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Unformatted text preview: its national network of spareparts inventories, IBM improved its service to customers
and reduced the capital tied up in inventory by over $250 million and saved an addi
tional $20 million per year through improved operational efﬁciency. Section 2.5 de
scribes the massive computer system used to control this inventory system. There are several basic considerations involved in determining an inventory pol
icy that must be reﬂected in the mathematical inventory model. These are illustrated in
the examples presented in the ﬁrst section and then are described in general terms in
Sec. 17.2. Section 17.3 develops and analyzes deterministic inventory models, i.e.,
models where the future demand for withdrawing the product from inventory is as
sumed to be known. Section 17.4 deals with stochastic models where this demand is a random variable. 17.1 Examples 757
17.1 /Exampla “M We present two examples in rather different contexts (a manufacturer and a wholesaler)
where an inventory policy needs to be developed. Example 1: Manufacturing Speakers for TV Sets A television manufacturing company produces its own speakers, which are used in the
production of its television sets. The television sets are assembled on a continuous
production line at a rate of 8,000 per month, with one speaker needed per set. The
speakers are produced in batches because they do not warrant setting up a continuous
production line, and relatively large quantities can be produced in a short time. There
fore, the speakers are placed into inventory until they are needed for assembly into
television sets on the production line. The company is interested in determining when
to produce a batch of speakers and how many speakers to produce in each batch.
Several costs must be considered: 1. Each time a batch is produced, a setup cost of $12,000 is incurred. This cost
includes the cost of “tooling up,” administrative costs, record keeping, and
so forth. Note that the existence of this cost argues for producing speakers in
large batches. 2. The unit production cost of a single speaker (excluding the setup cost) is
$10, independent of the batch size produced. (In general, however, the unit . production cost need not be constant and may decrease with batch size.) 3. The production of speakers in large batches leads to a large inventory. The
estimated holding cost of keeping a speaker in stock is $0.30 per month. This
cost includes the cost of capital tied up, storage space, insurance, taxes, pro
tection, and so on. The existence of a holding cost argues for producing small
batches. 4. Company policy prohibits deliberately planning for shortages of any of its
components. However, a shortage of speakers occasionally crops up, and it
has been estimated that each speaker that is not available when required costs
$1.10 per month. This shortage cost includes the cost of installing speakers
after the television set is fully assembled, storage space, delayed revenue,
record keeping, and so forth. 758
17 / Inventory Theory We will develop the inventory policy for this example with the help of the ﬁrst
inventory model presented in Sec. 17.3. Exainple 2: Wholesale Distribution of Bicycles A wholesale distributor of bicycles is having trouble with shortages of a popular model
(a small, onespeed girl’s bicycle) and is currently reviewing the inventory policy for
this model. The distributor purchases this model bicycle from the manufacturer
monthly and then supplies it to various bicycle shops in the western United States in
response to purchase orders. What the total demand from bicycle shops will be in any
given month is quite uncertain. Therefore, the question is, How many bicycles should
be ordered from the manufacturer for any given month, given the stock level leading
into that month? ‘ The distributor has analyzed her costs and has determined that the following are important: 1. The ordering cost, i.e., the cost of placing an order plus the cost of the
bicycles being purchased, has two components: The cost of paperwork in
volved in placing an order is estimated as $200, and the actual cost of each
bicycle is $35 for this wholesaler. 2. The holding cost, i.e., the cost of maintaining an inVentory, is $1 per bicycle
remaining at the end of the month. This cost represents the costs of capital
tied up, warehouse space, insurance, taxes, and so on. 3. The shortage cost is the cost of not having a bicycle on hand when needed.
Most models are easily reordered from the manufacturer, and stores usually
accept a delay in delivery. Still, although shortages are permissible, the dis
tributor feels that she incurs a loss, which she estimates to be $15 per bicycle.
This cost represents an evaluation of the cost of the loss of goodwill, addi
tional clerical costs incurred, and the cost of the delay in revenue received. On
a very few competitive (in price) models, stores do not accept a delay, which
results in lost sales. In this case, the cost of lost revenue must be included in
the shortage cost. We will return to this example again in Sec. 17.4. These examples illustrate that there are two possibilities for how a ﬁrm replen
ishes inventory, depending on the situation. One possibility is that the ﬁrm produces
the needed units itself (like the television manufacturer producing speakers). The other
is that the ﬁrm orders the units from a supplier (like the bicycle distributor ordering
bicycles from the manufacturer). Inventory models do not need to distinguish between
these two ways of replenishing inventory, so we will use such terms as producing and
ordering interchangeably. . Both examples deal with one specific product (speakers for a certain kind of
television set or a certain bicycle model). In most inventory models, just one product is
being considered at a time. Except for one subsection at the end of the chapter, all the
inventory models presented here assume a single product. Both examples indicate that there exists a tradeoff between the costs involved.
The next section discusses the basic cost components of inventory models for deter—
mining the optimal trade—off between these costs. 17.2 Components of Inventory Models Because inventory policies affect proﬁtability, the choice among policies depends upon
their relative proﬁtability. As already seen in Examples 1 and 2, some of the costs that
determine this proﬁtability are (1) the costs of ordering or manufacturing, (2) holding
costs, and (3) shortage costs. Other relevant factors include (4) revenues, (5) salvage
costs, and (6) discount rates. These six factors are described in turn below. The cost of ordering or manufacturing an amount 2 can be represented by a
function C(z). The simplest form of this function is one that is directly proportional to
the amount ordered or produced, that is, c  z, where c represents the unit price paid.
Another common assumption is that C(Z) is composed of two parts: a term that is
directly proportional to the amount ordered or produced and a term that is a constant K
for 2 positive. and is 0 for z = 0. For this case, C(Z) = cost of ordering or manufacturing 2 units
0 if z = 0,
K + (:7. if z > 0,
where K = setup cost and c = unit cost.
The constant K includes the administrative cost of ordering or, when manufactur
ing, the preliminary labor and other expenses of starting a production run.
There are other assumptions that can be made about the cost of ordering or
manufacturing, but this chapter is restricted to the cases just described.
In Example1, the speakers are manufactured, and the setup cost for a production run is $12,000. Furthermore, each speaker costs $10, so that the production cost is
given by C(z) = 12,000 + lOz, for z > 0. In Example 2, the distributor orders bicycles from the manufacturer, and the ordering
cost is given by " C(Z) = 200 + 35z, for z > 0. The holding cost (sometimes called the storage cost) represents all the costs
associated with the storage of the inventory until it is sold or used. Included are the cost
of capital tied up, space, insurance, protection, and taxes attributed to storage. These
costs may be a function of the maximum quantity held during a period, the average
amount held, or the cumulated excess of supply over the amount required (demand).
The last viewpoint is usually taken in this chapter. In the bicycle example, the holding cost is $1 per bicycle remaining at the end of
the month. This cost can be interpreted as the interest lost in keeping capital tied up in
an “unnecessary” bicycle for a month, the cost of extra storage space, insurance, and so forth.
The shortage cost—sometimes called the unsatisﬁed demand cost—is incurred when the amount of the commodity required (demand) exceeds the available stock.
This cost depends upon which of the following two cases applies. In one case, called backlogging, the excess demand is not lost, but instead is held
until it can be satisﬁed when the next normal delivery replenishes the inventory. For a
ﬁrm incurring a temporary shortage in supplying its customers (as for the bicycle 759 760
17/ Inventory Theory example in Sec. 17.1), the shortage cost then can be interpreted as the loss of custom
ers’ goodwill due to the delay, their subsequent reluctance to do business with the ﬁrm,
the cost of delayed revenue, and extra record keeping. For a manufacturer incurring a
temporary shortage in materials needed for production (such as a shortage of speakers
for assembly into television sets), the shortage cost becomes the cost associated with
delaying the completion of the production process. In the second case, called no backlogging, if any excess of demand over avail
able stock occurs, the distributor cannot wait for the next normal delivery to replenish
inventory. Either (1) the excess demand is met by a priority shipment, or (2) it is not
met at all. For situation 1, the shortage cost can be viewed as the cost of the priority
shipment. For situation 2, the shortage cost can be viewed as the loss in current revenue
from not meeting the demand, plus the cost of losing future business because of lost
goodwill. ‘ _ Revenue may or may not be included in the model. If both the price and the
demand for the product are established by the market and so are outside the control of
the company, the revenue from sales (assuming demand is met) is independent of the
ﬁrm’s inventory policy and may be neglected. However, if revenue is neglected in the
model, the loss in revenue must then be included in the shortage cost whenever the firm
cannot meet the demand and the sale is lost. Furthermore, even in the case where
demand is backlogged, the cost of the delay in revenue must also be included in the
shortage cost. With these interpretations, revenue will not be considered explicitly in
the remainder of this chapter. The salvage value of an item is the value of a leftover item when no further
inventory is desired. The salvage value represents the disposal value of the item to the
ﬁrm, perhaps through a discounted sale. The negative of the salvage value is called the
salvage cost. If there is a cost associated with the disposal of an item, the salvage cost
may be positive. We assume hereafter that any salvage cost is incorporated into the
holding cost. Finally, the discount rate takes into account the time value of money. When a
ﬁrm ties up capital in inventory, the ﬁrm is prevented from using this money for
alternative purposes. For example, it could invest this money in secure investments,
say, government bonds, and have a return on investment 1 year hence of, say, 7 percent.
Thus $1 invested today would be worth $1.07 in year 1, or alternatively, a $1 proﬁt 1
year hence is equivalent to a = $1/ 1.07 today. The quantity or is known as the discount
factor. Thus, in considering the proﬁtability of an inventory policy, the proﬁt or costs 1
year hence should be multiplied by a; in 2 years hence, by (12; and so on. (Units of time
other than 1 year also can be used.) In problems having short time horizons, a may be assumed to be 1 (and thereby
neglected) because the current value of $1 delivered during this short time horizon does
not change very much. However, in problems having long time horizons, the discount
factor must be included. In using quantitative techniques to seek optimal inventory policies, we use the
criterion of minimizing the total (expected) discounted cost. Under the assumptions
that the price and demand for the product are not under the control of the company and
that the lost or delayed revenue is included in the shortage penalty ,cost, minimizing
cost is equivalent to maximizing net income. Another useful criterion is to keep the ' inventory policy simple, i.e., keep the rule for indicating when to order and how much to order both understandable and easy to implement. Most of the policies considered in
this chapter possess this property. Inventory models are usually classiﬁed according to whether the demand for a
period is known (deterministic demand) or is a random variable having a known prob
ability distribution (nondeterministic or random demand). The production of batches of
speakers in Example 1 of Sec. 17.1 illustrates deterministic demand because the speak
ers are used in television assemblies at a ﬁxed rate of 8,000 per month. The bicycle
shops’ purchases of bicycles from the wholesale distributor in Example 2 of Sec. 17.1
illustrates random demand. This classiﬁcation is frequently coupled with whether or
not there exist time lags in the delivery of the items ordered or produced. In both
examples in Sec. 17.1, there was an implication that the items appeared immediately
after an order was placed. In fact, the production of speakers may require some time,
and similarly, the delivery of bicycles to the wholesaler may not be instantaneous, so
that time lags may have to be incorporated into the inventory model. However, for
simplicity, all the models considered in this chapter assume instantaneous deliver.l Another possible classiﬁcation relates to the way the inventory is reviewed,
either continuously or periodically. In continuous review, an order is placed as soon as
the stock level falls below the prescribed reorder point, whereas in periodic review the
inventory level is checked at discrete intervals, e.g., at the end of each week, and
' ordering decisions are made only at these times even if the inventory level dips below
the reorder point between the preceding and current review times. (In practice, a peri
odic review policy can be used to approximate a continuous review policy by making
the time interval sufﬁciently small.) 17.3 Deterministic Models 761 17.3 / Deterministic
Models This section is concerned with inventory problems where the actual demand in the
future is assumed to be known. Several models are considered, including the well
known economic lotsize formulation, considered ﬁrst. ' Continuous Review—Uniform Demand The most common inventory problem faced by manufacturers, retailers, and wholesal—
ers is that stock levels are depleted over time and then are replenished by the arrival of
new units. A simple model representing this situation is the following economic lot
size model. (It sometimes is also referred to as the economic order quantity model or, for short, the EOQ model.)
Units of the product under consideration are assumed to be withdrawn from inventory continuously at a known constant rate, denoted by a; that is, the demand is
0 units per unit time. It is further assumed that inventory is replenished when needed by
producing or ordering a batch of ﬁxed size (Q units), where all Q units arrive simulta
 neously at the desired time. The only costs to be considered are K = setup cost for producing or ordering one batch,
c = unit cost for producing or purchasing each unit,
h = holding cost per unit per unit of time held in inventory. 1 Results for the deliverylag case often can be obtained from the corresponding instantaneous delivery
model by a simple modiﬁcation in the calculation of some of the inventory costs. 762
17 / Inventory Theory The objective is to determine when and by how much to replenish inventory so as to
minimize the sum of these costs per unit time. We assume continuous review, so that inventory can be replenished whenever the
inventory level drops sufﬁciently low. We shall ﬁrst assume that shortages are not
allowed (but later we will relax this assumption). With the ﬁxed demand rate, shortages
can be avoided by replenishing inventory each time the inventory level drops to zero,
and this also will minimize the holding cost. Figure 17.1 depicts the resulting pattern of
inventory levels over time when we start at time 0 by producing or ordering a batch of
'Q units in order to increase the initial inventory level from 0 to Q. Example 1 in Sec. 17.1 (manufacturing speakers for TV sets) ﬁts this model and
will be used to illustrate the following discussion. SHORTAGES NOT PERMI'I'I‘ED: For the speaker example, a cycle can be viewed as
the time between production runs. Thus, if 24,000 speakers are produced in each
production run and are used at the rate of 8,000 per month, then the cycle length is ~ 24,000/ 8,000 = 3 months. In general, the cycle length is 'Q/a, as illustrated in Fig. 17.1. The total cost per unit time T is obtained from the following components.
Production or ordering cost per cycle = K + cQ. The average inventory level during a cycle is (Q + 0)/2 = Q/2 units per unit time, and
the corresponding cost is hQ/2 per unit time. Because the cycle length is Q/a, h 2
Holding cost per cycle = Q—.
2a
Therefore,
th
Total cost per cycle = K + cQ + —2——,
a
so the total cost per unit time is
K + + h 2/ 2 K h
T = ___C£__Q_<_a>_ = a_ + ac + _Q__
Q/a Q 2 The value of Q, say Q*, that minimizes Tis found by setting the ﬁrst derivative to‘
zero (and noting that the second derivative is positive). Inventory level
Q Batch size Q Figure 17.1 Diagram of inventory level as a function of time when no shortages are permitted. £_£+£0
dQ Q2 2 ’ so that Zak
Q  "T which is the wellknown economic lotsize formula.l Similarly, the time it takes to
withdraw this optimal value of Q*, say t*, is given by ”=21: 2—K
a a. It is interesting to observe that Q* and t* change in intuitively plausible ways
when a change is made in K, h, or a. As the setup cost K increases, both Q* and t*
increase (fewer setups). When the unit holding cost it increases, both Q* and t* de
crease (smaller inventory levels). As the demand rate a increases, (2* increases (larger batches) but t* decreases (more frequent setups).
These formulas for Q* and t* will now be applied to the speaker example. The appropriate parameter values from Sec. 17.1 are
K = 12,000, h = 0.30, a = 8,000, so that
i(2)(8,000)(12,000)
* = “— = 2
Q 0.30 5’298
and
25,298
* = = .2 .
t 8, 3 months Hence, the optimal solution is to set up the production facilities to produce speakers
once every 3.2 months and to produce 25,298 speakers each time. (The total cost curve
is rather ﬂat near this optimal value, so any similar production run that might be more
convenient, say 24,000 speakers every 3 months, would be nearly optimal.) SHORTAGES PERMI'r'rED: It may be worthwhile to pemiit small shortages to occur because the cycle length can then be increased with a resulting saving in setup costs.‘ However, this beneﬁt may be offset by the shortage cost, so a detailed analysis is
required. Let p = shortage cost per unit short per unit of time short,
S = inventory level just after a batch of Q units is added to inventory,
Q  S = shortage in inventory just before a batch of Q units is added. The resulting pattern of inventory levels over time is shown in Fig. 17.2 when one
starts at time 0 with an inventory level of S. 1 An interesting historical account of this model and formula. including a reprint of a 1913 paper that started
it all, is given by D. Erlenkotter, "Ford Whitman Harris and the ...
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 Fall '09
 MarinaEpelman

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