Modeling and empirical analysis of regional hub-and-spoke road-rail combined transport network based on uncertain cost-time-demand

3.1 Problem description

The performance of a hub-and-spoke RRCT system is generally characterized by two indicators, total transportation cost and total transportation time. The total transportation cost (including in-transit transportation cost, hub transshipment cost and hub construction cost) is related to the economic benefits of freight transportation enterprises. The total transportation time (including in-transit transportation time and hub transshipment time) is related to the quality of services provided by freight transportation enterprises. Generally speaking, there is a conflict between these two performance indicators, with the increase in one indicator meaning a decrease in the other. In other words, the scheme with the smallest total transportation cost usually does not result in the smallest total transportation time. In order to ensure the quality of freight transportation services and reduce the total transportation cost, a trade-off should be made between the economic objective (total transportation cost) and service quality (total transportation time).

In actual production practice, it is necessary to complete the design and optimization of a hub-and-spoke RRCT network before its operation. In the process of network design and optimization, the parameters like total transportation cost, total transportation time and flow of goods are not known, and they have great uncertainty. Therefore, under the requirement for balancing the transportation cost and transportation time when the total transportation cost, total transportation time and demand are uncertain, the main work of this research is to obtain the best scheme of a hub-and-spoke RRCT network that includes N nodes, road transportation mode, and rail transportation mode, thereby achieving the purpose of cost reduction and efficiency improvement of freight transportation.

The first thing we need to do is to deal with uncertain factors. The total transportation cost, total transportation time and demand in the RRCT network are influenced by various factors, such as infrastructure construction and operation conditions, weather conditions, road conditions, vehicle conditions, market factors and human factors, so that they cannot be exact values, or expressed by random probability values. Usually, their values are assigned a numerical range according to the experience and opinions of relevant experts in the field. In this research, the triangular fuzzy number method is used to characterize the above three uncertain parameters. The specific processing steps are as follows:

Total transportation cost. In our research, the total transportation cost includes the in-transit transportation cost, hub transshipment cost and hub construction cost. For example, C _ij is used to represent the fuzzy transportation cost of transporting goods from node i to node j, and C _ij  =  (c_ij,1, c_ij,2, c_ij,3) is used to represent the fuzzy transportation cost, which c_ij,1 represents the minimum estimated transportation cost, c_ij,2 represents the medium estimated transportation cost, and c_ij,3 represents the maximum estimated transportation cost.

Secondly, with uncertainties in the total transportation cost, total transportation time and demand, a hub-and-spoke RRCT network is constructed, with its schematic diagram shown in Fig. 1.

OPEN IN VIEWER

Fig. 1

Schematic diagram of a hub-and-spoke RRCT network considering the total transportation cost, total transportation time and demand.

The research on the design of a hub-and-spoke network with uncertain cost-time-demand looks fuzzy and abstract, but in fact we can also use the classic hub positioning-path allocation method to deal with such problem. In simple terms, the schematic diagram of the design process in this research is shown in Fig. 2.

OPEN IN VIEWER

Fig. 2

Schematic diagram of the design process of a hub-and-spoke RRCT network.

The symbols and model parameters used in this study are shown in Table 1.

In order to facilitate research, we make the following hypotheses about the design of the hub-and-spoke RRCT network, and build a mathematical model strictly according to the hypotheses:

Hypothesis 1. The transportation of goods from the starting node to the hub node and from the hub node to the end just can be completed by road transportation, and only rail transportation between the hubs.

Hypothesis 2. The transportation of goods between the starting and end nodes needs to pass through at least one hub, but at most two hubs, and the hub node only has the function of transshipment and does not serve as the end node of freight transportation.

Hypothesis 3. In order to reduce the computational complexity, the number of hubs to be located is given in advance, and this number can be estimated through the construction cost budget.

Hypothesis 4. The demand, in-transit transportation cost, hub transshipment cost, hub construction cost, in-transit transportation time and hub transshipment time are uncertain, and are represented by triangular fuzzy numbers and determined by the credibility measure.

The first three hypotheses are standardized hypotheses for modeling a hub-and-spoke RRCT network. Hypothesis 3 is reasonable because modeling a hub-and-spoke RRCT network is a strategic decision-making problem. The number of hubs can usually be predicted by the RRCT enterprise in advance according to the budget. The RRCT enterprise can change the number of hubs in the hub-and-spoke network according to the actual situation. Due to the historical data of the uncertain factors in the study, triangular fuzzy numbers are used to characterize the uncertain factors in Hypothesis 4.

Based on the above symbols, parameters and hypotheses, a hub-and-spoke RRCT network model based on uncertain cost-time-demand can be constructed at the macroscopic level.

(1) Decision variables

In the modeling of a hub-and-spoke RRCT network, the decision variables include the selection of the hub location, the distribution of other station nodes in each hub-and-spoke RRCT network, the transportation path between any two nodes and the transportation time limit required for the transportation of goods between any two nodes in the network. Therefore, the following five types of decision variables are defined.

X _kk : Decision variable of hub location; when the node k is a hub, it is taken as 1, otherwise 0;

X _ik : Decision variable of hub and spoke allocation; when other station node i is assigned to hub node k, it is taken as 1, otherwise 0;

Y _iklj : Decision variable of path establishment; when there is a transportation path starting from the node i, passing through the hub k and then through the hub l to finally reach the node j, it is taken as 1, otherwise 0;

δ _kl : Decision variable of hub connectivity; when there is a transportation path that first passes through the hub k and then through the hub l, it is taken as 0, otherwise 1;

Z: Decision variable of transportation time limit; it represents the maximum total transportation time for freight transportation between the starting and end nodes in the hub-and-spoke RRCT network.

(2) Basic constraints

In order to ensure that the transportation process of goods in the hub-and-spoke RRCT network conforms to the actual freight transportation process, the following constraints are defined: X ik ⩽ X kk , ∀ i , k ∈ N(1) ∑ k ∈ N ∑ l ∈ N Y iklj = 1 , ∀ i , j ∈ N(2)

∑ k ∈ N Y iklj = X jl , ∀ i , j , l ∈ N(3)

∑ l ∈ N Y iklj = X ik , ∀ i , j , k ∈ N(4)

{ if δ kl = 0 , Y iklj = 1 otherwise δ kl = 1 , Y iklj = 0(5)

∑ k ∈ N X kk = p(6)

Specifically, Equations (1)–(6) are the constraints on the selection of hub location and the allocation of other station nodes. Among them, constraint (1) means that other non-hub node i in the hub-and-spoke RRCT network can only be assigned to the hub k; constraint (2) ensures that each other non-hub node in the hub-and-spoke RRCT network can only be assigned to a single hub, in other words, there is only one path between the two nodes; constraints (3) and (4) ensure that the transportation path i → k → l → j is an effective path; constraint (5) represents the relationship between hub connectivity and path establishment; constraint (6) specifies the number of hubs in the hub-and-spoke RRCT network to be p.

(3) Objective function

The hub-and-spoke RRCT network model constructed based on uncertain cost-time-demand simultaneously takes the minimization of the total transportation cost and total transportation time between starting and end nodes as the dual objectives. For convenience, X =  (X _ik , X _kk ) , Y = (δ _kl , Y _iklj ) , Z = (Z) is defined as the set of the decision variables.

The first objective of the model is the economic objective, that is, to minimise the total transportation cost, including in-transit transportation cost between starting and end nodes, hub transshipment cost, and hub construction cost. The total transportation cost of the hub-and-spoke RRCT network can be calculated by:

∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N W ij [ δ kl ( C ik 1 + C kj 1 + C k ) + Y iklj ( C ik 1 + C kl 2 + C lj 1 + C k + C l ) ] + ∑ k ∈ N H k X kk(7)

Under the constraints of hypotheses, it is considered that there may be two types of paths for the goods in the network from the starting node to the end node. One is that the goods dock at a hub during the transportation process, and the other is that the goods need to dock at two hubs during the transportation process. If the goods only pass through one hub, the transportation path is i → k → j, at which time the in-transit transportation cost and hub transshipment cost are W ij ( C ik 1 + C kj 1 + C k ) . When the goods pass through two hubs, the transportation path is i → k → l → j, at which time the in-transit transportation cost and hub transshipment cost are W ij ( C ik 1 + C kl 2 + C lj 1 + C k + C l ) . These two situations need to be reflected in the objective function.

Because the above-mentioned total transportation cost is uncertain in the research, it needs to be expressed in the form of a fuzzy function. According to a previous research Liu B and Liu Y [14] introduced a expectation operator, and then the economic objective function can be expressed as: F 1 ( X , Y , Z ) = E [ ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N W ij [ δ kl ( C ik 1 + C kj 1 + C k ) + Y iklj ( C ik 1 + C kl 2 + C lj 1 + C k + C l ) ] + ∑ k ∈ N H k X kk ](8)

The second objective of the model is the service objective. Under the constraint of the maximum transportation time limit between the starting and end nodes, the transportation time is minimised. The maximum transportation time limit between the starting and end nodes of a single transportation path in the network can be expressed as: Maximise ∀ i , j , k , l ∈ N [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ](9)

However, since the value of the total transportation time cannot be clearly defined in the modeling of the hub-and-spoke RRCT network, the service objective function cannot be clearly limited to a certain range. Here, a credibility measure is introduced in the study. The value of the credibility measure can be used to control the quality of the service, so the service objective function can be expressed as: F 2 ( X , Y , Z ) = Minimise { Z | Cr [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ≤ Z ] ⩾ α , ∀ i , j , k , l ∈ N }(10) where 0 < α > 1 represents the credibility measure. The physical meaning of the parameter α can be interpreted as the on-time arrival rate that satisfies the service level.

According to the above basic constraints and objective function, the modeling of the hub-and-spoke RRCT network based on uncertain cost-time-demand can be expressed as the following problem of two-objective optimization: Minimise F ( X , Y , Z ) = { F 1 ( X , Y , Z ) , F 2 ( X , Y , Z ) }(11)

Sub to { Cr [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ⩽ Z ] ⩾ α , ∀ i , j , k , l ∈ N constraints ( 1 ) - ( 6 )(12)

Although the basis of this model is the fuzzy hub-and-spoke network model studied by Yang K et al. [15], our research object is a RRCT network, which contains two types of transportation modes. In addition, the uncertain factors in our model include three aspects, i.e. total transportation cost, total transportation time and demand, which increases the complexity of the problem. We will also improve the corresponding scheme algorithm, which will be described in detail later.

The linear weighted sum method is a common method when dealing with multi-objective modeling problems. Generally, there are an infinite number of Pareto optimal solutions for a multi-objective problem. Therefore, a multi-objective problem usually needs to combine user preferences in order to determine a single suitable solution. Wang R et al. [16] Used a priori correlation method that includes preferences, users indicate preferences before running the optimization algorithm and then allow the algorithm to determine a single scheme that may reflect this preference. In this study, using the prior clarity of the preferences, the preferences of each objective are calculated before modeling. Therefore, the new objective function is defined as a linear combination of the original objectives as follows: G ( X , Y , Z ) = ω 1 F 1 ( X , Y , Z ) + ω 2 F 2 ( X , Y , Z )(13)

Two weighting factors are used in the objective function G (X, Y, Z), namely ω₁ and ω₂ to balance the importance of each sub-objective function. In this study, the weighting factors ω₁ and ω₂ are determined by the operator by considering the difference between the total transportation cost and the total transportation time. For example: if we set ω₁ = 10 and ω₂ = 90, this makes the total transportation time be more important than the total transportation cost. The freight transportation enterprises can determine the value of each weighting factor according to the degree of independently-determined importance.

Through the processing of linear weighted sum method, the hub-and-spoke RRCT network model based on uncertain cost-time-demand can be transformed into the following single objective model: Minimise G ( X , Y , Z ) = ω 1 F 1 ( X , Y , Z ) + ω 2 F 2 ( X , Y , Z )(14)

Sub to { Cr [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ⩽ Z ] ⩾ α , ∀ i , j , k , l ∈ N constraints ( 1 ) - ( 6 )(15)

Obviously, the model is non-linear because there is an expectation operator in the objective function F₁ (X, Y, Z) and a credibility measure in the constraints.

In order to find an effective algorithm for solving the model, in-depth analysis is performed the mathematical properties and computational complexity of the proposed model. From this, some new ideas for solving model methods are proposed.

Theorem 1. It is assumed that demand, unit in-transit transportation cost, unit hub transshipment cost and hub construction cost are mutually independent triangular fuzzy number. Then, the economic objective function is equivalent to: F 1 ( X , Y , Z ) = E [ ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N { W ij [ δ kl ( C ik 1 + C kj 1 + C k ) + Y iklj ( C ik 1 + C kl 2 + C lj 1 + C k + C l ) ] } + ∑ k ∈ N H k X kk ] = f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7 + f 8(16)

f 1 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 3 × w ij , 2 × [ δ kl ( c ik , 2 1 + c kj , 2 1 + c k , 2 ) + Y iklj ( c ik , 2 1 + c kl , 2 2 + c lj , 2 1 + c k , 2 + c l , 2 ) ]

f 2 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 12 × w ij , 2 × [ δ kl ( c ik , 1 1 + c kj , 1 1 + c k , 1 ) + Y iklj ( c ik , 1 1 + c kl , 1 2 + c lj , 1 1 + c k , 1 + c l , 1 ) ]

f 3 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 12 × w ij , 2 × [ δ kl ( c ik , 3 1 + c kj , 3 1 + c k , 3 ) + Y iklj ( c ik , 3 1 + c kl , 3 2 + c lj , 3 1 + c k , 3 + c l , 3 ) ]

f 4 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 12 × w ij , 1 × [ δ kl ( c ik , 2 1 + c kj , 2 1 + c k , 2 ) + Y iklj ( c ik , 2 1 + c kl , 2 2 + c lj , 2 1 + c k , 2 + c l , 2 ) ]

f 5 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 12 × w ij , 3 × [ δ kl ( c ik , 2 1 + c kj , 2 1 + c k , 2 ) + Y iklj ( c ik , 2 1 + c kl , 2 2 + c lj , 2 1 + c k , 2 + c l , 2 ) ]

f 6 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 6 × w ij , 1 × [ δ kl ( c ik , 1 1 + c kj , 1 1 + c k , 1 ) + Y iklj ( c ik , 1 1 + c kl , 1 2 + c lj , 1 1 + c k , 1 + c l , 1 ) ] f 7 = ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N 1 6 × w ij , 3 × [ δ kl ( c ik , 3 1 + c kj , 3 1 + c k , 3 ) + Y iklj ( c ik , 3 1 + c kl , 3 2 + c lj , 3 1 + c k , 3 + c l , 3 ) ]

f 8 = ∑ k ∈ N 1 4 × ( h k , 1 + 2 × h k , 2 + h k , 3 )

Proof: assuming C ik 1 = ( c L 1 , c M 1 , c U 1 ) = ( c ik , 1 1 , c ik , 2 1 , c ik , 3 1 ) , C k = ( c L 1 , c M 1 , c U 1 ) = ( c k , 1 , c k , 2 , c k , 3 ) , C kl 2 = ( c L 2 , c M 2 , c U 2 ) = ( c kl , 1 2 , c kl , 2 2 , c kl , 3 2 ) , C l = ( c L , c M , c U ) = ( c l , 1 , c l , 2 , c l , 3 ) , C lj 1 = ( c L 1 , c M 1 , c U 1 ) = ( c lj , 1 1 , c lj , 2 1 , c lj , 3 1 ) , W ij = ( w L , w M , w U ) = ( w ij , 1 , w ij , 2 , w ij , 3 ) and H k = ( h L , h M , h U ) = ( h k , 1 , h k , 2 , h k , 3 ) are mutually independent triangular fuzzy number. According to the lemma of the expectation operator in the study of Liu Y K and Liu B [17], the total expected cost can be expressed as: F 1 ( X , Y , Z ) = E [ ∑ i ∈ N ∑ j ∈ N ∑ k ∈ N ∑ l ∈ N { W ij [ δ kl ( C ik 1 + C kj 1 + C k ) + Y iklj ( C ik 1 + C kl 2 + C lj 1 + C k + C l ) ] } + ∑ k ∈ N H k X kk ] = 1 2 ∫ 0 1 [ ( W ij × C ik ) α L + ( W ij × C ik ) α U ] d α × Y iklj + 1 2 ∫ 0 1 [ ( W ij × C k ) α L + ( W ij × C k ) α U ] d α × Y iklj + 1 2 ∫ 0 1 [ ( W ij × C kl ) α L + ( W ij × C kl ) α U ] d α × Y iklj + 1 2 ∫ 0 1 [ ( W ij × C l ) α L + ( W ij × C l ) α U ] d α × Y iklj + 1 2 ∫ 0 1 [ ( W ij × C lj ) α L + ( W ij × C lj ) α U ] d α × Y iklj + 1 4 ( h k , 1 + 2 × h k , 2 + h k , 3 ) × X kk(17)

Theorem 2. It is assumed that the in-transit transportation time and hub transshipment time are mutually independent triangular fuzzy number, then:

(1) If α ⩽ 1/2, then the credibility measure is given by:

Cr [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ⩽ Z ] ⩾ α is equivalent to: { ( 1 - 2 α ) × [ δ kl ( t ik , 1 1 + t kj , 1 1 + t k , 1 ) + Y iklj ( t ik , 1 1 + t kl , 1 2 + t lj , 1 1 + t k , 1 + t l , 1 ) ] } + { 2 α × [ δ kl ( t ik , 2 1 + t kj , 2 1 + t k , 2 ) + Y iklj ( t ik , 2 1 + t kl , 2 2 + t lj , 2 1 + t k , 2 + t l , 2 ) ] } ⩽ Z(18)

(2) If α > 1/2, the credibility measure is given by:

Cr [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ⩽ Z ] ⩾ α is equivalent to: { ( 2 - 2 α ) × [ δ kl ( t ik , 2 1 + t kj , 2 1 + t k , 2 ) + Y iklj ( t ik , 2 1 + t kl , 2 2 + t lj , 2 1 + t k , 2 + t l , 2 ) ] } + { ( 2 α - 1 ) × [ δ kl ( t ik , 3 1 + t kj , 3 1 + t k , 3 ) + Y iklj ( t ik , 3 1 + t kl , 3 2 + t lj , 3 1 + t k , 3 + t l , 3 ) ] } ⩽ Z(19)

Proof: We only prove Theorem (1), and Theorem (2) can be proved using a similar proof process. Assuming T ik 1 = ( t L 1 , t M 1 , t U 1 ) = ( t ik , 1 1 , t ik , 2 1 , t ik , 3 1 ) , T kj 1 = ( t L 1 , t M 1 , t U 1 ) = ( t kj , 1 1 , t kj , 2 1 , t kj , 3 1 ) ,

T kl 2 = ( t L 2 , t M 2 , t U 2 ) = ( t kl , 1 2 , t kl , 2 2 , t kl , 3 2 ) , T lj 1 = ( t L 1 , t M 1 , t U 1 ) = ( t lj , 1 1 , t lj , 2 1 , t lj , 3 1 ) , T k = ( t L , t M , t U ) = ( t k , 1 , t k , 2 , t k , 3 ) and T l = ( t L , t M , t U ) = ( t l , 1 , t l , 2 , t l , 3 ) are mutually independent triangular fuzzy number. For convenience, we define ψ iklj = δ kl ( T ik 1 + T kj 1 + T k ) + ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) . According to the linear characteristic of the triangular fuzzy number, ψ _iklj is also a triangular fuzzy number. Now we consider the credibility measure Cr { ψ _iklj  ⩽ Z } ⩾ α. If α ⩽ 1/2, there is the case of Cr { ψ iklj ⩽ Z } ⩾ α ⇔ Z ⩾ ( ψ iklj ) 2 α L , where ( ψ iklj ) 2 α L represents left pole of the 2α cut of ψ _iklj . Then, the above credibility measure is derived as: { ( 1 - 2 α ) × [ δ kl ( t ik , 1 1 + t kj , 1 1 + t k , 1 ) + ( t ik , 1 1 + t kl , 1 2 + t lj , 1 1 + t k , 1 + t l , 1 ) ] } + { 2 α × [ δ kl ( t ik , 2 1 + t kj , 2 1 + t k , 2 ) + ( t ik , 2 1 + t kl , 2 2 + t lj , 2 1 + t k , 2 + t l , 2 ) ] } ⩽ Z(20)

Then, Cr [ δ kl ( T ik 1 + T kj 1 + T k ) + Y iklj ( T ik 1 + T kl 2 + T lj 1 + T k + T l ) ⩽ Z ] ⩾ α can be expressed as an equivalent form: { ( 1 - 2 α ) × [ δ kl ( t ik , 1 1 + t kj , 1 1 + t k , 1 ) + Y iklj ( t ik , 1 1 + t kl , 1 2 + t lj , 1 1 + t k , 1 + t l , 1 ) ] } + { 2 α × [ δ kl ( t ik , 2 1 + t kj , 2 1 + t k , 2 ) + Y iklj ( t ik , 2 1 + t kl , 2 2 + t lj , 2 1 + t k , 2 + t l , 2 ) ] } ⩽ Z

According to Theorem 1 and Theorem 2, we will further deduce Inference 1 and Inference 2 to get the relationship between the objective and the parameters in the constructed model.

Inference 1. It is assumed that 0 < μ₁ < 1 and μ₂ > 1 represent the minimum ratio (left width coefficient) and maximum ratio (right width coefficient) of triangular fuzzy demand, fuzzy unit in-transit transportation cost, fuzzy unit hub transshipment cost and fuzzy hub construction cost, respectively. It follows the following rules.

(1) If μ₁ + μ₂ < 2, the economic objective value obtained by the model with uncertain factors is lower than that by the model with certain factors;

(2) If μ₁ + μ₂ > 2, the economic objective value obtained by the model with uncertain factors is higher than that by the model with certain factors.

Inference 2. The economic objective F₁ (X, Y, Z) has nothing to do with the value of the credibility measure α, and the service quality objective F₂ (X, Y, Z) is an increasing function of α.

Based on the above inferences, we can derive the equivalent model of the hub-and-spoke RRCT network based on uncertain cost-time-demand. Minimise G ( X , Y , Z ) = ω 1 F 1 ( X , Y , Z ) + ω 2 F 2 ( X , Y , Z )

Sub to { g ( Y iklj ) ⩽ Z , ∀ i , j , k , l ∈ N Constraints ( 1 ) - ( 6 )(21)

where g (Y _iklj ) is the piecewise linear function given by: g ( Y iklj ) = { { ( 1 - 2 α ) × [ δ kl ( t ik , 1 1 + t kj , 1 1 + t k , 1 ) + Y iklj ( t ik , 1 1 + t kl , 1 2 + t lj , 1 1 + t k , 1 + t l , 1 ) ] } + { 2 α × [ δ kl ( t ik , 2 1 + t kj , 2 1 + t k , 2 ) + Y iklj ( t ik , 2 1 + t kl , 2 2 + t lj , 2 1 + t k , 2 + t l , 2 ) ] } ⩽ Z α ⩽ 1 / 2 { ( 2 - 2 α ) × [ δ kl ( t ik , 2 1 + t kj , 2 1 + t k , 2 ) + Y iklj ( t ik , 2 1 + t kl , 2 2 + t lj , 2 1 + t k , 2 + t l , 2 ) ] } + { ( 2 α - 1 ) × [ δ kl ( t ik , 3 1 + t kj , 3 1 + t k , 3 ) + Y iklj ( t ik , 3 1 + t kl , 3 2 + t lj , 3 1 + t k , 3 + t l , 3 ) ] } ⩽ Z α > 1 / 2(22)

As presented in Equation (21), the hub-and-spoke RRCT network model based on uncertain cost-time-demand is obtained by changing the combination of the p-hub middle value problem and the p-hub median problem. This problem belongs to an NP-hard problem, our research also introduces triangular fuzzy numbers to represent uncertain factors, expands the traditional model, and increases the computational complexity. Our proposed hub-and-spoke RRCT network model based on uncertain cost-time-demand is difficult to be solved by traditional algorithms. The number of constraints and variables increases sharply as the hub-and-spoke RRCT network becomes larger, especially for large-scale networks. Even for small-scale networks, traditional methods such as branch and bound method and cutting plane method cannot solve the model effectively. So far, there are still no effective algorithms to obtain the optimal solution of this kind of problem. Therefore, in order to obtain the optimal (or near optimal solution) of the constructed model, this paper proposes an improved genetic algorithm that combines global search with local search to solve such problems.

5.1 Network node layout and parameter settings

The Yaan-Linzhi section of the Sichuan-Tibet Railway is scheduled to start construction in 2020. Located in the plateau alpine region where the transportation infrastructure is weak, in the study of Jin Z and Junxiang X [19], the project is planned to be finished in 6–8 years. The Yaan-Linzhi section plans to build 21 railway stations. In this research, 21 railway nodes and 62 road nodes along the line are selected according to the relevant planning and construction information of China Railway Academy. Based on the regional vector road network map, the node layout of the RRCT network for the Yaan-Linzhi section of the Sichuan-Tibet Railway is presented in Fig. 7. It is noted that the network objects selected our research are the real, but the relevant data is not presented in detail due to privacy protection. Besides, some data are reasonably assumed within a certain range.

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Fig. 7

Layout of actual network nodes.

Secondly, we set the parameters in the actual network topology diagram shown in Fig. 7. For each transportation path from node i to node j, the fuzzy demand on the transportation path is W _ij  = (μ₁w _ij , w _ij , μ₂w _ij ), where w _ij is the deterministic demand given in the actual network. Let μ₁ = 0.8 and μ₂ = 1.2. The value of these two parameters can be used to determine the minimum and maximum demand on the transportation path. In addition, the fuzzy hub construction cost, fuzzy unit in-transit transportation cost and fuzzy in-transit transportation time from node i to node j are indicated by H _k  = (μ₁h _k , h _k , μ₂h _k ), C _ij  = (μ₁c _ij , c _ij , μ₂c _ij ) and T _ij  = (μ₁t _ij , t _ij , μ₂t _ij ), respectively. Since the Yaan-Linzhi section of the Sichuan-Tibet Railway has not yet been completed, the unit in-transit transportation cost, hub transshipment cost, and in-transit transportation time are uncertain. Only the unit in-transit transportation costs of the road transportation in G317 and G318 of the Sichuan-Tibet region are available, so we use the cost ratio ɛ₁ to determine the unit in-transit transportation cost of the rail transportation.

Since the unit in-transit transportation cost of the rail transportation is generally lower than that of the road transportation, the cost ratio is set as ɛ₁ = 0.6. The unit in-transit transportation cost of the rail transportation in the actual RRCT network can be calculated by C ij 2 = ɛ 1 C ij 1 . The time ratio ɛ₂ is set here. Since the in-transit transportation time of the rail transportation is longer than that of the road transportation, the time ratio is set as ɛ₂ = 1.4. Then the in-transit transportation time of the rail transportation can be calculated by T ij 2 = ɛ 2 T ij 1 . Similarly, we can also express other fuzzy factors in this way. In the network topology along the Yaan-Linzhi section of the Sichuan-Tibet Railway, there are a total of 83 road and rail nodes, with 12, 14 and 16 hub nodes, credibility measures of 0.2, 0.4, 0.6 and 0.8, respectively, the weights of ω₁ = 10 and ω₂ = 90, respectively. The parameters of the improved genetic algorithm are set as follows: the population size is 200, the maximum number of iterations is 1000, and the probability of crossover and mutation is 0.5 and 0.3, respectively. The optimal number of hub locations, the location and the allocation of spokes and hub are determined by the algorithm. In order to facilitate the calculation, we also need to set the relevant parameters of cost and time. The distance data between any nodes is the real data on the map, as shown in Table 3.

In order to evaluate the performance of the hub-and-spoke RRCT network model and the improved genetic algorithm based on uncertain cost-time-demand, numerical analysis is performed on the 83-node network by setting different numbers of hubs, different credibility measures of service objective and different weight preferences. The results obtained under different values of ω₁ and ω₂ are shown in Tables 4 and 5. In fact, the weight values of ω₁ and ω₂ can be determined by freight transportation enterprises based on multiple attribute decision methods, such as fuzzy analytic hierarchy process and entropy method.

As can be seen from Tables 4 and 5, changing the value of α will change the value of the objective function. It can also be observed that the values of the economic and service objective functions will also change through exchanging the value of ω₁ and ω₂. After the value of ω₁ is adjusted from 10 to 90, the value of the economic objective function decreases, while the value of the service objective function corresponding to the weight increases. The influences of parameter values are analyzed in Section 5.3. Fig. 8 and Table 6 are used to demonstrate the No. 12 scheme in Table 5. It is noted that because the dataset has 83 nodes, it is impossible to directly show the optimization scheme of the hub-and-spoke RRCT network with a graph.

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Fig. 8

Hub location selection scheme when (ω₁, ω₂) = (90, 10),N = 83,p = 16 and α = 0.8.

It can be seen that our proposed model and algorithm can obtain the optimal layout of the RRCT network along the Yaan-Linzhi section of the Sichuan-Tibet Railway, and they can be utilized to help decision makers with different service and economic objectives preferences to make decision plans. However, since the rail transportation infrastructure nodes of the Sichuan-Tibet Railway are mostly located in the alpine regions of Tibet with complex and difficult environment, which has been reflected in the study of Zhang J et al. [20], there may be some hub nodes that do not meet the construction conditions. As a result, in the actual process, the relevant decision-making departments should also determine the scheme of hub location and allocation by considering the complex alpine region environment. Of course, we will further consider this issue in future research.

It can be known from Inference 2 that the value of α has nothing to do with the value of the economic objective function F₁, and it has an increasing relationship with the service objective function F₂. In order to further verify the reasonability of this inference, we conduct 200 sets of numerical analysis experiments on the value of α, F₁ and F₂, with the results shown in Fig. 9. From the experimental results, it can be observed that the economic objective F₁ has nothing to do with the value of the credibility measure α, because when α changes from 0.1 to 0.9, the maximum value of the economic objective function changes by only 0.20%. It can also be clearly observed that the service objective F₂ is an increasing function of α, with a maximum increase of 43.01%. As the credibility measurement is further increasing, freight transportation enterprises will set a longer time to meet the customer needs, and can get a high cost-effective (value of economic objective function unchanged) hub-and-spoke RRCT network.

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Fig. 9

Influence of credibility measure α on the objective function.

In the analysis of the influence of the weight value on the objective function value, we only need to investigate the influence of one weight value. This is because the weight value is related to preference, and the value of one weight has an opposite relationship with the value of the other. Here, the influence of the weight ω₁ of the economic objective function is investigated. A total of 200 sets of numerical analysis experiments are conducted and the analysis results are demonstrated in Fig. 10. It is found that the modeling of the hub-and-spoke RRCT network based on uncertain cost-time-demand can generate the scheme with the smallest total transportation cost, but may not generate the scheme with the smallest transportation time. Therefore, freight transportation enterprise can set objective weight values according to the preference of the objective function, so as to obtain the best scheme under the current decision conditions.

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Fig. 10

Influence of weight ω on the objective function.

Inference 1 gives the relationship between the left and right width coefficients (μ₁, μ₂) and the economic objective function under uncertain conditions and the economic objective function under certain conditions. In order to further verify the reasonability of this inference, 10 sets of numerical analysis experiment are conducted, with the results shown in Table 7. We take the condition of μ₁ = 1 and μ₂ = 1 as the certain condition. It can be seen that when μ₁ + μ₂2, the model with uncertain conditions have more cost advantages than the model with certain conditions. In other words, for economic objective, the objective value of the optimal scheme of the model with certain conditions is higher than that with uncertain conditions. Therefore, in the actual process, freight transportation enterprises can choose two parameters μ₁ and μ₂ according to their own conditions, so as to obtain a better hub-and-spoke RRCT network scheme.

In order to prove the applicability and effectiveness of the improved genetic algorithm proposed in this research, comparison is performed on the proposed algorithm, simulated annealing algorithm and standard genetic algorithm. In the simulated annealing algorithm, the parameters are set as follows: initial temperature of 1000 °C, end temperature of 0.05 °C, temperature reduction coefficient of 0.8, and step size of 5, It is stipulated that the iteration can be stopped when the temperature reaches 0.5 °C, and the maximum number of iterations is 1000. In the genetic algorithm, the parameters are set as follows: initial population size of 200, crossover probability of 0.8, mutation probability of 0.2, and maximum number of iterations of 1000. In the comparative analysis, we set a unified numerical analysis environment: N = 83, P = 16, α = 0.8 and (ω₁, ω₂) = (90, 10). According to Fig. 11, the performance of the improved genetic algorithm and the standard genetic algorithm is much better than the simulated annealing algorithm in the early iteration process (as indicated by the decline rate). Besides, the improved genetic algorithm is superior to the other two algorithms, due to smaller number of iterations, less running time and smaller value of the objective function.

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Fig. 11

Comparison of the iteration processes of different algorithms.