Homework 11

272 Chapter 11 • Managing Economies of Scale in a Supply Chain: Cycle Inventory

Time t

Q

Inventory

FIGURE 11-1 Inventory Profile of Jeans at Jean-Mart

In the rest of this chapter, we use the following notation:

Q: Quantity in a lot or batch size

D: Demand per unit time

Here, we ignore the impact of demand variability and assume that demand is stable. In Chapter 12, we introduce demand variability and its impact on safety inventory.

Let us consider the cycle inventory of jeans at Jean-Mart, a department store. The demand for jeans is relatively stable at D ! 100 pairs of jeans per day. The store manager at Jean-Mart currently purchases in lots of Q ! 1,000 pairs. The inventory profile of jeans at Jean-Mart is a plot depicting the level of inventory over time, as shown in Figure 11-1.

Because purchases are in lots of Q ! 1,000 units, whereas demand is only D ! 100 units per day, it takes 10 days for an entire lot to be sold. Over these 10 days, the inventory of jeans at Jean-Mart declines steadily from 1,000 units (when the lot arrives) to 0 (when the last pair is sold). This sequence of a lot arriving and demand depleting inventory until another lot arrives repeats itself every 10 days, as shown in the inventory profile in Figure 11-1.

When demand is steady, cycle inventory and lot size are related as follows:

(11.1)

For a lot size of 1,000 units, Jean-Mart carries a cycle inventory of Q/2 ! 500 pairs of jeans. From Equation 11.1, we see that cycle inventory is proportional to the lot size. A supply chain in which stages produce or purchase in larger lots has more cycle inventory than a supply chain in which stages produce and purchase in smaller lots. For example, if a competing department store with the same demand purchases in lot sizes of 200 pairs of jeans, it will carry a cycle inventory of only 100 pairs of jeans.

Lot sizes and cycle inventory also influence the flow time of material within the supply chain. Recall from Little’s law (Equation 3.1) that

For any supply chain, average flow rate equals demand. We thus have

For lot sizes of 1,000 pairs of jeans and daily demand of 100 pairs of jeans, we obtain

Average flow time resulting from cycle inventory = Q

2D =

1,000 200

= 5 days

Average flow time resulting from cycle inventory = cycle inventory

demand =

Q 2D

Average flow time = average inventory average flow rate

Cycle inventory = lot size

2 = Q 2

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The purchasing manager makes the lot-sizing decision to minimize the total cost the store incurs. He or she must consider three costs when deciding on the lot size:

• Annual material cost • Annual ordering cost • Annual holding cost

Because purchase price is independent of lot size, we have

The number of orders must suffice to meet the annual demand D. Given a lot size of Q, we thus have

(11.3)

Because an order cost of S is incurred for each order placed, we infer that

(11.4)

Given a lot size of Q, we have an average inventory of Q/2. The annual holding cost is thus the cost of holding Q/2 units in inventory for one year and is given as

The total annual cost, TC, is the sum of all three costs and is given as

Figure 11-2 shows the variation in different costs as the lot size is changed. Observe that the annual holding cost increases with an increase in lot size. In contrast, the annual ordering cost declines with an increase in lot size. The material cost is independent of lot size because we have assumed the price to be fixed. The total annual cost thus first declines and then increases with an increase in lot size.

From the perspective of the manager at Best Buy, the optimal lot size is one that minimizes the total cost to Best Buy. It is obtained by taking the first derivative of the total cost with respect to Q and setting it equal to 0 (see Appendix 11A at the end of this chapter). The optimal lot size

Total annual cost, TC = CD + aD Q bS + aQ

2 bhC

Annual holding cost = aQ 2 bH = aQ

2 bhC

Annual ordering cost = aD Q bS

Number of orders per year = D Q

Annual material cost = CD

Cost

Total Cost

Holding Cost

Ordering Cost

Material Cost

Lot Size FIGURE 11-2 Effect of Lot Size on Costs at Best Buy

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11.4 ECONOMIES OF SCALE TO EXPLOIT QUANTITY DISCOUNTS

We now consider pricing schedules that encourage buyers to purchase in large lots. There are many instances in business-to-business transactions in which the pricing schedule displays economies of scale, with prices decreasing as lot size increases. A discount is lot size–based if the pricing schedule offers discounts based on the quantity ordered in a single lot. A discount is volume based if the discount is based on the total quantity purchased over a given period, regardless of the number of lots purchased over that period. In this section, we will see that lot size–based quantity discounts tend to increase the lot size and cycle inventory in a supply chain. Two commonly used lot size–based discount schemes are

• All unit quantity discounts • Marginal unit quantity discount or multi-block tariffs

In order to investigate the impact of such quantity discounts on the supply chain, we must answer the following two basic questions:

1. Given a pricing schedule with quantity discounts, what is the optimal purchasing decision for a buyer seeking to maximize profits? How does this decision affect the supply chain in terms of lot sizes, cycle inventories, and flow times?

2. Under what conditions should a supplier offer quantity discounts? What are appropriate pricing schedules that a supplier seeking to maximize profits should offer?

We start by studying the optimal response of a retailer (the buyer) when faced with either of the two lot size–based discount schemes offered by a manufacturer (the supplier). The retailer’s objective is to select lot sizes to minimize the total annual material, order, and holding costs. Next, we evaluate the optimal lot size in the case of all unit quantity discounts.

All Unit Quantity Discounts

In all unit quantity discounts, the pricing schedule contains specified break points q0, q1, . . . , qr, where q0 ! 0. If an order placed is at least as large as qi but smaller than qi+1, each unit is obtained at a cost of Ci. In general, the unit cost decreases as the quantity ordered increases; that is,

For all unit discounts, the average unit cost varies with the quantity ordered, as shown in Figure 11-3. The retailer’s objective is to decide on lot sizes to maximize profits or, equivalently, to minimize the sum of material, order, and holding costs. The solution procedure eval- uates the optimal lot size for each price and picks the lot size that minimizes the overall cost.

Step 1: Evaluate the optimal lot size for each price as follows:

(11.10)Qi = C2DShCiCi,0 … i … r C0 Ú C1 Ú Á Ú Cr.

Quantity Purchased

C0

0 q1 q2 q3

C1

C2

A ve

ra ge

C os

t p er

U ni

t P ur

ch as

ed

FIGURE 11-3 Average Unit Cost with All Unit Quantity Discounts

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In Step 3, we obtain the total costs using Equation 11.11 as follows:

Observe that the lowest total cost is for i = 2. Thus, it is optimal for DO to order bottles per lot and obtain the discount price of $2.92 per bottle.

If the manufacturer in Example 11-7 sold all bottles for $3, it would be optimal for DO to order in lots of 6,324 bottles. The quantity discount is an incentive for DO to order in larger lots of 10,000 bottles, raising both the cycle inventory and the flow time. The impact of the discount is further magnified if DO works hard to reduce its fixed ordering cost to S = $4. Then, the optimal lot size in the absence of a discount is 1,265 bottles. In the presence of the all unit quantity discount, the optimal lot size is still 10,000 bottles. In this case, the presence of quantity discounts leads to an eightfold increase in average inventory as well as flow time at DO.

Pricing schedules with all unit quantity discounts encourage retailers to order in larger lots to take advantage of price discounts. This adds to the average inventory and flow time in a supply chain. This increase in inventory raises a question about the value that all unit quantity discounts offer in a supply chain. Before we consider this question, we discuss marginal unit quantity discounts.

Marginal Unit Quantity Discounts

Marginal (or incremental) unit quantity discounts are also referred to as multi-block tariffs. In this case, the pricing schedule contains specified break points q0, q1, . .., qr. It is not the average cost of a unit but the marginal cost of a unit that decreases at a breakpoint (in contrast to the all unit discount scheme). If an order of size q is placed, the first q1 ! q0 units are priced at C0, the next q2 ! q1 are priced at C1, and in general qi"1 ! qi units are priced at Ci. The marginal cost per unit varies with the quantity purchased, as shown in Figure 11-4.

Faced with such a pricing schedule, the retailer’s objective is to decide on a lot size that maximizes profits or, equivalently, minimizes material, order, and holding costs.

The solution procedure discussed here evaluates the optimal lot size for each marginal price Ci (this forces a lot size between qi and qi+1) and then settles on the lot size that minimizes the overall cost. A more streamlined procedure has been provided by Hu and Munson (2002).

For each value of i, , let Vi be the cost of ordering qi units. Define V0 # 0 and Vi for as follows:

(11.12)Vi = C0(q1 - q0) + C1(q2 - q1) + . . . + Ci- 1(qi - qi- 1)

0 … i … r 0 … i … r

Q2 * = 10,000

TC1 = a D Q1

* bS + a Q1*2 bhC1 + DC1 = $358,969; TC2 = $354,520

Quantity Purchased

C0

0 q1 q2

C1

C2

M ar

gi na

l C os

t p er

U ni

t P ur

ch as

ed

FIGURE 11-4 Marginal Unit Cost with Marginal Unit Quantity Discount

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The costs the retailer must consider when making this decision are material cost, holding cost, and order cost. Increasing the lot size Qd lowers the material cost for Cub Foods because it purchases more cans (for sale now and in the future) at the discounted price. Increasing the lot size Qd increases the holding cost because inventories increase. Increasing the lot size Qd lowers the order cost for Cub Foods because some orders that would otherwise have been placed are now not necessary. Cub Food’s goal is to make the trade-off that minimizes the total cost.

The inventory pattern when a lot size of Qd is followed by lot sizes of Q* is shown in Figure 11-5. The objective is to identify Qd that minimizes the total cost (material cost ! ordering cost ! holding cost) over the time interval during which the quantity Qd (ordered during the promotion period) is consumed.

The precise analysis in this case is complex, so we present a result that holds under some restrictions.3 The first key assumption is that the discount is offered once, with no future discounts. The second key assumption is that the retailer takes no action (such as passing on part of the trade promotion) to influence customer demand. The customer demand thus remains unchanged. The third key assumption is that we analyze a period over which the demand is an integer multiple of Q*. With these assumptions, the optimal order quantity at the discounted price is given by

(11.16)

In practice, retailers are often aware of the timing of the next promotion. If the demand until the next anticipated trade promotion is Q1, it is optimal for the retailer to order min{Q

d, Q1} Observe that the quantity Qd ordered as a result of the promotion is larger than the regular order quantity Q*. The forward buy in this case is given by

Even for relatively small discounts, the order size increases by a large quantity, as illustrated in Example 11-11.

Forward buy = Qd - Q*

Qd = dD

(C - d)h +

CQ*

C - d

I(t)

t

Q* Q* Q* Q* Q*

Qd

FIGURE 11-5 Inventory Profile for Forward Buying

3 See Silver, Pyke, and Petersen (1998) for a more detailed discussion.

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11.6 MANAGING MULTIECHELON CYCLE INVENTORY

A multiechelon supply chain has multiple stages and possibly many players at each stage. The lack of coordination in lot sizing decisions across the supply chain results in high costs and more cycle inventory than required. The goal in a multiechelon system is to decrease total costs by coordinating orders across the supply chain.

Consider a simple multiechelon system with one manufacturer supplying one retailer. Assume that production is instantaneous, so the manufacturer can produce a lot when needed. If the two stages are not synchronized, the manufacturer may produce a new lot of size Q right after shipping a lot of size Q to the retailer. Inventory at the two stages is as shown in Figure 11-6. In this case, the retailer carries an average inventory of Q/2 and the manufacturer carries an average inventory of about Q.

Overall supply chain inventory can be lowered if the manufacturer synchronizes its produc- tion to be ready just in time to be shipped to the retailer. In this case, the manufacturer carries no inventory and the retailer carries an average inventory of Q/2. Synchronization of production and replenishment allows the supply chain to lower total cycle inventory from about 3 Q/2 to Q/2.

For a simple multiechelon supply chain with only one player at each stage, ordering policies in which the lot size at each stage is an integer multiple of the lot size at its immediate customer have been shown to be quite close to optimal. When lot sizes are integer multiples, coordination of ordering across stages allows for a portion of the delivery to a stage to be cross- docked on to the next stage. The extent of cross-docking depends on the ratio of the fixed cost of ordering S and the holding cost H at each stage. The closer this ratio is between two stages, the higher is the optimal percentage of cross-docked product. Munson, Hu, and Rosenblatt (2003) provide optimal order quantities in a multiechelon setting with a single manufacturer supplying a single retailer.

If one party (distributor) in a supply chain supplies multiple parties (retailers) at the next stage of the supply chain, it is important to distinguish retailers with high demand from those with low demand. In this setting, Roundy (1985) has shown that a near-optimal policy results if

Manufacturer Inventory

Retailer lot is shipped Manufacturer lot arrives

Retailer Inventory

Time

Q

Time

Q

FIGURE 11-6 Inventory Profile at Retailer and Manufacturer with No Synchronization

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