Download Ch. 10 Single Period 10. One Period Decisions Safety stock is being

Ch. 10 Single Period
10. One Period Decisions
Safety stock is being held as a hedge against uncertainty. It can be viewed as the
result of trade-off between the costs resulting from too much inventory on the one
hand and the costs resulting from running out of inventory and losing sales on the
other. This trade-off is evident when considering a single-period decision process,
which takes place whenever somebody (say, a retailer) orders a product which
cannot be stocked from one period to the next. The classical examples include (i)
newsprint; where there is only one chance to order a daily or weekly publication,
(ii) seasonal items such as holiday toys, where a single order can be placed with
the manufacturer every year and (iii) perishable products such as food. In today’s
fast paced business environment, this framework is relevant for more and more
products since many goods, such as fashion items, go out of style or, like high
technology products, become obsolete in less time than the order lead time. In the
literature, this framework is sometimes referred to as “the newsboy” or the
“Christmas Tree” problem, for obvious reasons.
10.1 Example – The Data
Consider, a newsstand owner who has to order a weekly magazine “Logistics
2
Lullabies” (L ). The magazines have to be ordered at the beginning of every week
from the publisher. To decide how many to order, the owner would like to know
2
how many requests will there be next week for the L magazine. Unfortunately,
this future realization is unknown at order time. Thus, the owner has to forecast
the demand for next week’s issue.
To assist in the forecast, the owner has kept a record of the demand during the
last year, including both magazines sold and customers’ demands that were not
fulfilled. The total demand data for each of the last 52 weeks is the following:
The total number of magazines requested by customers was 4023, with an
average of 77.4 magazines per week and a standard deviation of 15.4
magazines/week. The lowest number of magazines requested was 51 and the
highest 113 per week. Thus, the owner may feel that if less than 51 magazines
are ordered every week, it is likely that the number of actual requests will be
higher, in which case all magazines will be sold, but the newsstand is likely to run
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2
out of L magazines frequently and many customers may be denied. On the other
hand, if more than 113 magazines are ordered, it is likely that the number of
requests will be lower than the number of magazines on hand -- all the demand
will be satisfied, but it is also likely that many magazines will be left unsold.
Figure 10.1 depicts a histogram of the data shown in Table 10.1. For each
possible weekly demand level the histogram shows the number of occurrences of
this demand level last year, N(X). The right hand axis depicts the relative
frequency of each occurrence, denoted Pr(X), where Pr(X) = 152()NX, since we
have 52 observations in all. It also depicts the cumulative frequency distribution of
these data,
In other words, Pr(X ≤ x) is the fraction of days where the demand, X was less
than x. This quantity is a measure of level of service -- it is the fraction of days
where all customers were served and there was no stock-out (in other words, the
probability that all demand will be satisfied). This level of service measure is also
known as cycle service since it is the fraction of order cycles in which all
customers are served. (Note that the actual level of service, from the customers’
point of view – the fill rate - will be significantly higher since even on days where
stock-out occurs, many customers are being served. The fill rate is the probability
of satisfying a random customer and it equals the ratio of expected sales to the
expected demand.)
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The histogram can also be accumulated in fewer bins so the general shape of
the underline distribution can be more evident, as shown in Fig 10.2.
1
The histogram shows
that if, for example, the
owner
would
have
ordered 100 magazines
every week last year,
demand would have
satisfied during 92% of
the weeks (48 out of 52
weeks). There would
have been four weeks
last year where demand
would have exceeded
the amount ordered.
10.2 Example – The Ordering Decision
The decision facing the owner is how much to order for next week. While the
owner cannot know the actual demand next week, such analysis can nevertheless
help him get a handle on some of the consequences of ordering a certain number
of magazines, assuming that next week will “behave like” last year. This is a very
important assumption and it underlies most algorithmic-based forecasting
methods. It assumes that there is no “structural” difference between what may
occur next week and what took place last year. With this assumption, Pr(X=x) can
be interpreted as the probability that the demand on a given week will be x, and
similarly Pr(X ≤ x) can be interpreted as the probability that the demand will be
less than or equal to x (see Eq. [10.1]). The complementary probability, Pr(X > x)
is the probability that the demand will be higher than x.
Assume, now, that each magazine ordered costs C, and the revenue derived from
each sale is R. If the owner orders Q magazines, the total profit will be a function
of the actual demand, x. The probability that the number of magazines demanded
during a given week will be x is Pr(X=x). Thus:
• If demand next week will be higher than the order (i.e., x > Q), the
owner will sell Q magazines and the profit each such a day will be:
Q•(R-C).
1
2
2
[10-2a]
The accumulation into fewer bins does involve a loss of information (about the distribution of values
within each bin) but it allows for a macro view of the data.
The expected profit will be Q•(R-C)•Pr(X=x)
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• If demand next week will be higher than the order (i.e., x ≤ Q), the
profit will be: [x•R-Q•C].
3
[10-2b]
Assuming a value of R = $15 and C = $8, and the histogram data shown in Fig.
10.2, the expected profit can be calculated using a spreadsheet as follows:
The first column in this spreadsheet is the demand in magazines/week and the
second is the probability of that demand. Possible orders are shown in the top row.
Each entry in the table is the profit given the relevant demand (row) and order
(column). Thus, for example, if 80 magazines were ordered and demand of 60
magazines was realized, the profit will be: 60$1580$8$260•−•=.
To get the expected profit given a particular order, each entry in the column for
that ordered is multiplied by the probability of the corresponding demand
realization and the column is summed. For example, the expected profit given an
order of 80 magazines (see column I in the spreadsheet) is given by the Excel
operation:
SUMPRODUCT(I8:I17,$B8:$B17)
This operation results in an expected profit of $464. This sum is shown, for each
column in the bottom row of the table. It is now clear from the table that the
highest profit, $482, can be achieved with an order of 80 magazines.
3
The expected profit will be [x•R-Q•C]•Pr(X=x).
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The expected profit as a function of the order size, Q, can be drawn as shown in
Figure 10.3.
As evident from the
spreadsheet and from
Figure 10.3, there is an
optimal
order
size
which maximizes the
weekly profit, at Q = 80
magazines/week.
Smaller sizes have
lower profit since too
many requests are not
met, while with larger
order sizes, the cost of
unsold inventory is
excessive. In fact, for very large order sizes it leads to a loss.
The graph also provides some additional insights into the nature of the ordering
decision. The expected profit function is relatively "flat" in the vicinity of the
optimum value, Q*, so that the actual expected profit is relatively insensitive to the
exact choice of Q, provided that it is sufficiently “near” the optimum value of Q*.
Note also that the "slope,” or marginal rate of change of the expected profit
function with regard to Q is approximately (R-C) dollars per unit for the first few
units (low orders). Since the first units are almost certain to be demanded, their
contribution to the expected profit is almost (R-C) dollars each. At the other end of
the graph, with this distribution of demand, every magazine over 100 or 110 units
ordered is almost certain not to be sold, so its contribution to expected profit is
almost (-C) dollars per magazine, meaning that the slope of the profit function for
large orders is approaching (-C) as the order grows.
10.3 The Optimal Order – Marginal Analysis
*
The optimal order size (at which the profit is the highest), Q , will be at a point
where the expected profit from an additional unit ordered will be approximately
*
st
zero, -- turning profit to loss. The expected profit from ordering the (Q +1)
magazine is R-C if it is sold and -C if it is unsold. The probability of selling the
*
st
*
(Q +1) magazine is the probability that the demand will be higher than Q , Pr(X >
*
*
Q ), and the probability of not selling it, is Pr(X ≤ Q ). The expected profit from
*
st
*
*
ordering the (Q +1) magazine is then: (R-C) • Pr(X > Q ) - C • Pr(X ≤ Q ) which
has to satisfy:
*
*
(R-C) • Pr(X > Q ) - C • Pr(X ≤ Q ) = 0
*
*
Since Pr(X > Q ) = 1 - Pr(X ≤ Q ), this equation can be written as:
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*
*
(R-C) • [1 - Pr(X ≤ Q )] - C • Pr(X ≤ Q ) = 0
This leads to the rule for optimal order:
[10.3]
In other words, the optimal order size is the one for which the cumulative
frequency distribution (see Fig. 10-1) equals (R-C)/R. For the example discussed
here, this “critical ratio” is
Using the cumulative distribution function in Fig. 10.1, the value of Q for which
*
Pr(X≤Q) = 0.47, can be determined to be: Q = 79 magazines. This result also
agrees with Fig. 10.3.
Thus, in this example, if the owner will use this ordering method, he will, in fact,
choose to stock out in over 53% of the weeks.
Note that if the owner would have been able to find alternative use for the unsold
paper, say, supply them to local dentist offices at $4 per copy, the cost of unsold
inventory would have been lower and the critical ratio would be (see Question
5.1):
This would lead to an order
of 87 magazines every
week and only 36% weeks
in which a stock-out occurs.
The expected profit as a
function of Q is depicted
for this example in Fig.
10.4.
Note that if there are
additional costs to dispose
unsold magazines, such as
environmental costs, the
order size will be smaller.
On the other hand, if the costs of running out were higher and in addition to lost
sales costs there was a penalty for poor level of service, the order size would
have been higher.
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10.4 The Profit Function
To build systematically the expected profit as a function of the order size, note that
in most cases there are four types of revenues and costs that the vendor in this
framework can experience:
• Revenue from sold items
• Revenue or costs associated with unsold items. These may include revenue
from salvage or cost associated with disposal.
• Costs associated with not meeting customers’ demand. The lost sales cost
can include lost of good will and actual penalties for low service.
• The cost of buying the merchandise in the first place.
Assuming that the demand follows a (continuous) probability density function
(PDF), f(x), with a cumulative distribution, F(x), the expected number of items sold
(expected sales) can be written as:
The first term on the right hand side of Eq. [10.5] is the expected number of items
sold when demand is greater than the order (i.e.,>xQ). The second item is the
expected sales when≤xQ.
The expected remaining inventory is what remains at the end of the selling
season when<xQ. The expected number of unsold items can be calculated as:
Appendix 1 in Section 10.13 shows details of the derivation.
Eq. [10.5b] can be optimized using Excel in order to determine the optimal order
quantity. In this case, since it can be solved analytically this is not necessary.
Sales are lost when>xQ. In this case, customer demand is higher than the order.
The expected number of lost sales can be expressed as:
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Again, Appendix 1 shows the details of the derivation.
Given these quantities one can calculate the expected fill rate as a function of the
order size, Q:
In the general case when all cost and revenue items are modeled, let R represent
the sale price of the item sold, let S represent the net salvage value (it can be
negative in case disposal expenses have to be incurred), L represent the net
costs of lost sales and C is the purchase cost of the item. Using these notations,
the expected profit as a function of Q can be expressed as:
The example analyzed in this section assumes no salvage value and no cost of
lost sales. Thus,
This expression is independent of the particular demand probability density
function (PDF) assumed. In fact, so far, no particular PDF was assumed.
To find the optimal value of Q (denoted Q* above), one can take the first derivative
of Eq. [10.6], set it equal to zero, and solve for Q. As shown in Appendix 2
(Section 10.13),
So the derivative of the profit function (Eq. [10.6]) is:
Leading to:
where F[Q] is the cumulative probability function evaluated at Q (i.e.,
-1
F[Q]=Pr(X≤Q], and F [•] is the inverse cumulative function. This result is identical
to the one obtained from the marginal analysis (see Eq. [10.4]).
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The same model can be developed for discrete PDFs. The application of discrete
distributions is warranted for low value of orders and expected sales since the
discrete nature of the items is more pronounced then.
Given a discrete probability density function, P(x), the relevant quantities are
given by:
The profit function can be set using these quantities and the optimal order size
found by taking first differences instead of differentiation (or using marginal
analysis, as done above).
10.5 Level of Service
The cycle level of service and the fill rate can be calculated for the example
depicted in this section. The cycle level of service is given simply by the
cumulative distribution (or the cumulative histogram). The fill rate can be
calculated from Eq. [10.9c] using the histogram data.
The result is depicted in Figure 10.5. As shown in the figure, the fill rate is
significantly higher, for any order, than the cycle service level. The reason is that
the fill rate is the level of service experienced by the customers, some of whom
will get served in during “failed cycles.”
10.6 Using Specific Distributions
In many real world cases, it may be difficult to come up with a forecasted
distribution of demand. Instead, one may have a parameter estimate of a
distribution. Thus, in the example shown in the preceding chapter, the entire
distribution may not be available and the analyst may have only distribution
parameters to work with.
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Normal Distribution
Assuming that the demand follows a Normal Distribution with mean μ and
standard deviation σ, the expected sales can be shown to be (see Appendix 3 in
section 10.13):
where z 
Q
is the standard Normal variable, ()zΦis the standard cumulative

Normal distribution and ()zφ is the standard Normal density function.
2
Consequently, for the L magazines example discussed in this chapter:
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The mean and variance of the data in Table 10.1 are 77.37 Mag/Wk and 15.38
Mag/Wk, respectively. Using these values, the NORMDIST function of Excel can
4
be used to calculate the expected profit function directly (see Eq. [10.9b]. The
results are depicted in Figure 10.6.
The optimal order size can be calculated directly from the distribution using the
critical ratio (see Eq. [10.7]):
The Normal distribution is a continuous one and therefore applicable to cases in
which the numbers (order size, demand realizations) are relatively large. For
small numbers, the continuous approximation is less relevant.
Poisson Distribution
As an example of a discrete problem, consider a retailer ordering flower
arrangements for the weekend. The wholesale cost to the retailer is $3.00 per
arrangement. The flower arrangements will sell during the weekend for $10.00
each. Any leftovers will be discarded. Demand during the season is assumed to
follow a Poisson distribution with a mean of three units. To determine how much
should the retailer order, one can develop a spreadsheet similar to table 10.2, with
the probabilities on the second column on the right given by the Poisson
probabilities with mean λ = 3 (i.e., Pr( X  x)  (e  x ) / x ! )
Table 10.3 Expected profit with Poisson Distribution
4
Note that working with the histogram (see Fig. 10.2 and table 10.2), rather than the actual data
involves an approximation. Working with the Normal distribution involves another type of
approximation and that is why the profit function shown Figure 10.5 differs slightly (especially for
high values of Q) from the result shown in Fig. 10.3. To get similar results to table 10.2 (and Fig.
10.3), the parameters of the Normal distribution have to be estimated from the histogram data.
This will yield estimated mean and standard deviation of 81.15 and 15.42 magazines/week,
respectively, leading to a profit function much closer to the one shown in Fig. 5.3 (and an optimal
order size of 80 magazines/week rather than the 76 depicted in Eq. [5-10]).
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From the bottom row, it is clear that the highest profit will be realized with an order
of Q* = 4 flower arrangements.
Note that to calculate the optimal order size from the critical ratio (Eq. [10.4])
alone (i.e., without developing the full spreadsheet), the values just lower and just
higher than the ratio have to be
checked. In this example, ((R C)/R)=0.7 . The cumulative Poisson
distribution with mean λ=3 is shown
in Figure 10.7. As it turns out the
critical ratio falls between Q=3 and
Q=4. Thus the expected profit has
to be checked at both values to
determine that the optimal order
quantity, Q*=4.
10.7 Incorporation of Fixed
Costs
In many situations there is a fixed cost associated with the single period inventory
ordering decision. This may be the setup cost associated with the vendor’s
production run which are expressed as a fixed order costs, marketing and
advertising expenses, or any other costs which will accrue independent of the size
of the order quantity or the quantity sold.
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Such a fixed cost will have no effect on the optimal order size. To see this, one
can add a fixed cost of -F dollars to the expected profit function in Eq. [10.5] or
[10.6], but since this fixed cost term is not a function of Q, the first derivative (or
the first difference) with respect to Q is zero, and hence it will not affect the
determination of Q*. This does not mean that the fixed cost is irrelevant; merely
that the value of F does
not affect the optimal
value of Q.
Consider the example
problem
previously
discussed.
Suppose
there was an additional
fixed cost of $300
associated with each
order. The resulting
expected profit function
is depicted in Figure
10.8. Now the total
expected profit for any
value of Q is $300
lower than previously
calculated:
The optimal order quantity remains unchanged, but this clearly makes the item
much less attractive to the retailer. Given that the net expected profit is still
positive, should the retailer go ahead and order the same amount? To answer this
question we should consider the inherent risk in the situation.
10.8 Analysis of Risk
The solution to the Single Period Inventory Problem is based on maximizing the
expected profit as the objective of an optimal solution. This maximization
balances the risk of “underage” (running out) with the risk of “overage” (having too
many). The cost of “overage” is C and the cost of “underage” is the lost sale, (R-C).
Consequently, the critical ratio can be interpreted to be:
The expected profit criterion is compatible with a firm’s attempt to maximize its
profits over the long run. The Single Period scenario, however, may be just that –
a single period -- where there is no “long run” to average outcomes over. How
appropriate is the criterion in this situation?
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If there are many “single period” items, over many selling seasons, and the
outcome on any one item is relatively minor to the firm, then these outcomes
will ”average out” across items and time, and the approach makes sense.
However, in case the order is relatively large or the item is relatively important to
the company, one would want to carefully consider the risks associated with the
decision. The expected value criterion is appropriate when the decision maker is
risk neutral, which is to say, in situations where a loss of n dollars is no more “bad”
than a comparable gain of n dollars is “good”. In many business situations,
however, this is not the case.
In general, the notion of risk includes the idea that there are many possible
outcomes which may vary widely from the expected value. Thus, the variance, or
the standard deviation of the profit might be used to measure the risk in the
inventory decision. In the business sense, however, the concept of “risk” generally
implies the probability of an adverse outcome (one seldom hears, for example,
“There’s a risk that we’ll make a lot of money”). This notion of adverse outcome is
not well captured by the standard deviation statistic, since its value is influenced
equally by outcomes where profit is higher than the mean as well as by outcomes
where profit is lower than the mean.
To illustrate, assume that the data in the sample problem (Table 10.2) comes from
a continuous distribution. A measure of business risk may be the probability of a
loss, or Pr(x•R ≤ Q•C+F). In other words, the probability that the revenue will be
lower than the cost of the magazines ordered plus any fixed ordering costs. This
probability can be calculated, for every order quantity, Q, as the cumulative
distribution function evaluated at(QgC+F)/g.
Assuming that the data in the sample problem come from a N(77.37,15.38)
distribution, the risk of loss becomes:
Figure 10.9 depicts the increased risk as the order quantity gets larger. The
right-hand curve is for the case of no fixed costs while the left curve is for the
same problem but with F = $300.
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While the risk of a loss is
negligible when ordering
the optimal amount (note
that this is not always the
case) for the case of F=0,
it is over 10% for the
case of F=$300.
When we have fixed
costs, the probability of
loss for very small order
sizes is 100% since the
profit on the order
(R-C)*Q is smaller than
the fixed costs. (In this
example,
an
order
smaller than $300/(15-8) = 43 items, will not earn enough to cover the fixed costs,
regardless of what the actual demand will be.) Once the fixed costs are covered,
however, the probability of loss is small since it is very likely that the entire order
will be sold. It then increases with order size since it is less and less likely that all
the magazines ordered will be sold and the probability of overage increases.
The probability of a loss is only one measure of risk one can also use other
metrics such as the standard deviation of the profit or the maximum possible loss.
By and large, all the measures of risk increase as the order size increases
(beyond the point that the fixed cost is covered). Note, for example, that the
expected profit, when the fixed cost is $300, of
ordering either 51 or 107 magazines is similar. The
risk associated with ordering 107 magazines,
however, is substantially higher (40% chance of
running a loss) than when ordering only 51
magazines (with only 1% chance of a loss).
This risk was not considered in any way in the construction of the optimal
inventory policy. A "risk constrained" inventory decision might deliberately
trade-off some of the expected profit to reduce the risk of loss by reducing the
inventory quantity to something less than Q*.
10.9 Consideration of Initial Inventory
In some situations the single period scenario includes a given level of initial
inventory; that is, the decision involves the opportunity to make one final purchase
to add to an existing inventory. The decision rule, then, is to “order up to”, but no
more than, the original optimal quantity, Q*. (See Eq. [10.4] or Table 10.2). In
other words, given an existing inventory level of Q0, the decision rule is:
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This rule follows naturally from the marginal analysis argument leading to Eq.
[10.4]; one should continue to “add” an additional unit to the inventory so long as
its expected marginal contribution to profit is positive. Note that the cost of the Q 0
original items need not be C dollars per unit. That cost is irrelevant; all that
matters is the marginal contribution of the incremental units.
As mentioned before, in many cases there is a fixed ordering cost of F dollars
associated with the opportunity to augment the existing inventory. Now the
analysis is slightly more complex. The retailer should compare the expected profit
associated with the pre-existing initial inventory to the maximum expected profit
associated with Q* less the additional fixed cost which would be incurred by
making the final order. It would “order up to” Q* only if the net expected profit
would be improved. In general, one can always determine a critical value for the
initial inventory, Qcrit , such that if the initial inventory is below Qcrit, one should
order up to Q*, and if the initial inventory is Qcrit or greater, one orders nothing. We
find Qcrit as the smallest value of Q such that:
and the decision rule becomes:
The procedure can be illustrated using the magazine retailer example in this
section. Recall that in this example, R=$15, C=$8, demand is assumed to be
given by the histogram in Fig. 10.2. Assume further that the fixed ordering costs
are given by F = $150.
Assume, for example, that the initial inventory is 40 magazines. Since Q*= 80
magazines the following considerations apply:
If the retailer orders 80 – 40 = 40 magazines, the additional profit will be the
difference between expected profits from having 80 magazines ($476) and the
expected profit associated with having only 40 magazines ($280). This difference
is $196. Since this expected profit is higher than the fixed costs of ordering ($150),
the extra 40 magazines should be ordered. To get the critical value, Eq. [10.14]
needs to be solved, by finding the initial inventory where the expected profit is
smaller than $476 - $150 = $326, yielding, in this example:
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Thus the retailer would order up to 80 magazines if its initial inventory is less than or
equal to 46 magazines and order nothing if it is 47 magazines or more. In the latter case
the fixed cost would wipe out the additional expected profits.
This procedure can be visualized in
the Fig. 10.10, which shows the
levels of expected profit associated
with each possible level of inventory.
The top curve represents values of
the expected profit function for all
inventory positions without fixed
costs. In this context the top curve
can be interpreted as the expected
profit which would be generated by
each level of initial inventory. The
bottom line represents the expected profit for the same inventory positions with the fixed
costs deducted, that is, at each point the bottom curve is lower by exactly $150.00. Thus,
the lower curve represents the expected profit which would be generated by a total
inventory position made up of initial inventory and a final order. Note that the maximum
profit on the bottom curve, $326, is attained at an inventory position of 80 magazines.
However, from the top curve we see that every initial inventory position of 47 magazines
and above will generate a higher expected profit, and hence Qcrit = 47 magazines. There is
no point in adding to an initial inventory position of 47 units or more.
Once again, it will be the case that the price paid for the initial inventory is not relevant to
the decision. We have apparently included in our analysis an assumption that we paid the
same $8.00 per magazine for the initial inventory as well as for units in the final buy, but
this is not really the case. Suppose, for example, that the retailer had received the 40 units
of initial inventory for free – this would have improved our estimate of the expected profit
from the initial inventory position by $320.00 (to a total of $500.00), however, it would also
improve the expected profit given a final purchase by exactly the same amount. Hence the
costs of the initial inventory are sunk with regard to this decision, and need not even be
known to arrive at the correct decision.
Note also that this procedure has focused on maximizing expected profit and has not
explicitly addressed the risk in the decision. In the case where only a small increase in
expected profit is achieved by a substantial final order, one would want to carefully
consider and compare the risks associated with the decision to add to the initial inventory
position.
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10.10 Elastic
Demand
The
Single
Period
Inventory
Model
described thus far can
be extended to include
the
relationships
between the retail price
and the demand. In
general,
we
would
expect that lowering the retail price would result in higher retail demand during the
period. The traditional Single Period Inventory Model does not explicitly consider
this trade-off; rather, the analysis proceeds as though the demand distribution and
the retail price are exogenous to the problem. Then the only issue is to determine
the order quantity which maximizes expected profit. A more general
merchandizing problem is to jointly determine what retail price to charge and how
many units to order so as to maximize the expected profit.
To see how this problem can be solved, consider the example discussed
throughout this chapter, that of selling L2 Magazines. Assume further that the
demand follows a Normal distribution. Instead of assuming the specific
parameters of this distribution, however, let the average demand be a function of
the price, p, as given by the following demand function:
Furthermore, assume that the distribution has a constant coefficient of variation.
In other words, the ratio of the standard deviation to the mean is constant. In this
example, let:
The expected profit can then be expressed as a function of the retail price. Let
Q*(p) be the optimal order quantity with price equal p. With this price, μ(p) and σ(p)
can be determined from Eqs. [10.16]. The optimal order quantity can now be
*
1
calculated as Q ( p)  F (( p  C ) / p) . Using Excel these calculations would take
the form:
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For each Q*(p) one can calculate the expected profit and thus get it as a function
of the price. This function is depicted for the example under consideration (with
C=$8) in Figure 10.11.
The optimal price can be found from the figure by inspection, or by numerical
optimization. Using Excel, the price cell can be turned into a variable and the
relationships between the demand and the price, the standard deviation and the
demand and the formula for the expected profit expressed as shown in Table
10.5.
Table 10.6 Optimal Order with Elastic Demand
solver, the optimal price in this case is p*=$22 and the highest profit attained is
$543.
An analytical expression for the optimal price is beyond the scope of this book. In
cases where the expected profit cannot be written explicitly (and a numerical
optimization is not available), it may be cumbersome to follow a similar procedure
(i.e., calculate the optimal order for every price using an excel table similar to
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Table 10.2, and then using the result as a single point on a curve of expected
profit, picking the optimum by inspection). Instead, the number of table
calculations can be minimized by calculating the optimal price under deterministic
conditions first. This can be done by setting the inverse demand, which in this
example would be: p(μ)=33-0.2•μ. The total revenue is then: μ•p(μ)=33•μ-0.2•μ2,
and the marginal revenue is (33-0.4•μ). The highest profit will be where the
marginal revenue equal to the marginal cost. Setting: 33-0.4•μ = 8, one gets:
μ*=62.5. Substituting this value back into the inverse demand function, one gets p*
= 20.5. Using the deterministic optimum price, one can calculate the expected
profit for several values of p in the vicinity of the deterministic optimum. For
example: p = 18, 19, 20, 21, 22 and 23, getting the optimum at p * = 22.
10.11 Summary
The single order problem demonstrates the tradeoff between ordering too much at
the risk of have unsold inventory and ordering too little and losing sales as a result.
In the example carried throughout this chapter the data was used sometimes
directly – to enter the critical ratio into the cumulative histogram in Figure 10.1 and
find the optimal order size. It also demonstrated the use of a histogram where it is
natural to use a spreadsheet for the calculations – in a fashion similar to the use of
discrete probability density function. Naturally, the aggregation involved in the
histogram introduces a small error. Similarly, the use of a statistical distribution
(Normal in this example) also involves a small error. In practice, one should use
the data as given and note that approximate solutions are fine – the input data is
typically not very accurate.
The optimal order size can be determined using the critical ratio which holds for
any demand distribution.
The initial model was extended in several ways. If there is initial inventory and
there is a fixed cost, the ordering rule turns into an (s,S) inventory policy: when the
inventory falls below a critical level, order up to the optimal order size. If demand
is elastic and is a function of the price, the optimal price can be determined by
maximizing the expected profit directly and the optimal order quantity can be
derived from the result.
10.12 References
10.13 Appendices
10.13.1. Key Expressions
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As stated in Eq. [10.5]:
To develop the expression for the remaining inventory, note that:
The expected lost sales can be developed as follows:
10.13.2. Differentiating the Profit Function
To see how the profit function can be differentiated, note that the derivative of the
expected sales with respect to the order quantity is given by:
Recall the formula for integration by parts:
Using this formula, the two terms in Eq. [10.A.3] can be differentiated as follows:
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Adding Eqs. [10.A.5] and [10.A.6] together, one gets:
The expected profit for the case of no salvage value and no cost for lost sales can
be written as:
Using the result in Eq. [10.7], one can take the first derivative of the expected
profit and equate it to zero, obtaining:
Giving the result:
For completeness, note that,
And,
10.13.3. Normal Distribution Calculations
If the distribution of demand is Normal with mean μ and standard deviation σ, the
following holds:
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Using the definition of the Normal distribution, insert it for f(x) in Eq. [10.A.10]:
Using the standard transformation z  x   , which means that x   gz   and

,
Eq.
[10.A.11]
becomes:
dx   gdz
Opening the parentheses:
The first expression in [10.A.13] is simply the product of μ and the standard
cumulative Normal distribution evaluated at Q   , or  g( Q   ) .


The second part of [10.A.13] can be transformed using y  z 2 / 2 . Since
dy  zgdz , this part can be written as:
Collecting the last two terms:
Eq. [10.10a] follows immediately:
In addition:
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