Rhodium-catalyzed borylative carbon monoxide reduction to gem-diborylmethane

Article abstract of DOI:10.1016/j.catcom.2020.106205

Herein, we developed a rhodium-catalyzed reduction of CO with bis(pinacolato)diboron (B₂pin₂) under atmospheric pressure of CO with silane as the hydride source, gem-diborylmethane [H₂C(Bpin)₂] as a versatile and fundamental C1 compound can be formed. Notably, this is the first example on transition metal-catalyzed borylation of CO.

Full text of DOI:10.1016/j.catcom.2020.106205

Scientia Iranica E (2011) 18 (3), 711–721
Sharif University of Technology
Scientia Iranica
Transactions E: Industrial Engineering
www.sciencedirect.com
A multi-stage stochastic programming model for dynamic pricing
and lead time decisions in multi-class make-to-order firm
S.K. Chaharsooghia,∗, M. Honarvara, M. Modarresb
a Department of Industrial Engineering, Tarbiat Modares University, Tehran, P.O. Box 14115-11, Iran
b Department of Industrial Engineering, Sharif University of Technology, Tehran, P.O. Box 14588-89694, Iran
Received 22 February 2010; revised 29 November 2010; accepted 4 January 2011
KEYWORDS
Abstract Make-to-order firms use different strategies, such as dynamic pricing and due date manage-
ment, to influence their performance. In these strategies, orders are segmented into classes based on their
sensitivity to lead time and price. Quoting different prices and lead times to different classes of customer
can increase a firm’s profit and its capacity utilization. Most research in this area does not consider the
effects of production constraints on price and lead time decisions. In this paper, we consider the role of
flexibility in dynamically choosing the price, lead time and segmentation of customers in make-to-order
environments with limited production capacity and multi-period horizon under a stochastic demand func-
tion. To reflect the dynamic variations of a system’s conditions, we propose a Multi-stage Stochastic Pro-
gramming (MSP) method to jointly determine prices, lead time and production. Furthermore, we assume
that demand is a linear function of price, lead time and time. Through numerical analyses, we indicate
the benefits of dynamic pricing and lead time decisions, based on different customer classes in various
environments.
Pricing;
Lead time;
Production;
Stochastic programming;
Multi-stage.
© 2011 Sharif University of Technology. Production and hosting by Elsevier B.V.
Open access under CC BY-NC-ND license.
1. Introduction
While revenue management techniques and dynamic pric-
ing policies have been successfully applied by airlines, hotels
and retail chains, it has great potential also for manufactur-
ing environments [2–4]. Some benefits of revenue management
and pricing include potential increases in profit and improve-
ments, such as a reduction in demand or production variability,
which results in more efficient supply chains.
In revenue management, segmentation of orders is based
on their sensitivity to price. Moreover, the segmentation and
quotation lead times, based on an order’s time sensitivity, is a
managerial challenge in make-to-order environments.
In Make-To-Order (MTO) environments, various attributes
of the product, such as price and lead time, are evaluated by
the buyer. Thus, for each customer, the firm should deter-
mine the due date and price based on customer preference,
available capacity and other potential orders that could de-
mand resources [5]. In fact, by using both lead time differ-
entiation and price differentiation in a revenue management
system, manufacturers can coordinate pricing and production
decisions to improve the firm’s performance, and increase the
overall capacity utilization by dynamically differentiating prod-
ucts. Dell Computers and Amazon.com are examples of com-
panies that separate their customers into different classes and
change prices based on parameters, such as demand variation,
inventory levels or production schedules [6].
Today’s world is more competitive than ever before. For
this reason, revenue management and pricing policies are
the most effective tools that managers can use to influence
market demand, and balance supply and demand. Revenue
management is the art of maximizing profit from a limited
capacity of products over a finite horizon. This is done by selling
each product to the right customer at the right time for the
right price [1]. The application of revenue management has
been most effective when demand can be segmented, and price
sensitivity varies across these segments [2].
∗
Corresponding author.
E-mail address: skch@modares.ac.ir (S.K. Chaharsooghi).
1026-3098 © 2011 Sharif University of Technology. Production and hosting by
Elsevier B.V.
Open access under CC BY-NC-ND license.
Peer review under responsibility of Sharif University of Technology.
doi:10.1016/j.scient.2011.05.018

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S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
Based on pricing and scheduling decisions, as proposed by
a pre-determined time interval. Sawik [31] proposed a dual-
objective problem of due-date setting over a rolling planning
horizon in make-to-order manufacturing. They considered the
total number of delayed products as a primary optimality
criterion, and the total or maximum delay of orders as a
secondary criterion. The results in the papers cited above
indicate that proper due-date management offers a much larger
improvement in performance than priority sequencing.
This literature also assumes that the demand process is
independent of any pricing and/or due-date decisions. Some
researchers considered that quoted lead times (or due dates)
and price affect the customers’ decisions on placing an order.
Duenyas and Hopp [32] were the first to analyze a queuing
model, where lead time quotation affects the demand rate. They
developed queuing models that allow customers to leave if the
due date offered by the firm is too late. The objective is to
maximize profit; the decisions involve sequencing and due date
setting. Duenyas [33] also developed an effective heuristic for
quoting due dates and sequencing orders.
Charnsirisakskul et al. [7], this paper proposes an extended
model that incorporates joint pricing and lead time control
problems in a stochastic production environment with multi-
ple customer segments, which are classified based on the pa-
rameters of willingness to pay and sensitivity to short delivery
time.
We model an MTO production system where demand is a
function of price, lead time and time. We take into account
that orders arrive within pre-determined intervals. Price and
due-date decisions over a planning horizon are assumed to be
made dynamically to upcoming orders in these time intervals,
based upon available capacity, operating costs associated with
the production of the order, tardiness and holding costs. The
orders in each time interval can be classified differently.
The firm bears holding costs and tardiness penalties incurred
for orders that are completed in advance of their due dates and
orders shipped after the preferred due date, respectively.
To reflect the dynamic variations of system conditions, we
propose Multi-stage Stochastic Programming (MSP) methods.
This paper is organized as follows: Section 2 discusses the
relevant literature. Section 3 describes the pricing and lead time
quotation problem. In Section 4, we describe the algorithm used
to approximate the stochastic demand process, and to refor-
mulate our problem in the scenario representation. Section 5
presents some numerical examples for our model. Conclusions
are summarized in Section 6.
Palaka et al. [34], So and Song [35] and Webster [36] studied
the optimal selection of price and delivery time, assuming a
fixed scheduling rule. They formulated the problem as a steady
state queuing model.
Keskinocak et al. [37] proposed several online and offline
algorithms for quoting lead times to different customer classes
for maximizing revenue, subject to the constraint that quoted
lead times are 100% reliable, when the processing time per
customer is known.
Easton and Moodie [38] developed a probabilistic model
that determined the optimal pricing and due date by setting
decisions with contingent orders. In their model, the proba-
bility that the customer will accept a quoted price/due-date
pair follows an S-shaped Logit model. Watanapa and Techaniti-
sawad [39] extended Easton and Moodie’s model by proposing a
model that incorporates the market segmentability by applying
different policies to different demand classes. In their model,
the multi-class policies are managed by the interrelationship of
the price, due date, competition level and quality requirement
in determining winning probability. Charnsirisakskul et al. [7]
expanded on previous research by formulating a deterministic
mixed-integer programming model for the problem of order se-
lection, due date setting and production scheduling. We modi-
fied this model as follows.
1. In Charnsirisakskul’s model, the demand function is deter-
ministic. Thus the production lot size is determined over the
production planning time. The significance of uncertainties
has prompted us to consider the multiple classes of orders
with stochastic demand. We assume that demand is a func-
tion of price, lead time and time as a linear-additive form
(defined as d(p, l, t, ξ) = D(p, l, t) + ξ, where d(p, l, t, ξ) is
demand and ξ is a zero-mean random variable that does not
depend on the price and lead time. In this case, the mean de-
mand is D(p, l, t), and the noise term, ξ, shifts the demand
randomly about this mean [1]). Whitin [8] was the first to
examine the linear-additive functional demand form. This
form of demand was also studied by Mills [9], Thowsen [14],
Federgruen and Heching [18], Dana and Petruzzi [40], Chen
and Simchi-Levi [19] and Yin and Rajaram [41].
2. Literature review
Joint pricing and inventory control strategies in a single
period (news vendor) manufacturing environment, assuming
static prices and demand, were first analyzed by Whitin [8].
Whitin’s work was extended by Mills [9] who studied the ef-
fect of uncertainty on a pricing policy under a linear-additive
functional form for demand. Static pricing with a multi-
plicative form of demand was formulated by Zabel [10] and
Karlin and Carr [11]. Additionally, Petruzzi and Dada [12] ex-
amined an extension of the news vendor problem. In this prob-
lem, demand depends on both price and inventory levels. Other
researchers considered multi-period, finite-horizon pricing
models [13–15]. The retail industry also uses dynamic pricing
techniques to match demand with capacity, maximize revenue
or achieve other strategic goals, as shown by Gallego and Van-
ryzin [16,17], Federgruen and Heching [18], Chen and Simchi-
Levi [19], You and Wu [20] and others. Chan et al. [21] and
Elmaghraby and Keskinocak [22] presented a thorough re-
view of both single and multi-period models combining pricing
and inventory strategies. Most research on integrating pricing
strategies with inventory control policies has ignored produc-
tion capacity limitations.
The second body of research related to our paper focuses on
due-date management in which the key decisions are due-date
setting and scheduling (e.g. [23–31]). Keskinocak and Tayur [5]
presented a review on due date management and dynamic
pricing. They categorized all due-date assignment methods
into two categories: exogenous (determined by the sales
department, without knowing the actual production schedule)
and endogenous (assigned internally by the scheduling model).
Some researchers used mathematical programming to solve
the due-date setting problem. For example, Chen et al. [27]
developed mixed-integer programming models for quantity
and due-date quoting to customer orders that arrive within
2. Under the time-dependent and stochastic demand assumed
above, we extend Charnsirisakskul’s model to dynamic
pricing and lead time decisions to match demand with
capacity. The actual orders arrive gradually over a planning
horizon. Therefore, a better solution method would track
actual demand and production and then adjust the next
production control policy, accordingly. This would be a
dynamic solution method.

S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
713
In this paper, we use multi-stage stochastic programming to
explicitly deal with uncertain demand, dynamic pricing and
lead time decision making. A standard approach for solving
multi-stage stochastic models uses scenario trees to model
the uncertainty of the relevant data. We developed a scenario
generation method for pricing and due date management.
To the best of our knowledge, there is no pricing and lead
time model based upon stochastic programming and scenario
generation methods.
The scenarios and their probabilities form a discrete approx-
imation of the probability distribution of the data process. There
are several approaches to generate scenario trees for stochastic
programs. Dupacova et al. [42] categorized them as:
commitment due date is considered late and incurs a tardiness
penalty proportional to the number of periods and the quantity
delay. Shortages are allowed, and unmet demands are lost.
The manufacturer’s objective is to maximize net profit,
which is the sum of revenues from accepted orders minus the
production, holding, tardiness and shortage costs, subject to
capacity, delivery time, and demand constraints. The notations
of the multi-stage stochastic pricing and lead time decision
model are defined as follows:
Ψ = {1 · · · I} set of order classes, classified based on
sensitivity to price and lead time;
T = {1 · · · T} set of planning periods;
(i) Bound-based constructions;
(ii) Monte Carlo-based schemes, or Quasi Monte Carlo-based
methods;
(iii) EVPI-based sampling and reduction within decomposition
schemes;
(iv) Moment-matching principles;
(v) Probability metric-based approximations.
R = {1 · · · R} set of stages;
Pr(i, r)
set of prices {pi1,r , . . . , pij,r , . . . , pni ,r } (per
r
i
unit), the manufacturer can charge for order
i ∈ Ψ in stage r ∈ R, where nri is the number of
prices offered to order class i in stage r ∈ R;
arrival time for customers in stage r ∈ R;
e(r)
Du(i, r)
set of due dates {li1,r , . . . , lik,r , . . . , lLi ,r }, the
r
manufacturer can charge for order ii∈ Ψ in
In this paper, a probability metric-based approximation for a
scenario tree construction approach, developed by Dupacova
et al. [43] and Heitsch and Romisch [44–46], is used.
With an initial set of discrete probability distributions, they
determined a scenario subset of prescribed cardinality and a
probability measure based on this set that is the closest to the
initial distribution in terms of a natural (or canonical) prob-
ability metric. They also used their algorithm to approximate
passenger demand and cancellation processes in multi-stage
stochastic programming for a revenue management model
in [47,48].
stage r ∈ R, where
e(r) ≤ li1,r < l2i ,r , . . . , < li ≤ T and Lir is the
r
L ,r
i
number of due dates offered to order class i in
stage r ∈ R;
Chi
Cai
third party holding cost per time period per
unit of order i ∈ Ψ ;
tardiness penalty per period per unit of order
i ∈ Ψ ;
Csi
shortage penalty per unit of order i ∈ Ψ ;
production cost per unit of order i ∈ Ψ in
period t ∈ T;
Cpit
3. Multi-stage stochastic pricing and lead time decision
model
Ktr
units of production capacity available within
the production period, t, at beginning of stage
r ∈ R, where t = e(r), . . . , T;
3.1. Problem description
dri,j,k
demand of order i ∈ Ψ in stage r ∈ R,
corresponding to price, pij,r ∈ Pr(i, r), and due
We model the MTO manufacturing facility as a single
machine. This might be the bottleneck in the system with
negligible setup times (and costs), where order preemption
is allowed. The planning horizon is assumed to be finite and
divided into periods of equal length. The capacity in each period
may differ. We assume that the demand dynamically arrives
in R stages and in each arrival (equivalently, stage) there are
different classes of orders, classified based on their sensitivity to
quoted price and due date. Demand for each order is stochastic
with a continuous function of price and lead time. Within stage
r, the price and lead time are defined for order class i, based on
stochastic demand and remaining capacities. A firm can accept
or reject any order. For accepted orders arriving at stage r,
and after realization of stochastic demand, the manufacturer
decides on a production schedule within a finite horizon. This,
in turn, affects the due-dates. Production scheduling must occur
within any time period between the arrival time of the order
and the end of the planning horizon. Scheduled orders should
be delivered in periods between the quoted due date and the
end of the planning horizon.
date, lik,r ∈ Du(i, r), expressed in units of
production capacity required.
Note: Demands at different stages are independent of each
other. We consider a linear additive demand function of the
following form:
dri,j,k = D(pij,r , lki ,r , r) + ζri = ari − bipji,r
− ci((lki ,r − e(r) + 1) − li0)2 + ζri, ait , bi,
ci > 0, i ∈ Ψ , r ∈ R,
where air represents the market size in stage r ∈ R for order
class i ∈ Ψ , bi and ci represent the customer price and lead time
sensitivity, l0i is the preferred lead time, ζri is a random variable
of the demand with PDF, f (.), and CDF, F(.). lik,r − e(r) + 1 is a
time interval between the arrival time of an order and quoted
due date, referred to as lead time. The decision variables are as
follows:
The production capacity in each period is finite and perish-
able if left unused. The capacity for periods before the current
time period is lost.
If an order is scheduled within any period prior to its
negotiated due date, it is stored in a third party warehousing
facility and incurs holding costs. An order shipped after the
r−1
Z
1 if price pij,r ∈ Pr(i, r), and due date
i,j,k
lik,r ∈ Du(i, r) are selected (quoted) for order
i ∈ Ψ at beginning of stage r (r = 1, . . . , R); 0,
otherwise;

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S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
r
i
r
i
′
t
n
L
T
− −
− −
xri,j,t,t quantity produced (in units of capacity) for
r−1
i,j,k
xri,j,t,t
≤
Z
dri,j,k
,
′
′
order i ∈ Ψ in production period t and
i
′
j=1 k=1
t=e(r)
t =l
1,r
delivered in period t within stage r if price pji,r
′
∀i ∈ Ψ , j = 1, . . . , nri , P-a.s.,
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
′
′
is selected: {t = e(r), . . . , t , t =
li1,r , . . . , T, r = 1, . . . , R};
r
n
T
i
− −
−
Uir
the total quantity produced for order i ∈ Ψ
within stage r(r = 1, . . . , R);
xri,j,t,t ≤ Ktr−1
,
′
i
i∈Ψ j=1
′
t =max{t,l
}
1,r
Hir
the total quantity-period inventory of order
i ∈ Ψ within stage r(r = 1, . . . , R);
the total quantity-period tardiness of order
i ∈ Ψ with quoted due date, lik,r ∈ Du(i, r),
t = e(r), . . . , T, P-a.s.,
Ari,k
r
n
i
T
− −
−
Ktr = Ktr−1
−
xri,j,t,t
,
′
within stage r(r = 1, . . . , R);
i
i∈Ψ j=1
′
t =max{t,l
}
1,r
r
ˆ
Y
the vector of the above decision variables at
r
r
r
r
r
ˆ
t = e(r), . . . , T, P-a.s.,
stage r Y = (Z , x , U , H , A ).




i
r
i
r
l
k,r −1
′
t
n
n
i
− − −
−
The stochastic input parameters are the unconstrained
r
r−1
i,j,k
xi,j,t,t ≤ M1 1 −
Z
,
′
demand components represented by a discrete time stochastic
process, ξ = (ξ0, . . . , ξR), on some probability space, (Ω, F, P),
where ξr := (ζr1, . . . , ζrI ). The first stage data, i.e. vector ξ0, are
deterministic.
′
j=1
t=e(r)
j=1
t =1
∀i ∈ Ψ , k = 1, . . . , Lri , P-a.s.,
r
′
t
n
T
i
− − −
Consider ξ
:= (ξ0, . . . , ξr ) to denote the history of the
[r]
′
Hir ≥
(t − t)xri,j,t,t
,
′
process up to stage r. We assume that the process is stagewise
i
j=1
′
t=e(r)
t =l
independent, i.e. ξr is stochastically independent of ξ
.
1,r
[r−1]
We model our problem with multi-stage stochastic pro-
gramming to explicitly deal with uncertain demand, dynamic
pricing and lead time decision making.
∀i ∈ Ψ , P-a.s.,
r
′
t
n
T
i
− − −
Ari,k
≥
(t − lik,r )xr
′
′
i,j,t,t
i
j=1
′
t=e(r)
t =l
3.2. Stochastic model
k,r


r
n
i
−
The multi-stage stochastic programming problem for multi-
period pricing and lead time decision making is formulated as:
r−1
i,j,k


+ M2
Z
− 1
,
j=1
∀i ∈ Ψ , k = 1, . . . , Lri , P-a.s.,
(3.10)
(3.11)
ˆ
ˆ
max f0(Y0) + E[max f1(Y1, ξ1)
r
ˆ
′
n
+ E[· · · + E[max fR(YR, ξR)]]],
T
t
i
− − −
Uir =
xri,j,t,t
,
′
where fr is the profit function at stage r, and defined as:
i
j=1
′
t=e(r)
t =l
1,r
f0 = 0,
(3.1)
(3.2)
∀i ∈ Ψ , P-a.s.,
S.T.
xri,j,t,t ≥ 0, ∀i ∈ Ψ , t = l1i ,t , . . . , T,
′
′
1
1
n
L
i
i
− −
Zi0,j,k ≤ 1, ∀i ∈ Ψ ,
t = e(r), . . . , T, P-a.s.,
Hir ≥ 0, ∀i ∈ Ψ , t = 1, . . . , T, P-a.s.,
j=1 k=1
and:
Ari,k ≥ 0, ∀i ∈ Ψ , k = 1, . . . , Lri , P-a.s.,
r
i
r
i
r−1
′
t
n
L
I
T
Z
∈ {0, 1}, ∀i ∈ Ψ , j = 1, . . . , nri ,
− − −
− −
i,j,k
pij,r
×
xri,j,t,t
ˆ
fr (Yr , ξr ) =
′
k = 1, . . . , Lri , P-a.s.
(3.12)
i
i=1 j=1 k=1
′
t=e(r)
t =l
1,r
r
r
′
t
ˆ
n
The values of decision vector Y , chosen at stage r, may depend
on the information (data), ξ0, . . . , ξr , available up to stage r,
but not on the results of future observations. This is the basic
requirement of nonanticipativity.
I
T
i
− − − −
−
xri,j,t,t × Cpit
′
i
i=1 j=1
′
t=e(r)
t =l
1
r
L
I
I
I
i
−
− −
−
At the beginning of stage 1, the only event is the price and
lead time definition for customers at stage 1, and there are not
any terms in function f0. Constraint 3–2 ensures that each order
is either rejected or accepted; only one price and one lead time
are chosen for each accepted order at the beginning of stage 1.
The five terms in Constraint 3–3 correspond to total revenue,
production cost, inventory holding cost, tardiness penalty and
shortage penalty, respectively. Constraint 3–4 is like Constraint
3–2 and ensures that each order is either rejected or accepted;
only one price and one lead time are chosen for each accepted
order at the beginning of stage r. Constraint 3–5 ensures that
−
×
Hr × Chi −
Ari,k × Cai −
Csi
i
i=1
i=1 k=1
i=1



r
i
r
i
n
L
− −
dri,j,k × Zr−1 − Ui
,
(3.3)
(3.4)
r



i,j,k
j=1 k=1
S.T.
r
r
i
n
L
i
− −
r−1
i,j,k
Z
≤ 1, ∀i ∈ Ψ , P-a.s.,
j=1 k=1

S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
715
if the price, pij,r , and due date, lki ,r , are selected for order i at
S.T.
stage r, at most, dri,j,k units must be produced and delivered for
order i.
r(n)+1 r(n)+1
n
L
i
i
−
−
Zi,j,k,n ≤ 1, ∀i ∈ Ψ , n ∈ N/D(T),
(4.2)
Constraint 3–6 are capacity constraints that ensure the
production capacity in each period is not exceeded. The remain-
ing capacity for each production period is determined dynami-
cally by Constraint 3–7. An order can be delivered between the
quoted due date and the end of the planning horizon only if the
order is accepted. Constraint 3–8 indicates that order i cannot
be delivered before lik,r if due date lki ,r is selected.
The total quantity-period inventory, total quantity-period
tardiness and total quantity produced by each order are defined
by Constraints 3–9, 3–10 and 3–11, respectively.
j=1
k=1
r(n)
′
L
i
T
t
−
−
−
xi,j,t,t ,n
≤
Zi,j,k,a(n)dn
,
i,j,k
′
i
′
k=1
t=e(r(n))
t =l
1,r(n)
∀i ∈ Ψ , j = 1, . . . , nri (n), n ∈ N/0,
(4.3)
(4.4)
(4.5)
r(n)
n
T
i
− −
−
xi,j,t,t ,n ≤ Kta(n)
,
′
i
i∈Ψ j=1
′
t =max{t,l
}
1,r(n)
t = e(r(n)), . . . , T, ∈ N/0,
4. Stochastic programming model in node representation
r(n)
i
n
T
− −
−
Ktn = Kta(n)
−
xi,j,t,t ,n
,
′
We assume random vector ξ = (ξ0, . . . , ξR) has a finite
discrete distribution defined by S realizations ξs = (ξ0s, . . . , ξRs)
called scenarios, and corresponding probabilities as πs, s =
1 · · · S. It is assumed the scenarios have a common root, i.e.
(ξ01 = ξ02 = · · · = ξ0S ). This information structure can be
represented in the form of a scenario tree where the nodes at
stage (or level) r of the tree indicate the information available
up to stage r.
i
i∈Ψ j=1
′
t =max{t,l
}
1,r(n)
t = e(r(n)), . . . , T, n ∈ N/0,
r(n)
i
l
−1
′
n
k,r(n)
t
i
− −
−
′
xi,j,t,t ,n
′
j=1
t=e(r(n))
t =1



r(n)
n
i
−

There are a finite set, N = {0, . . . , N}, of nodes and a total
of R stages in the scenario tree. We denote the stage (or level)
belonging to node n by r(n). The root node, n = 0, is the only
node at stage r = 0. Each node, n, of the scenario tree, except for
the root node, has a unique parent node denoted by a(n) and a
set C(n) of successor nodes. We assume that D(t) ⊂ N is the set
of nodes at stage r. The path from the root node to node n will be
denoted by Path(n). If n is a terminal (leaf) node, i.e. n ∈ D(T),
≤ M1 1 −
Zi,j,k,a(n)
,
j=1
∀i ∈ Ψ ,
k = 1, . . . , Lri (n)
,
n ∈ N/0,
(4.6)
(4.7)
r(n)
′
n
T
t
i
− −
−
′
′
Hi,n
≥
(t − t)xi,j,t,t ,n
,
i
j=1
′
t=e(r(n))
t =l
1,r(n)
then Path(n) corresponds to a scenario. For each node, n ∈ N,
∑
∀i ∈ Ψ ,
n ∈ N/0,
we obtain a probability, πn, where πn
=
m∈C(n) πm, and the
r(n)
′
n
probability of leaf nodes, n ∈ D(T), are the same probability of
T
t
i
− −
−
S
realized scenarios, {πn}n∈D(T) = {πs}s=1
.
′
i
′
Ai,k,n
≥
(t − lk,r(n))xi,j,t,t ,n
i
To solve our multi-stage stochastic model, we rewrite it
in node representation, according to the description of the
scenario tree. Thus we get:
j=1
′
t=e(r(n))
t =l
k,r(n)


r(n)
n
i
−


+ M2
Zi,j,k,a(n) − 1
,
r(n)
j=1
n
N
I
i
− − −
∀i ∈ Ψ , k = 1, . . . , Lri (n), n ∈ N/0,
(4.8)
(4.9)
max
πn × pij,r(n)
n=1 i=1 j=1
r(n)
i
′
n
T
t
− −
−
′
T
t
−
−
′
Ui,n
=
xi,j,t,t ,n
,
′
×
xi,j,t,t ,n
i
j=1
′
t=e(r(n))
t =l
1,r(n)
i
′
t=e(r(n))
r(n)
t =l
1,r(n)
∀i ∈ Ψ , n ∈ N/0,
′
n
i
N
I
T
t
− − − −
−
′
xi,j,t,t ,n ≥ 0, ∀i ∈ Ψ ,
−
πn
′
′
t = e(r(n)), . . . , t ,
t = li1,r(n), . . . , T,
i
n=1 i=1 j=1
′
t=e(r(n))
t =l
1,r(n)
j = 1, . . . , nri (n)
,
n ∈ N/0,
N
I
− −
× xi,j,t,t ,n × Cpi −
πn × Hi,n × Chi
′
Hi,n ≥ 0,
Ai,k,n ≥ 0,
t
n=1 i=1
∀i ∈ Ψ , k = 1, . . . , Lri (n), n ∈ N/0,
r(n)
i
L
N
I
− − −
Zi,j,k,n ∈ {0, 1}, ∀i ∈ Ψ , j = 1, . . . , nir(n)+1
,
−
πn × Ai,k,n × Cai − πn × Csi
k = 1, . . . , Lri (n)+1, n ∈ N/D(T).
(4.10)
n=1 i=1 k=1




r(n) r(n)
n
L
′
N
I
The decision variables xi,j,t,t ,n, Hi,n, Ai,k,n and Zi,j,k,n in Con-
straints 4-1 to 4-10 are variables defined in node n after real-
ization of random data, ξ, at stage r(n). Demand for order class
i
i
− −
− −
n


×
di,j,k × Zi,j,k,a(n) − Ui,n
,
(4.1)
n=1 i=1
j=1 k=1

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S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
i in node n is equal to:
Table 1: Characteristics of demand.
dni,j,k = D(pji,r(n), lik,r(n), r(n)) + ζni ,
Customer
class
Price sensitivity bi
Lead time
Preferred
sensitivity ci
lead time li0
with probability πn, where pij,r(n) and lik,r(n) are the price and
lead time quoted to customer class i in node a(n). This means
price and lead time are determined before the realization of
random data at each stage. ζni is realization of the random
parameter in node n for customer class i. Depending on
the number of realizations of ξ, the mixed-integer linear
programming problem denoted by Constraints 4-1 to 4-10
becomes (very) large in scale.
1
2
3
0.2
0.5
0.7
2
1
2
3
1
0.6
5. Numerical study
In this section, we perform a numerical study of our basic
model to investigate the effect of dynamic pricing and segmen-
tation of orders on the profitability of the firm. Also, we illus-
trate the advantage of our modeling approach compared to the
expected value solution approach.
One way to overcome this difficulty is to use decomposition
methods that exploit the special structures of the model [49].
Reducing the originally designed tree can also reduce the model
dimensions [43–46]. This approach makes use of probability
metrics, i.e. metric distances on spaces of probability measures,
where the metrics are selected such that the optimal values of
the original and approximate stochastic programs are similar
if the distance between the original probability distribution
and its approximation is small. We will briefly describe the
approach of Heitsch and Römisch [46] that is used for scenario
reduction in our model. They assume a stochastic process,
5.1. Flexibility in time, price and lead time
In our analysis, we consider three different policies:
1. Price flexibility,
2. Lead time flexibility,
3. Time flexibility.
ˆ
ξ, with finite and large sample sizes of individual scenarios,
{ξts}tt==T0 , with probabilities ps > 0, s = 1, . . . , S, and common
With price (lead time) flexibility, we charge different prices
(lead times) to different customer classes and classify cus-
tomers based on their sensitivity to changes in dominating fac-
tors, such as price and lead time. When there is no price (lead
time) flexibility, a single (fixed) price (lead time) is charged to
all customer classes [7]. With time flexibility, we change the
price and lead time dynamically to manage demand. We also
investigated the effect of demand variation over time in our
model. To this purpose, we consider combinations of two (low
and high) levels for demand variation over time (L–D, H–D).
We illustrate demand variation by varying market size, ait , with
time, t. We consider an example with three classes of customer
and a planning horizon, T = 4. Customer classes arrive at the
beginning of each period, so there are R = T = 4 stages. The
market size for each class at each stage is as follows:
root nodes, i.e. ξ01 = · · · = ξ0S , are given. They then determine
a stochastic process, ξtr , on (Ω, F, P) by bundling and deletion
processes, which have tree form scenarios and satisfy the
condition:
ˆ
‖ξ − ξtr ‖1 ≤ ε.
(4.11)
ε > 0 is a prescribed tolerance. The bundling and deletion
processes rely on computing and bounding the Kantorovich
distance, µˆ r (P, Q ), between the original probability distribu-
∑
S
tion given by the individual scenarios, P =
i=1 piδξi , and
their weights, and the reduced probability distribution, Q =
∑
S
qjδξj . J denotes some index set of deleted scenarios. The
j=1,j̸∈J
Kantorovich distance is given by Relation 4–12, where P and Q
are the fixed Borel probability measures on a closed subset of
Rs, Ω, i.e. P, Q ∈ P(Ω), and function c : Ω × Ω → R is given
by Relation 4–13.
t1
9
13
14
t2
10
12
13
t3
11
11
0
t4
12
0


class1
class2
class3
L−D =
H−D =

S
−
0
µˆ r (P, Q ) = min
cr (ξi, ξj)ηij : ηij ≥ 0,
t1
3
19
18
t2
8
12
9
t3
13
5
t4
18
0
i,j=1
j̸∈J


class1
class2
class3

S
S
−
−
ηij = qj,
ηij = pi
,
(4.12)
(4.13)
0
0
j=1
j̸∈J
i=1
Table 1 shows other demand function components for each
customer class in each time period.
r−1
cr (ξi, ξj) := max{1, |ξi − ξ0|
,
|ξj − ξ0| }|ξi − ξj|, ∀ξi, ξj ∈ Ω.
Also J ⊂ {1, . . . , N} and δξ ∈ P(Ω) denote the Dirac measures
r−1
The random perturbation, ζri, follows a discrete distribution
with probability mass function:
P(ζri = ζri,z ) = prri,z
We assume probability functions are the same at all stages and
for all classes; therefore, we can remove two indices, i and r. The
ζz and prz are presented in Table 2.
The holding and tardiness costs per period per unit and the
shortage costs per unit for each class of customer are as follows:
,
i ∈ Ψ ,
r = 1 · · · R.
that place the unit mass at ξ.
A set-covering problem represents the optimal choice of
an index set, J, for scenario reduction. It can be formulated
as a 0-1 integer program that is NP-hard. In [46], a heuristic
algorithm is approximated to determine a reduced probability
distribution, Q of ξ, in multi-stage stochastic programming.
Using the algorithm in [46], we determined a reduced stochastic
process of ξtr in scenario tree form, and employed it in a
stochastic programming model denoted by Constraints 4-1 to 4-
10. The details for the backward reduction algorithm are given
in the Appendix.
Ch1 = Ch2 = Ch3 = 1,
Cs1 = Cs2 = Cs3 = 15.
Ca1 = Ca2 = Ca3 = 4,
The production costs in all periods are 3.

S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
717
Table 2: Random perturbation for demand function.
Z
1
2
3
4
5
6
7
8
9
10
3
0.02
ζz
prz
−3
−2
−1.5
−0.75
−0.35
0.35
0.2
0.75
0.15
1.5
0.09
2
0.04
0.02
0.04
0.09
0.15
0.2
Figure 1a: Original scenario tree for customer class 1.
Figure 2a: Reduced scenario tree for customer class 1.
Figure 2b: Reduced scenario tree for customer class 2.
Figure 2c: Reduced scenario tree for customer class 3.
Figure 1b: Reduced scenario tree for customer class 2.
Figure 1c: Original scenario tree for customer class 2.
The production capacity in each period at the beginning of
ˆ
the planning horizon is K0 = (10, 10, 10, 10). By using discrete
distributions with finite realizations for each customer class
in each period from the data in Table 2, we get S
=
109
realizations or scenarios for the joint distribution over the total
time horizon, for all customers. The resulting original scenario
tree is illustrated in Figures 1a–1c for each respective customer
class.
In each figure, there are 144 scenarios, some of which
overlap. We are interested in simultaneously maximizing the
expected total profit, while determining the following decisions
dynamically:
Using the initial sets of scenarios and algorithms described
in [46], a scenario set consisting of 144 scenarios and 212
nodes is generated using a procedure implemented in MATLAB
1. Optimal price and lead time quoted to each class of order at
the beginning of each period;
2. Production amount for each class in each period according
to each realized scenario.
ˆ
software for the stochastic process, ξ, where ‖ξ − ξtr ‖1
≤
4.2026 in Relation 4–11. The reduced trees for each customer
class are shown in Figures 2a–2c.

718
S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
Figure 3a: Scenario tree for customer class 1 price.
Figure 3b: Scenario tree for customer class 2 price.
Figure 3c: Scenario tree for customer class 3 price.
Figure 4: Expected profit under different combinations of lead time and price
flexibility for two dynamic and static cases under high demand variability (H–D)
and low demand variability (L–D).
Table 4: Sensitivity analysis with respect to various combinations of
demand in the model on increased percentage profit of the policies over
the base case policy.
levels of
demand
variability
Policies
Expected
profit
Increased
percentage profit
over the base case
P1-Dynamic
P2-Dynamic
P3-Dynamic
P4-Dynamic
P1-Static
920.3819
60.01076
56.22865
55.59944
43.1452
38.15193
37.86702
0.207197
0
42.18995
31.52128
26.30027
13.73792
34.35121
27.08378
10.88685
0
898.6272
895.008
823.3712
794.6499
793.0111
576.3918
575.2
H–D
L–D
P2-Static
P3-Static
P4-Static
P1-Dynamic
P2-Dynamic
P3-Dynamic
P4-Dynamic
P1-Static
702.2992
649.6049
623.8175
561.77
663.5824
627.6874
547.6881
493.9162
P2-Static
P3-Static
P4-Static
decreasing the price during periods of low demand, the firms
can smooth the demand and improve their capacity utilization.
We consider different policies for price and lead time flexi-
bility and dynamic pricing as follows:
Table 3: The optimal prices and lead times of each customer class at the
beginning of the time horizon in the numerical example.
P1: price flexibility, lead time flexibility,
P2: price flexibility, no lead time flexibility,
P3: no price flexibility, lead time flexibility,
P4: no price flexibility, no lead time flexibility.
Optimal
price
Optimal
lead time
Class 1
Class 2
Class 3
Class 1
Class 2
Class 3
10
1
High demand variability
Low demand variability
22
16
25
16
13
2
2
1
2
2
These combinations are considered for two cases: Dynamic
pricing (D) and Static pricing (S).
The maximum expected profit of each policy for two demand
variation scenarios is represented in Figure 4. We also consider
the base case in which there is no flexibility of price or lead
time with static pricing. The percentage profit increases over
the base case for all policies under different combinations of
demand is summarized in Table 4.
We used the model expressed in Section 4 to solve this pricing
model.
Table 3 illustrates the price and lead time selected at the
beginning of period 1 for each customer class under two
considered demand variations. Figures 3a–3c illustrate the
scenario trees for the price selected at the beginning of each
period for each customer class.
The demand for class 1 increases with time; hence the price
is also increasing with time. For customer classes 2 and 3, the
price decreases with time because of the decreasing demand. In
fact, by increasing the price during periods of high demand and
The following inferences may be drawn from Table 4 and
Figure 4:
1. With high demand variability, dynamic pricing leads to
higher expected profit under all policies.
2. Under low demand variability, price and lead time flexibility
are more useful than dynamic pricing.
3. The percentage profit increases over the base case for P2-
Dynamic and P3-Dynamic are close to each other under both
demand environments, and are considered interchangeable.

S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
719
∗
¯ ¯
4. Under a no price flexibility-static pricing policy, lead time
flexibility can increase profit in a low demand variability en-
vironment.
Q − EEV, with EEV = E (Q (Z(ξ), ξ)). In general, a bigger VSS
ξ
0
indicates higher benefit from using the stochastic programming
approach.
5. In the high demand variation, the percentage profit increases
over the base case for dynamic pricing and price, and lead
time flexibility, are 43.14%, 37.86%, and 0.20%. Thus the man-
ufacturer chooses dynamic pricing and price flexibility in de-
scending order. But in an environment with low demand
variability, the ranking of policies, according to their per-
centage profit increases over the base case, is:
1. Price flexibility (27.083%),
In a maximization problem, the relation between the defined
values has been established by Madansky [51] as follows:
∗
EEV ≤ Q0 ≤ WS.
In this section, we compute these measures for five problem in-
stances. For each problem, we randomly choose the parameters,
ait , bi, ci, in customers demand functions and cost parameters
from a uniform distribution. To see how the lower and upper
bounds, and the Value of the Stochastic Solution (VSS) are repre-
sented under different probability distributions, the sensitivity
analysis, with respect to random data, is studied. To this pur-
pose, we solve generated examples considering four probabil-
ity distribution functions for random variable ξ in the demand
function as follows:
2. Dynamic pricing (13.73%),
3. Lead time flexibility (10.88%).
6. If the manufacturer can choose two types of flexibilities for
high demand variability, we rank the policies according to
their percentage profit increases over the base case as fol-
lows:
1. Price flexibility-dynamic pricing (56.23%),
2. Lead time flexibility-dynamic pricing (55.60%),
3. Lead time flexibility- price flexibility (38.15%),
and for low demand variability, we rank the policies as fol-
lows:
1. Lead time flexibility- price flexibility (34.35%),
2. Price flexibility-dynamic pricing (31.52%),
3. Lead time flexibility-dynamic pricing (26.30%).
1. Exponential distribution with a mean of 3: ξ ∼ exp(3),
2. Exponential distribution with a mean of 10: ξ ∼ exp(10),
3. Uniform distribution on interval [−3, 3] : ξ ∼ U(−3, 3),
4. Uniform distribution on interval [−10, 10]
:
ξ
∼
U(−10, 10).
With the initial set consisting of S = 10 500 scenarios and
the backward reduction algorithm described in [46], a scenario
set consisting of 300 scenarios is generated for the stochastic
process, ξ, by a procedure implemented in MATLAB software
Tables 5 and 6 show the WS, EVPI and VSS of five problem
instances, and the best objective value for the multi-stage
stochastic programming problem, which optimized with 300
scenarios. The column ‘‘LB’’ represents the lower bound for true
problems with 10 500 scenarios, and is defined as:
5.2. Value of the stochastic program
Stochastic programs are computationally difficult to solve.
Therefore, for real-world problems, people have a tendency
to solve much simpler versions. For example, researchers
may solve the deterministic program by replacing all random
variables with their expected values, or they may solve
deterministic programs, each corresponding to one particular
scenario, and then combine these different solutions by some
heuristic rule. The accuracy of such approaches is to be
evaluated by introducing two concepts, the Expected Value
of Perfect Information (EVPI) and the Value of the Stochastic
Solution (VSS) [50].
Expected Value of Perfect Information (EVPI): The EVPI concept
measures the maximum amount a decision maker would
be ready to pay in return for complete information about
the future. Let ξ be the random variable whose realizations
correspond to the various scenarios. Let Q0 be the optimal value
of the stochastic programming, and Q (ξ) the optimal value
∗
LB = E (Q (Z (ξ), ξ0)),
ξ
0
∗
where Z (ξ) is the optimal decision in the reduced stochastic
problem with 300 scenarios, and ξ0 is the random variable
consisting of S = 10 500 scenarios. We can obtain the relative


∗
Q
−LB
∗
optimality gap,
× 100 , of the solution, Z (ξ), using
LB
the lower bound estimate, (LB), and the objective function
value estimate from the reduced stochastic program (Q ). The
∗
∗
optimal objective value of the reduced stochastic program is
an upper bound for the true stochastic program, consisting of
initial scenarios. Therefore, a smaller gap indicates smaller error
from using the reduced stochastic programming approach.
As we can see from Tables 5 and 6, the values of stochastic
programming increase with increasing the exponential distri-
bution’s mean or increasing the uniform distribution’s interval
length. In fact, the solutions given by the deterministic models
would not be able to define best price and lead time for prob-
lem instances that had enormous variance in demand function.
This relatively large value for VSS justifies the use of more so-
phisticated modeling techniques and the extra computational
efforts.
As we can see from Table 5 and 6, all instances, except
instances with a U(−3, 3) distribution function, have high
EVPI, meaning that perfect information would be helpful to
substantially improve the objective function.
The small relative optimality gap, in almost all instances,
indicates that scenario reduction can be a good approximation
of the main problem.
¯
for the deterministic problem corresponding to one particular
scenario, ξ.
The wait-and-see value (WS), which corresponds to the
expected value of the optimal objective for each scenario, is
¯
WS = E (Q (ξ)). The expected value of perfect information
ξ
∗
(EVPI) is then defined by EVPI = WS − Q0 .
Value of the Stochastic Solution (VSS): The stochastic pro-
gramming approach considers the entire range of uncertain sce-
narios. On this score, it may be better than its deterministic
correspondents. However, it also increases computational com-
plexity dramatically. Therefore, the majority of people would
solve the deterministic problem by replacing the random vari-
ables by their corresponding expected values. The concept of
value of the stochastic programming solution (VSS) can be used
to justify whether the extra effort on modeling and solving
¯ ¯
stochastic programming is worthwhile. Let Z(ξ) be the optimal
decision of the first stage in the deterministic problem where
all random variables are replaced by their expected values. The
Value of the Stochastic Solution (VSS) is then defined as VSS =

720
S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
Table 5: Computational results for wait-and-see, EVPI, and lower bound of multi-stage stochastic programming under exponential distribution.
Exp(3)
EEV
Exp(10)
EEV
∗
∗
Q0
WS
EVPI
VSS
LB
Gap (%)
Q0
WS
EVPI
VSS
LB
Gap (%)
N1 1029.717 1079.370 49.653 1010.780 18.937 1025.045 0.456
N2 1184.674 1244.123 59.449 1171.930 12.744 1181.487 0.270
N3
N4 1215.623 1274.340 58.717 1198.700 16.923 1212.243 0.279
N5 361.905 387.670 25.765 351.870 10.035 358.825 0.858
1587.375 2131.380 544.005 1542.830
44.545 1580.133 0.458
1765.889 2257.620 491.731 1586.670 179.219 1752.174 0.783
826.122 1024.770 198.648 783.750 42.372 818.285 0.958
1594.970 1827.780 232.810 1501.670 93.300 1583.723 0.710
691.550 805.700 114.150 601.980 89.570 680.632 1.604
493.581
524.627 31.046
490.510
3.071
491.964 0.329
Table 6: Computational results for wait-and-see, EVPI, and lower bound of multi-stage stochastic programming under uniform distribution.
U[−3, 3]
U[−10, 10]
∗
∗
Q0
WS
EVPI
EEV
VSS
LB
Gap (%)
Q0
WS
EVPI
EEV
VSS
LB
Gap (%)
N1
N2
N3
715.550
926.339
324.916
763.006 47.456 711.040
970.880 44.541 900.870 25.469 917.560 0.957
4.510 712.545 0.422
674.290
906.869
301.095
775.050 100.760 531.240 143.050 658.763 2.357
992.360
358.070
85.491 695.990 210.879 889.278 1.978
328.870
3.954 322.760
4.766 993.040
2.689 236.700
2.156 323.051 0.577
9.064 999.249 0.286
2.161 238.748 0.047
56.975 215.400
56.525 900.500
48.890 190.580
85.695 289.367 4.053
69.105 960.031 0.997
11.200 198.456 1.675
N4 1002.104 1006.870
N5 238.861 241.550
969.605 1026.130
201.780
250.670
6. Conclusion
Sorting of:
{ckj : j = 1, . . . , N}, k = 1, . . . , N,
We have presented a multi-stage stochastic programming
approach to dynamically determine price and lead time for
MTO firms with multiple customer classes. We used an
additive form for the demand function, in which the stochastic
parameter is approximated by a scenario tree. The scenario
tree is generated by the forward reduction algorithm obtained
in [46]. Through numerical examples, we compared benefits of
price and lead time flexibility and dynamic pricing under two
demand environments.
[1]
cll := min clj, l = 1, . . . , N,
j̸=l
[1]
[1]
zl := plcll
,
l = 1, . . . , N,
[1]
[1]
l1 ∈ argl∈{1,...,N} min zl
Step i:
,
J
:= {l1}.
[i]
ckl := min ckj,
[i−1]
j̸=J
∪{l}
We can summarize the following findings from the results
of numerical examples:
[i−1]
l ̸∈ J
,
k ∈ J[i−1] ∪ {l},
−
[i]
[i]
[i−1]
,
zl :=
pkckl , l ̸∈ J
– Firstly, under both demand environments, price and lead
time flexibility leads to higher profits than no flexibility in
price and lead time.
– Secondly, the rankings for price, lead time flexibility and dy-
namic pricing are dependent on the demand environment.
Under high demand variability, the dynamic pricing is more
beneficial than price and lead time flexibility; under low de-
mand variability, price and lead time flexibility are more
beneficial than dynamic pricing.
– Thirdly, under a no price flexibility-static pricing policy, lead
time flexibility can increase profit in low demand variability,
but having lead time flexibility in a high demand variability
environment does not have a significant benefit.
[i−1]
j̸=J
∪{l}
[i]
[i]
li ∈ argj̸=J
min z ,
J
:= J[i−1] ∪ {li}.
[i−1]
l
Step N − n + 1. Redistribution by Eq. (A.1)
where:
−
¯
qj = pj +
pi, for each j ̸∈ J,
(A.1)
i∈J
j
Jj := {i ∈ J : j = j(i)},
and:
j(i) ∈ arg min c(ωi, ωj), for each i ∈ J,
(A.2)
(A.3)
j̸∈J
Scenario representation of pricing and lead time quotation
problems correspond to large scale mixed integer program-
ming. Future research may focus on using other methods, like
decomposition or Monte Carlo, to solve the resulting large scale
linear program, and compare the solutions with our results in
this paper.
and function c : Ω × Ω → R is given by:
c(ω, ω˜ ) = max {1, ‖ω − ω0‖ , ‖ω˜ − ω0‖} ‖ω − ω˜ ‖ ,
∀ω, ω˜ ∈ Ω.
References
Appendix. Simultaneous backward reduction [46]
[1] Talluri, K.T. and van Ryzin, G.J., The Theory and Practice of Revenue
Management, Kluwer Academic Publishers, New York (2004).
[2] Harris, F.H.d. and Pinder, J.P. ‘‘A revenue management approach to demand
management and order booking in assemble-to-order manufacturing’’,
Journal of Operations Management, 13(4), pp. 299–309 (1995).
let P be a fixed Borel probability measure on Ω, i.e. P ∈
P(Ω), with scenarios {ω1, ω2, . . . , ωN } and probability weights
{p1, p2, . . . , pN }. Thus the simultaneous backward reduction
algorithms, according to [46], are as in the following steps:
Step 1.
[3] Defregger, F. and Kuhn, H. ‘‘Revenue management for
a make-to-
order company with limited inventory capacity’’, OR Spectrum, 29(1),
pp. 137–156 (2007).
[4] Spengler, T., Rehkopf, S. and Volling, T. ‘‘Revenue management in make-
to-order manufacturing: an application to the iron and steel industry’’, OR
Spectrum, 29(1), pp. 157–171 (2007).
ckj := c(ωk, ωj), j = 1, . . . , N.

S.K. Chaharsooghi et al. / Scientia Iranica, Transactions E: Industrial Engineering 18 (2011) 711–721
721
[5] Keskinocak, P. and Tayur, S. ‘‘Due-date management policies’’, In Handbook
of Quantitative Supply Chain Analysis: Modeling in the E-Business Era,
D. Simchi-Levi, S.D. Wu and Z.J.M. Shen, Eds., In International Series
on Operations Research and Management Science, pp. 485–553, Kluwer
Academic Publishers, Norwell, MA (2004).
[36] Webster, S. ‘‘Dynamic pricing and lead time policies for make to order
systems’’, Decision Sciences, 33(4), pp. 579–599 (2002).
[37] Keskinocak, P., Ravi, R. and Tayur, S. ‘‘Scheduling and reliable lead time
quotation for orders with availability intervals and lead time sensitive
revenues’’, Management Science, 47(2), pp. 264–279 (2001).
[38] Easton, F.F. and Moodie, D.R. ‘‘Pricing and lead time decisions for make-
to-order firms with contingent orders’’, European Journal of Operational
Research, 116(2), pp. 305–318 (1999).
[6] Biller, S., et al. ‘‘Dynamic pricing and the direct to customer model in the
automotive industry’’, Electronic Commerce Research, 5(2), pp. 309–334
(2005).
[7] Charnsirisakskul, K., Griffin, P.M. and Keskinocak, P. ‘‘Pricing and
scheduling decisions with lead-time flexibility’’, European Journal of
Operational Research, 171(1), pp. 153–169 (2006).
[39] Watanapa, B. and Techanitisawad, A. ‘‘Simultaneous price and due date
settings for multiple customer classes’’, European Journal of Operational
Research, 166(2), pp. 351–368 (2005).
[40] Dana, J.D. and Petruzzi, N.C. ‘‘Note: the newsvendor model with
endogenous demand’’, Management Science, 47(11), pp. 1488–1497
(2001).
[41] Yin, R. and Rajaram, K. ‘‘Joint pricing and inventory control with a
Markovian demand model’’, European Journal of Operational Research,
182(1), pp. 113–126 (2007).
[42] Dupacova, J., Consigli, G. and Wallace, S.W. ‘‘Scenarios for multistage
stochastic programs’’, Annals of Operations Research, 100(1–4), pp. 25–53
(2000).
[43] Dupacova, J., Growekuska, N. and Romisch, W. ‘‘Scenario reduction in
stochastic programming’’, Mathematical Programming, Series A, 95(3),
pp. 493–511 (2003).
[44] Heitsch, H. and Romisch, W. ‘‘Scenario reduction algorithms in stochastic
programming’’, Computational Optimization and Applications, 24(2–3),
pp. 187–206 (2003).
[45] Heitsch, H. and Römisch, W. ‘‘Generation of multivariate scenario trees
to model stochasticity in power management’’, in: IEEE Power Tech.
Conference, St. Petersburg, Russia, 2005, pp. 1–7.
[46] Heitsch, H. and Römisch, W. ‘‘Scenario tree modeling for multistage
stochastic programs’’, Mathematical Programming, Series A and B, 118(2),
pp. 371–406 (2009).
[47] Möller, A., Romisch, W. and Weber, K. ‘‘A new approach to O&D revenue
management based on scenario trees’’, Journal of Revenue and Pricing
Management, 3, pp. 265–276 (2004).
[48] Möller, A., Romisch, W. and Wever, K. ‘‘Airline network revenue
management by multistage stochastic programming’’, Computational
Management Science, 5(4), pp. 355–377 (2008).
[49] Ruszczynski, A. and Shapiro, A. ‘‘Stochastic programming models’’,
In Handbooks in Operations Research and Management Science, A. Ruszczyn-
ski and A. Shapiro, Eds., pp. 1–63, Elsevier Science, Amsterdam (2003).
[50] Birge, J.R. and Louveaux, F., Introduction to Stochastic Programming,
Springer, New York (1997).
[51] Madansky, A. ‘‘Inequalities for stochastic linear programming problems’’,
Management Science, 6(2), pp. 197–204 (1960).
[8] Whitin, T.M. ‘‘Inventory control and price theory’’, Management Science,
2(1), pp. 61–68 (1955).
[9] Mills, E.S. ‘‘Uncertainty and price theory’’, The Quarterly Journal of
Economics, 73(1), pp. 116–130 (1959).
[10] Zabel, E. ‘‘Monopoly and uncertainty’’, Review of Economic Studies, 37(2),
pp. 205–219 (1970).
[11] Karlin, S. and Carr, C.R. ‘‘Prices and optimal inventory policy’’, In Studies
in Applied Probability and Management Science, K. Arrow, S. Karlin and H.
Scarf, Eds., pp. 159–172, Stanford University Press, Stanford, CA (1962).
[12] Petruzzi, N.C. and Dada, M. ‘‘Pricing and the newsvendor model: a review
with extensions’’, Operations Research, 47(2), pp. 183–194 (1999).
[13] Thomas, L. ‘‘Price production decision with deterministic demand’’,
Management Science, 16(11), pp. 747–750 (1970).
[14] Thowsen, G.T. ‘‘A dynamic, non-stationary inventory problem for
a
price/quantity setting firm’’, Naval Research Logistics Quarterly, 22(3),
pp. 461–476 (1975).
[15] Zabel, E. ‘‘Multi-period monopoly under uncertainty’’, Journal of Economic
Theory, 5(3), pp. 524–536 (1972).
[16] Gallego, G. and Vanryzin, G. ‘‘Optimal dynamic pricing of inventories
with stochastic demand over finite horizons’’, Management Science, 40(8),
pp. 999–1020 (1994).
[17] Gallego, G. and Vanryzin, G. ‘‘A multi-product dynamic pricing problem
and its applications to network yield management’’, Operations Research,
45(1), pp. 24–41 (1997).
[18] Federgruen, A. and Heching, A. ‘‘Combined pricing and inventory control
under uncertainty’’, Operations Research, 47(3), pp. 454–475 (1999).
[19] Chen, X. and Simchi-Levi, D. ‘‘Coordinating inventory control and pricing
strategies with random demand and fixed ordering cost: the finite horizon
case’’, Operations Research, 52(6), pp. 887–896 (2004).
[20] You, P.S. and Wu, M.T. ‘‘Optimal ordering and pricing policy for
an inventory system with order cancellations’’, OR Spectrum, 29(4),
pp. 661–679 (2007).
[21] Chan, L.M.A., et al. ‘‘Coordination of pricing and inventory decisions:
a survey and classification’’, In Handbook of Quantitative Supply Chain
Analysis: Modeling in the E-Business Era, D. Simchi-Levi, S.D. Wu and Z.J.M.
Shen, Eds., In International Series on Operations Research and Management
Science, pp. 335–392, Kluwer Academic Publishers, Norwell, MA (2004).
[22] Elmaghraby, W. and Keskinocak, P. ‘‘Dynamic pricing in the presence of
inventory considerations: research overview, current practices, and future
directions’’, Management Science, 49(10), pp. 1287–1309 (2003).
[23] Seidmann, A. and Smith, M. ‘‘Due date assignment for production
systems’’, Management Science, 27(5), pp. 101–413 (1981).
[24] Bertrand, J. ‘‘The effect of workload dependent due-dates on job shop
performance’’, Management Science, 29(7), pp. 799–816 (1983).
[25] Wein, L.M. ‘‘Due-date setting and priority sequencing in a multiclass
M/G/1 queue’’, Management Science, 37(7), pp. 834–850 (1991).
[26] Wein, L.M. and Chevalier, P. ‘‘A broader view of the job-shop scheduling
problem’’, Management Science, 38(7), pp. 1018–1033 (1992).
[27] Chen, C.Y., Zhao, Z.Y. and Ball, M.O. ‘‘Quantity and due date quoting
available to promise’’, Information Systems Frontiers, 3(4), pp. 477–488
(2001).
Seyyed Kamal Chaharsooghi is Associate Professor of Industrial Engineering at
the Industrial Engineering Department, Faculty of Engineering, Tarbiat Modares
University, Tehran, Iran. His research interests include Manufacturing Systems,
Supply Chain Management, Information Systems, Strategic Management,
International Marketing Strategy and Systems Theory. His work has appeared
in Decision Support Systems, European Journal of Operational Research,
International Journal of Production Economics, International Journal of
Advanced Manufacturing Technology, Applied Mathematics and Computation,
and Scientia Iranica. He obtained his Ph.D. from Hull University, England.
Mahboobeh Honarvar is a Ph.D. candidate of Industrial Engineering at the
Industrial Engineering Department, Faculty of Engineering, Tarbiat Modares
University, Tehran, Iran. She received her B.S. degree from Sharif University
of Technology and her M.S. degree from Tehran University, both in Industrial
Engineering. Her main research interests include Revenue Management and
Dynamic Pricing, Supply Chain Management, Production Management and
Meta-Heuristics.
[28] Hegedus, M.G. and Hopp, W.J. ‘‘Due date setting with supply constraints
in systems using MRP’’, Computers & Industrial Engineering, 39(3–4),
pp. 293–305 (2001).
[29] Moses, S., Grant, H., Gruenwald, L. and Pulat, S. ‘‘Real-time due-date
promising by build-to-order environments’’, International Journal of
Production Research, 42(20), pp. 4353–4375 (2004).
[30] Zorzini, M., Corti, D. and Pozzetti, A. ‘‘Due date (DD) quotation and capacity
planning in make-to-order companies: results from an empirical analysis’’,
International Journal of Production Economics, 112(2), pp. 919–933 (2008).
[31] Sawik, T. ‘‘Multi-objective due-date setting in a make-to-order environ-
ment’’, International Journal of Production Research, 47(22), pp. 6205–6231
(2009).
Mohammad Modarres is Professor of Industrial Engineering in the Department
of Industrial Engineering, Sharif University of Technology, Tehran, Iran. He
received his B.S. degree from the Department of Electrical Engineering at Tehran
University, Iran, in 1968 and his M.S. and Ph.D. degrees from the Department
of Engineering Systems at the University of California, Los Angeles, in 1973
and 1975, respectively. His main research interests include Stochastic Models,
Revenue Management, Supply Chain Management, Financial Engineering and
Mathematical Modeling. He has published several articles in journals, such
as European Journal of Operational Research, IEEE Transactions on Power
Systems, Naval Research Logistics Quarterly, Fuzzy Sets and Systems, Journal
of Operational Research Society, International Journal of Uncertainty, Fuzziness
Knowledge-Based Systems, International Journal Engineering Science, Journal
of Engineering, Iranian Journal of Science and Technology, Scientia Iranica,
Iranian Journal of Operations Research, Journal of Systems and Industrial
[32] Duenyas, I. and Hopp, W.J. ‘‘Quoting customer lead times’’, Management
Science, 41(1), pp. 43–57 (1995).
[33] Duenyas, I. ‘‘Single facility due date setting with multiple customer
classes’’, Management Science, 41(4), pp. 608–619 (1995).
[34] Palaka, K., Erlebacher, S. and Kropp, D.H. ‘‘Lead-time setting, capacity
utilization, and pricing decisions under lead-time dependent demand’’, IIE
Transactions, 30(2), pp. 151–163 (1998).
[35] So, K.C. and Song, J.S. ‘‘Price, delivery time guarantees and capacity
selection’’, European Journal of Operational Research, 111(1), pp. 28–49
(1998).
Engineering, International Journal of Industrial Engineering
Research and International Journal of Inquiry.
& Production