Acknowledgments
6.3 Stochastic Models
6.3.1 Newsvendor Problem
Consider any retailer who needs to make a single procurement decision for a perishable product that is sold over a single period during which demand is uncertain. There are several examples of such businesses. A newsvendor sells newspapers in a day and weekly magazines over a week. A retailer sells summer clothing over a summer season, or T-shirts and hats for the Super Bowl football event, and a manufacturer may design, produce, and sell winter fashion items such as ski jackets and coats over a winter season. The main characteristics of such businesses are: first, the products are perishable. That is, at the end of the selling period the excess inventory is not of any use in the current market; a day-old newspaper cannot be sold as newspaper anymore but can be disposed of as recycled paper or possibly sold to rural areas where paper is not delivered daily; summer clothing is not generally for sale in winter unless the excess stock is shipped to other parts of the world; T-shirts and hats for the 2006 Super Bowl are not in demand after the event. Similarly, winter fashion items are not generally sold after the season is over; they are either shipped off to discount stores or cleared through sales. Second, the procurement lead time is assumed to be too long to make second-ary procurements. Hence, there is only one procurement opportunity before the sales season and the retailer has to commit himself to a certain procurement quantity well in advance.
Based on realized demand from past sales, current economic conditions, and expert judgment, randomness in the demand, D, is assumed to follow a known product-specific demand distribution F(∙).
Our discussion in this section will assume continuous distributions unless otherwise stated. Products are procured at a per unit cost of c, sold at a per unit price of r. Due to the randomness in demand there could be excess inventory or demand at the end of the sales season. Excess inventory is assumed to be returned
to the supplier or salvaged at a per unit price of v, which is less than c, and excess demand is assumed to be lost causing not only a loss of the possible profit , r − c, but also a possible shortage cost of s dollars per unit that represents the loss of goodwill. Note that r > c > v; otherwise, the problem can trivially be solved by either ordering as much as necessary if v > c, or not ordering at all if c > r.
Since demand is random, the procurement decisions are very much dependent on the risk averseness of the retailer. In this chapter, we only consider the risk-neutral decision-makers. Hence, our risk-neutral retailer needs to determine a procurement quantity Q such that the single period expected total inventory ordering, holding, and shortage cost is minimized, or, equivalently, the single-period expected profit is maximized. That is, the retailer needs to solve
max ( )
Q Q
≥ ′
0 Π where the expected profit Π
(Q)
can be expressed asΠ( )Q =rE[min( , )]Q D −cQ vE+ [max( ,0Q D− )]−sE[max(00,D Q− )]
The operator E(∙) denotes the expectation. Each expectation, respectively, represents the expected sales, excess inventory, and excess demand for any given Q. This model is known as the newsvendor model (or, more commonly, newsboy problem). Note that, for any procurement quantity Q and any realization d of the random demand D,
min(Q, d) = d − max(0,d−Q) and Q = d + max(0,Q−d) − max(0,d−Q).
Hence,
E[min( , )]Q D =E D E[ ]− [max( ,0D Q− )]
and
Q E D E= [ ]+ [max( ,0Q D− )]−E[max( ,0D Q− )].
Substituting these identities in the given equation, the expected profit function can be rewritten as Π( )Q =E D r c[ ]( − − −) (c v E) [max( ,0Q D− )] (− + −r s c E) [mmax( ,0 D Q− )]. (6.5) Interpretation of this function is interesting by itself. The first term is the riskless profit for the equivalent certainty problem that experiences a known demand of E[D]. The second term represents the total expected holding cost, which is the per unit holding (overage) cost of c0 = c − v charged against every unit of excess inventory E[max(0, Q − D)]. And finally, the third term is the total expected shortage (underage) cost, which is the per unit shortage cost of cu = r + s − c (where r − c is the lost sales profit) charged against each unit of the excess demand E[max(0, D − Q)]. In the literature (see Silver and Peterson 1985) total expected cost
L Q( ) (= −c v E) [max( ,0Q D− )] (+ + −r s c E) [max( ,0D Q− )]] (6.6) is known as the single-period loss function. Since riskless profit E[D](r − c), which would occur in the absence of uncertainty, is independent of Q, maximizing Π(Q) is equivalent to minimizing L(Q). Before finding the optimal procurement policy, let us write L(Q) explicitly as
L Q c v Q x dF x r s c x Q
x Q
( ) (= − ) ( − ) ( ) (+ + − ) x Q( − )
= =
∫
0∫
∞ ddF x( ).Taking the derivative of L(Q) with respect to Q and applying Leibnitz’ Rule, the first-order optimality condition can be written as:
(c v− )Pr(D Q≤ ) (− + −r s c)Pr(D Q≥ )=0.
This condition suggests that the optimal procurement quantity S is such that the marginal cost of over-age, which is the probability of a shortage multiplied by the unit overage cost (c − v), is equal to the mar-ginal cost of underage, which is the probability of a shortage multiplied by the unit cost of a shortage (r + s − c).
Solving this equation for Q, the optimal procurement quantity S is found from the fractile formula
F S c
c c F S D S r s c
u r
u o
( )= ( ) Pr( )
+ = ≤ = + −
, that is, +ss v− .
The assumption r > c > v implies that the right-hand side of the formula is greater than 0 and less than 1, F(∙) is a continuous nondecreasing function, and hence a finite positive S always exists. Furthermore, the second derivative of L(Q), (c − v)f(Q) + (r + s − c) f(Q) ≥ 0 for all Q ≥ 0, implies the convexity of L(Q).
In addition, L(Q) has a negative slope at Q = 0, −(r + s − c), and a positive slope, c − v, as Q tends to ∞, imply-ing that L(Q) has a finite minimizer S over (0, ∞).
Sometimes customers order in bulk. In such cases, the number of customers might be low and their demand structure might not assume a continuous distribution. Also, some products such as planes, trains, and so on cannot be ordered in fractions. An airline can order an integral number of jumbo jets, but it does not quite make sense to order 0.11 planes! Hence, the assumption of a continuous demand distribu-tion might not make sense for all cases. Luckily, for the newsvendor model, this is not a problem. If demand distribution F is actually discrete, the above analysis follows similarly with a small adjustment.
The expectation terms in the loss function have to be explicitly represented by summations rather than the integrals. That is, let F be a discrete distribution with probability density function (pdf)
f dj q jj N qj ( )= , = , ,... , j = .
∑
N=1 2 1
and 1
Without loss of generality, one can assume that d1 = 0 and d1 < d2 < … < dN < ∞.
Then, the loss function for Q ∈ [dj, dj+1] for any j = 1, 2, . . . , N is
L Q c v Q d qi i r s c d Q q
i j
i i
i j
( ) (= − ) ( − ) + + −( ) ( − )
= = +
∑
1 11∑
N ,which is a piece-wise linear convex function of Q. Analyzing the first derivative of L(Q), this property can be easily observed:
L Q c v qi r s c q
i j
i i j
N
′ = − − + −
= = +
∑ ∑
( ) ( ) ( )
1 1
= −(c v)Pr(D Q≤ ) (− + −r s c)Pr(D Q> )
which is constant for all Q ∈ [dj, dj+1], meaning that L(Q) is linear over this range. For j = 1, that is, for all Q < d1, the derivative is a negative constant, −(r + s − c) < 0. Hence, L(Q) is a decreasing function at
Q = 0. As j increases, L′(Q) is nondecreasing (increasing if all qj > 0) because Pr(D ≤ Q), which multiplies the positive quantity (c − v), increases or stays the same, and Pr(D > Q), which multiplies the positive quantity (r + s − c), decreases or stays the same. Hence, L(Q) has a nondecreasing first derivative, and thus is a convex function. Since L(Q) is decreasing at Q = 0 and increasing at Q = dN, a minimizer of this function exists.
Finally, realize that when the demand distribution is discrete, the optimal quantity is equal to a possible demand point dj. Furthermore, this demand point is easily found by finding the smallest index such that L′(Q) > 0. Note that, as j increases, L′(Q) increases from a negative value −(r + s − c) to a positive value (c − v). Hence, the optimal procurement quantity S = dz where z is the smallest j such that
(c v) qi (r s c) q
i j
i i j
N
− − + − >
= = +
∑ ∑
1 1
0
There are several tacit assumptions in the earlier analysis: first, there is no initial inventory; second, there is no fixed ordering cost; third, the excess demand is lost; fourth, price is exogenous; and fifth, sal-vage value is guaranteed to be achieved. The first three of these assumptions can easily be dealt with by making some observations in the earlier analysis, but we will discuss the other two assumptions in more detail in the coming subsections.
Let us assume that before the retailer places an order, which costs her a setup cost of k dollars per order (paper work, labor etc.), she realizes that there are I units of the product in her warehouse. If the retailer would like to increase the inventory level to Q, the expected cost of procuring (Q − I) units is k − cI + L(Q), which is still minimized by S if we actually decide to procure any units at all. Setup cost k is only incurred if we decide to procure any item at all, and hence if we do not procure any units on top of I, k is not incurred. Under what conditions should the retailer decide to procure on top of the initial inventory I?
There are two cases: (i) if I > S, no units should be procured, and (ii) if I < S, then the retailer needs to compare the cost of procuring the extra S-I units, that is, k − cI + L(S), with the cost of not procuring any extra units at all, that is −cI + L(I). If k + L(S) < L(I), S-I units should be procured; otherwise, none should be procured.
If we let s be a value such that k + L(S) = L(s), the earlier discussion suggests that the optimal procurement policy is an (s, S) policy. That is, procure S-I if the initial inventory I is less than or equal to s; otherwise. do not procure. Quantity S is known as the order-up-to level, and s is known as the reorder point. Note that, if k = 0, s = S, this kind of a procurement policy is known as the base-stock policy. That is, if the initial inventory level I is less than S, procure S-I; otherwise, do not procure at all.
Let us now consider the case where the excess demand is not lost, but backordered, and the shortage cost not only reflects the loss of goodwill but also the emergency shipment costs. In this case, the single-period loss function is
L Q c v Q x dF x s c x Q dF
x Q
x Q
( ) (= − ) ( − ) ( ) (+ − ) ( − )
= =
∫
0∫
∞ (( ),xwhich is almost identical to the lost sales case except that the shortage cost, s − c, does not include the lost revenue anymore. Hence, the optimal procurement quantity is found from
F S s c ( )=s v− .
−
Example 1
A hot dog stand at Toronto SkyDome, home of the Blue Jays baseball club, sells hot dogs for $3.50 each on game days. Considering the labor, gas, rent, and material, each hot dog costs the vendor $2.00 each.
During any game day, based on the past sales history, the daily demand at SkyDome is found to be nor-mally distributed with mean 40 and standard deviation 10. If there are any hot dogs left at the end of the day, they can be sold at the entertainment district for $1.50 each. If the vendor sells out at SkyDome, she closes shop and calls it a day (lost sales).
(a) If the vendor buys the hot dogs daily, how many should she buy to maximize her profit?
The optimal procurement level S satisfies
F S r s c r s v ( ) = + −
+ −
where r = 3.50, c = 2.00, s = 0, v = 1.50, and F(∙) is normally distributed. That is, S satisfies P(D ≤ S) = 1.5/2.0 = 0.75. Standardizing the normal distribution, we have P(Z < (S − 40)/6) = 0.75. From the normal table or Microsoft Excel, z = 0.675 and S = 40 + 10(0.675) = 46.75. Rounding up, the vendor should procure 47 hot dogs with an expected profit of $53.64.
(b) If she buys 55 hot dogs on a given day, what is the probability that she will meet all day’s demand at SkyDome?
She needs to determine the probability that demand is going to be less than or equal to 55. This is easily done by calculating Pr(D ≤ 55) = Pr(Z ≤ (55 − 40)/10) = Pr(Z ≤ 1.5) = 0.9332. Hence, she has a 93.32%
chance that she will satisfy all the demand at SkyDome and have an expected profit of $51.92.
(c) If we assume that the vendor can purchase hot dogs from the next hot dog stand for $2.50 each in case she sells out her own stock (backorder case), how many hot dogs should she buy?
In the backorder case, the critical fractile is found as (s − c)/(s − v), where s = 2.50. Hence, Pr(D ≤ S) = (2.5 − 2)/(2.5 − 1.5) = 0.5. Standardizing the normal distribution, P(Z <(S − 40)/10) = 0.50. From the normal table or Microsoft Excel, z = 0.0 and S = 40 + 10(0.0) = 40. The vendor should procure 40 hot dogs with an expected profit of $65.98.