Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions

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Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions Jia-Cai Liu a,b, Wen-Jian Zhao c, Benjamin Lev d, Deng-Feng Li e, Jiuh-Biing Sheu f, Yong-Wu Dai a,∗ a

College of Management & College of Tourism, Fujian Agriculture and Forestry University, Fuzhou 350002, China College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China c Jinshan College, Fujian Agriculture and Forestry University, Fuzhou 350002, China d LeBow College of Business, Drexel University, Philadelphia, PA 19104, USA e School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 611731, China f Department of Business Administration, National Taiwan University, Taipei 10617, Taiwan, R.O.C. b

a r t i c l e

i n f o

Article history: Received 21 December 2018 Revised 23 August 2019 Accepted 13 September 2019 Available online xxx Keywords: Equal surplus division value Equal contribution division value Excess vector Logistics enterprise Profit distribution Least square method

a b s t r a c t Some sub-coalitions can not be formed or fail to satisfy the superadditivity in many realistic cooperative transferable utility (TU) games, which particularly exists in the logistics service industry. To exploit some novel solutions for addressing these TU games, we firstly propose again the equal surplus division value based on the least square method and players’ common excess vector, and then introduce the weighted equal surplus division value. Both of them belong to the family of the least square values. Inspired by the fact that many TU games base the profit distribution strategies not only on the egalitarian principle but also on the utility principle, the equal contribution division value and the weighted equal contribution division value based on the least square method and players’ contribution excess vector are spontaneously generated. An algorithm is described to make the four solutions proposed in this paper satisfy the property of individual rationality. Finally, to show the advantages, the practicability and the rationality of the four solutions, a practical example about the profit distribution strategy of a logistics enterprise coalition is illustrated and the contrastive analysis among them is given. © 2019 Published by Elsevier Inc.

1. Introduction To cope with the increasingly fierce market competition, some logistics enterprises must pay more attention to improve own competitiveness. Forming a coalition, beyond all doubt, is an effective measure. There are two important issues for logistics enterprise coalitions. One is whether the total cooperative profit is more than the sum of the profits when logistics enterprises work alone. The other is whether the cooperative profit distribution strategy is impartial and reasonable. Academically, the study on the profit distribution strategy of logistics enterprise coalitions can be regarded as an issue about the cooperative transferable utility game, which is called the TU game for short from now on. Theoretically, many eminent single-valued solutions for TU games can be applied to solve this issue, such as the Shapley value [25] and the Banzhaf value [19]. Nevertheless, this theoretical claim is proved not entirely true. In fact, some well-known solutions for TU games ∗

Corresponding author. E-mail address: [email protected] (Y.-W. Dai).

https://doi.org/10.1016/j.ins.2019.09.019 0020-0255/© 2019 Published by Elsevier Inc.

Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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seem inappropriate for the profit distribution problem of logistics enterprise coalitions. The Shapley value is an example in point. All coalitional values must be known beforehand for the Shapley value. That is to say, all sub-coalitions must be formed and satisfy the property of superadditivity. Sometimes, it is unpractical for logistics enterprise coalitions owing to the mismatch between the available vehicle types and the freight volume. The scale economic benefit is extremely important for logistics enterprise coalitions. It is impossible to realize the scale economic benefit and the total cooperative profit will not be satisfactory if the vehicle types do not well match the freight volume. Therefore, it is necessary to exploit some novel solutions for TU games to address the profit distribution strategies of logistics enterprise coalitions. The TU games theory provides a great diversity of practicable tools for solving the problems of profit distribution or cost sharing. Nowadays, an increasing number of cooperative coalitions base the distribution strategies of cooperative profits not only on players’ productivities but also on their contributions. The equal surplus division value proposed by Driessen and Funaki [8] is a typical profit distribution strategy based on players’ productivities. To be exact, each player is firstly allocated what he can obtain by itself, and then the cooperative surplus is divided equally among all players joining the coalition. Recently, many researchers have shown great interest in its axiomatizations [12,13,15] and the difference between it and other solutions, such as the Shapley value [5,9,10,27], the equal division value [5,27] and the Banzhaf value [2,14]. Academically, the equal surplus division value or some improved equal surplus division values could effectively address the profit distribution problem of logistics enterprise coalitions because it is not necessary to know in advance all the values of sub-coalitions. Several solutions for TU games are generated based on the idea of excess vector, such as the core [6,11], the (pre-) nucleolus [24,26], and the least square (pre-) nucleolus [20]. The relationship between the equal surplus division value and the excess vector seems very interesting. We may exploit some novel solutions for TU games, especially for logistics enterprise coalitions, if the equal surplus division value is related to the excess vector. The least square value and the excess vector are two important concepts for TU games. The least square method is usually combined with the excess vector to generate new solutions or obtain new conclusions for TU games. The prenucleolus and nucleolus based on the concept of excess vector are two well-known solutions. The least square prenucleolus and the least square nucleolus, which are regarded as prenucleolus-like and nucleolus-like solutions respectively, have attracted much attention. Through minimizing the variance of coalitions’ excess vector on the consequent payoffs, Ruiz et al. [20] introduced the least square prenucleolus and the least square nucleolus. These two solutions parallel the prenucleolus and the nucleolus. Furthermore, they described some nice properties and established some different axiomatizations for the least square prenucleolus. Ruiz et al. [21,22] obtained several new findings related to the least square values and proposed a formula for showing the process of the calculation, the properties, the characterizations and the axiomatizations. Molina and Tejada [18] proved that the least square nucleolus proposed by Ruiz et al. was a general nucleolus. What is more, they pointed out that the least square nucleolus and the lexicographical solution proposed by Sakawa and Nishizaki [23] selected the same imputation in every game with nonempty imputation set. Alonso-Meijide et al. [1] confirmed that the least square nucleolus was a normalized Banzhaf value. Dragan [7] drew the conclusion that any lease square value was the Shapley value of a game obtained from the given game by rescaling. Wang et al. [29] showed the relations between the family of ideal values and raised the center of gravity of imputation set from a new axiomatic angle. Li and Ye [16] introduced a simplified method for computing interval-valued equal surplus division values of interval-valued cooperative games. Béal introduced a new allocation rule called the sequential equal surplus division [3] and provided a strategic implementation of the sequential equal surplus division rule [4]. The study on the equal surplus division in this paper is quite different from many existing researches. We firstly propose again the equal surplus division value based on the least square method and players’ excess vector, which is called players’ common excess vector to differentiate from players’ contribution excess vector. Then, the weighted equal surplus division value is proposed. If the equal surplus division value satisfies the property of individual rationality, it can be regarded as one class of the least square nucleolus, and if not, as one class of the least square prenucleolus. Inspired by the equal surplus division value, another similar solution called the equal contribution division value is proposed. The equal contribution division value is introduced based on players’ marginal contributions. Taking into account the influence of players’ weights on the distribution strategy, the weighted equal contribution division value is put forward. To ensure that the four solutions proposed in this paper satisfy the property of individual rationality, one effective algorithm is described. It is clear that the four solutions proposed in this paper have many advantages against other similar solutions for TU games. For example, it is not necessary to know in advance all the values of sub-coalitions. The equal surplus division values is a good example, we just need to forecast the profits when they operate their businesses by themselves and the profit the grand cooperative coalition will yield. For the logistics service industry, some sub-coalitions can not be formed or fail to satisfy the superadditivity due to the constraints of all kinds of realistic conditions, such as the mismatch between the vehicle types and the freight volume and the incompatibility of business patterns. Even if all subcoalitions can be formed and satisfy the superadditivity, it is still difficult to forecast all the values sub-coalitions, especially when the number of players is large. Obviously, the larger the number of players is, the more highlighted the advantages of the four solutions proposed in this paper will be. This work’s main contributions are stated as follows:

(i) We validate the relationship between the equal surplus division value and the least square value and propose again the equal surplus division value based on the least square method and players’ common excess vector. Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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3

Table 1 Positioning and main contributions of recent state-of-the-art references and this work. Reference

Wang et al. (2019) Hu (2019) Kongo (2018) Li and Ye (2018) Hu and Li (2018) Zhao and Liu (2018) Zhao and Liu (2018) Li and Ye (2018) [16] Chen (2017) [6] Li and Liu (2017) Béal (2017) Li (2016) Béal (2015) AlonsoMeijide et al. (2015) This paper

Issue addressed

Showed the relations between the family of ideal values

Topic related Excess vector

Least square

●

●

● ● ●

Proposed axiomatic result of the equal surplus division value Introduced two axioms on players’ nullification Developed the least square prenucleolus of interval-valued cooperative games Introduced a new axiomatization of a class of equal surplus division values Developed the triangular fuzzy least square value

Equal surplus division value

●

● ●

●

●

Expanded the least square value to interval-valued cooperative games Exploited the interval-valued equal surplus division value

●

●

Studied the extension of the core

●

Proposed the interval-valued least square value with fuzzy coalitions Provided a strategic implementation of the sequential equal surplus division rule Expanded the least square value to interval-valued cooperative games Introduced a new allocation rule called the sequential equal surplus division Proved that the least square nucleolus is a normalized Banzhaf value

●

●

●

Prove that the (weighted) equal surplus division value is a least square value and propose the (weighted) equal contribution division value based on players’ contributions

●

●

●

● ●

●

● ●

●

(ii) Inspired by the equal surplus division value, a similar solution called the equal contribution division value is exploit. Furthermore, we introduce the weighted equal surplus division value and the weighted equal contribution division value. (iii) All the solutions proposed in this paper can be applied to address the TU games where some sub-coalitions can not be formed or fail to satisfy the superadditivity. The incompleteness of the values of sub-coalitions commonly exists in many fields, especially when the coalition is large. The logistics service industry is a good example. In these situations, some classic solutions appear powerless, such as the Shapley value [25], the least square (pre-) nucleolus [20], and the Banzhaf value [19]. Table 1 gives a quick overview of the positioning and main contributions of recent state-of-the-art references and this work. The remainder of this paper is organized as follows. In Section 2, some preliminary knowledge are introduced. Section 3 develops four solutions for TU games, where some sub-coalitions can not be formed or fail to satisfy the superadditivity. Furthermore, some properties of them are described and one effective algorithm is generated to make them satisfy the individual rationality. In Section 4, an example about the profit distribution strategy of logistics enterprise coalitions is illustrated and some comparisons are conducted with previous results. Section 5 is the conclusions and some expectation. 2. Preliminaries 2.1. TU games A TU game is denoted by a pair G = (N, υ ) in which N = {1, 2, . . . n} is a finite set of n players and υ : 2N → R, the characteristic function, is a function from the power set of N to real numbers. The grand coalition containing all players is expressed as N. 2N and GN are usually used to represent the family of all subsets of N and the vector space of 2n − 1 dimension, respectively. What is more, 2N \∅ denotes the family of all nonempty sub-coalitions of N. For an arbitrary coalition S (S⊆N), s (s = 1, 2, . . . , n) represents its cardinality and υ (S ) stands for the profit which it can obtain when all the players in it work together. In particular, υ (∅ ) = 0 and υ (S ) can be written as υ (i ) (i = 1, 2, . . . , n) for short when s = 1. Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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An arbitrary vector x=(x(1 ),x(2 ), . . . x(n ))T could be called a payoff vector for TU games without any constraint. An ordinary payoff vector usually fail to satisfy some excellent properties, such as the efficiency and the individually rationality.  x(i) (i = 1, 2, . . . , n) denotes the payoff that player i can obtain from coalition N and x(S ) = x(i ) (S⊆N) means the sum of i∈S

the payoffs of all players in coalition S. Generally, a payoff vector x is regarded as an efficient one or a preimputation as long as x(N ) = υ (N ) and an imputation if it satisfies x(i ) ≥ υ (i ) (i ∈ N) except x(N ) = υ (N ). Furthermore, TU games meet some important and frequently-used operations. For example, (υ1 + υ2 )(S ) = υ1 (S ) + υ2 (S ) and (αυ )(S ) = αυ (S ) (S⊆N), for any υ1 , υ2 , υ in GN and α ∈ R. 2.2. The concept of players’ excess vector Ruiz et al. [20] proposed the concept of coalitions’ excess vector as follows. For any payoff vector x ∈ Rn and any coalition S (S⊆N\∅), e(S, x ) := υ (S ) − x(S ) denotes the excess of coalition S on the resulting payoff vector x. θ (x ) := (e(S, x ))S⊆N is regarded as the excess vector of all coalitions S (S⊆N\∅). e(S, x) is usually considered as a scale of its dissatisfaction once the payoff vector x is determined as the consequent payoff vector. It is clear that the greater e(S, x) is, the more unfairly treated coalition S will feel. Referring to the coalitions’ excess vector proposed by Ruiz et al., in this paper, we introduce players’ common excess vector and players’ contribution excess vector, respectively. Based on them, we construct some new and practical quadratic programming models for addressing those TU games in which some sub-coalitions can not be formed or fail to satisfy the superadditivity. What follows is the concept of players’ common excess vector. For any payoff vector x ∈ Rn and any player i (i = 1, 2, . . . , n), e(i, x ) := υ (i ) − x(i ) denotes the common excess of player i on the resulting payoff vector x. ϕ (x ) := (e(i, x ))S⊆N is regarded as the common excess vector of player i. e(i, x) is usually considered as a scale of its dissatisfaction when the payoff vector x is determined as the consequent payoff vector. Similarly, for any payoff vector x ∈ Rn and any player i (i = 1, 2, . . . , n), eC (i, x ) := (υ (N ) − υ (N\i )) − x(i ) denotes the contribution excess of player i on the resulting payoff vector x, where (υ (N ) − υ (N\i )) means the contribution of player i to coalition N. ϕC (x ) := (eC (i, x ))S⊆N is regarded as the contribution excess vector of player i. eC (i, x) is usually considered as a scale of its dissatisfaction once the payoff vector x is determined as the consequent payoff vector. Through observing carefully the concepts and the interpretation of both players’ common excess vector and contribution excess vector, the conclusion can be drawn that the greater e(i, x) or eC (i, x) is, the more unfairly treated player i (i = 1, 2, . . . , n) will feel. 3. Novel equal division values The equal surplus division value, i.e., the center of the imputation set (CIS-vector) [8] of for TU games is expressed by

CISi (υ ) := υ (i ) + n−1 (υ (N ) −



υ ( j )) (i = 1, 2, . . . , n).

(1)

j∈N

CISi (υ ) is the unique solution which meets the equal treatment, efficiency and coalitional surplus equivalence (or monotonicity). The equal surplus division value allocates the cooperative profit based on each player’s productivity and the egalitarian principle. That is to say, player i (i = 1, 2, . . . , n) should first be allocated what he can obtain by itself. Then the cooperative  surplus (i.e., υ (N ) − υ ( j )) should be divided equally among all players who join the coalition. j∈N

In the following, we construct a quadratic programming model and generate again the equal surplus division value based on players’ common excess vector and the least square method. This successful attempt proves that seeking for novel solutions for TU games based on players’ excess vector and the least square method is feasible and significative. 3.1. The equal division values based on players’ common excess vector 3.1.1. The equal surplus division value To seek for the payoff vector where the square sum of the consequent excesses reaches the minimum, Problem 1 is constructed.  2 ( υ ( i ) − xE ( i ) ) Problem 1. Minimize i∈N  E s.t. x ( i ) = υ ( N ). i∈N

The Lagrangian of Problem 1 can be expressed by



L x ,λ E

E



=

   2 E E x (i ) − υ (N ) υ (i ) − x (i ) + λ

 i∈N

E

i∈N

Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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Except the constraint equation

i∈N

5

xE (i ) = υ (N ), another Lagrange condition can be shown as

−2(υ (i ) − xE (i )) + λE = 0 (i ∈ N ).

(2)

(xE∗ (1 ), xE∗ (2 ), . . . , xE∗ (n ))T

xE∗

= represents the payoff vector which satisfies both the two conditions mentioned above. What we should do next is the following mathematical derivation. According to Eq. (2), xE∗ (i ) = υ (i ) − 12 λE∗ , and then



xE∗ (i ) =

i∈N



n 2

υ ( j ) − λE∗ = υ (N ).

j∈N

Therefore,

λE∗ =

2 2 υ ( j ) − υ (N ) n n j∈N

and

 1 1 xE∗ (i ) = υ (i ) − λE∗ = υ (i ) − 2 2

2 2 υ ( j ) − υ (N ) n n

= υ (i ) + n−1 (υ (N ) −



j∈N

υ ( j ) ).

(3)

j∈N

So far, we have obtained the equal surplus division value, which looks entirely the same as the equal surplus division value proposed by Driessen and Funaki [8]. It is clear that the equal surplus division value satisfies some important and excellent properties, such as the uniqueness and efficiency, which was confirmed by Driessen and Funaki [8] and we do not repeat to prove them in this paper. 3.1.2. The weighted equal surplus division value In Section 3.1.1, we just consider the equal surplus division value without regard to players’ weights. However, in some situations, we must take into account the weights of players. The influence of players’ weights on the profit distribution strategy is important. It is clear that the greater the risk or the investment is, the higher the profit will be. Considering the influence of players’ weights on the final distribution strategy, the improved quadratic programming model can be constructed as follows:  1−ω (i ) 2 E  (1−ω ( j )) (υ (i ) − xω (i )) j∈N Problem 2. Minimize i∈N  s.t. xEω (i ) = υ (N ), i∈N

where ω(i) (i ∈ N) denotes the weight of player i. ω (i )  For the sake of brevity,  1(−1− ω ( j )) is replaced with ω (i). Therefore, Problem 2 can be rewritten as follows: j∈N

 i∈N

Problem 3. Minimize

s.t.

2

ω (i )(υ (i ) − xEω (i )) 

i∈N

xEω (i ) = υ (N ).

The Lagrangian of Problem 3 is





L xω , λω = E

E





2 ω (i )(υ (i ) − xEω (i )) + λEω

i∈N



 xω ( i ) − υ ( N ) . E

i∈N

In addition to the constraint equation, another Lagrange condition can be shown as

−2ω (i )(υ (i ) − xEω (i )) + λEω = 0 (i ∈ N ). xE∗

(4)

(xE∗ (1 ), xE∗ (2 ), . . . , xE∗ (n ))T

represents the payoff vector which satisfies both the two conditions mentioned above. ω = ω ω ω 1 E∗ According to Eq. (4), xE∗ ω (i ) = υ (i ) − 2ω (i ) λω , and



xE∗ ω (i ) =

i∈N



υ ( j ) − λE∗ ω

j∈N

Therefore,

j∈N



λω =  E∗

j∈N



1



1 2ω  ( j )

j∈N

1 = υ ( N ). 2ω  ( j )

υ ( j ) − υ (N )

Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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and

 xω ( i ) = υ ( i ) − E∗

1

λ = 2ω  ( i ) ω E∗

ω  (i )

υ (i ) +

−1



1

j∈N

ω ( j )

(υ (N ) −



υ ( j ) ).

(5)

j∈N

So far, we have obtained the weighted equal surplus division value. Compared to the equal surplus division value, the  large difference is that for the latter, the cooperative surplus (i.e., υ (N ) − υ ( j )) is divided based on players’ weights j∈N

instead of being allocated equally. 3.2. The equal division values based on players’ contribution excess vector In Section 3.1, we propose the equal surplus division value and the weighted equal surplus division value based on players’ common excess vector. Both of them can insure that all the players first gain what they can obtain by themselves, and then share the cooperative surplus equally or according to the weights. It is obvious that the (weighted) equal surplus division value just pays attention to the profit each player i (i = 1, 2, . . . , n) can obtain by itself, regardless of other key influence factors. For example, the contribution of player i to coalition N. Players’ contributions sometimes play an important role in the profit distribution strategy. From this perspective, the study on the (weighted) equal contribution division value based on players’ contribution excess vector seems interesting and practicable. 3.2.1. The equal contribution division value Taking into account the influence of players’ contributions on the process of profit distribution, the following quadratic programming model is constructed: 

2

[(υ (N ) − υ (N\i )) − xC (i )] Problem 4. Minimize  C s.t. x ( i ) = υ ( N ). i∈N

i∈N

The Lagrangian of Problem 4 is

L (x , λ ) = C

C



 2

[(υ (N ) − υ (N\i )) − x (i )] + λ C

i∈N

In addition to the constraint equation



 i∈N



C



 x (i ) − υ (N ) C

i∈N

xC (i ) = υ (N ), another Lagrange condition can be shown as

−2 (υ (N ) − υ (N\i )) − xC (i ) + λC = 0 (i ∈ N ). xC∗

(6)

(xC∗ (1 ), xC∗ (2 ), . . . , xC∗ (n ))T

= denotes the payoff vector which satisfies both the two conditions mentioned above. Through some mathematical derivation, the equal contribution division value based on players’ contribution excess vector can be obtained. Based on Eq. (6), xC ∗ (i) (i ∈ N) can be written as xC∗ (i ) = (υ (N ) − υ (N\i )) − 12 λC∗ and then



xC∗ (i ) = nυ (N ) −

i∈N



n 2

υ (N\ j ) − λC∗ = υ (N ).

j∈N

It is obvious that

2 n

λC∗ = (n − 1 )υ (N ) − For the sake of brevity, (i ∈ N), and then

2 υ (N\ j ). n j∈N

υ C (i )

denotes the contribution of player i (i ∈ N) to coalition N. Therefore, υ C (i ) = υ (N ) − υ (N\i )



x (i ) = (υ (N ) − υ (N\i )) − ( n − 1 )υ ( N ) −  = 1n υ (N ) − υ (N\i ) + n−1 υ (N \ j ) 1 2

C∗

2 n

j∈N

=

1 n



υ (N ) − (υ (N ) − υ C (i )) + n−1

2 n

 j∈N

υ (N \ j )

  C (υ (N ) − υ ( j )) = υ C (i ) + n−1 (υ (N ) − υ C ( j ) ),

j∈N

(7)

j∈N

which seems the similar as xE ∗ (i) in form. It is pretty clear from Eq. (7) that through the cooperation, player i (i = 1, 2, . . . , n) should first get the profit equal to its  C contribution to coalition N and then the cooperative surplus (i.e., υ (N ) − υ ( j )) should be divided equally among all the j∈N

players who join the cooperative coalition. Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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7

3.2.2. The weighted equal contribution division value To some extent, the equal contribution division value can lead to a feasible distribution strategy. However, the influence of players’ weights on the distribution strategy can not be underestimated. What follows is the quadratic programming model for the weighted equal contribution division value. 

2

ω (i )[(υ (N ) − υ (N\i )) − xCω (i )]

i∈N

Problem 5. Minimize

s.t.



i∈N

xCω (i ) = υ (N ),

 1−ω (i ) (1−ω ( j )) .

where ω (i ) =

j∈N

A closer examination reveals that Problem 5 can only be applied to the situation where the sum of players’ contributions is no more than υ (N ). Otherwise, Problem 5 should be rewritten as follows:  i∈N

Problem 6. Minimize



2

ω (i )[(υ (N ) − υ (N\i )) − xωC (i )]

s.t.



i∈N



xωC (i ) = υ (N ).

Taking Problem 6 for example, we give the solving process in detail and obtain the analytical expression of the weighted equal contribution division value. The Lagrangian of Problem 6 is C

C

L(xω , λω ) =





C

2

C

ω (i )[(υ (N ) − υ (N\i )) − xω (i )] + λω



i∈N



C

xω ( i ) − υ ( N )

i∈N

In addition to the constraint equation, another Lagrange condition can be shown as







−2 (υ (N ) − υ (N\i )) − xCω (i ) + λωC = 0 (i ∈ N ).  C∗

 C∗

 C∗

(8)

 C∗

 C∗

xω = (xω (1 ), xω (2 ), . . . , xω (n ))T represents the final payoff vector. According to Eq. (8), xω (i ) = (υ (N ) − υ (N\i )) −  1 λ C∗ , and 2ω ( i ) ω





xωC∗ (i ) = nυ (N ) −

i∈N





υ (N\ j )−λωC∗



j∈N

j∈N

1 = υ ( N ). 2ω ( j )

Therefore, 

λωC∗ =  j∈N

1 1

2ω ( j )

[ ( n − 1 )υ ( N ) −



υ (N\ j )]

j∈N

For the sake of brevity, υ C (i ) denotes the contribution of player i (i ∈ N) to coalition N, and then  C∗

xω ( i ) =

υ (i ) − (ω (i ) C

1

j∈N

ω( j)

 =





υ (i ) + ω (i ) C

)

−1

( n − 1 )υ ( N ) −

1

j∈N

ω( j)



(υ (N ) − υ ( j ))

j∈N

−1 





υ (N ) −



C



υ ( j) C

(9)

j∈N

So far, we have obtained the weighted equal contribution division value. Compared to the equal contribution division  C value, the large difference is that for the latter, the cooperative surplus (i.e., υ (N ) − υ ( j )) is divided based on players’ j∈N

weights instead of being allocated equally. Similarly, the unique solution of Problem 5 can be expressed as follows:



xω ( i ) = υ ( i ) + C∗

C

ω  (i )

−1 



1

j∈N

ω ( j )

υ (N ) −





υ ( j) . C

(10)

j∈N

3.3. Some properties of the proposed equal division values Not only the (weighted) equal surplus division value but also the (weighted) equal contribution division value proposed in this paper has some nice properties. Taking the weighted equal contribution division value for example, some properties are provided and the proofs are given. The properties of other values proposed in this paper are the similar to those of the weighted equal contribution division value. Therefore, they are not given one by one. Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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 C∗  C∗ Theorem 1 (Additivity). For any two TU games υ ∈ Gn and ν ∈ Gn , we have x C∗ ω ( υ + ν ) = x ω ( υ ) + x ω ( ν ).

Proof. Taking Eq. (9) as an example, it is obvious that 

xωC∗ (i )(υ + ν ) =

υ C (i ) + vC (i ) + (ω (i ) 

=

υ (i ) + ω (i ) C







1

ω( j) j∈N −1



1

j∈N

ω( j)

)−1 [υ (N ) + v(N ) − (

 j∈N

(υ (N ) −



υC ( j ) + 

 j∈N

υ ( j ) ) + v (i ) + ω (i ) C

j∈N

C

vC ( j ) )] −1 



1

j∈N

ω( j)



v (N ) −

 v ( j) . C

j∈N



Therefore, xωC∗ (υ + ν ) = xωC∗ (υ ) + xωC∗ (ν ). 



Theorem 2 (Symmetry). If i ∈ N and k ∈ N (i = k) are two symmetric players in a TU game υ ∈ Gn , then xωC∗ (i ) = xωC∗ (k ). Proof. For the players i ∈ N and k ∈ N (i = k), according to Eq. (9),



 C∗

xω ( i ) = υ ( i ) +

ω (i )

C

and

1

j∈N

ω( j)

  C∗

xω ( k ) = υ C ( k ) +

−1 



ω (k )

υ (N ) −



1

j∈N

ω( j)

υ ( j)

j∈N

−1 





C

υ (N ) −



 υC ( j )

j∈N

Due to the assumption that the players i and k are symmetric, it can be deduced that υ C (i ) = υ C (k ) and ω (i ) = ω (k ).   Obviously, xωC∗ (i ) = xωC∗ (k ). 



Theorem 3 (Anonymity). For any permutation σ on the set N and a TU game υ ∈ Gn , then xωC∗ (i )σ = σ # xωC∗ (i ). Proof. It can be easily proved according to Eq. (9) (omitted). 3.4. An improved algorithm not always necessary Generally speaking, the (weighted) equal surplus division value and the (weighted) equal contribution division value can be regarded as imputations if and only if the resulting payoff vector satisfies the property of individual rationality. Taking Problem 5 for example, the weighted equal contribution division value is an imputation if and only if the resulting payoff vector satisfies xCω (i ) ≥ υ (i ) (i ∈ N). Otherwise, it is just called a preimputation or said to be efficient. In order to guarantee the property of individual rationality, Problem 5 should be rewritten as the following form:  Problem 7. Minimize

i∈N

s.t.

2

ω (i )[(υ (N ) − υ (N\i )) − xIC ω (i )] 

i∈N

IC xIC ω (i ) = υ (N ) and xω (i ) ≥ υ (i ) (i ∈ N ).

It is clear that if the solution of Problem 5 is already an imputation, it is in perfect accordance with the solution of Problem 7. However, sometimes, both optimal solutions of Problems 5 and 7 are different from each other. Referring to the algorithm introduced by Ruiz et al. [20], an improved algorithm for Problem 7 is provided: Algorithm 1. Consider an ordered pair (xj (i), Mj ) (i ∈ N; j = 1, 2, . . .), where xj (i) is an efficient payoff vector and Mj is a subset of coalition N determined by the following steps: Step 1: x j (i ) = xCω (i ) (i ∈ N; j = 1); Step 2: M j = {i ∈ N/x j (i ) < 0} ( j = 1, 2, . . .); Step 3: The algorithm process stops once M j = ∅ ( j = 1, 2, . . .) and then xIC ω (i ) = x j (i ). Otherwise, please turn to Step 4;  x j−1 (i ) + (n − m j−1 )−1 x j−1 (k )(∀i ∈ / M j−1 ) k∈M j−1 Step 4: x j (i ) = { (i ∈ N; j = 2, 3, . . .), 0(∀i ∈ M j−1 ) where m j−1 ( j = 2, 3, . . .) denotes the cardinality of M j−1 . After Step 4 is completed, please switch back to Step 2. Obviously, the unique solution of Problem 7 has all the nice properties as that of Problem 5. To be exact, the solution of Problem 7 has the following nice properties: (1) Existence; (2) Uniqueness; (3) Efficiency; (4) Individual rationality; (5) Additivity; (6) Symmetry; (7) Anonymity. The process of proofs is the similar as that of Problem 5. The similar results and properties exist in other solutions proposed in this paper, which are omitted. Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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9

Table 2 Main parameters and their values. Sign

Definition

Unit

Note

Tv D M Pv Ci Cm n Ls S Pf Cf Sf Vi Vc Pb Ct Cc R Q

Available vehicle types Distance between the original and destination cities Estimated annual mileage Prices of vehicles Insurance charge per year Maintenance cost per year Number of tyres Service life of tyres Monthly salary of drivers Price of the fuel Fuel consumption per hundred kilometers Freeway toll standard Interior volume of the vehicles Volume of the carried goods Breakeven point of the route from the original city to the destination city Total transportation cost from the original city to the destination city Transportation cost of the carried goods Average profit rate of this industry Quotation of the carried goods Profit of the carried cargo

— km km yuan yuan yuan — year yuan yuan/L L yuan/km m3 m3 — yuan yuan — yuan yuan

I; II; III 320 320000 for I; 300000 for II; 260000 for III 120000 for I; 260000 for II; 380000 for III 10000 for I; 17000 for II; 22000 for III 8000 for I; 14000 for II; 18000 for III 6 for I; 8 for II; 18 for III 1 7000 per driver According to market price 20 for I; 30 for II; 40 for III 1.54 for I; 1.65 for II; 2 for III 15 for I; 60 for II; 80 for III According to the actual condition 65% when taking II for example Shown as Eq. (11) Cc = Ct Vc /Vi Pb 6% Q = Cc (1 + R ) Shown as Eq. (12)

υ (S )

Note: I, II and III represent the vehicles of 4.2 m, 9.6 m, and 13 m, respectively.

4. Illustrative example Theoretically, the (weighted) equal surplus division value and the (weighted) equal contribution division value proposed in this paper could be applied to address TU games with incomplete information no matter which fields the problems belong to. Taking the logistics service industry as an example, we consider a logistics enterprise coalition consisting of four logistics enterprises, which are called Enterprise 1, Enterprise 2, Enterprise 3, and Enterprise 4, respectively. N = {1, 2, 3, 4} represents the grand cooperative coalition and S (S⊆N) denotes the sub-coalitions contained in coalition N. To better withstand the market risk, expand the potential customers and finally increase the business revenue, the four logistics enterprises determine to form a coalition and work together. The cooperative profit of coalition S (S⊆N) raises sharply owing to the scale merit when the available vehicles well match the freight volume. On the contrary, it is possible that coalition S fails to satisfy the superadditivity if the available vehicle types do not well match the freight volume. Coalitions {12} and {13} are good examples. 4.1. Two necessary and practical assumptions Two necessary and practical assumptions in this example are shown as follows. Assumption 1. It is unpractical that the four logistics enterprises have all the possible vehicles. It is too expensive to temporarily lease the appropriate vehicles even if the freight volume does not well match their own vehicles. Only the vehicle types of 4.2 m, 9.6 m, and 13 m are available in this example. Assumption 2. The cooperation of the transportation from the destination city to the original city is not considered in this example. In fact, it is up to the logistics enterprises themselves. The profit of coalition S (S⊆N) is, obviously, regarded as the coalitional value of TU games for the logistics service industry. To clearly describe the accounting subjects of the logistics service industry, some parameters are introduced as shown in Table 2. The total transportation cost from the original city to the destination city is given by

Ct = Pv

D D D D D D + Ci + Cm + 2100n + 12S + C P + 0 .9 DS f 8M M M Ls M M 100 f f

(11)

Therefore, the profit, i.e., the coalitional value of S (S⊆N) can be shown as

υ (S ) = Q − Ct

(12)

4.2. Basic data and the distribution results from different solutions Table 3 gives some necessary information of all coalitions S (S⊆N). Table 3 shows that coalitions {12} and {13} can not be formed because they fail to satisfy the superadditivity. Therefore, some well-known solutions for TU games, such as the Shapley value and the Banzhaf value, can not be applied. However, Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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J.-C. Liu, W.-J. Zhao and B. Lev et al. / Information Sciences xxx (xxxx) xxx Table 3 Necessary information of all coalitions S (S⊆N). S

Vc

{1} {2} {3} {4} {12}

50 42 38 69 92

{13}

88

{14}

119

{23} {24}

80 111

{34}

107

{123}

130

{124}

161

{134}

157

{234}

149

{1234} 199

Vehicle type 9.6m 9.6m 9.6m 13m 4.2m 13m 4.2m 13m 9.6m 9.6m 13m 9.6m 9.6m 9.6m 9.6m 9.6m 13m 9.6m 9.6m 9.6m 13m 13m 13m 13m 9.6m 9.6m 13m

Ct 1144 1144 1144 1492 902 1492 902 1492 1144 1144 1492 1144 1144 1144 1144 1144 1492 1144 1144 1144 1492 1492 1492 1492 1144 1144 1492

Q

υ (S )

 i∈S

υ (i )

Super- additivity?

Coalition?

1555 1306 1181 2145 2861

411 162 37 653 467

411 162 37 653 573

— — — — No

Yes Yes Yes Yes No

2736

342

448

No

No

3700

1412

1064

Yes

Yes

2487 3451

995 1163

199 815

Yes Yes

Yes Yes

3326

1038

690

Yes

Yes

4042

1406

610

Yes

Yes

5006

1574

1226

Yes

Yes

4881

1897

1101

Yes

Yes

4632

1648

852

Yes

Yes

6187

2407

1263

Yes

Yes

Table 4 Profit distribution results from different solutions for TU games. Player Enterprise Enterprise Enterprise Enterprise

1 2 3 4

Eq. (3)

Eq. (5)

Eq. (7)

Eq. (9)

Equal surplus division value

Shapley value

Banzhaf value

697 448 323 939

793 408 324 882

585 336 659 827

690 312 720 685

697 448 323 939

Useless

Useless

the four equal division values proposed in this paper can effectively address this problem. Generally speaking, the weights of players are usually determined by the risk they take, the amount of investment they input and the technology they master. After adequate discussion, all players agree on the following results: ω (1 ) = 0.46, ω (2 ) = 0.16, ω (3 ) = 0.28, ω (4 ) = 0.1. Table 4 shows the distribution results from different solutions. 4.3. Comparison and discussion 4.3.1. The contrastive analysis of the four solutions proposed in this paper Different solutions proposed in this paper result in different distribution strategies. Most of TU games base the profit distribution strategies either on the egalitarian principle or on the utility principle. The equal surplus division value is mainly based on the egalitarian principle and on the contrary, the equal contribution division value on the utility principle. Furthermore, the weighted equal surplus division value considers the influence of players’ weights, which is actually a reflection of players’ utility. To some extent, it can be regarded as a solution based on both the egalitarian principle and the utility principle. The equal surplus division value (Eq. (3)) actually means each player has the equal weight (1/4). Enterprise 1 can get only 697 as the consequent profit based on Eq. (3). However, once considering players’ weights, the profit of Enterprise 1 will increase to 793 because its weight rises from 1/4 to 0.46. Inversely, the profit of Enterprise 4 will decrease to 882 with its weight dropping from 1/4 to 0.1. The similar conclusions could be drawn for both Enterprise 2 and Enterprise 3. In a word, the more player’s weight drops, the more player’s profit will decrease and vice versa, which is logical and intuitive. Many solutions for TU games, beyond all doubt, are based on the egalitarian principle. However, the utility principle is also widely used. According to Eq. (7), we can obtain a distinct-different distribution strategy from that based on Eq. (3). Taking Enterprise 4 as an example, due to the greatest contribution, it will get the most profit. Once considering players’ weights, its profit will decrease to 685 because its weight drops from 1/4 to 0.1. The similar conclusions are also tenable for other three enterprises. Please cite this article as: J.-C. Liu, W.-J. Zhao and B. Lev et al., Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions, Information Sciences, https://doi.org/10.1016/j.ins.2019.09.019

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4.3.2. The advantages of the proposed solutions against other similar solutions 1) All of them can be used to address many TU games, in which some well-known solutions appear powerless, such as the Shapley value and the Banzhaf value. It is not necessary to know in advance all the values of coalitions S (S⊆N) when using the four proposed solutions. Taking the (weighted) equal surplus division value as an example, we just need to forecast the profits when they operate the businesses by themselves and the profit coalition N will yield. In fact, many sub-coalitions can not be formed or fail to satisfy the superadditivity due to the constraints of all kinds of realistic conditions. The four proposed solutions are appropriate to the these situations. 2) Even if all sub-coalitions can be formed or satisfy the superadditivity, it is still difficult and labor-intensive to forecast all coalitional values, especially when coalition N is large. Obviously, the larger coalition N is, the more highlighted the advantages of the proposed solutions will be. Taking a TU game with 6 players as an example, 63 coalitional values should be known in advance if the Shapley value or the Banzhaf value is applied to determine the profit distribution strategy. However, only 7 and 13 coalitional values are necessary for the (weighted) equal surplus division value and the (weighted) equal contribution division value, respectively. In a word, the four solutions proposed in this paper can be applied to different distribution principles and successfully address the problems of TU games, where some sub-coalitions can not be formed or fail to satisfy the superadditivity. Furthermore, the workload of them is always less than some well-known solutions, such as the Shapley value and the Banzhaf value, especially when coalition N is large. 5. Conclusions and future researches In this paper, we propose four solutions based on players’ common excess vector and contribution excess vector, prove that all of them belong to the family of least square values for TU games and provide some nice properties of them. The (weighted) equal surplus division value based on players’ common excess vector and the (weighted) equal contribution division value based on players’ contribution excess vector usually result in distinct-different distribution strategies. To reach a compromise, the research on the equal division value based on both players’ common excess vector and contribution excess vector seems interesting. What is more, in the near further, maybe we could exploit the (weighted) equal contribution division value based on coalitions’ contribution excess vector, which also belongs to the family of least square values for TU games and has some nice properties. Besides, the classical solutions for TU games are usually extended to the fuzzy environment [17,28] to generate corresponding solutions for fuzzy TU games. Therefore, in the near future, the four solutions proposed in this paper may be extended to fuzzy TU games. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment Support was provided by the Social Science Planning Program of Fujian Province of China(grant no. FJ2018B014) and the Special Foundation Program for Science and Technology Innovation of Fujian Agriculture and Forestry University of China(grant nos. CXZX2018030, KFA17565A, and KCXRC621A). References [1] J.M. Alonso-Meijide, M. Álvarez-Mozos, M.G. Fiestras-Janeiro, The least square nucleolus is a normalized Banzhaf value, Optim. Lett. 9 (7) (2015) 1393–1399. [2] J.F. 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