Acknowledgments
6.3 Stochastic Models
6.3.2 Joint Pricing and Inventory Control in a Newsvendor Setting
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.
forces might not allow one to charge any price desired. Hence, the retailer needs to solve the nonlinear program
max ( , ) . Π Q r st r r r
UQ≥ ≥L
≥0
The expected single-period profit very much depends on the demand−price relationship. Each demand−price relationship scenario needs independent treatment in the lost sales case. However, a unified approach is possible in the backorder case.
6.3.2.1 Lost Sales Models Additive Demand−Price Relationship
In the joint pricing and procurement problem, minimizing the single-period loss function L(Q) is not equivalent to maximizing the single-period expected profit. Hence, the retailer needs to maximize her profit which is identical to the newsvendor profit in (6.5) except that demand D is replaced by D(r, ε), which is equal to a − br + ε.
Π( , )Q r E D r[ ( , )](r c) (c v) (Q x m r( ))
x
= − − − Q m − −
=
−
ε
0 (( )
( ) ( ) ( )( ( ) ) ( )
r
x Q m r
dF x r s c x m r Q dF x
∫
− + −∫
= − + −∆
As opposed to the exogenous price case, this expected profit function is not necessarily concave for all possible values of the parameters. However, it is shown by Karakul (2007) that if demand distribu-tion satisfies a weak condidistribu-tion, it is still a well-behaved funcdistribu-tion and it has a unique stadistribu-tionary point in the feasible region which is also the unique local maximum. That is, it is a unimodal function. To see this, we first introduce a change of variable u = Q − m(r) which is interpreted as a safety stock fac-tor representing the type 1 service level, that is, the probability of not stocking out. For given u, the service level is F(u), but the procurement quantity Q does not have this one-to-one correspondence with the service level: for given Q, the service level is F(Q − m(r)) and is dependent on the price.
Carrying out this change of variable, the expected profit in terms of u and r is:
Π( , )u r E D r[ ( , )](r c) (c v) (u x dF x) ( )
x
= − − − u − −
∫
=ε
0 ((r s c) (x u dF x) ( )
x u
+ − −
∫
=∆ ,where expectations are taken over the random variable ε. Now, consider the first-order conditions of this function with respect to r and u:
∂
∂
Π( , )u r ∆( ) ( )
r br a bc x u dF x
x u
= − + + + − − =
∫
=2 0 (6.7)
∂
∂
Π( , ) (u r )( ( )) ( ) .
u = + −r s v 1−F u − − =c v 0 (6.8) From Equation 6.7, optimal price r as a function of u is found as:
r u a bc x u dF x bx u
( ) ( ) ( )
.
=
+ + − −
∫
=∆2
Substituting this in (6.8) and assuming that the demand distribution F(∙) has a hazard rate z(∙) = f(∙)/
(1−F(∙)) such that 2z(x)2 + dz(x)/dx > 0 for all x ∈ (0, ∆),* Karakul (2007) shows that there is a unique solution that satisfies the first-order conditions and it corresponds to a local maximum.
Define ∏u = d∏(u, r(u))/du and consider its first and second derivatives
d du f u
b b r u s v F u
u z u
Π / ( ) ( ( ) ) ( ( ))
=− + − − −( )
2 2 1
= + − − −
,
/ ( )/ ( ) ( ( ) ) (
d du df u d u
b b r u s v
2 u 2
2 2 1
Π FF u
z u
f u F u bz u z u ( ))
( )
( )( ( ))
( ) ( )
− 1−
2 2 2 22+
dz u du( )/ .
Note that any stationary point of ∏u (not any stationary point of ∏) needs to satisfy the first-order condi-tion d∏u/du = 0, and hence
d du f u F u
bz u z u
u d udu
2 2
0 2
1
2 2
Π / |Π ( )( ( ))
( ) ( )
/ = = − − 22
0
+
<
dz u du( )/
if 2z(u)2 + dz(u)/du > 0 for all u ∈ (0, ∆). This suggests that all stationary points of ∏u (the total derivative of ∏) are local maxima, which means that it actually has a unique stationary point and it is a local maximum. This implies that ∏u can vanish at most twice over [0, Δ] and consequently, ∏ might have two stationary points, with the larger one being the local maximum over this range.
However, ∏u(0) = r(0) + s − c > 0 and hence ∏u equals zero at most once in (0, Δ), proving the uni-modality of ∏(u, r).
The optimal stocking factor and price (u*, r*) can be found by first solving the nonlinear equation
∂
∂ = + − − − − =
Π( , ( )) ( ( )u r u )( ( )) ( ) u r u s v 1 F u c v 0
with respect to u to obtain u* and then substituting u* in r(u) to obtain r*. The optimal procurement quantity is calculated as S = a − br* + u*.
Example 2
Consider the hot dog stand example. Assuming that excess demand is lost and there is not any competi-tion, the vendor would like to determine the best price and procurement level. Luckily, the vendor has an operations research background and she was able to figure out that the demand is a linear function of the price 100 − 10r + ε, where ε is a random variable with a normal distribution 40 and standard deviation 10.
What is her best price and procurement quantity?
Remember that c = 2, s = 0, and v = 1.5. Solving
∂
∂ = + − − − − =
Π( , ( )) ( ( )u r u . )( ( )) ( . )
u r u 2 1 5 1 F u 2 1 5 0
*Note that all log-concave distribution functions, that is, the distribution functions whose logarithms are concave, satisfy this condition (see An 1995 for a discussion of log-concave distributions which include normal, gamma, Erlang and many other well-known distributions).
for u, we find u* = 54.25. Note that F represents the normal distribution, and it is necessary to use a pack-age like Maple or Matlab to solve this nonlinear equation. The optimal price is r(54.25) = $7.98 and the optimal order quantity is S = 100 − 10 * 7.98 + 54.25 = 74.45. The closest integer value is 74, and hence the vendor should order 74 hot dogs and charge $7.98 each for a total profit of $350.62.
The rounding of the order quantity is not necessarily always up or down. Since, in this case, a continu-ous distribution is used to approximate a discrete one, the integer number that is closest to S is more likely to bring the highest profit. Note that the hot dogs would be quite expensive if there were not competition and the demand−price relationship were given by 100 − 10r. (What would the price be if the demand−price relationship was 100 − 15r?)
Multiplicative Demand−Price Relationship
In the case the demand and price have a multiplicative relationship, the change of variable is somewhat different. We define u = Q/m(r). Substituting D(r,ε) = m(r)ε for D and u = Q/m(r) in the objective func-tion of the newsvendor problem in Equafunc-tion 6.5, the single-period expected profit funcfunc-tion is
Π( , )u r E D r[ ( , )](r c) m r c v( ){( ) (u x d)
x
= − − − u −
∫
=ε
0 FF x r s c x u dF x m
x u
( ) (+ + − ) ( − ) ( )}
=
∫
=∆(( ) [ ](r E r c) (c v) (u x dF x) ( ) (r s c) (
x u
ε − − − − − + −
∫
=0 x u xx u dF x−
∫
=∆ ) ( ) .As in the additive case, this expected profit function is not necessarily concave or convex for all parameter values, but is unimodal for the demand distributions considered earlier. From the first-order condition that ∂∏(u, r)/∂r = 0, the optimal price r as a function of u can be obtained as:
r u bc b
b c v u x dF x s x u
x u
( ) x
( )
( ) ( ) ( ) ( )
= − +
− − + −
= =
∫
1
0 uu
x u
dF x
b x u dF x
∆
∆
∫
∫
− − −
=
( )
( 1 ) ( ) ( )
,
Substituting this into
∂
∂ = + − − − − =
Π( , )u r ( )[( )( ( )) ( )] , u m r r s v 1 F u c v 0
assuming that the demand distribution F(∙) has a hazard rate z(∙) = f(∙)/(1−F(∙)) such that 2z(x)2 + dz(x)/
dx > 0 for all x ∈ (0, ∆), and following similar ideas as in the proof for the additive case, one can show that d∏(u, r(u))/du is increasing at u = 0, decreasing at u = Δ, and is itself a unimodal function over [0, Δ] for b > 2. This proves that there is a unique solution that satisfies the first-order conditions and it corresponds to a local maximum (see Petruzzi and Dada 1999 for a proof). Hence, the optimal stocking factor and price (u*,r*) can be found by first solving the nonlinear equation:
dΠ( , ( ))u r ud ( ( ))[( ( ) )( ( )) (
u =m r u r u + −s c 1−F u − −c vv F u) ( )]=0
with respect to u to obtain u* and then substituting u* in r(u) to obtain r* and in u = Q/m(r(u)) to obtain the optimal procurement quantity S = u*ar*-b.
6.3.2.2 Backorder Models
The analysis of the joint pricing and procurement problem of a single product with random demand follows a different approach when it is assumed that the excess demand is backlogged rather than lost.
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.