Profit distribution of liner alliance based on shapley value

Abstract

Profit distribution plays an important role in the sustainable and stable development of liner alliances, this paper tries to solve the profit distribution issues in the liner alliance based on Shapley Value Method. Meanwhile, seeing that there is little consideration from the customer satisfaction, this paper establishes a new model by revising Shapley Value Method to distribute the profit of liner alliances from the perspectives of suppliers and customers and carry out verification through case analysis. The profit distribution method proposed in the paper is helpful to the reasonable profit distribution of liner alliance. It ensures the continuity and stability of liner alliance and provides a scientific decision-making basis for the profit distribution of liner alliance.

Keywords

Profit distribution liner alliance shapley value

1 Introduction

After the Global Financial Crisis, the competition of the liner transportation market became very fierce. Various liner companies were more inclined to reduce the cost and achieve the scale economy. As a result, the capacity of the liner transportation market grew rapidly. However, the demand growth rate was small and the transportation capacity was seriously oversupplied. Many liner companies went bankrupt and the market of small liner companies was annexed by large liner companies constantly, resulting in the enhancement of market concentration. In order to solve the cost problem of liner companies and improve their service quality to a certain extent, liner alliance became an important means for liner companies to cope with market competition, improve service network and reduce operation costs. At present, there are three major liner alliances in the market, including 2 M, OCEAN and THE. According to the latest data released by Alphaliner, with the transportation capacity share of nearly eighty percent by January 1st, 2021. The three liner alliances have become the leading force of the liner market and played an important role in the development and competition modes of the global liner market.

With the deepening of liner alliances, liner alliances have become a research focus in the field of shipping. Many scholars have carried out a lot of useful research in this aspect. The causes of the emergence and development of liner alliances were analyzed by Midoro and Pitto [9]. The three main characteristics of liner alliances were studied by Slack et al. [1]. The partner selection of liner alliances was studied by Ding [5]. The empirical analysis method was adopted by Das [10] to analyze the factors influencing the alliance or cooperation of liner companies. The structure, form and goal of liner alliances were deeply analyzed by Panayides and Wiedmer [7]. Lu et al. [4] conducted the empirical analysis from the two aspects of slot exchange and purchase planning of short sea services for liner carriers. The network design and capacity exchange of liner alliances with fixed and variable container demands was studied by Zheng et al. [6]. Huang [11] analyzed the key factors to form liner alliances. Hirata [3] analyzed the effect of liner alliances on the liner transportation market concentration.

However, when the actual profit failed to meet the expected profit of alliance members, liner alliances might tend to be restructured or disintegrated. As a result, the profit distribution mechanism was a key factor influencing the stability of liner alliances. On the basis of cooperative game analysis, it was proposed by Song and Panayides [2] to distribute profit according to the proportion of transport capacity provided by alliance members. Agarwal and Ergun [8] designed the profit distribution mechanism by making use of Mathematical Programming and Game Theory to guide alliance members to make the optimal cooperation strategy. Wang [12] studied the profit sharing and stable operation of liner alliances by making use of Game Theory and Shapley Value Method.

The existing profit distribution of liner alliance is studied from the perspective of suppliers. In the paper, according to the existing research results, it is proposed to distribute liner alliance profit from the perspectives of suppliers and customers so as to make up for the deficiency of the existing profit distribution mechanism with results relatively focused on the suppliers. It is conductive to guiding the practice of alliance operation and deepening the theory of liner alliances.

2 The design of profit distribution of model

Shapley Value Method gives full consideration to the contribution factors and profit of alliance members so as to reduce the weaknesses of average distribution and mobilize the enthusiasm of alliance members. Its definition is as follows.

Assume a set N = {1, 2, ... , n}, [I,v] is called the cooperative game among n members. v(S) practically means the profit created by liner alliance S, and is the maximum total profit among the members of S. The profit got by each member under liner alliance is called Shapley Value, Member i’s Shapley Value is φ_i [v], the Shapley Values of all members’ profit distributions are φ [v] = (φ₁ [v] , φ₂ [v] , . . . , φ_n [v]), which are calculated as follows. $φ_{l} (v) = \frac{(| S | - 1)! (n - | S |)!}{n!} [v (s) - v (S - i)]$ (1) $φ (v) = φ_{1} (v) + φ_{2} (v) + \dots + φ_{n} (v)$ (2)

Among them, |S| stands for the number of elements in the subset S; v(S-i) is the total profit of subset S removed out member i.

For the Shapley Value Method, it avoids average distribution in the process of profit distribution but ignores the effect of the proportions of resource input, risk and task. In the full consideration of the proportions of resource input, risks and tasks, Shapley Value Method is preliminarily revised from the perspective of suppliers.

Resource input: There are different resources about infrastructure, network facilities and information facilities among different liner companies. As a result, it is necessary to evaluate the proportion of resource input. Suppose that the resource input of member i is V_i, i = 1, 2, …, n, the proportion of the member i in the total resource input is V_i/∑V_i. After the resource input compensation factor g is introduced, the resource input compensation amount is: $φ_{i}^{V} (v) = g * φ (v) * (\frac{\underset{i}{v}}{\sum \underset{i}{v}} - \frac{1}{n})$ (3)

When V_i/∑V_i > 1/n, member i with more resource input shall be compensated by the other alliance members with less resource input.

Risk: The transportation service is affected by poor weather, strikes and other risks. Suppose that the risk coefficient of member i is R_i, i = 1, 2, . . . , n, ∑R_i = 1. After the risk sharing compensation factor of h is introduced, it is drawn that the risk compensation amount is: $φ_{i}^{R} (v) = h * φ (v) * (R_{i} - \frac{1}{n})$ (4)

Task: The amount of tasks is different in liner alliance. Suppose that the amount of tasks of member i is $\underset{i}{P}, i = 1, 2, \dots, n$ , the proportion of the member i in the total task is $\underset{i}{P} / \sum \underset{i}{P}$ . After the task compensation factor k is introduced, it is drawn that the task compensation amount is:

φ_{i}^{P} (v) = k * φ (v) * (\frac{\underset{i}{P}}{\sum \underset{i}{P}} - \frac{1}{n})

(5)

When P_i/∑P_i > 1/n, the task of member i is greater than the average value, member i shall be compensated by the other alliance members with less tasks.

According to the above analysis, the profit distribution of member i is: $φ_{i} {(v)}^{'} = φ_{i} (v) + φ_{i}^{V} (v) + φ_{i}^{p} (v) + φ_{i}^{R} (v)$ (6)

3 Modification of the profit distribution model

Profits are preliminarily distributed from the perspective of the proportions of resource input, risk and task. In fact, the performance of each task can affect the customer satisfaction. From the perspective of customers, the profit distribution model is revised again to obtain a reasonable and fair profit distribution finally.

The TOPSIS Method is adopted to calculate the customer satisfaction of tasks. Suppose that m liner companies are required to participate in tasks. There are n factors influencing the customer satisfaction of transportation service. The corresponding value of the factors influencing the customer satisfaction is $\underset{ij}{X}$ , the decision-making matrix is X = (X_ij) _m*n, ${(X_{ij})}_{m * n} = | \begin{matrix} X_{11} & \dots & X_{m 1} \\ \dots & \dots & \dots \\ X_{1 n} & \dots & X_{mn} \end{matrix} |$ (7)

After Equation 7 is normalized, then $X_{ij}^{'} = \frac{\underset{ij}{X}}{\sum_{i = 1}^{m} \underset{ij}{X}}, j = 1, 2, \dots, m$ (8)

The normalized decision-making matrix is: ${(X_{ij}^{'})}_{m * n} = | \begin{matrix} X_{11}^{'} & \dots & X_{1 m}^{'} \\ \dots & \dots & \dots \\ X_{1 n}^{'} & \dots & X_{mn}^{'} \end{matrix} |$ (9)

According to the normalized decision-making matrix, the information entropy and entropy weight for each customer satisfaction factor are: $\underset{j}{H} = - \begin{matrix} \end{matrix} \frac{1}{\overset{m}{ln}} \begin{matrix} \end{matrix} \sum_{i = 1}^{m} X_{ij}^{'} ln X_{ij}^{'}, j = 1, 2, \dots, n$ (10) $θ_{j} = \frac{1 - H_{j}}{\sum_{j = 1}^{n} (1 - H_{j})}, j = 1, 2, \dots, n$ (11)

Because the entropy weights of the factors influencing the customer satisfaction of liner companies are different, it is possible to weight the decision-making matrix,: $\underset{m^{*} n}{(X_{ij}^{″})} = \underset{j}{θ} \underset{m^{*} n}{(X_{ij}^{'})}$ (12)

The Euclidean Distance Method is adopted to calculate the positive and negative ideal solutions of the customer satisfaction of tasks completed by liner companies at each task. The distance of every feasible solution from the positive ideal solution and the negative ideal solution is: $d_{i}^{+} = \sqrt{\sum_{j = 1}^{n} {(X_{ij}^{″} - X_{j}^{+})}^{2}} i \in [1, m]$ (13) $d_{i}^{-} = \sqrt{\sum_{j = 1}^{n} {(X_{ij}^{″} - X_{j}^{-})}^{2}} i \in [1, m]$ (14) The relative degree of approximation is: $c_{i} = \frac{d_{i}^{-}}{d_{i}^{+} + d_{i}^{-}}, i = 1, 2, \dots, m$ (15)

The customer satisfaction of member i is: $λ_{i} = \frac{c_{i}}{\sum_{i = 1}^{m} c_{i}}, i = 1, 2, \dots, m$ (16)

The difference between the customer satisfaction of the member i and the average customer satisfaction is: $ψ_{i} = p_{i}^{*} - 1 / m, \sum_{i = 1}^{m} ψ_{i} = 0$ (17)

According to the difference between the customer satisfaction of the member i and the average customer satisfaction, the profit distribution results are revised again. $φ_{i} {(v)}^{″} = φ_{i} {(v)}^{'} + ψ_{i} * φ_{i} (v)$ (18)

4 Case analysis

Assume a liner alliance consists of three companies, A, B and C. If the three companies run separately, company A can realize a profit of 300 million dollars, B can get 200 million dollars while C get 100 million dollars. If A and B cooperate as a liner alliance, the profit will be 800 million dollars; if A and C cooperate, the profit is 600 million dollars; if B and C cooperate, the profit will be 400 million dollars. If A, B and C found a liner alliance, they would realize a 1000 million dollars profit. If the 3 companies share the profit, each will get 1000/3 million dollars.

4.1 The traditional shapely value method

The traditional Shapely Value Method is adopted to calculate the profit of Company A. The results are shown in Table 1.

Table 1
The profit distribution of company A based on the Shapely Value Method

S _A A A ∪ B A ∪ C A ∪ B ∪ C

v (s) 300 800 600 1000

v (s/A) 0 200 100 400

v (s) - v (s/A) 300 600 500 600

|s| 1 2 2 3

w (|s|) 1/3 1/6 1/6 1/3

w (|s|) ^* [v (s) - v (s/A)] 100 100 250/3 200

S _A	A	A ∪ B	A ∪ C	A ∪ B ∪ C
v (s)	300	800	600	1000
v (s/A)	0	200	100	400
v (s) - v (s/A)	300	600	500	600
\|s\|	1	2	2	3
w (\|s\|)	1/3	1/6	1/6	1/3
w (\|s\|) ^* [v (s) - v (s/A)]	100	100	250/3	200

According to the addition among the last-line values of Table 1, the profit of the member A is 1450/3 million dollars. In the same way, the profit of company B and C are 1000/3 million dollars and 550/3 million dollars respectively. According to the profit distribution results, the total profit of the three companies is 1000 million dollars and the profit of the three companies in liner alliance are greater than they run separately or bilateral cooperation.

4.2 Preliminary revision of the profit distribution from the perspective of suppliers

It is analyzed that the resource input proportions of the three companies are 0.38, 0.29 and 0.33 respectively. The risk proportions are 0.35, 0.32 and 0.33 respectively. The tasks proportions are 0.28, 0.4 and 0.32 respectively. By using a comprehensive AHP method we get the weight of each revised factor is 0.4, 0.35 and 0.25 respectively.

From the perspective of suppliers, the variable data and profit distribution results after the compensation for the influencing factors of resource input, risk and task are showed in Table 2.

Table 2
The profit distribution results after compensation

Results A B C

Shapely Value 1450/3 1000/3 550/3

Resource input compensation 56/3 –52/3 –4/3

Risk compensation 35/6 –14/3 –7/6

Task compensation 50/3 25/6 –125/6

Revision from the perspective of suppliers 524.5 315.5 160

Results	A	B	C
Shapely Value	1450/3	1000/3	550/3
Resource input compensation	56/3	–52/3	–4/3
Risk compensation	35/6	–14/3	–7/6
Task compensation	50/3	25/6	–125/6
Revision from the perspective of suppliers	524.5	315.5	160

There are different proportions of resource input, risk and task among different liner companies. It is possible to achieve the scientific and reasonable profit distribution from the perspective of suppliers.

4.3 Revision of the profit distribution from the perspective of customers

Through customer survey to score the customer satisfaction of liner companies, it is drawn that the index data of customer satisfaction are shown in Table 3.

Table 3
Index data of customer satisfaction

Factors A B C

Reliability 8.7 8.4 8.5

Assurance 8.9 9.1 9.2

Responsiveness 8.6 8.8 8.9

Tangibles 8.4 8.6 8.5

Empathy 8.8 8.9 8.6

Factors	A	B	C
Reliability	8.7	8.4	8.5
Assurance	8.9	9.1	9.2
Responsiveness	8.6	8.8	8.9
Tangibles	8.4	8.6	8.5
Empathy	8.8	8.9	8.6

Firstly, the index data of customer satisfaction in Table 3 are normalized according to Equation 8. Weights of indexes are calculated by Equation 10 and 11 and shown as follows. $θ = (0.2368, 0.2105, 0.2252, 0.1025, 0.2252)$

Secondly, The distance of every feasible solution from the positive ideal solution and the negative ideal solution is obtained according to Equation 13 and 14 and shown as follows. $\begin{matrix} d_{i}^{+} = (0.0037, 0.0030, 0.0032) \\ d_{i}^{-} = (0.0033, 0.0035, 0.0036) \end{matrix}$

Thirdly, The relative degree of approximation is determined according to Equation 15 and shown as follows. $\underset{i}{c} = (0.4714, 0.5412, 0.5306)$

Fourthly, The relative customer satisfaction of each member is calculated according to Equation 16. $\underset{i}{λ} = (0.3055, 0.3507, 0.3438)$

Finally, In combination of Equations 17 and 18, the final profit distribution results of Company A, B and C are 496.64, 332.89, 170.47.

After the comprehensive customer satisfaction evaluation of liner companies, the profit distribution result is revised from the perspective of customers. The profit distribution results of Company A, B and C in different distribution mechanisms are showed in Table 4.

Table 4

The profit of the company under different distribution mechanisms

Results	A	B	C
Shapely Value	1450/3	1000/3	550/3
Revision from the perspective of suppliers	524.5	315.5	160
Revision from the perspective of suppliers and customers	496.64	332.89	170.47

According to Table 4, when the profits are distributed from the perspective of suppliers, the profit of Company A is increased but the profit of Company B and C are reduced for the reason that the comprehensive value of the resource input, risk and task of Company A is higher than that of Company B and C in the process of transportation. From the perspective of customers, the profits of Company B and C are increased. Because the customer satisfaction of Company B and C is higher than that of Company A. In the process of transportation service, in order to get more profit, it is necessary for Company A to improve the overall transportation service quality.

5 Conclusion

The reasonable profit distribution makes liner alliances sustainable. In the paper, on the basis of the existing research results, it is proposed to distribute liner alliance profit from the perspectives of suppliers and customers so as to make up for the deficiency of the profit distribution with results relatively focused on the contribution of suppliers. From the perspective of suppliers, the profit distribution is preliminarily modified in combination of the proportions of resource input, risk and task. In addition, from the perspective of customers, the profit distribution is modified again according to the customer satisfaction of task so as to achieve the scientific and reasonable profit distribution of liner alliance.

References

Slack

, Comtois

and McCalla

, liner alliances in the container shipping industry: a global perspective, Maritime Policy and Management29(1) (2002), 65–76.

Song

D.W.

and Panayides

P.M.

, A conceptual application of cooperative game theory to liner shipping liner alliances, Maritime Policy and Management29(3) (2002), 285–301.

Hirata

, Contestability of container liner shipping market in alliance era, The Asian Journal of Shipping and Logistics33(1) (2017), 27–32.

H.A.

, Chen

S.L.

and Lai

, Slot exchange and purchase planning of short sea services for liner carriers, Journal of Marine Science and Technology18(5) (2010), 709–718.

Ding

J.F.

, Partner selection of liner alliance for a liner shipping company using extent analysis method of fuzzy AHP, Journal of Marine Science and Technology17(2) (2009), 97–105.

Zheng

, Gao

, Yang

and Sun

, Network design and capacity exchange for liner alliances with fixed and variable container demands, Transportation Science49(4) (2015), 886–899.

Panayides

P.M.

and Wiedmer

, liner alliances in container liner shipping, Research in Transportation Economics32(1) (2011), 25–38.

Agarwal

and Ergun

Ö.

, Network design and allocation mechanisms for carrier alliances in liner shipping, Operations Research58(6) (2010), 1726–1742.

Midoro

and Pitto

, A critical evaluation of liner alliances in liner shipping, Maritime Policy and Management27(1) (2000), 31–40.

10.

Das

S.S.

, To partner or to acquire? A longitudinal study of alliances in the shipping industry, Maritime Policy and Management38(2) (2011), 111–128.

11.

Huang

S.T.

, Key Factors Analysis of liner alliances in Container Liner Shipping Industry, Transport and Logistics16(38) (2016), 1–8.

12.

Wang

, Hu

, Zeng

and Li

, Profit sharing and the stability of shipping alliances based on game theory, Journal of Transport Economics and Policy50(3) (2016), 245–261.