) Cycle counting (p. 493) Shrinkage (p. 494) Pilferage (p. 494)

Discussion Questions 1. Describe the four types of inventory. 2. With the advent of low-cost computing, do you see alternatives to the popular ABC classifications? 3. What is the purpose of the ABC classification system? 4. Identify and explain the types of costs that are involved in an inventory system.

5. Explain the major assumptions of the basic EOQ model. 6. What is the relationship of the economic order quantity to demand? To the holding cost? To the setup cost? 7. Explain why it is not necessary to include product cost (price or price times quantity) in the EOQ model, but the quantity discount model requires this information.

516 P A R T 3 | M AN AG I N G O P ERAT I O N S 8. What are the advantages of cycle counting? 9. What impact does a decrease in setup time have on EOQ? 10. When quantity discounts are offered, why is it not necessary to check discount points that are below the EOQ or points above the EOQ that are not discount points? 11. What is meant by “service level”? 12. Explain the following: All things being equal, the production order quantity will be larger than the economic order quantity. 13. Describe the difference between a fixed-quantity (Q) and a fixed-period (P) inventory system.

14. Explain what is meant by the expression “robust model.” Specifically, what would you tell a manager who exclaimed, “Uh-oh, we’re in trouble! The calculated EOQ is wrong; actual demand is 10% greater than estimated.” 15. What is “safety stock”? What does safety stock provide safety against? 16. When demand is not constant, the reorder point is a function of what four parameters? 17. How are inventory levels monitored in retail stores? 18. State a major advantage, and a major disadvantage, of a fixed-period (P) system.

Using Software to Solve Inventory Problems This section presents three ways to solve inventory problems with computer software. First, you can create your own Excel spreadsheets. Second, you can use the Excel OM software that comes free with this text. Third, POM for Windows, also free with this text, can solve all problems marked with a P. CREATING YOUR OWN EXCEL SPREADSHEETS

Program 12.1 illustrates how you can make an Excel model to solve Example 8, which is a production order quantity model. Program

12.1

Using Excel for a Production Model, with Data from Example 8 =B5/B9 =SQRT((2*B5*B6)/(B7*(1-B10/B8)))

=B12*(1-B10/B8) =B16/2 =B5/B12 =B9/B18 =ROUND(B18*B6,2) =ROUND(B17*B7,2) =B21+B22

Program 12.2 illustrates how you can make an Excel model to solve Example 15, which is a single-period inventory model. Program

12.2

Using Excel for a SinglePeriod Inventory Model, with Data from Example 15

=B5-B6 =B6-B7 =B11/(B11+B12) =NORMSINV(B13) =B8+B14*B9

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X USING EXCEL OM

Excel OM allows us to easily model inventory problems ranging from ABC analysis, to the basic EOQ model, to the production model, to quantity discount situations. Program 12.3 shows the input data, selected formulas, and results for an ABC analysis, using data from Example 1. After the data are entered, we use the Data and Sort Excel commands to rank the items from largest to smallest dollar volumes.

Calculate the total dollar volume for each item. = B8*C8 Enter the item name or number, its sales volume, and the unit cost in columns A, B, and C.

Calculate the percentage of the grand total dollar volume for each item. = E8/E18

= SUM($F$8:F8) The cumulative dollar volumes in column G make sense only after the items have been sorted by dollar volume. Either use the copy and sort button, or, to sort by hand, highlight cells A7 through E17 and then use the Data and Sort commands.

= SUM(E8:E17)

Program

12.3

Using Excel OM for an ABC Analysis, with Data from Example 1 P USING POM FOR WINDOWS

The POM for Windows Inventory module can also solve the entire EOQ family of problems. Please refer to Appendix IV for further details.

Solved Problems

Virtual Office Hours help is available in MyOMLab.

SOLVED PROBLEM 12.1

SOLUTION

David Alexander has compiled the following table of six items in inventory at Angelo Products, along with the unit cost and the annual demand in units:

The item that needs strict control is 33CP, so it is an A item. Items that do not need to be strictly controlled are 3CPO, R2D2, and RMS; these are C items. The B items will be XX1 and B66.

IDENTIFICATION CODE

UNIT COST ($)

ANNUAL DEMAND (UNITS)

CODE

ANNUAL DOLLAR VOLUME 5 UNIT COST 3 DEMAND

XX1

5.84

1,200

B66

5.40

1,110

B66

$ 5,994.00

3CPO

1.12

896

3CPO

$ 1,003.52

33CP

74.54

1,104

33CP

$82,292.16

R2D2

2.00

1,110

R2D2

$ 2,220.00

RMS

2.08

961

RMS

$ 1,998.88

Use ABC analysis to determine which item(s) should be carefully controlled using a quantitative inventory technique and which item(s) should not be closely controlled.

XX1

$ 7,008.00

Total cost 5 $100,516.56 70% of total cost 5 $70,347.92

SOLVED PROBLEM 12.2

The Warren W. Fisher Computer Corporation purchases 8,000 transistors each year as components in minicomputers. The unit cost of each transistor is $10, and the cost of carrying one transistor in inventory for a year is $3. Ordering cost is $30 per order.

What are (a) the optimal order quantity, (b) the expected number of orders placed each year, and (c) the expected time between orders? Assume that Fisher operates on a 200-day working year.

518 P A R T 3 | M AN AG I N G O P ERAT I O N S SOLUTION

2(8,000) (30) 2DS = 400 units = 3 H DA 8,000 A = b) N = = 20 orders 400 Q* Number of working days 200 = = 10 working days c) Time between orders = T = 20 N With 20 orders placed each year, an order for 400 transistors is placed every 10 working days. a) Q* =

SOLVED PROBLEM 12.3

Annual demand for notebook binders at Meyer’s Stationery Shop is 10,000 units. Brad Meyer operates his business 300 days per year

and finds that deliveries from his supplier generally take 5 working days. Calculate the reorder point for the notebook binders.

SOLUTION

L = 5 days 10,000 = 33.3 units per day d = 300 ROP = d * L = (33.3 units per day) (5 days) = 166.7 units Thus, Brad should reorder when his stock reaches 167 units. SOLVED PROBLEM 12.4

Leonard Presby, Inc., has an annual demand rate of 1,000 units but can produce at an average production rate of 2,000 SOLUTION

2DS

Q* p = H =

units. Setup cost is $10; carrying cost is $1. What is the optimal number of units to be produced each time?

H¢1 -

Annual demand rate ≤ Annual production rate

2(1,000) (10) = A 1 3 1 - (1,000>2,000)4

20,000 A 1 > 2 = 240,000 = 200 units

SOLVED PROBLEM 12.5

Whole Nature Foods sells a gluten-free product for which the annual demand is 5,000 boxes. At the moment, it is paying $6.40 for each box; carrying cost is 25% of the unit cost; ordering costs are $25. A new supplier has offered to sell the same item for $6.00 if Whole Nature Foods buys at least 3,000 boxes per order. Should the firm stick with the old supplier, or take advantage of the new quantity discount?

Step 2 uses Equation (12-9) to compute the total cost for both of the possible best order quantities: 5,000 395 ($25) + (0.25)($6.40) + $6.40(5,000) 2 395 = $316 + $316 + $32,000 = $32,632

TC395 =

SOLUTION

And under the quantity discount price of $6.00 per box:

Step 1, under the lowest possible price of $6.00 per box: Economic order quantity, using Equation (12-10): 2(5,000)(25) Q*$6.00 = A (0.25)(6.00)

5,000 3,000 ($25) + (0.25)($6.00) + $6.00(5,000) 3,000 2 = $42 + $2,250 + $30,000

TC3,000 =

= $32,292

= 408.25, or 408 boxes Because 408 , 3,000, this EOQ is infeasible for the $6.00 price. So now we calculate Q* for the next-higher price of $6.40, which equals 395 boxes (and is feasible). Thus, the best possible order quantities are 395 (the first feasible EOQ) and 3,000 (the price-break quantity for the lower price of $6.00).

Therefore, the new supplier with which Whole Nature Foods would incur a total cost of $32,292 is preferable, but not by a large amount. If buying 3,000 boxes at a time raises problems of storage or freshness, the company may very well wish to stay with the current supplier.

SOLVED PROBLEM 12.6

DEMAND DURING LEAD TIME

Children’s art sets are ordered once each year by Ashok Kumar, Inc., and the reorder point, without safety stock (dL), is 100 art sets. Inventory carrying cost is $10 per set per year, and the cost of a stockout is $50 per set per year. Given the following demand probabilities during the lead time, how much safety stock should be carried? 

PROBABILITY

0

.1

50 ROP S 100 150

.2 .4 .2

200

.1 1.0

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SOLUTION INCREMENTAL COSTS SAFETY STOCK

CARRYING COST

STOCKOUT COST

TOTAL COST

0

0

50 3 (50 3 0.2 1 100 3 0.1) 5 1,000

$1,000

50

50 3 10 5 500

50 3 (0.1 3 50) 5 250

750

100

100 3 10 5 1,000

0

1,000

The safety stock that minimizes total incremental cost is 50 sets. The reorder point then becomes 100 sets 1 50 sets, or 150 sets. SOLVED PROBLEM 12.7

What safety stock should Ron Satterfield Corporation maintain if mean sales are 80 during the reorder period, the

standard deviation is 7, and Ron can tolerate stockouts 10% of the time?

SOLUTION

10% area under the normal curve Safety stock o = 80 udLT = 7

From Appendix I, Z at an area of .9 (or 1 2 .10) 5 1.28, and Equation (12-14): Safety stock = ZsdLT = 1.28(7) = 8.96 units, or 9 units SOLVED PROBLEM 12.8

The daily demand for 52” flat-screen TVs at Sarah’s Discount Emporium is normally distributed, with an average of 5 and a standard deviation of 2 units. The lead time for receiving a ship-

ment of new TVs is 10 days and is fairly constant. Determine the reorder point and safety stock for a 95% service level.

SOLUTION

The ROP for this variable demand and constant lead time model uses Equation (12-15): ROP = (Average daily demand * Lead time in days) + ZsdLT where

sdLT = sd 2Lead time

So, with Z 5 1.645, ROP = (5 * 10) + 1.645(2) 210 = 50 + 10.4 = 60.4 _ 60 TVs, or rounded up to 61 TVs The safety stock is 10.4, which can be rounded up to 11 TVs. SOLVED PROBLEM 12.9

The demand at Arnold Palmer Hospital for a specialized surgery pack is 60 per week, virtually every week. The lead time from McKesson, its main supplier, is normally

distributed, with a mean of 6 weeks for this product and a standard deviation of 2 weeks. A 90% weekly service level is desired. Find the ROP.

SOLUTION

Here the demand is constant and lead time is variable, with data given in weeks, not days. We apply Equation (12-16): ROP = (Weekly demand * Average lead time in weeks) + Z (Weekly demand) sLT where sLT = standard deviation of lead time in weeks = 2 So, with Z 5 1.28, for a 90% service level: ROP = (60 * 6) + 1.28(60) (2) = 360 + 153.6 = 513.6 _ 514 surgery packs

520 P A R T 3 | M AN AG I N G O P ERAT I O N S

Problems

Note: PX means the problem may be solved with POM for Windows and/or Excel OM.

Problems 12.1–12.6 relate to Managing

(cont’d)

Inventory

INVENTORY ITEM

• • 12.1 L. Houts Plastics is a large manufacturer of injectionmolded plastics in North Carolina. An investigation of the company’s manufacturing facility in Charlotte yields the information presented in the table below. How would the plant classify these items according to an ABC classification system? PX L. Houts Plastics’ Charlotte Inventory Levels ITEM CODE #

AVERAGE INVENTORY (UNITS)

VALUE ($/UNIT)

1289

400

3.75

2347

300

4.00

2349

120

2.50

2363

75

1.50

2394

60

2395 6782

Pepper Garlic powder

$ VALUE/CASE

# ORDERED/WEEK

3

3

11

3

Trash can liners

12

3

Table cloths

32

5

Fish filets

143

10

Prime rib roasts

166

6

Oil

28

2

Lettuce (case)

35

24

Chickens

75

14

Order pads

12

2

Eggs (case)

22

7

1.75

Bacon

56

5

30

2.00

Sugar

4

2

20

1.15

7844

12

2.05

8210

8

1.80

8310

7

2.00

9111

6

3.00

a) Which is the most expensive item, using annual dollar volume? b) Which are C items? c) What is the annual dollar volume for all 20 items? PX

• • 12.2 Boreki Enterprises has the following 10 items in inventory. Theodore Boreki asks you, a recent OM graduate, to divide these items into ABC classifications. COST/UNIT

• 12.4 Lindsay Electronics, a small manufacturer of electronic research equipment, has approximately 7,000 items in its inventory and has hired Joan Blasco-Paul to manage its inventory. Joan has determined that 10% of the items in inventory are A items, 35% are B items, and 55% are C items. She would like to set up a system in which all A items are counted monthly (every 20 working days), all B items are counted quarterly (every 60 working days), and all C items are counted semiannually (every 120 working days). How many items need to be counted each day?

ITEM

ANNUAL DEMAND

A2

3,000

B8

4,000

12

C7

1,500

45

D1

6,000

10

Problems 12.7–12.40 relate to Inventory Models for Independent Demand

E9

1,000

20

F3

500

500

• 12.7 William Beville’s computer training school, in Richmond, stocks workbooks with the following characteristics:

G2

300

1,500

H2

600

20

I5

1,750

10

J8

2,500

5

$ 50

Additional problems 12.5–12.6 are available in MyOMLab.

Demand D = 19,500 units>year Ordering cost S = +25>order

Holding cost H = +4>unit >year

a) Develop an ABC classification system for the 10 items. b) How can Boreki use this information? c) Boreki reviews the classification and then places item A2 into the A category. Why might he do so? PX • • 12.3 Jean-Marie Bourjolly’s restaurant has the following inventory items that it orders on a weekly basis: INVENTORY ITEM

$ VALUE/CASE

# ORDERED/WEEK

Ribeye steak

135

Lobster tail

245

3

23

12

3

2

Pasta Salt

3

Napkins

12

2

Tomato sauce

23

11

French fries

43

32 cont’d

a) Calculate the EOQ for the workbooks. b) What are the annual holding costs for the workbooks? c) What are the annual ordering costs?PX • 12.8 If D 5 8,000 per month, S 5 $45 per order, and H 5 $2 per unit per month, a) What is the economic order quantity? b) How does your answer change if the holding cost doubles? c) What if the holding cost drops in half?PX • • 12.9 Henry Crouch’s law office has traditionally ordered ink refills 60 units at a time. The firm estimates that carrying cost is 40% of the $10 unit cost and that annual demand is about 240 units per year. The assumptions of the basic EOQ model are thought to apply. a) For what value of ordering cost would its action be optimal? b) If the true ordering cost turns out to be much greater than your answer to (a), what is the impact on the firm’s ordering policy? • 12.10 Matthew Liotine’s Dream Store sells beds and assorted supplies. His best-selling bed has an annual demand of 400 units. Ordering cost is $40; holding cost is $5 per unit per year.

CHA P T E R 1 2 a) To minimize the total cost, how many units should be ordered each time an order is placed? b) If the holding cost per unit was $6 instead of $5, what would be the optimal order quantity? PX • 12.11 Southeastern Bell stocks a certain switch connector at its central warehouse for supplying field service offices. The yearly demand for these connectors is 15,000 units. Southeastern estimates its annual holding cost for this item to be $25 per unit. The cost to place and process an order from the supplier is $75. The company operates 300 days per year, and the lead time to receive an order from the supplier is 2 working days. a) Find the economic order quantity. b) Find the annual holding costs. c) Find the annual ordering costs. d) What is the reorder point?PX • 12.12 Lead time for one of your fastest-moving products is 21 days. Demand during this period averages 100 units per day. a) What would be an appropriate reorder point? b) How does your answer change if demand during lead time doubles? c) How does your answer change if demand during lead time drops in half? • 12.13 Annual demand for the notebook binders at Duncan’s Stationery Shop is 10,000 units. Dana Duncan operates her business 300 days per year and finds that deliveries from her supplier generally take 5 working days. a) Calculate the reorder point for the notebook binders that she stocks. b) Why is this number important to Duncan? • • 12.14 Thomas Kratzer is the purchasing manager for the headquarters of a large insurance company chain with a central inventory operation. Thomas’s fastest-moving inventory item has a demand of 6,000 units per year. The cost of each unit is $100, and the inventory carrying cost is $10 per unit per year. The average ordering cost is $30 per order. It takes about 5 days for an order to arrive, and the demand for 1 week is 120 units. (This is a corporate operation, and there are 250 working days per year.) a) What is the EOQ? b) What is the average inventory if the EOQ is used? c) What is the optimal number of orders per year? d) What is the optimal number of days in between any two orders? e) What is the annual cost of ordering and holding inventory? f ) What is the total annual inventory cost, including the cost of the 6,000 units? PX • • 12.15 Joe Henry’s machine shop uses 2,500 brackets during the course of a year. These brackets are purchased from a supplier 90 miles away. The following information is known about the brackets: Annual demand:

2,500

Holding cost per bracket per year:

$1.50

Order cost per order:

$18.75

Lead time:

2 days

Working days per year:

250

a) Given the above information, what would be the economic order quantity (EOQ)? b) Given the EOQ, what would be the average inventory? What would be the annual inventory holding cost?

|

I N V EN T O RY M AN AG EM EN T

521

c) Given the EOQ, how many orders would be made each year? What would be the annual order cost? d) Given the EOQ, what is the total annual cost of managing the inventory? e) What is the time between orders? f ) What is the reorder point (ROP)?PX • • 12.16 Abey Kuruvilla, of Parkside Plumbing, uses 1,200 of a certain spare part that costs $25 for each order, with an annual holding cost of $24. a) Calculate the total cost for order sizes of 25, 40, 50, 60, and 100. b) Identify the economic order quantity and consider the implications for making an error in calculating economic order quantity. PX • • • 12.17 M. Cotteleer Electronics supplies microcomputer circuitry to a company that incorporates microprocessors into refrigerators and other home appliances. One of the components has an annual demand of 250 units, and this is constant throughout the year. Carrying cost is estimated to be $1 per unit per year, and the ordering (setup) cost is $20 per order. a) To minimize cost, how many units should be ordered each time an order is placed? b) How many orders per year are needed with the optimal policy? c) What is the average inventory if costs are minimized? d) Suppose that the ordering (setup) cost is not $20, and Cotteleer has been ordering 150 units each time an order is placed. For this order policy (of Q 5 150) to be optimal, determine what the ordering (setup) cost would have to be.PX • • 12.18 Race One Motors is an Indonesian car manufacturer. At its largest manufacturing facility, in Jakarta, the company produces subcomponents at a rate of 300 per day, and it uses these subcomponents at a rate of 12,500 per year (of 250 working days). Holding costs are $2 per item per year, and ordering (setup) costs are $30 per order. a) What is the economic production quantity? b) How many production runs per year will be made? c) What will be the maximum inventory level? d) What percentage of time will the facility be producing components? e) What is the annual cost of ordering and holding inventory?PX • • 12.19 Radovilsky Manufacturing Company, in Hayward, California, makes flashing lights for toys. The company operates its production facility 300 days per year. It has orders for about 12,000 flashing lights per year and has the capability of producing 100 per day. Setting up the light production costs $50. The cost of each light is $1. The holding cost is $0.10 per light per year. a) What is the optimal size of the production run? b) What is the average holding cost per year? c) What is the average setup cost per year? d) What is the total cost per year, including the cost of the lights? PX • • 12.20 Arthur Meiners is the production manager of WheelRite, a small producer of metal parts. Wheel-Rite supplies CalTex, a larger assembly company, with 10,000 wheel bearings each year. This order has been stable for some time. Setup cost for Wheel-Rite is $40, and holding cost is $.60 per wheel bearing per year. Wheel-Rite can produce 500 wheel bearings per day. CalTex is a just-in-time manufacturer and requires that 50 bearings be shipped to it each business day. a) What is the optimum production quantity? b) What is the maximum number of wheel bearings that will be in inventory at Wheel-Rite?

522 P A R T 3 | M AN AG I N G O P ERAT I O N S c) How many production runs of wheel bearings will Wheel-Rite have in a year? d) What is the total setup 1 holding cost for Wheel-Rite?PX • • 12.21 Cesar Rego Computers, a Mississippi chain of computer hardware and software retail outlets, supplies both educational and commercial customers with memory and storage devices. It currently faces the following ordering decision relating to purchases of very high-density disks: D = 36,000 disks S = +25 H = +0.45 Purchase price = +.85 Discount price = +0.82 Quantity needed to qualify for the discount = 6,000 disks Should the discount be taken? PX • • 12.22 Bell Computers purchases integrated chips at $350 per chip. The holding cost is $35 per unit per year, the ordering cost is $120 per order, and sales are steady, at 400 per month. The company’s supplier, Rich Blue Chip Manufacturing, Inc., decides to offer price concessions in order to attract larger orders. The price structure is shown below. Rich Blue Chip’s Price Structure QUANTITY PURCHASED

PRICE/UNIT

1–99 units

$350

100–199 units

$325

200 or more units

$300

a) What is the optimal order quantity and the minimum annual cost for Bell Computers to order, purchase, and hold these integrated chips? b) Bell Computers wishes to use a 10% holding cost rather than the fixed $35 holding cost in (a). What is the optimal order quantity, and what is the optimal annual cost?PX • • 12.23 Wang Distributors has an annual demand for an airport metal detector of 1,400 units. The cost of a typical detector to Wang is $400. Carrying cost is estimated to be 20% of the unit cost, and the ordering cost is $25 per order. If Ping Wang, the owner, orders in quantities of 300 or more, he can get a 5% discount on the cost of the detectors. Should Wang take the quantity discount?PX • • 12.24 The catering manager of La Vista Hotel, Lisa Ferguson, is disturbed by the amount of silverware she is losing every week. Last Friday night, when her crew tried to set up for a banquet for 500 people, they did not have enough knives. She decides she needs to order some more silverware, but wants to take advantage of any quantity discounts her vendor will offer. For a small order (2,000 or fewer pieces), her vendor quotes a price of $1.80Ypiece. If she orders 2,001–5,000 pieces, the price drops to $1.60Y piece. 5,001–10,000 pieces brings the price to $1.40Ypiece, and 10,001 and above reduces the price to $1.25. Lisa’s order costs are $200 per order, her annual holding costs are 5%, and the annual demand is 45,000 pieces. For the best option: a) What is the optimal order quantity? b) What is the annual holding cost? c) What is the annual ordering (setup) cost?

d) What are the annual costs of the silverware itself with an optimal order quantity? e) What is the total annual cost, including ordering, holding, and purchasing the silverware?PX • • 12.25 Rocky Mountain Tire Center sells 20,000 go-cart tires per year. The ordering cost for each order is $40, and the holding cost is 20% of the purchase price of the tires per year. The purchase price is $20 per tire if fewer than 500 tires are ordered, $18 per tire if 500 or more—but fewer than 1,000—tires are ordered, and $17 per tire if 1,000 or more tires are ordered. a) How many tires should Rocky Mountain order each time it places an order? b) What is the total cost of this policy?PX • • 12.26 M. P. VanOyen Manufacturing has gone out on bid for a regulator component. Expected demand is 700 units per month. The item can be purchased from either Allen Manufacturing or Baker Manufacturing. Their price lists are shown in the table. Ordering cost is $50, and annual holding cost per unit is $5. ALLEN MFG. QUANTITY

BAKER MFG.

UNIT PRICE

QUANTITY

UNIT PRICE

1–499

$16.00

1–399

$16.10

500–999

15.50

400–799

15.60

1,0001

15.00

8001

15.10

a) What is the economic order quantity? b) Which supplier should be used? Why? c) What is the optimal order quantity and total annual cost of ordering, purchasing, and holding the component? PX • • • 12.27 Chris Sandvig Irrigation, Inc., has summarized the price list from four potential suppliers of an underground control valve. See the accompanying table. Annual usage is 2,400 valves; order cost is $10 per order; and annual inventory holding costs are $3.33 per unit. Which vendor should be selected and what order quantity is best if Sandvig Irrigation wants to minimize total cost? PX VENDOR A QUANTITY

VENDOR B PRICE

QUANTITY

PRICE

1–49

$35.00

1–74

$34.75

50–74

34.75

75–149

34.00

75–149

33.55

150–299

32.80

150–299

32.35

300–499

31.60

300–499

31.15

5001

30.50

5001

30.75

VENDOR C QUANTITY

VENDOR D PRICE

QUANTITY

PRICE

$34.25

1–99

$34.50

1–199

100–199

33.75

200–399

33.00

200–399

32.50

4001

31.00

4001

31.10

• • • 12.28 Emery Pharmaceutical uses an unstable chemical compound that must be kept in an environment where both temperature and humidity can be controlled. Emery uses 800 pounds per month of the chemical, estimates the holding cost to be 50% of the purchase price (because of spoilage), and estimates order costs to be $50 per order. The cost schedules of two suppliers are as follows:

CHA P T E R 1 2 VENDOR 1 PRICE/LB

1–499

$17.00

500–999 1,0001 

QUANTITY

PRICE/LB

1–399

$17.10

16.75

400–799

16.85

16.50

800–1,199

16.60



1,2001

16.25

a) What is the economic order quantity for each supplier? b) What quantity should be ordered, and which supplier should be used? c) What is the total cost for the most economic order size? d) What factor(s) should be considered besides total cost?PX • • • 12.29 Kim Clark has asked you to help him determine the best ordering policy for a new product. The demand for the new product has been forecasted to be about 1,000 units annually. To help you get a handle on the carrying and ordering costs, Kim has given you the list of last year’s costs. He thought that these costs might be appropriate for the new product. COST FACTOR

COST FACTOR

COST ($)

Taxes for the warehouse

2,000

Warehouse supplies

Receiving and incoming inspection

1,500

Research and development

New product development

2,500

Purchasing salaries & wages

30,000 12,800

Acct. Dept. costs to pay invoices

500

Warehouse salaries & wages

Inventory insurance

600

Pilferage of inventory

Product advertising

800

Purchase order supplies

Spoilage

750

Inventory obsolescence

Sending purchasing orders

800

Purchasing Dept. overhead

I N V EN T O RY M AN AG EM EN T

523

e) If the company produces 400 refrigerators at a time, what would the total annual setup cost and holding cost be? f ) If Bud Banis wants to minimize the total annual inventory cost, how many refrigerators should be produced in each production run? How much would this save the company in inventory costs compared to the current policy of producing 400 in each production run? PX

VENDOR 2

QUANTITY

|

COST ($)

280 2,750

800 500

Additional problems 12.31–12.40 are available in MyOMLab. Problems 12.41–12.50 relate to Probabilistic

Models and Safety Stock

• • 12.41 Barbara Flynn is in charge of maintaining hospital supplies at General Hospital. During the past year, the mean lead time demand for bandage BX-5 was 60 (and was normally distributed). Furthermore, the standard deviation for BX-5 was 7. Ms. Flynn would like to maintain a 90% service level. a) What safety stock level do you recommend for BX-5? b) What is the appropriate reorder point?PX • • 12.42 Based on available information, lead time demand for PC jump drives averages 50 units (normally distributed), with a standard deviation of 5 drives. Management wants a 97% service level. a) What value of Z should be applied? b) How many drives should be carried as safety stock? c) What is the appropriate reorder point?PX ••• 12.43 Authentic Thai rattan chairs (shown in the photo) are delivered to Gary Schwartz’s chain of retail stores, called The Kathmandu Shop, once a year. The reorder point, without safety stock, is 200 chairs. Carrying cost is $30 per unit per year, and the cost of a stockout is $70 per chair per year. Given the following demand probabilities during the lead time, how much safety stock should be carried?

300 1,000

• • • • 12.30 Emarpy Appliance is a company that produces all kinds of major appliances. Bud Banis, the president of Emarpy, is concerned about the production policy for the company’s bestselling refrigerator. The annual demand has been about 8,000 units each year, and this demand has been constant throughout the year. The production capacity is 200 units per day. Each time production starts, it costs the company $120 to move materials into place, reset the assembly line, and clean the equipment. The holding cost of a refrigerator is $50 per year. The current production plan calls for 400 refrigerators to be produced in each production run. Assume there are 250 working days per year. a) What is the daily demand of this product? b) If the company were to continue to produce 400 units each time production starts, how many days would production continue? c) Under the current policy, how many production runs per year would be required? What would the annual setup cost be? d) If the current policy continues, how many refrigerators would be in inventory when production stops? What would the average inventory level be?

Barry Render

He also told you that these data were compiled for 10,000 inventory items that were carried or held during the year. You have also determined that 200 orders were placed last year. Your job as a new operations management graduate is to help Kim determine the economic order quantity for the new product.

DEMAND DURING LEAD TIME

PROBABILITY

0

0.2

100

0.2

200

0.2

300

0.2

400

0.2

• • 12.44 Tobacco is shipped from North Carolina to a cigarette manufacturer in Cambodia once a year. The reorder point, without safety stock, is 200 kilos. The carrying cost is $15 per kilo per year, and the cost of a stockout is $70 per kilo per year. Given the following demand probabilities during the lead time, how much safety stock should be carried?

524 P A R T 3 | M AN AG I N G O P ERAT I O N S DEMAND DURING LEAD TIME (KILOS)

c) Explain what these two service levels mean. Which is preferable? PX

PROBABILITY

0

0.1

100

0.1

200

0.2

300

0.4

400

0.2

PX

• • • 12.45 Mr. Beautiful, an organization that sells weight training sets, has an ordering cost of $40 for the BB-1 set. (BB-1 stands for Body Beautiful Number 1.) The carrying cost for BB-1 is $5 per set per year. To meet demand, Mr. Beautiful orders large quantities of BB-1 seven times a year. The stockout cost for BB-1 is estimated to be $50 per set. Over the past several years, Mr. Beautiful has observed the following demand during the lead time for BB-1: DEMAND DURING LEAD TIME

PROBABILITY

40

.1

50

.2

60

.2

70

.2

80

.2

90

.1 1.0

• • • • 12.49 A gourmet coffee shop in downtown San Francisco is open 200 days a year and sells an average of 75 pounds of Kona coffee beans a day. (Demand can be assumed to be distributed normally, with a standard deviation of 15 pounds per day.) After ordering (fixed cost 5 $16 per order), beans are always shipped from Hawaii within exactly 4 days. Per-pound annual holding costs for the beans are $3. a) What is the economic order quantity (EOQ) for Kona coffee beans? b) What are the total annual holding costs of stock for Kona coffee beans? c) What are the total annual ordering costs for Kona coffee beans? d) Assume that management has specified that no more than a 1% risk during stockout is acceptable. What should the reorder point (ROP) be? e) What is the safety stock needed to attain a 1% risk of stockout during lead time? f ) What is the annual holding cost of maintaining the level of safety stock needed to support a 1% risk? g) If management specified that a 2% risk of stockout during lead time would be acceptable, would the safety stock holding costs decrease or increase? Additional problem 12.50 is available in MyOMLab.

The reorder point for BB-1 is 60 sets. What level of safety stock should be maintained for BB-1? PX • • 12.46 Chicago’s Hard Rock Hotel distributes a mean of 1,000 bath towels per day to guests at the pool and in their rooms. This demand is normally distributed with a standard deviation of 100 towels per day, based on occupancy. The laundry firm that has the linens contract requires a 2-day lead time. The hotel expects a 98% service level to satisfy high guest expectations. a) What is the safety stock? b) What is the ROP? PX •• 12.47 First Printing has contracts with legal firms in San Francisco to copy their court documents. Daily demand is almost constant at 12,500 pages of documents. The lead time for paper delivery is normally distributed with a mean of 4 days and a standard deviation of 1 day. A 97% service level is expected. Compute First’s ROP. PX • • • 12.48 Gainesville Cigar stocks Cuban cigars that have variable lead times because of the difficulty in importing the product: lead time is normally distributed with an average of 6 weeks and a standard deviation of 2 weeks. Demand is also a variable and normally distributed with a mean of 200 cigars per week and a standard deviation of 25 cigars. a) For a 90% service level, what is the ROP? b) What is the ROP for a 95% service level?

Problems 12.51–12.53 relate to Single-Period

Model

•• 12.51 Cynthia Knott’s oyster bar buys fresh Louisiana oysters for $5 per pound and sells them for $9 per pound. Any oysters not sold that day are sold to her cousin, who has a nearby grocery store, for $2 per pound. Cynthia believes that demand follows the normal distribution, with a mean of 100 pounds and a standard deviation of 15 pounds. How many pounds should she order each day? •• 12.52 Henrique Correa’s bakery prepares all its cakes between 4 a.m. and 6 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? ••• 12.53 University of Florida football programs are printed 1 week prior to each home game. Attendance averages 90,000 screaming and loyal Gators fans, of whom two-thirds usually buy the program, following a normal distribution, for $4 each. Unsold programs are sent to a recycling center that pays only 10 cents per program. The standard deviation is 5,000 programs, and the cost to print each program is $1. a) What is the cost of underestimating demand for each program? b) What is the overage cost per program? c) How many programs should be ordered per game? d) What is the stockout risk for this order size?

CASE STUDIES Zhou Bicycle Company Zhou Bicycle Company (ZBC), located in Seattle, is a wholesale distributor of bicycles and bicycle parts. Formed in 1981 by University of Washington Professor Yong-Pin Zhou, the firm’s primary retail outlets are located within a 400-mile radius of the distribution center. These retail outlets receive the order

from ZBC within 2 days after notifying the distribution center, provided that the stock is available. However, if an order is not fulfilled by the company, no backorder is placed; the retailers arrange to get their shipment from other distributors, and ZBC loses that amount of business.

CHA P T E R 1 2 The company distributes a wide variety of bicycles. The most popular model, and the major source of revenue to the company, is the AirWing. ZBC receives all the models from a single manufacturer in China, and shipment takes as long as 4 weeks from the time an order is placed. With the cost of communication, paperwork, and customs clearance included, ZBC estimates that each time an order is placed, it incurs a cost of $65. The purchase price paid by ZBC, per bicycle, is roughly 60% of the suggested retail price for all the styles available, and the inventory carrying cost is 1% per month (12% per year) of the purchase price paid by ZBC. The retail price (paid by the customers) for the AirWing is $170 per bicycle. ZBC is interested in making an inventory plan for 2016. The firm wants to maintain a 95% service level with its customers to minimize the losses on the lost orders. The data collected for the past 2 years are summarized in the following table. A forecast for AirWing model sales in 2016 has been developed and will be used to make an inventory plan for ZBC. Demands For Airwing Model MONTH

2014

2015

January February

6 12

7 14

FORECAST FOR 2016

8 15

MONTH

|

I N V EN T O RY M AN AG EM EN T

2014

2015

March April May

24 46 75

27 53 86

31 59 97

June July August

47 30 18

54 34 21

60 39 24

September October November

13 12 22

15 13 25

16 15 28

December Total

38 343

42 391

47 439

525

FORECAST FOR 2016

Discussion Questions 1. Develop an inventory plan to help ZBC. 2. Discuss ROPs and total costs. 3. How can you address demand that is not at the level of the planning horizon? Source: Professor Kala Chand Seal, Loyola Marymount University.

Parker Hi-Fi Systems Parker Hi-Fi Systems, located in Wellesley, Massachusetts, a Boston suburb, assembles and sells the very finest home theater systems. The systems are assembled with components from the best manufacturers worldwide. Although most of the components are procured from wholesalers on the East Coast, some critical items, such as LCD screens, come directly from their manufacturer. For instance, the LCD screens are shipped via air from Foxy, Ltd., in Taiwan, to Boston’s Logan airport, and the top-of-the-line speakers are purchased from the world-renowned U.S. manufacturer Boss. Parker’s purchasing agent, Raktim Pal, submits an order release for LCD screens once every 4 weeks. The company’s annual requirements total 500 units (2 per working day), and Parker’s per unit cost is $1,500. (Because of Parker’s relatively low volume and the quality focus—rather than volume focus— of many of Parker’s suppliers, Parker is seldom able to obtain quantity discounts.) Because Foxy promises delivery within 1 week following receipt of an order release, Parker has never had a shortage of LCDs. (Total time between date of the release and date of receipt is 1 week or 5 working days.)

Parker’s activity-based costing system has generated the following inventory-related costs. Procurement costs, which amount to $500 per order, include the actual labor costs involved in ordering, customs inspection, arranging for airport pickup, delivery to the plant, maintaining inventory records, and arranging for the bank to issue a check. Parker’s holding costs take into account storage, damage, insurance, taxes, and so forth on a square-foot basis. These costs equal $150 per LCD per year. With added emphasis being placed on efficiencies in the supply chain, Parker’s president has asked Raktim to seriously evaluate the purchase of the LCDs. One area to be closely scrutinized for possible cost savings is inventory procurement.

Discussion Questions 1. What is the optimal order number of LCDs that should be placed in each order? 2. What is the optimal reorder point (ROP) for LCDs? 3. What cost savings will Parker realize if it implements an order plan based on EOQ?

Video Case

Managing Inventory at Frito-Lay Frito-Lay has flourished since its origin—the 1931 purchase of a small San Antonio firm for $100 that included a recipe, 19 retail accounts, and a hand-operated potato ricer. The multibillion-dollar company, headquartered in Dallas, now has 41 products—15 with sales of over $100 million per year and 7 at over $1 billion in sales. Production takes place in 36 productfocused plants in the U.S. and Canada, with 48,000 employees. Inventory is a major investment and an expensive asset in most firms. Holding costs often exceed 25% of product value, but in Frito-Lay’s prepared food industry, holding cost can be much higher because the raw materials are perishable. In the food industry, inventory spoils. So poor inventory management is not only expensive but can also yield an unsatisfactory product that in the extreme can also ruin market acceptance. Major ingredients at Frito-Lay are corn meal, corn, potatoes, oil, and seasoning. Using potato chips to illustrate rapid inventory

flow: potatoes are moved via truck from farm, to regional plants for processing, to warehouse, to the retail store. This happens in a matter of hours—not days or weeks. This keeps freshness high and holding costs low. Frequent deliveries of the main ingredients at the Florida plant, for example, take several forms: ◆

◆ ◆

◆ ◆

Potatoes are delivered in 10 truckloads per day, with 150,000 lbs consumed in one shift: the entire potato storage area will only hold 7½ hours’ worth of potatoes. Oil inventory arrives by rail car, which lasts only 4½ days. Corn meal arrives from various farms in the Midwest, and inventory typically averages 4 days’ production. Seasoning inventory averages 7 days. Packaging inventory averages 8 to 10 days.

526 P A R T 3 | M AN AG I N G O P ERAT I O N S Frito-Lay’s product-focused facility represents a major capital investment. That investment must achieve high utilization to be efficient. The capital cost must be spread over a substantial volume to drive down total cost of the snack foods produced. This demand for high utilization requires reliable equipment and tight schedules. Reliable machinery requires an inventory of critical components: this is known as MRO, or maintenance, repair, and operating supplies. MRO inventory of motors, switches, gears, bearings, and other critical specialized components can be costly but is necessary. Frito-Lay’s non-MRO inventory moves rapidly. Raw material quickly becomes work-in-process, moving through the system and out the door as a bag of chips in about 112 shifts. Packaged finished products move from production to the distribution chain in less than 1.4 days.

Discussion Questions* 1. How does the mix of Frito-Lay’s inventory differ from those at a machine or cabinet shop (a process-focused facility)? 2. What are the major inventory items at Frito-Lay, and how rapidly do they move through the process? 3. What are the four types of inventory? Give an example of each at Frito-Lay. 4. How would you rank the dollar investment in each of the four types (from the most investment to the least investment)? 5. Why does inventory flow so quickly through a Frito-Lay plant? 6. Why does the company keep so many plants open? 7. Why doesn’t Frito-Lay make all its 41 products at each of its plants? *You may wish to view the video that accompanies this case before addressing these questions.

Video Case

Inventory Control at Wheeled Coach Controlling inventory is one of Wheeled Coach’s toughest problems. Operating according to a strategy of mass customization and responsiveness, management knows that success is dependent on tight inventory control. Anything else results in an inability to deliver promptly, chaos on the assembly line, and a huge inventory investment. Wheeled Coach finds that almost 50% of the cost of every ambulance it manufactures is purchased materials. A large proportion of that 50% is in chassis (purchased from Ford), aluminum (from Reynolds Metal), and plywood used for flooring and cabinetry construction (from local suppliers). Wheeled Coach tracks these A inventory items quite carefully, maintaining tight security/control and ordering carefully so as to maximize quantity discounts while minimizing on-hand stock. Because of long lead times and scheduling needs at Reynolds, aluminum must actually be ordered as much as 8 months in advance. In a crowded ambulance industry in which it is the only giant, its 45 competitors don’t have the purchasing power to draw the same discounts as Wheeled Coach. But this competitive cost advantage cannot be taken lightly, according to President Bob Collins. “Cycle



counting in our stockrooms is critical. No part can leave the locked stockrooms without appearing on a bill of materials.” Accurate bills of material (BOM) are a requirement if products are going to be built on time. Additionally, because of the custom nature of each vehicle, most orders are won only after a bidding process. Accurate BOMs are critical to cost estimation and the resulting bid. For these reasons, Collins was emphatic that Wheeled Coach maintain outstanding inventory control. The Global Company Profile featuring Wheeled Coach (which opens Chapter 14) provides further details about the ambulance inventory control and production process.

Discussion Questions* 1. Explain how Wheeled Coach implements ABC analysis. 2. If you were to take over as inventory control manager at Wheeled Coach, what additional policies and techniques would you initiate to ensure accurate inventory records? 3. How would you go about implementing these suggestions? *You may wish to view the video that accompanies this case before addressing these questions.

Additional Case Studies: Visit MyOMLab for these free case studies: Southwestern University (F): The university must decide how many football day programs to order, and from whom. LaPlace Power and Light: This utility company is evaluating its current inventory policies.

Endnotes 1. See E. Malykhina, “Retailers Take Stock,” Information Week (February 7, 2005): 20–22, and A. Raman, N. DeHoratius, and Z. Ton, “Execution: The Missing Link in Retail Operations,” California Management Review 43, no. 3 (Spring 2001): 136–141. 2. This is the case when holding costs are linear and begin at the origin—that is, when inventory costs do not decline (or they increase) as inventory volume increases and all holding costs are in small increments. In addition, there is probably some learning each time a setup (or order) is executed—a fact that lowers subsequent setup costs. Consequently, the EOQ model is probably a special case. However, we abide by the conventional wisdom that this model is a reasonable approximation. 3. The formula for the economic order quantity (Q*) can also be determined by finding where the total cost curve is at a minimum (i.e., where the slope of the total cost curve is zero). Using calculus, we set the derivative of the total cost with respect to Q* equal to 0. The calculations for finding the minimum of Q D TC = S + H + PD 2 Q

are

d(TC) 9DS H = ¢ 2 ≤ + + 0 = 0 2 dQ Q

Thus, Q* =

2DS A H

4. The number of units short, Demand–ROP, is true only when Demand–ROP is non-negative. 5. Equations (12-15), (12-16), and (12-17) are expressed in days; however, they could equivalently be expressed in weeks, months, or even years. Just be consistent, and use the same time units for all terms in the equations. 6. Note that Equation (12-17) can also be expressed as: ROP = Average daily demand * Average lead time + 2 . Z2(Average lead time * s d2) + d 2sLT

7. OM managers also call these continuous review systems.

MyOMLab

Main Heading

Review Material

THE IMPORTANCE OF INVENTORY

Inventory is one of the most expensive assets of many companies.

(pp. 490–491)

Concept Questions: 1.1–1.4

The objective of inventory management is to strike a balance between inventory investment and customer service.

VIDEO 12.1 Managing Inventory at Frito-Lay

The two basic inventory issues are how much to order and when to order. j

j

j j

MANAGING INVENTORY (pp. 491–495)

j

j j j

INVENTORY MODELS

j

(pp. 495–496)

j

INVENTORY MODELS FOR INDEPENDENT DEMAND

Raw material inventory—Materials that are usually purchased but have yet to enter the manufacturing process. Work-in-process (WIP) inventory—Products or components that are no longer raw materials but have yet to become finished products. MRO inventory—Maintenance, repair, and operating materials. Finished-goods inventory—An end item ready to be sold but still an asset on the company’s books. ABC analysis—A method for dividing on-hand inventory into three classifications based on annual dollar volume. Cycle counting—A continuing reconciliation of inventory with inventory records. Shrinkage—Retail inventory that is unaccounted for between receipt and sale. Pilferage—A small amount of theft.

Concept Questions: 2.1–2.4

Holding cost—The cost to keep or carry inventory in stock. Ordering cost—The cost of the ordering process.

Concept Questions: 3.1–3.4

Problems: 12.1–12.6 Virtual Office Hours for Solved Problem: 12.1

j

Setup cost—The cost to prepare a machine or process for production.

j

Setup time—The time required to prepare a machine or process for production.

VIDEO 12.2 Inventory Control at Wheeled Coach Ambulance

j

Economic order quantity (EOQ) model—An inventory-control technique that minimizes the total of ordering and holding costs:

Concept Questions: 4.1–4.4

Q* =

(pp. 496–507)

2DS A H

D Demand Expected number of orders = N = = Order quantity Q* Number of working days per year N Total annual cost = Setup (order) cost + Holding cost

Expected time between orders = T =

j

j

j

(12-1) (12-2)

Virtual Office Hours for Solved Problems: 12.2–12.5

(12-3)

ACTIVE MODELS 12.1, 12.2

(12-4)

Q D (12-5) TC = S + H Q 2 Robust—Giving satisfactory answers even with substantial variation in the parameters. Lead time—In purchasing systems, the time between placing an order and receiving it; in production systems, the wait, move, queue, setup, and run times for each component produced. Reorder point (ROP)—The inventory level (point) at which action is taken to replenish the stocked item.

ROP for known demand: ROP = Demand per day * Lead time for a new order in days = d * L j j

2DS H

j

(12-6)

Safety stock (ss)—Extra stock to allow for uneven demand; a buffer. Production order quantity model—An economic order quantity technique applied to production orders: 2DS Q*p = (12-7) A H 31 - (d > p)4 Q*p =

H a1 -

Annual demand rate b Annual production rate

(12-8)

Quantity discount—A reduced price for items purchased in large quantities: TC =

Q D S + H + PD 2 Q

(12-9)

2DS A IP

(12-10)

Q* =

Problems: 12.7–12.40

Rapid Review

12

Chapter 12 Rapid Review

12 Rapid Review

Chapter 12 Rapid Review continued

MyOMLab

Main Heading

Review Material

PROBABILISTIC MODELS AND SAFETY STOCK

Probabilistic model—A statistical model applicable when product demand or any other Concept Questions: 5.1–5.4 variable is not known but can be specified by means of a probability distribution. j Service level—The complement of the probability of a stockout. Problems: 12.41–12.50 ROP for unknown demand: Virtual Office Hours ROP = d * L + ss (12-11) for Solved Problems: 12.6–12.9 Annual stockout costs = The sum of the units short for each demand level (12-12) * The probability of that demand level * The stockout cost >unit * The number of orders per year

(pp. 508–513)

j

ROP for unknown demand and given service level: ROP = Expected demand during lead time + ZsdLT

(12-13)

Safety stock = ZsdLT

(12-14)

ROP for variable demand and constant lead time: ROP = (Average daily demand * Lead time in days) + ZsdLT

(12-15)

ROP for constant demand and variable lead time: ROP = (Daily demand * Average lead time in days) + Z * Daily demand * s LT (12-16) ROP for variable demand and variable lead time: ROP = (Average daily demand * Average lead time in days) + ZsdLT

(12-17)

In each case, sdLT = 2(Average lead time * s 2d) + d 2s2LT but under constant demand: s2d = 0, and under constant lead time: s2LT = 0.

SINGLE-PERIOD MODEL

j

Single-period inventory model—A system for ordering items that have little or no value at the end of the sales period:

(pp. 513–514)

Service level =

FIXED-PERIOD (P ) SYSTEMS

j j

(pp. 514–515) j

Cs C s + Co

(12-18)

Fixed-quantity (Q) system—An ordering system with the same order amount each time. Perpetual inventory system—A system that keeps track of each withdrawal or addition to inventory continuously, so records are always current. Fixed-period (P) system—A system in which inventory orders are made at regular time intervals.

Concept Questions: 6.1–6.4 Problems: 12.51–12.53

Concept Questions: 7.1–7.4

Self Test j

Before taking the self-test, refer to the learning objectives listed at the beginning of the chapter and the key terms listed at the end of the chapter.

LO 12.1 ABC analysis divides on-hand inventory into three classes, based on: a) unit price. b) the number of units on hand. c) annual demand. d) annual dollar values. LO 12.2 Cycle counting: a) provides a measure of inventory turnover. b) assumes that all inventory records must be verified with the same frequency. c) is a process by which inventory records are periodically verified. d) all of the above. LO 12.3 The two most important inventory-based questions answered by the typical inventory model are: a) when to place an order and the cost of the order. b) when to place an order and how much of an item to order. c) how much of an item to order and the cost of the order. d) how much of an item to order and with whom the order should be placed. LO 12.4 Extra units in inventory to help reduce stockouts are called: a) reorder point. b) safety stock. c) just-in-time inventory. d) all of the above.

LO 12.5 The difference(s) between the basic EOQ model and the production order quantity model is(are) that: a) the production order quantity model does not require the assumption of known, constant demand. b) the EOQ model does not require the assumption of negligible lead time. c) the production order quantity model does not require the assumption of instantaneous delivery. d) all of the above. LO 12.6 The EOQ model with quantity discounts attempts to determine: a) the lowest amount of inventory necessary to satisfy a certain service level. b) the lowest purchase price. c) whether to use a fixed-quantity or fixed-period order policy. d) how many units should be ordered. e) the shortest lead time. LO 12.7 The appropriate level of safety stock is typically determined by: a) minimizing an expected stockout cost. b) choosing the level of safety stock that assures a given service level. c) carrying sufficient safety stock so as to eliminate all stockouts. d) annual demand.

Answers: LO 12.1. d; LO 12.2. c; LO 12.3. b; LO 12.4. b; LO 12.5. c; LO 12.6. d; LO 12.7. b.

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