Research ArticleCooperative Strategies for Maximum-Flow Problem inUncertain Decentralized Systems Using Reliability Analysis

Hadi Heidari Gharehbolagh,1 Ashkan Hafezalkotob,1 Ahmad Makui,2 and Sadigh Raissi1

1School of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran 11518-63411, Iran2Department of Industrial Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran

Correspondence should be addressed to Ashkan Hafezalkotob; hafezalkotob@iust.ac.ir

Received 22 May 2016; Accepted 25 July 2016

Academic Editor: Arturo Pagano

Copyright © 2016 Hadi Heidari Gharehbolagh et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This study investigates amultiownermaximum-flow network problem, which suffers from risky events. Uncertain conditions effecton proper estimation and ignoring them may mislead decision makers by overestimation. A key question is how self-governingowners in the network can cooperate with each other to maintain a reliable flow. Hence, the question is answered by providinga mathematical programming model based on applying the triangular reliability function in the decentralized networks. Theproposed method concentrates on multiowner networks which suffer from risky time, cost, and capacity parameters for eachnetwork’s arcs. Some cooperative game methods such as 𝜏-value, Shapley, and core center are presented to fairly distribute extraprofit of cooperation. A numerical example including sensitivity analysis and the results of comparisons are presented. Indeed, theproposed method provides more reality in decision-making for risky systems, hence leading to significant profits in terms of realcost estimation when compared with unforeseen effects.

1. Introduction

Continuous development of technology in petrochemicalindustries, automobile manufacturing, water distributionnetworks, electricity industries, and transportation networkshas created complex, competitive, and decentralized environ-ments for suppliers [1]. This makes distribution companiesand transport networks consider their main variables suchas capacity, time, and cost as important elements in today’scompetitive environments and try to increase their servicelevels for customers and promote the products according tocustomers’ demand [2]. Therefore, designing an appropriatesystem in decentralized networks with the consideration ofuncertainty is one of the most important issues in today’scompetitive world.

In the design of suitable systems in decentralized net-works, quality survival and reliability in decision-makingswith nondeterministic circumstances are highly important.Since any failure in the network affects noticeably customerservices and causes heavy costs to suppliers, increasingnetwork reliability and network performance is extremely

essential. One of the key tools to improve network perform-ance in decentralized networks is game theory and coopera-tive games that can create optimal strategies for suppliers [3].

In uncertain circumstances, we have focused on realvalues of decision parameters such as time, cost, and capacitywhich may deploy from uncertain patterns. Such uncertaincircumstances may arise due to various sources of fluctua-tions in supply/demand patterns, politics, traffic, natural dis-asters, war, falling debris, dropping voltage, and so on.

Naturally the effects of such unforeseen events may mis-lead decision makers. Accordingly, uncertainty decision-making is an important issue in today’s competitive world. Inthis regard through this research, a novel mathematical pro-grammingmodel based on triangular probability distributionin nondeterministic decentralized systems is proposed tosolve maximum-flow problem. Applying reliability functionfor triangular density function helped us to estimate a morereliable value for uncertain parameters of traveling time,associated transportation cost, and the actual amount ofdisplacement capacity according to experts’ subjective com-ments.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 2349712, 9 pageshttp://dx.doi.org/10.1155/2016/2349712

2 Mathematical Problems in Engineering

The reminder of the paper is organized as follows:Literature review and research gap in decentralized networksare presented in Section 2. Through Section 3, the proposedmathematical model is presented. The properties of theconstructed model and further cooperative investigation areevaluated through a numerical example in Section 4. Finallyconcluding remarks are presented in the last section.

2. Literature Review

Themain related issues to the maximum-flow problems suchas logistic network, cooperative game theory, and reliabilityhave been discussed and the research gap is given as follows.

2.1. Logistic Network. Logistic network model is one of themost important models in mathematical programming andoperations research and includes network planning andinventory control, production planning, planning and projectcontrol, 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 mainissues in the maximum-flow problems [4]. Frisk et al. [5]worked on the collaboration between logistic companiesin the forest industries by investigating a number ofsharing mechanisms including nucleolus, Shapley value,shadow prices, separable and nonseparable costs, and volumeweights. Lehoux et al. [6] considered different cooperationtechniques such as the Shapley value, shadow prices, andnucleolus in logistic networks.

The purpose of the maximum-flow problem in the net-work is to reach the highest amount of transportation flowfrom the initial node to the terminal node by considering thecapacity of the arcs.

2.2. Cooperative GameTheory (CGT). In the last decades, thelogistic costs in distribution networks have been increaseddramatically due to a noticeable raise in customers’ expec-tations. To reduce the costs, various game theory methodsand horizontal cooperative games are used in the logisticnetworks in which the horizontal cooperative games haveled to getting higher payoff because of cooperation betweencompanies [7].

Vanovermeire and Sorensen [8] used the cooperationtechniques including the nucleolus and the Shapley valuemethods among shippers to increase their performance. Theresults indicated that the cooperation reduced the costs ofdistribution and delivery but the reduction is depending onthe flexibility of the companies for delivery of goods.

Saha et al. [9] used two price promotion policies, MIRand RS, and DDD as tools in a manufacturer-distributer-retailer channel to achieve improved individual profits andto eliminate channel conflict. Generalized Nash bargainingproduct could determine particular profit split in cooperativeenvironment although it needs negotiation powers of allchain members.

Networks are often controlled by multiple owners. As acase 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 networkand in fact a cooperative game in the network is built [5, 10].

Charles and Hansen [11] proposed a theoretical costsavingmechanism for cost saving assignment in an enterprisenetwork and global cost minimization by the help of CGT.The results showed that the cost allocations obtained throughthe activity based costing technique were stable and rational.

Zibaei et al. [12] proposed a multidepot vehicle routingproblem for minimizing the transportation costs when thereare multiple owners. The results indicate that the transporta-tion costs were declined that could lead to noticeable costsavings.Therefore, several methods based on the CGT theoryincluding 𝑠 value, Shapley value, least core, and equal costsaving method were proposed for a fair allocation of the costsavings between the owners.

Zhao et al. [13] recognized the game theory as a compre-hensive tool for studying strategies of supply chain elements.Bell [14] optimized a transportation model from origin todestination with five considered paths and four scenarios byzero-sum cooperative games and using linear programmingapproach in order tominimize paths’ costs. In the competitivemarket of emerging economies such as China and India, toomuchpressure on the global supply chain has causednew lim-itations for the countries’ transportation networks. Reyes [15]used Shapley method in cooperative games for optimizingthe transport network in order to stabilize the supply chainand reduce the excessive pressure. San Cristoba [16] opti-mized cost allocation between activities of networks by usinggame theory. Lozano et al. [7] introduced a mathematicalprogramming model to measure the benefits of merging thetransportation demands from different companies. Hafeza-lkotob and Makui [17] introduced a stochastic mathematicalprogramming model for a multiple-owner graph problem.Their model was based on the cooperative game theory inorder to indicate that the collaboration between independentowners of a logistic networkwill lead tomaintaining a reliablemaximumflow.McCain [18] concentrated on the cooperativegames in collaborating organizations in order to analyse theeffects of these games on the organization expansion andits profit. Esmaeili et al. [19] discussed guarantee serviceswith game theory approach in three different levels: manu-facturers, distributors, and customers. Guarantee is a servicecontract between manufacturers and customers and plays akey role in most of lawful business processes. Interactionbetween the players (manufacturers, distributors, and cus-tomers) is studied by noncooperative and semicooperativegames to obtain optimum results including selling price,period’s guarantee, guarantee’s fee for manufacturers, costs ofmaintenance, and repair cost for distributors.

2.3. Reliability. One of the most important aspects of relia-bility is network reliability. The network reliability is definedas a capability or probability that a network system has tocompletely fulfill customer-tailored communications tasksduring the stipulated successive operation procedure [20].

In the last decade, the reliability of the transport networkand 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

Mathematical Problems in Engineering 3

identify and improve the reliability of the transport networks.Also, reliability is highly important in special cases suchas poor weather conditions, disasters, road accidents, andterrorist attacks. Moreover, the increased economic activitiesall over the world have increased the importance of networksystems and value of the network [21].

Zhao et al. [22] proposed stochastic simulation methodsbased on Monte Carlo by considering the system reliabilityand component probabilistic importance of a road network.Then a new system that incorporates this proposed methodis developed. The system is a useful practical quantitativeanalysis tool to assist the decision-making for the roadmanagement departments, such as predicting the increasedsystem reliability of a road network when it adds a new link,finding the key components that need to be upgraded orimproved, and evaluating the system reliability of differentroad network planning schemes.

Hosseini andWadbro [23] studied the essential problemsof reliability and stability analysis in uncertain networks.They used uncertainty theory to make sure the arrival ofrelief materials and rescue vehicles to the disaster areas is intime. They then defined the new problems of 𝛼-most reliablepath (𝛼-MRP), which aims to minimize the pessimistic riskvalue of a path under a given confidence level 𝛼 and verymost reliable path (VMRP) in an uncertain traffic network.Amin et al. [24] introduced an approach to software reliabilityprediction based on time series modeling in order to showthe importance of reliability for quality systems. Therefore,with increasing attention to the quality, finding a way toenhance product’s reliability is considered widely, because,to stay in competition environments, product quality andthe associated costs have very important roles [25]. Manyreal-world systems such as power transmission systems canbe a multicast flow network in which each independentpart can have its own payoff and profit. Yeh et al. [26]calculated the reliability of networks in which the flow is runby multiplayers by using a new cut-based algorithm. Khalili-Damghani et al. [27] optimized the maximum reliabilityin the series-parallel systems by minimizing the weightand the value of the network with metaheuristic algorithmPSO. Hausken [28] measured the probability of the risk ofreliability in series, parallel, and combined networks by usinggame theory. Szeto [21] used cooperative game to measurethe reliability of conflicting reports on transport networksin order to minimize the cost of paths. Stackelberg andNash equilibrium methods were used in optimization. Theresults of the research indicated that noncooperative gamescan lead to worse situations. Prabhu Gaonkar et al. [29]employed fuzzy models to study reliability in transportationnetwork and Jiang et al. [30] assumed uncertain capacities ina transportation network design problem. Zhang et al. [31]dealt with a network resilience problem through studyingtopology in transportation networks.

2.4. Research Gap. To the best of the authors’ knowledge,no study has been done on the cooperative games amongdifferent players in decentralized systems in nondeterministiccircumstances with considering reliability.There ismain con-tribution in this study with regard to a mathematical model

k1

k2

kn

a1

a2

an

b1

b2

bn

c1

c2

cn

O D

Figure 1: A typical network which is controlled by 𝑛 players.

based on triangular probability distribution in decentralizedsystems in nondeterministic circumstances for coalitions ofowners/players. We have studied how cooperation amongthe multiple owners and the changes of flow parameters costand time in nondeterministic circumstances can increaseplayers’ payoff and the amount of reliability in the network.Cooperation value also ismeasured by effectiveness (synergy)index. To address the problem of allocating the cooperationvalue to the cooperating owners, we have considered severalmethods of cooperative game theory.

3. Material and Methods

3.1. The Proposed Mathematical Model. Themain frameworkof the considered problem is shown in Figure 1. Supposethat an interconnected decentralized system is controlled by𝑛 players at the same time where 𝑎

1, 𝑏1, 𝑐1, . . . , 𝑘

1nodes are

controlled by the 1st player and 𝑎2, 𝑏2, 𝑐2, . . . , 𝑘

2nodes are

controlled by the 2nd player and similarly 𝑎𝑛, 𝑏𝑛, 𝑐𝑛, . . . , 𝑘

𝑛

nodes are controlled by the 𝑛th player.In this network, the goal is maximizing players’ payoff in

the chain and determining the reliability of the entire chain byconsidering the uncertain amount of time, cost, and amountof flow by taking advantage ofmodeling in cooperative gamesin the nondeterministic circumstances. It should be notedthat 𝑎, 𝑏, 𝑐, . . . , 𝑘 are connecting nodes in the mentionednetwork. Let �� = (𝑉, 𝐴, Cap, Cost,

𝑇, 𝑂, 𝐷) be a networkwith𝑂, 𝐷 ∈ 𝑉 being the source and the destination, respectively.In the network there are 𝑛 nodes which arrange a finite nodeset 𝑉 = {1, 2, 3, . . . , 𝑛} and arc set of 𝐴 = {(𝑖, 𝑗) | 𝑖, 𝑗 ∈ 𝑉}.In our terminology the accent “∼” is used for emphasizinguncertain quantities. In the network, we consider that thereare also three nonnegative stochastic random variables ofCap, Cost, and

𝑇 defined as stochastic capacities, handlingcosts, and travel times for each arc in 𝐴, respectively.

3.2. Prerequisites and Assumptions. Prerequisites and as-sumptions are as follows:

(i) Each arc has a given and independent capacity indeterministic network, while it has different values inthe scenario-based model.

(ii) Each arc has a given and independent transportationcost in deterministic network, while it has differentvalues in the scenario-based model.

4 Mathematical Problems in Engineering

(iii) Each arc has a given and independent travel time indeterministic network, while it has different values inthe scenario-based model.

(iv) There is a budget limitation for each owner of thenetwork.

(v) There is a time window to start from the origin andend at the destination, whichmust be observed by thenetwork.

Notations. The indices, parameters, and decision variablesare explained before providing the objective functions andproposing the model.

Indices are as follows:

𝑖: index of initial nodes.

𝑗: index of terminal nodes.

𝑘: index of nodes.

𝑂: the source node.

𝐷: the sink node.

𝑚: index of coalitions.

𝑐𝑚: 𝑚th coalition of the network owners.

Parameters are as follows:

Cap𝑖𝑗: capacity of arc between node 𝑖 and 𝑗 ∈ V.

𝑇𝑖𝑗: travel time between node 𝑖 and 𝑗 ∈ V.

𝑇min: minimum allowed travel time in network 𝑁.

𝑇max: maximum allowed travel time in network 𝑁.

Cost𝑖𝑗: handing (shipping) cost per unit of flowing

items between node 𝑖 and 𝑗 ∈ V.

𝐵𝑐𝑚

: the total budget available for coalition.

Decision variables are as follows:

𝑦𝑖𝑗= {1, if arc between 𝑖 and 𝑗 is active; 0, elsewhere},

∀𝑖, 𝑗 ∈ 𝑉.

𝑥𝑖𝑗: flow between 𝑖 and 𝑗 ∈ V.

𝑓𝑐𝑚

: maximum-flow in the network controlled bycoalition 𝑚.

Here the main goal in the network is maximizing players’payoff in the entire network. Moreover, determining theoptimized reliability of the entire network with respectto the amount of transition time, cost, and capacity byusing nondeterministic collaborative games models is con-sidered. Therefore, this section presents mathematical linear

programming based on triangular probability distribution foreach possible coalition among players. Consider

Max 𝑧 = 𝑓𝑐𝑚

, (1)

Subject to: ∑

𝑗

𝑦𝑂𝑗

= 1, ∀𝑗; 𝑗 = 1, 2, . . . , 𝑛, (2)

∑

𝑖

𝑦𝑖𝐷

= 1, ∀𝑖; 𝑖 = 1, 2, . . . , 𝑛, (3)

∑

𝑗

𝑦𝑖𝑗

≤ 1, ∀𝑖, 𝑗 ∈ V, (4)

∑

𝑖

𝑦𝑖𝑗

≤ 1, ∀𝑖, 𝑗 ∈ V, (5)

∑

𝑖

𝑦𝑖𝑗

− ∑

𝑘

𝑦𝑗𝑘

= 0, ∀𝑖, 𝑗, 𝑘 ∈ V, (6)

𝑥𝑖𝑗

− 𝑦𝑖𝑗

⋅ 𝑅Cap𝑖𝑗

(Cap∗) ⋅ Cap∗ ≤ 0,

∀𝑖 = 𝑗 ∈ V,

(7)

𝑥𝑂𝑗

− 𝑦𝑂𝑗

⋅ 𝑅Cap𝑂𝑗

(Cap∗) ⋅ Cap∗ ≤ 0,

∀𝑗; 𝑗 = 1, 2, . . . , 𝑛,

(8)

𝑥𝑖𝐷

− 𝑦𝑖𝐷

⋅ 𝑅Cap𝑖𝐷

(Cap∗) ⋅ Cap∗ ≤ 0,

∀𝑖; 𝑖 = 1, 2, . . . , 𝑛,

(9)

∑

𝑖

𝑥𝑖𝑗

− ∑

𝑘

𝑥𝑗𝑘

= 0, ∀𝑖 = 𝑗 = 𝑘 ∈ V, (10)

𝑇min

≤ ∑

𝑖,𝑗

𝑦𝑖𝑗

⋅ [1 + (1 − 𝑅𝑇

𝑖𝑗(𝑇∗

))] ⋅ 𝑇∗

+ ∑

𝑗

𝑦𝑂𝑗

⋅ [1 + (1 − 𝑅𝑇

𝑂𝑗(𝑇∗

))] ⋅ 𝑇∗

+ ∑

𝑖

𝑦𝑖𝐷

⋅ [1 + (1 − 𝑅𝑇

𝑖𝐷(𝑇∗

))] ⋅ 𝑇∗

≤ 𝑇max

∀𝑖, 𝑗 ∈ V,

(11)

∑

𝑖,𝑗

𝑥𝑖𝑗

⋅ [1 + (1 − 𝑅Cost𝑖𝑗

(Cost∗))] ⋅ Cost∗

+ ∑

𝑗

𝑥𝑂𝑗

. [1 + (1 − 𝑅Cost𝑂𝑗

(Cost∗))]Cost∗

+ ∑

𝑖

𝑥𝑖𝐷

⋅ [1 + (1 − 𝑅Cost𝑖𝐷

(Cost∗))] ⋅ Cost∗

≤ 𝐵𝑐𝑚

∀𝑖, 𝑗 ∈ V,

(12)

Mathematical Problems in Engineering 5

𝑓𝑐𝑚

− 𝑥𝑖𝑗

− 𝑀 (1 − 𝑦𝑖𝑗

) ≤ 0, ∀𝑖, 𝑗 ∈ V, (13)

𝑦𝑖𝑗

∈ {0, 1} , ∀𝑖, 𝑗, (14)

𝑥𝑖𝑗

≥ 0, ∀𝑖, 𝑗. (15)

In this model, capacity, cost, and travel time for each arcare independent random variables and are considered to

deploy from a triangular density function with estimatedparameters derived from optimistic, pessimistic, and mostlikely values. Therefore, the reliability terms in model couldbe calculated based on the triangular cumulative densityfunction in exchange for target value for capacity, cost, andtravel time which are constraints (16), (17), and (18). In theseconstraints, respectively, nominated value is presented byCap∗, Cost∗, and 𝑇

∗. Consider

𝑅Cap𝑖𝑗

(Cap∗) =

{{{{{{{{

{{{{{{{{

{

1 −

(Cap∗ − 𝑎Cap𝑖𝑗

)

2

(𝑏Cap𝑖𝑗

− 𝑎Cap𝑖𝑗

) (𝑐Cap𝑖𝑗

− 𝑎Cap𝑖𝑗

)

, if: 𝑎Cap𝑖𝑗

≤ Cap∗ ≤ 𝑐Cap𝑖𝑗

(𝑏Cap𝑖𝑗

− Cap∗)2

(𝑏Cap𝑖𝑗

− 𝑎Cap𝑖𝑗

) (𝑏Cap𝑖𝑗

− 𝑐Cap𝑖𝑗

)

, if: 𝑐Cap𝑖𝑗

≤ Cap∗ ≤ 𝑏Cap𝑖𝑗

}}}}}}}}

}}}}}}}}

}

, (16)

𝑅Cost𝑖𝑗

(Cost∗) =

{{{{{{{{

{{{{{{{{

{

(Cost∗ − 𝑎Cost𝑖𝑗

)

2

(𝑏Cost𝑖𝑗

− 𝑎Cost𝑖𝑗

) (𝑐Cost𝑖𝑗

− 𝑎Cost𝑖𝑗

)

, if: 𝑎Cost𝑖𝑗

≤ Cost∗ ≤ 𝑐Cost𝑖𝑗

1 −

(𝑏Cost𝑖𝑗

− Cost∗)2

(𝑏Cost𝑖𝑗

− 𝑎Cost𝑖𝑗

) (𝑏Cost𝑖𝑗

− 𝑐Cost𝑖𝑗

)

, if: 𝑐Cost𝑖𝑗

≤ Cost∗ ≤ 𝑏Cost𝑖𝑗

}}}}}}}}

}}}}}}}}

}

, (17)

𝑅𝑇

𝑖𝑗(𝑇∗

) =

{{{{{{{{

{{{{{{{{

{

(𝑇∗

− 𝑎𝑇𝑖𝑗

)

2

(𝑏𝑇𝑖𝑗

− 𝑎𝑇𝑖𝑗

) (𝑐𝑇𝑖𝑗

− 𝑎𝑇𝑖𝑗

)

, if: 𝑎𝑇𝑖𝑗

≤ 𝑇∗

≤ 𝑐𝑇𝑖𝑗

1 −

(𝑏𝑇𝑖𝑗

− 𝑇∗

)

2

(𝑏𝑇𝑖𝑗

− 𝑎𝑇𝑖𝑗

) (𝑏𝑇𝑖𝑗

− 𝑐𝑇𝑖𝑗

)

, if: 𝑐𝑇𝑖𝑗

≤ 𝑇∗

≤ 𝑏𝑇𝑖𝑗

}}}}}}}}

}}}}}}}}

}

. (18)

Objective function (1) maximizes total flow of the network.Constraints (2)–(6) find the optimum path of the deter-ministic network. Constraints (7)–(9) are capacity-relatedconstraints.

Constraint (10) is a balance constraint. Constraints (11)and (12) are in charge of observing time windows and budgetlimitations. Constraint (13) measures the flow of the network𝑓𝑐𝑚

. Constraints (14) and (15) are related to the variables of theproblem.Constraints (16), (17), and (18) calculate reliability incontinuous triangular distribution.

The purpose of optimization of the uncertain approachwith continuous triangular distribution method is to opti-mize objective function at an acceptable level. Then, withsolving the proposed model the value of the objective func-tion is obtained. Furthermore, the reliability of the entiresystem and the value of the game in different coalitions arecomputed.

All ordinary linear programming packages such as Lingoor GAMS software can be used to solve this linear program-ming model. First, the basic idea is to solve this model byconcerning 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 shouldbe considered and this process continues until the grand

coalition is achieved. Due to super additive characteristic ofTU games, the utility of the network for any coalitional statusmust be greater than the sum of the utility of the network forthe coalition’s members; that is,

𝑓𝑐𝑚

= V (𝑐𝑚

) ≥ ∑

𝑝𝑖⊂𝑐𝑚

V (𝑝𝑖) , ∀𝑐

𝑚∈ 𝑃. (19)

Actually, the difference between the sum of the “separate”maximum utility and the “mixed” maximum utility showsextra utility EU(𝑐

𝑚) created by the coalition 𝑐

𝑚; that is,

EU (𝑐𝑚

) = V (𝑐𝑚

) − ∑

𝑝𝑖⊂𝑐𝑚

V (𝑝𝑖) , ∀𝑐

𝑚∈ 𝑃. (20)

According to the utility of the collaborating owners, the extrautility must be assessed. Therefore, the following criteria canbe more reliable measure for the synergy of a coalition:

Synergy (𝑐𝑚

) =

EU (𝑐𝑚

)

V (𝑐𝑚

)

. (21)

4. Results and Discussion

4.1. Illustrative Example. To verify and authenticate the valid-ity of the proposed model and to prove its functionality,

6 Mathematical Problems in Engineering

Table 1: Capacities, travel times, and transportation costs.

Pair Cap (m3) 𝑇 (h) Cost (dollars)𝑎 𝑐 𝑏 𝑎 𝑐 𝑏 𝑎 𝑐 𝑏

(1-2) 540 681 892 33 82 124 88 106 139(1-3) 596 842 886 40 68 130 84 93 126(1-4) 533 784 976 30 97 130 70 109 128(2-3) 638 651 919 33 78 133 76 114 130(2-7) 505 686 958 34 60 120 82 96 135(3-5) 536 834 891 43 62 119 71 108 134(3-6) 582 736 947 40 91 129 89 97 135(4-6) 535 674 941 33 78 130 86 101 134(4-7) 562 819 975 27 87 127 81 117 126(5-8) 548 674 991 40 79 129 81 110 131(6-8) 608 755 999 26 57 114 71 103 129(6-9) 555 803 944 43 106 139 74 95 134(7-6) 514 756 930 46 80 122 84 113 131(7-9) 552 680 938 28 53 132 75 104 134(8-9) 554 798 909 41 93 123 70 100 129

1

2

3

4

5

6

7

8

9

Arcs of owner 1Arcs of owner 2Arcs of owner 3

Figure 2: Assumed network and links [17].

a case study is used. For this purpose, a transmission networkwith three competitive suppliers is considered that is shownin Figure 2.Thefirst supplier (or the first player) is controllingthe set {(1, 3), (3, 5), (3, 6), (5, 8), (8, 9)}, the second one iscontrolling the set {(1, 4), (4, 6), (4, 7), (6, 8), (6, 9)}, and thethird supplier is controlling the set {(1, 2), (2, 3), (2, 7),(7, 6), (7, 9)}. It should be noted that it is assumed that thedata have normal distribution. The goal of the problem isto maximize flow from the original node to the destinationnode.

In Table 1, the amount of transmission capacity, time oftransition and distribution between the nodes, and transmis-sion costs in pessimistic cases (a), most likely cases (b), andthe most optimistic cases (c) are presented.

To solve the problem, 𝑇min and 𝑇max are considered to be100 h and 500 h, respectively. Moreover, the available budgetfor the first player is 5000 $, the available budget for thesecond player is 10,000 $, and the available budget for thethird player is 15,000 $. After solving the model by GAMS

Table 2: The values of reliability.

Arcs 𝑅(cap) 𝑅(𝑇) 𝑅(cost)(1-2) 0.794 0.974 0.848(1-3) 0.972 0.954 0.994(1-4) 0.895 0.922 0.977(2-3) 0.998 0.934 0.943(2-7) 0.774 0.993 0.930(3-5) 0.896 0.994 0.926(3-6) 0.938 0.933 0.918(4-6) 0.801 0.949 0.924(4-7) 0.941 0.958 0.978(5-8) 0.845 0.949 0.939(6-8) 0.981 1.000 0.976(6-9) 0.923 0.803 0.948(7-6) 0.840 0.980 0.924(7-9) 0.840 0.961 0.932(8-9) 0.913 0.967 0.979

Table 3: Characteristic function and optimal flows for risk behaviorof coalition’s member under different aggregation methods.

Coalition 𝑓𝑐𝑚

= 𝑉(𝑐𝑚

) EU (𝑐𝑚

) Synergy (𝑐𝑚

)

𝐶 {1} 27 0 0𝐶 {2} 76 0 0𝐶 {3} 119 0 0𝐶 {1, 2} 114 11 0.096𝐶 {1, 3} 159 13 0.081𝐶 {2, 3} 199 4 0.02𝐶 {1, 2, 3} 253 31 0.122

software, the results of reliability of links are shown in Table 2.Table 3 indicates the final output of the proposed model.

Mathematical Problems in Engineering 7

It should be mentioned that, in the network contract,optimal transmission capacity, time, and cost are consideredas follows: Cap∗ = 641, 𝑇

∗= 114, and Cost∗ = 123. For

example, consider a part of a pipeline has a random capacityover timewithin the range of [540–892] cubicmeters. Expertsbelieved that most of the time this part conducted 681m3.Here as a common sense we could consider that this part ofpipeline has a triangular random variable and according toconstraint (16), it has the chance of 79.4% to conduct 641m3.

4.2. Graph Collaborative Arrangement for the Multiple-OwnerGraph. When the utility of coalition of graph owners is com-puted, the problemof sharing the benefits of the collaborationamong variant owners should be considered. The problem isdifficult to solve because the contribution of each owner tothe utility of graph is ambiguous. Therefore, a theoreticallygrounded technique is needed and CGT can be the bestoption. In the first place, some basic concepts related to CGTare considered and then they are developed for multiple-owner graph in nondeterministic circumstances.

For each player, the set 𝑃 is 𝑃 = {1, 2, . . . , 𝑁}, and V(𝑃)

shows the available payoff when all players cooperate. Now,consider 𝑦

𝑖is a real number for each player 𝑖 = 1, 2, . . . , 𝑁,

with ∑𝑁

𝑖=1𝑦𝑖

≤ V(𝑃). If a vector �� = (𝑦1, 𝑦2, . . . , 𝑦

𝑛) is suitable

for individual and group rationality conditions, that is, 𝑦𝑖

≥

V(𝑖) for all 𝑖 ∈ 𝑃 and ∑𝑁

𝑖=1𝑦𝑖

= V(𝑃), then it will be consideredas an imputation.

Imputation is an allocation or a payoff vector and itmeanshow V(𝑃) is to be distributed to players so that no playerrejects the allocated payoff. The set of all imputations forthe competitive game is 𝑌 = {�� = (𝑦

1, 𝑦2, . . . , 𝑦

𝑛) | 𝑦𝑖

≥

V(𝑖), ∑𝑁

𝑖=1𝑦𝑖

= V(𝑃)}. Specifying the imputation form 𝑌,which yields a fair allocation of total payoff, is the maingoal in CGT. Several assignment methods based on differentinterpretation of fair allocation are developed that some ofthem are reviewed briefly [32, 33].

The excess of coalition 𝐶 ∈ 𝑃 for imputation �� ∈ 𝑌 isequal to 𝑒(𝐶, 𝑦) = V(𝐶) − ∑

𝑖∈𝐶𝑦𝑖. It indicates that the amount

of payoff allocated to the coalition 𝐶 varies from the payoffassociated with 𝐶. The core of the game can be defined asfollows:

core (0) = {�� ∈ 𝑌 | 𝑒 (𝐶, ��) ≤ 0, ∀𝐶 ⊂ 𝑃}

= {�� ∈ 𝑌 | V (𝐶) ≤

𝑁

∑

𝑖=1

𝑦𝑖, ∀𝐶 ⊂ 𝑃} .

(22)

As a matter of fact, the core is the set of all imputations sothat each coalition achieves least payoff associated with thecorresponding coalition. The game is called stable if the coreis nonempty. Besides, for the real number 𝜀, core(𝜀) can bedefined as follows:

core (𝜀)

= {�� ∈ 𝑌 | 𝑒 (𝐶, ��) ≤ 𝜀, ∀𝐶 ⊂ 𝑃, 𝐶 = 𝑃, 𝐶 = ⌀} .

(23)

Table 4: Assigning of the coalition payoff by variant methods.

Owner Shapley 𝜏-value Core center

Coalition

{1} 40 39.877 39.673{2} 84.5 84.585 84.688{3} 128.5 128.538 128.638

Stable Yes Yes Yes

(27, 107, 119)(58, 76, 119)

(27, 76, 150)

Figure 3: Core for themultiple-owner graph example under variantrisk attitudes.

An assignment method based on four axioms of efficiency,symmetry, additive, and dummy property has been formu-lated by Shapley in [34]. An imputation �� = (𝑦

1, 𝑦2, . . . , 𝑦

𝑛)

demonstrates Shapley value if

𝑦𝑖

= ∑

𝐶⊂Π𝑖

[V (𝐶) − V (𝐶 − {𝑖})]

(|𝐶| − 1)! (𝑁 − |𝐶|)!

𝑁!

,

𝑖 = 1, 2, . . . , 𝑛,

(24)

where Π𝑖 is the set of all coalitions 𝐶 ⊂ 𝑃 submitting player 𝑖

as a member and |𝐶| shows number of members in 𝐶.The imputations obtained by different TU game

approaches including the Shapley value, the 𝜏-value, and thecore center are presented in Table 4.

Figure 3 indicates that the imputation sets and corre-sponding cores in barycentric coordinates are calculated byTUGlab package [35].

The difference between allocated utility obtained fromgrand coalition and utility if the coalition 𝐶 performs inde-pendently is satisfaction of a coalition 𝐶 from imputation�� = (𝑦

1, 𝑦2, . . . , 𝑦

𝑛); that is, 𝐹

𝑠(𝐶, ��) = ∑

𝑖∈𝐶𝑦𝑖

− V(𝐶) =

−𝑒(𝐶, 𝑦). Table 4 indicates the corresponding satisfactionvalues 𝐹

𝑠(𝐶, ��), as well as relative values 𝐹

𝑠(𝐶, ��)/V(𝐶) for

imputations of Table 3. This table shows that when the sizeof the coalition increases absolute and relative satisfactionsof coalition will reduce. It shows that when a coalition getslarger, the obtained advantages of adding new members willdecline. In real situations, the complexity involved in collab-oration of multiple owners makes reaching these advantagesdifficult.

Moreover, Table 5 indicates the coalition satisfactions insituation. Thus, in this example, when the coalitions have a

8 Mathematical Problems in Engineering

Table 5: Coalition satisfactions for variant TU game approaches.

Coalition Shapley 𝜏-value Core center

Coalition

𝐶1

= {1}13

(48.1%)12.877(47.7%)

12.673(46.9%)

𝐶2

= {2}7.5

(10%)8.585(11.3%)

8.688(11.4%)

𝐶3

= {3}9.5(8%)

9.538(8%)

9.638(8.1%)

𝐶4

= {1, 2}10.5

(9.2%)10.462(9.1%)

10.361(9.1%)

𝐶5

= {1, 3}9.5(6%)

9.415(5.95%)

9.311(5.8%)

𝐶6

= {2, 3}14

(7%)14.123(7.1%)

14.326(7.2%)

high risk adverse attitude, the owners show less interest toparticipate in the coalition.

Table 5 illustrates the correlation measure for each pairof CGT methods. The solution of different CGT methodsleads to different solutions. The following observation andmanagerial insights are derived from the numerical examples:

(1) With regard to cooperation between the players,the synergy between the players is remarkable. Forexample, the synergy between the players in multi-lateral coalition is 0.122, which can serve as a goodincentive for cooperation between them in terms ofthe network.

(2) Table 3 indicates that combining cooperation be-tween the players leads to an increase in the reliabilityof the system. Therefore, the value of the game hasbeen increased.

(3) Table 5, according to the consent of the coalitionin the network, indicates that player’s considerationsregarding network reliability and uncertainty tend tohave a coalition together.

5. Conclusions

In traditional network games, rottenly analysis is performedunder deterministic conditions.Thus, estimates may be noisydue to lack of real attention to possible risky events. Ouroptimizationmethodproposed a novel approach to overcomerisky conditions on the maximum-flow problems on coop-erative circumstances. This method covers more reliabilityin decision-making under uncertain conditions and acts asa valid solution under full certainty. Another advantage ofthe proposed method is simplicity in using a well-knowntriangular probability distribution. Such methods help deci-sion makers to benefit from maximum experts’ viewpointsand there is no need to follow statistical distributions fittingmethod based on historical data. The method needs to haveoptimistic, pessimistic, and most likely estimates for anyrisk factors. The results of numerical example indicate thatthe mathematical model is efficient for cooperation amongplayers and increases reliability.

There are some directions and suggestions for futureresearch works. Researches may consider more risky factors,inventory management disciplines, and competition in theamount of sending flow to retailers. Moreover, it seems thatapplying our proposed method could satisfy the need forbuilding the cooperative strategies for stochastic networksmanaged by multiple owners/players under uncertain time,cost, and capacity parameters of the network’s arcs. Last butnot least, fuzzy-based methods can be employed in order toincrease network reliability in further research.

Competing Interests

The authors declare that they have no competing interests.

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