This study investigates a multiowner maximumflow network problem, which suffers from risky events. Uncertain conditions effect on proper estimation and ignoring them may mislead decision makers by overestimation. A key question is how selfgoverning owners in the network can cooperate with each other to maintain a reliable flow. Hence, the question is answered by providing a mathematical programming model based on applying the triangular reliability function in the decentralized networks. The proposed method concentrates on multiowner networks which suffer from risky time, cost, and capacity parameters for each network’s arcs. Some cooperative game methods such as
Continuous development of technology in petrochemical industries, automobile manufacturing, water distribution networks, electricity industries, and transportation networks has created complex, competitive, and decentralized environments for suppliers [
In the design of suitable systems in decentralized networks, quality survival and reliability in decisionmakings with nondeterministic circumstances are highly important. Since any failure in the network affects noticeably customer services and causes heavy costs to suppliers, increasing network reliability and network performance is extremely essential. One of the key tools to improve network performance in decentralized networks is game theory and cooperative games that can create optimal strategies for suppliers [
In uncertain circumstances, we have focused on real values of decision parameters such as time, cost, and capacity which may deploy from uncertain patterns. Such uncertain circumstances may arise due to various sources of fluctuations in supply/demand patterns, politics, traffic, natural disasters, war, falling debris, dropping voltage, and so on.
Naturally the effects of such unforeseen events may mislead decision makers. Accordingly, uncertainty decisionmaking is an important issue in today’s competitive world. In this regard through this research, a novel mathematical programming model based on triangular probability distribution in nondeterministic decentralized systems is proposed to solve maximumflow problem. Applying reliability function for triangular density function helped us to estimate a more reliable value for uncertain parameters of traveling time, associated transportation cost, and the actual amount of displacement capacity according to experts’ subjective comments.
The reminder of the paper is organized as follows: Literature review and research gap in decentralized networks are presented in Section
The main related issues to the maximumflow problems such as logistic network, cooperative game theory, and reliability have been discussed and the research gap is given as follows.
Logistic network model is one of the most important models in mathematical programming and operations research and includes network planning and inventory control, production planning, planning and project control, facilities location, and many other applications. Moreover, it can be used in airlines, railways, roads, pipelines, and so forth. Besides, logistic network is one of the main issues in the maximumflow problems [
The purpose of the maximumflow problem in the network is to reach the highest amount of transportation flow from the initial node to the terminal node by considering the capacity of the arcs.
In the last decades, the logistic costs in distribution networks have been increased dramatically due to a noticeable raise in customers’ expectations. To reduce the costs, various game theory methods and horizontal cooperative games are used in the logistic networks in which the horizontal cooperative games have led to getting higher payoff because of cooperation between companies [
Vanovermeire and Sörensen [
Saha et al. [
Networks are often controlled by multiple owners. As a case in point, gas pipeline, which is an international system, has established an integrated network in Europe. In this case, each country controls some parts of the distribution network and in fact a cooperative game in the network is built [
Charles and Hansen [
Zibaei et al. [
Zhao et al. [
One of the most important aspects of reliability is network reliability. The network reliability is defined as a capability or probability that a network system has to completely fulfill customertailored communications tasks during the stipulated successive operation procedure [
In the last decade, the reliability of the transport network and power distribution systems had been widely considered. The experience of incidents such as the Kobe earthquake, which occurred in Japan in 1995, made many researchers to identify and improve the reliability of the transport networks. Also, reliability is highly important in special cases such as poor weather conditions, disasters, road accidents, and terrorist attacks. Moreover, the increased economic activities all over the world have increased the importance of network systems and value of the network [
Zhao et al. [
Hosseini and Wadbro [
To the best of the authors’ knowledge, no study has been done on the cooperative games among different players in decentralized systems in nondeterministic circumstances with considering reliability. There is main contribution in this study with regard to a mathematical model based on triangular probability distribution in decentralized systems in nondeterministic circumstances for coalitions of owners/players. We have studied how cooperation among the multiple owners and the changes of flow parameters cost and time in nondeterministic circumstances can increase players’ payoff and the amount of reliability in the network. Cooperation value also is measured by effectiveness (synergy) index. To address the problem of allocating the cooperation value to the cooperating owners, we have considered several methods of cooperative game theory.
The main framework of the considered problem is shown in Figure
A typical network which is controlled by
In this network, the goal is maximizing players’ payoff in the chain and determining the reliability of the entire chain by considering the uncertain amount of time, cost, and amount of flow by taking advantage of modeling in cooperative games in the nondeterministic circumstances. It should be noted that
Prerequisites and assumptions are as follows:
Each arc has a given and independent capacity in deterministic network, while it has different values in the scenariobased model.
Each arc has a given and independent transportation cost in deterministic network, while it has different values in the scenariobased model.
Each arc has a given and independent travel time in deterministic network, while it has different values in the scenariobased model.
There is a budget limitation for each owner of the network.
There is a time window to start from the origin and end at the destination, which must be observed by the network.
Indices are as follows:
Parameters are as follows:
Decision variables are as follows:
Here the main goal in the network is maximizing players’ payoff in the entire network. Moreover, determining the optimized reliability of the entire network with respect to the amount of transition time, cost, and capacity by using nondeterministic collaborative games models is considered. Therefore, this section presents mathematical linear programming based on triangular probability distribution for each possible coalition among players. Consider
In this model, capacity, cost, and travel time for each arc are independent random variables and are considered to deploy from a triangular density function with estimated parameters derived from optimistic, pessimistic, and most likely values. Therefore, the reliability terms in model could be calculated based on the triangular cumulative density function in exchange for target value for capacity, cost, and travel time which are constraints (
Constraint (
The purpose of optimization of the uncertain approach with continuous triangular distribution method is to optimize objective function at an acceptable level. Then, with solving the proposed model the value of the objective function is obtained. Furthermore, the reliability of the entire system and the value of the game in different coalitions are computed.
All ordinary linear programming packages such as Lingo or GAMS software can be used to solve this linear programming model. First, the basic idea is to solve this model by concerning the network of each owner independently. Then, the model of all coalitions of two owners must be solved. In the next step, all coalitions of three of the owners should be considered and this process continues until the grand coalition is achieved. Due to super additive characteristic of TU games, the utility of the network for any coalitional status must be greater than the sum of the utility of the network for the coalition’s members; that is,
To verify and authenticate the validity of the proposed model and to prove its functionality, a case study is used. For this purpose, a transmission network with three competitive suppliers is considered that is shown in Figure
Assumed network and links [
In Table
Capacities, travel times, and transportation costs.
Pair  Cap (m^{3}) 

Cost (dollars)  









 
(12)  540  681  892  33  82  124  88  106  139 
(13)  596  842  886  40  68  130  84  93  126 
(14)  533  784  976  30  97  130  70  109  128 
(23)  638  651  919  33  78  133  76  114  130 
(27)  505  686  958  34  60  120  82  96  135 
(35)  536  834  891  43  62  119  71  108  134 
(36)  582  736  947  40  91  129  89  97  135 
(46)  535  674  941  33  78  130  86  101  134 
(47)  562  819  975  27  87  127  81  117  126 
(58)  548  674  991  40  79  129  81  110  131 
(68)  608  755  999  26  57  114  71  103  129 
(69)  555  803  944  43  106  139  74  95  134 
(76)  514  756  930  46  80  122  84  113  131 
(79)  552  680  938  28  53  132  75  104  134 
(89)  554  798  909  41  93  123  70  100  129 
To solve the problem,
The values of reliability.
Arcs 




(12)  0.794  0.974  0.848 
(13)  0.972  0.954  0.994 
(14)  0.895  0.922  0.977 
(23)  0.998  0.934  0.943 
(27)  0.774  0.993  0.930 
(35)  0.896  0.994  0.926 
(36)  0.938  0.933  0.918 
(46)  0.801  0.949  0.924 
(47)  0.941  0.958  0.978 
(58)  0.845  0.949  0.939 
(68)  0.981  1.000  0.976 
(69)  0.923  0.803  0.948 
(76)  0.840  0.980  0.924 
(79)  0.840  0.961  0.932 
(89)  0.913  0.967  0.979 
Characteristic function and optimal flows for risk behavior of coalition’s member under different aggregation methods.
Coalition 

EU 
Synergy 


27  0  0 

76  0  0 

119  0  0 

114  11  0.096 

159  13  0.081 

199  4  0.02 

253  31  0.122 
It should be mentioned that, in the network contract, optimal transmission capacity, time, and cost are considered as follows:
When the utility of coalition of graph owners is computed, the problem of sharing the benefits of the collaboration among variant owners should be considered. The problem is difficult to solve because the contribution of each owner to the utility of graph is ambiguous. Therefore, a theoretically grounded technique is needed and CGT can be the best option. In the first place, some basic concepts related to CGT are considered and then they are developed for multipleowner graph in nondeterministic circumstances.
For each player, the set
Imputation is an allocation or a payoff vector and it means how
The excess of coalition
The imputations obtained by different TU game approaches including the Shapley value, the
Assigning of the coalition payoff by variant methods.
Owner  Shapley 

Core center  

Coalition 

40  39.877  39.673 

84.5  84.585  84.688  

128.5  128.538  128.638  
Stable  Yes  Yes  Yes 
Figure
Core for the multipleowner graph example under variant risk attitudes.
The difference between allocated utility obtained from grand coalition and utility if the coalition
Moreover, Table
Coalition satisfactions for variant TU game approaches.
Coalition  Shapley 

Core center  

Coalition 

13 
12.877 
12.673 

7.5 
8.585 
8.688  

9.5 
9.538 
9.638  

10.5 
10.462 
10.361  

9.5 
9.415 
9.311  

14 
14.123 
14.326 
Table
With regard to cooperation between the players, the synergy between the players is remarkable. For example, the synergy between the players in multilateral coalition is 0.122, which can serve as a good incentive for cooperation between them in terms of the network.
Table
Table
In traditional network games, rottenly analysis is performed under deterministic conditions. Thus, estimates may be noisy due to lack of real attention to possible risky events. Our optimization method proposed a novel approach to overcome risky conditions on the maximumflow problems on cooperative circumstances. This method covers more reliability in decisionmaking under uncertain conditions and acts as a valid solution under full certainty. Another advantage of the proposed method is simplicity in using a wellknown triangular probability distribution. Such methods help decision makers to benefit from maximum experts’ viewpoints and there is no need to follow statistical distributions fitting method based on historical data. The method needs to have optimistic, pessimistic, and most likely estimates for any risk factors. The results of numerical example indicate that the mathematical model is efficient for cooperation among players and increases reliability.
There are some directions and suggestions for future research works. Researches may consider more risky factors, inventory management disciplines, and competition in the amount of sending flow to retailers. Moreover, it seems that applying our proposed method could satisfy the need for building the cooperative strategies for stochastic networks managed by multiple owners/players under uncertain time, cost, and capacity parameters of the network’s arcs. Last but not least, fuzzybased methods can be employed in order to increase network reliability in further research.
The authors declare that they have no competing interests.