Solving Multi-Objective Fuzzy Optimization in Wireless Smart Sensor Networks under Uncertainty Using a Hybrid of IFR and SSO Algorithm

Abstract

Wireless (smart) sensor networks (WSNs), networks made up of embedded wireless smart sensors, are an important paradigm with a wide range of applications, including the internet of things (IoT), smart grids, smart production systems, smart buildings and many others. WSNs achieve better execution efficiency if their energy consumption can be better controlled, because their component sensors are either difficult or impossible to recharge, and have a finite battery life. In addition, transmission cost must be minimized, and signal transmission quantity must be maximized to improve WSN performance. Thus, a multi-objective involving energy consumption, cost and signal transmission quantity in WSNs needs to be studied. Energy consumption, cost and signal transmission quantity usually have uncertain characteristics, and can often be represented by fuzzy numbers. Therefore, this work suggests a fuzzy simplified swarm optimization algorithm (fSSO) to resolve the multi-objective optimization problem consisting of energy consumption, cost and signal transmission quantity of the transmission process in WSNs under uncertainty. Finally, an experiment of ten benchmarks from smaller to larger scale WSNs is conducted to demonstrate the effectiveness and efficiency of the proposed fSSO algorithm.

1. Introduction

Wireless (smart) sensor networks (WSN) consist of smart sensors deployed and operated in wireless sensor networks, and have rapidly become an important design widely applied in many modern applications, such as the Internet of Things (IoT) [1,2], smart grids [3], healthcare and medical systems [4], wind energy systems [5], industrial automation [6], the smart transportation industry [7,8], the semiconductor industry [9] and smart cities [10]. The signal, information or flow cannot be successfully transmitted through the nodes, which are the sensors in the WSN, if the operation of the WSN system fails because nodes deplete their limited battery power. In other words, a critical restriction in any WSN system is energy consumption. Therefore, numerous investigations of WSNs have had a primary focus on energy consumption.

Optimizing energy efficiency (EE) in a WSN, which is defined as the ratio of output over energy consumption, has been the subject of a great number of studies. Mekonnen et al. [2] proposed a prototype of a WSN applied in a video surveillance system to optimize energy consumption. Trapasiya and Soni [11] addressed the goal of retransmission energy reduction in WSNs. Quang and Kim [12] proposed a gradient routing in an industrial WSN to optimize energy consumption. Setiawan et al. [13] came up with an energy management policy to maximize energy transfer efficiency for a WSN. Liu et al. [14] optimally designed a WSN to minimize energy consumption.

The minimization of energy consumption in WSNs has also been investigated by many researches [15,16,17,18]. Chanak et al. [19] discussed the balance of energy consumption among deployed sensors in a WSN. The energy-optimal routing problem in WSNs has recently been studied by several works [20,21,22]. The improvement of energy consumption by clustering method in a WSN has also been discussed by some works [23,24].

However, the measured values of energy consumption in real-life WSN systems are usually uncertain and imprecise. Fuzzy set theory can effectively resolve these uncertain and imprecise problems. In studies of energy consumption for WSNs, few researchers have presented fuzzy-based methods to solve uncertainty problems in such systems. Collotta et al. [25] considered the turning on/off of devices as the output of a Fuzzy Logic Controller (FLCs) because of the dynamical characteristics of the calculated distance from sensor nodes with regard to the Access Points (APs). They used a fuzzy-based technique to decide whether Wi-Fi access points should be switched off when they were underutilized to optimize the energy consumption of a WSN applied in a multimedia system. Kumar and Chaturvedi [26] considered the query dynamic, including the volume of the generated query, frequency of the query generation and the geographical distribution of the query, and aimed to optimize the energy efficiency in a WSN by treating the impact of uncertainties in the query generation process by a fuzzy method. Akram and Cho [27] considered the uncertainty of security attacks on sensor nodes, and adopted a fuzzy-based selection of the intermediate verification nodes for a WSN to optimize energy consumption. Also, there are many other recent work which have considered parameter optimization in WSN and the details can be found in [28,29,30].

In this study, a fuzzy-based algorithm is adopted to solve the uncertain characteristics of energy consumption in a WSN. A schematic picture of a WSN is provided in Figure 1, in which sensor nodes are expressed as circles in the WSN. Let the target sensor node represent the source node. Many paths can be chosen to transmit a signal from the source node to the sink node each node is expressed as a circle in Figure 1. For example, one chosen path sends the signal from the source node to the sink node through two linking sensor nodes which are dark colored in Figure 1.

To the best of the authors’ knowledge, this is the first work to use a fuzzy-based algorithm to effectively resolve uncertain problems of energy consumption measurement in a WSN system. In addition, arcs are used to represent the transmission process among the sensors in a WSN so that an activity on arcs (AOA) network can be implemented. This work, therefore, suggests a fuzzy-based algorithm to resolve the uncertain problems of energy consumption measurements in a WSN using an AOA network. Moreover, fuzzy cost and fuzzy signal transmission quantity are also considered in the proposed algorithm to enhance performance of the WSN system. Finally, an experiment is conducted to demonstrate that the proposed fuzzy-based algorithm can significantly and efficiently optimize WSN energy consumption.

The uncertain characteristics of energy consumption, cost and signal transmission quantity are briefly described as follows. Assume that Figure 2 is a WSN with the nodes set: {0, 1, 2, 3} and the arcs set: {a₁, a₂, a₃, a₄, a₅, a₆}. Node 0 and node 3 are the source node and sink node, respectively. Arcs a₁, a₂, a₄, a₅ and a₆ are designed as the directed arcs. Therefore, there are 4 paths (alternatives): P₁ = {0, 1, 3}, P₂ = {0, 2, 3}, P₃ = {0, 1, 2, 3} and P₄ = {0, 2, 1, 3}, which transmit signal from the source node to the sink node. The related information of fuzzy energy consumption, fuzzy cost and fuzzy signal transmission quantity of sensors in Figure 2 are presented in

Table 1. The fuzzy energy consumption, fuzzy cost and fuzzy signal transmission quantity of sensors in Figure 2.

	Arc	a1	a2	a3	a4	a5	a6
Criteria
Energy Consumption		(1, 4, 6)	(1, 3, 5)	(2, 3, 4)	(1, 2, 4)	(1, 3, 4)	(1, 4, 5)
Cost		(2, 3, 7)	(2, 4, 5)	(1, 3, 4)	(1, 4, 7)	(1, 3, 5)	(2, 4, 7)
Signal Transmission Quantity		(1, 4, 7)	(2, 3, 5)	(1, 3, 4)	(1, 2, 7)	(1, 3, 4)	(2, 4, 7)

.

As mentioned in the above discussion, a new multi-objective fuzzy problem arises involving three objectives: energy consumption, cost and signal transmission quantity in WSNs. Hence the goal of the proposed problem is to find a path, say path i, such that total utility value U(i) (see Equation (24) in Section 3.1) is maximal in sending signal between a pair of nodes under uncertainty, e.g., the change of the topologies, the changes in nodes’ energetic levels, sensor breakdowns, etc. in WSNs.

Since the 1990s, various forms of artificial intelligence have been applied to different optimization NP-problems which mean the running time increases dramatically as the number of nodes in the networks increases. [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,31,32,33,34,35,36,37,38]. For example, genetic algorithms [39], artificial bee colony algorithms [40], particle swarm optimization [34,37,41,42], simplified swarm optimization (SSO) [34,35,36,37,38,42,43,44,45], grey wolf [46], neural network [13,41,45,47], harmony search algorithm [47], Sugeno-Type Fuzzy Inference [48], etc. Among these algorithms, SSO proposed by Yeh is the most simple one. Also, Yeh’s SSO is most customizable amongst other algorithms to apply and solve relevant [34]. Hence, SSO is adapted here to create a fuzzy-based SSO to solve the proposed fuzzy optimization NP-problem.

The remainder of this paper is structured as follows. Section 2 introduces the fundamentals of fuzzy set theory, including arithmetic operations of fuzzy numbers, fuzzy criteria matrix with its addition as well as maximizing set and minimizing set. Section 3 introduces the inverse function-based fuzzy ranking (IFR) and simplified swarm optimization (SSO), which are the basis of the proposed algorithm. Numerical examples for these two methods are provided. The proposed fuzzy SSO (fSSO) for solving a WSN problem is presented in Section 4, and Section 5 examines the performance of the proposed algorithm. Conclusions and suggestions for future research are offered in Section 6.

2. Fuzzy Set Theory and the Proposed Fuzzy Criteria Matrix

Fuzzy set theory was first introduced by Zadeh in 1965 [31], and has become a popular and efficient method for solving uncertain and imprecise problems in decision-making. The fundamental concepts and the proposed Fuzzy criteria matrix are presented in this section.

2.1. Fuzzy Set and Arithmetic Operations

The XXXXX definitions of fuzzy set, α-cuts of a fuzzy set, triangular fuzzy number and arithmetic operations on fuzzy numbers are introduced below.

Definition 1.

A fuzzy set A is defined in Equation (1), where U denotes the universal discourse, x is a real number, −∞ ≤ x ≤ ∞, and u_A represents membership function of the fuzzy set. Let x be a number allocated to A. A corresponding real number can be found for the membership function of the fuzzy set u_A(x) belonging to [0,1], i.e., 0 ≤ u_A(x) ≤ 1 [49]:

A = {(x, u_A(x)) | x∈U}

(1)

Definition 2.

The α-cuts of a fuzzy set A are defined in Equation (2), where A_α^a and A_α^d are the lower bound and the upper bound of the α-cut, respectively:

A_α = {(x, u_A(x) ≥ α) | x∈U} = [A_α^a, A_α^d]

(2)

Definition 3.

A = (a, b, c), where a ≤ b ≤ c, is a triangular fuzzy number and can be defined as in Equation (3) and depicted in Figure 3:

$$u_A\left(x\right)=\{\begin{array}{ll}(x-a)/(b-a),& a\le x\le b,\\ (x-c)/(b-c),& b\le x\le c,\\ 0,& \mathrm{otherwise},\end{array}$$

(3)

Definition 4.

Suppose two triangular fuzzy numbers A₁ = (a₁, b₁, c₁) and A₂ = (a₂, b₂, c₂), the standard operations on triangular fuzzy numbers are as shown in Equations (4)–(7):

Fuzzy addition: A₁⊕A₂ = (a₁ + a₂, b₁ + b₂, c₁ + c₂)

(4)

Fuzzy subtraction: A₁ΘA₂ = (a₁ − c₂, b₁ − b₂, c₁ − a₂)

(5)

Fuzzy multiplication: A₁⊗A₂ = (a₁ × a₂, b₁ × b₂, c₁ × c₂)

(6)

Fuzzy division: A₁⊙A₂ = (a₁/c₂, b₁/b₂, c₁/a₂)

(7)

The concept of Chen’s maximizing set and minimizing set (1985) is introduced in Definitions 5–6 below:

Definition 5.

Let M and H be the maximizing set and the minimizing set defined as Equations (8) and (9):

$$f_M\left(x\right)=\{\begin{array}{ll}{[\frac{x-x_{\min }}{x_{\max }-x_{\min }}]}^k\hfill & x_{\min }\le x\le x_{\max }\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array},$$

(8)

$$f_H\left(x\right)=\{\begin{array}{ll}{[\frac{x_{\max }-x}{x_{\max }-x_{\min }}]}^k\hfill & x_{\min }\le x\le x_{\max }\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}$$

(9)

where x_min = inf S, x_max = sup S, S = ${\displaystyle {\cup }_{i=1}^n{S}_i}$, S_i = {x |$f_{A_i}(x)$>0}, and A_i is the fuzzy number of each alternative, i = 1, 2, …, n.

Suppose there are two fuzzy numbers A₁ and A₂. The maximizing set and minimizing set associated with A₁ and A₂ can be presented as shown in Figure 4.

Definition 6.

The right, left, and total utility value of alternative A_i are defined as U_R(i) and U_L(i) and U(i), respectively:

$$U_R(i)=\underset{x}{\sup }(f_M(x)\wedge f_{A_i}(x)),i=1,2,\dots ,n.$$

(10)

$$U_L(i)=\underset{x}{\sup }(f_H(x)\wedge f_{A_i}(x)),i=1,2,\dots ,n.$$

(11)

U(i) = U_R(i) + U_L(i)

(12)

2.2. The Proposed Fuzzy Criteria Matrix and Its Addition

The fuzzy criteria matrix is proposed to aggregate the three fuzzy criteria, energy consumption, cost and signal transmission quantity, into one matrix. In the fuzzy criteria matrix, each row represents the related criterion in a triangular fuzzy number. For example, in Figure 2, the fuzzy criteria matrix of sensor 1 is:

$$A_1=\left[\begin{array}{ccc}1& 4& 6\\ 2& 3& 7\\ 1& 4& 7\end{array}\right],$$

(13)

where the ith row represents the related triangular fuzzy number of the ith criterion, i.e., (2, 3, 7) is the triangular fuzzy number of the cost of sensor 1.

Similarly, the fuzzy criteria matrices of sensors 2–6 can be displayed as:

$$A_2=\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 5\\ 2& 3& 5\end{array}\right],A_3=\left[\begin{array}{ccc}2& 3& 4\\ 1& 3& 4\\ 1& 3& 4\end{array}\right],A_4=\left[\begin{array}{ccc}1& 2& 4\\ 1& 3& 7\\ 1& 2& 7\end{array}\right],A_5=\left[\begin{array}{ccc}1& 3& 4\\ 1& 3& 7\\ 1& 3& 7\end{array}\right],A_6=\left[\begin{array}{ccc}1& 4& 5\\ 2& 4& 7\\ 2& 4& 7\end{array}\right]$$

(14)

By Equation (4), the fuzzy criteria matrix of each path or arc can be obtained as:

$$A_1\oplus A_2=\left[\begin{array}{ccc}1& 4& 6\\ 2& 3& 7\\ 1& 4& 7\end{array}\right]+\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 5\\ 2& 3& 5\end{array}\right]=\left[\begin{array}{ccc}1+1& 4+3& 6+5\\ 2+2& 3+4& 7+5\\ 1+2& 4+3& 7+5\end{array}\right]=\left[\begin{array}{ccc}2& 7& 11\\ 4& 7& 12\\ 3& 7& 12\end{array}\right].$$

(15)

There are four paths (alternatives) in Figure 2, P₁ = {0, 1, 3}, P₂ = {0, 2, 3}, P₃ = {0, 1, 2, 3} and P₄ = {0, 2, 1, 3}, which transmit a signal from the source node to the sink node. Therefore, the fuzzy criteria matrices of these paths can be obtained using the proposed fuzzy criteria matrix addition as follows. For example, the matrix for P₁ is obtained by the following procedure:

$$P_1=A_1\oplus A_5=\left[\begin{array}{ccc}1& 4& 6\\ 2& 3& 7\\ 1& 4& 7\end{array}\right]+\left[\begin{array}{ccc}1& 3& 4\\ 1& 3& 7\\ 1& 3& 7\end{array}\right]=\left[\begin{array}{ccc}2& 7& 10\\ 3& 6& 14\\ 2& 7& 14\end{array}\right].$$

(16)

Similarly, the fuzzy criteria matrix addition of paths P₂, P₃ and P₄ can be produced as:

$$P_2=A_2\oplus A_6=\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 5\\ 2& 3& 5\end{array}\right]+\left[\begin{array}{ccc}1& 4& 5\\ 2& 4& 7\\ 2& 4& 7\end{array}\right]=\left[\begin{array}{ccc}2& 7& 10\\ 4& 8& 12\\ 4& 9& 12\end{array}\right]$$

(17)

$$P_3=A_1\oplus A_4\oplus A_6=\left[\begin{array}{ccc}1& 4& 6\\ 2& 3& 7\\ 1& 4& 7\end{array}\right]+\left[\begin{array}{ccc}1& 2& 4\\ 1& 3& 7\\ 1& 2& 7\end{array}\right]+\left[\begin{array}{ccc}1& 4& 5\\ 2& 4& 7\\ 2& 4& 7\end{array}\right]=\left[\begin{array}{ccc}3& 10& 15\\ 5& 10& 21\\ 4& 10& 21\end{array}\right]$$

(18)

$$P_4=A_2\oplus A_3\oplus A_5=\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 5\\ 2& 3& 5\end{array}\right]+\left[\begin{array}{ccc}2& 3& 4\\ 1& 3& 4\\ 1& 3& 4\end{array}\right]+\left[\begin{array}{ccc}1& 3& 4\\ 1& 3& 7\\ 1& 3& 7\end{array}\right]=\left[\begin{array}{ccc}4& 9& 13\\ 4& 10& 16\\ 4& 9& 16\end{array}\right].$$

(19)

3. The IFR, SSO and the Proposed Fitness Function

A multi-criteria multi-objective fuzzy optimization problem is considered in this work. In optimization problems, it is very important to solve for the best among all solutions regardless of the environment being certain or uncertain.

Chu and Yeh’s inverse function-based fuzzy number ranking method (IFR) is able to transform fuzzy numbers from a fuzzy multi-criteria decision-making model into crisp numbers [32]. Their method is more robust compared with some others. Meanwhile, Yeh’s simplified swarm optimization (SSO) is more easily customized to solve relevant problems than other algorithms [34]. Herein, an algorithm that combines the IFR and the SSO is used to solve the proposed problem. The methods of IFR and SSO, along with examples are presented in the following subsections, respectively. Moreover, the proposed fitness function based on the IFR is discussed.

3.1. The Inverse Function-Based Fuzzy Number Ranking

Assume that n alternatives P₁, P₂, …, P_n are required to be evaluated under n criteria F₁, F₂, …, F_n of which F₁, F₂, …, F_g are benefit criteria, and the rest are cost criteria. In IFR, each alternative, P_i, is represented by a triangular fuzzy number, ξ_i,j = (α_i,j, β_i,j, χ_i,j), for each criterion j. Fuzzy number ξ_i,j must be normalized to X_i,j = (a_i,j, b_i,j, c_i,j), of which values of elements in X_i,j fall into [0,1], and can be weighted by multiplying the weight W_j = (w_j,1, w_j,2, w_j,3) in order to obtain the fuzzy weighted normalized evaluation value G_i = (A_i, B_i, C_i) for each alternative. Note that the multiplication of two triangular fuzzy numbers can be approximated as a triangular fuzzy number. Hence, the fuzzy number G_i is still a triangular fuzzy number, where i = 1, 2, …, n; j = 1, 2, …, m.

The right utility U_R(i) of P_i is obtained from the right inverse function y_Ri of G_i and the inverse function of the maximizing set f_M(x); while the left utility U_L(i) of P_i is obtained from the left inverse function y_Li of G_i and the inverse function of the minimizing set f_H(x). The total utility U(i) is the sum of U_R(i) and U_L(i). A larger U(i) indicates that the corresponding alternative A_i is more favorable than the others. The procedure of IFR is described in the following steps [32]:

Step F0.

Find the weight W_j = (w_j,1, w_j,2, w_j,3) of criterion j, j = 1, 2, …, m.

Step F1.

Normalize the triangular fuzzy number of each alternative versus each criterion, ξ_i,j = (α_i,j, β_i,j, χ_i,j), i = 1, 2, …, n; j = 1, 2, …, m, to X_i,j = (a_i,j, b_i,j, c_i,j) to make evaluation values across criteria in a comparable scale. Herein Equation (20) is used for normalization [32,33]:

$$X_{i,j}=(\frac{{\alpha }_{i,j}-\underset{i}{\min }{\alpha }_{i,j}}{\underset{i}{\max }{\chi }_{i,j}-\underset{i}{\min }{\alpha }_{i,j}},\frac{{\beta }_{i,j}-\underset{i}{\min }{\alpha }_{i,j}}{\underset{i}{\max }{\chi }_{i,j}-\underset{i}{\min }{k}_{i,j}},\frac{{\chi }_{i,j}-\underset{i}{\min }{\alpha }_{i,j}}{\underset{i}{\max }{\chi }_{i,j}-\underset{i}{\min }{k}_{i,j}}).$$

(20)

Step F2.

Calculate the weighted normalized triangular fuzzy number G_i,j = (A_i,j, B_i,j, C_i,j) = (a_i,jw_i,1, b_i,jw_i,2, c_i,jw_i,3) and the aggregated triangular fuzzy number G_i = (A_i, B_i, C_i) for alternative i, i = 1, 2, …, n, as shown in Equation (21):

$$G_i=\left(A_i,B_i,C_i\right)={\displaystyle \sum _{j=1}^g(W_j\otimes X_{i,j})}–{\displaystyle \sum _{j=g+1}^n(W_j\otimes X_{i,j})},$$

(21)

where G_i = G_i,1 + G_i,2 − G_i,3; A_i = A_i,1 + A_i,2 − A_i,3; B_i = B_i,1 + B_i,2 − B_i,3; C_i = C_i,1 + C_i,2 − C_i,3.

Step F3.

Calculate the right inverse function y_Ri and the left inverse function y_Li as shown in Equations (22) and (23):

U_R(i) = (C_i − x_min)/(x_max − x_min − B_i + C_i)

(22)

U_L(i) = (A_i − x_max)/(x_min − x_max − B_i + A_i).

(23)

Step F4.

Calculate total utility value U(i) of alternative i based on Equation (12) for i = 1, 2, …, n:

U(i) = (B_i − C_i)∙U_R(i) + (B_i − A_i) U_L(i) + C_i + A_i.

(24)

Step F5.

Find the best alternative which has the largest inverse function-based total utility value. The flow chart of the above algorithm is depicted in Figure 5:

3.2. The SSO and an Example

The proposed fSSO is developed based on the SSO. In 2009, Yeh first developed the SSO algorithm, initially called discrete PSO, to overcome the weakness of particle swarm optimization (PSO) in solving discrete problems [34]. Since then, the SSO has become a famous swarm intelligence-based random optimization algorithm. The SSO has also played a very significant solution role in relevant studies of artificial intelligence. Furthermore, the SSO has been applied by many papers to solve different types of problems in various fields [35,36,37,38,43].

The parameters c_g, c_p, c_w and c_r are the probabilities of the new variable value generated from a global search, a global search together with a local search, a local search, and a random number in SSO, respectively, where c_g + c_p + c_w + c_r = 1. The update mechanism of the simple, efficient and agile SSO algorithm is presented as Equation (25):

$$x_{t,i,j}=\{\begin{array}{l}{p}_{gBest,j}\hfill \\ p_{i,j}\hfill \\ x_{t-1,i,j}\hfill \\ x\hfill \end{array}\begin{array}{l}\mathrm{if}\mathsf{\rho }\in [0,C_g)\hfill \\ \mathrm{if}\mathsf{\rho }\in [C_g,C_p)\hfill \\ \mathrm{if}\mathsf{\rho }\in [C_p,C_w)\hfill \\ \mathrm{if}\mathsf{\rho }\in [C_w,1]\hfill \end{array},$$

(25)

where the number of generations is denoted as t, t = 1, 2, …, N_gen; the number of solutions is denoted as i, i = 1, 2, …, N_sol; the number of variables is denoted as j, j = 1, 2, …, N_var; x_t,i,j and x_t−1,i,j are the ith solution of the jth variable at generations t and t − 1, respectively; P_gBest,j and P_i,j are the jth variable of the temporary global best of all solutions and the temporary personal (local) best of the ith solution, respectively; ρ belongs to uniform distribution [0,1]; C_g = c_g, C_p = C_g + c_p, C_w = C_p + c_w; x belongs to uniform distribution [l_i, u_i].

Let C_g = c_g = 0.5, C_p = C_g + c_p = 0.5 + 0.2 = 0.7, C_w = C_p + c_w = 0.7 + 0.25 = 0.95, X_10,8 = (1.1, 1.8, 3.5, 2.2, 0.1) which is the 8th solution of the 11th generation, P₈ = (2.5, 2.0, 1.2, 1.9, 5.0), and P_gBest = (3.3, 2.8, 1.2, 4.5, 5.6). An example is shown below to explain how SSO is implemented to update X_10,8 to X_11,8 in

Table 2. An example of simplified swarm optimization (SSO) algorithm.

	i	1	2	3	4	5
Coordinate
li		0	1	0	2	0
ui		8	10	6	8	6
X10,8		1.1	1.8	3.5	2.2	0.1
P8		2.5	2.0	1.2	1.9	5.0
PgBest		3.3	2.8	1.2	4.5	5.6
ρi		0.4	0.8	0.96	0.25	0.6
x11,8		3.3	1.8	5.8 *	4.5	5.0
Remark		0 < ρ < Cg	Cp < ρ < Cw	Cw < ρ	Cp < ρ < Cw	Cg < ρ < Cp

* a new feasible value generated randomly from [l_i, u_i].

. The lower bound l_i, the upper-bound u_i, and the value of ρ_i for variable x_10,8,i are listed in the 2nd, 3rd and 7th row of Table 2, respectively.

3.3. The Proposed Fitness Function

In IFR, all alternatives (i.e., all paths here) are represented by fuzzy numbers and these fuzzy numbers must be normalized to a comparable scale. The normalization procedure is conducted using maximal and minimal elements among those alternatives versus each criterion. However, each alternative is updated from generation to generation, i.e., the maximal and minimal elements among alternatives in generation i may be different to those in generation j for all i < j. The above situation means that in the gBest in generation i is not better than some solutions in generation j after using the new maximal and minimal elements obtained from generation j. This is contrary to the meaning of gBest.

To fix the above problem, both the minimal x_min and the maximal x_max are redefined by the following equations:

$$x_{\min }=\underset{i}{\min }\{a_{i,j}\}$$

(26)

$$x_{\max }=\underset{i}{\max }\left\{c_{i,j}\right\}$$

(27)

where i = 1, 2, …, n and j = 1, 2, …, m. The values of x_min and x_max are always fixed and all fitness values of solutions are also fixed after the redefinition.

Example 1.

Suppose three criteria including energy consumption, cost and signal transmission quantity are considered in Figure 2, and are denoted by symbol j = 1, 2, 3.

Solution:

Step F0.

The triangular fuzzy weight, denoted as W_j = (w_j,1, w_j,2, w_j,3) obtained from the analytic hierarchy process (AHP) for each criterion j, j = 1, 2, 3, is provided as shown in

Table 3. Triangular fuzzy weights of the three criteria.

	Weight	Wj = (wj,1, wj,2, wj,3)
Criteria j
1 (Energy Consumption)		(0.775, 0.875, 1.000)
2 (Cost)		(0.775, 0.875, 1.000)
3 (Signal Transmission Quantity)		(0.775, 0.875, 1.000)

:

Step F1.

By the addition of the fuzzy criteria matrix presented in Section 2.2, triangular fuzzy values of the paths (i.e., alternatives): P₁ = {0, 1, 3}, P₂ = {0, 2, 3}, P₃ = {0, 1, 2, 3} and P₄ = {0, 2, 1, 3} versus different criteria can be shown as in

Table 4. Triangular fuzzy values ξ_i,j = (α_i,j, β_i,j, χ_i,j) in Figure 2.

	j	1	2	3
i
1		(2, 7, 10)	(3, 6, 14)	(2, 7, 14)
2		(2, 7, 10)	(4, 8, 12)	(4, 9, 12)
3		(3, 10, 15)	(5, 10, 21)	(4, 10, 21)
4		(4, 9, 13)	(4, 10, 16)	(4, 9, 16)

:

The values of $\underset{i}{\min }{\alpha }_{i,j}$ and $\underset{i}{\max }{\chi }_{i,j}$ for each criterion are indicated in bold and underlined in Table 3, respectively, and are presented in

Table 5. Values of $\underset{i}{\min }{\alpha }_{i,j}$ and $\underset{i}{\max }{\chi }_{i,j}$ for each criterion.

j	1	2	3
$\underset{i}{\min }{\alpha }_{i,j}$	2	3	2
$\underset{i}{\max }{\chi }_{i,j}$	15	21	21
$\underset{i}{\max }{\chi }_{i,j}-\underset{i}{\min }{\alpha }_{i,j}$	13	18	19

:

By Equation (20), normalize triangular fuzzy numbers X_i,j = (a_i,j, b_i,j, c_i,j) for paths (alternates), i = 1, 2, 3, 4, versus criteria, j = 1, 2, 3, can be obtained as presented in

Table 6. Normalized triangular fuzzy numbers X_i,j = (a_i,j, b_i,j, c_i,j).

	j	1	2	3
i
1		(0.000, 0.385, 0.615)	(0.000, 0.167, 0.500)	(0.000, 0.263, 0.474)
2		(0.000, 0.385, 0.615)	(0.056, 0.278, 0.500)	(0.105, 0.263, 0.526)
3		(0.077, 0.615, 1.000)	(0.111, 0.444, 1.000)	(0.105, 0.421, 1.000)
4		(0.154, 0.538, 0.846)	(0.056, 0.389, 0.611)	(0.105, 0.368, 0.579)

:

Step F2.

Values of G_i,j = (A_i,j, B_i,j, C_i,j), i = 1, 2, 3, 4 and j = 1, 2, 3, can be obtained by (A_i,j, B_i,j, C_i,j) = (a_i,jw_i,1, b_i,jw_i,2, c_i,jw_i,3) as shown in

Table 7. Weighted normalized triangular fuzzy numbers of G_i,j = (A_i,j, B_i,j, C_i,j).

	j	1	2	3
i
1		(0.000, 0.337, 0.615)	(0.000, 0.146, 0.500)	(0.000, 0.230, 0.474)
2		(0.000, 0.337, 0.615)	(0.043, 0.243, 0.500)	(0.082, 0.230, 0.526)
3		(0.060, 0.538, 1.000)	(0.086, 0.389, 1.000)	(0.082, 0.368, 1.000)
4		(0.119, 0.471, 0.846)	(0.043, 0.340, 0.611)	(0.082, 0.322, 0.579)

. Values of G_i = (A_i, B_i, C_i) can be obtained from Equation (21) as listed in

Table 8. Aggregated triangular fuzzy numbers G_i = (A_i, B_i, C_i).

i	1	2	3	4
Ai	−0.4737	−0.4833	−0.8527	−0.4167
Bi	0.2521	0.3493	0.5589	0.4891
Ci	1.1154	1.0338	1.9184	1.3757

:

Step F3.

The right inverse function U_R(i) and the left inverse function U_L(i) based on Equations (22) and (23) can be produced as shown in

Table 9. The y_Ri and y_Li.

i	1	2	3	4
UR(i)	0.5986	0.6137	0.8131	0.7292
yLi	0.8539	0.8094	0.7684	0.7434

:

Step F4.

The inverse function-based total utilities U(i) of alternatives are listed in

Table 10. The U(i).

i	1	2	3	4
U(i)	0.7447	0.8043	1.0447	0.9858

:

Step F5.

From Table 10, it is clear that path P₁ is the best transmission path because it has the lowest total utility value, i.e., U(1) < U(2) < U(4) < U(3).

4. The Proposed Fuzzy SSO

The proposed fSSO is a population, all-variable and stepwise-function-based soft computing method, i.e., there are a fixed number of solutions in each generation and all variables must be updated based on the stepwise function with each solution.

4.1. The Flexible-Length Structure without Targets Solution Structure

Each solution in the proposed fSSO is a path, which records a sequence of nodes from the source node to the sink node, and is also called an alternative in the multi-criteria decision-making problem. For example, both P₁ = (0, 1, 3) and P₂ = (0, 1, 2, 3) are paths from nodes 0 to 3 in Figure 2.

The lengths of all paths are different, e.g., there are two and three arcs in P₁ and P₂, respectively. Herein, the flexible-length solution structure is used. In addition, only non-target nodes are recoded sequentially in each solution to save memory space and processing time. For example, P₁ = (0, 1, 3) and P₂ = (0, 1, 2, 3) in Figure 2 can be simplified to P₁ = (1) and P₂ = (1, 2) without showing the source and sink nodes. Note that the processing time and the memory space can be reduced by up to (2/N_var)∙N_sol∙N_gen and (2/N_var) times of those results without using this concept, respectively.

4.2. The Novel Update Mechanism

The proposed fSSO has a fixed population in each generation and all variables of each solution must be updated based on the stepwise function, which is based on the original stepwise function listed in Equation (25), and is modified to fit the proposed problem as follows:

$$x_{i,j}=\{\begin{array}{l}{p}_{gBest,j}\hfill \\ p_{i,j}\hfill \\ x_{i,j}\hfill \\ x\hfill \end{array}\begin{array}{l}{\mathrm{if}\mathsf{\rho }}_{[0,1]}\in [0,C_g)\mathrm{and}j<〚X_{gBest}〛\hfill \\ {\mathrm{if}\mathsf{\rho }}_{[0,1]}\in [C_g,C_p)\mathrm{and}j<〚P_i〛\hfill \\ {\mathrm{if}\mathsf{\rho }}_{[0,1]}\in [C_p,C_w)\hfill \\ {\mathrm{if}\mathsf{\rho }}_{[0,1]}\in [C_w,1)\hfill \end{array}$$

(28)

where ${\mathsf{\rho }}_{[0,1]}$ is the same as that defined in Equation (25) and $〚X〛$ < N_var is defined to be the length of X, e.g., the number of variables in X. Note that the length of the new update X_i is 20% longer than the longest among old X_i before it is modified to be a feasible solution. For example, let X₅ = (2, 4, 6, 8, 10), P₅ = (1, 3, 5, 7) and P_gBest = G = (5, 2, 7, 4, 6, 3, 1). The new length of the new X₅ is set to the smallest integer that is larger than 7 · (1 + 20%) = 8.5, i.e., 9 since $〚P_5〛=4<〚X_5〛=5<〚G〛=7$. Assume C_g = 0.35, C_p = 0.55, C_w = 0.65, ρ = (ρ₁, ρ₂, …, ρ₉) = (0.3, 0.5, 0.1, 0.5) for this X₅. From Equation (28), the process to update X₅ is displayed in

Table 11. The example of the update in the proposed fuzzy simplified swarm optimization (fSSO).

	Length of X5	1	2	3	4	5	6	7	8	9
Coordinate
Old X5		2	4	6	8	10
P5		1	3	5	7
G		5	2	7	4	6	3	1	1	1
ρ		0.34	0.42	0.27	0.57	0.16	0.13	0.79	0.25	0.39
New X5		5	3	7	8	6	2 *	5 *	7 *	1 *

Note that the numbers marked with “*”are generated randomly.

.

4.3. The Solution Repair Procedure

From the example shown in Section 4.2, the obtained solutions may have duplicated nodes, e.g., 5 appears in the first and seventh coordinates of the new X₅. Any solution with duplicate nodes is infeasible. Hence, duplicate nodes must be removed randomly to repair an infeasible solution to a feasible solution, which represents a simple routing. For example, the new X₅ = (5, 3, 7, 8, 6, 2, 5, 7, 1), and it can be (3, 8, 6, 2, 5, 7, 1), (5, 3, 7, 8, 6, 2, 5, 1), (5, 3, 7, 8, 6, 2, 1) or (5, 3, 8, 6, 2, 7, 1) after the repair.

4.4. The Pseudo Code and Flowchart of the Proposed fSSO

The details of fSSO are described in the following steps, and the flowchart of fSSO is depicted in Figure 6.

Step 0.

Generate X_0,i = P_i randomly, calculate U(X_0,i), find gBest, and let t = 1, where i = 1, 2, …, N_sol.

Step 1.

Let i = 1.

Step 2.

Update X_i based on the above equation, repair X_i if possible, and calculate U(X_t,i).

Step 3.

If U(X_i) is better than U(P_i), then let P_i = X_i. Otherwise, go to Step 5.

Step 4.

If U(P_i) is better than U(P_gBest), then let gBest = i.

Step 5.

If i < N_sol, let i = i + 1 and go to Step 2.

Step 6.

If t < N_gen, then let t = t + 1 and go back to Step 1. Otherwise, halt.

5. Numerical Experiments

The proposed fSSO for this problem is only run immediately once a signal request for any pair of nodes to overcome the uncertainty is received. The answer obtained from the proposed algorithm thus corresponds to the real-life and real-time WSNs under uncertain environment and situations including sensor breakdowns, the change of network topology, the changes in nodes’ energetic levels and the network topologies, etc.

5.1. Experimental Setting and Metrics Derivation

To verify the performance of the proposed fSSO, the fSSO are compared with SSO [34] and Ant colony optimization (ACO) algorithms [30] by two metrics following.

• Statistical analyses of the minimum, maximum, average and standard deviation of the fitness values.
• Statistical analyses of the minimum, maximum, average and standard deviation of the running time.

The proposed fSSO, SSO [34] and ACO [30] are coded using DEV C++ with 64-bit Windows 10, implemented on an Intel Core i7-6650U CPU @ 2.20 GHz 2.21 GHz notebook with 64 GB memory. The proposed fSSO, SSO [34] and ACO [30] are executed on ten benchmarks including N_var = 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000 sensor nodes in the WSN, respectively. The number of runs, generations, and solutions of each benchmark are equal to 30, 100, and 50, respectively, i.e., N_run = 30, N_gen = 100, and N_sol = 50 for three algorithms. After the process of trial and error, the parameters of the proposed fSSO and SSO [34] are C_g = 0.35, C_p = 0.55, and C_w = 0.65, i.e., c_g = 0.35, c_p = 0.20, c_w = 0.10, and c_r = 0.35. The parameters of ACO are adapted from [30]. The related performance includes the fitness values of the inverse function-based total utility values described in Equation (28).

5.2. Analysis of Results

Statistical analyses of the minimum, maximum, average and standard deviation of the fitness values from the ten benchmarks are displayed in Figure 7, Figure 8, Figure 9 and Figure 10, where the y-axis and x-axis are the related fitness values and the number of sensors N_var, respectively. The best solutions, e.g., U(P_gBest), of the minimum, maximum, average and standard deviation among the ten benchmarks are indicated in bold in

Table 12. Minimum of fitness values of fSSO compared with SSO and Ant colony optimization (ACO) for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.00452306	0.00452306	0.00452306
200	0.00180625	0.00166242	0.00166242
300	0.00178568	0.00156124	0.00156124
400	0.00105628	0.00101487	0.00101487
500	0.00110300	0.00096829	0.00137581
600	0.00075973	0.00052838	0.00052838
700	0.00103935	0.00061055	0.00061055
800	0.00075122	0.00061571	0.00084433
900	0.00049162	0.00036236	0.00060928
1000	0.00040413	0.00038784	0.00038437

,

Table 13. Maximum of fitness values of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.01410044	0.00526698	0.01906664
200	0.02178184	0.00248900	0.01138107
300	0.02547567	0.00203804	0.01598114
400	0.02105131	0.00155256	0.02413952
500	0.03444513	0.00114329	0.03025675
600	0.02261785	0.00133658	0.01146645
700	0.01617311	0.00090024	0.02024985
800	0.02766630	0.00086589	0.02537341
900	0.01323276	0.00056249	0.01392113
1000	0.01992036	0.00052185	0.01938993

,

Table 14. Average of fitness values of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.00674068	0.00482380	0.00823440
200	0.00709806	0.00216483	0.00402226
300	0.00851326	0.00181031	0.00577248
400	0.00686126	0.00130268	0.00552441
500	0.00667798	0.00114677	0.00720863
600	0.00595977	0.00073681	0.00371094
700	0.00458019	0.00072653	0.00582587
800	0.00863629	0.00072723	0.00531023
900	0.00381658	0.00048270	0.00408860
1000	0.00566787	0.00045867	0.00642156

and

Table 15. Standard deviation of fitness values of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.00292774	0.00020802	0.00368675
200	0.00510935	0.00026014	0.00283857
300	0.00762143	0.00013432	0.00379149
400	0.00558120	0.00015241	0.00532354
500	0.00647559	0.00009293	0.00696904
600	0.00511571	0.00008784	0.00273727
700	0.00384812	0.00007331	0.00556464
800	0.00719715	0.00007417	0.00478121
900	0.00311042	0.00005469	0.00374546
1000	0.00530407	0.00003940	0.00503124

, respectively.

From Table 12, Table 13, Table 14 and Table 15, performance of the proposed fSSO compared with SSO and ACO in the WSN is concluded as follows:

• The proposed fSSO possesses the ability to solve multi-objective problems in a WSN.
• The proposed fSSO can be applied to not only small scale problems involving 100 sensor nodes, but also big scale problems involving 1000 sensor nodes in a WSN.
• The proposed fSSO can effectively find the best configuration of the number of sensor nodes in a WSN.
• According to Table 12, the configuration of 500, 800 and 900 sensor nodes in a WSN has the best solutions of minimum fitness values obtained by the proposed fSSO compared with those found by SSO and ACO for the proposed multi-objective problem including energy consumption, cost and signal transmission quantity.
• According to Table 13, Table 14 and Table 15, the configuration of ten benchmarks of 100, 200,…, 1000 sensor nodes in a WSN has the best solutions of maximum, average and standard deviation of fitness values obtained by the proposed fSSO compared with those found by SSO and ACO for the proposed multi-objective problem including energy consumption, cost and signal transmission quantity.

Moreover, statistical analyses of the minimum, maximum, average and standard deviation of the running time from the ten benchmarks are displayed in Figure 11, Figure 12, Figure 13 and Figure 14, where the x-axis and y-axis are the related running times based on CPU seconds for each benchmark. The best solutions of the minimum, maximum, average and standard deviation among the ten benchmarks are indicated in bold in

Table 16. Minimum of running time of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.00000000	0.00000000	0.00000000
200	0.01500000	0.00000000	0.00000000
300	0.03100000	0.00000000	0.00000000
400	0.04700000	0.00000000	0.00653549
500	0.12500000	0.00000000	0.00000000
600	0.20299999	0.00000000	0.00000000
700	0.26499999	0.00000000	0.00000000
800	0.31200001	0.00000000	0.00000000
900	0.34400001	0.01500000	0.00000000
1000	0.42100000	0.01500000	0.00000000

,

Table 17. Maximum of running time of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.03100000	0.01600000	0.01600000
200	0.04700000	0.01600000	0.01600000
300	0.06300000	0.01600000	0.01600000
400	0.07900000	0.01600000	0.01400000
500	0.20299999	0.01600000	0.01600000
600	0.26600000	0.01600000	0.01600000
700	0.36000001	0.01600000	0.01600000
800	0.43700001	0.03200000	0.01600000
900	0.45400000	0.03200000	0.03100000
1000	0.59299999	0.03200000	0.03200000

,

Table 18. Average of running time of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.01455172	0.00103333	0.00210000
200	0.03073333	0.00206667	0.00260000
300	0.04633333	0.00310000	0.00313333
400	0.07186667	0.00466667	0.00600000
500	0.15366667	0.00730000	0.00623333
600	0.23696667	0.01040000	0.00676667
700	0.30730000	0.01406667	0.00936667
800	0.38020000	0.01613333	0.00990000
900	0.41666667	0.01926667	0.01093333
1000	0.49946667	0.02290000	0.01456667

and

Table 19. Standard deviation of running time of fSSO compared with SSO and ACO for ten benchmarks.

Number of Nodes	SSO	fSSO	ACO
100	0.00570299	0.00393467	0.00544787
200	0.00773230	0.00536228	0.00591666
300	0.00653549	0.00631009	0.00637740
400	0.00891234	0.00725560	0.00361033
500	0.01587089	0.00794442	0.00777123
600	0.01537011	0.00749068	0.00787700
700	0.02315637	0.00479176	0.00778807
800	0.03159048	0.00659014	0.00767149
900	0.02767089	0.00667953	0.00834982
1000	0.04112699	0.00792356	0.00579348

. From Table 16, Table 17, Table 18 and Table 19, performance of running time obtained by the proposed fSSO compared with those found by SSO and ACO in the WSN is concluded as follows:

• According to Table 16 and Table 17, the configuration of ten benchmarks of 100, 200,…, 1000 sensor nodes in a WSN has the best solutions of minimum and maximum running time obtained by either the proposed fSSO or ACO compared with those found by SSO for the proposed multi-objective problem including energy consumption, cost and signal transmission quantity.
• According to Table 18, the configuration of 100, 200, 300 and 400 sensor nodes in a WSN has the best solutions of average running time obtained by the proposed fSSO compared with those found by SSO and ACO for the proposed multi-objective problem including energy consumption, cost and signal transmission quantity. And the configuration of 500, 600, …, 1000 sensor nodes in a WSN has the best solutions of average running time obtained by ACO.
• According to Table 19, the configuration of 400, 500 and 1000 sensor nodes in a WSN has the best solutions of standard deviation running time obtained by ACO compared with those found by SSO and the proposed fSSO for the proposed multi-objective problem including energy consumption, cost and signal transmission quantity. And the configuration of 100, 200, 300, 600, 700, 800 and 900 sensor nodes in a WSN has the best solutions of standard deviation running time obtained by the proposed fSSO.

6. Conclusions

Optimization of the transmission process with respect to a multi-objective of energy consumption, cost and signal transmission quantity in a WSN was studied in this paper. The sensor nodes are usually set up in remote, inaccessible or hazardous environments, which be negatively affected by various external factors such as heavy rain, sunlight, wind, snow and earthquake etc., and this may result in uncertainty of energy consumption, cost and signal transmission quantity of the sensor nodes in a WSN. To resolve the problems of uncertainty and ranking of transmission paths in consideration of the multi-objective of energy consumption, cost and signal transmission quantity in a WSN, a fuzzy simplified swarm optimization algorithm (fSSO) is proposed. To the best of the authors’ knowledge, this is the first work which effectively resolves the uncertainty problems of energy consumption, cost and signal transmission quantity in a WSN using a fuzzy-based algorithm.

The methods of inverse function-based fuzzy number ranking (IFR) and SSO were applied to the proposed fSSO algorithm to defuzzify the fuzzy characteristics of the problem, and to transform the multi-objective problem into a single objective problem in order to solve optimization in the WSN. Furthermore, two rising operators including the flexible-length structure without targets solution structure and a novel SSO update mechanism were introduced to the developed fSSO algorithm. An experiment of ten benchmarks from smaller scale to larger scale including 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000 sensor nodes in a WSN was successfully conducted to demonstrate the effectiveness and efficiency of the proposed fSSO algorithm. When compared with the SSO [31] and ACO [49], the proposed fSSO algorithm shows the improvement in solution quality. For future studies, more objectives of the transmission process in the WSN by the proposed fSSO algorithm, the energy used in diverse computations (e.g., data aggregation, data compression, encryption, etc.), and the effects of the random parameters in the algorithms, will all be taken into account.