Multicast routing is an effective way to transmit messages to multiple hosts in a network. However, it is vulnerable to intermittent connectivity property in mobile ad hoc network (MANET) especially for multimedia applications, which have some quality of service (QoS) requirements. The goal of QoS provisioning is to well organize network resources to satisfy the QoS requirement and achieve good network delivery services. However, there remains a challenge to provide QoS solutions and maintain endtoend QoS with user mobility. In this paper, a novel penalty adjustment method based on the rough set theory is proposed to deal with pathdelay constraints for multicast routing problems in MANETs. We formulate the problem as a constrained optimization problem, where the objective function is to minimize the total cost of the multicast tree subject to QoS constraints. The RPGA is evaluated on three multicast scenarios and compared with two stateoftheart methods in terms of cost, success rate, and time complexity. The performance analyses show that this approach is a selfadaptive method for penalty adjustment. Remarkably, the method can address a variety of constrained multicast routing problems even though the initial routes do not satisfy all QoS requirements.
Multicasting is a service method in which a source node can deliver copies of messages to multiple recipients at different locations in a communication network. Multicasting techniques play a critical role in many applications such as video conference, internet games, and webbased learning. In this paper, multicast routing problems mainly focus on finding a minimum Steiner tree and satisfying qualityofservice (QoS) requirements. Unfortunately, the problem of finding a Steiner tree is known to be a NPcomplete problem [
The multicast tree in mobile ad hoc networks (MANETs) is vulnerable to intermittent connectivity property during the transmission period [
The battery limitation of a mobile node is a critical constraint while developing multicast routing protocols. Genetic algorithm (GA) presents a potential solution for the multiconstrained multicast routing problem [
Different from several related researches [
The rest of paper is organized as follows. Section
At a certain time period in a MANET, we assume that the service provider knows
The cost of multicast tree
The bottleneck bandwidth of path
The delay of path
The delay of multicast tree
Figure
An example of a network graph, link parameters, and a Steiner tree.
The optimal multicast tree
The objective function is to minimize the total cost of the multicast tree. The pathdelay constraint enforces that the total delay of each OD pair must be smaller than or equal to its delay bound
Since there are so many candidate paths between two nodes in the network graph
An example of routing table for OD pair (1, 7) in Figure
Routing table of path from node 1 to node 7  

Route no.  Route path  Cost of the path  Delay along the path  Bottleneck bandwidth 
0  147  10  5  8 
1  167  11  6  10 
2  1457  12  8  8 






12387  24  10  12 
The proposed RPGA adopts a RP method to enhance the searching abilities of original GAs for handling constrained multicasting routing problems. To enhance the exploration ability, the RPGA adopts the RST to enlarge the genetic diversity by releasing inefficient constraints and also enforcing efficient ones when the generation number is odd. The flow chart of RPGA (in Figure
Flow chart of the RPGA.
In this paper, the encoding method is based on a routing representation for multicast trees. The RPGA maintains a population of chromosomes, which represent a candidate set of Steiner trees for the multicast routing problem. Given a source node
Representation of chromosome, genes, and routing table.
The fitness value of each chromosome represents the quality of the corresponding multicast tree (i.e.,
The proposed RP method adjusts penalty terms according to both violation magnitude and evolution time. To solve constrained optimization problems effectively, each individual in generation
A selection operation uses fitness to determine the solution quality and to select highquality chromosomes for the recombination operation [
The crossover operation represents the mixing of genetic material from two selected parents to produce one child chromosome. The RPGA proposes a therapeutic crossover that incorporates a genetherapy method with a conventional crossover scheme to enhance the exploitation ability and speed up the convergence rate [
Each time the selection operation chooses two crossover parents from the population. The therapeutic crossover gives each gene locus an equal chance of being a crossover point (i.e., belongs to a therapeutic genome
According to their relative merit, these two genomes combine to generate a new genome for their offspring. Therefore, offspring inherit more genetic material from the superior genome than the inferior one. We depict the pseudocode of the therapeutic crossover in Pseudocode
Pick chromosomes
Randomly select
Record path
Record path
Interchange the subroutes (s, m) or (m, d) in a and b;
Record path
Record path
Interchange the subroutes (s, m) or (m, d) in a and b
A mutation operation used in GAs can increase population diversity to enhance its exploration ability [
The proposed RPGA adopts a replacementwithelitism method to prevent best solutions from being lost through a selection process. A successive population is produced from three sources:
To address the multicast routing problem with QoS constraints, the challenge is how to optimize the objective function value against its constraint violations. Traditional GAs are ineffective in searching feasible solutions because genetic operations do not always preserve feasibility [
The novel RP method has been proposed in our previous work for numerical constrained problems [
The pseudocode of the RP method is depicted in Pseudocode
Create information table
//Divide
//Find characteristic from
//Find RP coefficient
//Modify RP penalty exponent
This work uses information granulation as a key function for implementing a divideandconquer strategy. Elementary information granules are indiscernibility classes of constraint violations. The information system is an information table of attribute values containing rows labeled by objects and columns labeled by attributes [
A partition granularity
A decision system is an IS with the form
Because the penalty multiplier should be adjusted according to the region of its constraint violation, this work enlarges the penalty multiplier when its violation level increases. In the illustration in Figure
Penalty multiplier classification.
Based on the concept of attribute reduction, attributes may not be equally important, and some of them can be eliminated from a decision table without degrading information quality. Attribute reduction can be generalized by introducing attribute evaluation, which can express the merit of each attribute in the information table [
Decision attribute
In this paper, the proposed RPGA is evaluated by solving multicast routing problems in MANETs. We use the wellknown network generation tool [
A randomly generated network with 40 nodes and average degree four.
In this paper, the performance metrics of solution algorithms consist of
It is well known that the performance of GAs significantly depends on the configuration of its operating parameters. To investigate the impact of various parameter settings in the RP method, this study experiments on two parameters: the partition granularity (
Firstly, this study conducts 30 runs to find the appropriate partition granularity (
The results obtained by different partition granularities (
Partition ( 
Cost  Success rate  CPU time 

4  1032.9  0.966666667  1.089747617 
6  1006.4  0.98245614  1.075849966 
8  1021.5  0.970175439  1.085799014 
Comparison with different partition granularities (in percentage relative to
Secondly, when the partition granularity is given (
The results obtained by different penalty coefficients (
Coefficient ( 
Cost  Success rate  CPU time 

10  1022.833333  0.989473684  1.084000564 
1.1  1031.966667  0.978947368  1.068427877 
1.01  1006.4  0.98245614  1.075849966 
1.001  1031.7  0.975438596  1.065008048 
1.0001  1022.333333  0.973684211  1.059503391 
Comparison with different penalty coefficients (in percentage relative to
The performance of the proposed RPGA is compared with two wellknown penalty methods, which are the Wang’s penalty (WP) method [
The results of different penalty methods.
Penalty methods  Cost (mean)  Cost (st. dev.)  Success rate (mean)  Success rate (st. dev.)  CPU time 

WP  1010.366667  82.23745406  0.907017544  0.077799684  1.074814229 
DP  1016.5  101.1191683  0.975438596  0.033095275  1.00755095 
RPGA  1006.4  83.13910238  0.98245614  0.028772225  1.075849966 
For the mean cost in Table
For comparison, all the results are normalized as percentages relative to the results of the RPGA (in Figure
Comparison with different penalty methods (in percentage relative to the RPGA).
The evolution curves of all three methods on the total cost and path delay are shown in Figures
Evolution curves of three methods on the performance metrics: (a) cost and (b) delay.
In the second test scenario, we randomly generate 8 test networks with the numbers of nodes from 10 to 80 to mimic the stress test for these three methods. In those tests, all delay constraint bounds are 60 msec and the multicast group size is 50% of network nodes. When nodes number increases, the network overhead increases obviously and endtoend delay increases at the same time. The experimental results depicted in Figure
Comparison with three methods for different numbers of network nodes in percentage relative to the RPGA on (a) cost, (b) success rate, and (c) CPU time.
In the third test scenario, we change the delaybound requirements from 20 msec to 90 msec in a test network, which has 40 nodes and its multicast group size is 20. The comparisons between the success rates and the delay bounds are shown in Figure
Comparison with three methods on the success rates for different delay bounds.
The proposed RP method cooperates with GAs for dealing with QoSbased multicast routing problems. The principle of the RP method is that the RPGA releases/enforces some penalties on inefficient/efficient constraints during evolution. Importantly, this approach can find the optimum or nearoptimal solutions even though the initial population includes infeasible solutions. The performance of the proposed algorithm was measured using three kinds of test scenarios and compared with two stateoftheart methods. Experimental results indicate that the proposed RPGA can find nearoptimal solutions and outperforms two existing methods for constrained multicast routing problems. The proposed algorithm is also robust in obtaining feasible solutions of all the test functions even though the WP method has smaller success rates for some difficult problems. In conclusion, the performance assessment also demonstrates that the proposed RP method has a remarkable capability to balance the objective function and the constraint violations as an effective and efficient method for solving a variety of QoSbased multicast routing problems.
We have observed that the computing effort of the RPGA is higher than that of other two penalty methods in smallscale networks. In the future, we will study the scalability of the proposed RPGA in finding multicast routes for dynamic MANETs with largescale dimension. Since MANETs allow ubiquitous service access without any fixed infrastructure, developing a distributed algorithm for high mobility environments is also our future work.