1 Introduction

Recently, the area of ad-hoc networks [1–5] has increased rapidly. It includes some of wireless nodes that self-organize to form a network without a recognized or centralized infrastructure. This network can be used anytime and anywhere without any fixed infrastructure. Due to being rapidly deployable and needing negligible configuration, it has several purposes such as emergency situations, rescue operation, military applications, conference meeting, disaster management, collaborative computing [6–8], etc. Each node cooperates with one another to forward data packet to their hop node or their destination node in a distributed manner [9]. According to battery energy and hardware complexity this network is classified as “Homogeneous” and “Heterogeneous”. In homogeneous networks, all nodes are identical, whereas in heterogeneous all nodes are different in terms of battery energy and hardware complexity. The diagram of both types of networks is shown in Fig. 1.

Fig. 1
figure 1

Types of ad-hoc network

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The ad-hoc nodes [10, 11] are powered by limited capacity of the battery. So they can function for limited time without changing or refilling their energy resources. Because of this reason, nodes might fail to achieve many goals. An example is e-commerce [12] which includes Custom data reporting, E-mail marketing, Payment processing, etc. During transaction in e-commerce session connection is lost because of limited energy. Then it has to be rolled back to a safe state for setting up new route and the entire session is re-started. The primary reasons for energy efficiency issues are given below:

  1. a.

    Selection of the best transmission power: Higher transmission power results in high energy utilization and high intervention among nodes [13].

  2. b.

    Lack of base station: This network works as a distributed network. Because of this reason, some nodes may work as relay nodes, when relay traffic is heavy and the power consumption is high [14].

  3. c.

    Difficulties in replacing the batteries: In emergency situations, battery replacement in military application is not possible [15].

  4. d.

    Limitation on the battery source: The capacity of batteries is too small. So, it is stores limited energy [16].

Multicast routing is a communication between one or multiple sources and multiple destination nodes. It helps to enhance efficiency by controlling network traffic and decreases CPU loads as well as server loads. It also helps to optimize performance of the network by eliminating traffic redundancy.

During communication in network, several changes occur in energy resources, data traffic and high bit error rate in the wireless channel due to uncertain environment and dynamic topology. So network suffers from two kinds of difficulties such as randomness and fuzziness in multicast routing. These difficulties help to influence some paradigms of collisions shown in Fig. 2. These difficulties are main reasons of four vital network issues such as data dropped, node dispute, data retransmissions, and jamming. Due to these reasons performance of the network degrades.

Fig. 2
figure 2

Some paradigms of collisions

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1.1 Motivation

Many multicast routing protocols [17–21] have been proposed which enhance the quantitative performance of the network. But major concern is the qualitative performance that depends on the quality of the links. For instance, if energy of batteries is low for most of the nodes in a path then chances of path failures are high. In a comparable way, most of the links suffer from huge amount of packet loss due to insufficient energy. However, it is indeterminate that a link is a weak at the time of low battery because this types of information is imprecise. To handle this unpredictable situation the proposed multicast routing protocol is used which has intellectual capabilities based on fuzzy logic. In this protocol, at the time of join query processing, decision maker fuzzify two parameters such as residual energy and distance of previous as well as subsequent nodes with the help of analytical model of fuzzy inference system and reduce interference among nodes. Hence, it helps to enhanced qualitative performance of network.

1.2 Contributions

The proposed Intellectual Energy Efficient Multicast routing protocol (IE2M) is proposed to overcome the limitations (e.g. high packet overheads, low packet delivery ratio, and high average end-to-end delay) of ODMRP when multicast receivers and multicast traffics are increases.

The key contributions of this paper are as follows:

  1. a.

    Construct mesh by using packets join query and join reply by additional information such as energy, distance, and reward.

  2. b.

    Fuzzify two input parameters (i.e. energy and distance) for previous node as well as subsequent node with the help of membership functions.

  3. c.

    Evaluates these two parameter in fuzzy inference system to get fuzzy value of output parameter reward.

  4. d.

    Construct routing path and assign reward to each path for packet delivery.

  5. e.

    It helps to distinguish different multicast route and reduce the effect of mutual interference among routes.

  6. f.

    It helps to reduces energy consumption, network overhead and end-to-end delay.

  7. g.

    It also helps to preserve routes completely by handling node failure and link failure before they crashed.

1.3 Organization of the paper

The remainder of the paper is organized as follows. In Section 2, the related works are presented and commented. In Section 3, offer a general idea of the basic of multicast routing, fuzzy logic and energy efficiency. The concise description of the proposed protocol is given in Section 4. Section 5, demonstrates the performance evaluation including simulation set-up and performance metrics. Finally, Section 6 provides the concluding remarks and lists future scope.

2 Related works

In this section, a concise discussion based on related work has been illustrated such as: [20] is an extension of AODV protocol. But AODV is a unicast protocol where Multicast Ad-hoc On-demand Distance Vector (MAODV) is a multicast routing protocol. This protocol contains multiple groups with sequence number and a unique address. If any node desires to connect to a group then it becomes the leader of that group and joins it. After joining the group node is liable for preserving the whole group. Basically, it is a tree base multicast routing protocol. The limitation of this protocol is that it cannot repair the routes partially or quickly due to a single route between source to multicast groups.

In On-Demand Multicast Routing Protocol (ODMRP) [21], more than one path between a source node to a multicast group is present. In this protocol, if source node wishes to transmit data to its receiver then it floods J-Q message within the network. Every intermediate node that receives J-Q messages record two attributes named as, source ID and sequence number in its catch to determine the path.

Su et al. [22] offered the mesh based Fuzzy logic Modified AODV Routing (FMAR) protocol for multicast routing. The foremost aspire of this proposal is a dynamic evaluation of the agile paths using fuzzy logic approach. It helps to supervise partial bandwidths of wireless links. But, it has an inferior performance as it does not evaluates rating of every path, rather it selects few paths only. So, routes which are more useful cannot be determined.

Biradar et al. [23] proposed a mesh based routing protocol to find a secure multicast route between the source and receivers. There are two type of acceptances namely, route request and route reply which are used to create multi-cast mesh. These two acknowledgements also use database information that is stored in cache of each node. It focused on mesh strategy for designing multicast tree. But it has a drawback that it cannot discuss about route repairing.

Torkestani and Meybodi [24] proposed a multicast routing algorithm. This algorithm approximates the usual relative mobility of each mobile node. The algorithm also demonstrates that how virtual multicast backbone can be outlined using Steiner connected dominating method. It uses a stochastic way to evaluate multi-cast route within weighted graph. It also used relative mobility for each mobile node. Finally, it solves a routing problem with the help of learning automata. It is not only difficult, but also not suitable for simple application network.

Jahanshahi and Barmi [25] make a comprehensive study on the multi-cast routing protocols. In this survey they mentioned that there are different types of multicast routing methods which have been examined based on different channels. They also present a relative swot up on multiple channels and radios. In [26] a ring based multi-cast routing protocol. And a comprehensive swot of dissimilar multi-cast communication methods in IEEE 802.11aa is offered in [27].

Hui Cheng and Shengxiang Yang [28] proposed a multicast routing protocol based on tabu search, genetic algorithm and simulated annealing. In this routing protocol every node has a channel conflict and current forwarding node may not have dynamic transmission. Therefore, this protocol processes nosiness moderately not completely.

Zeng et al. [29] proposed two deterministic methodologies which are level channel assignment and multichannel multicast. In multicast tree construction both methods are considered as an allocation of channel in a separate step. In these methodologies, multi-cast tree is constructed with the help of breadth first search algorithm. At the same time multicast tree construction helps to reduce the hop count and relay nodes with breadth first search algorithm. The authors illustrate that multichannel multicast outperforms level channel assignment. It is surprising that the author did not discussed about energy efficiency which is a vital parameter.

He et al. [30] proposed a protocol to solved the issue of data propagation in wireless sensor and actor networks with the help of actor. Actor is a network entity which take decisions and perform specific operation based on decision making. In this protocol, during transmission, source actor broadcasts data packet to sensors and sensors forward it to other actors within range at lowest communication cost and limited time bound.

Jahanshahi et al. [31] proposed a tree based multicast routing protocol using automata theory. In this protocol there are two phases: tree construction and determining optimized tree. In another work of the same author [32] is a cross layer design for an allotment of channel issue and creation of multi-cast tree with the help of integer programming. In another research [33], the authors propose multi objective genetic algorithm method to interpret multi-cast routing problem. In this work authors uses non-dominated sorting algorithm for designing multicast tree and channel allocation.

Tehuang Liu and Wanjiun Liao [34] proposed an algorithm for multicast routing. Basically, this algorithm design for dynamic traffic model by the help of integer linear programming. The drawback of the algorithm is network not adjustable. So, the authors [35] design this multicast routing using same integer linear programming. The foremost assignment of specifying a routing protocol is avoided multicast tree construction and encourages the channel task as main dilemma.

Nguyen (H. L) and Nguyen (U. T) [36] proposed a method for multi-cast routing within a wireless mesh network. This algorithm works on multicast tree and it cannot be fabricated the most favorable result. So in [37] the authors initiated a method to calculate the services of connections. The computation of this method tolerates a momentous overhead. Hence, in [38] another method is proposed named minimum number of transmissions. This method illustrates the reduction of data packets which replicate within different channels of the node.

Farzinvash and Dehghan [39] proposed an algorithm for multicast routing. This algorithm consists of two strategies for tree creation. Firstly, each branch of the tree consists of less number of relay nodes. Secondly, select different shortest path that are connected to different receivers.

Li et al. [40] planned a routing protocol based on multicast method. This routing method involves two metrics named as: flow load multicast metric and reliable flow load multicast metric. The second method is an extension of the first method. Searching path between source to group in terms of reduced intrusion is the main aim of the first method. Intrusion is reduced in two ways like intra-flow and inter-flow. It utilizes channel range for enhancing bandwidth performance. At last, the authors evaluate both methods in the context of multicast routing.

Jahanshahi et al. [41] planned a routing protocol for multi-cast based on cross-layer design. This is tree based multicast routing method. The authors used integer programming as a meta-heuristic approach to construct tree and find the best path from the source node to each multicast group.

However, none of these above mentioned methods handle uncertainties in the ad-hoc network with parameters, energy along with distance and network lifetime issues together with the respect of multi-cast routing. Hence, in proposed protocol, both the issues have been considered. Assigning reward to each multi-cast route is the foremost advantage of this proposed protocol. Hence, before any route crashes it assists to renovate that route perfectly.

3 Preliminaries: concepts and definitions

Some prefaces are discussed in this section which serves as a foremost character to outline the proposed protocol. Short illustrations of these preliminaries are stated given below:

3.1 Fuzzy logic

Fuzzy logic is a part of soft computing [42, 43] which helps to control the uncertainty. The control mechanism based on linguistic values is used by the decision maker when coming up with an estimate. It is different from crisp logic because crisp logic deals with exactly 0 or 1 but it deals with between 0 and 1. Many authors [44, 45] apply it within ad-hoc network to solve routing problems and came up with good approximate solutions.

3.2 Energy efficiency

The Ad-hoc network is dynamic in nature because wireless nodes are enabled with finite capacity of the battery. An account of battery lifetime is too finite, which is insufficient during any mission. Various sources of energy expenditure in wireless node exists. There are various reasons for energy expenditures namely, the node is in nap mode, indolent modes, receiving mode, and sending mode. Energy failure of nodes affects only the node itself as well as whole networks. Hence, energy-efficiency signifies to reduce the capacity of energy required to provide some operations in the network.

3.3 Multicast routing

Multicast routing is a one to many or many to many communications within sender and receivers. So, it is a technique of creating the best possible path between the source and destination node. There are two attributes named as, source IP address (S) and group addresses (G) which exist in a single source node. The combination of both these addresses is signified as (S, G). Some time they are commonly used as (*, G) in the case of multiple source for a particular group. The router creates a table with the entries (S, G) and (*, G) to establish routes between source and destination node. It helps to enhance reliability of packet communication, since a lone data packets is replicated with more than one data packet.

4 IE2M: intellectual energy efficient multicast routing protocol

4.1 Overview

In this section, FLS is used to design proposed protocol. There are total five phases used in this protocol. Each and every phase plays an imperative role in proposed protocol. In the first phase, the network information model is defined for the proposed protocol with two zones. In the second phase, multicast mesh is created with the help of three packets named as J-Q, J-R, and HELLO. In the third phase, energy aware route is evaluated with the help of two basic network parameters such as energy and distance. In the fourth phase, analytical model of intellectual Fuzzy Inference System (FIS) is defined for proposed protocol. The main aim of this phase is to generate output of FIS. Finally, in the fifth phase, the energy efficient multicast route is uniquely selected by using two parameters energy and distance. Illustrations of these phases are given below:

4.2 The network information model

In this network model, G is an undirected graph as G = (V, P). The set of vertices (nodes) and edges (links) are represented as V and P respectively. Each link is incident with two nodes. These two nodes may or may not be within a single communication range. The set of neighbor nodes are signified by ξ and (x i , y i ) is the location of node N i . This location denoted by Γ i . The distance between two nodes j and k is signified by |Γ(j)−Γ(k)|. The source node located at Γ(s) and group address located at Γ(g). The value of Γ(s) and Γ(g) are given in Eqs. 1 and 2. The c 1, c 2, c 3 and c 4 are the set of four points which denoted as a rectangle area (Δ). This rectangular area is a network zone of the proposed model where the source and multicast groups are situated. The values of different points are c 1 = (x 1, y 1), c 2 = (x 1, y 2), c 3 = (x 2, y 2), and c 4 = (x 2, y 1).

$$ \Gamma_{(s)} = s (x_{s}, \; y_{s}), \;\;\; where \; x_{s}=x_{1}, \; y_{s}=x_{2}. $$
(1)
$$ \Gamma_{(g)} = g (x_{g}, \; y_{g}), \;\;\; where \; x_{g}=y_{1}, \; y_{g}=y_{2}. $$
(2)

The transmission area of the proposed network model is divided into two zones with V vertices and P links named as, “Safe zone” and “Unsafe zone” as shown in Fig. 3. The safe zone located within the radio range and it is divided into n virtual tracks based on the chronological order of the linguistic nature of energy, distance, and transmission range of the nodes as shown in Fig. 4. The requested track with respect to group nodes (g) in track 1, 2, 3, . . . . n, denoted by T i (s, g) → (1 ≤ i ≤ n). Relay nodes are the nodes that are located in the safe zone. These nodes are free from any type of natural disaster such as flood, storm etc. So, this area is considered as a stable area because of being free from any type of hindrances. The end of transmission range is the location of unsafe zone. The probability of link breakage is high in this zone due to lack of transmission range. The node of unsafe zone is called unsafe node because of frequent link failure and Denial of Service (DoS) attacks [46]. So, these nodes cannot participate in exchanging information. This zone is unstable due to lack of radio range and safety. Therefore, in emergency situation few nodes change their place from unsafe zone to safe zone.

Fig. 3
figure 3

The transmission zone area

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Fig. 4
figure 4

The safe zone network model

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4.3 The multicast mesh creation

In this phase, multicast mesh is created with deploying few mobile nodes within the network zone. At this time, initial energy (E i n i t ) is defined on each and every mobile node randomly. Multi-cast mesh creation involves four types of packets named as J-Q, J-R, Join Error (J-E) and HELLO. The J-Q packet contains five fields such as S r c_a d d (source IP address), S r c_s e q # (source sequence no.), B i d (broadcast id), G r p_a d d (group address), G r p_s e q # (group sequence no.). Every node keeps a routing table for storing routing information such as group address, next hop address, hop-count, sequence no., type, status, route list and two input parameters such as residual energy (E) and distance (D), and output parameter is reward (R ϕ ). The type field stores category of different classes. The categories of different classes are distributed as g r o u p 1, g r o u p 2, g r o u p 3 and so on. Status field stores 1 if the node located is in safe zone otherwise 0 for unsafe zone. The route list field stores different route between source and multicast group. The link between two nodes N i and N j if and only if there is no other node N l ∈ N ξ that is closer to either N i or N j . Formally, the distance d(N i , N j ) between N i and N j is the Euclidean distance as shown in Eq. 3. Residual energy of each node is calculated from Eq. 4. The two input parameters E and D are updated during transmission in the form of cumulative. The R ϕ is calculated based on both input parameters when J-Q arrived at multicast group. At the initial stage, source node checks for route in its routing table when it wants to send data packet to the multicast group. If the source node contains route then simply sends the data packet. Otherwise, it will reserve the data packet and generate HELLO message for indicating neighbor nodes. The purpose of this HELLO message is used to initiate the J-Q process by spreading J-Q message to its neighbor. When the optimal path for multicast group is originated then J-R packet is forwarded. The optimal path is a path having highest R ϕ . The strategy of J-Q and J-R is shown in Fig. 5. By this way, the proposed protocol designs a mesh for sending the data from source node to a multicast group. Forwarding nodes help to transmit data from one hop to another hop during data transmission.

$$ max \{d(N_{i}, \; N_{l}), \; d(N_{j}, N_{l})\} < d(N_{i}, N_{j}). $$
(3)
$$ E_{res} = E_{init} - E_{con}, $$
(4)

where E i n i t is initial energy and E c o n is energy consumed of a node at time t. E c o n can be calculated from Eq. 5.

$$ E_{con} = En_{t} + En_{r}, $$
(5)

where E n t indicates energy consumption at the time of transmitting packets and E n r indicates energy consumption at the time of receiving packets.

Fig. 5
figure 5

The J-Q and J-R strategy for mesh creation

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In the proposed protocol, the decision maker can select relatively two nodes disjoint routes with maximum reward. The second route is used when the first route is unexpectedly defective. This route is used to enhance the reliability of the network. In this work, we enhanced ODMRP and introduce the fuzzy logic for allotting R ϕ to each selected route in the routing table. This method is useful for evaluating R ϕ with the help of a dissimilar energy level and distances under the fuzzy environment. The control flow diagram of the basic operation of the proposed protocol is given in Fig. 6 and detail description of energy aware route evaluation is given in next phase.

Fig. 6
figure 6

Control flow diagram

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4.4 The Energy-aware route evaluation

In this phase, application of fuzzy logic is illustrated with the help of two phrases: multiple intermediate nodes and route evaluation which are stated as follows:

4.4.1 The application of fuzzy logic to multiple intermediate nodes

Ad-hoc networks are associated with various uncertainties (i.e., resource, mobility, channel, traffic load, routing, etc.) due to a number of stochastic processes such as unexpected dissimilarities in network load, continuously changing propagation channels, escalating number of access points, and random mobility of user. As a result of which topology of the network is dynamic. In order to manage the uncertainties, fuzzy logic is used. It represents uncertainty and imprecise knowledge efficiently. In the proposed protocol, it handles imprecision and uncertainty with the help of input parameters (i.e., energy and distance) and output parameter reward. The input and output parameters are varied based on the linguistic nature of fuzzy logic. During communication, this variation copes with the help of FIS. At the time of communication between two nodes N i and N j energy and distance should be updated as a cumulative format in routing tables of two nodes. Hence, stored energy and distance of routing table N j is more than N i . This process repeats continuously until J-Q reaches the multicast group. When J-Q reaches multicast group then final energy and distance process with the help of different steps of FIS and generates output reward for each multicast route. Based on finest route J-R is send. The detailed description of route evaluation using fuzzy logic given in the next phase.

4.4.2 The application of fuzzy logic for route evaluation

In this section, application of fuzzy logic is illustrated for energy aware route evaluation. In proposed network model, each and every node is used to send the information on defined area is Δ. Sufficient energy and suitable distance are linguistic variables which are not defined precisely. Therefore, ξ is a set of dynamic nodes is given by:

$$\xi = \{{N_{1}}, {N_{2}}, {N_{3}}, {N_{4}}, {N_{5}}, {\ldots} {N_{\xi}}\}\\ $$

The equivalence class of each node corresponding to different energies (E V L , E L , E M , E H and E V H ) are given by:

$$\begin{array}{@{}rcl@{}} E_{VL} &=& \{ e_{1}, e_{2}, e_{3}, e_{4}, {\ldots} e_{i} \}\\ E_{L} &=& \{ e_{i+1}, e_{i+2}, e_{i+3}, e_{i+4}, {\ldots} e_{j} \}\\ E_{M} &=& \{ e_{j+1}, e_{j+2}, e_{j+3}, e_{j+4}, {\ldots} e_{k} \}\\ E_{H} &=& \{ e_{k+1}, e_{k+2}, e_{k+3}, e_{k+4}, {\ldots} e_{l} \}\\ E_{VH} &=&\{ e_{l+1}, e_{l+2}, e_{l+3}, e_{l+4}, {\ldots} e_{m} \}\\ \end{array} $$

where E V L , E L , E M , E H and E V H are pairwise disjoint sets and E V L ∪ E L ∪ E M ∪ E H ∪ E V H = ξ.

The equivalence class of each node corresponding to different distances (D V S , D S , D M , D L and D V L ) are given by:

$$\begin{array}{@{}rcl@{}} D_{VS} &=& \{ d_{1}, d_{2}, d_{3}, d_{4}, {\ldots} d_{i} \}\\ D_{S} &=& \{ d_{i+1}, d_{i+2}, d_{i+3}, d_{i+4}, {\ldots} d_{j} \}\\ D_{M} &=& \{ d_{j+1}, d_{j+2}, d_{j+3}, d_{j+4}, {\ldots} d_{k} \}\\ D_{L} &=& \{ d_{k+1}, d_{k+2}, d_{k+3}, d_{k+4}, {\ldots} dl \}\\ D_{VL} &=& \{ d_{l+1}, d_{l+2}, d_{l+3}, d_{l+4}, {\ldots} d_{m} \} \end{array} $$

where D V S , D S , D M , D L and D V L are pairwise disjoint sets and D V S ∪ D S ∪ D M ∪ D L ∪ D V L = ξ.

Here, G B is used as a complete bipartite graph given as G B = (V i , V j , P) for the above equivalence classes. This graph resolves energy efficient routing issue shown in Fig. 7. The graph G B consist of two subsets of vertices as V i and V j , where V i is a set of different energies and V j is a set of different distances. P is a set of different links. In this graph, v i ∈ V i and v j ∈ V j where v i v j is an edge in P. X and Y is the size of two partitions named as, energy set and distance set denoted by X= ∣V i ∣ and Y= ∣V j ∣. Table 1, summarizes the notations used in this graph G B to help the readers to understand this paper.

Fig. 7
figure 7

Equivalence classes of input parameters

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Table 1 Notations of the proposed complete bi-partite graph
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Exchange of data packet within one node to another node based on finest situation. Here, finest situation is not properly defined because of uncertainty related to mobile nodes. This finest situation actually indicates route having shortest distance with high energy. The nature of distance and energy are linguistic which have different membership values. For an example, the value of X and Y are assumed as 5 for assigning different membership function of the input parameters. In this proposed protocol, the triangular membership function is used for E and D which are defined in Tables 2 and 3. The output parameter R ϕ is evaluated based on both parameters. In Tables 2 and 3, range of linguistic values are E V L−, E V L+, E L−, E L+, D V S−, D V S+, D S−, D S+ and so on defined as increasing in nature. Detail description of FIS for proposed protocol is given in the next phase.

Table 2 Triangular membership functions for energy
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Table 3 Triangular membership functions for distance
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4.5 Analytical model of FIS

In IE2M, FIS plays the role of the intellectual system. Basically, three steps of FIS model are involved to deal with this proposed protocol named as fuzzification, inference engine, and defuzzification.

4.5.1 Fuzzification

In fuzzification, actual values (crisp) of input parameters (E and D) are changed into the Degree of Membership (DOM). This DOM provides different set of fuzzy input with the help of Membership Functions (MFs). A MFs is a non-linear curve signified by μ(x) that defines the manner in which each value in the crisp input space is mapped to a DOM. The range of the DOM is R D O M , where R D O M ∈ [0, 1]. There are several membership functions available such as Triangular MFs, Trapezoidal MFs, Gaussian MFs, Piecewise linear MFs, Singleton MFs. According to a survey [47], it is clear that Triangular MFs provide better results than other MFs in real life scenarios. Hence, the proposed protocol used Triangular MFs for fuzzification of input parameters. The Triangular MFs are specified by three parameters {a, b, c} is shown in Eq. 6. In Fig. 8, overall control flow diagram of FIS for IE2M is illustrated. In this figure, x i is an input parameter, where i=1 for E, and i=2 for D. Now first input parameter (x 1) is divided into a range of fuzzy sets by using Triangular MFs such as Very Low (VL), Low (L), Medium (M), High (H) and Very High (VH). Here, second input parameter (x 2) is not defined, but it illustrates as same x 1. In this MFs, x i no longer jumps suddenly from one fuzzy set to the next. Instead, x i loses value is one membership function and gains value in the next as it changes. If x 1 = λ 1 then x 1 is VL, which has a membership value α 2; x 1 is L, which has a membership value 1; x 1 is M, which has a membership value α 1. In other words, we are mapping x 1 = λ 1 into a set of membership values (α 1, α 2, 1).

$$ triangle(x;a,b,c) = \left\{\begin{array}{ll} 0 &\text{ if } x \leq a. \\ \frac{x - a}{b - a} &\text{ if } a \leq x \leq b. \\ \frac{c - x}{c - b} &\text{ if } b \leq x \leq c. \\ 0 &\text{ if } c \leq x. \\ \end{array}\right. $$
(6)
Fig. 8
figure 8

Overall control flow diagram of intellectual FIS modeling for IE2M

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By using min and max, we have an alternative expression of the preceding equation is shown in Eq. 7:

$$ triangle(x;a,b,c) = max\left[min\left[\frac{x - a}{b - a}, \frac{c - x}{c - b}\right], 0\right]. $$
(7)

The parameters {a, b, c} (where a < b < c) determine the x coordinates of the three corners of the underlying Triangular MF.

4.5.2 Fuzzy implication and fuzzy inference

Fuzzy implication or fuzzy rules consist of logical rules that determine relationships between input and output parameters. The form of fuzzy implication is shown in Eq. 8:

$$ IF \; antecedent, \; THEN \; consequence . $$
(8)

In this implication, antecedent (β 1) is composed of one or more fuzzy input parameter with/without a connection operator. The consequence (β 2) is a fuzzy output parameter. These fuzzy implications provided by experts those are often based on common sense and are consistent with some logic. The rule base of FIS in proposed protocol can be accessed formally in the form of Eq. 9.

$$ R_{i}: \; IF \; \overbrace{(x_{1} \; is \; A \; and \; x_{2} \; is \; B)}^{\beta_{1}} \; THEN \; \overbrace{(R_{\phi} \; is \; C) ,}^{\beta_{2}} $$
(9)

where x 1 and x 2 are two input parameters for E and D, R ϕ is the output parameter, A is the fuzzy sets of input parameter E, B is the fuzzy sets of another input parameter D, and C is the fuzzy sets of the output parameter R ϕ .

Fuzzy inference engine processes the fuzzy implications by devising a set of the given input fuzzy set to a single output sets. It first evaluates fuzzy implication then determines their firing strength. Mamdani method [48] is one of the most common and efficient method for defining the firing strength of a rule. In this method, firing strength of the rule is generated by β 1. For illustration, proposed protocol has 25 implication based on linguistic natures of x i is given in Table 4.

Table 4 Fuzzy inference table for route evaluation
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Before calculating output parameter reward (R ϕ ), states (S ϕ ) is calculated for each and every route by Eq. 10. All possible states for 25 rules are given in Table 5.

$$ S_{\phi}=S_{ij} = \frac{Ratio\; of\; energy}{Ratio\; of\; distance}{,} $$
(10)

where i∈1 ≤ i≤ 5 for E V L , E L , E M , E H , E V H and j∈1 ≤ j≤ 5 for D V S , D S , D M , D L , D V L .

Table 5 State of different route
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There are three possible categories are found in Table 5, such as C1{S6, S10, S11, S15, S16, S20}, C2{S1, S5, S7, S8, S9, S12, S13, S14, S17, S18, S19, S21, S25} and C3{S2, S3, S4, S22, S23, S24}. Every category contains few duplicate state that are increases energy inefficiency for different routes. For reducing duplicate state reward is calculated based on Eq. 11. All possible rewards for 25 rules are given in Table 6.

$$ R_{\phi}=R_{k} = \frac{Mean \; of \; energy}{Mean \; of \; distance}{,} $$
(11)

where k∈1 ≤ k≤ 25 for different linguistic nature in chronological order.

Table 6 Reward of different route
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4.5.3 Defuzzification

Fuzzy inference engine produces a set of fuzzy output set. These are combined in the composition phase to receive single modified fuzzy output. Defuzzification method is used to convert fuzzy output to crisp output. There are several defuzzification methods available such as Centroid, Center of Gravity, Center of Gravity for Singleton, Left Most Maximum, Right Most Maximum. The most frequently used and more computational method is Centroid. So this method is used for defuzzification.

4.6 The Energy-aware route selection

In this phase, optimal route is selected with the help of fuzzy implications. In Table 4, a fuzzy rule base containing 25 rules with two input parameters of G B are energy (V i ) and distance (V j ) are given. The coverage of both the parameters lies between 1 to 5 i.e. 1 ≤ (i,j) ≤ 5. This coverage helps to demonstrate the Fuzzy rule matrix (F r m ) shown in Eq. 12. In this matrix, rows are labeled by different linguistic natures of first input parameter energy and columns are labeled by different linguistic natures of second input parameter distance. Based on F r m rewards are arranged in non-increasing order with their linguistic natures given in Table 7.

$$\begin{array}{@{}rcl@{}} && {\kern3pt}E/D{\kern13pt} D_{VS}{\kern5pt} D_{S}{\kern10pt} D_{M}{\kern9pt} D_{L}{\kern9pt} D_{VL}\\ F_{rm}&=&\begin{array}{l} E_{VL}\\ E_{L}\\ E_{M}\\ E_{H}\\ E_{VH} \end{array} \left( \begin{array}{llllll} R_{1} & R_{2} & R_{3} & R_{4} & R_{5}\\ R_{6} & R_{7} & R_{8} & R_{9} & R_{10}\\ R_{11} & R_{12} & R_{13} & R_{14} & R_{15}\\ R_{16} & R_{17} & R_{18} & R_{19} & R_{20}\\ R_{21}\phantom{0} & R_{22}\phantom{0} & R_{23}\phantom{0} & R_{24}\phantom{0} & R_{25} \end{array}\right) \end{array} $$
(12)
Table 7 Linguistic nature of different reward
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The natures of these rewards are linguistic like Very Excellent, Excellent, Less Excellent, Very Good, Good, Mild Good, Medium, Mild Poor, Poor, Very Poor, Mild Bad, Bad, Very Bad and so on. The series of the rewards in descending order is given below:

$$\begin{array}{@{}rcl@{}} &&R_{21} > R_{16} {>} R_{11} {>} R_{22} {>} \{R{_{6}}, R_{17}\}{>}R_{12} {>} R_{23} {>} R_{18}\\ &&>R_{24} {>} \{R_{1}, R_{7}, R_{13}, R_{19}, R_{25}\}{>} R_{20} {>} R_{14} > R_{15} {>} R_{8}\\ &&>\{R_{2}, R_{9}\}{>} R_{10} {>} R_{3}{>} R_{4}{>} R_{5} \end{array} $$

Therefore, reward, R21 (Very short distance and Very high energy) is the best choice and R5 (Very long distance and Very low energy) is the worst choice in multicast routing. Other selected route as R16, R11 and R22 work as an alternative route at the time of node or link failures. Hence, these routes perform the operation of route maintenance if any route is failing to transmit the data.

5 Performance Evaluation

In this section, IE2M and other existing protocols (MAODV [14], ODMRP [15] and FMAR [16]) are evaluated based on different network metrics. For a fair comparison, the proposed protocol is compared with the existing multicast routing protocols.

5.1 Simulation model and parameters

The simulation structure is based on NS-2 environment. In simulated structure 100 wireless nodes were scattered randomly in areas of 1000 X 1000 m 2. Maximum speed of wireless node is 20 m/s. Total simulation time required for simulate this protocol is 1000s. The software requirement specifications are given in Table 8. The details of simulation constraints are described in Table 9. The following metrics are considered to measure the performance of the proposed protocol.

Table 8 Software requirement specification
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Table 9 Simulation environment
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5.2 Performance metrics

The performance is evaluated based on packet delivery ratio, packet overhead, average end-to-end delay on number of multicast receivers, and multicast traffic. The multicast receivers are varied as 10, 15, 20, 25, 30, 35, 40, 45, and 50. The multicast traffic with packet per second is 512 and packet size is varied as 5, 10, 15, 20, 25, 30, and 35.

Packet delivery ratio (PDR): It is the ratio of the number of packets received to the number of packets sends. It is calculated from Eq. 13.

$$ PDR= \frac{{\sum}Number \; of \; packets \; received}{{\sum} Number \; of \; packets \; send}{.} $$
(13)

Packet overhead:

The packet overhead is calculated as the ratio of all the transmitted routing query packets over the number of successfully received messages.

Average end-to-end delay (AED):

It is average time taken using a data packet to arrive at the destination. It also includes the delay caused by route discovery process and the queue in the data packet transmission. It is calculated from Eq. 14.

$$ AED= \frac{{\sum}(arrive \; time - send \; time)}{{\sum}Number \; of \; connections} . $$
(14)

In Table 10, some of the characteristics of the proposed protocol with other existing protocols is given. It can be noted that delay and routing overhead of the proposed protocol are less than other protocols. Packet delivery ratio, scalability, bandwidth, robustness, handling high mobility, handling traffic load, and handling multipath of the proposed protocol is better than other protocols due to the reduced effect of mutual interference between routes. The proposed protocol has better QoS support and the cost of joining a new group is much more cheaper compared to other protocols since the proposed protocol has a property FIS which handle the high unpredictability and uncertainty of the network.

Table 10 Comparison of proposed protocol with other protocols
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To validate the proposed protocol, the performance metrics are evaluated on multicast group size (multicast receivers) and multicast traffic load.

5.3 Outcome based on multicast group size

This section demonstrates the performance of the proposed protocol with existing protocols in terms of packet delivery ratio, packet overhead, and average end-to-end delay on multicast group size.

The multicast group size is varied from 10 to 50 nodes. Figure 9 evaluates the performance of scatter the four routing protocols in terms of scatters of packet delivery ratio. It can be easily seen that the performance of the proposed protocol is better compared to other existing protocol since the route discovery and maintenance are efficiently managed in IE2M.

Fig. 9
figure 9

Packet delivery ratio vs. Number of multicast receivers

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Figure 10 shows the performance activities of four routing protocols as a role of packet overhead for diverse values of multicast group size. IE2M demonstrates comparable performance in terms of packet overhead as contrasted to other three protocols. But it quite improves the routing overhead. There are two reasons for the improvement of routing overhead. First reason is efficiently packet forwarding mechanism because when the number of multicast receivers increases, then the ratio of received packets over the transmitted packets also increases. The second reason is when the packet arrival rate is high, then delay over the corresponding nodes also increases because the network overcrowding occurs when a node is carrying more data.

Fig. 10
figure 10

Packet overhead vs. Number of multicast receivers

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Figure 11 represents the performance behavior of four routing protocols with respect to the average end-to-end delay and number of multicast receivers. Average end-to-end delay increases as the multicast group size is increased. IE2M shows better average end-to-end delays cause of to their more efficient forwarding mechanism. It demonstrates the lowest average end-to-end delay because multicast routing selects based on the route having higher energy and shortest distance. The proposed protocol avoids the longest distance with the lowest energy consumption routes due to large delay, which results in higher average end-to-end delay during the simulation time.

Fig. 11
figure 11

Average end-to-end delay vs. Number of multicast receivers

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5.4 Outcome based on multicast traffic load

Figures 12–14 symbolize the outcome of the traffic load on network performance. In this scenario, one multicast source and multiple receivers in the multicast group are used to evaluate the performance of different matrices such as a packet delivery ratio, packet overhead and average end-to-end delay in respect to multicast traffic. Multicast traffic is generated by varying packet size with constant packet sending rate. Figure 12 demonstrated packet delivery ratio decreases as the packet sending rate (multicast traffic) increases while the packet overhead increases. Therefore, the results accomplished from simulation testing that IE2M outshines the existing protocols at the time packet size increases due to shunning concentrated flooding of query messages compared to ODMRP.

Fig. 12
figure 12

Packet delivery ratio vs. Multicast traffic

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Figure 13 illustrates packet overhead with respect to multicast traffic. Packet overhead increases based on packet sending rate. Packet overhead of proposed protocol is constantly lower than other existing protocols because it reduces the flooding mechanism during the sending and receiving packet.

Fig. 13
figure 13

Packet over head vs. Multicast traffic

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Figure 14 illustrates the average end-to-end delay in respect to multicast traffic. Average end-to-end delay is also increases based on packet sending rate. The proposed protocol accomplishes the minimum average end-to-end delay comparable to other existing protocols.

Fig. 14
figure 14

Average end-to-end vs. Multicast traffic

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6 Conclusion and future scope

In this paper, IE2M protocol is proposed as a mesh based multicast routing protocol for ad-hoc network. Its main advantages are enhanced energy efficiency and lifetime of ad-hoc network. Therefore, energy efficient multicast routing indicates selecting route of each multicast group that require the shortest distance, instead of longer distance with sufficient energy. It also improves network performance in terms of several network metrics. It selects the best path based on residual energy. Here, residual energy actually indicates the lifetime of the network. Route lifetime of ad-hoc network is unpredictable due to its dynamic nature and it cannot be scientifically resultants. Hence, to predict this situation, the reward is formulated by intellectual FIS for each and every route. FIS is used to evaluate reward by using two input parameters energy and distance. It reduces the effect of mutual interference between routes. The simulation results of the packet delivery ratio, packet overheads and average end-to-end delay are evaluated with respect to number of multicast receivers and multicast traffic that illustrates the effectiveness of the proposed protocol compared to existing protocols such as FMAR, ODMRP and MAODV. Future scope includes, but not limited to (i) the mathematical analysis of the proposed protocol in which we wish to compare the analytical results with simulation results (ii) implementation of a new energy efficient multicast routing protocol which is a combination of mesh and tree based structure.