mu

M.y, P896

ProductionTwo-stage stochastic programming

ue-rsrmry hconlimcha

based approach for stochastic demand. Through numerical analyses, we indicate the benets of exibilityin delivery, price and lead-time in various environments.

ce, prie effece suphe art oa prod

orders time sensitivity is a managerial challenge in make-to-orderenvironment.

the rm should determine these parameters for upcoming orders,based upon its available capacity, the operating costs associatedwith the production of the order, holding costs and tardiness pen-alties incurred for the orders that are completed in advance of theirdue-dates and orders shipped after the preferred due-date,respectively.

We assume prices and due-dates (lead-times) are set at thebeginning of the horizon. However, the production decision ismade after the arrival of customer orders as well as more denitiveinformation becomes available. As a real world case of this

q This manuscript was processed by Area Editor Joseph Geunes. Corresponding author. Tel.: +98 21 82883345.

E-mail addresses: skch@modares.ac.ir (S.K. Chaharsooghi), mahboobeh_honarvar@yahoo.com (M. Honarvar), Modarres@sharif.edu (M. Modarres),nakhai@modares.ac.ir (I.N. Kamalabadi).

1 Tel.: +98 21 44209944.2 Tel.: +98 21 66165719.

Computers & Industrial Engineering 61 (2011) 10861097

Contents lists availab

Computers & Indus

journal homepage: www.e3 Tel.: +98 21 82884387.by selling each product to the right customer at the right time forthe right price (Talluri & van Ryzin, 2004). Revenue management ismost effective when the demand can be segmented and price sen-sitivity varies across market segments.

While revenue management techniques and dynamic pricingpolicies have been used widely in the airline and hotel industries,retail and manufacturing companies are also employing pricingstrategies such as dynamic pricing and target pricing for their dif-ferent classes of customers. In revenue management system, seg-mentation of orders is based on their sensitivity to price.Moreover, the segmentation and quoting lead-times based on an

ses and change their prices based on parameters such as demandvariation, inventory levels, or production schedules (Biller, Chan,Simchi-levi, & Swann, 2005).

Based on price and scheduling decisions as proposed byCharnsirisakskul, Grifn, and Keskinocak (2006), this paper pro-poses an extended model that incorporates joint pricing andlead-time control problems in a production environment with sto-chastic demand. Customers are classied into different segments,based on their reservation price and sensitivity to delivery time.

We model a MTO (make-to-order) production system, wherestochastic demand is a function of both price and lead-time and1. Introduction

In todays competitive marketplanue management techniques are thinuence market demand and balannue management can be dened as tgenerated from a limited capacity of0360-8352/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.06.022 2011 Elsevier Ltd. All rights reserved.

cing policies and reve-ctive components thatply and demand. Reve-f maximizing the protuct over a nite horizon

In make-to-order environments, various attributes of the prod-uct, such as its price and lead-time, are evaluated by the buyer.Therefore, for each new customer, the rm should determine adue-date and price based on the customers preferences, the avail-able capacity, and other potential orders that could demand thoseresources. Companies such as Dell Computers and Amazon.com areexamples of rms that separate their customers into different clas-PricingLead-time amount jointly in each period. The difculty of continuous distributions is avoided by using a scenario-Developing a two stage stochastic prograand lead-time decision problem in the m

S. Kamal Chaharsooghi a,1, Mahboobeh Honarvar a,,aDepartment of Industrial Engineering, School of engineering, Tarbiat Modares UniversitbDepartment of Industrial Engineering, Sharif University of Technology, P.O. Box 14588-

a r t i c l e i n f o

Article history:Received 13 March 2010Received in revised form 15 May 2011Accepted 27 June 2011Available online 3 July 2011

Keywords:

a b s t r a c t

Pricing coordination and dketplace. Segmenting ordeand price can increase a production policy, inventodecisions. In this paper, weorder environment, whereWe develop a two-stage stoll rights reserved.ming model of the pricelti-class make-to-order rmq

Modarres b,2, Isa Nakhai Kamalabadi a,3

.O. Box 14115-11, Tehran, Iran94, Tehran, Iran

date management are managerial challenges in todays competitive mar-into classes and allocating resources based on their sensitivity to times prot and its capacity utilization. In addition, other parameters such asolding and delivery system should be considered in pricing and due-datesider the role of exibility in price, lead-time and delivery in the make-to-ited production capacity under a stochastic demand function is allowed.stic programming model to determine the price, lead-time and production

le at ScienceDirect

trial Engineering

lsev ier .com/ locate/caie

strategies with inventory control policies has ignored productioncapacity limitations.

dusThe second body of research related to our paper focuses ondue-date management in which the key decisions are due-date set-ting and scheduling (e.g., Bertrand, 1983; Chen, Zhao, & Ball, 2001;Hegedus & Hopp, 2001; Moses, Grant, Gruenwald, & Pulat, 2004;Sawik, 2009; Seidmann & Smith, 1981; Wein, 1991; Wein &Chevalier, 1992; Zorzini, Corti, & Pozzetti, 2008). The research ondue-date management is reviewed by Keskinocak and Tayur(2004). They categorized all due-date assignment methods intotwo categories: exogenous (determined by the sales department,without knowing the actual production schedule) and endogenous(assigned internally by the scheduling model). Some researchershave used mathematical programming in solving the due-date set-ting problem. For example, Chen et al. (2001) developed mixed-integer programming models for quantity and due-date quotationto customer orders that arrive within a pre-determined time inter-val. Sawik (2009) proposed a dual-objective problem of due-datesetting over a rolling planning horizon in make-to-order manufac-turing. They considered the total number of delayed products as aprimary optimality criterion, and the total or maximum delay ofproblem, one can name the internet-based retail system of carindustry. Automotive industry is pushed towards producingcustomized cars. Thus, they have adopted make-to-order system.Potential customers make their purchase decisions by optimallyconsider both non-negotiable price and delivery dates. Then, theyoptimize their objective by trading off between price and deliverydate.

Stochastic programming can be an appropriate approach tomodel this problem, in which decisions need to be made prior toobtaining complete information. This motivated us to use stochas-tic programming approach to formulate the integrated problem ofpricing and lead-time quotation.

This paper is organized as follows. We discuss the related liter-ature in Section 2. In Section 3, we describe the pricing and lead-time quotation problem. In Section 4, we explain the algorithmused to approximate the stochastic demand process and reformu-late our problem in scenario representation. Section 5 presents thenumerical examples and sensitivity analysis for our model. Finally,Section 6 is the conclusion of the paper.

2. Literature review

Joint pricing and inventory control strategies in a single period(newsvendor) manufacturing environment were rst stated byWhitin (1955). Whitins work was later extended by Mills (1959)who studied effect of uncertainty on a pricing policy under alinear-additive functional form for demand. Static pricing withmultiplicative form of demand was formulated by Zabel (1970)and Karlin and Carr (1962). In addition, Petruzzi and Dada (1999)examined an extension of the newsvendor problem in whichdemand depends on both price and inventory level. Some otherresearchers have considered multi-period, nite-horizon pricingmodels (e.g., see Thomas, 1970; Thowsen, 1975; Zabel, 1972).The retail industry has also used dynamic pricing techniques tomatch demand with capacity, maximize revenue or achieve otherstrategic goals, as shown by several authors (Chen & Simchi-Levi,2004; Federgruen & Heching, 1999; Gallego & van Ryzin, 1994,1997, and others). Elmaghraby and Keskinocak (2003) and Chan,Shen, Simchi-Levi, and Swann (2004) provided a thorough reviewof both single and multi-period models combining pricing andinventory strategies. Most prior research on integrating pricing

S.K. Chaharsooghi et al. / Computers & Inorders as a secondary criterion. The results in the papers citedabove indicate that proper due-date management offers a muchlarger improvement in performance than priority sequencing.In the papers reviewed above, they all assume that the demandprocess is independent of any price and/or lead-time quoted tocustomers. Some researchers have considered whether the quotedlead-times (or due-dates) and price affect customers decisions onplacing an order. Duenyas and Hopp (1995) were the rst to ana-lyze a queueing model in which lead-time quotation affects the de-mand rate. They developed queueing models that allow customersto leave if the due-date offered by the rm is too late. The objectiveis to maximize prot by making the optimal decisions on sequenc-ing and due-date setting. Duenyas (1995) also developed an effec-tive heuristic for quoting due-dates and sequencing orders.

Palaka, Erlebacher, and Kropp (1998), So and Song (1998) andWebster (2002) studied the optimal selection of price and deliverytime assuming a xed scheduling rule, and formulate the problemas a steady state queueing model.

Keskinocak, Ravi, and Tayur (2001) proposed several online andofine algorithms for quoting lead-times to different customerclasses to maximize revenue, subject to the constraint that quotedlead-times are 100% reliable when the processing time per cus-tomer is known. Moodie (1999) considered the negotiation processin price and due-date setting, where the number of quotationsfrom customer is dependent on past delivery service level. Eastonand Moodie (1999) developed a probabilistic model to determineoptimal pricing and due-date setting decisions with contingentorders. In their model, the probability that the customer will accepta quoted price/due-date pair follows an S-shaped logit model.Watanapa and Techanitisawad (2005) extended Easton andMoodies model proposing a model that incorporates the marketsegmentability, in which different policies can be applied to differ-ent demand classes. In their model, price and lead-time are denedfor a single job in a single-period production system. Extendingprevious research, Charnsirisakskul et al. (2006) formulated adeterministic mixed-integer programming model for the problemof order selection, due-date setting and production scheduling inmultiple nite periods with customers arriving at different times.In this paper, we extend the model of Charnsirisakskul et al.(2006) in which the demand function is assumed to be determin-istic. Therefore, all production lot sizes are deterministic. Due touncertainty of data, we consider the case of multiple classes oforders with stochastic demand.

Substituting the uncertain demand by an expected value can beconsidered a modeling error that cause an increase in cost andresult in lost sale and unsatised customers. Therefore, we assumethat demand is stochastic and function of both price and lead-time.

We consider a linear-additive form dened as D(p, l, n) =d(p, l) + n for demand, where D(p, l, n) is the demand and n is a ran-dom variable that does not depend on the price and lead-time; inthis case the mean demand is d(p, l), and the noise term n shifts thedemand randomly about this mean (Talluri & van Ryzin, 2004).Whitin (1955) was the rst to examine a linear-additive functionaldemand form. Other related works that considered this form of de-mand include (Chen & Simchi-Levi, 2004; Dana & Petruzzi, 2001;Federgruen & Heching, 1999; Mills, 1959; Yin & Rajaram, 2007;Zabel, 1972).

An alternative way to incorporate more information about thedemand uncertainty into the model is by formulating a stochasticlinear program.

Stochastic programming (SP) models were originated byDantzig (1955) and Beale (1955). They proposed a stochastic viewto replace the deterministic one, where the unknown coefcientsor parameters are random, with assumed probability distributionthat is independent of the decision variables. Advances in compu-tational methods to solve large scale problems made SP techniques

trial Engineering 61 (2011) 10861097 1087applicable to real-world problems. Stochastic programming hasbeen successfully applied to several optimization problems, suchas assetliability management (Consigli & Dempster, 1998; Sodhi,

Birge and Louveaux (1997) and Ruszczynski and Shapiro (2003)for basic references on stochastic programming with recourse.

ndusA standard approach to solve the stochastic models is by using anite number of scenarios to model uncertainty of relevant data.The scenarios and their probabilities represent an approximationof the probability distribution given by the random data and alsoavoid the difculty of continuous distributions. In the present pa-per, we investigate a scenario tree-based stochastic programmingapproach to the pricing and due-date management problem. In thismethod, a scenario tree consisting of a nite number of scenariosapproximates the stochastic demand process.

There are several ways to generate scenario trees for stochasticprograms. Heitsch and Rmisch (2009) categorized them as (i)bound-based constructions, (ii) Monte Carlo-based schemes orQuasi Monte Carlo-based methods, (iii) EVPI-based (expected va-lue of perfect information) sampling and reduction within decom-position schemes, (iv) the moment-matching principle, (v)probability metric-based approximations.

In this paper, the scenario tree construction approach devel-oped by Dupacova, Growe-Kuska, and Rmisch (2003) and Heitschand Rmisch (2003) is used. Given an initial set of discrete proba-bility distribution, they determined a scenario subset of prescribedcardinality and a probability measure based on this set that is theclosest to the initial distribution in terms of a natural (or canonical)probability metric. Mller, Rmisch, and Weber (2004, 2008) alsoconsidered their algorithm to approximate the evolution of passen-ger demand and the cancellation process in a multistage stochasticprogram for revenue management model.

To thebest of our knowledge froma reviewof the literature, thereis no existing joint pricing and lead-time decisionmodel based uponstochastic programming and scenario generation method.

3. Problem description: Notation and assumptions

We model the MTO manufacturing facility as a single machine,which may be the bottleneck in a system with negligible setuptimes (and costs), where order preemption is allowed.

The planning horizon is divided into periods of equal length,and the capacity in each period may differ. The rm denes a priceand lead-time for each customer class at the beginning of the plan-ning horizon, and customer classes arrive with a stochastic de-mand that is a function of the price and quoted lead-time. Themanufacturer has the option of not presenting price and lead-timefor some customer classes. The accepted order (the order that priceand lead-time is presented for him) must be shipped to the cus-tomer between the commitment due-date and the end of the plan-ning horizon. For accepted orders, after the realization of the2005), supply chain planning (Azaron, Brown, Tarim, & Modarres,2008; El-Sayed, Aa, & El-Kharbotly, 2010; Santoso, Ahmed,Goetschalckx, & Shapiro, 2005), capacity planning (Ahmed et al.,2003; Chen, Li, & Tirupati, 2002), production planning (Fleten &Kristoffersen, 2008; Gupta, Maranas, & McDonald, 2000), etc.

Within the stochastic programming, the nature of the decisionalprocess suggests that we consider a two-stage paradigm. Here, thedecision variables are partitioned into two different subsets: therst-stage decisions that are made before the random variablesare observed, and the second-stage decisions that depend on therealization of the rst-stage random vector. In our problem, deci-sions on the price and lead-time represent rst-stage actions,whereas decisions on production planning and order schedulingare second-stage decisions since they are inuenced by the uncer-tain demands. We refer the reader to Kall and Wallace (1994),

1088 S. K. Chaharsooghi et al. / Computers & Istochastic demand, the manufacturer decides on a productionschedule in a nite horizon, which, in turn, affects the due-dates.Production scheduling must occur after the release time. The re-lease time is the earliest time the production can be started. A pro-duced order can be shipped to customer between the commitmentdue-date and the end of the planning horizon. If an order is sched-uled in any period prior to its commitment and negotiated due-date, it is stored in a third party warehousing facility and incursholding costs. An order shipped after the commitment due-dateis considered late and incurs a tardiness penalty proportional tothe number of periods and the quantity delay. Shortages are al-lowed, and unmet demands are lost.

The manufacturers objective is to maximize the net prot,which is the sum of revenues from accepted orders minus produc-tion, holding, tardiness and shortage costs, subject to capacity,delivery time, and demand constraints.

3.1. Notation

In the rest of this paper, we shall use the following notation.

W = {1, . . . , N} set of customer classes classied based onsensitivity to price and lead-time,T = {1, . . . , T} set of planning periods,Pi fpi1; . . . ; pij; . . . ; pinig set of prices (per unit) the manufac-turer can charge for customer class i eW, where ni is the num-ber of prices offered to customer class i,e(i) earliest start time for producing customer class i eW,Dui fli1; . . . ; lij; . . . ; liLig set of due-dates the manufacturercan charge for customer class i eW, where ei li1 0, where Ai, Bi and Ci repre-sent the market size, customer price and lead-time sensitivity forcustomer class i, respectively, and ni is a random variable withPDF (probability density function) f() and CDF (cumulative distri-bution function) F(). As we can see, lik ei 1, is the time intervalbetween the arrival time of the customer and the quoted due-date,which we called the lead-time. The vector n = (n1, . . . , nN) repre-sents the random data vector, where ni is the stochastic inputparameter in the demand function for customer class i eW.

We will formulate this problem as a two-stage stochastic re-course model. Such model is characterized by an initial rst stagedecision corresponding to the price and lead-time quoted to eachclass of customer, after which the true values of the random eventsare realized, and a second-stage or recourse decision is made. Thesecond stage decisions correspond to future compensation or re-course actions of scheduling and production planning in each timeperiod. The rst and second stage decision variables are as follows:

First stage decision variables:

i i

trial Engineering 61 (2011) 10861097Zi,j,k 1 if price pj 2 Pi and due-date lk 2 Dui are selected(quoted) for customer class i, 0, otherwise (j = 1, . . . , ni, k =1, . . . , Li, i eW),

MaxX

psXX

pij X X

xsi;j;t;t0@ AXpsXXX

dus Second stage decision variables:

The second stage decision variables are the following stochasticvariables:

xi;j;t;t0 quantity produced (in units of capacity) for customer class iin period t and delivered in period t0 if price pij 2 Pi is selectedt ei; . . . ; t0; t0 li1; . . . ; T; i 2 W.

Ui total quantity produced for customer class i, (i eW).Hi total quantity-period inventory of customer class i, (i eW).Vi,k total quantity-period tardiness of customer class i withquoted due-date lik, (k = 1, . . . , Li, i eW).

The above bold face letters are used to denote random variables(random vectors), in order to distinguish them from their particu-lar realizations.

3.2. Stochastic model

The two-stage stochastic programming problem with randomrecourse for pricing and lead-time decisions is formulated asfollows:

Max QZ 3:1Xnij1

XLik1

Zi;j;k 1 8i 2 W 3:2

Zi;j;k 2 f0;1g 8i 2 W; j 1; . . . ;ni; k 1; . . . ; Li; 3:3

where Constraints (3.2) ensure that each order is either rejected oraccepted and only one price and one lead-time are chosen for theorder, Q(Z) is the expected second-stage recourse function denedas Q(Z) = EnQ(Z, n), and Q(Z, n) is the optimal value of the followingmodel:

MaxXNi1

Xnij1

pij XTt0li1

Xt0tei

xi;j;t;t0

0@

1AXN

i1

Xnij1

XTt0li1

Xt0tei

xi;j;t;t0 Cpit

0@

XNi1

Hi Chi XNi1

XLik1

Vi;k Cai

XNi1

Csi Xnij1

XLik1

Ai Bipij Cilik ei 1 ni Zi;j;k !

Ui !!

3:4

S.t.

XTt0li1

Xt0tei

xi;j;t;t0 6XLik1

Zi;j;kDi;j;k 8i2W; j1; . . . ;ni 3:5

Xi2Wjei6t

Xnij1

XTt0maxft;li1g

xi;j;t;t0 6Kt t1; . . . ;T 3:6

Xnij1

Xlik1t01

Xt0t1

xi;j;t;t0 6M1 1Xnij1

Zi;j;k

!8i2W; k1; . . . ;Li 3:7

HiPXnij1

XTt0li1

Xt0tei

t0 txi;j;t;t0 8i2W 3:8

Vi;kPXnij1

XTt0lik

Xt0tei

t0 likxi;j;t;t0 M2Xnij1

Zi;j;k1 !

8i2W; k1; . . . ;Li 3:9

Ui Xnij1

XTt0li1

Xt0tei

xi;j;t;t0 8i2W 3:10

xi;j;t;t0P 0 8i2W; j1; . . . ;ni; t ei; . . . ;t0; t0 li1; . . . ;T;HiP0 8i2W;

S.K. Chaharsooghi et al. / Computers & In3:11Vi;kP0 8i2W; k1; . . . ;Lis1 i1 j1 t0li1 tei s1 i1 j1 t0li1

Xt0tei

xsi;j;t;t0 Cpit XSs1

psXNi1

Hsi Chi XSs1

psXNi1

XLik1

Vsi;k

Cai XSs1

psXNi1

Csi

Xnij1

XLik1

Ai Bipij Ci lik ei 1

nsi

Zi;j;k !

Usi !

4:1After rewriting the stochastic programming model (3.2)(3.11), ourscenario-based constraints are as follows:

Xnij1

XLik1

Zi;j;k 6 1 8i 2 W 4:2

XTt0li1

Xt0tei

xsi;j;t;t0 6XLik1

Zi;j;kDsi;j;k 8i 2 W and j 1; . . . ;ni; s

1; . . . S 4:3

X Xni XTsTo solve the stochastic recourse model, assume a random vectorn = (n1, . . . , nN) is given, with a nite number of scenariosns ns1; . . . ; nsN some Euclidean space RN with probabilities ps,s = 1, . . . , S. In this case QZ EnQZ; n

PspsQZ; ns. So we

can express the expected value as weighted sum (4.1), where vari-ables xsi;j;t;t0 ; H

si ; V

si;k; U

si are production quantity, total quantity-

period inventory, total quantity-period tardiness, total quantityproduced under scenario s, respectively.

S N ni T t00 1

S N ni TThe ve terms in (3.4) correspond to the total revenue, produc-tion cost, holding cost, tardiness penalty and shortage penalty,respectively. Constraint (3.5) ensures that if price pij and due-datelik are selected for customer class i, at most Di,j,k units must beproduced and delivered for customer class i. Constraint (3.6) isa capacity constraint that ensures that the production capacityin each period is not exceeded. An order can be delivered be-tween the quoted due-date and the end of the planning horizon,but only if the order is accepted. So, we have Constraint (3.7), inwhich customer class i cannot be delivered before lik if lead-timelik is selected.

The total quantity-period inventory, total quantity-period tardi-ness and total quantity produced of each order are dened by Con-straints (3.8)(3.10), respectively. The parameters M1 and M2 aresufciently large numbers.

As we can see, the model has so-called complete xed recourse;that is, forany feasiblerst-stagesolution, thesecond-stageproblemis feasible (the recourse function is nite), since x^ 0; I^ 0; H^ 0; A^ 0 is always a feasible second-stage solution.We assume thatthe possible realizations of demand are sufciently high such thatthelowerboundLBforQ(Z) is0(Q(Z, n) > 1)andaccordingtothede-mandandcapacityconstraints,Q(Z, n)

ent groups to carry out this numerical study. For each group of

sitivity to price, and customer class N has the greatest sensitivity to

ndusXnij1

Xlik1t01

Xt0t1

xsi;j;t;t0 6 M11Xnij1

Zi;j;k 8i 2 W; k

1; . . . ; Li; s 1; . . . ; S 4:5

Hsi PXnij1

XTt0li1

Xt0tei

t0 txsi;j;t;t0 8i 2 W; s 1; . . . ; S 4:6

Vsi;k PXnij1

XTt0lik

Xt0tei

t0 likxsi;j;t;t0 M2Xnij1

Zi;j;k 1 8i

2 W; k 1; . . . ; Li; s 1; . . . ; S 4:7

Usi Xnij1

XTt0li1

Xt0tei

xsi;j;t;t0 8i 2 W; s 1; . . . ; S 4:8

xsi;j;t;t0 P 0 8i 2 W; j 1; . . . ; ni; t ei; . . . ; t0; t0 li1; . . . ; THsi P 0 8i 2 W; s 1; . . . ; SVsi;k P 0 8i 2 W; k 1; . . . ; Li; s 1; . . . ; SZi;j;k 2 f0;1g 8i 2 W; j 1; . . . ;ni; k 1; . . . ; Li

4:9The constraints (4.3)(4.8) are demand constraint, capacity con-straint, delivery constraint, total quantity-period inventory, totalquantity-period tardiness and total quantity produced for customerclasses under scenario s. The constraint (4.2) is non-anticipativityconstraint linking the separate scenarios.

Depending on the number of realizations of n, this linear pro-gram may become (very) large in scale. One way to overcome thisdifculty is using decomposition methods that exploit specialstructures of the model (Ruszczynski & Shapiro, 2003). Anothermethod is to reduce the model dimension; it might be desirableto reduce the originally designed tree (Dupacova et al., 2003;Heitsch & Rmisch, 2003). These approaches make use of probabil-ity metrics, i.e., of metric distances on spaces of probability mea-sures, where the metrics are selected such that the optimalvalues of the original and approximate stochastic programs areclose if the distance between the original probability distributionand its approximation is small.

We will briey describe the approach of Heitsch and Rmisch(2003), where the bundling and deletion process relies on comput-ing and bounding the Kantorovich distance l^rP;Q of the originalprobability distribution given by the individual scenarios

P PSi1pidni and their weights, and the distributions of theapproximate trees Q PSj1;jRJqjdnj . The Kantorovich distancel^rP;Q is given by (4.10), where P and Q are xed Borel probabilitymeasures on a closed subsetX of RN, i.e., P, Q e P(X), and a functionc: X X? R is given by (4.11).

l^rP;QMinXSi;j1jRJ

crni;njgij :gijP0;XSi1gijqj;

XSj1jRJ

gijpi

8>>>:

9>>=>>; 4:10

crn;~n :Max 1;knn0kr1;k~nn0kr1n o

k~nnk; 8n; ~n2X 4:11

Also, J {1, . . . , S}, and dn e P(X) denotes the Dirac measure placingunit mass at n. The distances l^r; r P 1, between the multivariateprobability distributions given by MongeKantorovich (mass)transportation problems are relevant for the stability of two-stage

1090 S. K. Chaharsooghi et al. / Computers & Imodels (Heitsch & Rmisch, 2007; Rmisch, 2003).The optimal choice of an index set J for scenario reduction

represents a set-covering problem. It may be formulated as a 01lead-time.The other parameters are considered to be xed in all instances

as follows:Without the loss of generality, we assume that all customers

arrive at the beginning of the time horizon. With this assumption,the lead-time and the due-date will be equivalent. The holding andtardiness costs per unit of time for each customer of class i areChi = 5, Cai = 20, and the production cost in each period t for eachcustomer of class i is Cpit 10. Prices are selected for each cus-tomer class from a bounded set of integer values, P(i) = {pi:10 6 pi 6 60}. The due-date for all customers is chosen from setproblems, the production capacity of each period (K) is categorizedas high, medium, and low capacity. Table 1 presents the character-istics of different groups of problems.

For each group of problems, we randomly generate the param-eters, Ai, Bi and Ci, in demand functions, from a uniform distribu-tion. The intervals are determined such that B1P B2P P BN,C1 6 C2 6 6 CN. Therefore, customer class 1 has the greatest sen-integer program and is NP-hard. The simultaneous backwardreduction algorithm is approximated by Heitsch and Rmisch(2003) to determine a reduced probability distribution Q of n suchthat the set of deleted scenarios has maximal cardinality and thatl^rP;Q 6 e holds.

Using their backward reduction algorithm, we determine a re-duced probability distribution of the random vector ~n ~n1; . . . ; ~nNand use it in the stochastic programming model ((4.1)(4.9)). Thedetails for backward reduction algorithm and a small instance ofconstructing the scenario tree by this algorithm are given inAppendix A.

5. Numerical study

In this section, we investigate the benets of exibility in deliv-ery, price and lead-time in various environments. Also, we illus-trate the advantage of our modeling approach compared with theexpected value solution approach.

5.1. Flexibility in delivery, price and lead-time

In this section we perform a numerical study to compare howthree different types of exibility affect the protability of therm:

(1) Price exibility(2) Lead-time exibility(3) Delivery exibility

With price (lead-time) exibility, we quote different prices(lead-times) to different customers. When there is no price (lead-time) exibility, a single (xed) price (lead-time) is quoted to allcustomers (Charnsirisakskul et al., 2006). With delivery exibility,we can ship orders after their quoted lead-time.

We consider different policies of price and lead-time exibility:P1 (price exibility, lead-time exibility), P2 (price exibility, nolead-time exibility), P3 (no price exibility, lead-time exibility),and P4 (no price exibility, no lead-time exibility). These combi-nations are considered in two cases: delivery exibility (D1) and nodelivery exibility (D2).

Since the number of customer classes (N) and the number oftime periods (T) vary, we have divided the problems into six differ-

trial Engineering 61 (2011) 10861097Du(i) = {1, 2, . . ., T 1, T}.The random perturbation nj in the demand function is exponen-

tially distributed with a mean of ve. With the initial set consisting

of S = 2000 scenarios and the backward reduction algorithmdescribed in Heitsch and Rmisch (2003), a scenario set consistingof 50 scenarios is generated for the stochastic process n by a proce-dure implemented in MATLAB software, where l^rP;Q 5:7035for problem with N = 6. This value will be reduced with a decreaseof N.

We use the model ((4.1)(4.9)) to solve this pricing model. Withdened intervals for prices and a reduced scenario tree, the modelcontains 51NT binary variables and 2550N TT12

50NT 50N

non-negative variables. For the case of a 12-period time horizonand 5-customer class problem, the number of binary decisionvariables will be 3060 and the number of non-negative variableswill be 844,750. The total number of model constraints (excludingnon-negativity and binary constraints) is 2600N + 150NTconstraints. This corresponds to 22,000 constraints for a 12-periodtime horizon and a 5-customer class problem.

The generated instances solved for all policies and inferencesare obtained based on the results. Our problem instances weresolved by ILOG CPLEX 9.0 on a PC Quad Core 2.83-GHz processor.

The maximum expected prot under each policy and three de-scribed capacity levels for generated instances are represented inFigs. 1a1f.

To compare how the protability of the rm changes accordingto different policies, we consider the base case in which there is noexibility of price, lead-time or delivery and compare the percent-age of prot increases over base case policy (P4-D2) for otherpolicies. The results are summarized in Table 2.

Also, we use the average value of prices and lead-times ob-tained from eighteen generated instances in Figs. 2 and 3. In theseFigures, we only present the prices and lead-times for two classesof customers: customer class 1 has the greatest sensitivity to priceand customer class N has the greatest sensitivity to lead-time.

The main conclusions one can draw from Table 2 and Figs.1a1f, 2 and 3 are as follows.

, pric

Table 1Specications of different groups of problems.

Problemnumber

T N K

N1-1 6 4 400N1-2 6 4 100N1-3 6 4 50N2-1 6 3 400N2-2 6 3 100N2-3 6 3 50N3-1 5 3 400N3-2 5 3 100N3-3 5 3 50N4-1 12 5 400N4-2 12 5 100N4-3 12 5 50N5-1 6 6 600N5-2 6 6 100N5-3 6 6 50N6-1 4 3 400N6-2 4 3 100N6-3 4 3 50

S.K. Chaharsooghi et al. / Computers & Industrial Engineering 61 (2011) 10861097 1091Fig. 1a. Expected prot under different combinations of lead timeFig. 1b. Expected prot under different combinations of lead time, pric

Fig. 1c. Expected prot under different combinations of lead time, price and delivery exibility in generated instances 1-1, 1-2 and 1-3.

e and delivery exibility in generated instances 2-1, 2-2 and 2-3.e and delivery exibility in generated instances 3-1, 3-2 and 3-3.

Fig. 1d. Expected prot under different combinations of lead time, price and delivery exibility in generated instances 4-1, 4-2 and 4-3.

Fig. 1e. Expected prot under different combinations of lead time, price and delivery exibility in generated instances 5-1, 5-2 and 5-3.

Fig. 1f. Expected prot under different combinations of lead time, price and delivery exibility in generated instances 6-1, 6-2 and 6-3.

Fig. 2. Average value of prices under different combinations of lead time, price and delivery exibility for each customer class in generated instances.

Fig. 3. Average value of lead times under different combinations of lead time, price and delivery exibility for each customer class in generated instances.

1092 S. K. Chaharsooghi et al. / Computers & Industrial Engineering 61 (2011) 10861097

nexdes

aluePP4

dus1. As we can see in Figs. 1a1f and Table 2, delivery exibilityleads to higher expected prot under all policies. This is alsoobtained by Charnsirisakskul et al. (2006).

2. When the production capacity is high, the manufacturer candetermine a lead-time equal to 1 for all customers andchange the demand and its prot by changing the price.Thus, under both delivery exibility and no delivery exibil-ity, price exibility is more useful than lead-time exibility.With decreasing the production capacity, the lead-time ex-ibility is more benecial because the manufacturer cannotproduce all orders in the rst period and should determinedifferent lead-times for different customer classes.

3. In the medium and low production capacity, although underexibility or with no exibility in delivery, the lead-timeexibility is more useful than the price exibility, but this

Table 2The percentage of prot increases over base case policy (P4-D2) for other policies V

Problem number P1-D1 P2-D1

N1-1 17.677 9.652N1-2 99.926 13.739N1-3 125.854 22.577N2-1 22.928 22.928N2-2 70.191 70.186N2-3 72.828 72.828N3-1 12.538 12.538N3-2 83.371 12.347N3-3 84.62 12.785N4-1 23 5.269N4-2 113.417 18.857N4-3 135.07 31.236N5-1 35.792 30.106N5-2 117.879 18.857N5-3 155.102 31.236N6-1 12.569 12.569N6-2 84.434 11.865N6-3 101.811 19.170

Average High capacity 20.75 15.51Medium capacity 94.87 24.309Low capacity 112.548 31.6394.

5.S.K. Chaharsooghi et al. / Computers & Indifference is more obvious in the no delivery exibility. Inthe absence of delivery and lead-time exibility, all ordersmust be held in the inventory and be delivered at once. Thisleads to high holding cost that the price exibility cannotcompensate.The average percentages of prot increase over the base casefor price, lead-time and delivery exibility, at medium pro-duction capacity, are 6.101%, 38.206%, and 20.099% in. Thesevalues are changed to 2.874%, 41.646%, and 26.847% at lowproduction capacity. Thus, the manufacturer chooses lead-time, delivery and price exibility in decreasing order. Theseorders may be changed due to the problems parameters.The range of holding and tardiness costs for each customerclass considered in the problems is usually high relative tothe price ranges. Therefore, the manufacturer prefers notto hold inventories. In delivery exibility, the orders madeafter the quoted due-date, are delivered immediately. Thus,delivery exibility is an advantage. This order changes toprice, lead-time and delivery exibility at a high productioncapacity because the manufacturer can produce all orders inthe rst period and deliver them in the same period.Under both medium and low production capacities, if themanufacturer can choose two types of exibility, we canrank the policies according to their percentage of protincreases over the base case as follows:(1) Price exibilitylead-time exibility,(2) Lead-time exibilitydelivery exibility,

7.

8.

9.

10.

11.

12.demand rate is high compared with the production capacity,other exibilities.6. According to Figs. 2 and 3, if the base level of demand orthe rst period for all customers and deliver orders in thet periods with a low tardiness cost. According to thiscription, the exibility in delivery is more useful than the(3) Price exibilitydelivery exibility.The two exceptions are examples N2-2 and N2-3 where theabove order was changed as follows:

(4) Price exibilitydelivery exibility,(5) Lead-time exibilitydelivery exibility,(6) Price exibilitylead-time exibility.

Unlike the other examples, the tardiness cost is considered to beone in these examples which is low compared with other costs.Thus, the manufacturer can determine a lead-time equal to thatof

4 D2D2 100

.

P3-D1 P4-D1 P1-D2 P2-D2 P3-D2

4.218 0.827 16.638 8.265 3.33144.933 11.797 87.848 4.169 42.46962.454 21.517 83.965 1.06 44.16913.015 13.015 21.907 3.239 13.19570.191 66.571 44.253 2.312 26.57866.959 66.959 40.041 1.811 27.1910.476 0.476 12.347 12.347 0

30.925 4.138 79.641 10.578 28.31231.966 6.15 71.49 5.191 22.4198.196 2.114 20.03 3.294 8.564

52.826 16.386 106.722 5.888 44.49378.15 28.333 122.948 4.031 59.73722.048 19.61 34.534 29.257 21.97857.581 16.386 107.825 5.887 57.58178.15 28.333 121.007 4.031 59.7370.339 0.339 12.095 12.095 0

34.363 5.319 81.932 7.770 29.80137.053 9.793 90.065 1.119 36.623

8.049 6.063 19.592 11.416 7.84548.47 20.099 84.704 6.101 38.20659.122 26.847 88.253 2.874 41.646

trial Engineering 61 (2011) 10861097 1093the sale price and lead-time for customers should be sethigher.According to Fig. 2, under both delivery exibility and nodelivery exibility policies, when we have price exibility,the price charges for customer class 1 (customers who aresensitive to price) is less than that for other customerclasses.With price or lead-time exibility, the manufacturer canobtain more prot while charging lower prices to customerclass 1.Comparing the two policies, P3 and P4, we can see that withlead-time exibility, the manufacturer can obtain moreprot while charging lower prices to all customer classes.According to Fig. 3, when there is high production capacity,the manufacturer can assign a lead-time equal to one forapproximately all customer classes.According to Fig. 3, under both delivery exibility and nodelivery exibility policies, when we have price exibility(policies P1 and P2), the lead-time charges for customerclass N (customers who are sensitive to lead-time) are lessthan the ones for the other customer class.As we can see from Fig. 3, in some examples, when there isno exibility in price (P3 policy), the lead-time charges forcustomer class N are higher than the ones for other customerclass. Demand for customer class N is almost high due to lowcommon price in P3 policy. The high lead-time charged forthis customer class is for balancing the demand and supply.

13. According to Fig. 3, under delivery exibility, the lead-timecharged for customers in each policy is lower than that withno delivery exibility. With delivery exibility, the lead-time quoted to customers will be low and the orders canbe produced and delivered after the quoted due-date. Withno delivery exibility, the orders are delivered only at thequoted due date. Thus, to compensate for the productioncapacity, the higher lead-time (or due-date) is charged.

5.2. Value of the stochastic program

Stochastic programs are computationally difcult to solve.Therefore, for real-world problems, people have a tendency tosolve much simpler versions. For example, researchers may solvethe deterministic program by replacing all random variables withtheir expected values, or they may solve deterministic programs,each corresponding to one particular scenario, and then combinethese different solutions with some heuristic rule. The accuracyof such approaches can be evaluated by introducing two concepts,the Expected Value of Perfect Information (EVPI) and the Value of theStochastic Solution (VSS), (Birge & Louveaux, 1997).

The Expected Value of Perfect Information (EVPI): The EVPI con-cept measures the maximum amount a decision maker would beready to pay in return for complete information about the future.Let n be the random variable whose realizations correspond tothe various scenarios. Let Q 0 be the optimal value of the stochasticprogramming and Qn be the optimal value for the deterministicproblem corresponding to one particular scenario n.

The wait and see value (WS), which corresponds to the expectedvalue of the optimal objective for each scenario is WS EnQn.

The expected value of perfect information (EVPI) is then denedas EVPI WS Q 0.

The 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 deterministiccorrespondents. However, it also dramatically increases computa-tional complexity. Therefore, the majority of people would solvethe deterministic problem by replacing the random variables withtheir corresponding expected values. The concept of the value ofthe stochastic programming solution (VSS) can be used to justifywhether the putting extra effort into modeling and solving sto-chastic programming is worthwhile. Let Zn be the optimal deci-sion of the rst stage in deterministic problem where all randomvariables are replaced by their expected values. The value of thestochastic solution (VSS) is then dened as VSS Q 0 EEV , withEEV EnQZn; n. In general, a bigger VSS indicates a higherbenet from using the stochastic programming approach.

In this section, we compute these two measures for probleminstances.

Table 3 shows the WS, EVPI, and VSS of 18 problem instances,and the best objective value for the two-stage stochastic program-ming problem which was optimized with 50 scenarios. The lastcolumn represents the lower bound for the true problem with2000 scenarios and dened as LB En0 QZn; n0, where Z*(n)is the optimal decision of the rst stage in a reduced stochasticproblemwith 50 scenarios and n0 is the random variable consistingof S = 2000 scenarios. We can obtain the optimality gap of the solu-tion Z*(n) using the lower bound estimate and the objective func-tion value estimate from the reduced stochastic program. Asmaller gap indicates a smaller error resulting from using the re-duced stochastic programming approach.

to

93963

18735

1094 S. K. Chaharsooghi et al. / Computers & Industrial Engineering 61 (2011) 10861097Table 3Computational results for wait-and-see and the stochastic programming solutions.

Problem number Q0 WS EVPI

N1-1 11078.630 11118.380 39.750N1-2 8717.539 8823.588 106.049N1-3 5850.263 5917.060 66.797N2-1 19343.57 19694.66 351.094N2-2 16059.658 16158.650 98.992N2-3 10503.853 10708.37 204.517N3-1 5700.87 5708.931 8.061N3-2 4660.653 4768.656 108.003N3-3 3074.317 3080.8 6.483N4-1 14044.294 14079.25 34.96N4-2 9904.212 10014.94 110.728N4-3 6127.792 6321.719 193.927N5-1 15528.61 15573.47 44.85319N5-2 10111.321 10200.56 89.239N5-3 6649.984 6794.502 144.518N6-1 7204.229 7263.448 59.219N6-2 5934.107 6007.378 73.270N6-3 4145.437 4280.57 135.133

Table 4Computational results for VSS and the stochastic programming solutions with respect

EXP(5)

Q0 EEV VSS

N1-1 11078.63 11076.88 1.74N1-2 8717.539 8707 10.5N1-3 5850.263 5831 19.2N2-1 19343.57 19343.46 0.11N2-2 16059.66 15984.14 75.5N2-3 10503.85 10460.88 42.9N3-1 5700.87 5695.365 5.50

N3-2 4660.653 4625.733 34.9204N3-3 3074.317 3021.947 52.37EEV VSS LB Gap

11076.88 1.75 11058.529 20.18707 10.539 8696.896 20.6435809 41.263 5841.703 8.56

19343.46 0.11 19337.32 6.2515984.14 75.518 16053.95 5.71310460.88 42.973 10499.03 4.1035695.365 5.505 5689.43 11.444625.733 34.92 4649.061 11.5923021.947 52.37 3062.255 12.062

14044.29 0.004 14035.8 8.4949801.78 102.432 9894.518 9.6946093 34.792 6120.316 7.476

15527.81 0.804 15489.43 39.18410071.13 40.191 9993.639 117.6826613 36.984 6591.82 58.1647191.283 12.946 7195.814 8.4155903.011 31.09639 5934.107 04135.264 10.173 4142.677 2.76

different probability distributions (exponential distribution).

EXP(20)

Q0 EEV VSS

11877.1 11875.21 1.898923.519 8847.576 75.9435544.539 4494.584 1049.955

20152.05 20150.05 1.99816110.6 15697.65 412.9519933.915 9824.367 109.5486155.888 6090.374 65.514

4736.073 4698.346 37.7273111.648 2952.644 159.004

First, price and lead-time exibility lead to higher prots thanno exibility in price and lead-time under all the values consideredfor the model parameters.

Second, the ranking of price, lead-time and delivery exibilityare dependent on our environment, and under some parametervalues lead-time exibility is more benecial than price exibility.

Third, with lead-time or price exibility, we can obtain more

N3-1 5508.81 5468.66 40.15 5445.64 5320.069 125.571

dustrial Engineering 61 (2011) 10861097 1095As we can see from Table 3, all instances except instances N3-1and N3-3 have a high EVPI meaning that perfect information wouldbe helpful to substantially improve the objective function.

Instances N2-2, N3-3 and N4-2 have relatively a high VSS indi-cating that a stochastic programming approach is justied.

The optimality gaps for instances N5-2 and N5-3 are relativelyhigh indicating that a scenario reduction with more precision mustbe done for these examples.

In almost all instances with a high production capacity, the VSSis low indicating that decision making in these problems is basedon expected values of random variables.

To see how the above results are represented under differentprobability distributions, a sensitivity analysis is performed withrespect to the random data. To this purpose, we solve the rst nineexamples described in Table 1 considering three new probabilitydistribution functions for random variable n in the demand func-tion as follows:

1. Exponential distribution with a mean of 20: n EXP(20),2. Uniform distribution on interval [20, 20]: n U(20, 20),3. Uniform distribution on interval [40, 40]: n U(40, 40).

As done in the last section, for each distribution, a scenario setconsisting of 50 scenarios is derived from the initial set consistingof S = 2000 sample scenarios.

Tables 4 and 5 show the best objective value for the two-stagestochastic programming problem and the VSS of rst nine probleminstances according to the considered probability distributions. Aswe can see from Tables 4 and 5, the value of the stochastic program-ming increaseswith themeanof theexponentialdistributionorwiththe coefcient of variationofdemandwhen thedemand is uniformlydistributed. In fact, the solutions given by the deterministic modelswould not be able to dene the best price and lead-time for problem

N3-2 4408.073 4213.197 194.876 4144.49 3833.521 310.969N3-3 2873.865 2828.401 45.464 2676.009 2510.491 165.518Table 5Computational results for VSS and the stochastic programming solutions with respectto different probability distributions (uniform distribution).

U(20, 20) U(40, 40)Q0 EEV VSS Q

0 EEV VSS

N1-1 10726.15 10666.16 59.989 10608.65 10521.31 87.34N1-2 8279.43 8116.736 162.694 7884.611 7550.477 334.134N1-3 5526.559 5236.163 290.396 4922.105 3713.238 1208.867N2-1 19095.85 19077.55 18.3 19073.88 19050.15 23.73N2-2 15772.69 15136.06 636.634 15440.59 14113.05 1327.538N2-3 10059.6 9795.167 264.434 9412.807 9016.401 396.406

S.K. Chaharsooghi et al. / Computers & Ininstances that had enormous variance in demand function. This rel-atively large value for VSS justies the use of more sophisticatedmodeling techniques and the extra computational effort.

6. Conclusion

We have presented a stochastic programming approach formake-to-order rms with multiple customer classes to simulta-neously determine the price and lead-time. We used an additiveform for the demand function, in which the stochastic parameteris approximated by a scenario tree. The scenario tree is generatedby the backward reduction algorithm obtained by Heitsch andRmisch (2003). Through numerical examples and a sensitivityanalysis, we compared the benets of price, lead-time and deliveryexibility with changes in the values of the model parameters. As aresult from sensitivity analysis, we can summarize the followingndings:benet by charging a lower price for some customer classes, thanwe can in the case where we have no exibility in price or lead-time. The relatively large value for VSS and EVPI in some examplesjusties the use of more sophisticated modeling techniques andextra computational efforts.

There are several other results that suggest possible further re-search. First, we have considered static price and lead-time quota-tions where price and lead-time are determined for all customersat the beginning of the time horizon. Dynamic pricing and lead-time quotations can be more benecial than static pricing in someenvironments in which the demand function changes with time.

Second, the scenario representation of the pricing and lead-timequotation problem corresponds to large scale mixed integer pro-gramming. Therefore, a future work will focus on using decompo-sition methods to solve the resulting large scale linear program.

Appendix A

A.1. Simultaneous backward reduction

Let P be a xed Borel probability measure on X, i.e., P e P(X)with scenarios {n1, n2, . . . , nS} in some Euclidean space RN and prob-ability weights {p1, p2, . . . , pS}. N is the number of customer classes.Thus the simultaneous backward reduction algorithm according toHeitsch and Rmisch (2003) is as follows:

step 1 : ckj : cni; nj; j 1; . . . ; S;Sorting of fckj : j 1; . . . ; Sg; k 1; . . . ; S;c1ll : minjl clj; l 1; . . . ; S

z1l : plc1ll ; l 1; . . . ; S;l1 2 argl2f1;...;Sgmin z1l ; J1 : fl1g:

step i : cikl : minjJi1[flg

ckj; l R Ji1; k 2 Ji1 [ flg

zil :X

k2Ji1[flgpkc

i

kl; l R J

i1;

li 2 argjJi1 min zil ; Ji : Ji1 [ flig:

A:1

Step S s + 1: redistribution by (A.2)where

qj pj Xi2Jj

pi; for each j R J A:2

Jj : fi 2 J : j jig and ji 2 arg minjRJ

cni; nj for each i 2 J A:3

and the function c: X X? R is given bycn; ~n maxf1; kn n0k; k~n n0kgr1kn ~nk; 8n; ~n 2 X A:4

Table A.1The scenarios nj and probabilities pj.

Class 1 Class 2 Class 3 Probability

n1 3.5 3.5 3.75 0.125n2 2 2.5 3 0.200n3 1 1.5 1.5 0.350

n4 1 1 1.5 0.200n5 2 2.5 3 0.125

e co

ndusFig. A.1. Initial scenario tre

1096 S. K. Chaharsooghi et al. / Computers & ITo show how the backward reduction algorithm works, we considera small instance consisting of N = 3 customer classes. We consider ascenario tree with the total number of scenarios S = 5. The scenariosnj and probabilities pj are presented in Table A.1.

The initial generated scenario tree is presented in Fig. A.1Substituting r = 1 in Eq. (A.3), the cost function cn; ~n is deter-

mined by the following matrix:

Using the reduction algorithm, a scenario set consisting of threescenarios is generated for the stochastic process n. The two scenar-ios n1 and n5 are deleted of which J = {1, 5}, j(1) = 2 and j(5) = 4. So,the new probabilities for the remaining scenarios are q2 = p1 + p2 =0.325, q3 = p3 = 0.35 and q4 = p4 + p5 = 0.325. The new scenario treeis shown in (Fig. A.2).

References

Ahmed, S., King, A. J., & Parija, G. (2003). A multi-stage stochastic integerprogramming approach for capacity expansion under uncertainty. Journal ofGlobal Optimization, 26, 324.

Azaron, A., Brown, K. N., Tarim, S. A., & Modarres, M. (2008). A multi-objectivestochastic programming approach for supply chain design considering risk.International Journal of Production Economics, 116, 129138.

Beale, E. (1955). On minimizing a convex function subject to linear inequalities.Journal of the Royal Statistical Society B, 17, 173184.

Bertrand, J. W. M. (1983). The effect of workload dependent due-dates on job shopperformance. Management Science, 29, 799816.

Fig. A.2. Reduced scenario tree cnsisting of ve scenarios.

trial Engineering 61 (2011) 10861097Biller, S., Chan, L. M. A., Simchi-levi, D., & Swann, J. (2005). Dynamic pricing and thedirect-to-customer model in the automotive industry. Electronic CommerceResearch, 5, 309334 (special issue on Dynamic Pricing).

Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York:Springer Verlag.

Chan, L. M. A., Shen, Z. J. M., Simchi-Levi, D., & Swann, J. (2004). Coordination ofpricing and inventory decisions: A survey and classication. In D. Simchi-Levi, S.D. Wu, & Z. J. M. Shen (Eds.), Handbook of quantitative supply chain analysis:Modeling in the e-business era, International series on operations research andmanagement science (pp. 335392). Norwell, MA: Kluwer Academic Publishers.

Charnsirisakskul, K., Grifn, P., & Keskinocak, P. (2006). Pricing and schedulingdecisions with lead-time exibility. European Journal of Operational Research,171, 153169.

Chen, Z. L., Li, S., & Tirupati, D. (2002). A scenario-based stochastic programmingapproach for technology and capacity planning. Computers & OperationsResearch, 29, 781806.

Chen, X., & Simchi-Levi, D. (2004). Coordinating inventory control and pricingstrategies with random demand and xed ordering cost: The nite horizon case.Operations Research, 52, 887896.

Chen, C. Y., Zhao, Z. Y., & Ball, M. O. (2001). Quantity and due-date quoting availableto promise. Information Systems Frontiers, 3, 477488.

Consigli, G., & Dempster, M. A. H. (1998). Dynamic stochastic programming forassetliability management. Annals of Operations Research, 81, 131161.

Dana, J. D., & Petruzzi, N. C. (2001). Note: The newsvendor model with endogenousdemand. Management Science, 47, 14881497.

Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science,1, 197206.

Duenyas, I. (1995). Single facility due-date setting with multiple customer classes.Management Science, 41, 608619.

Duenyas, I., & Hopp, W. J. (1995). Quoting customer lead times.Management Science,41, 4357.

Dupacova, J., Growe-Kuska, N., & Rmisch, W. (2003). Scenario reduction instochastic programming: An approach using probability metrics. MathematicalProgramming: Series A, 95, 493511.

Easton, F. F., & Moodie, D. R. (1999). Pricing and lead time decisions for make-to-order rms with contingent orders. European Journal of Operational Research,116, 305318.

Elmaghraby, W., & Keskinocak, P. (2003). Dynamic pricing in the presence ofinventory considerations: Research overview, current practices, and futuredirections. Management Science, 49, 12871309.

El-Sayed, M., Aa, N., & El-Kharbotly, A. (2010). A stochastic model for forwardreverse logistics network design under risk. Computers & Industrial Engineering,58, 423431.

onsisting of three scenarios.

Federgruen, A., & Heching, A. (1999). Combined pricing and inventory control underuncertainty. Operations Research, 47, 454475.

Fleten, S. E., & Kristoffersen, T. K. (2008). Short-term hydropower productionplanning by stochastic programming. Computers & Operations Research, 35,26562671.

Gallego, G., & van Ryzin, G. (1994). Optimal dynamic pricing of inventories withstochastic demand over nite horizons. Management Science, 40, 9991020.

Gallego, G., & van Ryzin, G. (1997). A multi-product dynamic pricing problem andits applications to network yield management. Operations Research, 45, 2441.

Gupta, A., Maranas, C. D., & McDonald, C. M. (2000). Midterm supply chain planningunder demand uncertainty: Customer demand satisfaction and inventorymanagement. Computers and Chemical Engineering, 24, 26132621.

Hegedus, M. G., & Hopp, W. J. (2001). Due-date setting with supply constraints insystems using MRP. Computers & Industrial Engineering, 39, 293305.

Heitsch, H., & Rmisch, W. (2003). Scenario reduction algorithms in stochasticprogramming. Computational Optimization and Applications, 24, 187206.

Heitsch, H., & Rmisch, W. (2009). Scenario tree modeling for multistage stochasticprograms. Mathematical Programming, 118, 371406.

Heitsch, H., & Rmisch, W. (2007). A note on scenario reduction for two-stagestochastic programs. Operations Research Letters, 35, 731738.

Kall, P., & Wallace, S. W. (1994). Stochastic programming. Chichester, England: JohnWiley and Sons.

Karlin, S., & Carr, C. R. (1962). Prices and optimal inventory policy. In K. Arrow, S.Karlin, & H. Scarf (Eds.), Studies in applied probability and management science(pp. 159172). Stanford, CA: Stanford University Press.

Keskinocak, P., Ravi, R., & Tayur, S. (2001). Scheduling and reliable lead-timequotation for orders with availability intervals and lead-time sensitiverevenues. Management Science, 47, 264279.

Keskinocak, P., & Tayur, S. (2004). Due-date management policies. In D. Simchi-Levi,S. D. Wu, & Z. J. M. Shen (Eds.), Handbook of quantitative supply chain analysis:Modeling in the e-business era, international series on operations research andmanagement science (pp. 485553). Norwell, MA: Kluwer Academic Publishers.

Mills, E. S. (1959). Uncertainty and price theory. The Quarterly Journal of Economics,

Petruzzi, N. C., & Dada, M. (1999). Pricing and the newsvendor model: A review withextensions. Operations Research, 47, 183194.

Rmisch, W. (2003). Stability of stochastic programming problems. In A.Ruszczynski & A. Shapiro (Eds.), Stochastic programming. Handbooks inoperations research and management science (pp. 483554). Amsterdam:Elsevier Science.

Ruszczynski, A., & Shapiro, A. (2003). Stochastic programming. In A. Ruszczynski &A. Shapiro (Eds.), Stochastic programming. Handbooks in operations research andmanagement science (pp. 163). Amsterdam: Elsevier Science.

Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochasticprogramming approach for supply chain network design under uncertainty.European Journal of Operational Research, 167, 96115.

Sawik, T. (2009). Multi-objective due-date setting in a make-to-order environment.International Journal of Production Research, 47, 62056231.

Seidmann, A., & Smith, M. L. (1981). Due-date assignment for production systems.Management Science, 27, 571581.

So, K. C., & Song, J. S. (1998). Price, delivery time guarantees and capacity selection.European Journal of Operational Research, 111, 2849.

Sodhi, M. M. S. (2005). LP modeling for assetliability management: A survey ofchoices and simplications. Operations Research, 53, 181196.

Talluri, K. T., & van Ryzin, G. J. (2004). The theory and practice of revenuemanagement. New York: Kluwer Academic Publishers.

Thomas, J. (1970). Price production decisions with deterministic demand.Management Science, 16, 747750.

Thowsen, G. T. (1975). A dynamic, non-stationary inventory problem for a price/quantity setting rm. Naval Research Logistics Quarterly, 22, 461476.

Watanapa, B., & Techanitisawad, A. (2005). Simultaneous price and due-datesettings for multiple customer classes. European Journal of Operational Research,166, 351368.

Webster, S. (2002). Dynamic pricing and lead-time policies for make-to-ordersystems. Decision Sciences, 33, 579599.

Wein, L. M. (1991). Due-date setting and priority sequencing in a multiclass M/G/1queue. Management Science, 37, 834850.

Wein, L. M., & Chevalier, P. B. (1992). A broader view of the job-shop schedulingproblem. Management Science, 38, 10181033.

S.K. Chaharsooghi et al. / Computers & Industrial Engineering 61 (2011) 10861097 109773, 116130.Mller, A., Rmisch, W., & Weber, K. (2004). A new approach to O&D revenue

management based on scenario trees. Journal of Revenue and PricingManagement, 3, 265276.

Mller, A., Rmisch, W., & Weber, K. (2008). Airline network revenue managementby multistage stochastic programming. Computational Management Science, 5,355377.

Moodie, D. R. (1999). Due-date demand management: negotiating the trade-offbetween price and delivery. International Journal of Production Research, 37,9971021.

Moses, S., Grant, H., Gruenwald, L., & Pulat, S. (2004). Real-time due-date promisingby build-to-order environments. International Journal of Production Research, 42,43534375.

Palaka, K., Erlebacher, S., & Kropp, D. H. (1998). Lead-time setting, capacityutilization, and pricing decisions under lead-time dependent demand. IIETransactions, 30, 151163.Whitin, T. M. (1955). Inventory control and price theory. Management Science, 2,6168.

Yin, R., & Rajaram, K. (2007). Joint pricing and inventory control with a Markoviandemand model. European Journal of Operational Research, 182, 113126.

Zabel, E. (1970). Monopoly and uncertainty. The Review of Economic Studies, 37,205219.

Zabel, E. (1972). Multi-period monopoly under uncertainty. Journal of EconomicTheory, 5, 524536.

Zorzini, M., Corti, D., & Pozzetti, A. (2008). Due-date (DD) quotation and capacityplanning in make-to-order companies: results from an empirical analysis.International Journal of Production Economics, 112, 919933.

Developing a two stage stochastic programming model of the price and lead-time decision problem in the multi-class make-to-order firm1 Introduction2 Literature review3 Problem description: Notation and assumptions3.1 Notation3.2 Stochastic model

4 A scenario representation of the stochastic model5 Numerical study5.1 Flexibility in delivery, price and lead-time5.2 Value of the stochastic program

6 ConclusionAppendix AA.1 Simultaneous backward reduction

References