The multicast routing problem is a well-known problem in combinatorial optimization. It is defined as finding the route between two nodes in the weighted graph where that path is the shortest, and shortest means the path with a minimum summation of weights, where an edge between any two nodes always has a certain weight. The problem is to find accordingly the shorter path between a source and a destination in computer networks.

Gen et al. [1] considered the problem of searching the shortest paths with two conflicting objectives of minimizing cost and maximizing flow, as a bicriteria network design problem. They proposed a multi-objective hybrid genetic algorithm (GA) to solve it.

Granat et al. [2], presented an interactive method to analyze the multicriteria shortest path problem by the reference point approach. The multi-objective problem was converted into a parametric single-objective problem. The algorithm succeeded to find the Pareto-optimal shortest path according to the specified preferences.

There are many applications such as multimedia conferencing, distant learning, and video on demand to encourage the network service provider to adapt their network to support additional multicast traffic. The multicast routing problem is the problem of searching a multicast tree that spans all vertices in a communication network [3]. Searching low-cost multicast tree or low delay multicast tree are discussed in [3–5].

To serve the penalty number of users and satisfy quality-of-service (QoS) in real-time applications, this problem is taken into consideration as NP-Complete [6]. Many optimization algorithms based on GA have been proposed to solve the QoS multicast routing (QMR) problem (with different types of QoS constraints) [6–10].

Authors in [11–13] discussed and solved the QoS with multiple constraints like bandwidth, delay, and packet loss rate. An ant colony based heuristic presented by Chu et al. [14] to search minimum cost multicast tree within the case of considering QoS metrics, like bandwidth, delay, delay jitter, and packet loss rate. While, Huang et al. [15], discussed low-cost multicast tree problem subject to delay constraints and ASDLMA (Ant system for delay-constrained low-cost multicast routing algorithm) was constructed to solve it.

It is known that GA is one of the heuristic algorithms that can solve many problems, network design problems [16], real road network problems [17], and unicast routing [18]. Also, GAs used to solve the multicast routing problem [19,20]. In addition, there is a constrained QoS problem [21–27] and [4].

In the case of considering more than one constraint like the cost of the tree, hop count, bandwidth utilization, the problem is considered as a Multi-Objective Problem (MOP) [28].

Ant colony optimization (ACO) is a meta-heuristic approach that has been applied to QoS multicast routing problems [29,30].

Younes et al. [29] presented an AC based algorithm to search a multicast tree with low-cost, minimum delay, and a minimum number of hops. The problem is treated as a multi-objective multicast tree problem.

In this paper, an algorithm based on GA is proposed to solve the multi-objective multicast tree problem. The experimental results prove that the solutions found by the proposed GA are better than those obtained by using AC presented by Younes et al. [30].

The rest of the paper is organized as follows: Section 2 presents the problem description and formulation. Sections 3 describe GA components. The entire GA algorithm is given in Section 4. Studied cases are presented in Section 5. Section 6 gives the conclusion.

Let G = (N, E) is a weighted directed graph with N vertices and E edges represents a network with |N| nodes and |E| links. The multicast tree from the source node n₀ to the set of destination nodes U={u1,u2,…,,um} denotes a set of destination nodes. Let X={n0,u1,u2,…,,um}∈N be a set of from source to destination nodes of the multicast tree. Multicast tree T = (N_T , E_T ), where NT⊆N and ET⊆E , there exists the path P_T (n₀ , d) from source node n₀ to each destination node d∈U in T. Three non-negative real value functions are associated with each link e ( e∈E ): C(e), D(e), and H(e). The link cost function, C(e), may be either monetary cost or any measure of resource utilization. The link delay functions, D(e), define the criteria. The link hop is the number of hops, H(e) = 1.

The cost of the path P_T is defined as the sum of the cost of all links in that path and can be given by

(1) C(PT)=∑e∈PT⁡C(e)

The total cost of the tree T is defined as the sum of the cost of all links in that tree and can be given by

(2) C(T)=∑e∈ET⁡C(e)

The total delay of the path P_T(n₀,d) is simply the sum of the delay of all links along with P_T(n₀,d):

(3) D(PT)=∑e∈PT(n0,d)⁡D(e),d∈U

The delay of multicast tree T is the maximum value of delay in the path from source node n₀ to each destination node d ∈ U.

(4) D(T)=max(∑e∈PT(n0,d)⁡D(PT),d∈U)

The hop of the path P_T is defined as the sum of the hop of all links in that path and can be given by

(5) H(PT)=∑e∈PT⁡H(e)

The hop of multicast tree is defined as the sum of the hop of all links in that tree and can be given by:

(6) H(T)=∑e∈T⁡H(e)

The vector SW(P_T) of the path P_T consists of the vector sum of the vectors corresponding to arcs.

(7) SW(PT)=C(PT)+D(PT)+H(PT);

The objective of the presented problem is to find a multicast routing tree (T) such that minimizes the cost C (T) , the delay D(T) , and the hop H(T) . The problem can be formulated as follows:

(8) MinimizeW(T)=∑e∈ET⁡(C(T)+D(T)+H(T))

where W(T) is the weight of a multicast routing tree (T). The cost C (T) , the delay D(T) , and the hop H(T) are defined as follows:

(9) C(T)=∑e∈ET⁡C(e)

(10) D(T)=max(∑e∈PT⁡D(PT))

(11) H(T)=∑e∈T⁡H(e)

If the given network has N nodes, then the candidate path is represented by a chromosome x of N fields, each field represents a node in the network. At least two fields have none zero values to consider the candidate path to be a real path (we called here the reality condition).

The following steps show how the presented GA solves the multi-objective multicast routing tree problem of a given network.

The presented GA is implemented using Borland C++ Ver. 5.5, where pop_size, max_gen, P_m, and P_c equals to 25, 50, 0.95 and 0.02 respectively. Two networks with 10 and 20 nodes are studied to show the efficiency of the proposed GA. Also, the results are compared with the AC algorithm presented in Younes et al. [30].

A multi-objective multicast routing problem subject to cost, hop, and delay is presented and formulated as a minimization problem. Furthermore, an approach based on GA is proposed to solve the presented problem. The experimental results illustrated that the proposed GA is efficient in solving this problem and searching the minimum W(T) in a few seconds. In addition, the results obtained by the proposed GA are better than those obtained by AC algorithm presented by Hamed et al. [30].