A Genetic Algorithm to Solve Capacity Assignment Problem in a Flow Network

Computer networks and power transmission networks are treated as capacitated flow networks. A capacitated flow network may partially fail due to maintenance. Therefore, the capacity of each edge should be optimally assigned to face critical situations—i.e., to keep the network functioning normally in the case of failure at one or more edges. The robust design problem (RDP) in a capacitated flow network is to search for the minimum capacity assignment of each edge such that the network still survived even under the edge’s failure. The RDP is known as NP-hard. Thus, capacity assignment problem subject to system reliability and total capacity constraints is studied in this paper. The problem is formulated mathematically, and a genetic algorithm is proposed to determine the optimal solution. The optimal solution found by the proposed algorithm is characterized by maximum reliability and minimum total capacity. Some numerical examples are presented to illustrate the efficiency of the proposed approach.

The analysis of a network structure has an effect on the capacity assignment strategy, as this analysis helps determine the critical nodes (the reliability value of an SFN equals zero when the critical node has zero capacity). Consequently, the best solution to the CAP must be found in order to decrease the number of critical nodes. Chen et al. [Chen and Lin (2008)] studied the CAP for SFNs with node failure-where the node has several states or capacities and may fail. They also studied the reliability of an SFN in the case of an existing critical node. Later, Chen et al. [Chen and Lin (2010)] defined the CAP as a robust design problem for an SFN and proposed an algorithm to solve the problem. Chen [Chen (2012)] discussed the robust design problem for an SFN in the case of each edge having several capacities and the potential to fail, proposing an algorithm to determine the minimum capacity assignment of each edge so that the network can still function. Chen [Chen (2012)] also stated that the problem is an NP-hard problem and proposed an exact algorithm to solve it. The capacity assignment in a stochastic-flow network is known to be NP-hard [Ball (1986); Chen and Lin (2010)], a relatively fast optimization algorithm would be beneficial. A genetic algorithm (GA) is a heuristic search method used in optimization problems. To solve reliability optimization problems, many based GA approaches were proposed such as [Younes and Hassan (2011)]. Also, GA is used to solve multiple objective optimization problems, [Taboada, Espiritu and Coit (2008) (2018)]. Therefore, this paper presents a GA to solve the capacity assignment problem. The objective is to minimize the total capacities while meeting the network reliability requirement. The rest of this paper is organized as follows. Section 2 sets forth the preliminaries. Section 3 presents the problem formulation. Then, Section 4 explains the proposed genetic algorithm (GA). Section 5 provides the steps of the entire algorithm. Section 6 includes several examples demonstrating the usability of the proposed approach. Finally, Section 7 draws conclusions and outlines possibilities for future work.

Network reliability
Given the demand and the set of minimal paths ( ), the system reliability Rd is defined as Lin [Lin (2001) for each i = 1, 2, … , n

Critical edge
The arc is said to be critical if and only if the network reliability , is zero, where , is the reliability of the given SFN when zero capacity is assigned to (failed).

Assumptions
The following assumptions should be satisfied for the given SFN: (i) Each node is perfect and has an infinite capacity.
(ii) The capacity of each arc has an integer-valued random variable.
(iii) The arcs are perfect and unlimited in capacity.
(iv) Flow of the given SFN satisfies the flow-conservation law [Ford and Fulkerson (1962)].
(v) The arcs are statistically independent.
3 Problem formulation Let = ( 1 , 2 , … , ) as assigned capacities to the set of edges ( 1 , 2 , … , ). The mathematical formulation of the problem is: is reliability corresponding to the assigned capacities under demand d and 0 is the network reliability requirement. ranges from 1 to , except for critical edges [Lin (2008)], = .

The genetic algorithm (GA)
The following subsections describe the different components of the presented GA.

Representation, crossover and mutation
The chromosome M is represented by a string of length n, where n is the number of edges, as follows: ( 1 , 2 , … , ) Figure 1: Chromosomal representation The one-cut point crossover is used to generate two new offspring, and a simple mutation process is used to mutate the offspring. Figs. 2 and 3 show the crossover process and the mutation process, respectively.

Fitness function
The following penalty function is used as a fitness function [Coit and Smith (1996);Altiparmak, Dengiz and Smith (1997); Gen and Cheng (2000)]: where Mmax is the maximum capacity of M, and The fitness function is where ℂ ( ) is the maximum value of Eq. (9) for the current population.

Selection process
The algorithm uses the roulette wheel selection mechanism to select new parents [Hassan (2020)], and the selection of a chromosome is based on its fitness value.

Conclusions
The capacity assignment problem was discussed in this paper and was formulated mathematically. A genetic algorithm was proposed to solve the problem. The presented algorithm successfully determined the optimal capacities with minimum total capacities and maximum reliability. The critical edges were treated in a special manner by being assigned capacity values equal to the demand. In comparison with Chen [Chen 2012], this paper finds the best capacity distribution to all studied cases. The proposed solution approach may be applied to various problem related to the capacity assignment problem.