A unified cooperative model for environmental costs in supply chains: the Shapley value for the linear case
Abstract
This paper provides a normative framework based on cooperative game theory aimed at studying the problem of pollution responsibility allocation across multitier supply chains. The model is further developed with reference to the case of a linear supply chain, by using three responsibility principles (namely: Upstream, Downstream and Local Responsibility). Allocation rules are derived; also, desirable properties in terms of fairness, efficiency and transparency are introduced, in order to characterize such rules. Furthermore, a stability concept for efficient allocations is formulated. An example of a possible application of the introduced cost allocation rules is provided.
Keywords
Supply chains Pollution responsibility allocation Game theory Shapley value1 Introduction
The promotion of environmental sustainability has become crucial in the implementation and daytoday functioning of complex and global supply networks. A growing number of large multinational enterprises are implementing more stringent environmental practices; this has also an impact on small and mediumsized enterprises (SMEs) which are often involved in their supply networks (UN Global 2011).
The promotion of tighter environmental standards also implies the adoption of benchmarking approaches for comparing the sustainability performances of supply chains against industry standards. Within this context, Life Cycle Assessment (LCA) methodologies allow the estimation of cumulative environmental impacts across the entire supply chain and against a wide range of indicators, adopting a full product life cycle perspective. Within the current context of growing environmental sustainability, performing an allocation of environmental impacts to the different actors in the supply chain, with the aim of understanding which of them should be deemed responsible for polluting activities and related taxes or abatement costs, is a relevant and timely issue, if proper mitigation measures need to be implemented.
The interest in the topic of environmental pollution responsibility started due to the need to establish mechanisms to perform pollution burden sharing actions across countries (Zhou and Wang 2016); the related literature has then evolved in order to address allocation problems at different levels. According to an analysis of the current literature (see, for instance, Zhou and Wang 2016), most of the available approaches such problems at a macrolevel; the application of pollution responsibilities approaches to contemporary multitier and multistakeholders supply chains is often overlooked.
This paper aims at addressing this research gap, by providing a normative framework based on cooperative game theory aimed at studying the problem of pollution responsibility allocation across multitier supply chains.
The paper is organised as follows. After a review of the literature concerned with pollution responsibility allocation, and the identification of relevant gaps (Sect. 2), the paper presents a cooperative game theoretical responsibility model for pollution allocation (Sect. 3). In Sect. 4, the paper details the generic model in the case of a linear supply chain. In Sect. 5, a practical application of the introduced cost allocation rules is provided; finally, some conclusions are drawn.
2 Literature review
The problem of associating pollution responsibilities with economic actors traces its roots in the international trade literature (Leontief and Ford 1970; Proops et al. 1993; Wyckoff and Roop 1994; Imura and Moriguchi 1995; Subak 1995; Bosquet 2000; Korhonen 2002; Bastianoni et al. 2004; Lenzen and Murray 2010). While the interest in the topic started because of the need to establish mechanisms to perform pollution burden sharing actions across countries, the related literature has then evolved in order to address allocation problems at different levels. These include the decomposition of national objectives into regional ones and the distribution of environmental pollution responsibility across partner firms in a given supply chain.
One of the fundamental research questions in the field of environmental pollution responsibility has been represented by the allocation principle to be followed. Scholars have developed different allocation criteria that can be mainly categorized into fairness and efficiencyinspired principles. The concept of “fairness” is generally linked to equitable distribution issues (Rose 1990); the one of “efficiency” is more related to the minimization of costs related to abatement and mitigation measures. Many emissions allocation methods have been proposed. According to the extensive review performed by Zhou and Wang (2016), these can be classified into multiple categories. One of them is the socalled indicator approach. Through the development of specific indicators, pollution responsibilities, targets or permits are worked out in relation to these measures. Another stream of studies is devoted to the use of optimization approaches, which is based on mathematical programming frameworks for pollution allocation, which try to minimize costs for pollution abatements. For example, Ridgley (1996) combed the use of composite indicators and an optimization method for deriving suitable pollution responsibility allocations at a country level. This rapid scan of the extant literature, coherently with findings from Zhou and Wang (2016), highlights that most of the developed approaches deal with the pollution responsibility problem at a macrolevel. Indeed, most of the methodologies tackle the problem at a country or regional level.
While some firmlevel approaches can be retrieved, the supply chain perspective has been, so far, largely ignored, with the few available approaches being characterized by very simple and naïve models, mainly dealing with dyadic configurations; the application of pollution responsibilities approaches to contemporary multitier and multistakeholders supply chains is often overlooked.
Pollution allocation mechanisms often include interactions involving multiple actors. As such, pollution allocation mechanisms may be represented in a very effective way through approaches based on Game Theory. As stated by Zhou and Wang (2016), game theoretic approaches seem to be naturally suited for this purpose. Such allocations result from equilibria among different parties’ interactions or from values, which may satisfy enjoyable properties imposed by a central authority. Compared to the traditional approaches for pollution costs allocation, game theoretic approaches can be deemed as more sophisticated and less immediate. The use of game theoretic approaches is underexploited, especially when dealing with complex and multitier supply chains. Concepts from cooperative game theory may assume a normative value, i.e. a value that is imposed by an external authority.
Chander and Tulkens (1995) and Filar and Gaertner (1997) developed the first adaptation of Game Theory to the problem in order to investigate the allocation of GHGs emission reductions quota among countries. A similar approach, at an international level, was developed by Eyckmans and Tulkens (2003), while Germain and Steenberghe (2003) utilized a dynamic game framework. Viguier et al. (2006) deployed a twolevel game for solving a similar allocation problem. At a firm level, MacKenzie et al. (2008, 2009) utilized, respectively, rankorder contests and incomplete information games for allocating pollution allowances. Chung et al. (2013) utilized dynamic games to evaluate the response of companies to environmental pollution taxes in a spatially distributed supply chain. Ren et al. (2015) proposed a Stackelberg game for studying CO2 reduction targets in a very simple dyadic supply chain.
Surprisingly, the development of normative approaches to pollution cost allocation is less extensive. A practical applications of the Shapley value approach was developed by Liao et al. (2015) for allocating emission allowances across energy producers in Shanghai.
2.1 Contribution of the paper
The next section presents a unified cooperative game theoretical approach to pollution responsibility allocation in a multitier supply chain. In Sect. 4, the paper details the generic model in the case of a linear supply chain, by selecting three different responsibility principles (namely: Upstream, Downstream and Local Responsibility). Also, fairness, efficiency and transparency properties are introduced, as desirable characteristics for allocation rules; furthermore, a stability concept for efficient allocations is formulated. In particular, here we present the Shapley value for this unified cooperative game theoretical responsibility model. In doing this, we find interesting connections between the socalled river problem introduced by Dong et al. (2012) in the pollution games literature and our framework. The Shapley allocation satisfies many of the mentioned desirable properties.
In Sect. 5, a practical application of the introduced cost allocation rules is provided, with the reference to the context of dairy supply chains. Finally, some conclusions are drawn.
3 Cooperative responsibility models for environmental supply chains
In a supply chain context, it can be fair assuming that companies cooperate (by forming a grand coalition) in order to coordinate the activities of the production system. This way, they will incur in different individual production costs because of initial binding agreements among partners; also, they will face costs related to the mitigation of the environmental impacts of their production activities. Cooperative game theory can provide solutions to allocate environmental pollution impacts across each member of the production system, by allowing transferable payments among companies.
In this study, the main aim is to identify a vector \( x = \left( {x_{i} } \right) \in R_{ + }^{\left N \right} \) representing a solution to the cooperative model (2), which assigns the cost \( x_{i} \) to each company \( i \).
An allocation is responsiblecompatible when, if \( B_{i} \) is empty, then \( {\text{x}}_{\text{i}} \) is null. In the following, we will always refer to allocations which respect the responsiblecompatible principle.
An allocation is efficient if the sum of all costallocations, i.e. \( \sum\nolimits_{{{\text{i}} = 1}}^{{\left {\text{N}} \right}} {x_{i} } \), is equal to the sum of all costs, i.e. \( \sum\nolimits_{i = 1}^{\left N \right} {\sum\nolimits_{j = 1}^{\left N \right} {c_{ij} } } \).^{2}
3.1 Desirable properties for cost allocation rules
Cost allocation rules should respect some desirable properties that have been described in the environmental pollution responsibility literature. In the following, we list and discuss each of these properties; we remark that some of these have been already listed in Gopalakrishnan et al. (2016), while and some of them are novel.
Property 1: Equivalence
If two companies \( i \) and \( j \) are responsible for the same processes, then they must have the same cost allocation. Therefore, if \( B_{i} = B_{j} \) then \( x_{i} = x_{j} \) must be satisfied for any \( i \) and \( j \in N. \)
Property 2: Equal sharing of extra pollution
This property provides fairness to the allocation of extra costs that might arise as the result of additional polluting activities (that could happen over time). If there is an increase in total pollution costs, it is required that companies responsible for this increase should be equally affected by extra burdens.
As a preliminary definition, it can be stated that C′ ≥ C if and only if c_{ij}′ ≥ c_{ij} for any \( i, j \in N \). We say that C′ > C if C′ ≥ C and there exists a process \( \left( {i, j} \right) \in P \) such that c_{ij}′ > c_{ij} Let \( G = \left( {N, P, B, C, v} \right) \), \( G^{\prime} = \left( {N, P,B,C^{\prime},v} \right) \) be the two associated cooperative models where \( {\text{C}} \) and C′ are two sets of environmental costs with C′ ≥ C. If \( i, j \in N \) are responsible for any process \( p \), that is \( b_{i, p} = b_{j,p} = 1 \) where \( p \in P \) is characterized by c_{p}′ > c_{p}. Then the following equality \( x_{i} \left( {G^{\prime}} \right)  x_{i} \left( G \right) = x_{j} \left( {G^{\prime}} \right)  x_{j} \left( G \right) \) must be satisfied.
Property 3: No free riding
The equal sharing of extra pollution property (introduced above as Property 2 ) does not guarantee the efficiency of such net cost increase reallocation. In other words, the property does not require that the sum of extra charges for responsible companies is equal to the net increase of the pollution costs across the supply chain. As such, Property 3 provides an additional efficiency requirement to the abovementioned situation. This property requires that if the total pollution costs increase, but, at the same time, for some firms the pollution of the processes they are responsible for remains unchanged, the allocation for these firms should remain the same.
Let \( G = \left( {N, P, B, C, v} \right) \) and \( G^{\prime} = \left( {N,P,B,C^{\prime},v} \right) \) be the two associated cooperative models where \( {\text{C}} \) , C′ are two sets of environmental costs with C′ ≥ C. Let \( i \in N \) ; suppose that \( c_{p}^{\prime } = c_{p} \) for each process \( p \) satisfying \( b_{i, p} = 1 \) . Then, the equality \( x_{i } \left( {G^{\prime}} \right) = x_{i} \left( G \right) \) must be satisfied.
Property 4: Process independence
Suppose that a single process \( p \) increases its pollution cost \( c_{p} \) ; as a consequence, cost allocations across the supply chain will change. In other words, cost allocations might be affected by the new cost distribution over the set of all processes even if just one single process increases its cost. If this does not happen, it means that the variation of cost allocations solely depends on the process whose cost has increased; then, we say that cost allocations are process independent. As a consequence, this property implies that the firms can transparently relate their cost variations to the processes with increased costs; as such, they may make an official grievance against firms involved in those more costly processes.
Let \( G = \Big( {N, P, B, C, v} \Big) \),\( G^{\prime} = \Big( {N, P, B,C^{\prime},v} \Big) \),\( \tilde{G} = \Big( {N,P,B,\tilde{C},v} \Big) \),\( \tilde{G}^{\prime} = \Big( N,P,B,\tilde{C^{\prime}},v \Big) \) be the cooperative models associated to the cost structures \( C,C^{\prime},\tilde{C},\tilde{C}^{\prime}. \) We assume that \( C^{\prime} \ge C \), \( \tilde{C}^{\prime} \ge \tilde{C} \) and \( c_{p}^{\prime } = c_{p} \), \( \tilde{c}_{p}^{\prime } = \tilde{c}_{p} \) for \( p \in M{ \setminus } \left\{ q \right\} \). We assume that \( \tilde{c}_{q}^{\prime } = c_{q}^{\prime } \) and \( c_{q} = {\tilde{\text{c}}}_{q} \). Then, \( x_{i} \left( {G^{\prime}} \right)  x_{i} \left( G \right) = x_{i} \left( {\tilde{G^{\prime}}} \right)  x_{i} \left( {\tilde{G}} \right) \) must hold for each \( i \in N \).
Property 5: (Weak) unilateral disaggregation stability
Lenzen et al. (2007) and Rodrigues and Domingos (2008) originally discussed the stability of pollution responsibility allocations due to demerging of companies over supply chains. Let \( G = \left( {N, P, B, C, v} \right) \) be our cooperative model. As a preliminary step, we formalize disaggregated supply chains with reference to the original model.
Let \( P_{ } \left( i \right) \) \( \{ j \in N \left( {i, j} \right) \in P\} \) and \( R_{ } \left( i \right) \) \( \{ j \in N \left( {j, i} \right) \in P\} \) be the subsets including firms, who are supplied by \( i \) or supply firm \( i \), respectively. Similarly, let \( P_{i} \) and \( R_{i} \) be the subsets including processes in which firm \( i \) is a supplier or firm \( i \) is supplied by, respectively. Assume that firm \( i \) disaggregates its activities into two companies \( i_{1} \) and \( i_{2} \). After disaggregation, the set of firms is modified as \( N^{\prime} = \left\{ {i_{1} , i_{2} } \right\} \cup N\backslash \left\{ i \right\} \). Then, the new set of processes is composed by \( P^{\prime} = R\left( i \right) \times \left\{ {i_{1} } \right\} \cup \left\{ {i_{2} } \right\} \times P\left( i \right) \cup (i_{1} ,i_{2} ) \cup P\backslash (P_{i} \cup R_{i} ) \). The new set of environmental costs \( C^{\prime} \) is composed as it follows. First of all, we define \( c_{{ji_{1} }}^{\prime } : = c_{ji} \) for \( j \in R\left( i \right) \) and \( c_{{i_{2} j }}^{\prime } ,c_{{i_{1} i_{2} }}^{\prime } \ge 0 \) where \( c_{ij} : = c_{{i_{2} j}}^{\prime } + c_{{i_{1} i_{2} }}^{\prime } \) for each \( j \in P\left( i \right) \). The last equality is named costsplitting property. For the remaining cases, we simply require that \( c_{hk }^{\prime } : = c_{hk} . \) It is straightforward to think that disaggregated responsibility situations need a new responsibility matrix, that is \( B^{\prime}. \) First of all, such a new responsibility matrix \( B^{\prime} \) must have N + 1 rows, that is the new number of firms, and P + 1 columns, that is the new number of processes. Such a new responsibility matrix might be defined in several ways. We require some reasonable assumption on it. Let \( h \) be a firm participating in the original game and in the disaggregated game. Then it is immediate to see that \( h \) cannot be firms \( i_{1} ,i_{2} ,i \). For such a firm, we require some consistency property related to the original responsibility matrix \( B \). Firstly, \( B_{hp} = 1 \) with \( p = \left( {{\text{i}}, {\text{j}}} \right) ^{ }_{ } \) implies \( B_{{{\text{h}}p^{\prime}}}^{\prime } = 1 \) with \( p^{\prime} = \left( {i_{2} ,{\text{j}}} \right) \); \( B_{hp} = 1 \) with \( p = \left( {{\text{j}}, {\text{i}}} \right) ^{ }_{ } \) implies \( B_{{{\text{h}}p^{\prime}}}^{\prime } = 1 \) with \( p^{\prime} = \left( {j,i_{1} } \right) \). Finally \( B_{{{\text{h}}p}}^{\prime } = B_{{{\text{h}}p}} \) holds if the process \( p \) does not contain \( i_{1} ,i_{2} \).
The above conditions ensure that, if a firm is responsible for processes involving the firm \( i \) in the original game, is also responsible for associated processes involving \( i_{1} ,i_{2} \) after the disaggregation. The latter means that the disaggregation does not create externalities on others’ responsibilities for processes. In addition, we require that \( \left( {i_{1} ,i_{2} } \right) \notin B^{\prime}_{{i_{1} }} \cap B^{\prime}_{{i_{2} }} \); this means that, at most, one of the new companies is responsible for the internal process \( \left( {i_{1} ,i_{2} } \right), \) due to the disaggregation. If \( B^{\prime} \) satisfies both conditions we say that a responsibility matrix \( B^{\prime} \) is \( B^{ } \)compatible.
We have defined disaggregated models \( G^{\prime} = \left( {N^{\prime}, P^{\prime}, B^{\prime}, C^{\prime}, v} \right) \) from the original model \( {\text{G}} = \left( {N^{ } , P^{ } , B^{ } , C^{ } ,v} \right) \). The multiplicity of disaggregated models depends on the multiple ways to split costs and/or on the multiple ways to implement new pollution responsibilities over supply chains.
Within this context, we say that \( {\text{G}} = \left( {N^{ } , P^{ } , B^{ } , C^{ } , v} \right)^{ } \) satisfies the unilateral disaggregation stability property if there exists a \( B \)compatible responsibility matrix \( B^{\prime} \) and, then, there exists a firm \( i \) such that \( x_{{i_{1} }} \left( {G^{\prime}} \right) + x_{{i_{2} }} \left( {G^{\prime}} \right) > x_{i} \left( G \right) \) for any disaggregated model \( G^{\prime} = \left( {N^{\prime}, P^{\prime}, B^{\prime}, C^{\prime}, v} \right) \) (Unilateral Disaggregation Stability). We say that firm \( i \) is the aggregating company and \( B^{\prime} \) is the aggregating responsibility matrix. We say that \( G^{ } \) satisfies weak unilateral disaggregation stability if \( x_{j} \left( G \right) \ge x_{j} \left( {G^{\prime}} \right) \) (Weak Unilateral Disaggregation Stability) for each company \( j \) different from i.
The last formula implies that \( x_{h} \left( G \right) > x_{h} \left( {G^{\prime}} \right) \) for some companies \( h. \) This means that there exists, at least, one firm who might consider the disaggregation option as a profitable one. If each company can get a benefit from disaggregation, then the whole supply chain can be considered as weakly stable, even if the aggregating firm prefers not to disaggregate its activities. In addition, if the cost of original game is strictly superior to the cost of the disaggregated game, there are internal incentives along supply chains to cause disaggregation. Even if cost allocations satisfy the unilateral disaggregation stability property, internal resources might still provide incentives to companies, which do not directly demerge, to collude and cause the disaggregation of the aggregating firm.
3.2 Additional remarks
The first two properties (Equivalence and Equal Sharing of Extra Pollution) respect fairness by definition. It must be highlighted that, according to an egalitarianism principle, each company across a supply chain is granted the same right to pollute (Zhou and Wang 2016). As such, the total pollution across a supply chain might be increased due to unilateral actions from individual partners. However, in this case, an egalitarian perspective does not necessarily imply distributive justice concerning pollution responsibility allocation deriving from extra pollution (Rose 1990). Within this context, the equal sharing of extra pollution, no free riding and process independence properties provide additional commitment to substantive distributive justice.
4 Linear supply chains and pollution responsibility
Here, we specify our model \( \left( {N, P, B, C, v} \right) \) in (2) where \( \upsilon \) has been already defined in (1). The specifications deal with the structure of the process set \( P \) and of the responsibility matrices \( B \). A generic process \( \left( {i, j} \right) \) represents the production of goods by company \( i \) and the supply of such goods to company \( j \). In the linear supply chain, the set \( P_{i} \) is reduced to a singleton for each company, at most. This implies that company \( i \) is supplying goods or services to a single supply chain partner. Then a process \( \left( {i, j} \right) \) can be identified as an element of the set of companies. In addition, each process has an environmental cost \( c_{i,i + 1}^{ } \), which can be simply expressed by \( c_{i } \). In the context, we can assume that existence of a final company N + 1, which is the company such that \( P_{\left N \right + 1} = \emptyset . \) From company N, a process starts and supplies company N + 1. We assume that the final company has no responsibility, that is characterized by the following assumption \( B_{\left N \right + 1} = \emptyset . \) Because allocations are responsible compatible by our initial choice, we know that \( x_{\left N \right + 1} = 0 \). Then, we disregard the last company \( \left N \right + 1 _{ } \) because its cost allocation is null. Therefore the number of processes \( \left P \right_{ } \) is equal to the number of pertinent firms is \( \left N \right = n_{ } \).
Here, we define how the responsibility matrix \( B_{ } \) is characterized according to environmental supply chain philosophies. In this context, three principles can be employed to solve the allocation problem. A Local Responsibility principle (LR), according to which each company \( i \) is strictly responsible for the pollution costs, related to the production activities strictly happening at its premises. An Upstream Responsibility principle (UR), stating that upstream suppliers (dealing with raw material extraction, subcomponent manufacturing and other energy intensive activities) are responsible not only for pollution happening at their premises, but can also influence the environmental performance of downstream partners. A Downstream Responsibility principle (DR), stating that downstream partners in the supply chain are responsible for the polluting activities happening at upstream suppliers’ premises.
5 The Shapley allocation for linear supply chains
This particular allocation can be called Downstream Equal Sharing (DES) allocation method. Here, we verify if LRS, DES and UES allocation rules satisfy the desirable properties introduced in Sect. 3.
Proposition 1
LRS, DES and UES allocation rules satisfy the equivalence property.
Proposition 2
LRS, DES and UES allocation rules satisfy the equal sharing of extra pollution property.
Proposition 3
LRS, DES and UES allocation rules satisfy the no free riding property.
Proposition 4
LRS, DES, UES cost allocations satisfy the process independence property.
Because the Shapley value is an efficient allocation, then UES cost allocations are efficient.
Proposition 5
LRS allocation rule does not satisfy the unilateral disaggregation stability property. We assume that \( c_{n  1 } \ne 0 \) and \( n > 2. \) Then, UES allocation rule satisfies unilateral disaggregation stability property. UES allocation rule satisfies weak unilateral disaggregation stability.
6 Applications

Feed producer, producing animal feed from raw agricultural products;

Milk Producer, managing lactating animals (generally cows) and obtaining raw milk;

Processor, performing important operations for converting milk into a product fit for human consumption (heat treatments and pasteurization) and transforming it into a final product to be sold onto the market (also taking care of the packaging element);

Distributor, taking care of the logistical processes throughout the supply chain, involving cold chain operations;

Retailer, making the product available to the final consumer.

Consumers, buying and consuming the final product.
Environmental impacts across a typical dairy supply chain
Emissions (Kg CO_{2}eq per Litre)  

Feed producer  0.284 
Milk producer  0.727 
Processor  0.129 
Distributor  0.108 
Retailer  0.091 
Final consumer  0.000 
Total  1.337 
Environmental impact allocation according to LRS, UES, DES rules
Stakeholder  Emissions share (%)  LRS (%)  DES (%)  UES (%) 

Feed producer  21.21  21.24  3.54  55.03 
Milk producer  54.34  54.38  14.42  33.78 
Processor  9.61  9.65  16.83  6.60 
Distributor  8.05  8.08  19.52  3.38 
Retailer  6.79  6.81  22.92  1.36 
Consumers  0.00  0.00  22.92  0.00 
Total  100.00  100.00  100.00  100.00 
Hybrid Allocation Rules
Stakeholder  Emissions share (%)  LRS + DES (50–50) (%)  LRS + DES (80–20) (%) 

Feed producer  21.21  12.39  17.70 
Milk producer  54.34  34.40  46.38 
Processor  9.61  13.24  11.08 
Distributor  8.05  13.80  10.37 
Retailer  6.79  14.86  10.03 
Consumers  0  11.46  4.58 
7 Conclusions and future works
This study shed light on normative frameworks for the problem of pollution responsibility allocation across multitier supply chains through cooperative game theory concepts. After a thorough literature review, the paper has presented a cooperative game theoretical responsibility model for pollution allocation; the paper has detailed the generic model in the case of a linear supply chain, by selecting three different responsibility principles (namely: Upstream, Downstream and Local Responsibility) and developing some associated pollution responsibility allocation rules. These rules have been tested against some desirable properties that have been proposed. Furthermore, a practical application of the introduced cost allocation rules has been provided in the context of dairy supply chains.
The work proposed in this paper can be further extended in future researches, in order to overcome some of the limitations that characterize the current approaches. First of all, the model has been implemented on a simple linear supply chain, in which each company supplies and receives goods from a single partner; future work could extend the framework to more complex supply networks, in which multiple companies operate at each tier. Also, different pollution responsibility schemes and different supply chains structures (including, for instance, reverse elements, such as theorized by Jacobs and Subramanian 2012) might be employed in our model. A new set of normative properties could be defined, with the specific aim to assess the stability of international supply chains, especially against pollution costs differently perceived and taxed in nonhomogeneous countries.
Footnotes
 1.
The environmental cost function can be also defined as \( v\left( S \right) = \sum\nolimits_{s \in S} {\sum\nolimits_{{\left( {i,j} \right) \in B_{s} /{ \sqcup }_{t = 1}^{s  1} B_{t} }} {c_{ij} } } \) in an equivalent way.
 2.
We will denote an allocation \( x_{i} \) of the cooperative model \( G \) by \( x \left( G \right) = \left( {x_{i} \left( G \right)} \right) \in R_{ + }^{\left N \right} \).
 3.
Notes
Acknowledgements
This research was partially supported by the project “Promoting Sustainable Freight Transport in Urban Contexts: Policy and DecisionMaking Approaches (ProSFeT)”, funded by the H2020MSCARISE2016 programme (Grant Number: 734909). The authors are extremely grateful to the two anonymous referees; their very useful observations helped us lifting the quality of our paper in a very significant way.
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