A supply chain-transport supernetwork equilibrium model with the behaviour of freight carriers

Transportation Research Part E 47 (2011) 887–907

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

A supply chain-transport supernetwork equilibrium modelwith the behaviour of freight carriers

Tadashi Yamada a,⇑, Koji Imai b, Takamasa Nakamura a, Eiichi Taniguchi a

a Department of Urban Management, Kyoto University, C-1 Kyotodaigaku Katsura Nishikyo, Kyoto 615-8540, Japanb Ministry of Land, Infrastructure, Transport and Tourism, 2-1-3 Kasumigaseki Chiyoda, Tokyo 100-8918, Japan

a r t i c l e i n f o

Article history:Received 11 November 2010Received in revised form 22 February 2011Accepted 19 April 2011

Keywords:Supply chain network modellingTransport network modellingSupernetworkNetwork equilibriumFreight carriers’ behaviourFreight transport measures

1366-5545/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.tre.2011.05.009

⇑ Corresponding author. Tel.: +81 75 383 3230; faE-mail address: [email protected]

a b s t r a c t

This paper presents a supernetwork equilibrium model integrating supply chain networkswith a transport network, namely, a supply chain-transport supernetwork equilibriummodel. The model takes into account the behaviour of freight carriers and transport net-work users to endogenously determine the transport costs generated in the supply chainnetworks. The interaction between transport network and supply chain networks can alsobe examined. Results of the numerical tests reveal that the improvement of transport net-work could enhance the efficiency of supply chain networks. The paper makes contribu-tions to modelling of supply chain networks as well as to that of transport networks.

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1. Introduction

A supply chain constitutes a network of linkages between various economic actors for the passage of products fromproduction to consumption. In the supply chain, several different actors are engaged in a variety of activities, such asmanufacturing, transaction, transport and pickup/delivery. Recently, supply chain management (SCM) has become a cruciallong-term strategy for many companies, as consumer demands have become more diversified and international salescompetition has intensified (e.g., Mentzer et al., 2001). Although the definition of SCM and its concept can somewhat vary(e.g., Ellram, 1991; Scott and Westbrook, 1991; Cooper et al., 1997; Christopher, 1998; La Londe, 1998; Handfield andNichols, 1999; Simchi-Levi et al., 2000; Shapiro, 2001; Mentzer, 2001; Muckstadt et al., 2001), its fundamental objectiveis to develop the effective network between companies and/or organizations, that is, the efficient formation of supply chainnetwork (SCN) (e.g., Ellram, 1991; Lee and Billington, 1992; Towill et al., 1992; Cooper and Ellram, 1993; Lee and Ng, 1997;Kopczak, 1997).

At present, as the role of SCM is becoming more and more important for companies to remain competitive, the decisionsrelating to goods distribution and freight transport are typically made looking over the entire SCN. Therefore, accurate com-prehension of what happens on the SCN, namely, to precisely describe the behaviour of economic actors in the SCN and theresulting flow of products, allows administrators and planners to understand the mechanism of the generation of goodsmovement as well as to investigate the effects of transport and logistics-related measures. These can also enable companiesto recognise the necessity and effectiveness of such measures.

A range of freight transport measures have been implemented at intra-urban, inter-urban or international scale. However,the impacts of these measures are not only on the areas targeted but on other areas in the SCN. Hence, the effects of any

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888 T. Yamada et al. / Transportation Research Part E 47 (2011) 887–907

proposed freight transport measures should be assessed over the entire SCN. The characteristics of goods movement and ofthe behaviour of actors in the SCN must be investigated for identifying the factors of such widespread effects. As such, amethod is required to explicitly explore the effects of freight transport measures on SCNs as well as to fully describe whatoccurs in the entire multi-tiered SCNs with multiple decision-makers.

A recent trend of modelling supply chain is to incorporate decentralized decision-making by multiple agents on an SCNand their behavioural interactions (e.g., Geunes and Pardalos, 2003; Chaib-draa and Müller, 2006). Lee and Billington(1993) emphasised the importance of such an approach as well as of establishing network models. Lederer and Li(1997) modelled the competition between the firms that provide products or services to the customers who are sensitiveto delay-time. Corbett and Karmarkar (2001) studied a supply chain network consisting of several tiers of decision-makersand estimated the number of enterprises in the equilibrium conditions in an oligopolistic market. Game theory has alsobeen increasingly applied to supply chain modelling and analysis. Cachon and Netessine (2004) overviewed the applica-tions of non-cooperative game, while Nagarajan and Sosic (2008) introduced the cases and views on the application ofcooperative game. However, except for the supply chain network equilibrium (SCNE) models mentioned below, therehas been very little effort to study the methodology for comprehensively illustrating what comes about in the entireSCN, as focused in this paper.

The existing SCNE models can provide several important outputs for a multi-tiered SCN, such as the amount of prod-ucts produced by manufacturers and transacted between the agents involved in the SCN, and the price of products, withthe decentralized decision-making by such agents and behavioural interactions between them being taken into account.Nagurney et al. (2002a) developed a supply chain network equilibrium model comprising three tiers of decision-makerson an SCN with manufacturers, retailers, and consumers, and established a finite-dimensional variational inequality for-mulation, which represents the governing equilibrium conditions reflecting the optimality conditions of these decision-makers. The SCNE model has then been expanded to consider the consumers’ demand being assumed to be random witha known probability distribution (Dong et al., 2004), electronic commerce (Nagurney et al., 2005; Zhao and Nagurney,2008), reverse supply chains (Nagurney and Toyasaki, 2005; Hammond and Beullens, 2007), corporate social responsibil-ity (Cruz, 2008) and production capacity constraints (Qiang et al., 2009), and has been developed into a dynamic modelusing evolutionary variational inequalities (Daniele, 2010). Furthermore, the SCNE models have also been proved to beable to reformulate and solve as traffic network equilibrium problems (Nagurney, 2006; Nagurney et al., 2007). Yet,these SCNE studies have not dealt with the endogenous decision-making process for determining transport fares (i.e.,carriage) and costs, and therefore, their direct effects on traffic conditions in a transport network have not properly beenidentified. Consequently, the existing SCNE models are not proper to be applied to the case where the mutual effectsbetween the behavioural changes in the SCN and the traffic conditions in the transport network should be taken intoconsideration. These models would therefore be inappropriate for the impact analysis of freight transport measures tobe implemented.

Traffic network equilibrium models are typical procedures for representing traffic conditions in a transport network.Patriksson (1994), Florian and Hearn (1995) and Boyce (2007) provide a comprehensive overview of such models. The trafficnetwork equilibrium models originated with the Wardrop equilibrium (i.e., user equilibrium) by Wardrop (1952) and basi-cally deal with fixed origin–destination (OD) traffic demands, which were first formulated by Jorgenson (1963) and called‘‘user equilibrium traffic assignment with fixed origin–destination flows’’. The traffic assignment method adopted in this pa-per, where OD traffic demands vary in response to the service levels of the transport network, can be called ‘‘user equilibriumtraffic assignment with elastic demands’’, which Beckmann et al. (1955) typically formulated. This study also incorporates amulticlass user equilibrium traffic assignment technique (e.g., Yang, 1998) with multiple types of vehicles being simulta-neously taken into account.

There have also been several researches on modelling the relationship between traffic flow and goods movement, such asFriesz et al. (1983), Harker and Friesz (1986), Crainic et al. (1990), Guelat et al. (1990), Fernandez et al. (2003), and Yamadaet al. (2009). However, these merely indicate the interaction between the behaviour of shippers and that of freight carriersand do not highlight the SCN as a whole.

This paper proposes a supernetwork equilibrium model integrating SCNs with a transport network (see Nagurney andDong (2002) for the detail of supernetwork), namely, a supply chain-transport supernetwork equilibrium (SC-T-SNE) model,which allows administrators and planners to investigate the effects of freight transport measures as well as to understandthe mechanisms of goods movement on the SCNs. The model would also help companies to identify the efficiency and use-fulness of such transport measures. As Fig. 1 demonstrates, a supernetwork handled in this model is made up of several SCNsdealing with different kinds of products and a transport network, with the interaction between the traffic conditions in thetransport network and the events happened within the SCNs being considered. The model takes into account the behaviourof manufacturers, wholesalers, retailers, freight carriers and consumers as well as that of transport network users. This is asort of SCNE model, and the subsequent formulation is undertaken being based on the conventional SCNE models. However,the model explicitly represents the relationship between the events occurred on the SCNs (e.g., production, transaction andtransport of the products) and the traffic conditions in the transport network (e.g., path travel time) by incorporating thebehaviour of freight carriers within it.

Comparing the SC-T-SNE model with the existing SCNE models (e.g., Nagurney et al., 2002a,b, 2007; Dong et al., 2004;Nagurney and Toyasaki, 2005; Nagurney, 2006; Hammond and Beullens, 2007; Cruz, 2008; Qiang et al., 2009; Daniele,2010), the SC-T-SNE model has several unique features mentioned below.

Fig. 1. Supply chain-transport supernetwork.

T. Yamada et al. / Transportation Research Part E 47 (2011) 887–907 889

– The SCNE model incorporating the behaviour of manufacturers and consumers (Nagurney et al., 2002b) can examine theinfluence of transport costs on an SCN. However, this is only allowed using transport cost functions given in advance, andthe behaviour of freight carriers is not explicitly embedded. Nagurney (2006) and Nagurney et al. (2007) demonstratedthe equivalence of SCNE and transport network equilibrium, but both networks are not necessarily integrated into amodel. The SC-T-SNE model encompasses freight carriers’ behaviour within an SCNE model, which enables not only toendogenously determine the transport costs (i.e., carriage) but to identify the interaction between the transport networkand the SCNs, specifically the effects of traffic conditions in the transport network on the behaviour of each agent on theSCNs and vice versa.

– The behaviour of transport network users is also incorporated within the SC-T-SNE model. Accordingly, the model cansimultaneously describe the behaviour undertaken in both the SCNs and transport network.

– The routes (i.e., paths) used by freight carriers are identified on the road network. In the existing SCNE models, theamount of products transacted among the distinct tiers of decision-makers are only differentiated by the combinationof the decision-makers involved. The SC-T-SNE model further allows for the estimation of the amount of products accord-ing to the paths travelled within the transport network. Thus, the paths on the transport network must be determined andlisted.

– To enhance the applicability of the model, wholesalers’ behaviour is also embedded within the SC-T-SNE model as well asthe facility costs incurred by manufacturers, wholesalers, retailers and freight carriers.

– The CIF (Cost, Freight and Insurance) price system typically practiced in Japan is taken into account. Therefore, the SC-T-SNE model is formulated such that the decision-makers upstream in the SCNs bear transaction and transport costs, whichare then reflected in the price of products sold by the decision-makers being located downstream in the SCNs.

In addition, the numerical studies to be conducted hereafter hold the following characteristics:

– The model is validated and calibrated using the data actually observed.– The efficiency of the supernetwork is assessed on the basis of total surplus being estimated as the sum of producer surplus

and consumer surplus.

The rest of the paper is organised as follows. In the following section, the formulation of the SC-T-SNE model is given, deriv-ing the optimality conditions for the decision-makers, and then the governing equilibrium conditions are presented. In Sec-tion 3, the qualitative properties of the solutions are provided, with their existence and uniqueness being proved. The solutionprocedures to the variational inequality problems formulated in the previous section are also represented. The model is thentested and applied to a hypothetical supernetwork in Section 4, where several case studies are carried out for the improve-ment of road network. Finally, in Section 5, the methodologies, results, and analyses in the paper are summarised.

2. Supply chain-transport supernetwork equilibrium model

An SC-T-SNE model is formulated for a supernetwork in this section, where SCNs for various different products are in-volved in a transport network. The following assumptions are provided for the modelling.


– Consider a transport network G (V, A) with the set of nodes V and that of links A, as Fig. 2 illustrates, Y kinds of oligopolisticSCNs with the same distribution channel lie on the transport network of G (V, A), each providing product y (y = 1, . . . , Y).

- An SCN for product y consists of my manufacturers, with a typical manufacturer denoted by iy; ny wholesalers, with a typ-ical wholesaler denoted by jy; oy retailers, with a typical retailer denoted by ky; uy freight carriers, with a typical freightcarrier denoted by hy, and consumers associated with r demand markets, with a typical demand market denoted by l, asdepicted in Fig. 2.

- As can be seen from Fig. 2, the manufacturers iy(iy = 1, . . . , my) on the SCN for product y is involved in its production,which can then be purchased by the wholesalers jy(jy = 1, . . . , ny), who, in turn, sell that to the retailers ky(ky = 1, . . . , oy).Then, the retailers make the product available to the consumers at demand markets l(l = 1, . . . , r). Transport of the prod-uct is undertaken by the freight carriers hy(hy = 1, . . . , uy). The links in the supply chain networks represent those fortransport/transaction.

– Manufacturers, wholesalers, retailers and demand markets in Y kinds of SCNs exist on the nodes in the transport network.Freight vehicles trips are therefore generated from and attracted at each node in it, since the products are traded betweensuch decision-makers.

– Other traffic than the freight vehicles can also be generated from and attracted at each node in the transport network.– Set of origins for all the traffic is represented as R # V, and the set of destinations as S # V.– Every node in the transport network allows both the generation and attraction of traffic. The generation and attraction of

the traffic other than the freight vehicles is also possible at the nodes where the freight vehicles generate and attract.– No more than one decision-maker dealing with the same kind of product can be located at a single node in the transport

network.

2.1. The behaviour of manufacturers and their optimality conditions

The total costs incurred by a manufacturer iy are equal to the sum of his production cost, facility cost, transaction cost andtransport cost. His revenue, in turn, is equal to the price that the manufacturer charges for the product (and the wholesalersare willing to pay) times the total quantity of the product obtained/purchased from the manufacturer by the wholesalers.

Let E1yð¼ Eiyjy Þ be the set of paths for transporting product y between manufacturer iy and wholesaler jy on the transportnetwork, and dimp1y = e1y is given to path p1yð¼ piyjy Þ 2 E1y between OD pair (iy, jy)(iy 2 R, jy 2 S). The behaviour of manufac-turer iy dealing with product y is formulated with the following criterion of profit maximisation. Here, equilibrium solutionsare denoted by ‘‘⁄’’.

Maxqiy

Xny

jy¼1

q1�iy jyXuy

hy¼1

Xp1y2E1y

qp1y

hyiyjy� fiyðQ

1yÞ � giy ðQ1yÞ �

Xny

jy¼1

ciyjy ðQ1yÞ �

Xuy

hy¼1

Xny

jy¼1

q5�hyiyjy

Xp1y2E1y

qp1y

hyiyjyð1Þ

s:t: qp1y

hyiyjyP 0 8hy

; jy; p1y ð2Þ

Fig. 2. Modelled supply chain network.


where

q1iyjy : price charged for product y by manufacturer iy to wholesaler jy,

qp1y

hyiy jy: amount of product y transacted/transported from manufacturer iy to wholesaler jy by freight carrier hy using path p1y,

qiy : uynye1y-dimensional vector with component hyjyp1y denoted by qp1y

hyiyjyrepresenting production output of product y by

manufacturer iy,Q1y: uymynye1y-dimensional vector with component hyiyjyp1y denoted by qp1y

hyiyjyrepresenting shipments of product y

between manufacturers and wholesalers,fiy ðQ

1yÞ: production cost of manufacturer iy for product y,giy ðQ

1yÞ: facility cost of manufacturer iy,ciyjy ðQ

1yÞ: transaction cost for product y incurred between manufacturer iyand wholesaler jy (excluding transport costincurred between iy and jy),q5

hyiyjy : carriage charged by freight carrier hyfor transporting product y between manufacturer iy and wholesaler jy.

Let Q1 be an S1-dimensional vector with components: Q11, . . . , Q1Y, where S1 ¼PY

y¼1ðuymynye1yÞ� �

. Assuming that the pro-duction cost functions, facility cost functions and transaction cost functions for each manufacturer are continuously differ-entiable and convex as well as that the manufacturers compete in a noncooperative fashion (see Nagurney et al., 2002a), theoptimality conditions for all manufacturers for all kinds of products can simultaneously be expressed as the following var-iational inequality: determine Q1� 2 RS1

þ satisfying:

XY

y¼1

Xuy

hy¼1

Xmy

iy¼1

Xny

jy¼1

Xp1y2E1y

@fiy ðQ1y�Þ

@qp1y

hyiyjy

þ @giy ðQ1y�Þ

@qp1y

hyiyjy

þ@ciyjy ðQ

1y�Þ@qp1y

hyiyjy

þ q5�hyiyjy � q1�

iyjy

24

35� qp1y

hyiyjy� qp1y�

hyiyjy

h iP 0 8Q 1 2 RS1

þ ð3Þ

In this derivation, as in the subsequent derivation of inequalities (7), (11), (14) and (23), the prices charged are not consid-ered variables. Instead, they can be treated as endogenous variables in the complete equilibrium model (i.e., inequality (29))(see Nagurney et al., 2002a; Dong et al., 2004; Hammond and Beullens, 2007).

2.2. The behaviour of wholesalers and their optimality conditions

The wholesalers are involved in transactions both with the manufacturers and with the retailers. Let E2yð¼ Ejyky Þ denotethe set of paths for transporting product y between wholesaler jy and retailer ky on the transport network, andp2yð¼ pjyky Þ 2 E2yðdimp2y ¼ e2yÞ be the path travelled between OD pair (jy, ky)(jy 2 R, ky 2 S) in it. The behaviour of wholesalerjy dealing with product y is formulated below as a profit maximisation problem.

Maxqiy ;qjy

q2�jyXuy

hy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky � cjy ðQ1yÞ � gjy ðQ

1yÞ �Xoy

ky¼1

cjyky ðQ 2yÞ

�Xuy

hy¼1

Xoy

ky¼1

q6�hyjyky

Xp2y2E2y

qp2y

hyjyky �Xmy

iy¼1

q1�iy jyXuy

hy¼1

Xp1y2E1y

qp1y

hyiyjyð4Þ

s:t:Xuy

hy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky 6

Xuy

hy¼1

Xmy

iy¼1

Xp1y2E1y

qp1y

hyiyjyð5Þ

qp1y

hyiyjyP 0 8hy

; iy; p1y; qp2y

hyjyky P 0 8hy; ky

;p2y ð6Þ

where

q2jy : sales price charged for product y by wholesaler jy to retailers,

qp2y

hyjyky : amount of product y transacted/transported between wholesaler jy to retailer ky by freight carrier hy using path p2y,qjy : uyoye2y-dimensional vector with component hykyp2y denoted by qp2y

hyjyky representing shipments of product y by whole-saler jy,Q2y: uynyoye2y-dimensional vector with component hyjykyp2ydenoted by qp2y

hyjyky representing shipments of product ybetween wholesalers and retailers,cjy ðQ

1yÞ: handling/inventory costs of wholesaler jy,gjy ðQ

1yÞ: facility cost of wholesaler jy,cjyky ðQ 2yÞ: transaction cost for product y incurred between wholesaler jy and retailer ky (excluding transport cost incurredbetween jy and ky),q6

hyjyky : carriage charged by freight carrier hy for transporting product y between wholesaler jy and retailer ky.


Objective function (4) represents that the difference between the revenues minus the handling/inventory costs, facilitycost, transaction cost, transport cost and the payout to the manufacturers should be maximised. Constraint (5) simply ex-presses that retailers cannot purchase more of the product from a wholesaler than is available in stock.

Let Q2 be an S2-dimensional vector with components: Q21, . . . , Q2Y, where S2 ¼PY

y¼1ðuynyoye2yÞ� �

. The handling/inventorycost functions, facility cost functions and transaction cost functions are assumed to be continuously differentiable and con-vex. The wholesalers are also assumed to compete with one another in a noncooperative manner, seeking to determine theiroptimal shipments from the manufacturers and to the retailers. The optimality conditions for all wholesalers for all kinds ofproducts simultaneously coincide with the solution of the following variational inequality: determine Q1�; Q2�; c�

� �2 RS1þS2þnY

þ which satisfies:

XY

y¼1

Xuy

hy¼1

Xmy

iy¼1

Xny

jy¼1

Xp1y2E1y

@cjy ðQ1y�Þ

@qp1y

hyiyjy

þ@gjy ðQ

1y�Þ@qp1y

hyiyjy

þ q1�iyjy � c�jy

24

35� qp1y

hyiyjy� qp1y�

hyiyjy

h i

þXY

y¼1

Xuy

hy¼1

Xny

jy¼1

Xoy

ky¼1

Xp2y2E2y

�q2�jy þ

@cjykyðQ 2y�Þ@qp2y

hyjyky

þ q6�hyjyky þ c�jy

24

35� qp2y

hyjyky � qp2y�hyjyky

h i

þXY

y¼1

Xny

jy¼1

Xuy

hy¼1

Xmy

iy¼1

Xp1y2E1y

qp1y�hyiyjy�Xoy

ky¼1

Xp2y2E2y

qp2y�hyjyky

0@

1A

24

35� cjy � c�jy

h iP 0 8ðQ 1;Q2; cÞ 2 RS1þS2þnY

þ ð7Þ

Here, the term cjy is the Lagrange multiplier associated with constraint (5), and cy is an n-dimensional vector with com-ponent jy denoted by cjy , whereas c is an nY-dimensional vector with component y denoted by cy.

2.3. The behaviour of retailers and their optimality conditions

The retailers, in turn, are involved in transactions with the wholesalers, since they wish to obtain the product for theirretail outlets, as well as with the consumers who are the ultimate purchasers of the product. Given pathp3yð¼ pkylÞ 2 E3yðdimp3y ¼ e3yÞ used between OD pair of (ky, l)(ky 2 R, l 2 S) on the transport network, where E3yð¼ EkylÞ isthe set of paths between retailer ky and demand market l, the behaviour of retailer ky who deals with product y and seeksa maximum profit can be formulated as follows:

Maxqjy ;qky

q3�ky

Xuy

hy¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl� cky ðQ 2yÞ � gky ðQ 2yÞ �

Xr

l¼1

ckylðQ3yÞ

�Xuy

hy¼1

Xr

l¼1

q7�hykyl

Xp3y2E3y

qp3y

hykyl�Xny

jy¼1

q2�jyXuy

hy¼1

Xp2y2E2y

qp2y

hyjyky ð8Þ

s:t:Xuy

hy¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl6

Xuy

hy¼1

Xny

jy¼1

Xp2y2E2y

qp2y

hyjyky ð9Þ

qp2y

hyjyky P 0 8hy; jy

; p2y; qp3y

hykylP 0 8hy

; l; p3y ð10Þ

where

q3ky : sales price charged for the product y by retailer ky to consumers,

qp3y

hykyl: amount of product y transacted/transported from retailer kyto demand market l by freight carrier hy using path p3y,

qky : uyrey-dimensional vector with component kylp3y denoted by qp3y

hykylrepresenting shipments of product y by retailer ky,

Q3y: uyoyrey-dimensional vector with component hykylp3y denoted by qp3y

hykylrepresenting shipments of product y between

retailers and demand markets,cky ðQ2yÞ: handling/inventory costs of retailer ky,gky ðQ2yÞ: facility cost of retailer ky,ckylðQ

3yÞ: transaction cost for product y incurred between retailer ky and demand market l (excluding transport costincurred between ky and l),q7

hykyl: carriage charged by freight carrier hy for transporting product y between retailer ky and demand market l.

Objective function (8) indicates that the difference between the revenues minus all the costs generated and the payout tothe wholesalers should be maximised. Constraint (9) also represents that consumers cannot purchase more of the productfrom a retailer than is available in stock.

Let Q3 denote an S3-dimensional vector with components: Q31, . . . , Q3Y, where S3 ¼PY

y¼1ðuyoyre3yÞ� �

. Assuming that han-dling/inventory cost functions, facility cost functions and transaction cost functions are continuously differentiable and con-vex as well as that the retailers compete with one another in a noncooperative manner, seeking to determine their optimal


shipments from the wholesalers and to the demand markets, the optimality conditions for all retailers for all kinds of prod-ucts can simultaneously be formulated as the following variational inequality problem: determine Q 2�; Q3�; d�

� �2 RS2þS3þoY

þsatisfying :

XY

y¼1

Xuy

hy¼1

Xny

jy¼1

Xoy

ky¼1

Xp2y2E2y

@cky ðQ 2y�Þ@qp2y

hyjyky

þ @gky ðQ 2y�Þ@qp2y

hyjyky

þ q2�jy � d�ky

24

35� qp2y


h i

þXY

y¼1

Xuy

hy¼1

Xoy

ky¼1

Xr

l¼1

Xp3y2E3y

�q3�ky þ @ckylðQ

3y�Þ@qp3y

hykyl

þ q7�hykyl þ d�ky

" #� qp3y

hykyl� qp3y�

hykyl

h i

þXY

y¼1

Xoy

ky¼1

Xuy

hy¼1

Xny

jy¼1

Xp2y2E2y

qp2y�hyjyky �

Xr

l¼1

Xp3y2E3y

qp3y�hykyl

0@

1A

24

35� dky � d�ky

� �P 0 8ðQ 2;Q 3; dÞ 2 RS2þS3þoY

þ ð11Þ

Here, the term dky is the Lagrange multiplier associated with constraint (9), and dy is an o-dimensional vector with com-ponent ky denoted by dky , while d is an oY-dimensional vector with component y denoted by dy.

2.4. The consumers in the demand markets and the equilibrium conditions

The behaviour of consumers located at the demand markets is described in this section. The consumers take into ac-count the price charged for the product by the retailers in making their consumption decisions. It is assumed that thedemand function is continuous as well as that the following equilibrium (complementarity) conditions hold for demandmarket l.

q3�ky

¼ q4y�l if qp3y�

hykyl> 0

P q4y�l if qp3y�

hykyl¼ 0

8<: ð12Þ

dyl ðq4y�Þ

¼Puy

hy¼1

Poy

ky¼1

Pp3y2E3y

qp3y�

hykylif q4y�

l > 0

6Puy

hy¼1

Poy

ky¼1

Pp3y2E3y

qp3y�

hykylif q4y�

l ¼ 0

8>>>><>>>>:

ð13Þ

where

q4yl : market price of product y at demand market l,

q4y: r-dimensional vector for product y with component l denoted by q4yl ,

dyl ðq4yÞ : demand function of product y at demand market l.

Conditions (12) state that, in equilibrium, if the consumers at demand market l purchase the product y from retailerky, then the price charged by the retailer for the product y is equal to the price that the consumers are willing to pay forit. If the price exceeds the price the consumers are willing to pay at the demand market, then there will be no trans-action between the retailer and demand market pair. Conditions (13) state, in turn, that if the equilibrium price theconsumers are willing to pay for the product y at the demand market is positive, then the quantities purchased ofthe product from the retailers will be precisely equal to the demand for that product at the demand market. If theequilibrium price at the demand market is zero, then the shipments to that demand market may exceed the actualdemand.

Let q4 be an rY-dimensional vector for product y with component ly denoted by q4yl . In equilibrium, conditions (12) and

(13) will have to hold for all demand markets for all kinds of products, and these, in turn, can also be expressed as a vari-ational inequality problem, and given by: determine Q 3�; q4�

� �2 RS3þrY

þ such that

XY

y¼1

Xuy

hy¼1

Xoy

ky¼1

Xr

l¼1

Xp3y2E3y

q3y�k � q4y�

l

h i� qp3y

hykyl� qp3y�

hykyl

h iþXY

y¼1

Xr

l¼1

Xuy

hy¼1

Xoy

ky¼1

Xp3y2E3y

qp3y

hykyl� dy

l ðq4y�Þ

24

35� q4y

l � q4y�l

h i

P 0 8ðQ 3;q4Þ 2 RS3þrYþ ð14Þ

2.5. The behaviour of freight carriers and their optimality conditions

The freight carriers are not only decision-makers in the SCNs but transport network users. A road network is assumedas transport network with two kinds of user groups: freight vehicles operated by the freight carriers on the SCNs and


other vehicles (this can simply be called ‘‘passenger car traffic’’ or ‘‘passenger car’’ hereafter). The node for the generationof passenger cars (i.e., node of their origin) is expressed with r 2 R, and those for their attraction (i.e., node of their des-tination) with s 2 S. Path prs 2 Ers between OD pair (r, s), where Ers is the set of paths between r and s, is given asdimprs = e4.

Travel time on a path (i.e., path cost) varies in response to the traffic volume for each OD pair and the amount of the prod-ucts transacted (i.e., the amount of the products produced, distributed or transported). The path travel time affects theamount of the products transported on the SCNs through freight carriers’ decision-making, and consequently, the demandof the freight transport fluctuates.

The freight carriers are also profit-maximisers, and the optimisation problem for freight carrier hy is given asbelow:

Maxqiy ;qjy ;qky

Xmy

iy¼1

Xny

jy¼1

q5�hyiyjy

Xp1y2E1y

qp1y

hyiyjyþXny

jy¼1

Xoy

ky¼1

q6�hyjyky

Xp2y2E2y

qp2y

hyjyky þXoy

ky¼1

Xr

l¼1

q7�hykyl

Xp3y2E3y

qp3y

hykyl� ghyðQ 1y;Q 2y;Q 3yÞ ð15Þ

�Xmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjyCp1y

hyiyjyðQ 1;Q 2;Q3;X�Þ �

Xny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky Cp2y

hyjykyðQ 1;Q2;Q 3;X�Þ

�Xoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykylCp3y

hykylðQ 1;Q2;Q 3;X�Þ

s:t: qp1y

hyiyjyP 0 8iy

; jy; p1y; qp2y

hyjyky P 0 8jy; ky

; p2y; qp3y

hykylP 0 8ky

; l; p3y ð16Þ

where

ghy ðQ 1y;Q 2y;Q3yÞ: facility cost of freight carrier hy,

Cp1y

hyiyjyðQ 1;Q 2;Q 3;XÞ: unit operation cost (per transport volume) of freight carrier hy for transporting products y from man-

ufacturer iy to wholesaler jy using path p1y,Cp2y

hyjyky ðQ1;Q2;Q3;XÞ: unit operation cost (per transport volume) of freight carrier hy for transporting products y fromwholesaler jy to retailer ky using path p2y,

Cp3y

hykylðQ 1;Q 2;Q 3;XÞ: unit operation cost (per transport volume) of freight carrier hy for transporting products y from retai-

ler ky to demand market l using path p3y,X:e5e6e4-dimensional vector with component rsprs denoted by xprs

rs ,xprs

rs : traffic volume of passenger cars between r and s using path prs,e5: number of origin nodes for passenger cars,e6: number of destination nodes for passenger cars.

The facility cost is incurred for the operation, improvement and maintenance of facilities owned by the freight carriers.The operation cost is generated by the operation of freight vehicles, including their fixed cost. The unit operation costs can beformulated as follows:

Cp1y

hyiyjyðQ 1;Q 2;Q3;X�Þ ¼

gtp1y

iyjyðQ 1;Q2;Q 3;X�Þ

ijð17Þ

Cp2y

hyjyky ðQ 1;Q 2;Q3;X�Þ ¼gtp2y

jyky ðQ1;Q 2;Q 3;X�Þij

ð18Þ

Cp3y

hykylðQ 1;Q 2;Q 3;X�Þ ¼

gtp3y

kylðQ1;Q 2;Q 3;X�Þ

ijð19Þ

where

g: operation cost for a freight vehicle per unit of time,

tp1y

iyjyð�Þ; tp2y

jyky ð�Þ; tp3y

kylð�Þ: travel time on path p1y, p2y and p3y, respectively,

i: capacity of a freight vehicle,j: average loading factor of a freight vehicle.


The path travel times can also be derived as below:

tp1y

iyjyðQ1;Q 2;Q 3;X�Þ ¼

Xa2A

taðxaÞdiyjy

a;p1y ð20Þ

tp2y

jyky ðQ1;Q 2;Q3;X�Þ ¼Xa2A

taðxaÞdjyky

a;p2y ð21Þ

tp3y

kylðQ 1;Q 2;Q3;X�Þ ¼

Xa2A

taðxaÞdkyla;p3y ð22Þ

where

diy jy

a;p1y : binary value of 1 if link a is contained in path p1y between iy and jy; 0 if it is otherwise,

djyky

a;p2y : binary value of 1 if link a is contained in path p2y between jy and ky; 0 if it is otherwise,

dkyla;p3y : binary value of 1 if link a is contained in path p3y between ky and l; 0 if it is otherwise,

ta(xa): travel time on link a,xa : traffic volume on link a (see Eq. (27)).

If the facility cost functions and operation cost functions are continuously differentiable and convex, the optimality con-ditions for all freight carriers for all kinds of products can simultaneously be formulated as the following variational inequal-ity problem: determine Q1�; Q2�; Q3�

� �2 RS1þS2þS3

þ satisfying:

XY

y¼1

Xuy

hy¼1

Xmy

iy¼1

Xny

jy¼1

Xp1y2E1y

@ghy ðQ 1y�;Q2y�;Q3y�Þ@qp1y

hyiyjy

þCp1y

hyiy jyðQ 1�;Q 2�;Q 3�;X�Þþqp1y

hyiyjy

@Cp1y

hyiyjyðQ 1�;Q 2�;Q 3�;X�Þ

@qp1y

hyiyjy

24

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y

hyjykyðQ 1�;Q 2�;Q 3�;X�Þ

@qp1y

hyiyjy

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y

hykylðQ1�;Q2�;Q3�;X�Þ@qp1y

hyiyjy

�q5�hyiyjy

35� qp1y

hyiyjy�qp1y�

hyiyjy

h i

þXY

y¼1

Xuy

hy¼1

Xny

jy¼1

Xoy

ky¼1

Xp2y2E2y

@ghy Q 1y�;Q2y�;Q3y��

@qp2y

hyjyky

þCp2y

hyjyky Q 1�;Q 2�;Q 3�;X��

þqp2y

hyjyky

@Cp2y

hyjyky Q 1�;Q 2�;Q 3�;X�� @qp2y

hyjyky

24

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp2y

hyjyky

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y

hykylðQ1�;Q 2�;Q 3�;X�Þ@qp2y

hyjyky

�q6�hyjyky

35� qp2y

hyjyky�qp2y�hyjyky

h i

þXY

y¼1

Xuy

hy¼1

Xoy

ky¼1

Xr

l¼1

Xp3y2E3y

@ghyðQ 1y�;Q 2y�;Q3y�Þ@qp3y

hykyl

þCp3y

hykylðQ 1�;Q 2�;Q 3�;X�Þþqp3y

hykyl

@Cp3y

hykylðQ1�;Q2�;Q 3�;X�Þ@qp3y

hykyl

"

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y

hyjyky ðQ 1�;Q 2�;Q 3�;X�Þ

@qp3y

hykyl

�q7�hykyl

35

� qp3y

hykyl�qp3y�

hykyl

h iP0 8ðQ1;Q 2;Q 3Þ2RS1þS2þS3

þ ð23Þ

2.6. The passenger car traffic on road network and the equilibrium conditions

The behaviour of passenger cars is assumed to follow the user equilibrium traffic conditions with variable demand. Thus,the demand of passenger cars in a given OD pair fluctuates, depending on the travel cost (i.e., travel time) on its shortestpath. In this case, the behaviour of passenger cars in the road network is formulated as below:

tprsrs ðQ

1�;Q2�;Q 3�;X�Þ ¼ c�rs if xprs�rs > 0

tprsrs ðQ

1�;Q2�;Q 3�;X�ÞP c�rs if xprs�rs ¼ 0

(ð24Þ

drs c�rs

� � ¼P

prs2Ers

xprs�rs if c�rs > 0

6P

prs2Ers

xprs�rs if c�rs ¼ 0

8><>: ð25Þ


where

tprsrs ðQ

1;Q2;Q 3;XÞ ¼Xa2A

taðxaÞdrsa;p ð26Þ

xa ¼X

prs2Ers

drsa;pxprs

rs þ mX

p1y2E1y

diyjy

a;p1y

qp1y

hyiyjy

ijþX

p2y2E2y

djyky

a;p2y

qp2y

hyjyky

ijþX

p3y2E3y

dkyla;p3y

qp3y

hykyl

ij

0@

1A ð27Þ

where

tprsrs ðQ

1;Q2;Q3;XÞ: travel time on path prs,drs

a;prs: binary value of 1 if link a is contained in path prs between r and s; 0 if it is otherwise,

c�rs: minimum travel cost incurred between r and s (i.e., shortest path travel time),drs c�rs

� �: traffic demand function between r and s,

m: passenger car equivalent.

Conditions (24) represent the equilibrium conditions followed by Wordrop’s first principle. Path costs (i.e., path traveltimes) spent on all the paths used in a given OD are equal and lower than or equivalent to the path costs on any unused path.Conditions (25) show the requirements to be fulfilled for OD traffic demand.

Traffic volume on a link is obtained by adding the passenger car volume on the link and the converted value of the prod-ucts transacted among the actors on the SCNs into the passenger car volume (see Eq. (27)). As Eqs. (26) and (27) indicate, thepath travel time of passenger cars is subject to the path traffic volume of freight vehicles. Objective function (15), Eqs. (17)–(22), (26) and (27) demonstrate that the amount of the products distributed in the SCNs (i.e., the amount of the productsproduced, transacted or transported) is influenced by the traffic volume of passenger cars. The model likewise allows thefreight vehicles and passenger cars to be treated as multiclass users.

In equilibrium, conditions (24) and (25) must hold for all OD pairs. Hence, these conditions are equivalent to obtainingX�; c�rs

� �2 Re5e6e4þe5e6

þ which satisfies:

Xr2R

Xs2S

Xprs2Ers

tprsrs ðQ

1�;Q 2�;Q3�;X�Þ�c�rs

h i� xprs

rs �xprs�rs

� �þXr2R

Xs2S

Xprs2Ers

xprs�rs �drs c�rs

� �" #� crs�c�rs

� �P 0 8ðX;crsÞ 2Re5e6e4þe5e6

þ

ð28Þ

2.7. The equilibrium conditions of the supply chain-transport supernetwork

In equilibrium, all the optimality conditions (see (3), (7), (11) and (23)) and all the equilibrium conditions (see (14) and(28)) must hold simultaneously, and the product flows (as well as that transported) between the distinct tiers of the deci-sion-makers on each SCN coincide. This is explicitly stated in the following definition:

Definition 1. (Supply chain-transport supernetwork equilibrium). The equilibrium state of the supply chain-transportsupernetwork is one where: all manufacturers have achieved optimality for all kinds of products (cf. (3)); all wholesalershave achieved optimality for all kinds of products (cf. (7)); all retailers have achieved optimality for all kinds of products (cf.(11)); all freight carriers have achieved optimality for all kinds of products (cf. (23)); equilibrium conditions for all demandmarkets hold for all kinds of products (cf. (14)), and, finally, equilibrium conditions for all OD pairs of passenger car traffic hold(cf. (28)).

Under this definition, since the product flows (as well as that transported), market prices, path traffic volume, and pathcosts (i.e., path travel time) will have to satisfy the sum optimality conditions (3), (7), (11) and (23) and conditions (14) and(28), the following theorem can be established.

Theorem 1. (Variational inequality formulation). The equilibrium conditions governing the supply chain-transport supernetworkmodel are equivalent to the solution to the variational inequality problem given by: determine

Q 1�;Q 2�;Q3�; c�; d�;q4�;X�; c�rs

� �2 RS1þS2þS3þuyþoyþrþe5e6e4þe5e6

þ satisfying:

XY

y¼1

Xuy

hy¼1

Xmy

iy¼1

Xny

jy¼1

Xp1y2E1y

@fiy ðQ1y�Þ

@qp1y

hyiyjy

þ @giy ðQ1y�Þ

@qp1y

hyiyjy

þ@ciyjy ðQ

1y�Þ@qp1y

hyiyjy

þ@cjy ðQ

1y�Þ@qp1y

hyiyjy

þ@gjyðQ

1y�Þ@qp1y

hyiyjy

þ @ghy ðQ1y�;Q 2y�;Q 3y�Þ@qp1y

hyiyjy

24

þ Cp1y

hyiyjyðQ 1�;Q2�;Q 3�;X�Þ þ qp1y

hyiyjy

@Cp1y

hyiyjyðQ 1�;Q2�;Q 3�;X�Þ

@qp1y

hyiyjy

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y

hyjyky ðQ 1�;Q 2�;Q3�;X�Þ

@qp1y

hyiyjy


þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y

hykylðQ 1�;Q 2�;Q3�;X�Þ@qp1y

hyiyjy

� c�jy

35� qp1y

hyiyjy� qp1y�

hyiyjy

h i

þXY

y¼1

Xuy

hy¼1

Xny

jy¼1

Xoy

ky¼1

Xp2y2E2y

@cky ðQ2y�Þ@qp2y

hyjyky


hyjyky

þ@cjyky ðQ2y�Þ@qp2y

hyjyky

þ @ghy ðQ 1y�;Q 2y�;Q 3y�Þ@qp2y

hyjyky

þ Cp2y

hyjyky ðQ 1�;Q2�;Q 3�;X�Þ

24

þ qp2y

hyjyky

@Cp2y


@qp2y

hyjyky

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y

hyiyjyðQ 1�;Q 2�;Q3�;X�Þ

@qp2y

hyjyky

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y


hyjyky

þ c�jy � d�ky

35� qp2y


h i

þXY

y¼1

Xuy

hy¼1

Xoy

ky¼1

Xr

l¼1

Xp3y2E3y

@ckylðQ3y�Þ

@qp3y

hykyl


hykyl

þ Cp3y

hykylðQ 1�;Q2�;Q 3�;X�Þ þ qp3y

hykyl

@Cp3y


hykyl

24

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y


@qp3y

hykyl

þ d�ky � q4y�l

35

� qp3y

hykyl� qp3y�

hykyl

h iþXY

y¼1

Xny

jy¼1

Xuy

hy¼1

Xmy

iy¼1

Xp1y2E1y

qp1y�hyiyjy�Xoy

ky¼1

Xp2y2E2y

qp2y�hyjyky

0@

1A

24

35� cjy � c�jy

h i

þXY

y¼1

Xoy

ky¼1

Xuy

hy¼1

Xny

jy¼1

Xp2y2E2y

qp2y�hyjyky �

Xr

l¼1

Xp3y2E3y

qp3y�hykyl

0@

1A

24

35� dky � d�ky

� �

þXY

y¼1

Xr

l¼1

Xuy

hy¼1

Xoy

ky¼1

Xp3y2E3y

qp3y�hykyl� dy

l ðq4y�Þ

24

35� q4y

l � q4y�l

h iþXr2R

Xs2S

Xprs2Ers

tprsrs ðQ

1�;Q2�;Q 3�;X�Þ � c�rs

h i� xprs

rs � xprs�rs

� �

þXr2R

Xs2S

Xprs2Ers

xprs�rs � drs c�rs

� �" #� crs � c�rs

� �P 0 8ðQ 1;Q 2;Q 3; c; d;q4;X; crsÞ 2 RS1þS2þS3þnYþoYþrYþe5e6e4þe5e6

þ

ð29Þ
The proof to Theorem 1 can be similar to that established by Nagurney et al. (2002a) and Hammond and Beullens (2007).
With some algebraic manipulation, it can be seen that variational inequality (29) is indeed the sum of the inequalities (3),(7), (11), (14), (23) and (28). The converse also needs to be shown if the solution to (29) is in fact an equilibrium as per Def-inition 1. This can be shown as follows: To inequality (29) add the term þq1�

iy jy � q1�iyjy and þq5�

hyiyjy � q5�hyiyjy to the term in the

first set of brackets preceding the multiplication sign, add the term þq2�jy � q2�

jy and þq6�hyjyky � q6�

hyjyky to the term preceding thesecond multiplication sign, and add the term þq3�

ky � q3�ky and þq7�

hykyl � q7�hykyl to the term preceding the third multiplication

sign. The variational inequality supplemented with these terms turns into the sum of inequalities (3), (7), (11), (14), (23)and (28) without changing the value of variational inequality (29).

2.8. Retrieving the price variables

The price variables q1�iy jy can be retrieved from the eventual solution by Eq. (30), setting qp1y

hyiy jy> 0 in inequality (3).

q1�iyjy ¼

@fiy ðQ1y�Þ

@qp1y

hyiyjy

þ @giy ðQ1y�Þ

@qp1y

hyiyjy

þ@ciyjy ðQ

1y�Þ@qp1y

hyiyjy

þ q5�hyiyjy ð30Þ

The equilibrium solutions of c and d can be obtained from inequality (29), and the prices q2�jy can also be obtained by find-

ing a qp2y

hyjyky > 0 in inequality (7) as follows:

q2�jy ¼

@cjyky ðQ2y�Þ@qp2y

hyjyky

þ q6�hyjyky þ c�jy ð31Þ

Also, if qp3y

hykyl> 0 in inequality (11), the prices q3�

ky can be derived as Eq. (32).

q3�ky ¼ @ckylðQ

3y�Þ@qp3y

hykyl

þ q7�hykyl þ d�ky ð32Þ


Likewise, the carriage charged by freight carrier hy for transporting product y can be obtained from inequality (23) asfollows:

q5�hyiyjy ¼

@ghy ðQ 1y�;Q2y�;Q3y�Þ@qp1y

hyiyjy

þ Cp1y

hyiyjyðQ1�;Q 2�;Q 3�;X�Þ þ qp1y

hyiyjy

@Cp1y

hyiyjyðQ1�;Q 2�;Q 3�;X�Þ

@qp1y

hyiyjy

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y


@qp1y

hyiyjy

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y


hyiyjy

ð33Þ

q6�hyjyky ¼ @ghy ðQ 1y�;Q 2y�;Q3y�Þ

@qp2y

hyjyky

þ Cp2y

hyjyky ðQ 1�;Q2�;Q 3�;X�Þ þ qp2y

hyjyky

@Cp2y


@qp2y

hyjyky

þXmy

iy¼1

Xny

jy¼1

Xp2y2E2y

qp1y

hyiyjy

@Cp1y

hyiyjyðQ1�;Q 2�;Q 3�;X�Þ

@qp2y

hyjyky

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y

hykylðQ 1�;Q 2�;Q 3�;X�Þ@qp2y

hyjyky

ð34Þ

q7�hykyl ¼

@ghy ðQ 1y�;Q 2y�;Q 3y�Þ@qp3y

hykyl

þ Cp3y


hykyl

@Cp3y


hykyl

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y


@qp3y

hykyl

ð35Þ

3. Qualitative properties and solution algorithms

Firstly, the existence of a solution to inequality (29) is examined in this section. For ease

of reference, assuming that Z � ðQ 1;Q 2; Q 3; c; d;q4;X; crsÞ, FðZÞ � Fp1y

hyiy jy; Fp2y

hyjyky ; Fp3y

hykyl; Fjy ; Fky ; Fy

l ; Fprsrs ;

�Frs

�; hy¼1;...;uy ; iy¼1;...;my ; jy¼1;...;ny ; ky¼1;...;oy ; l¼1;...;r; y¼1;...;Y; p1y¼1;...;e1y ; p2y¼1;...;e2y ; p3y¼1;...;e3y ; r¼1;...;e5 ; s¼1;...;e6 ; prs¼1;...;e4 , as well as that the spe-

cific components of F are given by the functional terms preceding the multiplication signs in (29), inequality (29) canbe rewritten in standard variational inequality form: determine Z⁄ 2 K, where

K � ðQ1;Q 2;Q 3; c; d;q4;X; crsÞ��ðQ 1;Q 2;Q 3; c; d;q4;X; crsÞ 2 RS1þS2þS3þnYþoYþrYþe5e6e4þe5e6

þ

n oð36Þ

satisfying

hFðZ�Þ; Z � Z�iP 0; 8Z 2 K ð37Þ

Here, the term h�,�i represents the inner product in N-dimensional Euclidean space.The feasible region K is not always compact, even if F in inequality (37) is continuous. However it is possible to impose a

weak condition on K to guarantee the existence. Let

Kb ¼ ðQ 1;Q 2;Q 3;c;d;q4;X;crsÞ��06Q1

6 b1; 06Q26 b2; 06Q3

6 b3; 06 c6 b4; 06 d6 b5; 06q46 b6; 06X6 b7; 06 crs 6 b8

n oð38Þ

where b = (b1, b2, b3, b4, b5, b6, b7, b8) P 0, and Q16 b1; Q2

6 b2; Q36 b3; c 6 b4; d 6 b5; q4

6 b6, X 6 b7, crs 6 b8 means thatqp1y

hyiyjy6 b1, qp2y

hyjyky 6 b2, qp3y

hykyl6 b3, cjy 6 b4, dky 6 b5, q4y

l 6 b6; xprsrs 6 b7; crs 6 b8 or all i, j, k, l, h, r, s, p1y, p2y, p3y and prs. Kb is

a bounded, closed convex subset of RS1þS2þS3þnYþoYþrYþe5e6e4þe5e6

þ . Consequently, the following variational inequality admitsat least one solution of Zb 2 Kb, as F is continuous:

hFðZbÞ; Z � ZbiP 0 8Zb 2 Kb ð39Þ

Following Theorem 4.2 in Kinderlehrer and Stampacchia (1980) (see also Theorem 1.5 in Nagurney (1999)), Lemma 1 canhold as below:

Lemma 1. Variational inequality (37) (and (29) as well) admits a solution if and only if there exists a b > 0 such that variationalinequality (39) admits a solution in Kb with

Q1 < b1; Q 2 < b2; Q 3 < b3; c < b4; d < b5; q4 < b6; X < b7; crs < b8 ð40Þ

Under the conditions in Theorem 2 described below, the existence of a solution to the original variational inequality prob-lem is guaranteed by Lemma 1 (see Nagurney and Zhao, 1993; Hammond and Beullens, 2007).


Theorem 2. Suppose there exists positive constants M, N and R (M < R) such that,

@fiy ðQ1yÞ

@qp1y

hyiyjy

þ @giyðQ1yÞ

@qp1y

hyiyjy

þ@ciyjy ðQ

1yÞ@qp1y

hyiyjy

þ@cjy ðQ

1yÞ@qp1y

hyiyjy

þ@gjy ðQ

1yÞ@qp1y

hyiyjy

þ @ghy ðQ 1y;Q2y;Q 3yÞ@qp1y

hyiyjy

þ Cp1y

hyiyjyðQ 1;Q2;Q 3;XÞ

þ qp1y

hyiyjy

@Cp1y

hyiyjyðQ 1;Q 2;Q 3;XÞ

@qp1y

hyiyjy

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y

hyjykyðQ 1;Q2;Q 3;XÞ

@qp1y

hyiyjy

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y

hykylðQ 1;Q 2;Q 3;XÞ@qp1y

hyiyjy

P R 8Q 1 with qp1y

hyiyjyP N 8hy

; iy; jy

; p1y ð41Þ

@cky ðQ2yÞ@qp2y

hyjyky

þ @gky ðQ 2yÞ@qp2y

hyjyky

þ@cjyky ðQ 2yÞ@qp2y

hyjyky

þ @ghy ðQ1y;Q 2y;Q 3yÞ@qp2y

hyjyky

þ Cp2y

hyjyky ðQ 1;Q 2;Q3;XÞ

þ qp2y

hyjyky

@Cp2y

hyjyky ðQ 1;Q2;Q 3;XÞ

@qp2y

hyjyky

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y

hyiyjyðQ 1;Q2;Q 3;XÞ

@qp2y

hyjyky

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y

hykylðQ 1;Q 2;Q 3;XÞ@qp2y

hyjyky

P R 8Q 2 with qp2y

hyjyky P N 8hy; jy

; ky; p2y ð42Þ

@cky lðQ3yÞ

@qp3y

hy ky l

þ@ghy ðQ 1y;Q2y;Q 3yÞ@qp3y

hykyl

þCp3y

hykylðQ 1;Q 2;Q 3;XÞþqp3y

hy ky l

@Cp3y

hyky lðQ1;Q2;Q3;XÞ@qp3y

hykyl

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiy jy

@Cp1y

hyiy jyðQ 1;Q 2;Q3;XÞ

@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hy jyky

@Cp2y

hy jyky ðQ1;Q2;Q3;XÞ

@qp3y

hykyl

PR 8Q3 with qp3y

hykylPN 8hy

; ky; l; p3y ð43Þ

dyl ðq4yÞ 6 N 8q4y with q4y

l P M; 8l; y ð44Þ

tprsrs ðQ

1;Q 2;Q 3;XÞP R 8X with xprsrs > N; 8r; s;prs ð45Þ

drs c�rs

� �6 N 8crs P M; 8r; s ð46Þ

then variational inequality (29) (and (37) as well) admits at least one solution.

Proof. Choose b1 = b2 = b3 = b7 = b > N, b6 > N1, b8 > N2, where

N1 ¼ maxhy ;ky ;l;p3y ;Q1

6b;Q26b;Q3

6b;X6b

@ckylðQ3y�Þ

@qp3y

hykyl

þ @ghyðQ 1y�;Q 2y�;Q 3y�Þ@qp3y

hykyl

þ Cp3y

hykylðQ 1�;Q2�;Q 3�;X�Þ

(

þ qp3y

hykyl

@Cp3y

hykylðQ 1�;Q2�;Q 3�;X�Þ@qp3y

hykyl

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y


@qp3y

hykyl

9=; ð47Þ

N2 ¼ maxr;s;prs ;Q

16b;Q2

6b;Q36b;X6b

tprsrs Q1�;Q 2�;Q 3�;X�� n o

ð48Þ

If the bounds b4, b5 for c, d are constructed large enough, the existence of a solution to inequality (29) (and (37) as well) isensured through Lemma 1, since Q 1b < b; Q2b < b; Q 3b < b, q4b < b6; Xb < b; cb

rs < b8 are assured by the followingstatements.

1. Suppose there exists a hy, iy,jy and p1y such that Q1b = b > N. This would then require


@fiyðQ1y�Þ

@qp1y

hyiyjy

þ @giy ðQ1y�Þ

@qp1y

hyiy jy

þ@ciyjyðQ

1y�Þ@qp1y

hyiyjy

þ@cjy ðQ

1y�Þ@qp1y

hyiyjy

þ@gjy ðQ

1y�Þ@qp1y

hyiyjy


hyiyjy

þ Cp1y


þ qp1y

hyiyjy

@Cp1y


@qp1y

hyiyjy

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y

hyjyky ðQ1�;Q 2�;Q 3�;X�Þ

@qp1y

hyiyjy

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y


hyiyjy

6 0

in inequality (29). However, this contradicts assumption (41).2. Suppose also there exists a hy, jy,ky and p2y such that Q2b = b > N. This would then require

@cky ðQ 2y�Þ@qp2y

hyjyky


hyjyky

þ@cjyky ðQ 2y�Þ@qp2y

hyjyky


hyjyky

þ Cp2y

hyjyky ðQ 1�;Q 2�;Q3�;X�Þ þ qp2y

hyjyky

@Cp2y


@qp2y

hyjyky

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp2y

hyjyky

þXoy

ky¼1

Xr

l¼1

Xp3y2E3y

qp3y

hykyl

@Cp3y


hyjyky

6 0

in inequality (29). However, this contradicts assumption (42).3. Suppose there exists a hy, ky, l and p3y such that Q3b = b > N. According to inequalities (29) and (43), this would require

R <@ckylðQ

3y�Þ@qp3y

hykyl


hykyl

þ Cp3y

hykylðQ 1�;Q 2�;Q 3�;X�Þ þ qp3y

hykyl

@Cp3y


hykyl

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y


@qp3y

hykyl

6 q4y;bl

Inequality (29) would then also requirePuy

hy¼1

Poy

ky¼1

Pp3y2E3y qp3y�

hykyl� dy

l ðq4y�Þ 6 0, i.e., dyl ðq4y;bÞ > N. By assumptions (44) and

M < R, it leads to q4y;bl < M < R. This contradicts R < q4y;b

l mentioned above.4. Accordingly, Q3b < b holds for all hy, ky, l and p3y. Then,

q4y;bl 6

@ckylðQ3y�Þ

@qp3y

hykyl


hykyl

þ Cp3y


hykyl

@Cp3y


hykyl

þXmy

iy¼1

Xny

jy¼1

Xp1y2E1y

qp1y

hyiyjy

@Cp1y


@qp3y

hykyl

þXny

jy¼1

Xoy

ky¼1

Xp2y2E2y

qp2y

hyjyky

@Cp2y


@qp3y

hykyl

8l

is derived through inequality (29), and q4y;bl 6 N1 < b6 holds for all l under the definition of N1 in assumption (47).

5. Suppose there exists a r, s and prs such that Xb = b > N. Following inequalities (29) and (45), this would requireR 6 tprs

rs Q1�;Q 2�;Q 3�;X��

6 c�rs. Then, inequality (29) leads toP

prs2Ersxprs�

rs � drs c�rs

� �6 0, i.e., drs c�rs

� �> N. By assumptions

(46) and M < R; cbrs < M < R must then hold. However, this contradicts the above-mentioned condition of R 6 cb

rs.6. Hence, Xb < b holds for all r,s and prs; and c�rs 6 tprs

rs ðQ1�;Q 2�;Q 3�;X�Þ is provided for all r and s by inequality (29). Then, by

assumption (48), cbrs 6 N2 < b8 holds for all r and s.

The proof is complete. h

The existence of the positive constants M, N and R is reasonable from an economics perspective. If any of the shipments islarge, then the marginal costs of each would be expected to exceed some positive lower bound. Also, when the price a markethas to pay is high, the demand for the product is expected to be low at that consumer market.

According to Kinderlehrer and Stampacchia (1980) and Nagurney (1999), a solution to variational inequality (29) is un-ique if the vector function F(Z) is strictly monotone. As Lemma 2 in Nagurney et al. (2002a) indicates, it can easily be dem-onstrated that the function F(Z) is monotone; when fiy ; giy ; ciyjy , cjy ; gjy ; ghy , cky ; gky and cjyky are convex functions,Cp1y

hyiyjy; Cp2y

hyjyky ; Cp3y

hykylare non-decreasing convex functions, dy

l and drs are monotone decreasing functions, and tprsrs is a mono-

tone increasing function. In addition, following Lemma 3 in Nagurney et al. (2002a), if one of the families of these convexfunctions is a family of strictly convex functions, and dy

l ; drs and tprsrs are strictly monotone; the function F(Z) is strictly mono-

tone. The strict monotonicity and the uniqueness based on it can be easily proved as Lemma 2, Lemma 3, and Theorem 3 inNagurney et al. (2002a).

A modified projection method can be used to solve variational inequality (29) (e.g., Nagurney et al. (2002a)). This methodis based on Lipschitz continuity condition for F(Z) and involves complicated procedures where the step size must be set in


advance with unknown Lipschitz constant. This study therefore applies the solution procedures proposed by Meng et al.(2007) as outlined below to avoid such difficulties.

(i) As Z is non-negative, variational inequality (29) (and (37) as well) can be converted into an equivalent complemen-tarity problem: determine Z⁄ satisfying

Z� P 0; FðZ�ÞP 0; hFðZ�Þ; Z�iP 0; 8Z 2 K ð49Þ

(ii) Using Fischer–Burmeister function (Fischer, 1992) of /: R2 ? R+ (see Eq. (50)), a non-negative real-valued function isdefined as Eq. (51).

/ðw; vÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2 þ v2

p� ðwþ vÞ

h i2ð50Þ

wðZÞ ¼XY

y¼1

Xuy

hy¼1

Xmy

iy¼1

Xny

jy¼1

Xp1y2E1y

/ qp1y

hyiyjy; Fp1y

hyiyjyðZÞ

� �þXY

y¼1

Xuy

hy¼1

Xny

jy¼1

Xoy

ky¼1

Xp2y2E2y

/ qp2y

hyjyky ; Fp2y

hyjykyðZÞ� �

þXY

y¼1

Xuy

hy¼1

Xoy

ky¼1

Xr

l¼1

Xp3y2E3y

/ qp3y

hykyl; Fp3y

hykylðZÞ

� �þXY

y¼1

Xny

jy¼1

/ðcjy ; Fjy ðZÞÞ þXY

y¼1

Xoy

ky¼1

/ dky ; Fky ðZÞð Þ

þXY

y¼1

Xr

l¼1

/ q4y

l ; Fyl ðZÞ

� �þXr2R

Xs2S

Xprs2Ers

/ X; Fprsrs ðZÞ

� �þXr2R

Xs2S

/ðcrs; FrsðZÞÞ ð51Þ

(iii) Complementarity conditions (49) can also be converted into an equivalent unconstrained nonlinear optimisationproblem of min w(X).

(iv) The optimisation problem is solved using the Quasi-Newton method.

As stated in Meng et al. (2007), the existence of a solution to this optimisation problem can be verified with Theorem 2described above. Geiger and Kanzow (1996) demonstrated that any stationary point of the unconstrained minimisationproblem is its global minimum under the conditions where F(Z) is monotone and continuously differentiable. The existenceof an accumulation point can be guaranteed based on the function form of (50) and (51), and hence, the solution obtainedusing the Quasi-Newton method turns into a global minimum.

1 5 9

62

4 8

Manufacturer 1(Product 1)

Wholesaler 1(Product 1)

Passenger Car OD

Retailer 2(Product 2)

3 7


Market 1

1

2

3

4

5

6

78

910

11

1213

14

15

16

17

1819

20

21

22

23

24

25

26

2728

2930

31

32

33

34

35

36



Passenger Car OD






Passenger Car OD


Market 2 Market 3

Fig. 3. Test supernetwork.


4. Numerical examples

4.1. Problem definition and the base case

The SC-T-SNE model is then numerically tested to validate its performance as well as to investigate the impact of freighttransport measures on SCNs, such as the establishment of new links on the road network. These are undertaken using ahypothesised supply chain-transport supernetwork as shown in Fig. 3, where it is assumed that the transport network inves-tigated is an urban road network, and the SCNs are located in that urban area; namely, the economic activities from produc-tion to consumption on them are assumed to be achieved within this area, and all products are assumed to be transported bytrucks travelled on the road network.

The network illustrated in Fig. 3 is composed of 9 nodes and 36 links. The link distances are set to 1 (for link numbers 3, 4,7, 8, 9, 10, 15, 16, 23, 24, 27, 28, 29, 30, 33 and 34), 2 (for link numbers 11, 12, 19 and 20) and

ffiffiffi2p

(for the remaining links),respectively. Here, two kinds of products (i.e., product 1 and 2) are only assumed to be manufactured, transacted and trans-ported. The SCN for each product comprises two manufacturers, two wholesalers, two retailers, three demand markets andone freight carrier. The demand markets are assumed to be located at nodes 3, 5 and 7. The SCN for product 1 has the man-ufacturers at nodes 1 and 9, wholesalers at nodes 2 and 8, and retailers at nodes 4 and 6; whereas that for product 2 containsthe manufacturers at nodes 1 and 9, wholesalers at nodes 4 and 6, and retailers at nodes 2 and 8. These nodes are also originsand destinations for trucks, whilst the nodes 1, 5 and 9 are for the generation and attraction of passenger cars.

Both the functional forms and parameter values of fiy ; cjy ; cky ; ciyjy ; cjyky , ckyl; giy ; gjy ; gky , ghy ; dyl ; i; j; g; m and ta are deter-

mined so that the existence and uniqueness of solutions are ensured. These settings are also based on the existing studies(e.g., Nagurney et al., 2002a; Patriksson, 1994). The two kinds of products are assumed to be independently consumed witheach other, and for simplicity, their functional forms and parameter values are set identically (i.e., the same functional formsand parameter values are used for both y = 1 and y = 2).

fiy ¼ 100X1

hy¼1

X2

jy¼1

Xp1y2E1y

qp1y

hyiyjy

0@

1Aþ 0:2

X1

hy¼1

X2

jy¼1

Xp1y2E1y

qp1y

hyiyjy

0@

1A

2

ð52Þ

cjy ¼ 0:1X1

hy¼1

X2

iy¼1

Xp1y2E1y

qp1y

hyiyjy

0@

1A

2

; cky ¼ 0:1X1

hy¼1

X2

jy¼1

Xp2y2E2y

qp2y

hyjyky

0@

1A

2

ð53Þ

ciyjy ¼ 5X1

hy¼1

Xp1y2E1y

qp1y

hyiy jy

0@

1A; cjyky ¼ 5

X1

hy¼1

Xp2y2E2y

qp2y

hyjyky

0@

1A; ckyl ¼ 5

X1

hy¼1

Xp3y2E3y

qp3y

hykyl

0@

1A ð54Þ

giy ¼ 20X1

hy¼1

X2

jy¼1

Xp1y2E1y

qp1y

hyiyjy; gjy ¼ 20

X1

hy¼1

X2

iy¼1

Xp1y2E1y

qp1y

hyiyjy; gky ¼ 20

X1

hy¼1

X2

jy¼1

Xp2y2E2y

qp2y

hyjyky ð55Þ

ghy ¼ 0:4X2

iy¼1

X2

jy¼1

Xp1y2E1y

qp1y

hyiyjyþX2

jy¼1

X2

ky¼1

Xp2y2E2y

qp2y

hyjyky þX2

ky¼1

X2

l¼1

Xp3y2E3y

qp3y

hykyl

0@

1A ð56Þ

dyl ¼ 1000� 3:0q4y

l ð57Þ

The other parameters are set as i = 8; j = 0.43; g = 5; m = 2; C0,a = 120 for all a; t0,a = 0.8 for a = 3, 4, 7, 8, 9, 10, 15, 16, 23, 24,27, 28, 29, 30, 33 and 34; t0,a = 1.6 for a = 11, 12, 19 and 20; and t0,a = 1.13 for a = 1, 2, 5, 6, 13, 14, 17, 18, 21, 22, 25, 26, 31, 32,35 and 36. For the OD demand of passenger car traffic, drs ¼ 120� 1:0c�rs is given for all OD pairs. The following modified BPRfunction is employed as a link cost function ta, which is required for estimating the path travel time on the road network.

taðxaÞ ¼ t0;af1þ 2:62ðxa=CaÞ5g ð58Þ

where

t0,a: free travel time on link a,Ca: traffic capacity on link a.

Passenger car equivalents (PCE) for trucks are utilised for estimating link travel times. Here, this is set to 2 as commonlyapplied in Japan to the traffic conditions, where the percentage of heavy vehicle traffic is 30% of the total in a two-lane roadwith a gradient of 3% or less. The average loading factor of the trucks is set to 43%, based on an annual statistical report onroad-based freight transport in Japan. It is almost impossible to enumerate all possible paths for all OD pairs on the transportnetwork due to the considerable number of combinations of links. Thus, the paths to be used in the calculation are limited tothose with a length not exceeding 1.5 times that of the shortest path for each OD pair.

Since the model requires many functional forms and parameters, unrealistic results might be obtained depending ontheir settings. Therefore, in order to make the hypothetical numerical examples represent the situation as realistic as

0%

2%

4%

6%

8%

10%

Observed Estimated

Fig. 4. Ratio of logistics costs to total sales for each type of industry.


possible, the functional forms and parameter values used in the model are calibrated using the results of an interviewsurvey to a logistics company and the data on logistics costs observed by type of industry (JILS, 2006) as well as on roadtraffic flow (MLIT, 2005).

As can be seen in Fig. 4, with the adjusted functional forms and parameter values (hereafter, referred to as Case 0),resultant estimates of the ratio of logistics costs (i.e., the sum of handling/inventory costs and carriage) to total salesexhibit a good agreement with that actually observed for all types of industries. The ratio of facility cost to operationcost for the freight carriers is estimated to be 1:10 in Case 0, though the results of the interview survey to a logisticscompany in Japan reveal that it is 1:11. The percentage of freight vehicles on the transport network is averagely calcu-lated to be 44% for all the links in Case 0, whilst the road traffic census in Japan (MLIT, 2005) reports that those are 47%on national expressways, 41% on urban expressways, and 33% on national highways, respectively. The average conges-tion rate (i.e., volume–capacity ratio) of all the links is estimated to be 0.68 in this case, whereas those are reported bythat census as 0.78 for urban expressways and 0.72 for the national highways (MLIT, 2005). These results do not nec-essarily verify that the case study is capable of fully representing realistic situations, but such attempts are crucial toenhance the reliability of the results to be estimated.

The amount of the products transacted (i.e., that transported or distributed) in Case 0 is listed in Table 1, which shows thesame results for both products 1 and 2 with the total amount of the products being 266. The producer surplus gained in bothSCNs is 16,393, and 3939 is the consumer surplus estimated for both SCNs, making the total surplus be 20,332. The producersurplus can be computed as the sum of the profits gained by the decision-makers on the SCNs excluding the demand mar-kets, whilst the consumer surplus can be calculated as follows:
Z q4y;0
l

q4y�l

dyl q4y

l

� �dq4y

l ð59Þ

where q4y;0l represents the market price of product y at demand market l when dy

l q4yl

� �¼ 0. The total travel time in the

transport network is estimated to be 6435 in Case 0, including 3724 for passenger cars and 2711 for trucks, respectively.

Table 1Amount of products transacted (Case 0).

Wholesaler

1 2

Manufacturer 1 107 262 26 107

Retailer

1 2

Wholesaler 1 59 742 74 59

Market

1 2 3

Retailer 1 0 45 882 88 45 0

Table 2Amount of products and surpluses (Case 0 and Case 1).

Case 0 Case 1

Link capacity (%) 100 150 200 250 300Amount of products 266 277 282 284 285Producer surplus 16,393 16,925 17,182 17,256 17,288Consumer surplus 3939 4258 4417 4485 4513Total surplus 20,332 21,183 21,599 21,741 21,801


4.2. Increase in capacity for congested links

Links 3 and 34 are found to be the most congested links in Case 0. Therefore, the capacity of these links is expanded from100% to 300% (Case 1). This can be considered as the case where the transport measures for increasing the capacity of con-gested links on the road network are implemented like road widening.

Table 2 indicates the changes in the surpluses and the total amount of products manufactured (i.e., that transacted, trans-ported or distributed between each tier of the SCN) as a result of link capacity expanded. It can be seen from Table 2 that theamount of products manufactured and the surpluses increase as the link capacity increases. The increase in total surplus meansthe enhancement of the efficiency of SCNs. Accordingly, the capacity expansion of congested links would improve the efficiencyof SCNs.

In general, the total travel time in the transport network can decrease as the link capacity increases. When the capacity oflinks 3 and 34 is 200% (i.e., twice as much as that in Case 0), the total travel times decrease to 73% from Case 0 for passengercars, to 71% for trucks, and to 72% for all kinds of vehicles, respectively, with the average congestion rate of all links on theroad network being 0.66. In the case of 300%, these reduce to 69% from Case 0 for passenger cars, to 66% for trucks, and to 67%for all kinds of vehicles, respectively. The reduction in total travel time and congestion rate implies the improvement of thetraffic conditions on the road network.

4.3. Addition of new links

Here, Case 2 assumes to be the case where links 11, 12, 19 and 20 are deleted from Case 0. In other words, Case 0 cor-responds to the case with the establishment of additional new links in Case 2. Therefore, the effects of newly constructedlinks in the road network on the SCNs can be examined by comparing the results of both cases. The addition of new links

Table 4Amount of products transacted (Case 2).

Wholesaler

1 2

Manufacturer 1 125 02 0 125

Retailer

1 2

Wholesaler 1 125 02 0 125

Market

1 2 3

Retailer 1 84 41 02 0 41 84

Table 3Amount of products and surpluses (Case 0 and Case 2).

Case 0 Case 2

Amount of products 266 250Producer surplus 16,393 16,239Consumer surplus 3939 3493Total surplus 20,332 19,731


can be considered as the establishment of new bypasses in an urban area, which would be particularly useful and practicalfor urban freight transport network planning in developing countries.

The results for the amount of products and the surpluses in Table 3 demonstrate that both increase in Case 0 as comparedto Case 2. This suggests that newly constructed links would enhance the efficiency of SCNs. The total travel times in the roadnetwork in Case 0 decrease to 87% from Case 2 for passenger cars, to 94% for trucks, and to 90% for all kinds of vehicles,respectively. Comparison of the congestion rates in Case 0 with those in Case 2 clarifies that the congestion in links otherthan those added is significantly mitigated. It is found that the average congestion rate for these links declines from 0.78in Case 2 to 0.56 in Case 0.

The amount of the products transacted between the decision-makers in each SCN in Case 2 are shown in Table 4, whereboth products 1 and 2 achieve the same results. These are smaller than those estimated in Case 0. The distribution channel inCase 2 is found to be substantially changed as compared to Case 0. Here, the network displayed in Fig. 3 is divided into halvesaround node 5: the left segment consisting of nodes1, 2, 3, 4 and 5, and the right segment consisting of nodes 5, 6, 7, 8 and 9.In Case 0, the products produced by the manufacturers in the left segment are provided to the markets through the whole-salers and retailers not only in the left segment but in the right segment as well. In Case 2, however, the products producedby the manufacturers in the left segment are offered to the markets only through the wholesalers and retailers in the leftsegment, and those produced in the right segment are similarly restricted to being handled by the wholesalers and retailersin the right segment.

Without the establishment of new links (i.e., in Case 2), the products are distributed only within the segmental bound-aries of the SCNs, and the trucks use links 7, 8, 9, 10, 27, 28, 29 and 30. However, the construction of new links (i.e., Case 0)brings about the change in distribution channel, and hence, the products in the SCNs can be distributed freely regardless ofthe boundaries. This leads to that the use of these links become less intense as well as that the congestion rate of existinglinks reduces with the establishment of new links.

The results indicate that the establishment of new links would not only contribute to the improvement of the efficiency ofSCNs and the traffic conditions on road networks but also lead to the change in distribution channel on the SCNs, eventhough the effect might vary depending on the location of additional links.

5. Conclusions

Supply chain analyses have been interdisciplinary undertaken, since they involve manufacturing, transportation, logistics,as well as retailing/marketing. However, except for the SCNE models (e.g., Nagurney and Dong, 2002), there have been fewmethods developed for administrators and planners to understand the mechanisms of the generation of goods distributionand investigate the effects of freight transport measures to be implemented, as well as for companies to appreciate the influ-ence of such measures. This paper proposed a supply chain-transport supernetwork equilibrium (SC-T-SNE) model, which isa sort of SCNE model, allowing the interaction of behavioural changes of agents in the SCNs and traffic conditions in thetransport network to be taken into account. The contributions of the paper to the literature are summarised as follows:

– The SC-T-SNE model deals with a supernetwork integrating SCNs with a transport network, while the existing SCNE mod-els (Nagurney et al., 2002a,b, 2007; Dong et al., 2004; Nagurney and Toyasaki, 2005; Nagurney, 2006; Hammond andBeullens, 2007; Cruz, 2008; Qiang et al., 2009; Daniele, 2010) only handles SCNs. This model can therefore address themutual effects between the behavioural changes in the SCNs and the transport network, taking into account the endog-enous decision-making process for determining transport fares (i.e., carriage) and costs.

– Both the behaviour of freight carriers and that of transport network users are incorporated within the SC-T-SNE model.This makes it possible to explicitly describe the behaviour undertaken in the supernetwork (i.e., SCNs and transport net-work). Thus, the effects of traffic conditions in the transport network on the activities of each SCN actor and vice versa canbe investigated using the model. Furthermore, in order to enhance the applicability of the model, the behaviour of whole-salers is also embedded within it as well as the facility costs incurred by manufacturers, wholesalers, retailers and freightcarriers. These are not encompassed within the existing SCNE models.

– The SC-T-SNE model allows the relationship between traffic flow and goods movement to be identified, looking over theentire SCNs (i.e., the behaviour of manufacturers, wholesalers, retailers, freight carriers, demand markets and transportnetwork users), although the existing transport network models (Friesz et al., 1983; Harker and Friesz, 1986; Crainicet al., 1990; Guelat et al., 1990; Fernandez et al., 2003; Yamada et al., 2009) merely deals with the behaviour of two agentsof shippers and freight carriers.

The model was then applied to a supernetwork, not only to validate the performance of the model but to investigate theeffects of road-related transport measures, where the efficiency of the SCNs was assessed on the basis of total surplus beingestimated as the sum of producer surplus and consumer surplus. These analyses have not been conducted in the existingSCNE studies.

Results showed that capacity expansion of congested links would improve the efficiency of SCNs, where total amount ofproducts manufactured (i.e., that transacted, transported or distributed between each tier of the SCNs), profits gained by theSCN actors, consumer surplus and total surplus all increase; whilst total travel time in the transport network decreases. It is


also indicated that the establishment of new links would bring about similar effects as an expansion of link capacity, causingthe significant change in distribution channel on the SCNs.

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A supply chain-transport supernetwork equilibrium model with the behaviour of freight carriers

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