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Inventory is one of the most expensive and important assets of many companies, representing as much as 50% of total invested capital. Managers have long recognized that good inventory control is crucial. On one hand, a firm can try to reduce costs by reducing onhand inventory levels. On the other hand, customers become dissatisfied when frequent inventory outages, called stockouts, occur. Thus, companies must make the balance between low and high inventory levels. As you would expect, cost minimization is the major factor in obtaining this delicate balance.

Inventory is any stored resource that is used to satisfy a current or future need. Raw materials, work-in-process, and finished goods are examples of inventory. Inventory levels for finished goods, such as clothes dryers, are a direct function of market demand. By using this demand information, it is possible to determine how much raw materials (for example, sheet metal, paint, and electric motors in the case of clothes dryers) and work-in-process are needed to produce the finished product. Every organization has 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 other important items. State and federal governments, schools, and virtually every manufacturing and production organization are concerned with inventory planning and control. Studying how organizations control their inventory is equivalent to studying how they achieve their objectives by supplying goods and services to their customers. Inventory is the common thread that ties all the functions and departments of the organization together. Figure 12. illustrates the basic components of an inventory planning and control system. The planning phase involves primarily what inventory is to be stocked and how it is to be acquired (whether it is to be manufactured or purchased). This information is then used in forecasting demand for the inventory and in controlling inventory levels. The feedback loop in Figure 12. 1 provides a way of revising the plan and forecast based on experiences and observation. Through inventory planning, an organization determines what goods and/or services are to be produced.

In cases of physical products, the organization must also determine whether to produce these goods or to purchase them from another manufacturer. When this has been determined, the next step is to forecast the demand. As discussed in Chapter 11, many mathematical techniques can be used in forecasting demand for a particular product. The emphasis in this chapter is on inventory control—that is, how to maintain adequate inventory levels within an organization to support a production or procurement plan that will satisfy the forecasted demand.

In this chapter, we discuss several different inventory control models that are commonly used in practice. For each model, we provide examples of how they are analyzed. Although we show the equations needed to compute the relevant parameters for each Inventory is any stored resource that is used to satisfy a current or future need. FIGURE 12. 1 Inventory Planning and Control Planning on What Inventory to Stock and How to Acquire It Forecasting Parts/Product Demand Controlling Inventory Levels Feedback Measurements to Revise Plans and Forecasts 12. : Importance of Inventory Control 12-3 ?MODELING IN THE REAL WORLD Defining the Problem Using an Inventory Model to Reduce Costs for a Hewlett-Packard Printer In making products for different markets, manufacturing companies often produce basic products and materials that can be used in a variety of end products. Hewlett-Packard, a leading manufacturer of printers, wanted to explore ways of reducing material and inventory costs of its Deskjet line of printers. One specific problem is that different power supplies are required in different countries. FORMULATION Developing a Model

The inventory model investigates inventory and material requirements as they relate to different markets. An inventory and materials flow diagram was developed that showed how each Deskjet printer was to be manufactured for various countries requiring different power supplies. Acquiring Input Data The input data consisted of inventory requirements, costs, and product versions. Different Deskjet versions are needed for the U. S. market, European markets, and Far East markets. The data included estimated demand in weeks of supply, replenishment lead times, and various cost data. Developing a Solution SOLUTION

The solution resulted in tighter inventory control and a change in how the printer was manufactured. The power supply was to be one of the last components installed in each Deskjet during the manufacturing process. Testing the Solution Testing was done by selecting one of the markets and performing a number of tests over a two-month period. The tests included material shortages, downtime profiles, service levels, and various inventory flows. INTERPRETATION Analyzing the Results and Sensitivity Analysis The results revealed that an inventory cost savings of 18% could be achieved by using the inventory model. Implementing the Results

As a result of the inventory model, Hewlett-Packard decided to redesign how its Deskjet printers are manufactured to reduce inventory costs in meeting a global market for its printers. Source: H. Lee, et al. “Hewlett-Packard Gains Control of Inventory and Service through Design for Localization,” Interfaces 23, 4 (July–August 1993): 1–11. model, we use Excel worksheets (included on the CD-ROM that accompanies this texbook) to actually calculate these values. 12. 2 IMPORTANCE OF INVENTORY CONTROL Inventory control serves several important functions and adds a great deal of flexibility to the operation of a firm.

Five main uses of inventory are as follows: There are five main uses of inventory. 1. The decoupling function 2. Storing resources 12-4 CHAPTER 12 Inventory Control Models 3. Irregular supply and demand 4. Quantity discounts 5. Avoiding stockouts and shortages Decoupling Function Inventory can act as a buffer. One of the major functions of inventory is to decouple manufacturing processes within the organization. If a company did not store inventory, there could be many delays and inefficiencies. For example, when one manufacturing activity has to be completed before a second activity can be started, it could stop the entire process.

However, stored inventory between processes could act as a buffer. Storing Resources Resources can be stored in work-in-process. Agricultural and seafood products often have definite seasons over which they can be harvested or caught, but the demand for these products is somewhat constant during the year. In these and similar cases, inventory can be used to store these resources. In a manufacturing process, raw materials can be stored by themselves, as work-inprocess, or as finished products. Thus, if your company makes lawn mowers, you might obtain lawn mower tires from another manufacturer.

If you have 400 finished lawn mowers and 300 tires in inventory, you actually have 1,900 tires stored in inventory. Three hundred tires are stored by themselves, and 1,600 (= 4 tires per lawn mower ? 400 lawn mowers) tires are stored on the finished lawn mowers. In the same sense, labor can be stored in inventory. If you have 500 subassemblies and it takes 50 hours of labor to produce each assembly, you actually have 25,000 labor hours stored in inventory in the subassemblies. In general, any resource, physical or otherwise, can be stored in inventory. Irregular Supply and Demand

Inventory helps when there is irregular supply or demand. When the supply or demand for an inventory item is irregular, storing certain amounts in inventory can be important. If the greatest demand for Diet-Delight beverage is during the summer, the Diet-Delight company will have to make sure there is enough supply to meet this irregular demand. This might require that the company produce more of the soft drink in the winter than is actually needed in order to meet the winter demand. The inventory levels of Diet-Delight will gradually build up over the winter, but this inventory will be needed in the summer.

The same is true for irregular supplies. Quantity Discounts Purchasing in large quantities may lower unit costs. Another use of inventory is to take advantage of quantity discounts. Many suppliers offer discounts for large orders. For example, an electric jigsaw might normally cost $10 per unit. If you order 300 or more saws at one time, your supplier may lower the cost to $8. 75. Purchasing in larger quantities can substantially reduce the cost of products. There are, however, some disadvantages of buying in larger quantities.

You will have higher storage costs and higher costs due to spoilage, damaged stock, theft, insurance, and so on. Furthermore, if you invest in more inventory, you will have less cash to invest elsewhere. Avoiding Stockouts and Shortages Inventory can help avoid stockouts. Another important function of inventory is to avoid shortages or stockouts. If a company is repeatedly out of stock, customers are likely to go elsewhere to satisfy their needs. Lost goodwill can be an expensive price to pay for not having the right item at the right time. 12. 4: Economic Order Quantity: Determining How Much to Order 12-5 12. INVENTORY CONTROL DECISIONS Even though there are literally millions of different types of products manufactured in our society, there are only two fundamental decisions that you have to make when controlling inventory: 1. How much to order 2. When to order The purpose of all inventory models is to minimize inventory costs. The purpose of all inventory models is to determine how much to order and when to order. As you know, inventory fulfills many important functions in an organization. But as the inventory levels go up to provide these functions, the cost of storing and holding inventory also increases.

Thus, we must reach a fine balance in establishing inventory levels. A major objective in controlling inventory is to minimize total inventory costs. Some of the most significant inventory costs are as follows: 1. 2. 3. 4. 5. Cost of the items Cost of ordering Cost of carrying, or holding, inventory Cost of stockouts Cost of safety stock, the additional inventory that may be held to help avoid stockouts Components of total cost. The inventory models discussed in the first part of this chapter assume that demand and the time it takes to receive an order are known and constant, and that no quantity discounts are given.

When this is the case, the most significant costs are the cost of placing an order and the cost of holding inventory items over a period of time. Table 12. 1 provides a list of important factors that make up these costs. Later in this chapter we discuss several more sophisticated inventory models. 12. 4 ECONOMIC ORDER QUANTITY: DETERMINING HOW MUCH TO ORDER The economic order quantity (EOQ) model is one of the oldest and most commonly known inventory control techniques. Research on its use dates back to a 1915 publication by Ford W. Harris. This model is still used by a large number of organizations today. This technique

TABLE 12. 1 Inventory Cost Factors ORDERING COST FACTORS Developing and sending purchase orders Processing and inspecting incoming inventory Bill paying Inventory inquiries Utilities, phone bills, and so on for the purchasing department Salaries and wages for purchasing department employees Supplies such as forms and paper for the purchasing department CARRYING COST FACTORS Cost of capital Taxes Insurance Spoilage Theft Obsolescence Salaries and wages for warehouse employees Utilities and building costs for the warehouse Supplies such as forms and papers for the warehouse 12-6 CHAPTER 12 Inventory Control Models IN ACTION

Implementing Speed and Quality in the Production Run at Milton Bradley Not getting the correct number of parts and pieces is very frustrating for customers. It can also be time-consuming, expensive, and frustrating for Milton Bradley to supply the extra parts or get returned toys or games. If shortages are found during the assembly stage, the entire production run can be stopped until the problem is corrected. Counting parts by hand or machine was always problematic and not always accurate. As a result, Milton Bradley decided to weigh the pieces and complete games to determine whether the correct number of parts had been included.

If the weight is not exactly correct, there is a problem that needs to be resolved before the game or toy is packaged or shipped. Using highly accurate digital scales, Milton Bradley has been able to get the right parts to the right production line at the right time. Without this simple implementation approach, the most sophisticated production run results would be meaningless. Source: D. Smock. “Games Tip the Scale at Milton Bradley,” Plastics World (March 1997): 22–26. Milton Bradley, a division of Hasbro, Inc. , has been manufacturing toys for more than 100 years.

Founded by Milton Bradley in 1860, the company started by making a lithograph of Abraham Lincoln. Using his printing skills, Bradley developed games and toys, 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. When Milton Bradley has determined the optimal quantities for its production runs, it must implement these quantities. 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 companies, including Wal-Mart, can require 20,000 or more perfectly assembled games delivered to their warehouses in a matter of days. is relatively easy to use, but it makes a number of assumptions. Some of the more important assumptions follow: Assumptions of the EOQ model. 1. Demand is known and constant. 2. The lead time—that is, the time between the placement of the order and the receipt of the order—is known and constant. . The receipt of inventory is instantaneous. In other words, the inventory from an order arrives in one batch, at one point in time. 4. Quantity discounts are not possible. 5. The only variable costs are the cost of placing an order, ordering cost, and the cost of holding or storing inventory over time, carrying, or holding, cost. 6. If orders are placed at the right time, stockouts and shortages can be avoided completely. The inventory usage curve has a sawtooth shape in the EOQ model. With these assumptions, inventory usage has a sawtooth shape, as in Figure 12. 2.

Here, Q represents the amount that is ordered. If this amount is 500 units, all 500 units arrive at one time when an order is received. Thus, the inventory level jumps from 0 to 500 units. In general, the 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 line in Figure 12. 2. ) Another order is placed such that when the inventory level reaches 0, the new order is received and the inventory level again jumps to Q units, represented by the vertical lines.

This process continues indefinitely over time. Ordering and Inventory Costs The objective of most inventory models is to minimize the total cost. With the assumptions just given, the significant costs are the ordering cost and the inventory carrying cost. All other costs, such as the cost of the inventory itself, are constant. Thus, if we minimize the sum of the ordering and carrying costs, we also minimize the total cost. 12. 4: Economic Order Quantity: Determining How Much to Order FIGURE 12. 2 Inventory Usage over Time Inventory Level 12-7 Q Order Quantity = Q = Maximum Inventory Level

Minimum Inventory 0 Time FIGURE 12. 3 Total Cost as a Function of Order Quantity Cost Curve for Total Cost of Carrying and Ordering Minimum Total Cost Carrying Cost Curve Ordering Cost Curve Optimal Order Quantity (Q *) Order Quantity in Units The objective of the simple EOQ model is to minimize ordering and carrying costs. The average inventory level is one-half the maximum level. To help visualize this, Figure 12. 3 graphs total cost as a function of the order quantity, Q. As the value of Q increases, the total number of orders placed per year decreases. Hence, the total ordering cost decreases.

However, as the value of Q increases, the carrying cost increases because the firm has to maintain larger average inventories. The optimal order size, Q*, is the quantity that minimizes the total cost. Note in Figure 12. 3 that Q* occurs at the point where the ordering cost curve and the carrying cost curve intersect. This is not by chance. With this particular type of cost function, the optimal quantity always occurs at a point where the ordering cost is equal to the carrying cost. Now that we have a better understanding of inventory costs, let us see how we can determine the value of Q* that minimizes the total cost.

In determining the annual carrying cost, it is convenient to use the average inventory. Referring to Figure 12. 2, we see that the on-hand inventory ranges from a high of Q units to a low of zero units, with a uniform 12-8 CHAPTER 12 Inventory Control Models rate of decrease between these levels. Thus, the average inventory can be calculated as the average of the minimum and maximum inventory levels. That is, Average inventory level = (0 + Q)/2 = Q/2 (12-1) We multiply this average inventory by a factor called the annual inventory carrying cost per unit to determine the annual inventory cost. Finding the Economic Order Quantity

We pointed out that the optimal order quantity, Q*, is the point that minimizes the total cost, where total cost is the sum of ordering cost and carrying cost. We also indicated graphically that the optimal order quantity was at the point where the ordering cost was equal to the carrying cost. Let us now define the following parameters: Q* = Optimal order quantity (i. e. , the EOQ) D = Annual demand, in units, for the inventory item Co = Ordering cost per order Ch = Carrying or holding cost per unit per year P = Purchase cost per unit of the inventory item The unit carrying cost, Ch, is usually expressed in one of two ways, as follows: 1.

As a fixed cost. For example, Ch is $0. 50 per unit per year. I is the annual carrying cost expressed as a percentage of the unit cost of the item. 2. As a percentage (typically denoted by I) of the item’s unit cost or price. For example, Ch is 20% of the item’s unit cost. In general, Ch = I ? P (12-2) For a given order quantity Q, the ordering, holding, and total costs can be computed using the following formulas:1 Total ordering cost = (D /Q ) ? C o Total carrying cost = (Q / 2) ? C h (12-3) (12-4) Total cost = Total ordering cost + Total carrying cost + Total purchase cost (12-5) = (D /Q) ? Co + (Q / 2) ? C h +P ?

D Observe that the total purchase cost (i. e. , P ? D) does not depend on the value of Q. This is so because regardless of how many orders we place each year, or how many units we order each time, we will still incur the same annual total purchase cost. The presence of Q in the denominator of the first term makes Equation 12-5 a nonlinear equation with respect to Q. Nevertheless, because the total ordering cost is equal to the total carrying cost at the optimal value of Q, we can set the terms in Equations 12-3 and 12-4 equal to each other and calculate the EOQ as Q * = (2DCo /C h ) (12-6) Total cost is a nonlinear function of Q.

We determine Q* by setting ordering cost equal to carrying cost. 1 See a recent operations management textbook such as J. Heizer and B. Render. Operations Management, 8/e. Upper Saddle River, NJ: Prentice Hall, 2006, for more details of these formulas (and other formulas in this chapter). 12. 4: Economic Order Quantity: Determining How Much to Order 12-9 Sumco Pump Company Example Let us now apply these formulas to the case of Sumco, a company that buys pump housings from a manufacturer and distributes to retailers. Sumco would like to reduce its inventory cost by determining the optimal number of pump housings to obtain per order.

The annual demand is 1,000 units, the ordering cost is $10 per order, and the carrying cost is $0. 50 per unit per year. Each pump housing has a purchase cost of $5. How many housings should Sumco order each time? To answer these and other questions, we use the ExcelModules program. Using ExcelModules for Inventory Model Computations Excel Notes I The CD-ROM that accompanies this textbook contains a set of Excel worksheets, bundled together in a software package called ExcelModules. Appendix B describes the procedure for installing and running this program, and it gives a brief description of its contents.

I The CD-ROM also contains the Excel file for each sample problem discussed here. The relevant file name is shown in the margin next to each example. I For clarity, all worksheets for inventory models in ExcelModules are color coded as follows: I Input cells, where we enter the problem data, are shaded yellow. I Output cells, which show results, are shaded green. Although these colors are not apparent in the screenshots shown in the textbook, they are seen in the Excel files on the CD-ROM. We use Excel worksheets to do all inventory model computations.

When we run the ExcelModules program, we see a menu option titled ExcelModules in the main menu bar of Excel. We click ExcelModules and then click Inventory Models. The choices shown in Screenshot 12-1A are displayed. From these choices, we select the appropriate model. When we select any of the inventory models in ExcelModules, we are first presented with a window that allows us to specify several options. Some of these options are common for all models, whereas others are specific to the inventory model selected. For example, SCREENSHOT 12-1A Inventory Models Submenu in ExcelModules

This choice appears in Excel’s main menu bar when ExcelModules is run. Main menu in ExcelModules. ExcelModules options. See Appendix B for details. Inventory Models submenu in ExcelModules 12-10 CHAPTER 12 Inventory Control Models Screenshot 12-1B shows the options window that appears when we select the Economic Order Quantity (EOQ) model. The options here include the following: 1. Title of the problem. The default value is Problem Title. 2. Graph. Checking this box results in a graph of ordering, carrying, and total costs versus order quantity. 3. Holding Cost. This is either a fixed amount or a percentage of unit purchase cost. . Reorder Point. Checking this box results in the calculation of the reorder point, for a given lead time between placement of the order and receipt of the order. We discuss the reorder point in section 12. 5. This option is available only for the EOQ model. Using ExcelModules for the EOQ Model Screenshot 12-2A shows the options we select for the Sumco Pump Company example. When we click OK on this screen, we get the worksheet shown in Screenshot 12-2B on page 12-12. We now enter the values for the annual demand, D, ordering cost, Co, carrying cost, Ch, and unit purchase cost, P, in cells B6 to B9, espectively. Excel Notes I File: 12-2. xls, sheet: 12-2B I The worksheets in ExcelModules contain formulas to compute the results for different inventory models. The default value of zero for the input data causes the results of these formulas to initially appear as #N/A, #VALUE! , or #DIV/0!. However, as soon as we enter valid values for these input data, the worksheets display the formula results. Once ExcelModules has been used to create the Excel worksheet for a particular inventory model (e. g. , EOQ), the resulting worksheet can be used to compute the results with several different input data.

For example, we can enter different input data in cells B6:B9 of Screenshot 12-2B and compute the results without having to create a new EOQ worksheet each time. SCREENSHOT 12-1B Sample Options Window for Inventory Models in ExcelModules Default problem title Check here to get plot of costs. Check here to compute reorder point (see section 12. 5). This specifies how carrying or holding cost is entered. 12. 4: Economic Order Quantity: Determining How Much to Order SCREENSHOT 12-2A Options Window for EOQ Model in ExcelModules 12-11 Problem title Carrying cost specified as fixed amount

The worksheet calculates the EOQ (shown in cell B12 of Screenshot 12-2B). In addition, the following output measures are calculated and reported: I Maximum inventory (= Q*), in cell B13 I Average inventory (= Q*/2), in cell B14 I Number of orders (= D/Q*), in cell B15 I Total holding cost (= Ch ? Q*/2), in cell B17 I Total ordering cost (= Co ? D/Q*), in cell B18 I Total purchase cost (= P ? D), in cell B19 I Total cost (= Ch ? Q*/2 + Co ? D/Q* + P ? D), in cell B20 File: 12-2. xls, sheet: 12-2C As you might expect, the total ordering cost of $50 is equal to the total carrying cost. (Refer to Figure 12. 3 on page 12-7 again to see why. You may wish to try different values for the order quantity Q, such as 100 or 300 pump housings. (Plug in these values one at a time in cell B12. ) You will find that the total cost (in cell B20) has the lowest value when Q is 200 units. That is, the EOQ, Q*, for Sumco is 200 pump housings. The total cost, including the purchase cost of $5,000, is $5,100. If requested, a plot of the total ordering cost, total holding cost and total cost for different values of Q is drawn by ExcelModules. The graph, shown in Screenshot 12-2C, is drawn on a separate worksheet. 12-12 CHAPTER 12 Inventory Control Models EOQ Model for Sumco Pump

SCREENSHOT 12-2B Input data EOQ is 200 units. 1 Average inventory = – Maximum inventory 2 Holding cost = Ordering cost Data for graph, generated and used by ExcelModules Purchase Cost of Inventory Items We can calculate the average inventory value in dollar terms. It is often useful to know the value of the average inventory level in dollar terms. We know from Equation 12-1 that the average inventory level is Q/2, where Q is the order quantity. If we order Q* (the EOQ) units each time, the value of the average inventory can be computed by multiplying the average inventory by the unit purchase cost, P.

That is, Average dollar value of inventory = P ? (Q*/2) (12-7) Calculating the Ordering and Carrying Costs for a Given Value of Q Recall that the EOQ formula is given by Equation 12-6 as Q* = (2DCo / Ch ) In using this formula, we assumed that the values of the ordering cost Co and carrying cost Ch are known constants. In some situations, however, these costs may be difficult to estimate precisely. For example, if the firm orders several items from a supplier simultaneously, it may be difficult to identify the ordering cost separately for each item.

In such cases, we can use the EOQ formula to compute the value of Co or Ch that would make a given order quantity the optimal order quantity. 12. 4: Economic Order Quantity: Determining How Much to Order SCREENSHOT 12-2C Plot of Costs versus Order Quantity for Sumco Pump 12-13 Total cost is lowest when Holding cost = Ordering cost. Total cost Ordering cost Holding cost For a given Q, we compute a Co or Ch that makes Q optimal. To compute these Co or Ch values, we can manipulate the EOQ formula algebraically and rewrite it as follows: Co = Q 2 ? Ch/(2D) and Ch = 2DCo/Q 2 (12-9) (12-8) where Q is the given order quantity.

We illustrate the use of these formulas in Solved Problem 12-1 at the end of this chapter. Sensitivity of the EOQ Formula If any of the input data values change, the EOQ also changes. Due to the nonlinear formula for EOQ, changes in Q* are less severe than changes in input data values. The EOQ formula in Equation 12-6 assumes that all input data are known with certainty. What happens if one of the input values is incorrect? If any of the values used in the formula changes, the optimal value of Q* also changes. Determining the magnitude and effect of these changes on Q* is called sensitivity analysis.

This type of analysis is important in practice because the input values for the EOQ model are usually estimated and hence subject to error or change. Let us use the Sumco example again to illustrate this issue. Suppose the ordering cost, Co, is actually $15, instead of $10. Let us assume that the annual demand for pump housings is still the same, namely, D = 1,000 units, and that the carrying cost, Ch, is $0. 50 per unit per year. If we use these new values in the EOQ worksheet (as in Screenshot 12-2B), the revised EOQ turns out to be 245 units. (See if you can verify this for yourself. That is, when the ordering cost increases by 50% (from $10 to $15), the optimal order quantity increases 12-14 CHAPTER 12 Inventory Control Models only by 22. 5% (from 200 to 245). This is because the EOQ formula involves a square root and is, therefore, nonlinear. We observe a similar occurrence when the carrying cost, Ch, changes. Let us suppose that Sumco’s annual carrying cost is $0. 80 per unit, instead of $0. 50. Let us also assume that the annual demand is still 1,000 units, and the ordering cost is $10 per order. Using the EOQ worksheet in ExcelModules, we can calculate the revised EOQ as 158 units.

That is, when the carrying cost increases by 60% (from $0. 50 to $0. 80), the EOQ decreases by only 21%. Note that the order quantity decreases here because a higher carrying cost makes holding inventory more expensive. 12. 5 REORDER POINT: DETERMINING WHEN TO ORDER Now that we have decided how much to order, we look at the second inventory question: when to order. In most simple inventory models, it is assumed that receipt of an order is instantaneous. That is, we assume that a firm waits until its inventory level for a particular item reaches zero, places an order, and receives the items in stock immediately.

In many cases, however, the time between the placing and receipt of an order, called the lead time, or delivery time, is often a few days or even a few weeks. Thus, the when to order decision is usually expressed in terms of a reorder point (ROP), the inventory level at which an order should be placed. The ROP is given as The ROP determines when to order inventory. ROP = (Demand per day) ? (Lead time, in days) =d ? L (12-10) Figure 12. 4 shows the reorder point graphically. The slope of the graph is the daily inventory usage. This is expressed in units demanded per day, d. The lead time, L, is the time that it takes to receive an order.

Thus, if an order is placed when the inventory level reaches the ROP, the new inventory arrives at the same instant the inventory is reaching zero. Let’s look at an example. FIGURE 12. 4 Reorder Point (ROP) Curve Inventory Level (Units) Q* Slope = Units/Day = d ROP (Units) Lead Time = L Time (Days) 12. 5: Reorder Point: Determining When to Order 12-15 Sumco Pump Company Example Revisited Recall that we calculated an EOQ value of 200 and a total cost of $5,100 for Sumco (see Screenshot 12-2B on page 12-12). These calculations were based on an annual demand of 1,000 units, an ordering cost of $10 per order, an annual carrying cost of $0. 0 per unit, and a purchase cost of $5 per pump housing. Now let us assume that there is a lead time of 3 business days between the time Sumco places an order and the time the order is received. Further, let us assume there are 250 business days in a year. To calculate the ROP, we must first determine the daily demand rate, d. In Sumco’s case, because there are 250 business days in a year and the annual demand is 1,000, the daily demand rate is 4 (= 1,000/250) pump housings. Using ExcelModules to Compute the ROP We can include the ROP computation in the EOQ worksheet provided in ExcelModules.

To do so for Sumco’s problem, we once again choose the EOQ option from the Inventory Models submenu in ExcelModules (refer to Screenshot 12-1A). The only change in the options window (see Screenshot 12-2A) is that we now check the box labeled Reorder Point. The worksheet shown in Screenshot 12-3 is now displayed. We enter the input data as before (see Screenshot 12-2B). Note the additional input entries for the daily demand rate in cell B10 and the lead time in cell B11. In addition to all the computations shown in Screenshot 12-2B, the worksheet now calculates and reports the ROP of 12 units (shown in cell B24).

Hence, when the inventory stock of pump housings drops to 12, an order should be placed. The order will arrive three days later, just as the firm’s stock is depleted to zero. It should be mentioned that this calculation assumes that all the assumptions listed earlier for EOQ are valid. When demand is not known with complete certainty, these calculations must be modified. This is discussed later in this chapter. To compute the ROP, we need to know the demand rate per period. File: 12-3. xls SCREENSHOT 12-3 EOQ Model with ROP for Sumco Pump Input data for computing ROP EOQ is 200 units.

ROP is 12 units. 12-16 CHAPTER 12 Inventory Control Models IN ACTION Inland Steel Uses Systems Contracts to Control Inventory Costs equipment. To overcome these problems, she developed a comprehensive inventory ordering system that took advantage of standardization and contract buying. The result was a contract ordering system that provided superior equipment at substantial savings. Most of the equipment was leased or rented. The new system provided low monthly rates for office equipment, free installation, and a 30-day free trial. Another advantage was a floating systems contract.

With this type of contract, there is no termination date, which helps reduce the time and costs of maintaining leasing agreements. The bottom line is that a systems contract approach allowed Inland Steel to order goodquality office equipment for fewer dollars. Sound inventory control involves much more than computing the EOQ. In most cases, other practical and financial considerations must be taken into account to minimize total inventory costs and to provide tighter control on inventory levels. Both practical and financial considerations led Inland Steel to consider several inventory policies, including systems contracts.

Inland Steel produces approximately 5. 5 million tons of steel each year. The steel mill has two blast furnaces that supply steel to four casting operations. Yet, steel inventory is not the company’s only inventory concern. For many large corporations, office equipment, such as scanners, printers, and fax machines can represent a substantial investment. Furthermore, all steel-processing facilities are controlled through computers, which are considered office equipment by Inland Steel. Tricia Wynn, a project buyer for Inland Steel, was concerned about high costs and a lack of standardization for office Source: K.

Evans-Correia. “All Systems Go,” Purchasing (March 23, 1989): 106–107. 12. 6 ECONOMIC PRODUCTION QUANTITY: DETERMINING HOW MUCH TO PRODUCE In the EOQ model, we assumed that the receipt of inventory is instantaneous. In other words, the entire order arrives in one batch, at a single point in time. In many cases, however, a firm may build up its inventory gradually over a period of time. For example, a firm may receive shipments from its supplier uniformly over a period of time. Or, a firm may be producing at a rate of p per day and simultaneously selling at a rate of d per day (where p > d). Figure 12. shows inventory levels as a function of time in these situations. Clearly, the EOQ model is no longer applicable here, and we need a new model to calculate the optimal order (or production) quantity. Because this model is especially suited to the production environment, it is also commonly known as the production lot size model or the economic production quantity (EPQ) model. We refer to this model as the EPQ model in the remainder of this chapter. In a production process, instead of having an ordering cost, there will be a setup cost. This is the cost of setting up the production facility to manufacture the desired product.

It normally includes the salaries and wages of employees who are responsible for setting up the equipment, engineering and design costs of making the setup, and the costs of paperwork, supplies, utilities, and so on. The carrying cost per unit is composed of the same factors as the traditional EOQ model, although the equation to compute the annual carrying cost changes. Inventory Level The EPQ model eliminates the instantaneous receipt assumption. FIGURE 12. 5 Inventory Control and the Production Process Part of Inventory Cycle During Which Production Is Taking Place

There Is No Production During This Part of the Inventory Cycle Maximum Inventory t Time 12. 6: Economic Production Quantity: Determining How Much to Produce 12-17 In determining the annual carrying cost for the EPQ model, it is again convenient to use the average on-hand inventory. Referring to Figure 12. 5, we can show that the maximum on-hand inventory is Q ? (1 d/p) units, where d is the daily demand rate and p is the daily production rate. The minimum on-hand inventory is again zero units, and the inventory decreases at a uniform rate between the maximum and minimum levels.

Thus, the average inventory can be calculated as the average of the minimum and maximum inventory levels. That is, This is the formula for average inventory in the EPQ model. Average inventory level = [0 + Q ? (1 – d/p)]/2 = Q ? (1 – d/p)/2 (12-11) Analogous to the EOQ model, it turns out that the optimal order quantity in the EPQ model also occurs when the total setup cost equals the total carrying cost. You should note, however, that making the total setup cost equal to the total carrying cost does not always guarantee optimal solutions for models more complex than the EPQ model.

Finding the Economic Production Quantity Let us first define the following additional parameters: Q* = Optimal order or production quantity (i. e. , the EPQ) Cs = Setup cost per setup For a given order quantity, Q, the setup, holding, and total costs can now be computed using the following formulas: Total setup cost = (D /Q) ? C s Total carrying cost = [Q(1 ? d /p)/2 ]? C h Total cost = Total setup cost + Total carrying cost + Total production cost = (D /Q) ? C s + [Q(1 ? d /p )/ 2] ? C h + P ? D (12-14) (12-12) (12-13) Here is the formula for the optimal production quantity.

Notice the similarity to the basic EOQ model. As in the EOQ model, the total production (or purchase, if the item is purchased) cost does not depend on the value of Q. Further, the presence of Q in the denominator of the first term makes the total cost function nonlinear. Nevertheless, because the total setup cost should equal the total ordering cost at the optimal value of Q, we can set the terms in Equations 12-12 and 12-13 equal to each other and calculate the EPQ as Q * = 2DC s /[C h (1 ? d /p)] (12-15) Brown Manufacturing Example Brown Manufacturing produces mini-sized refrigeration packs in batches.

The firm’s estimated demand for the year is 10,000 units. Because Brown operates for 167 business days each year, this annual demand translates to a daily demand rate of about 60 units per day. It costs about $100 to set up the manufacturing process, and the carrying cost is about $0. 50 per unit per year. When the production process has been set up, 80 refrigeration packs can be manufactured daily. Each pack costs $5 to produce. How many packs should Brown produce in each batch? As discussed next, we determine this value, as well as values for the associated costs, by using ExcelModules.

Using ExcelModules for the EPQ Model We select the Economic Production Quantity (EPQ) option from the Inventory Models submenu in ExcelModules (refer to Screenshot 12-1A). The options for this procedure are similar to those for the EOQ model (see Screenshot 12-2A). The only change is that the ROP option is not available here. After we enter the title and other options for this problem, we get the worksheet shown in File: 12-4. xls, sheet: 12-4A 12-18 CHAPTER 12 Inventory Control Models EPQ Model for Brown Manufacturing SCREENSHOT 12-4A Carrying cost is specified as a fixed amount.

EPQ is 4,000 units. Holding cost = Setup cost Data for graph, generated and used by ExcelModules Screenshot 12-4A. We now enter the values for the annual demand, D, setup cost, Cs, carrying cost, Ch, daily production rate, p, daily demand rate, d, and unit production (or purchase) cost, P, in cells B7 to B12, respectively. The worksheet calculates and reports the EPQ (shown in cell B15), as well as the following output measures: I Maximum inventory (= Q*[1 – d/p]), in cell B16 I Average inventory (= Q*[1 – d/p]/2), in cell B17 I Number of setups (= D/Q*), in cell B18 I Total holding cost (= Ch ?

Q*[1 – d/p]/2), in cell B20 I Total setup cost (= Cs ? D/Q*), in cell B21 I Total purchase cost (= P ? D), in cell B22 I Total cost (= Ch ? Q*[1 – d/p]/2 + Cs ? D/Q* + P ? D), in cell B23 Here again, as you might expect, the total setup cost is equal to the total carrying cost ($250 each). You may wish to try different values for Q, such as 3,000 or 5,000 pumps. (Plug these values, one at a time, into cell B15. ) You will find that the minimum total cost occurs 12. 7: Quantity Discount Models SCREENSHOT 12-4B Plot of Costs versus Order Quantity for Brown Manufacturing 2-19 Total cost Setup cost Holding cost File: 12-4. xls, sheet: 12-4B when Q is 4,000 units. That is, the EPQ, Q*, for Brown is 4,000 units. The total cost, including the production cost of $50,000, is $50,500. If requested, a plot of the total setup cost, holding cost, and total cost for different values of Q is drawn by ExcelModules. This graph, shown in Screenshot 12-4B, is drawn on a separate worksheet. Length of the Production Cycle Production cycle is the length of each manufacturing run. Referring to Figure 12. , we see that the inventory buildup occurs over a period t during which Brown is both producing and selling refrigeration packs. We refer to this period t as the production cycle. In Brown’s case, if Q* = 4,000 units and we know that 80 units can be produced daily, the length of each production cycle will be Q* / p = 4,000 / 80 = 50 days. Thus, when Brown decides to produce refrigeration packs, the equipment will be set up to manufacture the units for a 50-day time span. 12. 7 QUANTITY DISCOUNT MODELS To increase sales, many companies offer quantity discounts to their customers.

A quantity discount is simply a decreased unit cost for an item when it is purchased in larger quantities. It is not uncommon to have a discount schedule with several discounts for large orders. A typical quantity discount schedule is shown in Table 12. 2. As can be seen in Table 12. 2, the normal cost for the item in this example is $5. When 1,000 to 1,999 units are ordered at one time, the cost per unit drops to $4. 80, and when the quantity ordered at one time is 2,000 units or more, the cost is $4. 75 per unit. As always, A discount is a reduced price for an item when it is purchased in large quantities. 2-20 CHAPTER 12 Inventory Control Models DISCOUNT NUMBER 1 2 3 DISCOUNT QUANTITY 0 to 999 1,000 to 1,999 2,000 and over DISCOUNT COST $5. 00 $4. 80 $4. 75 TABLE 12. 2 Quantity Discount Schedule DISCOUNT 0% 4% 5% management must decide when and how much to order. But with quantity discounts, how does a manager make these decisions? As with other inventory models discussed so far, the overall objective is to minimize the total cost. Because the unit cost for the third discount in Table 12. 2 is lowest, you might be tempted to order 2,000 units or more to take advantage of this discount.

Placing an order for that many units, however, might not minimize the total inventory cost. As the discount quantity goes up, the item cost goes down, but the carrying cost increases because the order sizes are large. Thus, the major trade-off when considering quantity discounts is between the reduced item cost and the increased carrying cost. Recall that we computed the total cost (including the total purchase cost) for the EOQ model as follows (see Equation 12-5): Total cost = Total ordering cost + Total carrying cost + Total purchase cost = ( D /Q ) ? Co + (Q / 2) ? Ch +P? D

Next, we illustrate the four-step process to determine the quantity that minimizes the total cost. However, we use a worksheet included in ExcelModules to actually compute the optimal order quantity and associated costs in our example. We calculate Q* values for each discount. Four Steps to Analyze Quantity Discount Models 1. For each discount price, calculate a Q* value, using the EOQ formula (Equation 12-6). In quantity discount EOQ models, the unit carrying cost, Ch, is typically expressed as a percentage (I) of the unit purchase cost (P). That is, Ch = I ? P, as discussed in Equation 12-2.

As a result, the value of Q* will be different for each discounted price. 2. For any discount level, if the Q* computed in step 1 is too low to qualify for the discount, adjust Q* upward to the lowest quantity that qualifies for the discount. For example, if Q* for discount 2 in Table 12. 2 turns out to be 500 units, adjust this value up to 1,000 units. The reason for this step is illustrated in Figure 12. 6. As seen in Figure 12. 6, the total cost curve for the discounts shown in Table 12. 2 is broken into three different curves. There are separate cost curves for the first (0 ? Q ? 999), second (1,000 ?

Q ? 1,999), and third (Q ? 2,000) discounts. Look at the total cost curve for discount 2. The Q* for discount 2 is less than the allowable discount range of 1,000 to 1,999 units. However, the total cost at 1,000 units (which is the minimum quantity needed to get this discount) is still less than the lowest total cost for discount 1. Thus, step 2 is needed to ensure that we do not discard any discount level that may indeed produce the minimum total cost. Note that an order quantity computed in step 1 that is greater than the range that would qualify it for a discount may be discarded. . Using the total cost equation (Equation 12-5), compute a total cost for every Q* determined in steps 1 and 2. If a Q* had to be adjusted upward because it was below the allowable quantity range, be sure to use the adjusted Q* value. 4. Select the Q* that has the lowest total cost, as computed in step 3. It will be the order quantity that minimizes the total cost. Next, we adjust the Q* values. The total cost curve is broken into parts. Next, we compute total cost. We select the Q* with the lowest total cost. 12. 7: Quantity Discount Models FIGURE 12. Total Cost Curve for the Quantity Discount Model Total Cost $ TC Curve for Discount 1 TC Curve for Discount 3 12-21 TC Curve for Discount 2 Q* for Discount 2 0 1,000 Order Quantity 2,000 Brass Department Store Example This is an example of the quantity discount model. Brass Department Store stocks toy cars. Recently, the store was given a quantity discount schedule for the cars, as shown in Table 12. 2. Thus, the normal cost for the cars is $5. For orders between 1,000 and 1,999 units, the unit cost is $4. 80, and for orders of 2,000 or more units, the unit cost is $4. 75.

Furthermore, the ordering cost is $49 per order, the annual demand is 5,000 race cars, and the inventory carrying charge as a percentage of cost, I, is 20%, or 0. 2. What order quantity will minimize the total cost? We use the ExcelModules program to answer this question. Using ExcelModules for the Quantity Discount Model We select the Quantity Discount option from the Inventory Models submenu in ExcelModules (refer back to Screenshot 12-1A). The window shown in Screenshot 12-5A is displayed. The option entries in this window are similar to those for the EOQ model (see Screenshot 12-2A).

The only additional choice is the box labeled Number of price ranges. The specific entries for Brass Department Store’s problem are shown in Screenshot 12-5A. When we click OK on this screen, we get the worksheet shown in Screenshot 12-5B. We now enter the values for the annual demand, D, ordering cost, Co, and holding cost percentage, I, in cells B7 to B9, respectively. Note that I is entered as a percentage value (e. g. , enter 20 for the Brass Department Store example). Then, for each of the three discount ranges, we enter the minimum quantity needed to get the discount and the discounted unit price, P.

These entries are shown in cells B12:D13 of Screenshot 12-5B. The worksheet works through the four-step process and reports the following output measures for each discount range: I EOQ value (shown in cells B17:D17), computed using Equation 12-6 I Adjusted EOQ value (shown in cells B18:D18), as discussed in step 2 of the four-step File: 12-5. xls, sheet: 12-5B process I Total holding cost, total ordering cost, total purchase cost, and overall total cost, shown in cells B20:D23 SCREENSHOT 12-5A Options Window for Quantity Discount Model in ExcelModules 3 discount ranges Holding cost is specified as I ? P. SCREENSHOT 12-5B

Quantity Discount Model for Brass Department Store 20% is entered as 20 here. Q* for each price range Adjusted Q* value Lowest cost option Data for graph, generated and used by ExcelModules 12. 8: Use of Safety Stock SCREENSHOT 12-5C Plot of Total Cost versus Order Quantity for Brass Department Store 12-23 Cost curve for range 1 Cost curve for range 2 Cost curve for range 3 Minimum cost File: 12-5. xls, sheet: 12-5C In the Brass Department Store example, observe that the Q* values for discounts 2 and 3 are too low to be eligible for the discounted prices. They are, therefore, adjusted upward to 1,000 and 2,000, respectively.

With these adjusted Q* values, we find that the lowest total cost of $24,725 results when we use an order quantity of 1,000 units. If requested, ExcelModules will also draw a plot of the total cost for different values of Q. This graph, shown in Screenshot 12-5C, is drawn on a separate worksheet. 12. 8 USE OF SAFETY STOCK Safety stock is additional stock that is kept on hand. 2 If, for example, the safety stock for an item is 50 units, you are carrying an average of 50 units more of inventory during the year. When demand is unusually high, you dip into the safety stock instead of encountering a stockout.

Thus, the main purpose of safety stock is to avoid stockouts when the demand is higher than expected. Its use is shown in Figure 12. 7. Note that although stockouts can often be avoided by using safety stock, there is still a chance that they may occur. The demand may be so high that all the safety stock is used up, and thus there is still a stockout. Safety stock helps in avoiding stockouts. It is extra stock kept on hand. 2 Safety stock is used only when demand is uncertain, and models under uncertainty are generally much harder to deal with than models under certainty. 12-24

CHAPTER 12 Inventory Control Models FIGURE 12. 7 Use of Safety Stock Inventory on Hand Q Time Stockout Inventory on Hand Q + SS Safety Stock, SS 0 Units Stockout Is Avoided Time One of the best ways of maintaining a safety stock level is to use the ROP. This can be accomplished by adding the number of units of safety stock as a buffer to the reorder point. Recall from Equation 12-10 that Reorder point (ROP) = d ? L where d is the daily demand rate and L is the order lead time. With the inclusion of safety stock (SS), the reorder point becomes Safety stock is included in the ROP.

ROP = d ? L + SS (12-16) How to determine the correct amount of safety stock is the only remaining question. The answer to this question depends on whether we know the cost of a stockout. We discuss both of these situations next. 12. 8: Use of Safety Stock 12-25 IN ACTION Telephone Companies Analyze Price Quotations and Quantity Discounts One major inventory item needed by the Bell client companies is modular circuit boards, so managers of these firms inquired how they could purchase the boards from Bellcore under business volume discounts.

The result was a quantitative model, called the Procurement Decision Support System (PDSS), that determines the optimal ordering policy based on the most economical purchase of items under business volume discounts. PDSS was written to run on personal computers. The program allowed Bell client companies to move away from quantity discounts toward business volume discounts. What is the result of using business volume discounts? PDSS now controls inventory and products worth more than $600 million. The savings for Bell client companies have ranged from about $5 million to $15 million per year. Source: P. Katz, et al. Telephone Companies Analyze Price Quotations with Bellcore’s PDSS Software,” Interfaces 24 (January–February 1994): 50–63. In many cases, companies buy inventory supplies from several suppliers. This was the case with Bellcore, formed in 1984 to allow the regional Bell operating companies to share common resources. The operating companies, which included Ameritech, Bell Atlantic, BellSouth Telecommunications, NYNEX, Pacific Bell, Southwestern Bell, and US West are often referred to as Bell client companies. By pooling their resources, the Bell client companies have considerable power over their suppliers.

As a result, they decided to select suppliers of required raw materials and inventory based on the availability and amount of quantity discounts and business volume discounts. Whereas a traditional quantity discount is based on the amount of a particular inventory item that is ordered, a business volume discount is based on the total dollar value of all items purchased. With a business volume discount, the supplier typically discounts each item in an order by the same amount. Safety Stock with Known Stockout Costs When the EOQ is fixed and the ROP is used to place orders, the only time a stockout can occur is during the lead time.

Recall that the lead time is the time between when an order is placed and when it is received. In the procedure discussed here, it is necessary to know the probability of demand during lead time (DDLT) and the cost of a stockout. In what follows, we assume that DDLT follows a discrete probability distribution. This approach, however, can be easily modified when DDLT follows a continuous probability distribution. What factors should we include in computing the stockout cost per unit? In general, we should include all costs that are a direct or indirect result of a stockout.

For example, let us assume that if a stockout occurs, we lose that specific sale forever. Thus, if there is a profit margin of $1 per unit, we have lost this amount. Furthermore, we may end up losing future business from customers who are upset about the stockout. An estimate of this cost must also be included in the stockout cost. When we know the probability distribution of DDLT and the cost of a stockout, we can determine the safety stock level that minimizes the total cost. We illustrate this computation using an example. ABCO Example ABCO, Inc. , uses the EOQ model and ROP analysis (which we saw in sections 12. and 12. 5, respectively) to set its inventory policy. The company has determined that its optimal ROP is 50 (= d ? L) units, and the optimal number of orders per year is 6. ABCO’s DDLT is, however, not a constant. Instead, it follows the probability distribution shown in Table 12. 3. 3 Loss of goodwill must be included in stockout costs. Note that we have assumed that we already know the values of Q* and ROP. If this is not true, the values of Q*, ROP, and safety stock would have to be determined simultaneously. This requires a more complex solution. 3 12-26

CHAPTER 12 Inventory Control Models NUMBER OF UNITS 30 40 ROP > 50 60 70 PROBABILITY 0. 2 0. 2 0. 3 0. 2 0. 1 1. 0 TABLE 12. 3 Probability of Demand During Lead Time for ABCO, Inc. We use a decision making under risk approach here. Stockout and additional carrying costs will be zero when ROP = demand during lead time. If ROP < DDLT, total cost = total stockout cost. If ROP > DDLT, total cost = total additional inventory carrying cost. File: 12-6. xls, sheet: 12-6A File: 12-6. xls, sheet: 12-6B Because DDLT is uncertain, ABCO would like to find the revised ROP, including safety stock, which will minimize total expected cost.

The total expected cost is the sum of expected stockout cost and the expected carrying cost of the additional inventory. When we know the unit stockout cost and the probability distribution of DDLT, the inventory problem becomes a decision making under risk problem. (Refer to section 8. 5 in Chapter 8 for a discussion of such problems, if necessary. ) For ABCO, the decision alternatives are to use an ROP of 30 (alternative 1), 40 (alternative 2), 50 (alternative 3), 60 (alternative 4), or 70 (alternative 5) units. The outcomes are DDLT values of 30 (outcome 1), 40 (outcome 2), 50 (outcome 3), 60 (outcome 4), or 70 (outcome 5) units.

Determining the economic payoffs for any decision alternative and outcome combination involves a careful analysis of the stockout and additional carrying costs. Consider a situation in which the ROP equals the DDLT (say, 30 units each). This means that there will be no stockouts and no extra units on hand when the new order arrives. Thus, stockouts and additional carrying costs will be zero. In general, when the ROP equals the DDLT, total cost will be zero. Now consider what happens when the ROP is less than the DDLT. For example, say that ROP is 30 units and DDLT is 40 units.

In this case we will be 10 units short. The cost of this stockout situation is $2,400 (= 10 units short ? $40 per stockout ? 6 orders per year). Note that we have to multiply the stockout cost per unit and the number of units short by the number of orders per year (6, in this case) to determine annual expected stockout cost. Likewise, if the ROP is 30 units and the DDLT is 50 units, the stockout cost will be $4,800 (= 20 ? $40 ? 6), and so on. In general, when the ROP is less than the DDLT, the total cost is equal to the total stockout cost.

Finally, consider what happens when the ROP exceeds the DDLT. For example, say that ROP is 70 units and DDLT is 60 units. In this case, we will have 10 additional units on hand when the new inventory is received. If this situation continues during the year, we will have 10 additional units on hand, on average. The additional carrying cost is $50 (= 10 additional units ? $5 carrying cost per unit per year). Likewise, if the DDLT is 50 units, we will have 20 additional units on hand when the new inventory arrives, and the additional carrying cost will be $100 (= 20 ? $5).

In general, when the ROP is greater than the DDLT, total cost will be equal to the total additional carrying cost. Using the procedures described previously, we can easily set up a spreadsheet to compute the total cost for every alternative and state of nature combination. The formula view for this spreadsheet is shown in Screenshot 12-6A. The results of the analysis are shown in Screenshot 12-6B. The expected monetary values (EMV) in column G show that the best reorder point for ABCO is 70 units, with an expected total cost of $110. Recall that ABCO had determined its optimal ROP to be 50 units if DDLT was a constant.

Hence, the results in Screenshot 12-6B imply that due to the uncertain nature of DDLT, ABCO should carry a safety stock of 20 (= 70 – 50) units. 12. 8: Use of Safety Stock SCREENSHOT 12-6A Formula View of Safety Stock Computation for ABCO, Inc. 12-27 Cost = Stockout cost, if ROP < DDLT Cost = Additional holding cost, if ROP > DDLT Cost = 0, if ROP = DDLT Expected monetary value of each decision alternative SCREENSHOT 12-6B Safety Stock Computation for ABCO, Inc. Input data Probability of each DDLT value Best alternative is ROP = 70. Safety Stock with Unknown Stockout Costs

Determining stockout costs may be difficult or impossible. When stockout costs are not available or if they are not relevant, the preceding type of analysis cannot be used. Actually, there are many situations in which stockout costs are unknown or extremely difficult to determine. For example, let’s assume that you run a small bicycle shop that sells mopeds and bicycles with a one-year service warranty. Any adjustments made within the year are done at no charge to the customer. If the customer comes in for maintenance under the warranty, and you do not have the necessary part, what is the stockout cost?

It cannot be lost profit because the maintenance is done free of charge. Thus, the major stockout cost is the loss of goodwill. The customer may not buy another bicycle from your shop if you have a poor service record. In this situation, it could be very difficult to determine the stockout cost. In other cases, a stockout cost may simply not apply. What is the stockout cost for life-saving drugs in a hospital? The drugs may only cost $10 per bottle. Is 12-28 CHAPTER 12 Inventory Control Models the stockout cost $10? Is it $100 or $10,000? Perhaps the stockout cost should be $1 million.

What is the cost when a life may be lost as a result of not having the drug? In such cases, an alternative approach to determining safety stock levels is to use a service level. In general, a service level is the percentage of the time that you will have the item in stock. In other words, the probability of having a stockout is 1 minus the service level. That is, An alternative to determining safety stock is to use service level and the normal distribution. Service level = 1 – Probability of a stockout or Probability of a stockout = 1 – Service level (12-17)

To determine the safety stock level, it is only necessary to know the probability of DDLT and the desired service level. Here is an example of how the safety stock level can be determined when the DDLT follows a normal probability distribution. Hinsdale Company Example Hinsdale Company carries an item whose DDLT follows a normal distribution, with a mean of 350 units and a standard deviation of 10 units. Hinsdale wants to follow a policy that results in a service level of 95%. How much safety stock should Hinsdale maintain for this item? Figure 12. 8 may help you to visualize the example.

We use the properties of a standardized normal curve to get a Z value for an area under the normal curve of 0. 95 = (1 – 0. 05). Using the normal table in Appendix C on page 637, we find this Z value to be 1. 645. As shown in Figure 12. 8, Z is equal to (X – µ)/? , or SS/?. Hence, SS is equal to Z ? ?. That is, Hinsdale’s safety stock for a service level of 95% is (1. 645 ? 10) = 16. 45 units (which can be rounded off to 17 units, if necessary). We can calculate the safety stocks for different service levels in a similar fashion. Let’s assume that Hinsdale has a carrying cost of $1 per unit per year.

What is the carrying cost for service levels that range from 90% to 99. 99%? To compute this cost, we first compute the safety stock for each service level (as discussed earlier) and then multiply the safety stock by the unit carrying cost. The Z value, safety stock, and total carrying cost for different service levels for Hinsdale are summarized in Table 12. 4. A graph of the total carrying cost as a function of the service level is given in Figure 12. 9. We find the Z value for the desired service level. A safety stock level is determined for each service level. FIGURE 12. 8 Safety Stock and the Normal Distribution % Area of Normal Curve SS µ = 350 X =? µ = Mean Demand = 350 ? = Standard Deviation = 10 X = Mean Demand + Safety Stock SS = Safety Stock = X – µ Z = X–µ ? 12. 8: Use of Safety Stock TABLE 12. 4 Cost of Different Service Levels SERVICE LEVEL 90% 91% 92% 93% 94% 95% 96% 97% 98% 99% 99. 99% Z VALUE FROM NORMAL CURVE TABLE 1. 28 1. 34 1. 41 1. 48 1. 55 1. 65 1. 75 1. 88 2. 05 2. 33 3. 72 SAFETY STOCK (UNITS) 12. 8 13. 4 14. 1 14. 8 15. 5 16. 5 17. 5 18. 8 20. 5 23. 3 37. 2 12-29 CARRYING COST $12. 80 $13. 40 $14. 10 $14. 80 $15. 50 $16. 50 $17. 50 $18. 80 $20. 50 $23. 20 $37. 20 FIGURE 12. 9 ($)

Service Level versus Annual Carrying Costs 40 35 Inventory Carrying Costs ($) 30 25 20 15 10 90 91 92 93 94 95 96 97 98 99 99. 99 (%) Service Level (%) The relationship between service level and carrying cost is nonlinear. Note from Figure 12. 9 that the relationship between service level and carrying cost is nonlinear. As the service level increases, the carrying cost increases at an increasing rate. Indeed, at very high service levels, the carrying cost becomes very large. Therefore, as you are setting service levels, you should be aware of the additional carrying cost that you will encounter.

Although Figure 12. 9 was developed for a specific case, the general shape of the curve is the same for all service-level problems. 12-30 CHAPTER 12 Inventory Control Models File: 12-7. xls, sheet: 12-7A Using ExcelModules to Compute the Safety Stock We select the Safety Stock (Normal DDLT) option from the Inventory Models submenu in ExcelModules (refer to Screenshot 12-1A). The options for this procedure include the problem title and a box to specify whether we want a graph of carrying cost versus service level. After we specify these options, we get the worksheet shown in Screenshot 12-7A.

We now enter values for the mean DDLT (µ), standard deviation of DDLT (? ), service level desired, and carrying cost, Ch, in cells B6 to B9, respectively. The worksheet calculates and displays the following output measures: I Safety stock, SS (= Z ? ?), in cell B12 I Reorder point (= µ + Z ? ?), in cell B13 I Safety stock carrying cost (= Ch ? Z ? ?), in cell B15 File: 12-7. xls, sheet: 12-7B If requested, ExcelModules will draw a plot of the safety stock carrying cost for different values of the service level. This graph, shown in Screenshot 12-7B, is drawn on a separate worksheet.

As expected, the shape of this graph is the same as that shown in Figure 12. 9. SCREENSHOT 12-7A Safety Stock (Normal DDLT) Model for Hinsdale 95% is entered as 95 here. ROP = µ + SS Data for graph, generated and used by ExcelModules 12. 9: ABC Analysis SCREENSHOT 12-7B Plot of Safety Stock Cost versus Service Level for Hinsdale 12-31 Cost increases sharply at this level. Curve is nonlinear. IN ACTION Inventory Modeling at the San Miguel Corporation in the Phillippines ice cream balances ordering, carrying, and stockout costs while considering delivery frequency constraints and minimum order quantities.

Results showed that current safety stocks of 30–51 days could be cut in half for dairy and cheese curd. Even with the increased use of expensive airfreight, SMC saved $170,000 per year through the new policy. Another SMC product, beer, consists of three major ingredients: malt, hops, and chemicals. Because these ingredients are characterized by low expediting costs and high unit costs, inventory modeling pointed to optimal policies that reduced safety stock levels, saving another $180,000 per year. Source: E. Del Rosario. “Logistical Nightmare,” OR/MS Today (April 1999): 44–45. Copyright © 1999. Reprinted with permission.

In a typical manufacturing firm, inventories comprise a big part of assets. At the San Miguel Corporation (SMC), which produces and distributes more than 300 products to every corner of the Philippine archipelago, raw material accounts for about 10% of total assets. The significant amount of money tied up in inventory encouraged the company’s Operations Research Department to develop a series of cost-minimizing inventory models. One major SMC product, ice cream, uses dairy and cheese curd imported from Australia, New Zealand, and Europe. The normal mode of delivery is sea, and delivery frequencies are limited by supplier schedules.

Stockouts, however, are avoidable through airfreight expediting. SMC’s inventory model for 12. 9 ABC ANALYSIS So far, we have shown how to develop inventory policies using quantitative decision models. There are, however, some very practical issues, such as ABC analysis, that should be incorporated into inventory decisions. ABC analysis recognizes the fact 12-32 CHAPTER 12 Inventory Control Models ARE QUANTITATIVE INVENTORY CONTROL TECHNIQUES USED? Yes In some cases No TABLE 12. 5 Summary of ABC Analysis INVENTORY GROUP A B C DOLLAR USAGE 70% 20% 10% INVENTORY ITEMS 10% 20% 70% The items in the A group are critical.

The B group items are important but not critical. The C group items are not as important as the others in terms of annual dollar value. that some inventory items are more important than others. The purpose of this analysis is to divide all of a company’s inventory items into three groups: A, B, and C. Then, depending on the group, we decide how the inventory levels should be controlled. A brief description of each group follows, with general guidelines as to which items are A, B, and C. The inventory items in the A group are critical to the functioning of the company. As a result, their inventory levels must be closely monitored.

These items typically make up more than 70% of the company’s business in monetary value but only about 10% of all inventory items. That is, a few inventory items are very important to the company. As a result, the inventory control techniques discussed in this chapter should be used where appropriate for every item in the A group (see Table 12. 5). The items in the B group are important to the firm but not critical. Thus, it may not be necessary to monitor all these items closely. These items typically represent about 20% of the company’s business in monetary value and constitute about 20% of the items in inventory.

Quantitative inventory models should be used only on some of the B items. The cost of implementing and using these models must be carefully balanced with the benefits of better inventory control. Usually, less than half of the B group items are controlled through the use of inventory control models. The items in the C group are not as important to the operation of the company. These items typically represent only about 10% of the company’s business in monetary value but may constitute 70% of the items in inventory. Group C could include inexpensive items such as bolts, washers, screws, and so on.

They are usually not controlled using inventory control models because the cost of implementing and using such models would far exceed the value gained. We illustrate the use of ABC analysis using the example of Silicon Chips, Inc. Silicon Chips, Inc. , Example Silicon Chips, Inc. , maker of super-fast DRAM chips, has organized its 10 inventory items on an annual dollar-volume basis. Table 12. 6 shows the items (identified by the item number and part number), their annual demands, and unit costs. How should the company classify these items into groups A, B, and C?

As discussed next, we use the worksheet provided in ExcelModules to answer this question. Using ExcelModules for ABC Analysis We select the ABC Analysis option from the Inventory Models submenu in ExcelModules (refer to Screenshot 12-1A). The options for this procedure include the problem title and boxes to specify the number and names of the items we want to classify. After we specify these options for the Silicon Chips example, we get the worksheet shown in Screenshot 12-8. We now enter the volume and unit cost for each item in cells B7:C16 of this worksheet. File: 12-8. xls TABLE 12. Inventory Data for Silicon Chips, Inc. ITEM NUMBER Item 1 Item 2 Item 3 Item 4 Item 5 Item 6 Item 7 Item 8 Item 9 Item 10 PART NUMBER 01036 01307 10286 10500 10572 10867 11526 12572 12760 14075 ANNUAL VOLUME (UNITS) 100 1,200 1,000 1,000 250 350 500 600 1,550 2,000 UNIT COST $ $ 8. 50 0. 42 $ 90. 00 $ 12. 50 $ 0. 60 $ 42. 86 $154. 00 $ 14. 17 $ 17. 00 $ 0. 60 SCREENSHOT 12-8 ABC Analysis for Silicon Chips, Inc. Click this button after entering all input data. Input data A B 2 items 3 items C 5 items Items are sorted in descending order of percentage $ volume. Sorted values of percentage $ volume 2-34 CHAPTER 12 Inventory Control Models IN ACTION Inventory Modeling at Teradyne Input data to the inventory models included actual planned inventory levels, holding costs, observed demand rates, and estimated lead times. The outputs included service levels and a prediction of the expected number of late part shipments. The first inventory model showed that Teradyne could reduce late shipments by over 90% with just a 3% increase in inventory investment. The second model showed that the company could reduce inventory by 37% while improving customer service levels by 4%.

Teradyne, a huge manufacturer of electronic testing equipment for semiconductor plants worldwide, recently asked the Wharton School of Business to evaluate its global inventory parts system. Teradyne’s system is complex because it stocks over 10,000 parts with a wide variety of prices (from a few dollars to $10,000) because its customers are dispersed all over the world, and because customers demand immediate response when a part is needed. The professors selected two basic inventory models they felt could be used to improve the current inventory system effectively.

An important consideration in using basic inventory models is their simplicity, which improved the professors’ communication with Teradyne executives. In the field of modeling, it is very important for managers who depend on the models to thoroughly understand the underlying processes and a model’s limitations. Source: M. A. Cohen, Y. Zheng, and Y. Wang. “Identifying Opportunities for Improving Teradyne’s Service Parts Logistics System,” Interfaces (July–August 1999): 1–18. Items are sorted in descending order of percentage dollar volume.

When we enter the input data, the worksheet computes the dollar volume and percentage dollar volume (based on total dollar volume) for each item. These values are shown in cells E7:F16 of Screenshot 12-8. After entering the data for all items, we click the Analyze button. The worksheet now sorts the items, in descending order of percentage dollar volume. These values are shown in descending order in cells F21:F30 of Screenshot 12-8. The sorted results for the Silicon Chips, Inc. , problem are shown in cells A21:C30 of Screenshot 12-8. Items 3 and 7, which constitute only 20% (= 2/10) of the total number of items, account for 71. 7% of the total dollar volume of all items. These two items should therefore be classified as group A items. Items 9, 6, and 4, which constitute 30% (= 3/10) of the total number of items, account for 23. 20% (= 95. 17 – 71. 97) of the total dollar volume of all items. These three items should therefore be classified as group B items. The remaining items constitute 50% (= 5/10) of the total number of items. However, they account for only 4. 83% (= 100 – 95. 17) of the total dollar volume of all items. These five items should therefore be classified as group C items. SUMMARY

This chapter presents several inventory models and discusses how we can use ExcelModules to analyze these models. The focus of all models is to answer the same two primary questions in inventory planning: (1) how much to order and (2) when to order. The basic EOQ inventory model makes a number of assumptions: (1) known and constant demand and lead times, (2) instantaneous receipt of inventory, (3) no quantity discounts, (4) no stockouts or shortages, and (5) the only variable costs are ordering costs and carrying costs. If these assumptions are valid, the EOQ inventory model provides optimal solutions.

If these assumptions do not hold, more complex models are needed. For such cases, the economic production quantity and quantity discount models are necessary. We also discuss the computation of safety stocks when demand during lead time is unknown for two cases: (1) cost of stockout is known and (2) cost of stockout is unknown. Finally, we present ABC analysis to determine how inventory items should be classified based on their importance and value. For all models discussed in this chapter, we show how Excel worksheets can be used to perform the computations. Solved Problems 12-35 GLOSSARY

ABC Analysis. An analysis that divides inventory into three groups: Group A is more important than group B, which is more important than group C. Average Inventory. The average inventory on hand. Computed as (Maximum inventory + Minimum inventory)/2. Carrying Cost. The cost of holding one unit of an item in inventory for one period (typically a year). Also called holding cost. Demand During Lead Time (DDLT). The demand for an item during the lead time, between order placement and order receipt. Economic Order Quantity (EOQ). The amount of inventory ordered that will minimize the total inventory cost.

It is also called the optimal order quantity, or Q*. Economic Production Quantity (EPQ) Model. An inventory model in which the instantaneous receipt assumption has been eliminated. The inventory build-up, therefore, occurs over a period of time. Instantaneous Inventory Receipt. A system in which inventory is received or obtained at one point in time and not over a period of time. Lead Time. The time it takes to receive an order after it is placed. Ordering Cost. The cost of placing an order. Purchase Cost. The cost of purchasing one unit of an inventory item. Quantity Discount.

The cost per unit when large orders of an inventory item are placed. Reorder Point (ROP). The number of units on hand when an order for more inventory is placed. Safety Stock. Extra inventory that is used to help avoid stockouts. Safety Stock with Known Stockout Costs. An inventory model in which the probability distribution of DDLT and the unit stockout cost are known. Safety Stock with Unknown Stockout Costs. An inventory model in which the probability distribution of DDLT is known. The stockout cost is, however, not known. Service Level. The chance, expressed as a percentage, that there will not be a stockout.

Service level = 1 – Probability of a stockout. Setup Cost. The cost to set up the manufacturing or production process. Stockout. A situation that occurs when there is no inventory on hand. Total Cost. The sum of the total ordering, total carrying, and total purchasing costs. SOLVED PROBLEMS Solved Problem 12-1 Patterson Electronics supplies microcomputer circuitry to a company that incorporates microprocessors into refrigerators and other home appliances. Currently, Patterson orders a particular component in batches of 300 units from one of its suppliers.

The annual demand for this component is 2,000. a. If the carrying cost is estimated at $1 per unit per year, what would the ordering cost have to be to make the order quantity optimal? b. If the ordering cost is estimated to be $50 per order, what would the carrying cost have to be to make the order quantity optimal? Solution a. Recall from Equation 12-8 that the ordering cost can be computed as Co = Q 2 ? Ch /(2 D) In Patterson’s case, D = 2,000, Q = 300, and Ch = $1. Substituting these values, we get an ordering cost, Co, of $22. 50 per order. b.

Recall from Equation 12-9 that the carrying cost can be computed as Ch = 2 DCo /Q 2 In Patterson’s case, D = 2,000, Q = 300, and Co = $50. Substituting these values, we get a carrying cost, Ch, of $2. 22 per unit per year. 12-36 CHAPTER 12 Inventory Control Models SCREENSHOT 12-9 EPQ Model for Flemming Accessories Input data EPQ is 1632 units. Total cost is $68,740. 56. Solved Problem 12-2 Flemming Accessories produces paper slicers used in offices and in art stores. The minislicer has been one of its most popular items: Annual demand is 6,750 units. Kristen Flemming, owner of the firm,

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