Acknowledgments
6.3 Stochastic Models
6.3.3 Multiple Period Models
As in the discussion of the backorder case in the newsvendor problem, the per unit shortage cost is now represented by s and it does not consider the loss of profit (r − c). Note that s does not only represent the loss of goodwill but also the cost of fulfilling the unmet demand with an emergency order and s > c is a reasonable assumption. Furthermore, by defining h = h+ − v as the per unit adjusted holding cost (which can be a negative value because it is defined as the real holding cost h+ minus the salvage v) and realizing that the expected sales is equal to the expected demand, the single-period profit is:
Π( , )Q r =E rD r[ ( , )]ε −cQ E h− [ max( ,0Q D r− ( , ))ε +smaxx( , ( , )0 D rε −Q)].
For some specific demand−price relationships, further analysis is possible. Let the demand function satisfy D(r, ε) = αm(r) + β, where ε = (α, β), α is a non-negative random variable with E[α] = 1 and β is a random variable with E[β] = 0. By scaling and shifting, the assumptions E[α] = 1 and E[β] = 0 can be made without loss of generality. Furthermore, assume that m(r) is continuous and strictly decreasing, and the expected revenue R(d) = dm−1(d) is a concave function of the expected demand d. Note that D(r, ε) = a − br + β (a > 0, b > 0) and D(r, ε) = α ar−b (a > 0, b > 1) are special cases that satisfy these conditions.
Since there is a one-to-one correspondence between the selling price r and the expected demand d, the single-period expected profit function can be equivalently expressed as:
Π( , )Q d =R d( )−cQ E h− [ max( ,0Q−αd− +β) smax( ,0αd+β −−Q)]
Observing that h max(0, y) + s max(0, −y) is a convex function of y, one can see that h max (0, Q − αd − β) + s max(0, αd + β − Q) is a convex function of (Q, d) for any realization of α, β (see Bazaraa et al. 1993, page 80). Furthermore, taking the expectation over α and β preserves convexity and hence, H(Q) = E[h max(0, Q − αd − β) + s max(0, αd + β − Q)] is convex in (Q, d). This proves that ∏(Q, d) is a concave function and the optimal expected demand, d*, and procurement quantity, Q*, can be obtained from the first-order conditions. Optimal price is determined as r* = m-1(d*). In the existence of initial inventory, it is shown by Simchi-Levi et al. (2005) that the optimal procurement quantity is deter-mined by a base-stock policy. That is, if the initial inventory I is less than the optimal procurement level S, then we replenish our stock to bring the inventory level up to S; otherwise, we do not order. The optimal price is determined as a nonincreasing function of the initial inventory level.
There are several extensions to the given single-period joint pricing and inventory control problems with stochastic demand. Karakul and Chan (2004) and Karakul (2007) consider the case in which the excess inventory is not salvaged for certain, but they are sold at a known discounted price to a group of clients who exhibit a discrete demand distribution for this excess stock. This case is known as the newsvendor problem with pricing and clearance markets. Cachon and Kok (2007) analyze the importance of estimating the salvage price correctly. Karakul and Chan (2007) consider the product introduction problem of a company which already has a similar but inferior product in the market. Authors consider a single-period model that maximizes the expected profit from the optimal procurement of these two products and the optimal pricing of the new product. A detailed review of the inventory control of substi-tutable products that include the seats in flights, hotel rooms, technologically improved new products, fashion goods, etc. can be found there.
Although the analysis can be carried out for time-varying demand distributions, for the sake of sim-plicity, we assume that demand at each period D is independent and identically distributed following the continuous distribution F(∙) defined over a bounded non-negative region (0, Δ). We focus on backorder models in this section.
Although the costs involved in this model are very similar to the newsvendor model, they might have a different interpretation. Initially, let us assume that setup cost k is zero. There is a non-negative holding cost h for each unit of the excess inventory at the end of each period; this can be thought of as the capital, insurance, and handling cost per unit carried in inventory. For each unit of demand that is not met at the end of the period, the retailer incurs s dollars of backorder penalty cost.
Since the price is exogenous, the retailer needs to determine the optimal procurement quantities Qt for t = 0, 1, 2 . . . , N − 1 that minimize the total expected cost
TC(Q)→ = cQt+hEmax It+Qt−D sE D
( ,0 ) + max( ,0 −− −
{
}
=
∑
− I Qt t tN
) .
0 1
The most natural and appropriate technique to solve this problem is Dynamic Programming (DP). The appropriate DP algorithm has the following cost-to-go function:
J It
( )
t =minQt≥0 cQt+H I(
t+Qt)
+E J t+1(
It+Qt−D)
{
}
, (6.9)where
H y( )=hEmax( ,y D− ) sE max( ,D y)
+ −
0 0 .
The cost-to-go function represents the minimum expected cost from periods t, t + 1, . . . , N − 1 for an initial inventory of It at the beginning of period t and optimal procurement quantities Qj, j = t, t + 1, . . . , N − 1. Note that the inventory at the beginning of period t + 1 is found as It+1 = It + Qt − d, where d is a realization of the demand variable D. Assuming that any excess inventory at the end of period N is worth nothing, the DP algorithm has the boundary condition:
J IN
( )
N =0.A change of variables is useful in analyzing (6.9). We introduce the variable yt = It + Qt that represents the inventory level immediately after the order in period t is placed. With this change of variable, the right-hand side of Equation 6.9 can be rewritten as:
miny xt≥t
{
cyt+H y( )
t +E J[ t+1(yt−D)]}
−cIt.The function H is easily seen to be convex because, for each realization of D, max(0, y − D) and max(0, D − y) are convex in y and taking the expectation over D preserves convexity. If we can prove that Jt+1 is convex, the function in the curly brackets, call it Gt(yt), is convex as well. Then the only result that remains to be shown is lim|y|→∞Gt(y) = ∞ which proves the existence of an unconstrained minimum St. If these properties are proven, which we will do shortly, then a base-stock policy is optimal. That is, if St is the unconstrained minimum of Gt(yt) with respect to yt, then considering the constraint yt = It, a mini-mizing yt equals St if It≤ St and equals It otherwise. Using the reverse transformation Qt = yt − It, the mini-mum of the DP Equation 6.9 is attained at Qt = St − It if I t≤ St, and at Qt = 0 otherwise. Hence, an optimal policy is determined by a sequence of scalars {S0, S1, …, SN−1} and has the form
Q I S I I
I S
t t t t
t t
∗( ) ,
= ,− <
≥
t St
if if 0
(6.10)
where each St t = 0, 1, . . . , N − 1 solves
Gt(y) = cy + H(y) + E[Jt+1(y − D)].
The earlier-discussed convexity and existence proofs are done inductively. We have JN = 0, so it is convex.
Since s > c and the derivative of H(y) tends to −s as y → −∞, GN−1(y) = cy + H(y) has a negative derivative as y → −∞ and a positive derivative as y → ∞. Therefore, lim|y| → ∞GN−1(y) = ∞ and the optimal policy for period N − 1 is given as:
Q I S I I
N N N N
− − = − − − <
1 1 1 1
0
∗ ( ) , if 1 S 1
, N N
− −
if IN− ≥SN−
1 1 ,
where SN-1 minimizes GN-1(y). From the DP Equation 6.9 we have
J I c S I H S I S
N N N N N N N
− − = − − − + − − < −
1( 1) ( 1 1) ( 1), if 1 11
H I( N−1), if IN−− ≥ −
1 SN 1 ,
which is a convex function because: first, both H(IN−1) and c(SN−1 − IN−1) + H(SN−1) are convex; second, it is continuous; and, finally, at IN−1 = SN−1 its left and right derivatives are both equal to −c. For IN-1 < SN−1, JN-1 is a linear function with slope −c and, as IN-1 approaches SN-1 from the right-hand-side, its derivative is −c because SN-1 minimizes the convex function cy + H(y) whose derivative c + H′(y) vanishes at y = SN-1 (see Fig. 6.5).
Note that if the initial inventory at the beginning of period N − 1 is greater than the unconstrained minimizer SN-1, we do not order any more and hence do not incur any extra procurement cost but, rather, face the possible holding or shortage cost H. On the contrary, if the initial inventory is less than the unconstrained minimizer SN-1, then we procure enough to increase the on-hand inventory level to SN-1. Hence, we incur not only the procurement cost c(SN-1 - xN-1) but also the possible holding or shortage cost H(SN-1).
Hence, given the convexity of JN, we proved that JN-1 is convex and lim|y|→∞ JN−1(y) = ∞. This argument can be repeated to show that if Jt+1 is convex for t = N − 2, N − 3, . . . , 0, lim|y|→∞Jt+1(y) = ∞, and lim|y|→∞Gt(y) = ∞, then
J I c S I H S E J S D
t t
t t t t t
( ) ( ) ( ) ( ) ,
= − + + −
+1 if II S
H I E J I D
t t
t t t
<
+ −
( ) +1( ) , if It≥St
,
FIGuRE 6�5 Structure of the cost-to-go function.
–cy SN–1
H(y) cSN–1
cy+H(y)
y –cy SN–1
JN–1(IN–1)
IN–1
where St minimizes cy + H(y) + E[Jt+1(y − D)]. Furthermore, Jt is convex, lim|y|→∞Jt(y) = ∞, and lim|y|→∞Gt−1(y) = ∞. This completes the proof that Jt is convex for all t = 0, 1, . . . , N = 1 and a base-stock policy is optimal.
Analysis is more complicated in the existence of a positive setup cost k.
6.3.3.1 Positive Setup Cost
If there is a setup cost for any non-negative procurement quantity Qt, then the procurement cost is:
C Q k cQ Q
( ) , Q
= + >.
=
if if ,
0
0 0
The DP algorithm has the following cost-to-go function:
J It
( )
t =minQt≥0 C Q( )t +H I Q(
t+ t)
+E J t+1(
I Q Dt+ −t)
{
}
, (6.11)with the boundary condition JN(IN) = 0.
Considering the functions Gt(y) = cy + H(y) + E[Jt+1(y − D)] and the piecewise linear procurement cost function C(Q),
Jt(It) = min{Gt(It), min Qt > 0[k + Gt(It + Qt)]} − cIt, or by the change of variable yt = It + Qt,
Jt(It) = min{Gt(It), min y1 > It[k + Gt(yt)]}−cIt.
If Gt can be shown to be convex for all t = 0, 1, …, N − 1, then it can be easily seen that an (s, S) policy will be optimal. That is,
Q I S I I
I s
t t t t
t t
∗( )= − , <
≥ if if
s
, t t
0
would be optimal, where St minimizes Gt(y) and st is the smallest y value such that Gt(y) = k + Gt(St).
Unfortunately, if k > 0, it is not necessarily true that Gt is convex. However, it can be shown that Gt is still a well-behaved function that satisfies the property
k G z y G y z G y G y b
t t t bt
+ + ≥ + − −
( ) ( ) ( ) ( ) ,, for all z≥0,b>0, .y
Since the proof is mathematically involved, we skip the proof and refer the interested readers to Bertsekas (2000). Functions that satisfy the stated property are known as K-convex functions. There are several properties of K-convex functions, which we provide in the next lemma without its proof [for proofs, see Bertsekas (2000), pp. 159−160], that help us show that the (s, S) policy is still optimal in the existence of a non-negative fixed ordering cost.