An improved NSGA2 to solve a bi-objective optimization problem of multi-state electronic transaction network
Introduction
An electronic transaction network (ETN), a kind of communication network, plays a critical role in the banking industry because it transmits transaction data processed by branches to the headquarters (HQ) for verification, storage, analysis, or other applications every day. In other words, a breakdown of the ETN may result in a great transaction loss [35]. For example, on February 25, 2013, the ETN provided by CHIEF, an Internet service provider, experienced a fire accident and needed to be repaired; thus, it could not provide real-time transmission service for several banks. Hence, many investors and banks suffered great losses. It is clear that ETN stability is crucial for personal and business financial services provided by banking companies [7], [38].
ETN stability can be enhanced by maximizing transmission reliability, which is defined as the probability that the transaction data can be successfully transmitted from multiple branches to the HQ. Many studies [19], [23], [26], [30] have explored the transmission reliability evaluation and optimization problems. In particular, these studies are related to single-objective optimization problems, such as reliability maximization subjected to limited resources and cost minimization subjected to reliability constraints. However, increasing transmission reliability means increasing costs [24], [25]. Therefore, an extended and important problem is how to determine the trade-off between cost minimization and transmission reliability maximization.
Several studies [1], [12], [16], [18], [28], [34] have discussed the multi-objective problems. These include reliability maximization and cost minimization from the perspective of network topology optimization. However, the problems that involve network restructuring consume a substantial amount of time and cost in real-life society [33]. In other words, most supervisors may not adopt a network restructuring strategy to enhance ETN stability when the ETN structure is already in place. A way to enhance the ETN stability without changing its structure is by improving the bandwidth or capacity allocation based on the specified demand of data transmission. For instance, Lin and Yeh [24], [25] adopted a multi-state resource allocation strategy to maximize transmission reliability and minimize allocation cost without restructuring the network, in which each multi-state resource comprised several binary-state components. However, Yeh and Fiondella [36] pointed out that the strategy of multi-state resource allocation may suffer from surplus capacity on an arc and then result in extraneous and unnecessary expenses.
To avoid surplus capacity, this study uses a binary-state physical line allocation strategy for allocation cost minimization and transmission reliability maximization, in which the ETN structure is given and represented by arcs and nodes. The binary-state physical line allocation focuses on determining an adequate allocation level for each arc, i.e., to determine the number of physical lines allocated to each arc in parallel based on the maximal demand flow. For instance, an arc with allocation level 3 is combined with three binary-state physical lines and owns four states including no line fails, one fails, two fails, and all fail. The ETN under such an allocation should be a typical multi-state flow network [22] as each arc consists of multiple states. Moreover, the cost of binary-state physical line allocation is counted in terms of the allocated physical lines. In particular, an allocated physical line that needs to be maintained or repaired owing to an accident or a disaster may cause the other lines allocated to the same arc to be partially or fully suspended. Such a phenomenon is called correlated failure, and it should be considered in transmission reliability calculation [20].
The addressed problem is a bi-objective optimization involving physical line allocation and reliability calculation, and thus, it should be NP-hard [4], [8]. Several recent multi-objective optimization problems related to the concept of physical line allocation [2], [11], [13], [17], [31] have been solved by the non-dominated sorting genetic algorithm II (NSGA2). The NSGA2 was proposed by Deb et al. [5], and it is a popular and powerful multi-objective evolutionary algorithm. It utilizes rank and crowding distance to represent the fitness value of a chromosome (i.e., solution) in a population. In its evolution process, a tournament selection operator, a crossover operator (such as single-point crossover, two-point crossover, and uniform crossover), and a mutation operator (such as one-point mutation and uniform mutation) are integrated to generate better offspring. Salazar et al. [32] pointed out that the NSGA2 can reserve elite solutions with low similarity and thus can avoid premature convergence. Cheng et al. [3] mentioned that the NSGA2 is easy to use because of its characteristic of modularization. Nojima et al. [29] suggested that the NSGA2 search diversity can be reinforced by avoiding selecting similar solutions in the tournament selection. Deb and Jain [6] extended the NSGA2 to develop NSGA3. However, the NSGA3 focuses on handling many-objective (having four or more objectives) optimization problems. Hence, it is not considered in this paper.
Considering the above discussion, this study proposes an improved NSGA2 (iNSGA2), which integrates the NSGA2 and a k-means algorithm, to solve the addressed problem. The k-means is utilized to partition a set of solutions into several subsets, in which the solutions in a subset are similar to each other, yet dissimilar to those in other subsets [14]. Accordingly, the k-means can reinforce the search diversity of the NSGA2 while implementing the selection operator. Because iNSGA2 involves transmission reliability evaluation for each binary-state physical line allocation, the concept of minimal path (MP, [37]), recursive sum of disjoint products (RSDP, [39]), and correlated binomial distribution model [20] are integrated to calculate transmission reliability. Moreover, a set of Pareto solutions (non-dominated solutions) is generated by the iNSGA2, and thus, it depends on supervisors to obtain the best compromise alternative from the Pareto set and put the optimal allocation in use. A technique for order preference by similarity to an ideal solution (TOPSIS) is adopted in this study to determine the best compromise alternative.
The remainder of this paper is organized as follows. Assumptions and problem formulation are described in Section 2. A multi-state ETN model with correlated failures and a transmission-reliability calculation algorithm are built in Section 3. Section 4 introduces the iNSGA2. A real ETN and a larger random network are adopted to demonstrate the usability and computational efficiency of the iNSGA2, while comparing it to NSGA2, and show the integration of iNSGA2 and TOPSIS to determine the best compromise alternative in Section 5. The conclusion of this paper is summarized in Section 6, along with future research.
Section snippets
Assumptions and problem formulation
An ETN is denoted by (A, N), where A = {ai|1 ≤ i ≤ n} denotes a set of arcs and N denotes a set of nodes consisting of several medium nodes, q source nodes (i.e., a set of branches: {Sw|w = 1, 2, …, q}), and a sink node (i.e., HQ, T). For each pair of Sw and T, there exists mw MP including ηw,1, ηw,2, …, and ηw,mw, where ηw,j denotes the jth MP connecting Sw and T. The length of ai is denoted by li for i = 1, 2, …, n. The capacity of a physical line is represented as μ. The probability that a
Multi-state electronic transaction network model and reliability measurement
Under allocation X, an ETN should be multi-state. This section describes the multi-state ETN model with correlated failures and proposes a transmission reliability calculation algorithm.
Development of iNSGA2
This section develops iNSGA2 for solving the bi-objective binary-state physical line allocation problem. In general, an optimization problem including minimal and maximal objectives is solved by transforming it into a multi-objective minimization problem or a multi-objective maximization one. Subsequently, a multi-objective GA is utilized to generate a set of Pareto solutions, where a solution is said to be a Pareto solution if its objective vector is not dominated by the objective vectors of
Numerical analysis
In this section, a real ETN and a larger random network are utilized to demonstrate the usability and computational efficiency of the iNSGA2. Under a given demand d, the goal is to determine the optimal allocation, with maximal transmission reliability and minimal allocation cost, for both networks. Fig. 2 shows the random network with 54 arcs, 13 MPs, and 8 branches. The length data assumed in Fig. 2 are listed in Table 1. Suppose an accident occurs in the area marked with a circle in Fig. 2;
Conclusions
This study attempts to find the optimal binary-state physical line allocation with maximal transmission reliability and minimal allocation cost for an ETN with correlated failures. We propose the iNSGA2 to solve this bi-objective optimization problem and then utilize the TOPSIS to decide the best compromise alternative. The iNSGA2 is a strengthened version of the NSGA2 and uses the k-means algorithm to explore the solution space more extensively. For two cases, a real ETN and a larger random
Acknowledgements
This work was supported in part by the Ministry of Science and Technology, Taiwan, Republic of China, under Grant No. MOST 105-2410-H-128-018.
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