TWO-STAGE NEWSBOY MODEL WITH BACKORDERS AND
INITIAL INVENTORY
Ali Cheaitou† , Christian van Delft‡ , Yves Dallery†‡ and Zied Jemai†
†
Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes,
92295 Chatenay-Malabry Cedex, France, [email protected], [email protected], [email protected]
‡
HEC Paris, 1 Rue de la Libération, 78351 Jouy-en-Josas Cedex, France, [email protected]
Abstract: In this paper, we develop a general two-stage newsboy model. Each period decision
induces specific costs. In addition to the usual decision variables for such models, we consider
that, at the beginning of the decision process, an initial inventory is available and some preliminary fixed orders are to be delivered at each period. The unsatisfied demands during a period
are backlogged to be satisfied in the future. The model is solved by a dynamic programming
approach. We then provide insight regarding this type of two-stage inventory decision process
with the help of numerical examples.
Keyword: Supply Chain Planning; Dynamic Programming; Demand Management; Modelling
Methodologies.
1
Introduction
The single-stage newsboy model has received a lot of attention in operations research and operations management literature. Basically, the numerous versions of this model determine, under
different assumptions, the orders and inventory quantities that optimally satisfy an uncertain
future demand. Given the intrinsic simplicity of this class of inventory model, the optimal
solutions can often be explicitly given under an analytic form.
In many real-life applications, such simple single-stage models do not really apply because several correlated decisions have to be sequentially taken. It is thus quite natural to consider two-period newsboy models to analyze the structure of optimal decisions in such multiperiod decision processes. This class of models is characterized by several features: the structure of demand uncertainty, the structure of the decision process and related costs and the way
unsatisfied demands at a given period are dealt with.
In this paper, we consider style-goods type products with a short life cycle. For this kind of
product, a two-period decision model appears quite naturally. The induced costs are purchasing
costs, inventory holding costs and backorder costs. The demands at the first and second period
are described by independent random variables, with known probability distributions. We assume that at the end of the season, the remaining inventory can be sold to a specific market with
a given salvage value.
In the literature about style-goods production and inventory problem, most of the models
are a newsboy single-period problems (Khouja 1999). However, several two-stage extensions
have been developed. All these papers exploited the two-period horizon in order to improve the
inventory management process facing the demand uncertainty. Such two-period decision processes permit one to adapt the inventory levels to the demand variability. In other words, using a
single period it is ordered only once, at the beginning of the season before information about the
effective demand is available. On the contrary, in a two-period model, after the first order, the
realized demand of the first period can be observed and a second order is made, which clearly
exploits this information. Several authors have considered such models. First, Hillier Lieberman (2001) analyzed a two-period model with uniformly distributed independent demands. Via
a dynamic programming approach, these authors analytically solved this model and proposed
an explicit optimal order-up-to policy. Lau Lau (1997,1998) developed lost sales two-period
models and proposed numerical solutions via dynamic programming. Bradford Sugrue (1990)
proposed another class of model in which the second period demand is correlated to the first
period demand. A bayesian update for the second period’s demand forecast can thus be used
after having observed the value of the first period demand. These authors determined a conditional order-up-to policy for the second period and an optimal order quantity for the first period.
Another important two-period model has been proposed by Fisher Raman (1996). In this paper
the demand of the whole horizon and the demand of the first period are characterized via a joint
probability density function. Furthermore, the order size for the second period is constrained
by a limited amount. Gurnani Tang (1999) considered a two-period model with a first period
demand equal to zero. In their model, the dynamic structure concerns available information for
the sequential decisions: at the end of the first period, exogenous information is collected which
permits one to update the forecast for the second period demand. Choi, et al. (2003) proposed a
quite similar two-stage newsboy model with an update of the forecast of the second-period demand via some market information. Donohue (2000) applies a similar approach for developing
supply contracts.
The contributions of our model are the followings:
• First, the periodic ordering process is quite general in the sense that at each time period
orders can be made for the different subsequent periods, possibly with different costs,
• Second, the periodic selling process is quite general, in the sense that, in addition to the
classical selling process, it is possible, at the beginning of each period, to sell a part of
the available inventory to a parallel market, at a given salvage value,
• Third, the data are dynamic : the selling prices, costs, salvage values and demand probability distributions are period-dependent,
• Fourth, the model includes initial inventory and initially fixed order quantities to be delivered in the different periods.
The remaining part of this paper is structured as follows: the second section describes the model
(namely the complete decision process, the information structure and the costs and profits structure), the third section details the objective function and the dynamic programming approach.
Numerical examples are solved in section four. The last section is dedicated to the conclusion.
2
2.1
Decision process and information structure
Demand processes description
Define D1 and D2 as the demand at the first and the second period respectively. These random
variables are characterized by probability distributions F1 (·) and F2 (·) and probability density
functions f1 (·) and f2 (·).
2.2
Decision process description
First, the state variables of the model are the inventory level at the beginning of each period,
X1 and X2 and the inventory level at the end of each period I1 and I2 (I0 being the given initial
inventory for the problem). The decision variables of the model are as follows. First, we define
Qts as the quantity ordered at the beginning of period t to be received at the beginning of period
s (with t ≤ s and t, s = 1, 2, 3). Then we introduce, for each period, the variable St , the
quantity that is salvaged (to the parallel market) at the beginning of period t (with t = 1, 2, 3).
We will show that the decision variables Q33 and S3 can be optimally chosen directly as explicit
functions of the other variables.
Figure 1 presents the structure of the decision process and demand realization, which is
the following:
the available inventory at the beginning of the first period, before current orders are decided and
demand occurs, is
X1 = I0 + Q01 ,
where, in fact, I0 and Q01 can be considered as data. Then decision variables Q11 , Q12 and S1
are fixed. Then, the demand D1 occurs and the available inventory at the end of the first period
is given by
I1 = X1 + Q11 − S1 − D1 .
The decision structure for the second period is similar, with the initial inventory given by
X2 = I1 + Q02 + Q12 ,
and the final inventory given by the expression
I2 = X2 + Q22 − S2 − D2 .
The terminal decision process is then as follows: after demand occurs in the second period, it
is optimal to order Q33 = −I2 units to satisfy backlogged demand (when I2 < 0) or to sell
S3 = I2 units with a salvage value to eliminate the remaining inventory (when I2 ≥ 0).
2.3
Costs and profits structure and assumptions
In each period, any demand is charged at a price Pt , even if not immediately delivered. The
unit order cost of Qts is cts . In the case of a positive inventory at the end of a period, an
inventory holding cost ht is paid. Unsatisfied orders in period t are backlogged to the next
period, with a penalty bt . We will show that under the assumptions of this paper, it is optimal
that all backlogged orders in a given period be satisfied at the beginning of the next period. The
unit salvage value at the beginning of period t is given by st . It is necessary to introduce some
-D
-S
+Q )
I
I
(s )
X
I
(P
-D
h
b
-S
(c
)
)
+Q +Q
(c
I
I
(s )
X
I
+Q
(P
h
b
-S (s )
I
)
+Q (c
)
+Q (c
)
Figure 1: Decision process
assumptions about the different periodical costs in order to guarantee the coherence and interest
of our model. These assumptions could be classified in three categories:
2.3.1
Type 1 assumptions:
c11 <c22 + b1 , c11 <c12 + b1 , c12 <c33 + b2 and c22 <c33 + b2 .
These constraints aim at avoiding situations with systematic backlogs of demands to the next
period. For example, if the first constraint is not satisfied, the optimal policy will consist of
backlogging the first period demand to the second period and to satisfy this demand with a
second period order (with c22 as unit order cost).
2.3.2
Type 2 assumptions:
s2 <c11 + h1 , s3 <c12 + h2 , s3 <c11 + h1 + h2 and s3 <c22 + h2 .
These constraints aim at avoiding situations where it would be profitable to order at a given
period in order to sell to the parallel market at a salvage price. For example, if the first constraint
is not satisfied, the optimal policy will consist of ordering an infinite Q11 quantity in the first
period and selling it at a salvage price s2 in the second period.
2.3.3
Type 3 assumptions:
s1 <c11 , s2 <c22 , s2 <c12 and s3 <c33 .
These constraints aim at avoiding other situations where it would be profitable to order at a given
period and to sell at the delivery period to the parallel market at the corresponding salvage price.
For example, if the first constraint is not satisfied, the optimal policy will consist of ordering an
infinite quantity in the first period and selling it at a salvage price s1 in the same period.
2.4
Terminal conditions
The optimal value of the decision variables Q33 and S3 can be shown to be explicit functions of
the state variable I2 as follows (Cheaitou, et al.,2005):
if I2 ≤ 0 ⇒ Q∗33 = −I2 and S3∗ = 0,
if I2 ≥ 0 ⇒ Q∗33 = 0 and S3∗ = I2 .
3
3.1
(1)
(2)
The dynamic programming approach
The model
First, we recall the equilibrium equations of the system,
X1 = I0 + Q01
(3)
I1 = X1 + Q11 − D1 − S1
(4)
X2 = I1 + Q02 + Q12 = X1 + Q11 − D1 − S1 + Q02 + Q12
(5)
I2 = X2 + Q22 − D2 − S2
(6)
Introduce Π(I0 , Q01 , Q02 , Q11 , Q12 , Q22 , Q33 , S1 , S2 , S3 ) as the expected profit with respect to
the random variables D1 and D2 . This expected profit Π(·) is formulated as follows,
Π(I0 , Q01 , Q02 , Q11 , Q12 , Q22 , Q33 , S1 , S2 , S3 ) =
Z ∞
D1 f1 (D1 ) dD1 + s1 S1 − c11 Q11 − c12 Q12
P1
0
Z X1 +Q11 −S1
− h1
(X1 + Q11 − S1 − D1 )f1 (D1 ) dD1
0
Z ∞
− b1
(D1 − X1 − Q11 + S1 )f1 (D1 ) dD1
X1 +Q11 −S1
Z ∞
+ P2
D2 f2 (D2 ) dD2 + s2 S2 − c22 Q22
0
Z X2 +Q22 −S2
− h2
(X2 + Q22 − S2 − D2 )f2 (D2 ) dD2
0
Z ∞
− b2
(D2 − X2 − Q22 + S2 )f2 (D2 ) dD2
(7)
X2 +Q22 −S2
X2 +Q22 −S2
Z
+ s3
(X2 + Q22 − S2 − D2 )f2 (D2 ) dD2
Z0
∞
− c33
(D2 − X2 − Q22 + S2 )f2 (D2 ) dD2
X2 +Q22 −S2
3.2
Problem decomposition
Using a dynamic programming approach, this problem can be decomposed into two one-period
subproblems that are the followings. The first subproblem is associated to the second period.
The solution of this problem is optimal value of the second-period decision variables, namely
Q∗22 and S2∗ . These variables are expressed as a function of the state variable X2 and are computed as the solution of the optimization problem
max {Π2 (X2 , ξ2 (X2 ))} ,
ξ2 (X2 )
(8)
where we formally have ξ2 (X2 ) = (Q22 (X2 ), S2 (X2 )).
Then, the second subproblem exploits ξ2∗ (X2 ) in order to find the optimal policy for the
first period, namely ξ1∗ (X1 ) = (Q∗11 (X1 ), Q∗12 (X1 ), S1∗ (X1 )). This optimal policy is obtained as
the solution of the problem
max {Π1 (X1 , ξ1 (X1 )) + ED1 {Π∗2 (X2 , ξ2∗ (X2 ))}} ,
ξ1 (X1 )
(9)
where Π1 (X1 , ξ1 (X1 )) is the expected first period profit function, while the second term is the
expectation, with respect to D1 , of the second period profit function, under the optimal policy
ξ2∗ (X2 ).
3.3
Second-period subproblem.
The objective function of the second period is defined by the following expression:
½ Z ∞
Π2 (X2 , Q22 , S2 ) =
P2
D2 f2 (D2 ) dD2 + s2 S2 − c22 Q22
0
Z X2 +Q22 −S2
−h2
(X2 + Q22 − S2 − D2 )f2 (D2 ) dD2
Z 0∞
−b2
(D2 − X2 − Q22 + S2 )f2 (D2 ) dD2
X2 +Q22 −S2
X2 +Q22 −S2
Z
+s3
(X2 + Q22 − S2 − D2 )f2 (D2 ) dD2
0
Z
¾
∞
−c33
(D2 − X2 − Q22 + S2 )f2 (D2 ) dD2
(10)
X2 +Q22 −S2
This class of model has been analyzed by Cheaitou, et al. (2005). These authors have shown
that the objective function defined by (10) is a concave function of Q22 and S2 . Furthermore,
the following properties have been proven.
∗
Property 1: The optimal values of the two decision variables Q∗22 and S22
can not be
simultaneously positive.
Property 2: If I1 < 0 and X2 < 0, the optimal quantity Q∗22 satisfies
X2 + Q∗22 ≥ 0.
Optimal policy: From Cheaitou, et al. (2005) the second period optimal policy is given
by
½
if X2 <
Y1∗
⇒
Q∗22 = Y1∗ − X2 ,
S2∗ =
0,
½
if
Y1∗
Q∗22 = 0,
S2∗ = 0,
(12)
Q∗22 =
0,
∗
S2 = X2 − Y2∗ .
(13)
Y2∗
≤ X2 ≤
(11)
⇒
and
½
if X2 >
Y2∗
⇒
These conditions amount to
Q∗22 = max (Y1∗ − X2 ; 0) and S2∗ = max (X2 − Y2∗ ; 0)
µ
with
Y1∗
=
F2−1
b2 + c33 − c22
b2 + c33 + h2 − s3
µ
¶
and
Y2∗
=
F2−1
b2 + c33 − s2
b2 + c33 + h2 − s3
Note that it is easily seen that under the assumptions of this paper, one has
Y1∗ < Y2∗ .
(14)
¶
.
(15)
3.4
First period subproblem.
Using the results of the second period subproblem, it is possible to numerically solve the first
period subproblem and compute the optimal values of the decision variables of the first period,
Q11 , Q12 and S1 . The total expected profit function Π(·), under the optimal second-period
policy ξ2∗ (X2 ) becomes:
Π(I0 , Q01 , Q02 , Q11 , Q12 , S1 ) = Π1 (X1 , Q11 , Q12 , S1 ) + ED1 {Π∗2 (X2 , Q∗22 (X2 ), S2∗ (X2 ))} (16)
The optimization problem to solve for this subproblem is then the following:
Π∗ (I0 , Q01 , Q02 ) =
=
max
Q11 ,Q12 ,S1
{Π(I0 , Q01 , Q02 , Q11 , Q12 , S1 )}
½ Z ∞
P1
D1 f1 (D1 ) dD1 + s1 S1 − c11 Q11 − c12 Q12
max
0
Z X1 +Q11 −S1
−h1
(X1 + Q11 − S1 − D1 )f1 (D1 ) dD1
0
Z ∞
−b1
(D1 − X1 − Q11 + S1 )f1 (D1 ) dD1
Q11 ,Q12 ,S1
X1 +Q11 −S1
+ED1 {Π∗2 (X2 , Q∗22 (X2 ), S2∗ (X2 ))}}
(17)
By using the optimal values Q∗22 (X2 ) and S2∗ (X2 ) defined by (14), it is easily seen that the
total expected profit function Π(I0 , Q01 , Q02 , Q11 , Q12 , S1 ) is concave w.r.t. Q11 , Q12 and S1 ,
in such a way that the optimization problem described in equation (17) has a unique maximum.
However, no closed form formula exists for general demand probability distributions and the
problem has to be numerically solved.
4
Numerical examples
In this section we show, via some numerical applications, how our model gives some insight
regarding this type of general two-stage inventory decision process. In the first example we
show the behavior of the first period optimal policy in function of the initial inventory and of
the salvage value s1 . In the second example we show the impact of the variability and of the
difference between the costs on the optimal value of Q12 and the expected optimal value of
Q22 . For the two following examples we will suppose normal distributed demands for the two
periods.
4.1
Example 1
Q11
S1
800
700
600
Q 11, S 1
500
400
300
200
100
0
100
500
900
1300
1700
2100
2500
2900
Initia l inventory I 0
Figure 2: First period optimal policy - low s1 value
In the first example we consider the following parameter values: D1 = N [1000, 200],
D2 = N [1000, 200], h1 = 5, h2 = 5, P1 = 100, P2 = 100, b1 = 25, b2 = 25, c11 = 50,
c12 = 30, c22 = 50, c33 = 50, s2 = 20, s3 = 20, Q01 = 0 and Q02 = 0, with N[µ,σ] a normal
distribution with mean of µ and standard deviation σ. In this example, the optimal policy is
numerically computed as a function of the initial inventory I0 . In the first part (Figure 2) we
have s1 = 20 and in the second part (Figure 3) we have s1 = 29. It is clearly seen, from this
Q11
S1
1800
1600
1400
Q 11, S 1
1200
1000
800
600
400
200
0
100
500
900
1300
1700
2100
2500
2900
Initial inventory I 0
Figure 3: First period optimal policy - high s1 value
numerical example, that the first period optimal policy has the same structure as the second
period, for which the explicit form is known. From figures 2 and 3 we deduce that there are
two thresholds. For the values of I0 between these two thresholds, the optimal values for the
decision variables Q11 and S1 are both equal to zero. Below the low threshold only the optimal
value of Q11 is positive and above the high threshold only the optimal value of S1 is positive.
The higher threshold value depends on s1 (see Figure 2 and Figure 3). When s1 increases the
higher threshold value decreases. It can be seen that the low threshold value depends on c11 ,
while the difference between the two thresholds is proportional to the difference c11 − s1 .
4.2
Example 2
Q11
Q12
Q22
3000
Q 11, Q 12 , Q 22
2500
2000
1500
1000
500
0
23
29
35
41
47
53
59
65
71
The cost c12
Figure 4: Early and late reorder modes: optimal values with low D1 variability
In the second example, we have the following parameter values : D2 = N [1000, 200],
h1 = 5, h2 = 5, P1 = 100, P2 = 100, b1 = 25, b2 = 25, c11 = 50, c22 = 50, c33 = 50, s1 = 20,
s2 = 20, s3 = 20, I0 = 0, Q01 = 0 and Q02 = 0. In this example, the optimal values are
computed as a function of the unit ordering cost c12 . For the first part of this example (Figure 4)
we consider the first period demand with a low variability, namely we have D1 = N [1000, 200].
For the second part (Figure 5) we suppose a higher demand variability, i.e. D1 = N [1000, 400]
It can be seen that the c12 values can be divided into three intervals.
• The first interval corresponds to small c12 values, for which the optimal value of Q12 is
positive and the expected optimal value of Q22 is equal to zero.
Q11
Q12
Q22
3000
Q 11, Q 12 , Q 22
2500
2000
1500
1000
500
0
23
29
35
41
47
53
59
65
71
The cost c12
Figure 5: Early and late reorder modes: optimal values with high D1 variability
• The second region corresponds to mean c12 for which the optimal values of Q12 and Q22
are both positive.
• The third region corresponds to the situation c12 > c22 , where the optimal value of Q12 is
equal to zero.
Another insight can be seen with respect to variability of D1 . When D1 variability increases
several phenomenons occur, see figures 4 and 5:
• The width of the second region (for medium c12 values) increases toward the low c12
values side. In fact Q22 is decided after observing the realization of D1 and therefore this
decision benefits from the realization information. Q12 is fixed before the realization of
D1 . So when the variability of D1 increases it is profitable to wait for realization of the
first period demand and then to use this information to optimize the reorder, and this even
if the cost (c22 ) is high.
• For low c12 values, the optimal value of Q12 increases with demand variability while the
optimal value of Q11 decreases. This corresponds to the different numerical costs values
that we have defined. In our case, for small c12 values, the differences between c11 and
c12 and between c12 and c22 are important, in such a way that for high D1 variability,
it becomes more profitable to order a small Q11 value and a bigger Q12 quantity, and if
necessary reorder a Q22 quantity in period 2. So in this case Q12 increases to face the first
period variability and to satisfy the second period demands.
• For medium c12 values, the optimal Q11 and Q12 values decrease while the expected
optimal Q22 value increases to face the variability of the first period demand and to satisfy
the second period demand.
• For high c12 values, the optimal value of Q12 is equal to zero (c12 > c22 ). When the
variability of D1 increases, the optimal Q11 value increases and the expected optimal Q22
value decreases. This is associated to the fact that c11 is equal to c22 in our example, and
that the holding inventory cost in the first period is very small with respect to the shortage
cost b1 . So in this case, Q11 faces the high D1 variability and satisfies a part of the second
period orders.
5 Conclusion
In this paper, we have analyzed a general two-period newsboy model, including initial inventory,
pre-fixed orders and periodic salvage opportunities. Via a dynamic programming approach, we
characterized the structure of the optimal policy and provided a numerical approach in order to
compute this optimal policy in a general setting. Via the numerical examples, we gave some
insight into the effect of the parameters values on the solution.
6
Biography
Ali Cheaitou studied mechanical engineering at the Lebanese University(Beirut). In 2004 he
prepared a DEA in industrial engineering at Ecole Centrale Paris(France). Since October 2004,
he is PhD student and teaching assistant in the Laboratoire Génie Industriel of the Ecole Centrale Paris. He is mainly interested in production planning and stochastic optimization.
Christian van Delft is an associate professor in the department of Industrial Management
and Logistics, at HEC-Paris School of Management (France), where he teaches and researches
in operations management, operations research and quality control. He has published in IEEE
Transactions on Robotics and Automation, Annals of Operations Research, European Journal
on Operations Research, Optimal Control: Applications and Methods and others.
Yves Dallery is a Professor of Industrial Engineering at /Ecole Centrale Paris/. His research interests are in production management, supply chain management and service operations management, with a special emphasis on modelling and optimization.
Zied JEMAI is an assistant Professor in the department of Industrial Engineering, at Ecole
Central Paris, where he teaches and researches in stochastic models and supply chain management. He has published in IIE Transactions and European Journal of Operational Research.
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