Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 2349712, 9 pages

http://dx.doi.org/10.1155/2016/2349712

## Cooperative Strategies for Maximum-Flow Problem in Uncertain Decentralized Systems Using Reliability Analysis

^{1}School of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran 11518-63411, Iran^{2}Department of Industrial Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran

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 Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study investigates a multiowner maximum-flow 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 self-governing 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 -value, Shapley, and core center are presented to fairly distribute extra profit of cooperation. A numerical example including sensitivity analysis and the results of comparisons are presented. Indeed, the proposed method provides more reality in decision-making for risky systems, hence leading to significant profits in terms of real cost estimation when compared with unforeseen effects.

#### 1. Introduction

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 [1]. This makes distribution companies and transport networks consider their main variables such as capacity, time, and cost as important elements in today’s competitive environments and try to increase their service levels for customers and promote the products according to customers’ demand [2]. Therefore, designing an appropriate system in decentralized networks with the consideration of uncertainty is one of the most important issues in today’s competitive world.

In the design of suitable systems in decentralized networks, quality survival and reliability in decision-makings 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 [3].

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 decision-making 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 maximum-flow 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 2. Through Section 3, the proposed mathematical model is presented. The properties of the constructed model and further cooperative investigation are evaluated through a numerical example in Section 4. Finally concluding remarks are presented in the last section.

#### 2. Literature Review

The main related issues to the maximum-flow problems such as logistic network, cooperative game theory, and reliability have been discussed and the research gap is given as follows.

##### 2.1. Logistic Network

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 maximum-flow problems [4]. Frisk et al. [5] worked on the collaboration between logistic companies in the forest industries by investigating a number of sharing mechanisms including nucleolus, Shapley value, shadow prices, separable and nonseparable costs, and volume weights. Lehoux et al. [6] considered different cooperation techniques such as the Shapley value, shadow prices, and nucleolus in logistic networks.

The purpose of the maximum-flow 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.

##### 2.2. Cooperative Game Theory (CGT)

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 [7].

Vanovermeire and Sörensen [8] used the cooperation techniques including the nucleolus and the Shapley value methods among shippers to increase their performance. The results indicated that the cooperation reduced the costs of distribution and delivery but the reduction is depending on the flexibility of the companies for delivery of goods.

Saha et al. [9] used two price promotion policies, MIR and RS, and DDD as tools in a manufacturer-distributer-retailer channel to achieve improved individual profits and to eliminate channel conflict. Generalized Nash bargaining product could determine particular profit split in cooperative environment although it needs negotiation powers of all chain members.

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 [5, 10].

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

Zibaei et al. [12] proposed a multidepot vehicle routing problem for minimizing the transportation costs when there are multiple owners. The results indicate that the transportation costs were declined that could lead to noticeable cost savings. Therefore, several methods based on the CGT theory including value, Shapley value, least core, and equal cost saving method were proposed for a fair allocation of the cost savings between the owners.

Zhao et al. [13] recognized the game theory as a comprehensive tool for studying strategies of supply chain elements. Bell [14] optimized a transportation model from origin to destination with five considered paths and four scenarios by zero-sum cooperative games and using linear programming approach in order to minimize paths’ costs. In the competitive market of emerging economies such as China and India, too much pressure on the global supply chain has caused new limitations for the countries’ transportation networks. Reyes [15] used Shapley method in cooperative games for optimizing the transport network in order to stabilize the supply chain and reduce the excessive pressure. San Cristóba [16] optimized cost allocation between activities of networks by using game theory. Lozano et al. [7] introduced a mathematical programming model to measure the benefits of merging the transportation demands from different companies. Hafezalkotob and Makui [17] introduced a stochastic mathematical programming model for a multiple-owner graph problem. Their model was based on the cooperative game theory in order to indicate that the collaboration between independent owners of a logistic network will lead to maintaining a reliable maximum flow. McCain [18] concentrated on the cooperative games in collaborating organizations in order to analyse the effects of these games on the organization expansion and its profit. Esmaeili et al. [19] discussed guarantee services with game theory approach in three different levels: manufacturers, distributors, and customers. Guarantee is a service contract between manufacturers and customers and plays a key role in most of lawful business processes. Interaction between the players (manufacturers, distributors, and customers) is studied by noncooperative and semicooperative games to obtain optimum results including selling price, period’s guarantee, guarantee’s fee for manufacturers, costs of maintenance, and repair cost for distributors.

##### 2.3. Reliability

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 customer-tailored communications tasks during the stipulated successive operation procedure [20].

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 [21].

Zhao et al. [22] proposed stochastic simulation methods based on Monte Carlo by considering the system reliability and component probabilistic importance of a road network. Then a new system that incorporates this proposed method is developed. The system is a useful practical quantitative analysis tool to assist the decision-making for the road management departments, such as predicting the increased system reliability of a road network when it adds a new link, finding the key components that need to be upgraded or improved, and evaluating the system reliability of different road network planning schemes.

Hosseini and Wadbro [23] studied the essential problems of reliability and stability analysis in uncertain networks. They used uncertainty theory to make sure the arrival of relief materials and rescue vehicles to the disaster areas is in time. They then defined the new problems of *α*-most reliable path (*α*-MRP), which aims to minimize the pessimistic risk value of a path under a given confidence level *α* and very most reliable path (VMRP) in an uncertain traffic network. Amin et al. [24] introduced an approach to software reliability prediction based on time series modeling in order to show the importance of reliability for quality systems. Therefore, with increasing attention to the quality, finding a way to enhance product’s reliability is considered widely, because, to stay in competition environments, product quality and the associated costs have very important roles [25]. Many real-world systems such as power transmission systems can be a multicast flow network in which each independent part can have its own payoff and profit. Yeh et al. [26] calculated the reliability of networks in which the flow is run by multiplayers by using a new cut-based algorithm. Khalili-Damghani et al. [27] optimized the maximum reliability in the series-parallel systems by minimizing the weight and the value of the network with metaheuristic algorithm PSO. Hausken [28] measured the probability of the risk of reliability in series, parallel, and combined networks by using game theory. Szeto [21] used cooperative game to measure the reliability of conflicting reports on transport networks in order to minimize the cost of paths. Stackelberg and Nash equilibrium methods were used in optimization. The results of the research indicated that noncooperative games can lead to worse situations. Prabhu Gaonkar et al. [29] employed fuzzy models to study reliability in transportation network and Jiang et al. [30] assumed uncertain capacities in a transportation network design problem. Zhang et al. [31] dealt with a network resilience problem through studying topology in transportation networks.

##### 2.4. Research Gap

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.

#### 3. Material and Methods

##### 3.1. The Proposed Mathematical Model

The main framework of the considered problem is shown in Figure 1. Suppose that an interconnected decentralized system is controlled by players at the same time where nodes are controlled by the 1st player and nodes are controlled by the 2nd player and similarly nodes are controlled by the th player.