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OPTIMAL LOCATION AND SIZE OF LOGISTICS PARKS IN A REGIONAL LOGISTICS NETWORK WITH ECONOMIES OF SCALE AND CO2 EMISSION TAXES

This paper proposes a model to address the design problem of a regional logistics network. In the proposed model, the decision variables include the location and size of logistics parks. The interaction between the logistics authority and logistics users as well as the effects of economies of scale and CO2 emission taxes on the logistics network design are explicitly considered. The proposed model is formulated as a bi-level formulation, in which the upper level aims to maximize total social welfare of the system by determining the optimal location and size of logistics parks with CO2 emission taxes consideration, whereas the lower level describes the logistics users’ choices for service routes. A heuristic solution algorithm is presented to solve the proposed model, and a numerical example is given to illustrate the applications of the proposed model and solution algorithm. The findings show that the optimal location and size of logistics parks depend on the realized logistics demand and the level of the economies of scale. The CO2 emission taxation can help to improve the total social welfare of the system and drive the logistics users to choose greener transportation modes.


Introduction
Environmental issues have recently attracted considerable public attention around the world. There is a wide consensus that freight transportation is a major contributor to climate change and global warming is due to various pollution emissions. It has been shown that freight transportation contributes to about 5.5% of global greenhouse gas emissions (McKinnon 2010). The report on CO 2 emissions in 25 European countries during 1990-2005 also indicated that about 26% of total CO 2 emissions in the air come from transportation. During logistics service activities, the CO 2 emissions from transportation amounts to 93% of total pollution emissions, while the warehousing only covers 7% (Decker 2011). It is, therefore, very important and urgent to create an environmentally sustainable logistics system.
Green logistics focuses on improvement of logistics service efficiency, decrease in logistics cost, and reduction in environmental externalities (e.g. CO 2 ) so as to achieve a sustainable balance among economic, en-vironmental and social objectives (Dekker et al. 2012;McKinnon et al. 2010). Aronsson and Brodin (2006) showed that logistics efficiency and cost were not only related to the structure of supply chains, but also the logistics network design and logistics infrastructure. As an important component of a regional logistics system (or urban logistics system), logistics network design is a strategic issue, involving logistics facility planning and sustainable logistics management policy-making (Lindholm, Behrends 2012).
Various logistics centres have recently been established in large cities for quickly distributing freight. However, this raises many important issues, such as traffic congestion, air pollution and high energy consumption. To efficiently cope with these issues, it is proposed to combine multiple logistics centres into a logistics park. A logistics park, which is also referred to as 'logistics village' in Germany, 'distribution park' in Japan and 'logistics platform' in Spain, is a specially important component of the regional logistics network. A logistics park implies a spatial concentration area for grouping various activities, such as transportation, distribution, warehousing, commercial trade, and other related services (such as maintenance and repair). It is also an intersection of different transport modes, and an interface between local traffic and long-distance traffic (Rodrigue et al. 2009;Wagener 2008).
Logistics parks play an important role in environmental effects (e.g. reducing CO 2 emissions and air pollution) in Germany (Nobel 2010). Owing to the successful practices of logistics park operations in Germany and Japan, there is a growing trend in introducing logistics parks in some developing countries. For example, in China, the number of logistics park projects has more than tripled from 207 in 2006 to 754 in 2012 according to the survey data by the China Federation of Logistics & Purchasing (CFLP 2012). However, there are some important problems in the planning and operation of the logistics park projects in China. For example, logistics users' behavioural responses to the project investment were seldom considered in the design of logistics parks, leading to a low usage rate of the logistics parks. Since the construction of a logistics park consumes a great deal of money and land, the number and size of the logistics parks should be carefully designed.
There are a number of studies on logistics network design in the literature. They can be classified into two major classes in terms of modelling methodology: Hub-Spoke location models based on classical location theory, and network equilibrium models based on spatial price equilibrium theory. As far as the Hub-Spoke location models are concerned, O'Kelly (1987) formulated the hub location problem as a quadratic integer programming model. Sender and Clausen (2011) presented a network model of wagonload traffic, which aimed to determine the hub location and size considering total cost and efficiency of the network system. Alumur et al. (2012) investigated a hub location problem from the perspective of network design, jointly considering the transportation cost and travel time, and proposed a mixed integer programming formulation. Tang et al. (2013) presented an optimization model for the location planning problem of logistics parks with variable capacity. The goal of their model was to determine the optimal locations and to allocate the customers to the logistics parks using a hybrid heuristic algorithm. Crainic et al. (2012) addressed a two-tiered freight distribution system in a big city. They presented a location-routing model to determine the optimal locations of logistics facilities (primary facilities and secondary facilities), and optimal sizes and routes for different vehicle fleets.
As far as the network equilibrium modelling approach is concerned, Friesz et al. (1983) presented some important freight network models for predicting intercity freight movement, and provided a case study about the US coal industry. Harker and Friesz (1986) reviewed the major modelling techniques that had been applied in analysing intercity freight network equilibrium, and pointed out the shortcomings of the spatial price equi-librium models. Taniguchi et al. (1999) proposed a bilevel model to determine the optimal size and location of public logistics terminals and solved the model using queuing theory and nonlinear programming techniques. Nagurney (2010b) investigated a multi-modal multicommodity problem from the perspective of supply chain equilibrium, and obtained a Nash-Cournot equilibrium solution based on Variational Inequality (VI) theory. Yamada et al. (2011) presented a network equilibrium model to integrate supply chain networks with transportation networks, taking into account the behaviour of freight carriers. Bauer et al. (2010) addressed a multi-commodity, capacitated intermodal freight transportation network planning problem, which considered greenhouse gas emissions as the primary objective. Up to now, the freight network equilibrium modelling approach has been widely applied in practical logistics network design (Crainic 2000;Crainic, Laporte 1997;Crainic et al. 2010;Crainic, Rousseau 1986;Fernandez et al. 1994;Guélat et al. 1990;Li et al. 2010). Table 1 further summarizes the previous related studies on logistics network design problems (Alumur et al. 2012;Bauer et al. 2010;Crainic 2000;Crainic et al. 2001;Crainic, Rousseau 1986;Ham et al. 2005;Li et al. 2012;Lin, Chen 2008;O'Kelly, Bryan 1998;Powell, Sheffi 1989;Sender, Clausen 2011;Yamada et al. 2011Yamada et al. , 2009. It indicates that the existing related studies mainly focused on minimizing transportation cost or time, and little attention has been paid to environment-related costs, and the effects of economies of scale. Economy of scale for the construction of logistics parks refers to the phenomenon that the average construction cost per unit area for a logistics park decreases as the size of the logistics park increases (Berechman, Giuliano 1985;Li et al. 2012;Zhang 2006). Meanwhile, the economies of scale for the operation of logistics parks refers to the phenomenon that average operating cost per unit of shipment decreases as the size of the logistics park increases due to the clustering and synergetic effects among the different logistics service providers (Winkler, Seebacher 2012). It is of great importance to consider these effects in the design of the logistics parks particularly in an era of (capital and land) resource shortage and climate change.
We introduce the parameter ρ ( ) < ρ ≤ 0 1 to capture the effects of the scale economics of construction of logistics parks, as shown in Fig. 1. It can be seen in Fig. 1 that when ρ =1.0 , the total fixed cost is a linear function of the scale of logistics park area, implying that the effects of the scale economies of the logistics park construction is not considered. When < ρ < 0 1, the fixed cost that results from an additional unit construction area (i.e. the marginal cost) decreases as the total size of the logistics parks increases. In particular, for a given logistics park, the smaller the value of ρ, the larger the scale economies, and vice versa. Hence, the parameter ρ ( ) < ρ ≤ 0 1 can indeed capture the effects of the scale economies of the logistics parks' construction and operations (Berechman, Giuliano 1985;Li et al. 2012;Zhang 2006).
In view of the above, the problems to be investigated in this paper are as follows: given the alternative sites for construction of logistics parks in a region logistics network, how to decide where to locate these logistics parks, and what size each park should be? What are the effects of economies of scale on the number and size of the logistics parks? The main contributions made in this paper are twofold. First, we develop a model for simultaneously determining the optimal number and size of logistics parks in regional logistics networks, as well as the optimal CO 2 emission taxes. In the proposed model, the interaction between the logistics authority and logistics users is explicitly considered, which is formulated as a bi-level model. Second, we disclose the effects of the economies of scale on the logistics network design.
The remainder of this paper is organized as follows. Section 1 describes basic components of the model, including the regional logistics network representation and model assumptions. Section 2 presents the model formulation and solution algorithm. Section 3 presents a numerical example for illustration of the model applications. Last section concludes the paper.

Network Representation
In order to model the regional logistics services, Fig. 2 show the logistics demand network, logistics service network, service route, and physical logistics network, respectively. In Fig. 2a, there are different types of logistics demands (such as industrial, commercial, and agricultural logistics) between a given logistics Origin-Destination (OD) pair. These demands are served by the logistics service network, as shown in Fig. 2b, c.
It consists of a sequence of logistics nodes on the service route and is differentiated by type of logistics services (such as transportation mode, service cost and service time). A logistics leg is a non-stop component of a logistics service route that is defined by itinerary, service cost and time, and transportation mode.
The logistics physical network, as shown in Fig. 2d, is composed of a set of logistics nodes (logistics parks, distribution centers, and freight terminals) and a set of logistics links or arcs, which represent physical links, such as road segments, rail tracks, river segments or sea lines. We denote a logistics physical network as where N p is the set of logistics nodes (or transfer nodes) and A p is the set of logistics links. All transfers take place at logistics transfer nodes. In this paper, two different types of logistics transfer nodes are considered: one is the logistics park with economy of scale effects, and the other is a general transfer node with a small capacity, such as a distribution center.
In order to facilitate the construction of the model, we introduce a virtual arc to represent a logistics transfer activity (i.e. change of transportation mode). For example, Fig. 3 represents a logistics service from node A to B, via transferring from railway to highway at transfer node H. In this paper, for presentation purpose, we denote ⊂

Assumptions
To facilitate the presentation of the essential ideas without loss of generality, the following basic assumptions are made in this paper: A1. The planning period is assumed to be one week. The model proposed in this paper is thus mainly used for strategic planning and/or policy evaluation purposes. A2. In the regional logistics system, the locations of logistics parks and CO 2 emission taxes are determined by the logistics authority. A3. Logistics users make their logistics service route choices based on their own perceptions of the service disutility. The disutility of logistics service is measured by service time and monetary cost. The latter consists of logistics fare and CO 2 emission taxes for using this service. A4. An elastic demand function is used to capture the responses of logistics users to the disutility of logistics service.
A5. The logistics users can use a single (pure) mode or a combination of several modes (called combined mode). The single/pure mode includes Heavy Goods Vehicle (HGV), Light Goods Vehicle (LGV), railway or waterway.

Notations
Sets: M -set of all transport modes in regional logitics service market, =   Discrete integer decision variable (logistics authority): y i -size of a potential logistics park to be constructed; Yvector of variable y i , Binary decision variable (logistics authority): x i -0-1 variable, equals to 1 if the potential logistics node i is chosen as a park and 0 otherwise; Constants: e m -average CO 2 emission per unit turnover by transport mode m ∈ M [kg/ton·km]; z -emission taxes per unit CO 2 emission [$/kg]; ρ -parameter for capturing the effects of economies of scale for logistics parks; β -demand dispersion parameter in the elastic demand function; θ -parameter for representing the perception variation of logistics users on logistics service disutility.

Model Formulation
In the logistics system, there are two interrelated players, namely the logistics authority and users (i.e. shippers).
The goal of the logistics authority is to determine the optimal number, size and location of logistics parks so as to maximize the total social welfare of the logistics system. The decision of the logistics authority will affect the logistics users' service choices. The logistics users make their route and mode choices based on their own perceived logistics service disutility, which will further affect the logistics authority's decision. The interaction between the logistics authority and the users can be formulated as a bi-level model, as shown in Fig. 4. In the following, we in turn formulate the decisional behaviour of the logistics authority and the users.

Logistics Users' Service Choice Equilibrium
According to A3, the logistics users' disutility consists of the logistics service time, transportation cost, and CO 2 emission taxes (if any), which can be expressed as: where: m wr C , m wr T and m wr G represent the transportation cost, logistics service time and CO 2 emission taxes on service route r by transport mode m between OD pair w, respectively; τ vot is the Value Of Time (VOT).
The transportation cost and service time on a route can be expressed as the sum of the transportation costs and service times on all the arcs along that route, in- Considering the difference in the attributes of different transportation modes, the logistics service time on link a for different modes should be estimated by different functions, as shown in Eq. (4). For the transportation mode of HGV or LGV, the Bureau of Public Roads (BPR) US type function can be used to estimate the service time (Lam et al. 1999a). For railway or waterway, the average transport service time and departure interval time should be considered, i.e.: where: 0 m a t is the free-flow transport service time; t md is the average shift interval for transport mode m.
The CO 2 emission taxes on route r between OD pair w can be expressed as the sum of the emission taxes on all the arcs along that route, i.e.: According to A3, the logistics service mode/route choices can be calculated by a logit-based Stochastic User Equilibrium (SUE). The flow m wr h on route r between OD pair w ∈ W served by mode m can thus be given by: where the parameter θ represents the variation of perception of logistics users on logistics service disutility. The higher the value of θ, the smaller the variation in logistics users' perception, and vice versa (Huang, Li 2007;Lam et al. 1999b;Li et al. 2007a).
In order to capture the logistics users' responses to logistics service disutility, an elastic travel demand function is introduced. It is assumed that the total resultant demand q w between OD pair w is a continuous and monotonically decreasing function, ⋅ ( ) w D , of the expected minimal service disutility λ w for the logistics OD pair concerned (Sheffi 1985), i.e., in this paper, we adopt the following demand function: where: w q is the potential logistics service demand between OD pair w; β is the demand dispersion parameter that reflects the demand sensitivity to the logistics service disutility by OD pair w. According to random utility theory, λ w can be measured by the following log-sum formula shown as Eq. (9) (Li et al. 2012;Oppenheim 1995): According to the theory of equilibrium network flow, the following constraints should be satisfied (Friesz et al. 1983;Sheffi 1985): where: Eq. (10) is the modal demand conservation constraint; Eq. (11) is the route flow conservation constraint for the pure mode or combined mode; Eq. (12) is the network flow conservation constraint regarding logistics transfer node i. Eq. (13) shows the relationship between the route flow and arc flow in the network; Eq. (14) shows the non-negativity constraints for the route flows and OD pairs flow respectively. The network equilibrium for a multimodal logistics network with elastic demand can be defined as follows.

Equivalent Variational Inequality Model
The multimodal logistics network equilibrium conditions can be formulated as an equivalent VI model. h h h q q . The proof of Proposition 1 is given in the Appendix.

Social Welfare Maximization Model
As previously stated, the logistics authority aims to determine the optimal number and size of the logistics parks and the optimal CO 2 emission tax so as to maximize the total social welfare of the system. The total Social Welfare (SW) is defined as the sum of Consumer Surplus (CS) and producer surplus, i.e.: The objective function (16) includes three components: the first bracket represents CS, whereas the sum of the second term (i.e. the revenue at logistics nodes) and the third term (i.e. the revenue of logistics services on arcs) is the producer surplus.
is the inverse function of the logistics demand function. Constraint (17) expresses a binary variable with regard to the location of a logistics park, where '1' represents the decision to construct a logistics park at node i, and 0 otherwise. Constraint (18) represents the candidate sizes of logistics park i.
According to Eq. (8), one can obtain the inverse function of the logistics demand function as below: The CS can thus be represented as:

Solution Algorithm
In this section, we develop a solution algorithm to solve the above bi-level decision model, which is based on implicit enumeration approach and the Method of Successive Average (MSA) (Geoffrion 1969;Li et al. 2012;Nguyen et al. 1998). Given the location and size of the logistics parks, then logistics users make their logistics service mode and route choices to minimize their own logistics service disutility according to Algorithm 1. On the other hand, given logistics users' decisions about logistics services, the authority can determine the location and size of the logistics parks according to Algorithm 2.

Solving the Logistics User Sub-Problem at the Lower Level of the Model (Algorithm 1)
Step 1 Step 2. Calculating the disutility. Calculate the disutility of all logistics service routes m wr u according to Eq. (1). Compute the expected minimum disutility λ w between logistics OD pair w based on Eq. (9).
Step 3. Calculating logistics demand. Calculate the resultant demand for each OD pair w based on Eq. (8).
Step 4. Computing the logistics service demand, mode share and transfer flows. Use Eqs (8) Step 5. Logit-based SUE assignment. Conduct logit-based SUE assignment to get auxiliary link flow ( ) n a g , a ∈ A based on Eq. (13).
Step 6. MSA. Let ( ) ( ) ( ) ( ) Step 7 Step 9 and output all results as the final solution, where ξ is a predetermined parameter.
Step 8. Set n = n + 1, compute all link transport costs and route transport costs on the basis of current link flows, then go to Step 2.

Solving the Logistics Authority Sub-Problem at the Upper Level of the Model (Algorithm 2)
Let Ω be the set of decisions of the authority about the logistics park locations and CO 2 emission taxes, i.e.
Step 1. Initialization. Set = −∞ SW (i.e., the lower bound of the objective function SW in Eq. (16)). Choose an initial feasible plan from government decision set Ω.
Step 2. First loop operation (logistics authority implicit enumeration loop). Perform a complete logistics author-ity decision in decision set Ω. Set the logistics authority decision counter to t = 1.
Step 3. Second loop operation (logistics service route flow assignment loop). Given the logistics authority de- x y and the logistics service fare and service time. Solve the logistics service choice sub-model (1)-(9) using the above solution Algorithm 1 to obtain the corresponding logistics demand ( ) Step 4. Update the objective function value. If Step 5

Input Data
In the following, a numerical example is used to illustrate the applications of the proposed model and solution algorithm. The example regional logistics network is shown in Fig. 6. It consists of two logistics demand OD pairs, i.e. OD pair 1 (1→8) and OD pair 2 (1→7), eight nodes and fifteen links which are denoted by arc i (i = 1, 2, 3, …, 15). In this figure, nodes 3, 4, and 5 are assumed to be three candidate logistics parks.
Figs 6 and 7 show the original and modified logistics service networks, respectively. For some links, different vehicle types are available, and one link that is served by different vehicle types can thus be expanded as different links. For example, link 1 in the original logistics service network (i.e. Fig. 6) can be expanded as arcs 1 and 16, as shown in the modified logistics service network (i.e. Fig. 7). The basic data for the logistics service network are given in Table 2.  LGV -Light Goods Vehicle. Suppose that the logistics OD pair 1 (from node 1 to node 8) serves agricultural goods with a potential demand of 1000 units per week, whereas the logistics OD pair 2 (from node 1 to node 7) serves industrial goods with a potential demand of 1200 units per week. The parameters θ and β are 0.0006 and 0.02, respectively. The VOT is $10 per hour (De Jong et al. 2014;Guan, Kazuo 2000). The transfer cost and time at the logistics transfer node are shown in Table 3 (Qu et al. 2014;Yong 2011). The other parameters are shown in Table 4 (GB/T 21334:2008; Yong 2011). The average emissions of different transportation modes are different. Specifically, the average emissions of HGV, LGV, railway, and waterway are 0.132, 0.283, 0.022, and 0.016 (kg/ton·km) respectively Qu et al. 2014). The parameter z on CO 2 emission taxes is 0.09 $/kg. In the following analysis, unless specifically stated otsherwise, these input data are considered as the base case.

Numerical Results and Discussions
The size of the optimization problem is analysed in Ta i i x y i I . The proposed solution algo-rithm was coded in Microsoft Visual C++ 6 and run on a laptop Dell N5040 with an Intel Pentium 2.13 GHz CPU and 4.00 GB RAM. The CPU time for searching the corresponding optimal location and size under the different number of logistics parks is shown in Fig. 8. Applying the proposed model and solution algorithm to the example network, the results are summarized as follows. LGV 1.5 (5) 0 (0) 3 (11) 1.5 (13)

Comparison of the Optimal Solutions with and without Logistics Parks
We first compare the solutions of the models with and without logistics parks, as shown in Table 6. It can be seen in this table that for the base case, the optimal number of the logistics parks is one (i.e. node 5), with a size of 800 tons per week. Compared to the doing-nothing case (i.e. no logistic parks), the introduction of the logistics park (node 5) can lead to an increase in the logistics demand by 35 tons per week (from 1450 to 1485 tons per week), in the total social welfare by $3988 per week (from $2178095 to $2218083 per week), and in market share of the combined mode by 11% (from 63 to 74%), but a decrease in average CO 2 emissions per unit of shipment by 19.50 kilogram per ton (from 166.79 to 147.29 kilogram per ton). In addition, it can also be seen in Table 6 that the introduction of the logistics park can change the flow distribution on the arcs of the network through changing the logistics users' route/mode choice behaviour.

Effects of Economy of Scale Parameter ρ
We now look at the impacts of the economy of scale parameter ρ on the optimal number and size of the logistics parks. Fig. 9 shows that the optimal number and size of the logistics parks decrease with the increase of the value of ρ. Specifically, when the parameter ρ is in the range [0.70, 0.75], all three candidate logistics parks (i.e. nodes 3, 4, and 5) are chosen. Their corresponding sizes are 800 for logistic park 3, 1200 for logistic park 5, respectively, while the size of logistic park 4 is 1000 before the critical point 0.71, changing to 800 after this point. When the parameter ρ is in the range [0.76, 0.82], two candidate logistics parks are selected (i.e. nodes 4 and 5), and their corresponding sizes are 800 and 1200, respectively. However, When ρ falls in the range [0.83, 0.85] and [0.86, 1.00] only the logistics park 5 is selected, and the corresponding optimal size of is 1000, 800 respectively. Fig. 10 shows that as the economy of scale parameter ρ increases, the total social welfare decreases, but the average CO 2 emissions per unit of shipment increase. This is because a larger value of ρ means a higher cost for constructing and operating the logistics park. Therefore, fewer logistics parks are introduced into the system, as shown in Fig. 9. As a result, the market share of the combined mode with low CO 2 emissions decreases, and thus the average CO 2 emissions per unit of shipment in the network increase. Accordingly, total social welfare decreases.   Fig. 11 shows the optimal numbers, locations and size of the logistics parks with various combinations of the demands of OD pairs 1 and 2 when the economy of scale parameter ρ is fixed as 0.90.

Effects of Logistics Demand on Optimal Number of Logistics Parks
In figure, the different types of the markers represent different optimal numbers of logistics parks introduced, while the different colours of the same marker implies different size combinations for the same number. The diamond, circles, asterisk and upward-pointing triangle in this figure denote to introduce zero, one, two and three logistics parks, respectively.
In figure, the area in colourful circles represents to introduce only one logistics park (i.e. node 5), in which different colours implies different sizes. The blue, green and cyan circle indicate that the size of node 5 is 800, 1000, 1200 tons/week, respectively.

Effects of CO 2 Emission Taxes on the Logistics System
In the following, we examine the effects of the CO 2 emission taxes on the total social welfare, CO 2 emissions per unit of shipment and the share of the combined mode. Fig. 12 shows that as the tax per unit of CO 2 emission increases, the total social welfare first increases and then decreases, and reaches the maximum ($2218112 per week) at the tax of $0.11 per gram CO 2 emission, which implies that charging the appropriate CO 2 emission tax can help to improve the total social welfare. Fig. 13 shows that as the CO 2 emission taxes increase, the CO 2 emissions per unit of shipment decrease, whereas the market share of the combined transportation mode increases. This is because the CO 2 emission tax policy increases the cost of the logistics users due to an extra payment. Table 7 further shows the effects of the CO 2 emission taxes on the logistics network performance in terms of the total resultant logistics demand, total social welfare, customer surplus, producer surplus, market share of combined transport mode, and the CO 2 emissions per unit of shipment. It can be seen in Table 7 that the introduction of the CO 2 emission taxes can lead to a decrease in the total resultant logistics demand and the CO 2 emissions per unit of shipment, but an increase in the total social welfare, the producer surplus, and the market share of the combined transport mode. Specifically, the total resultant logistics demand decreases from 1493 to 1483 tons per week, and the CO 2 emissions per unit of shipment decreases from 161.65 to 144.37 kilograms per ton. However, the total social welfare increases from $2210845 to $2218112 per week, the producer surplus increases from $58301 to $80888 per week, and the market share of the combined transport mode increases from 70 to 75%. This implies the policy of CO 2 emission taxes will improve the social welfare and induce logistics users to choose greener transport modes.

Conclusion and Future Studies
This paper proposed a new model for simultaneous optimization of the location and size of logistics parks with economies of scale and CO 2 emission taxes in a multimodal regional logistics network. It was assumed that the logistics service can be completed by a pure transport mode or a combined transport mode. The logistics demand elasticity regarding logistics service level was also explicitly considered. The proposed model incorporated the interaction between the logistics authority and logistics users and also explicitly took into account the impacts of the economy of scale of the logistics parks. The proposed model was described as a bi-level model. The upper level of the model for the logistics authority was formulated as a mixed integer programming problem which can be solved by an implicit enumeration algorithm, and the lower level was formulated as a logit-based SUE problem. Some new insights and important findings have been obtained, as follows. First, the introduction of a logistics park is closely related to the logistics demand. When the OD demand reaches some certain level, introducing a logistics park can improve the efficiency of the system in terms of social welfare and CO 2 emissions per unit of shipment. Different OD demand levels can significantly affect the number and size of the logistics parks introduced. Second, the economy of scale parameter also has an important effect on the optimal number and size of the logistics parks and CO 2 emissions per unit of shipment as well as the social welfare of the system. Specifically, the smaller the economy of scale parameter, the bigger the optimal number of the logistics parks due to a decrease in the construction/operating cost of the logistics system, and vice versa. Therefore, it is important for the authority to consider the effects of the economies of scale on the logistics system design. Third, the optimal emission tax policy can reduce the total amount of CO 2 emissions and enhance the market share of the combined transportation mode, and thus contribute to an environmentally sustainable regional logistics service system.
It should be pointed out that, although the numerical results that are presented in this paper can be explained logically, case studies on large and realistic logistics networks are necessary to further justify the findings of this paper and the performance of the proposed model. The research directions for future studies may also focus on an extension of the proposed model to logistics network design with uncertainty in demand/supply sides, the pricing issue for a combined transportation mode in a multi-modal logistics network, the multi-period investment problem in a logistics network, and the design of an effective solution algorithm particularly for a large-scale logistics network.  Combining Eqs (A.5), (A.9) and (A.18) with conservation Eq. (A.2), we can get: