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

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.


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 Energies 2018, 11, 2385 2 of 23 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.
Energies 2018, 11, x FOR PEER REVIEW 2 of 23 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  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: {a 1 , a 2 , a 3 , a 4 , a 5 , a 6 }.Node 0 and node 3 are the source node and sink node, respectively.Arcs a 1 , a 2 , a 4 , a 5 and a 6 are designed as the directed arcs.Therefore, there are 4 paths (alternatives): P 1 = {0, 1, 3}, P 2 = {0, 2, 3}, P 3 = {0, 1, 2, 3} and P 4 = {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.
Energies 2018, 11, x FOR PEER REVIEW 3 of 23 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.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.
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 functionbased fuzzy ranking (IFR) and simplified swarm optimization (SSO), which are the basis of the  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.
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.

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.

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]: 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: 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: Energies 2018, 11, x FOR PEER REVIEW 4 of 23 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.

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.

Fuzzy Set and Arithmetic Operations
The definitions of fuzzy set, α-cuts of a fuzzy set, triangular fuzzy number and arithmetic operations on fuzzy numbers are introduced below.1), where U denotes the universal discourse, x is a real number, −∞ ≤ x ≤ ∞, and uA 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 uA(x) belonging to [0,1], i.e., 0 ≤ uA(x) ≤ 1 [49]:

Definition 1. A fuzzy set A is defined in Equation (
Definition 2. The α-cuts of a fuzzy set A are defined in Equation ( 2), where Aα a and Aα d are the lower bound and the upper bound of the α-cut, respectively: 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:  4)-( 7):  4)-( 7): Energies 2018, 11, 2385 5 of 23 Fuzzy subtraction: Fuzzy multiplication: Fuzzy division: 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): where x min = inf S, x max = sup S, S = ∪ n i=1 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 A 1 and A 2 .The maximizing set and minimizing set associated with A 1 and A 2 can be presented as shown in Figure 4. Fuzzy multiplication: Fuzzy division: A1⊙A2 = (a1/c2, b1/b2, c1/a2) ( 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): ( ) ( ) where xmin = inf S, xmax = sup S, S = and Ai 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

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: 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: By Equation ( 4), the fuzzy criteria matrix of each path or arc can be obtained as: There are four paths (alternatives) in Figure 2, P 1 = {0, 1, 3}, P 2 = {0, 2, 3}, P 3 = {0, 1, 2, 3} and P 4 = {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 P 1 is obtained by the following procedure: Similarly, the fuzzy criteria matrix addition of paths P 2 , P 3 and P 4 can be produced as:

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.

The Inverse Function-Based Fuzzy Number Ranking
Assume that n alternatives P 1 , P 2 , . . ., P n are required to be evaluated under n criteria F 1 , F 2 , . . ., F n of which F 1 , F 2 , . . ., 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]: , Step F2.Calculate the weighted normalized triangular fuzzy number G i,j = (A i,j , B i,j , C i,j ) = (a i,j w i,1 , b i,j w i,2 , c i,j w 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): where Step F3.Calculate the right inverse function y Ri and the left inverse function y Li as shown in Equations (22) and ( 23): Step F4.Calculate total utility value U(i) of alternative i based on Equation ( 12) for i = 1, 2, . . ., n: 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: 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: Find all W j = (w j,1 , w j,2 , w j,3 ).

Calculate U(i).
Find the best alternative.
Normalize ξ i,j to X i,j = (a i,j , b i,j , c i,j ).

Halt
Calculate G i and G.
Calculate U R (i) and U L (i).

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 intelligencebased 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 cg, cp, cw and cr 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 cg + cp + cw + cr = 1.The update mechanism of the simple, efficient and agile SSO algorithm is presented as Equation ( 25): where the number of generations is denoted as t, t = 1, 2, ..., Ngen; the number of solutions is denoted as i, i = 1, 2, …, Nsol; the number of variables is denoted as j, j = 1, 2, …, Nvar; xt,i,j and xt−1,i,j are the ith solution of the jth variable at generations t and t − 1, respectively; PgBest,j and Pi,j are the jth variable of

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): 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 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, P 8 = (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.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.
* a new feasible value generated randomly from [l i , u i ].

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: 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: Step F1.By the addition of the fuzzy criteria matrix presented in Section 2.2, triangular fuzzy values of the paths (i.e., alternatives): P 1 = {0, 1, 3}, P 2 = {0, 2, 3}, P 3 = {0, 1, 2, 3} and P 4 = {0, 2, 1, 3} versus different criteria can be shown as in Table 4: The values of min i α i,j and max i χ i,j for each criterion are indicated in bold and underlined in Table 3, respectively, and are presented in Table 5: 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: 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,j w i,1 , b i,j w i,2 , c i,j w i,3 ) as shown in Table 7. Values of G i = (A i , B i , C i ) can be obtained from Equation (21) as listed in Table 8: 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: Step F4.The inverse function-based total utilities U(i) of alternatives are listed in Table 10: Step F5.From Table 10, it is clear that path P 1 is the best transmission path because it has the lowest total utility value, i.e., U(1) < U(2) < U(4) < U(3).

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.

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 P 1 = (0, 1, 3) and P 2 = (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 P 1 and P 2 , 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, P 1 = (0, 1, 3) and P 2 = (0, 1, 2, 3) in Figure 2 can be simplified to P 1 = (1) and P 2 = (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.

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: where ρ [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 X 5 = (2, 4, 6, 8, 10), P 5 = (1, 3, 5, 7) and P gBest = G = (5,2,7,4,6,3,1).The new length of the new X 5 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 = (ρ 1 , ρ 2 , . . ., ρ 9 ) = (0.3, 0.5, 0.1, 0.5) for this X 5 .From Equation ( 28), the process to update X 5 is displayed in Table 11.Note that the numbers marked with "*"are generated randomly.

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 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.

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.

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.

1.
Statistical analyses of the minimum, maximum, average and standard deviation of the fitness values.

2.
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).

Analysis of Results
Statistical analyses of the minimum, maximum, average and standard deviation of the fitness values from the ten benchmarks are displayed in Figures 7-10, 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 Tables 12-15, respectively.Moreover, statistical analyses of the minimum, maximum, average and standard deviation of the running time from the ten benchmarks are displayed in Figures 11-14, where the x-axis and yaxis 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 Tables 16-19.From Tables 16-19, performance of running time obtained by the proposed fSSO compared with those found by SSO and ACO in the WSN is concluded as follows: 1.According to Tables 16 and 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 multiobjective problem including energy consumption, cost and signal transmission quantity.From Tables 12-15, performance of the proposed fSSO compared with SSO and ACO in the WSN is concluded as follows:

According to
1.
The proposed fSSO possesses the ability to solve multi-objective problems in a WSN.

2.
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.

3.
The proposed fSSO can effectively find the best configuration of the number of sensor nodes in a WSN.

4.
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.

5.
According to Tables 13-15, 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 Figures 11-14, 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 Tables [16][17][18][19].From Tables 16-19, performance of running time obtained by the proposed fSSO compared with those found by SSO and ACO in the WSN is concluded as follows: 1.
According to Tables 16 and 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.

2.
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.

3.
According to

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 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

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

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.

Figure 1 .
Figure 1.A schematic picture of a wireless (smart) sensor network (WSN).

Figure 1 .
Figure 1.A schematic picture of a wireless (smart) sensor network (WSN).

Figure 4 .
Figure 4.The maximizing set and minimizing set.

Figure 4 .
Figure 4.The maximizing set and minimizing set.

Energies 2018, 11
; 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 ].

Figure 7 .
Figure 7. Minimum of the fitness values for ten benchmarks.

Figure 7 .
Figure 7. Minimum of the fitness values for ten benchmarks.

Figure 7 .
Figure 7. Minimum of the fitness values for ten benchmarks.

Figure 10 .Figure 9 .
Figure 10.Standard deviation of the fitness values for ten benchmarks.

Figure 9 .
Figure 9. Average of the fitness values for ten benchmarks.

Figure 10 .
Figure 10.Standard deviation of the fitness values for ten benchmarks.

Figure 13 .
Figure 13.Average of the running time for ten benchmarks.

Figure 14 .
Figure 14.Standard deviation of the running time for ten benchmarks.

Figure 13 . 23 Figure 13 .
Figure 13.Average of the running time for ten benchmarks.

Figure 14 .
Figure 14.Standard deviation of the running time for ten benchmarks.

Figure 14 .
Figure 14.Standard deviation of the running time for ten benchmarks.

Table 1 .
The fuzzy energy consumption, fuzzy cost and fuzzy signal transmission quantity of sensors in Figure2.

Table 1 .
The fuzzy energy consumption, fuzzy cost and fuzzy signal transmission quantity of sensors in Figure2.

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

Table 3 .
Triangular fuzzy weights of the three criteria.

Table 5 .
Values of min i α i,j and max i χ i,j for each criterion.

Table 6 .
Normalized triangular fuzzy numbers X i,j = (a i,j , b i,j , c i,j ).

Table 7 .
Weighted normalized triangular fuzzy numbers of G i,j = (A i,j , B i,j , C i,j ).

Table 8 .
Aggregated triangular fuzzy numbersG i = (A i , B i , C i ).

Table 9 .
The y Ri and y Li .

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

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

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

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.Standard deviation of the fitness values for ten benchmarks.

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

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

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

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.Minimum of the running time for ten benchmarks.Minimum of the running time for ten benchmarks.Maximum of the running time for ten benchmarks.

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

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

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

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