Pom Lecture (31)

Unit 2 Management of Conversion System Chapter 10: Inventory Management Lesson 30 - Quantity discount models Learning Objectives After reading this lesson you would be able to understand Role of quantity discount in inventory decisions Quantity discount model Continuous review systems Periodic review systems Good Morning students, today we are going to introduce the concept of what is known as quantity discount model and the various types of inventory review systems. Well friends, we started off with the concept of inventory management during the previous lecture and were able to cover quite a bit of ground. Today we shall start our discussions with Quantity discount models.

To increase sales, many companies offer quantity discounts to their customers. A quantity discount is simply a reduced price (P) for the item when it is purchased in larger quantities. It is common to have a discount schedule with several discounts for large orders. As always, management must decide when and how much to order. But with quantity discounts, how does the operations manager make these decisions? As with other inventory model, the overall objective will be to minimize the total cost. Placing an order for the quantity with the greatest discount price might not minimize the total inventory cost. As the discount quantity goes up, the product cost goes down, but the holding cost increases because the orders are large. Thus, the major trade-off when considering quantity discounts is between the reduced product

cost and the increased holding cost. When we include the cost of the product, the equation for the total annual inventory cost becomes: Total cost = Setup cost + Holding cost + Product cost Or Tc = (D / Q)S + (QH / 2) + PD Where Q = Quantity ordered D = Annual demand in units S = Ordering or setup cost per order or per setup P = Price per unit H = Holding cost per unit per year Now, we have to determine the quantity that will minimize the total annual inventory cost. Because there are several discounts, this process involves four steps:

Figure 30.1 Total cost curve for the quantity discount model

Step 1. For each discount, calculate a value for Q*, using the following equation: Q* = √(2DS / IP) You should note that the holding cost is IP instead of H. Because the price of the item is a factor in annual holding cost, we cannot 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 (I) as a percentage of unit price (P) instead of as a constant cost per unit per year, H. Step 2. For any discount, if the order quantity is too low to qualify for the discount, adjust the order quantity upward to the lowest quantity that will qualify for the discount. This reasoning may not be obvious. If the order quantity is below the quantity range that will qualify for a discount, a quantity within this range may still result in the lowest total cost. Step 3. Using the total cost equation above, compute a total cost for every Q* determined in steps 1 and 2. If you had to adjust Q* upward because it was below the allowable quantity range, make sure to use the adjusted value for Q*. Step 4. Select that Q* that has the lowest total cost as computed in step 3. It will be the quantity that will minimize the total inventory cost. It’s now the right time to consider the practical applicability of the aforesaid concept. Let us see how this procedure can be applied with an example.

Example Toyseras stocks toy racecars. Recently, they have been given a quantity discount schedule for the cars. This quantity schedule was shown in Table 10.1. Thus, the normal cost for the toy racecars is Rs5. For orders between 1000 and 1999 units, the unit cost is Rs4.80; and for orders of 2000 or more units, the unit cost is Rs4.75. Furthermore, the ordering cost is Rs49 per order, the annual demand is 5000 race cars, and the inventory carrying charge as a percentage of cost, I, is 20% or .2. What order quantity will minimize the total inventory cost? Table 10.1 Quantity discount schedule Discount

Discount

Discount

Discount

Number

Quantity

(%)

Price (P)

1

0 to 999

0

Rs5.00

2

1000 to

4

Rs4.80

3

1999

5

Rs4.75

2000 and over Friends, do it yourself first and then tally with the solution given here. Solution The first step is to compute Q* for every discount in Table 10.1. This is done as follows: Q*1 = √(2 (5000)(49) / ( (.2)(5.00) ) = 700 cars order Q*2 = √(2 (5000)(49) / ( (.2)(4.80) ) = 714 cars order Q*3 = √(2 (5000)(49) / ( (.2)(4.75) ) = 718 cars order

The second step is to adjust upward those values of Q* that are below the allowable discount range. Since Q*1 is between 0 and 999, it does not have to be adjusted. Q*2 is below the allowable range of 1000 to 1999, and therefore, it must be adjusted to 1000 units. The same is true for Q*3. It must be adjusted to 2000 units. After this step, the following order quantities must be tested in the total cost equation: Q*1 = 700 Q*2 = 1000 – adjusted Q*3 = 2000 - adjusted The third step is to use the total cost equation and compute a total cost for each of the order quantities. This is accomplished with the aid of Table 10.2 Table 10.2 Total cost computations for Toyseras’ discount store Discount Unit

Order

Number

Quantity Product

Price

Annual Cost

Annual

Annual

Total

Ordering Holding Cost

Cost

1

Rs5.00

700

Rs25,000 Rs350

Rs350

Rs25,700

2

Rs4.80

1000

Rs24,000 Rs245

Rs480

Rs24,725

3

Rs4.75

2000

Rs23,750 Rs122.50 Rs950

Rs24,822.50

The fourth step is to select that an order quantity with the lowest total cost. Looking at Table 10.2, you can see that an order quantity of 1000 toy race cars will minimize the total cost. It should be recognized, however; that the total cost for ordering 2000 cars is only slightly greater than the total cost for ordering 1000 cars.

Let us now move over to what is known as a Continuous review (Q) System Continuous review (Q) System A continuous review (Q) system, sometimes called a reorder point (ROP) system or fixed order-quantity system, tracks the remaining inventory of an item each time a withdrawal is made to determine whether it is time to reorder. In practice, these reviews are done frequently (e.g,, daily) and often continuously (after each withdrawal). The advent of computers and electronic cash registers linked to inventory records has made continuous reviews easy. At each review a decision is made about an item’s inventory position. If it is judged to be too low, the system triggers a new order. The inventory position (IP) measures the item’s ability to satisfy future demand. It includes scheduled receipts (SR), which are orders that have been placed but not yet received, plus on-hand inventory (OH) minus backorders (BO). Sometimes scheduled receipts are called open orders. More specifically, Inventory position = On-hand inventory + Scheduled receipts – Backorders IP = OH + SR – BO When the inventory position reaches a predetermined minimum level, called the reorder point (R), a fixed quantity Q of the item is ordered. In a continuous review system, although the order quantity Q is fixed, the time between orders can vary. Hence, Q can be based on the EOQ, a price break quantity (the minimum lot size that qualifies for a quantity discount), a container size (such as truckload), or some other quantity selected by management.

We now focus our attention to:Two-Bin system The concept of a Q system can be incorporated in a visual system, that is, a system that allows employees to place orders when inventory visibly reaches a certain marker. Visual systems are easy to administer because records are not kept on the current inventory position. The historical usage rate can simply be reconstructed from past purchase orders. Visual systems are intended for use with low-volume items that have a steady demand, such as nuts and bolts or office supplies. Overstocking is common, but the extra inventory holding cost is minimal because the items have relatively little value. A visual system version of the Q system is the two-bin system in which an item’s inventory is stored at two different locations. Inventory is first withdrawn from one bin. If the first bin is empty, the second bin provides backup to cover demand until a replenishment order arrives. An empty first bin signals the need to place a new order. Premade order forms placed near the bins let workers send one to purchasing or even directly to the supplier. When the new order arrives, the second bin is restored to its normal level and the rest is put in the first bin. The two-bin system operates like a Q system, with the normal level in the second bin being the reorder point R. The system also may be implemented with just one bin by marking the bin at the reorder point level. Now that we have understood the concept of Continuous review (Q) System, let us cross over to:Periodic review system An alternative inventory control system is the periodic review (P) system, sometimes called a fixed interval reorder system or periodic reorder system, in which an item’s inventory position is reviewed periodically rather than continuously. Such a system

can simplify delivery scheduling because it establishes a routine. A new order is always placed at the end of each review, and the time between orders (TBO) is fixed at P. Demand is a random variable, so total demand between reviews varies. In a P system, the lot size, Q, may change from one order to the next, but the time between orders is fixed. An example of a periodic review system is that of a soft-drink supplier making weekly rounds of grocery stores. Each week, the supplier reviews the store’s inventory of soft drinks and restocks the store with enough items to meet demand and safety stock requirements until the next week. Four of the original EOQ assumptions are maintained: that there are no constraints on the size of the lot, that the relevant costs are holding and ordering costs, that decisions for one item are independent of decisions for other items, and that there is no uncertainty in lead times or supply. However, demand uncertainty is again allowed for. Figure 10.4 shows the periodic review system under these assumptions. The downward-sloping line again represents on-hand inventory. When the predetermined time, P, has elapsed since the last review, an order is placed to bring the inventory position, represented by the dashed line, up to the target inventory level, T. The lot size for the first review is Q1, or the difference between inventory position IP1 and T. As with the continuous review system, IP and OH differ only during the lead time. When the order arrives, at the end of the lead time, OH and IP again are identical. Figure 10.4 shows that lot sizes vary from one order cycle to the next. Because the inventory position is lower at the second review, a greater quantity is needed to achieve an inventory level of T.

Figure 30.2 P system when demand is uncertain Time to supplement our understanding with an example. Example 10.6 There is a backorder of five 36-inch colour TV sets at a distribution center. There is no on-hand inventory, and now is the time to review. How much should be reordered if T = 400 and there are no scheduled receipts? Solution IP = OH + SR – BO = 0 + 0 – 5 = -5 sets T – IP = 400 – (-5) = 405 sets That is, 405 sets must be ordered to bring the inventory position up to T sets. Let us quickly take up the concept of:-

Selecting the time between reviews To run a P system, managers must make two decisions: • The length of time between reviews, P • The target inventory level, T The time between reviews, P, can be any convenient interval, such as each Friday or every other Friday. Another option is to base P on the cost trade-offs of the EOQ. In other words, P can be set equal to the average time between orders for the economic order quantity. Because demand is variable, some orders will be larger than the EOQ and some will be smaller. However, over an extended period of time, the average lot size should equal the EOQ. If other models are used to determine the lot size, we divide the lot size chosen by the annual demand, D, and use this ratio as P. It will be expressed as the fraction of a year between orders, which can be converted into months, weeks, or days as needed. A question coming up now:Could you please tell us something about SELECTING THE TARGET INVENTORY LEVEL

SELECTING THE TARGET INVENTORY LEVEL Now let us consider how to calculate the target inventory level, T. Figure 10.4 reveals that an order must be large enough to make the inventory position, IP, last beyond the next review, which is P time periods away. The checker must wait P periods to revise, correct, and reestablish the inventory position. Then, a new order is placed, but it does not arrive until after the lead time, L. Therefore, as Figure 10.4 shows, a protection interval of P + L periods is needed, or the time interval for which inventory must be planned when each new order is placed. A fundamental difference between the Q and P systems is the length of time needed

for stockout protection. A Q system needs stockout protection only during the lead time because orders can be placed as soon as they are needed and will be received L periods later. A P system, however, needs stockout protection for the longer P + L protection interval because orders are placed only at fixed intervals and the inventory isn't checked until the next designated review time. As with the Q system, we need to develop the appropriate distribution of demand during the protection interval to specify the system fully. In a P system, we must develop the distribution of demand for P + L time periods. The target inventory level T must equal the expected demand during the protection interval of P + L periods, plus enough safety stock to protect against demand uncertainty over this same protection interval. We use the same statistical assumptions that we made for the Q system. Thus, the average demand during the protection interval is d(P + L), or T = d(P + L) + (Safety stock for protection interval) We compute safety stock for a P system much as we did for the Q system. However, the safety stock must cover demand uncertainty for a longer period of time. When using a normal probability distribution, we multiply the desired standard deviations to implement the cycle-service level, z, by the standard deviation of demand during the protection interval, σP+L. The value of z is the same as for a Q system with the same cycle-service level. Thus, Safety stock = ZσP+ L

Based on our earlier logic for calculating σL we know that the standard deviation of the distribution of demand during the protection interval is

σP+L = σt √P+L

Because a P system requires safety stock to cover demand uncertainty over a longer time period than a Q system, a P system requires more safety stock; that is, σP+L exceeds σL. Hence, to gain the convenience of a P system requires that overall inventory levels be somewhat higher than those for a Q system. You got it. Good. Let’s now learn to calculate total p system costs. CALCULATING TOTAL P SYSTEM COSTS. The total costs for the P system are the sum of the same three cost elements as for the Q system. The differences are in the calculation of the order quantity and the safety stock. Referring to Figure 10.4, the average order quantity will be the average consumption of inventory during the P periods between orders. Consequently, Q = dP. Total costs for the P system are C = ( dp /2 ) H + ( D / dp ) S + H z σ P+L

POM in Practice 10.1 shows how Hewlett-Packard implemented a periodic review system for many of their business units. The next logical question should be:What exactly is a SINGLE-BIN SYSTEM SINGLE-BIN SYSTEM The concept of a P system can be translated into a simple visual system of inventory control. In the single-bin system, a maximum level is marked on the storage shelf or bin on a measuring rod, and the inventory is brought up to the mark periodically-say, once a week. The single bin may be, for example, a gasoline storage tank at a service station or a storage bin for small parts at a manufacturing plant. Now, I would request all of you to focus on:-

POM in practice 10.1 – Implementing a Periodic Review Inventory System at Hewlett-Packard Here, we go. Hewlett-Packard manufactures computers, accessories, and a wide variety of instrumentation devices in more than 100 separate businesses, each responsible for its own product designing, marketing. and manufacturing processes as well as the required inventories to service its customers. At most HP businesses, inventory-driven costs (which include currency devaluation, obsolescence. price protection, and financing) are now the biggest control lever that the manufacturing organization has on business performance, measured in terms of return on assets or economic value added. Inventory is a major cost driver and the most variable element on the balance sheet Most of HP's business units were inefficient, carrying more inventory than needed in order to achieve a desired level of product delivery performance. They often used simplified approaches such as ABC analysis to determine their safety stocks for independent demand items, ignoring supply or demand uncertainty, part commonality, desired part availability, or cost The solution was to develop a periodic review system that used part availability targets and included as many uncertainties as possible. The system, although in principle similar to the P system discussed in this chapter, uses complex equations to determine the review interval and target inventory parameters. The complexity arises from considering uncertainties in supply as well as demand in the determination of the safety stocks. Even though the system could be shown to reduce inventories and improve customer

service, no benefits would be realized until the planning and procurement staff actually used it Since each business unit had some unique characteristics, the results had to be easily understandable and credible, and the system had to be easily configurable to each situation. Consequently, HP developed a software wizard that allows the user to enter product data and costs in a friendly environment, develops the equations for the periodic review system, and then translates the results to the user's format requirements. The wizard is programmed in Excel, which allows users access to all of Excel's functionality for conducting their own analyses. The periodic review system and the software wizard have been very successful. At HP's Integrated Circuit Manufacturing Division, for example planners cut inventories by $1.6 million while simultaneously improving on-time delivery performance from 93 percent to 97 percent Other benefits included less expediting, fewer disagreements about operating policy, and more control of the production system. The system is used across a wide variety of product lines and geographies worldwide. HP believes that without exception, the product lines now have more efficient operations. Source: Operations Management Strategy and Analysis (L. E. Krajewski and L. P. Ritzman) Prentice Hall You know I’m not very much for comparing people (inferiority complex? ), however at this point in our discussion, I feel it would be profitable to understand the:-

COMPARATIVE ADVANTAGES OF THE Q AND P SYSTEMS Neither the Q nor P system is best for all situations. Three P-system advantages must

be balanced against three Q-system advantages. The advantages of one system are implicitly disadvantages of the other one. The primary advantages of P systems are the following: 1. Administration of the system is convenient because replenishments are made at fixed intervals. Employees can regularly set aside a day or part of a day to concentrate on this particular task. Fixed replenishment intervals also allow for standardized pickup and delivery times. 2. Orders for multiple items from the same supplier may be combined into a single purchase order. This approach reduces ordering and transportation costs and may result in a price break from the supplier. 1. The inventory position, IP, needs to be known only when a review is made (not continuously, as in a Q system). However, this advantage is moot for firms using computerized record-keeping systems, in which a transaction is reported upon each receipt or withdrawal. When inventory records are always current, . the system is called a perpetual inventory system. Let’s see the main:Advantages of Q systems The primary advantages of Q systems are the following: 1. The review frequency of each item may be individualized. Tailoring the review frequency to the item can reduce total ordering and holding costs. 2. Fixed lot sizes, if large enough, may result in quantity discounts. Physical limitations such as truckload capacities, materials handling methods, and furnace capacities also may require a fixed lot size. 3. Lower safety stocks result in savings. In conclusion, the choice between Q and P systems is not clear cut. Which one is better depends on the relative importance of its advantages in various situations.

Management must weigh each alternative carefully in selecting the best system. It’s now the turn of:-

HYBRID SYSTEMS Various hybrid inventory control systems merge some but not all the features of the P and Q systems; we briefly examine two such systems: optional replenishment and base stock. You have any idea of OPTIONAL REPLENISHMENT SYSTEM? Quite an important concept. You decide. Here goes.

OPTIONAL REPLENISHMENT SYSTEM Sometimes called the optional review, min-max, or (s, S) system, the optional replenishment system is much like the P system. It is used to review the inventory position at fixed time intervals and, if the position has dropped to (or below) a predetermined level, to place a variable-sized order to cover expected needs. The new order is large enough to bring the inventory position up to a target inventory, similar to T for the P system. However, orders are not placed after a review unless the inventory position has dropped to the predetermined minimum level. The minimum level acts as reorder point R does in a Q system. If the target is 100 and the minimum level is 60, the minimum order size is 40 (or 100 - 60). The optional review system avoids continuous reviews and so is particularly attractive when both review and ordering costs are significant. The next in line is:BASE-STOCK SYSTEM

In its simplest form, the base-stock system issues a replenishment order, Q, each time a withdrawal is made, for the same amount as the withdrawal. This one-for-one replacement policy maintains the inventory position at a base-stock level equal to expected demand during the lead time plus safety stock. The base-stock level, therefore, is equivalent to the reorder point in a Q system. However, order quantities now vary to keep the inventory position at R at all times. Because this position is the lowest IP possible that will maintain a specified service level, the base-stock system may be used to minimize cycle inventory. More orders are placed, but each is smaller. This system is appropriate for very expensive items, such as replacement engines for jet airplanes. No more inventory is held than the maximum demand expected until a replacement order can be received. The base-stock system is used in just-in-time systems. With that, we have come to the end of today’s discussions. I hope it has been an enriching and satisfying experience. Points to ponder