A profit distribution model for the multimodal transport alliance considering participation rate and communication structure

Abstract

The rationality of profit distribution affects the stability of the multimodal transport alliance. In the multimodal transport alliance, the participation rate of the carrier and the communication structure of the alliance are important influencing factors of profit distribution results. To get a reasonable profit distribution scheme, this paper constructs a profit distribution model considering the characteristics of multimodal transport, called the Choquet Cloud Gravity Center AT model. Firstly, considering the communication structure of the alliance, the Cloud Gravity Center Average Tree method is used as the base model for profit distribution. Secondly, considering the multimodal transport alliance is a fuzzy coalition, the profit for each alliance subset in the base model is calculated by the Choquet integral. Then, the profit distribution model considering participation rate and communication structure is obtained. Finally, a numerical example is given to illustrate the applicability of the model, and comparative analysis is conducted to verify the rationality of the model. This study provides a suitable profit-sharing model for multimodal transport alliances, which is conducive to the stable and efficient operation of alliances.

Keywords

Multimodal transport profit distribution fuzzy coalition communication structure Choquet integral

1 Introduction

Due to the uneven resource endowment of countries and the global industrial division of labor, the global layout of supply chains has further deepened, and international trade has become more frequent. Multimodal transport plays an important function in international transportation and has become an essential support for the global layout of supply chains. Multimodal transport has the apparent characteristics of a dynamic alliance organization and is a particular dynamic alliance composed of transport enterprises [1]. The stability of the alliance is affected by the rationality of profit distribution. Designing a reasonable profit distribution scheme is one of the critical issues that the alliance must address.

The profit distribution of transport alliances has been widely concerned by scholars, and scholars have carried out studies on profit distribution methods. Yilmaz et al. [2] studied the cooperation among carriers, and proposed a profit distribution scheme with the help of game theory. Lozano et al. [3], based on the cooperative game method, proposed a method to allocate the excess profits generated by carriers through cooperation. Krajewska et al. [4] studied the cooperation and the way of profit distribution among carriers, then proposed to apply the Shapley value method for getting the profit distribution scheme. Agarwal et al. [5] designed membership mechanisms for the rational profit distribution of transport alliances. Guajardo et al. [6] described cost-sharing methods for collaborative transportation, including the Shapley value method, proportional methods, the Nucleolus method, etc. The methods described can also be applied to profit-sharing studies. Methods proposed by the above scholars can be used for ordinary transport alliances, and provide solutions for the profit distribution of multimodal transport alliances.

When the methods mentioned above are applied to the design of profit distribution schemes for multimodal transport alliances, although the profit distribution scheme can be obtained, it does not reflect the characteristics of multimodal transport well and may not be the most appropriate. In order to get a reasonable profit allocation scheme, scholars studied the relevant contents of profit allocation based on the characteristics of multimodal transport. Including the performance evaluation system [7], incentive schemes [8], the formation of alliances [9], the interaction mechanism [10], and profit distribution models. Currently, many studies are focusing on profit distribution models. Methods used in constructing profit distribution models include the Shapley Value method, Myerson Value method, Average Tree Solution method, and Raiffa Solution method. However, the above methods have drawbacks, and when constructing a profit distribution model based on the above methods, the model needs to be adjusted. For example, Guo et al. [11] constructed a profit distribution model for the liner alliance based on the Shapley Value method and improved the model by considering differences in resources invested, the risk assumed, and the business operation volume of carriers. To realize the reasonable profit distribution of multimodal transport alliances, scholars constructed profit distribution models based on the above methods while adjusting models by considering features of multimodal transport. Algaba et al. [12] proposed profit-sharing methods for the multimodal public transport system, namely the Coloured Egalitarian Solution and the Coloured Cost Proportional Solution, both of which can provide a reasonable profit distribution scheme. He et al. [13] applied the ant colony labor division model for profit distribution, and penalty factors were introduced to punish destructive behaviors of carriers, like midway dropout. Jia et al. [14] analyzed the similarity between an ant colony and carriers, and constructed the ant colony labor division model for profit distribution. Duan et al. [15] built the profit distribution model based on the Myerson Value method, and the difference in resources invested by carriers was considered. Ma et al. [16] used the Nash bargaining utility function to simulate the bargaining process of carriers, and built a profit-sharing game model for profit distribution. Zhu et al. [17] established the profit distribution model based on the Shapley Value method, and improved the model by considering the dynamic nature of the alliance and the contribution degree and risk undertaken by carriers. Li et al. [18] used the improved Raiffa Solution method to calculate the profit distribution values of carriers. Zhu et al. [1] introduced contribution degree, risk-bearing ratio, dynamic penalty factor, and time reward factor as correction factors, improved the Shapley value method, and proposed a profit distribution model.

When scholars studied profit distribution models of the multimodal transport alliance, they comprehensively considered influencing factors, such as differences in risk assumed, level of effort, and resource investment of carriers. However, existing studies lack the following two considerations: (1) In multimodal transport alliances, carriers may only be able to participate at a specific participation rate due to resource and other factors. In this case, the alliance is a fuzzy coalition. (2) Multimodal transport is mostly long-distance across regions and countries, and carriers are located in different regions or countries. Due to geographical and other factors, the alliance has a communication structure. The profit of the alliance is affected by the participation rates of carriers, and the importance of carriers is affected by the communication structure of the alliance, so these two factors should be considered for profit distribution. Aubin proposed the concept of the fuzzy coalition, in which participation rates of participants were represented as a real number between [0,1]. In this paper, considering that the exact participation rate is challenging to obtain, an interval number between [0,1] is used. Considering communication structure, cooperation games with communication structure are used to represent the cooperation of carriers.

In this paper, when constructing the model for profit distribution, the proportion of various factors borne by carriers, participation rates of carriers, and the communication structure of the alliance are considered. Design ideas of the model are as follows: (1) The Choquet integral is used to calculate the profit of each alliance subset. (2) The Average Tree Solution method is adopted to calculate the initial profit distribution values of carriers. (3) In response to the differences in resource investment, firm status, level of effort, and risk assumed of carriers, the Cloud Gravity Center method is adopted to calculate the weight of carriers, then profit distribution correction values are calculated. Finally, initial values are corrected to obtain the final profit distribution scheme.

The proposed model can effectively get the profit distribution values of carriers and make the profit distribution scheme more reasonable. It is named the Choquet Cloud Gravity Center AT model. Compared with other models, the proposed model is more in line with the characteristics of multimodal transport. When getting the profit distribution scheme, not only the differences in resource investment, firm status, level of effort, and risk assumed of carriers, but also the communication structure of the alliance and participation rates of carriers are taken into account. The proposed model is applicable to the solution of profit allocation schemes for multimodal transport alliances, especially when the alliance has the communication structure, and carriers in the alliance have different participation rates.

The rest of this paper is organized as follows: Section 2 introduces the basic concepts used in this paper. The construction and detailed solution steps of the model are proposed in Section 3. A numerical example is provided to prove the feasibility and effectiveness of the model in Section 4. Lastly, the conclusion is drawn in Section 5.

2 Preliminaries

In this section, basic concepts are introduced, precisely the interval number and operations, the Choquet integral, the average tree solution, and the Cloud Gravity Center method.

2.1 The interval number and operations

Define R to be the set of real numbers. The interval $\tilde{a} = [a^{-}, a^{+}]$ is a finite closed interval in R, where a^- and a⁺ are the left and right endpoints of $\tilde{a}$ . If a^- = a⁺, it means that $\tilde{a}$ is an exact number. The entire set of intervals in R is denoted as IR.

Definition 1 [19]. For any $\tilde{a} = [a^{-}, a^{+}]$ , $\tilde{b} = [b^{-}, b^{+}]$ , $\tilde{a}, \tilde{b} \in IR$ , λ is a real number. Rules of operations are as follows:

$\tilde{a} + \tilde{b} = [a^{-} + b^{-}, a^{+} + b^{+}]$

$\tilde{a} - \tilde{b} = [a^{-} - b^{+}, a^{+} - b^{-}]$

$λ \tilde{b} = [\min {λ b^{-}, λ b^{+}}, \max {λ b^{-}, λ b^{+}}]$

Since the inverse operation is not satisfied, define the interval algorithm as follows: $\tilde{a} ⊖ \tilde{b} = [a^{-} - b^{-}, a^{+} - b^{+}]$ [20]. This algorithm is taken in this paper.

2.2 Choquet integral

Definition 2 [21]. Let X be a non-empty and monotonic set, and P(X) is the power set of X. Function g : P (X) → R, if the following conditions are satisfied: g (ø) = 0; if A, B ∈ P (X), g (A) ⩽ g (B) whenever A ⊆ B. Then g is a fuzzy measure on X.

Definition 3 [22]. If g is a fuzzy measure on X, the function f : X → R. If X ={ x₁, x₂, …, x_n }, the function f can be expressed in discrete form, which is {f (x₁), f (x₂), f (x₃), . . . , f (x_n)}. Rearrange {f (x₁), f (x₂), f (x₃), . . . , f (x_n)} in increasing order as ${f (x_{1}^{*}), f (x_{2}^{*}), f (x_{3}^{*}), ..., f (x_{n}^{*})}$ . The Choquet integral of f with respect to g is defined by Equation (1). $\begin{matrix} C_{g} (f) = \sum_{i = 1}^{n} [f ({x_{i}}^{*}) - f ({x_{i - 1}}^{*})] g \\ ({{x_{i}}^{*}, {x_{i + 1}}^{*}, \dots, {x_{n}}^{*}}) \end{matrix}$ (1) where, f (x₀^*) = 0.

2.3 The average tree solution

In graph game (N, v, L), N is the set of players, N ={ 1, 2, …, n }; v is the payoff function of players, v : 2^N → R; L is the communication edge set of all players, L ⊆ {{ i, j } |i ≠ j, i, j ∈ N}. Directly or indirectly connected players can form alliances, and p (N) is the set of all alliances. Suppose an alliance is labeled s, s ∈ p (N). If s cannot form a larger connected subset by adding any player j, where j ∈ (N ∖ S), then s is called a connected branch, and all connected branches form a set ${\hat{C}}^{L (S)} (S)$ .

In figure (N,L), the n-tuple B ={ B₁, B₂, …, B_n } composed of the subset of N is called an admissible n-tuple if the following conditions are satisfied: for any i ∈ N, there is i ∈ B_i; there is j ∈ N, such that B_j = N; for any i ∈ N and $s \in {\hat{C}}^{L} {B_{i} ∖ {i}}$ , exist j ∈ N, for any {i, j } ∈ L, there is s = B_j. All admissible n-tuples constitute the set B^L.

Definition 4 [23]. In graph game (N, v, L), The n-dimensional average tree solution AT (N, v, L) is as follows. $\begin{matrix} A T_{i} (N, v, L) = \frac{1}{| B^{L} |} \\ (\sum_{B \in B^{L}} (v (B_{i}) - \sum_{s \in {\hat{C}}^{L} {B_{i} ∖ {i}}} v (s))) \end{matrix}$ (2) where: i = 1, 2, …, n, |B^L| is the number of elements in B^L. v (s) means the profit obtained by s, and v (B_i) means the profit obtained by B_i.

2.4 The Cloud Gravity Center method

The Cloud Gravity Center method is developed based on the normal cloud method, which can solve the problem of excessive subjectivity in the evaluation process, and is a powerful tool for information processing. The expected value E_x, the entropy value E_n and the hyper-entropy value H_e denote the position of the gravity center, the expected width, and the thickness of the normal cloud, respectively. The cloud gravity center can be expressed as T = a * b, where a = E_x, and b represents the height of the cloud gravity center.

3 The Choquet Cloud Gravity Center AT model

Suppose there is a multimodal transport alliance, and members of the alliance form a set N. Carriers are labeled i, i ={ 1, 2, …, n }, including shipping, railway, and road transportation enterprises. The multimodal transport operator is acted by carriers, and multimodal transport is organized in a collaborative manner. Due to the constraint of capital and other resources, carriers join the alliance at an ambiguous participation rate. The alliance has a communication structure due to geographical and other factors. In addition, the proportion of various factors borne by each carrier varies, such as the risk borne. Under the constraint of communication structure, carriers can form the set s, s ∈ p (N), and p(N) is the set of all alliance subsets. For modeling, the following assumptions are made: (1) Profit of the carrier joining the alliance is greater than the profit of its independent operation, and carriers are wholly rational and pursue profit maximization. (2) The optimal profit for the alliance is predictable, and the participation rate of each carrier is 100% at this point. (3) Carriers are in a perfectly competitive market. To obtain a reasonable profit distribution scheme for the alliance, the Choquet Cloud Gravity Center AT model is proposed, and a detailed procedure is as follows.

Phase 1: Calculate initial values of profit distribution for carriers.

Step 1: Evaluate the resource input of carriers. Capital, facilities, human, technology, and marketing resources are labeled j in order, and j ={ 1, 2, 3, 4, 5 }. Experts are invited to evaluate resources invested by carrier i. Experts are labeled Z, Z ={ 1, 2, …, K }.

Capital investment $y_{i 1}^{'^{z}}$ . In addition to their own business, the capital is invested in the alliance to develop logistics hubs and information technology construction. It is difficult to accurately determine the amount of capital invested in the alliance by carriers. Expert Z is invited to analyze the financial statements of carrier i to obtain $y_{i 1}^{'^{z}}$ .

Facilities input $y_{i'}^{2^{z}}$ . The label of equipment invested by carrier i is m_i, m_i ={ 1, 2, …, L }. Similar to capital investment, it is difficult to accurately determine the number of facilities resources invested in the alliance. Expert Z is invited to calculate $y_{i'}^{2^{z}}$ with Equation (3). $y_{i'}^{2^{z}} = \sum_{m_{i} = 1}^{L} v_{m_{i}} \times t_{m_{i}}$ (3)

where: v_{m
_i} is the average value created per unit of working time of equipment m_i, and t_mi is the actual working time of equipment m_i.

Human resource input $y_{i'}^{3^{z}}$ . It is not easy to obtain quantitative results of human resource input, so the five-grade scaling method is used for evaluation. The evaluation set is shown in Table 1. The value corresponding to the evaluation result of expert Z on human resource input of enterprise i is $y_{i'}^{3^{z}}$ .

Investment in technology resource $y_{i'}^{4^{z}}$ . After joining the alliance, carriers will invest technology resources, and the investment is evaluated by the five-grade scaling method. Expert Z evaluates technology resources invested by enterprise i, and the value corresponding to the evaluation result is $y_{i'}^{4^{z}}$ .

Marketing resource input $y_{i'}^{5^{z}}$ . The marketing resource invested mainly includes advertising, marketing personnel, and marketing channels. The investment is evaluated by the five-grade scaling method. Expert Z evaluates the marketing resource invested by enterprise i, and the value corresponding to the evaluation result is $y_{i'}^{5^{z}}$ .

Table 1

The evaluation set

Comments	Very poor	Poor	Middle	Good	Very good
Value range	[0,0.2)	[0.2, 0.4)	[0.4, 0.6)	[0.6, 0.8)	[0.8, 1]

Step 2: Calculate the resources investment rates of carriers. The capital investment rate of enterprise i is y_i1, $y_{i 1} = y'_{i 1}^{-} ∕ y ″_{i 1} - y'_{i 1}^{+} ∕ y ″_{i 1}$ , where $y_{i 1}^{' -}$ and $y_{i 1}^{' +}$ are the minimum and maximum values of $y_{i 1}^{'^{z}}$ respectively, and $y_{i 1}^{″}$ is the expected capital investment value of carrier i when the alliance reaches optimal profit. The facilities investment rate of enterprise i is y_i2, $y_{i 2} = y'_{i 2}^{-} ∕ y ″_{i 2} - y'_{i 2}^{+} ∕ y ″_{i 2}$ , where $y_{i 2}^{' -}$ and $y_{i 2}^{' +}$ are the minimum and maximum values of $y_{i_{2}}^{'^{z}}$ respectively, and $y_{i 2}^{″}$ is the expected facilities input value of carrier i when the alliance reaches optimal profit. The human resource investment rate of enterprise i is y_i3, $y_{i 3} = (\sum_{z = 1}^{k} y_{i'}^{3^{z}}) / k$ . The investment rate of technology resources of enterprise i is y_i4, $y_{i 4} = (\sum_{z = 1}^{k} y_{i'}^{4^{z}}) / k$ . The marketing resource investment rate of enterprise i is y_i5, $y_{i 5} = (\sum_{z = 1}^{k} y_{i 5}^{'^{z}}) / k$ .

Step 3: Determine the weight of each type of resource. Determine the weight based on queuing theory. Formulas are Equations (4)–(5). $\begin{array}{l} w'_{j} = \frac{1}{2} + \sqrt{- 2 \ln (\frac{2 (q_{j} -1)}{p})} /6, \\ 1 < q_{j} \leq \frac{p+1}{2} \end{array}$ (4) $\begin{matrix} {w'}_{j} = \frac{1}{2} - \sqrt{- 2 ln (2 - \frac{2 (q_{j} - 1)}{p})} / 6, \\ \frac{p + 1}{2} < q_{j} ⩽ p \end{matrix}$ (5)

where: w₁′=1; p = 5, is the total number of indicators; q_j is the queuing order of resource j, and the smaller the number means that the resource is queued higher and more critical. Normalize w_j′ to obtain the weight w_j of resource j.

Step 4: Calculate the participation rates of carriers. The participation rate is calculated from the resources investment rate. The participation rate of carrier i is x_i, calculated by Equation (6). $x_{i} = \sum_{j = 1}^{5} y_{ij} * w_{j}$ (6)

Step 5: Calculate the profit of the set s. The participation rate x_i is an interval number, $x_{i} = [x_{i}^{-}, x_{i}^{+}]$ . For any alliance s, s ∈ p (N), assume that the carrier label in alliance s is si, let $s^{-} = {x_{si}^{-} | si \in N}$ , $s^{+} = {x_{si}^{+} | si \in N}$ . d (s^-) and d (s⁺) are the total number of elements in s^- and s⁺ respectively. Rearrange elements in s^- and s⁺ in increasing order of $h_{1^{*}}^{-} ⩽ h_{2^{*}}^{-} ⩽ \dots ⩽ h_{d^{*} (s^{-})}^{-}$ and $h_{1^{* *}}^{+} ⩽ h_{2^{* *}}^{+} ⩽ \dots ⩽ h_{d^{* *} (s^{+})}^{+}$ respectively. The profit value of alliance s is $\tilde{tv} (s)$ , $\tilde{tv} (s) = [t v^{-} (s), t v^{+} (s)]$ . $\tilde{tv} (s)$ is calculated by Equations (7)–(8). $t v^{-} (s) = \sum_{m^{*} = 1}^{d (s^{-})} v ({[s]}_{h_{m^{*}}^{-}}) * (h_{m^{*}}^{-} - h_{m^{*} - 1}^{-})$ (7) $t v^{+} (s) = \sum_{m^{* *} = 1}^{d (s^{+})} v ({[s]}_{h_{m^{* *}}^{+}}) * (h_{m^{* *}}^{+} - h_{m^{* *} - 1}^{+})$ (8) where: $h_{0^{*}}^{-} = h_{0^{* *}}^{+} = 0$ , ${[s]}_{h_{m^{*}}^{-}} = {s i^{*} \in N | h_{s i^{*}}^{-} ⩾ h_{m^{*}}^{-}}$ , ${[s]}_{h_{m^{* *}}^{+}} = {s i^{* *} \in N | h_{s i^{* *}}^{+} ⩾ h_{m^{* *}}^{+}}$ , $v ({[s]}_{h_{m^{*}}^{-}})$ and $v ({[s]}_{h_{m^{* *}}^{+}})$ are profit of ${[s]}_{h_{m^{*}}^{-}}$ and ${[s]}_{h_{m^{* *}}^{+}}$ respectively.

Step 6: Calculate initial values of profit distribution for carriers. The initial value of profit distribution for carrier i is $A T^{'}_{i}$ , $A T'_{i} = [A T'_{i}^{-}, A T'_{i}^{+}]$ . Calculation formulas are Equations (9)–(10). $\begin{array}{l} A T'_{i} - (N, \tilde{t v}, L) = \frac{1}{| B^{L} |} \\ (\sum_{B \in B^{L}} (t v^{-} (B_{i}) - \sum_{S \in {\hat{C}}^{L} {B_{i} \ {i}}} t v^{-} (S))) \end{array}$ (9) $\begin{array}{l} A T'_{i} (N, \tilde{t v}, L) = \frac{1}{| B^{L} |} \\ (\sum_{B \in B^{L}} (t v^{+} (B_{i}) - \sum_{S \in {\hat{C}}^{L} {B_{i} \ {i}}} t v^{+} (S))) \end{array}$ (10)

Phase 2: Calculate profit allocation correction values for carriers.

Step 1: Calculate the weight bo_i that carrier i should theoretically bear for factors. The bo_i is calculated based on the participation rate of carrier i. Calculation formulas are Equations (11)–(12). $b o'_{i} = (x_{i}^{-} + x_{i}^{+}) / 2$ (11) $b o_{i} = b o'_{i} / \sum_{i = 1}^{n} b o'_{i}$ (12)

Step 2: Calculate the actual weight η_i of carrier i

(1) Assessment of the actual situation for carrier i.

Evaluate carriers from four factors: resource investment, firm status, level of effort, and risk assumed. Factors are labeled l in order, and l ={ 1, 2, 3, 4 }. Among factors, resource investment is a quantitative factor. The evaluation result of resource investment for expert Z to carrier i is ${ac}_{i 1}^{Z}$ , which is calculated according to the evaluation result in step 1, phase 1 Section 3. The calculation formula is Equation (13). ${ac}_{i 1}^{Z} = \sum_{j = 1}^{2} ((y_{ij}^{' z} / y_{ij}^{″}) * w_{j}) + \sum_{j = 3}^{5} (y_{ij}^{iZ} * w_{j})$ (13)

The firm status, level of effort, and risk assumed are difficult to obtain quantitative results. The five-grade scaling method is used for evaluation. Experts are invited to evaluate carriers. Evaluate the firm status of the carrier considering its transportation market share, brand and goodwill, and corporate substitutability. The level of effort is evaluated in terms of efforts made to improve the level of transportation services, flexible operation efforts, and efforts made to cooperate with the market strategy of the alliance. The risk assumed is assessed from aspects of opportunity risk, information leakage risk, and transportation market competition risk. The value corresponding to the evaluation result of expert Z on factor l of firm i is ${ac}_{il}^{Z}$ .

(2) Determine the weight of each type of factor. The weight is determined based on queuing theory. Formulas are Equations (14)–(15). ${W_{l}}^{'} = \frac{1}{2} + \sqrt{- 2 ln (\frac{2 (Q_{l} - 1)}{p})} / 6, 1 < Q_{l} ⩽ \frac{p + 1}{2}$ (14) $\begin{array}{l} W'_{l} = \frac{1}{2} - \sqrt{- 2 \ln (2 - \frac{2 (Q_{l} -1)}{p})} /6, \\ \frac{p+1}{2} < Q_{l} \leq p \end{array}$ (15)

where: W₁′=1; p = 4; Q_l is the queuing order of factor l. Normalize W_l′ to obtain the weight W_l of factor l.

(3) Calculation of expected values of quantitative factor evaluation results. The expected value of factor l for carrier i is E_x^il. For quantitative factors, E_x^il is calculated by Equation (16). ${E_{x}}^{il} = (\sum_{Z = 1}^{k} ({ac}_{il}^{Z -} + {ac}_{il}^{Z +}) / 2) / k$ (16) where: ${ac}_{il}^{Z -}$ and ${ac}_{il}^{Z +}$ are the lower and upper bounds of ${ac}_{il}^{Z}$ .

(4) Calculation of expected values of qualitative factor evaluation results. Qualitative factors are expressed using a set of comments. Comments and their corresponding numerical intervals are shown in Table 1. The expected value E_x^t and the entropy value E_n^t of comment t are calculated by Equations (17)–(18). ${E_{x}}^{t} = (c_{\min}^{t} + c_{\max}^{t}) / 2$ (17) ${E_{n}}^{t} = (c_{\max}^{t} - c_{\min}^{t}) / 6$ (18) where: t is the marker for comments, t ={ 1, 2, 3, 4, 5 }. $c_{\min}^{t}$ and $c_{\max}^{t}$ are the lower and upper bounds of the numerical interval corresponding to comment t.

For qualitative factors, E_x^il is calculated by Equation (19). ${E_{x}}^{il} = (\sum_{z = 1}^{k} {E_{x}}^{ilz} {E_{n}}^{ilz}) / (\sum_{z = 1}^{k} {E_{n}}^{ilz})$ (19)

where: E_x^ilz and E_n^ilz are the expected value and the entropy value corresponding to the evaluation result given by expert z for factor l of carrier i, respectively.

(5): T^iG calculation. The cloud gravity center of carrier i is Tⁱ, $T^{i} = (T_{1}^{i}, T_{2}^{i}, T_{3}^{i}, T_{4}^{i})$ , where $T_{l}^{i} = a_{l}^{i} * W_{l}$ , $a_{l}^{i}$ is E_x^il, and W_l is the weight of factor l. Similarly, under an ideal scenario, the cloud gravity center of the carrier is Tⁱ⁰, $T^{i 0} = (T_{1}^{0}, T_{2}^{0}, T_{3}^{0}, T_{4}^{0})$ . Normalize $T_{l}^{i}$ to obtain $T_{l}^{iG}$ , $T^{iG} = (T_{1}^{iG}, T_{2}^{iG}, T_{3}^{iG}, T_{4}^{iG})$ , normalization formulas are Equations (20)–(21). $T_{l}^{iG} = (T_{l}^{i} - T_{l}^{0}) / T_{l}^{0}, T_{l}^{i} < T_{l}^{0}$ (20) $T_{l}^{iG} = (T_{l}^{i} - T_{l}^{0}) / T_{l}^{i}, T_{l}^{i} ⩾ T_{l}^{0}$ (21)

(6): Calculate η_i of carrier i. η_i is obtained according to Equations (22)–(23). $θ_{i} = \sum_{l = 1}^{4} W_{l} * T_{l}^{iG}$ (22) $η_{i} = θ_{i} / (\sum_{i = 1}^{n} θ_{i})$ (23)

Step 3: Calculate profit allocation correction values. The correction value of carrier i is Δφ_i, and $Δ φ_{i} = [Δ φ_{i}^{-}, Δ φ_{i}^{+}]$ . Δφ_i is calculated by Equations (24)–(25). $Δ φ_{i}^{-} = t v^{-} (n) * (b o_{i} - η_{i})$ (24) $Δ φ_{i}^{+} = t v^{+} (n) * (b o_{i} - η_{i})$ (25)

where: tv^- (n) and tv⁺ (n) are the lower and upper bounds of profit when all carriers cooperate.

Phase 3: Calculate final profit allocation values. The final value of carrier i is AT_i, and $A T_{i} = [{AT}_{i}^{-}, {AT}_{i}^{+}]$ . AT_i is calculated by Equations (26)–(27). ${AT}_{i}^{-} (N, \tilde{tv}, L) = {AT}_{'}^{i} - (N, \tilde{tv}, L) + Δ φ_{i}^{-}$ (26) ${AT}_{i}^{+} (N, \tilde{tv}, L) = {AT}_{i}^{'} + (N, \tilde{tv}, L) + Δ φ_{i}^{+}$ (27)

The above process is the entire content of the Choquet Cloud Gravity Center AT model. The model is mainly divided into three phases: Phase 1, Phase 2 and Phase 3. Within each phase, there are many steps to accomplish the goal of the phase. A reasonable profit distribution scheme for the multimodal transport alliance can be obtained through the model.

4 The numerical example and comparative analysis of results

In this section, a numerical example is given to show the validity of the proposed method. Then the comparative analysis is described to prove the advantage of the proposed method.

4.1 The numerical example

Suppose that three carriers cooperate to complete multimodal transport operations. Carriers are labeled carrier 1, carrier 2, and carrier 3. N is the set of carriers. Due to geographical constraints, carrier 1 and carrier 3 cannot be directly allied. p (N) is the set of all alliance subsets, and p (N) = ({ 1 } , { 2 } , { 3 } , { 1, 2 } , { 2, 3 } , { 1, 2, 3 }). Assume that the maximum profit of alliance {1, 2, 3} is $ 28 million, and each carrier has a participation rate of 100% in this situation. Each carrier has a 100% investment rate in capital, facilities, human, technology, and marketing resources. If the same resource investment is taken, the profit of alliance {1,2} is $ 12 million, and the profit of alliance {2,3} is $ 10 million. When carriers are operating individually. The profit of carriers 1, 2, and 3 is $ 6 million, $ 4 million, and $ 3 million, respectively. However, in practice, carriers are limited by capital, equipment, and other resources and can only join the alliance at a certain participation rate. The participation rate is difficult to be obtained precisely and is usually expressed as an interval number. In addition, the proportion of various factors borne by each carrier varies, such as the risk borne. Next, the proposed model will be used to calculate profit distribution values, and the profit distribution scheme will be obtained.

Phase 1: Calculate initial values of profit distribution for carriers.

Step 1: Evaluate the resource input of carriers. Four experts are invited to evaluate resource investment according to the method in step 1, phase 1 Section 3. Assume obtained results are shown in Tables 2–4.

Table 2
Resources input evaluation for carrier 1

Resources Capital Facilities Human resource Technology resource Marketing resource

Expert 1 0.4 0.5 Middle Poor Middle

Expert 2 0.44 0.575 Middle Middle Middle

Expert 3 0.48 0.55 Middle Poor Good

Expert 4 0.44 0.625 Poor Poor Good

Resources	Capital	Facilities	Human resource	Technology resource	Marketing resource
Expert 1	0.4	0.5	Middle	Poor	Middle
Expert 2	0.44	0.575	Middle	Middle	Middle
Expert 3	0.48	0.55	Middle	Poor	Good
Expert 4	0.44	0.625	Poor	Poor	Good

Table 3

Resources input evaluation for carrier 2

Resources	Capital	Facilities	Human resource	Technology resource	Marketing resource

Expert 1	0.46	0.625	Middle	Middle	Middle
Expert 2	0.48	0.75	Middle	Middle	Good
Expert 3	0.52	0.7	Good	Middle	Good
Expert 4	0.52	0.7	Good	Poor	Good

Table 4

Resources input evaluation for carrier 3

Resources	Capital	Facilities	Human resource	Technology resource	Marketing resource
Expert 1	0.5	0.625	Middle	Middle	Middle
Expert 2	0.56	0.75	Good	Middle	Good
Expert 3	0.6	0.7	Good	Middle	Good
Expert 4	0.56	0.7	Good	Good	Good

Step 2: Calculate the resources investment rates of carriers. The rate is obtained according to step 2, phase 1 Section 3. Results are shown in Table 5.

Table 5

Resources investment rates of carriers

Resources	Capital	Facilities	Human resource	Technology resource	Marketing resource
Carrier 1	(0.4,0.48)	(0.5,0.625)	(0.35,0.55)	(0.25,0.45)	(0.5,0.7)
Carrier 2	(0.46,0.52)	(0.625,0.75)	(0.5,0.7)	(0.35,0.55)	(0.55,0.75)
Carrier 3	(0.5,0.6)	(0.625,0.75)	(0.55,0.75)	(0.45,0.65)	(0.55,0.75)

Step 3: Determine the weight of each type of resource. Each kind of resource is ranked in order of importance, and the weight is calculated by Equations (4)–(5). Results are shown in Table 6.

Table 6

Weight of resources

Resources	Capital	Facilities	Human resource	Technology resource	Marketing resource
Queuing order	1	2	3	4	5
Weight	0.33	0.24	0.21	0.13	0.09

Step 4: Calculate the participation rates of carriers. Equation (6) is applied to calculate the participation rates of carriers, and the results are shown in Table 7.

Table 7

Participation rates of carriers

Carriers	1	2	3
Participation rate	(0.4,0.55)	(0.5,0.64)	(0.54,0.69)

Step 5: Calculate the profit of the set s. Equations (7)–(8) are applied to calculate the profit of the set s. Results are shown in Table 8.

Table 8

The profit of alliance subsets

Sets	1	2	3	{1, 2}	{2, 3}	{1, 2, 3}
Profit (MUSD)	(2.4,3.3)	(2.0,2.56)	(1.62,2.07)	(5.2,6.96)	(5.12,6.55)	(12.32,16.45)

Step 6: Calculate initial values of profit distribution for carriers. Equations (9)–(10) are applied to calculate initial profit distribution values. Results are shown in Table 9.

Table 9

Initial profit distribution values of carriers

Carriers	1	2	3
Profit (MUSD)	(4.0,5.5)	(4.867,6.407)	(3.453,4.543)

Phase 2: Calculate profit allocation correction values for carriers.

Step 1: Calculate the weight bo_i that carrier i should theoretically bear for factors. Calculation formulas are Equations (11)–(12). Results are shown in Table 10.

Table 10

The weight bo_i that carrier i should theoretically bear

Carriers	1	2	3
The weight	0.29	0.34	0.37

Step 2: Calculate the actual weight η_i of carrier i.

(1) Assessment of the actual situation for carrier i. The actual situation is assessed according to step 2(1), phase 2 Section 3. Results are shown in Tables 11–13.

Table 11

Actual situation for carrier 1

Factors	Resource investment	Firm status	Level of effort	Risk assumed
Expert 1	(0.4,0.48)	Poor	Middle	Middle
Expert 2	(0.46,0.54)	Poor	Middle	Good
Expert 3	(0.46,0.54)	Middle	Middle	Good
Expert 4	(0.42,0.5)	Middle	Poor	Good

Table 12

Actual situation for carrier 2

Factors	Resource investment	Firm status	Level of effort	Risk assumed
Expert 1	(0.47,0.56)	Poor	Middle	Middle
Expert 2	(0.53,0.61)	Middle	Middle	Good
Expert 3	(0.57,0.66)	Middle	Middle	Good
Expert 4	(0.55,0.63)	Middle	Middle	Good

Table 13

Actual situation for carrier 3

Factors	Resource investment	Firm status	Level of effort	Risk assumed
Expert 1	(0.49,0.57)	Middle	Middle	Good
Expert 2	(0.6,0.68)	Middle	Middle	Good
Expert 3	(0.6,0.68)	Middle	Middle	Good
Expert 4	(0.61,0.7)	Middle	Good	Good

(2) Determine the weight of each type of factor. Each kind of factor is ranked in order of importance, and the weight is calculated by Equations (14)–(15). Results are shown in Table 14.

Table 14

Weight of factors

Factors	Resource investment	Firm status	Level of effort	Risk assumed
Queuing order	1	4	3	2
Weight	0.40	0.12	0.20	0.28

(3) Calculation of expected values of quantitative factor evaluation results. Among all factors, resource investment is a quantitative factor. E_xⁱ¹ is obtained by Equation (16).

(4) Calculation of expected values of qualitative factor evaluation results. Among all factors, the firm status, level of effort, and risk assumed are quantitative factors. E_xⁱ², E_xⁱ³, E_xⁱ⁴ are obtained by Equations (17)–(19). Results are shown in Table 15.

Table 15

The expected value of evaluation results of factors for carriers

Factors	Resource investment	Firm status	Level of effort	Risk assumed
Carrier 1	0.47	0.4	0.45	0.65
Carrier 2	0.57	0.45	0.5	0.65
Carrier 3	0.62	0.5	0.55	0.7

(5) T^iG calculation. Adopt the method in step 2(5), phase 2 Section 3 to obtain T¹ = (0.19, 0.05, 0.09, 0.18) , and T¹⁰ = (0.4, 0.12, 0.20, 0.28) . Then, T¹ is normalized to get T^1G = (- 0.53, - 0.60, - 0.55 , - 0.35) according to Equations (20)–(21). Similarly, T^2G and T^3G are obtained. T^2G = (- 0.43, - 0.55, - 0.50 , - 0.35), and T^3G = (- 0.38, - 0.50, - 0.45 , - 0.30)

(6) Calculate η_i of carrier i. Applying Equation (22) to obtain θ_i of carrier i. Getting θ₁ = -0.49, θ₂ = -0.44, and θ₃ = -0.39. Applying Equation (23) to obtain η_i of carrier i. Getting η₁ = 0.374, η₂ = 0.331, and η₃ = 0.295.

Step 3: Calculate profit allocation correction values. Assume that the profit of alliance {1, 2, 3} is $ 16.45 million with current resource input, correction value Δφ_i of carrier i is obtained by Equations (24)–(25). Getting Δφ₁ = $ -1.43million, Δφ₂ = $0.20million, and Δφ₃ = $1.23million.

Phase 3: Calculate final profit allocation values.

The final values are calculated by Equations (26)–(27), and AT₁ = $4.07million, AT₂ = $6.607million, AT₃ = $5.773million. The profit distribution scheme is ($ 4.07 million, $ 6.607 million, $ 5.773 million).

4.2 Comparative analysis

In this section, in order to verify the reasonableness of the proposed model, profit distribution schemes without considering participation rate and communication structure are calculated separately, and results are analyzed in comparison with the scheme in section 4.1.

4.2.1 The profit distribution scheme without considering the participation rate

Participation rates of carriers are the same when they are not considered. When the profit of alliance {1, 2, 3} is $ 16.45 million, the participation rate of each carrier is 0.588. Calculate the profit value of each alliance subset, initial values, correction values, and final values of profit distribution for carriers. Results are shown in Tables 16 and 17.

Table 16
The profit of alliance subsets

Sets 1 2 3 12 23 123

Profit (MUSD) 3.525 2.35 1.763 7.05 5.875 16.45

Sets	1	2	3	12	23	123
Profit (MUSD)	3.525	2.35	1.763	7.05	5.875	16.45

Table 17

Initial values, correction values, and final values of profit distribution for carriers

Carriers	1	2	3
Initial values (MUSD)	5.875	6.267	4.308
Correction values (MUSD)	-1.43	0.20	1.23
Final values (MUSD)	4.445	6.467	5.538

Comparing calculation results, it is found that the profit distribution value decreases by $ 0.375 million for carrier 1, and increases by $ 0.14 million for carrier 2 and $ 0.235 million for carrier 3 when considering the participation rate. Through comparison of data, the above change results from the change in initial profit distribution values, and the change in initial values is caused by the change in the profit of each alliance subset. To justify that it is more reasonable to consider participation rate, carrier 1 is used as an example. The formula for calculating the initial profit distribution value of carrier 1 is as follows. $\begin{matrix} A T_{1} (N, v, L) \\ = \frac{1}{3} [v ({1, 2, 3}) - v ({2, 3}) \\ + (v ({1}) - 0) + (v ({1}) - 0)] \end{matrix}$ (28)

The initial profit distribution value directly relates to v ({ 1, 2, 3 }), v ({ 2, 3 }), and v ({ 1 }). When considering the participation rate, the participation rate for carrier 1 is 0.55, carrier 2 is 0.64, and carrier 3 is 0.69. When the participation rate is not considered, it is 0.588 for each carrier. v ({ 1, 2, 3 }) is constant at $ 16.45 million, regardless of whether the participation rate is considered. When the participation rate is considered, the participation rates of carriers 2 and 3 are higher than 0.588. The profit created by carriers 2 and 3 is higher than that without considering the participation rate, and v ({ 2, 3 }) increases by $ 0.675 million. The participation rate of carrier 1 is less than 0.588, so the profit created by carrier 1 is less than that without considering the participation rate. v ({ 1 }) is reduced by $ 0.225 million. As can be seen, it is more realistic to consider the participation rate, with the profit of each alliance subset being calculated based on actual participation rates for carriers.

4.2.2 The profit distribution scheme without considering the communication structure

Calculate the profit value of each alliance subset, initial values, correction values, and final values of profit distribution for carriers. Results are shown in Tables 18 and 19.

Table 18
The profit of alliance subsets

Sets 1 2 3 12 23 13 123

Profit (MUSD) 3.3 2.56 2.07 6.96 6.55 5.37 16.45

Sets	1	2	3	12	23	13	123
Profit (MUSD)	3.3	2.56	2.07	6.96	6.55	5.37	16.45

Table 19

Initial values, correction values, and final values of profit distribution for carriers

Carriers	1	2	3
Initial values (MUSD)	5.683	5.904	4.863
Correction values (MUSD)	–1.43	0.20	1.23
Final values (MUSD)	4.253	6.104	6.093

Comparing calculation results, it is found that profit distribution value increases by $ 0.503 million for carrier 2, decrease by $ 0.183 million for carrier 1, and $ 0.32 million for carrier 3 when considering communication structure. That is because the importance of carriers changes when a communication structure exists in the alliance. Taking carrier 2 as an example, when considering communication structure, if carrier 1 and carrier 3 want to ally, it can only be combined with carrier 2 to form the alliance {1, 2, 3}. Therefore, carrier 2 has an increased status and should receive a more significant profit share. It can be known that considering communication structure is more realistic and can reflect the position of carriers better.

Through comparative analysis, the profit distribution scheme obtained by the proposed model is more reasonable and realistic.

5 Conclusion

In multimodal transport alliances, the motivation of carriers and the stability of the alliance will be affected by the reasonableness of profit distribution. The purpose of this paper is to establish a suitable profit distribution model for multimodal transport alliances, so as to improve the rationality of the profit distribution scheme. Then the stability of the alliance and the motivation of carriers will be improved, and the profit of the alliance may also be further increased.

In order to achieve reasonable profit distribution, scholars have conducted studies, mainly on the profit distribution model. In previous studies of profit allocation models for multimodal transport alliances, scholars have considered factors affecting profit distribution, such as the resource input of carriers. Scholars have generally recognized that the differences in resource investment, firm status, level of effort, and risk taken of carriers affect the profit distribution values of carriers. Therefore, in many studies, the above factors are taken into account when solving for the profit distribution scheme. However, previous studies lacked consideration of participation rate and communication structure. In practice, they have an impact on profit distribution results, and they should be taken into account. This paper proposes the Choquet Cloud Gravity Center AT model. Firstly, considering the communication structure of the alliance and the proportion of various factors borne by carriers, the Cloud Gravity Center Average Tree method is used as the base model for profit distribution. Secondly, considering the multimodal transport alliance is a fuzzy coalition, the profit for each alliance subset in the base model is calculated by the Choquet integral. Then, the profit distribution model considering participation rate and communication structure is obtained. Finally, the numerical example is given to confirm that the model is effective and can calculate the profit distribution values of carriers. The comparative analysis of results shows that the model can calculate the profit distribution values of carriers based on the consideration of participation rate and communication structure. Comparing the profit distribution scheme obtained by the proposed model with the scheme without considering the participation rate, it is found that more profit is allocated to carriers with higher participation rates by the proposed model. Comparing the scheme obtained by the proposed model with the scheme without considering communication structure, it is found that more profit is distributed to high-status carriers through this model. The proposed model is reasonable, and the profit distribution scheme aligns with the situation. In general, the proposed method is more consistent with the actual situation, and can obtain a more reasonable profit distribution scheme.

However, although the model proposed in this paper is very suitable for the multimodal transport alliance, it also has drawbacks due to the complexity of the problem. There is the area for improvement. Due to the lack of a method for assessing the participation rates of carriers, the participation rate is calculated through the resource input rate in this paper. Although the calculation method in this paper can obtain the participation rate of carriers, it lacks consideration of other factors. It cannot fully reflect the participation of carriers in the alliance. More appropriate methods can be considered in future studies to improve the model. Then, the profit distribution scheme obtained can be more reasonable, and be more consistent with the actual situation.

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A profit distribution model for the multimodal transport alliance considering participation rate and communication structure

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 The interval number and operations

2.2 Choquet integral

3 The Choquet Cloud Gravity Center AT model

4.1 The numerical example

Table 2 Resources input evaluation for carrier 1 Resources Capital Facilities Human resource Technology resource Marketing resource Expert 1 0.4 0.5 Middle Poor Middle Expert 2 0.44 0.575 Middle Middle Middle Expert 3 0.48 0.55 Middle Poor Good Expert 4 0.44 0.625 Poor Poor Good

4.2.1 The profit distribution scheme without considering the participation rate

Table 16 The profit of alliance subsets Sets 1 2 3 12 23 123 Profit (MUSD) 3.525 2.35 1.763 7.05 5.875 16.45

Table 18 The profit of alliance subsets Sets 1 2 3 12 23 13 123 Profit (MUSD) 3.3 2.56 2.07 6.96 6.55 5.37 16.45

References

Table 2
Resources input evaluation for carrier 1

Resources Capital Facilities Human resource Technology resource Marketing resource

Expert 1 0.4 0.5 Middle Poor Middle

Expert 2 0.44 0.575 Middle Middle Middle

Expert 3 0.48 0.55 Middle Poor Good

Expert 4 0.44 0.625 Poor Poor Good

Table 16
The profit of alliance subsets

Sets 1 2 3 12 23 123

Profit (MUSD) 3.525 2.35 1.763 7.05 5.875 16.45

Table 18
The profit of alliance subsets

Sets 1 2 3 12 23 13 123

Profit (MUSD) 3.3 2.56 2.07 6.96 6.55 5.37 16.45