**4.1 Proposed optimization control flow model**

**Figure 1** shows the fuzzy DE control flow model for optimizing the transmission cost (TC) metric. In general the DE key operators are fixed. In order to overcome the premature convergence which usually occurs in DE, a fuzzy DE approach is implemented wherein the scaling factor (S) and cross over constants (CR) are adapted using the fuzzy inference engine (FIS) [12].

As the network size increases the load of the network, transmission power and the distance between the router and client varies in different instances. The proposed FDE is a knowledge based system which dynamically selects the best search parameters from the fuzzy set. The Mamdani's fuzzy inference method is used to map the output function. The output of each rule is a fuzzy set and the output fuzzy set is the aggregation of all the sets. The sample rule set is constructed and given as follows for the fuzzy inputs CR, S and output F(x) using IF-THEN structure.


The categorical value for CR includes {low, average, high}, the categorical values for S include {low, average, high}. The decision attribute F(x) i.e. output has four categorical values that include {very low, low, medium, high, very high}. A total number of possible fuzzy inference rules will be 9(3\*3), hence there are two linguistic states. The fuzzy rules are given in **Table 1**. An example membership plot of the input variables CR, S and output variable F(x) using triangular membership function is shown in **Figure 2**.

Defuzzification is used to get the crisp values from the fuzzy inference rules. The input fuzzy set 'μ' is defuzzified into crisp value 'c' using centroid technique.

For example the linguistic values of CR (low = 0.3, average = 0.6, high = 1) and S

As the network control parameters like the minimum distance (Dmin), Traffic Load (TL) and Transmission power (Pt) are uncertain parameters, a fuzzy inference method is used to map the input parameters with output cost metric function TC which refers only the energy consumption of mesh nodes. A sample rule set is

1. IF (Dmin is short) AND (TL is low) AND (Pt is low) THEN TC is low.

2. IF (Dmin is medium) AND (TL is medium) AND (Pt is low) THEN TC is

*C out* ð Þ¼ ∑*xi μ*ð Þ *xi =*∑*μ*ð Þ *xi* (10)

**Low Average High**

Low VL L M Average L M M High M H VH

*Energy Aware Router Placements Using Fuzzy Differential Evolution*

(low = 0.2, average = 0.6, high = 1) the crisp output can be calculated by

where *xi* is the CR or S and *μ*(*xi*) is the linguistic value.

*Membership plot of DE inputs CR, S and output F(x) with surface view plot.*

given by:

**139**

**Figure 2.**

acceptable.

**CR S**

*DOI: http://dx.doi.org/10.5772/intechopen.83747*

*VL, very low; L, low; medium, M; H, high.*

*Fuzzy rules-DE control inputs F(x).*

**Table 1.**

### *Energy Aware Router Placements Using Fuzzy Differential Evolution DOI: http://dx.doi.org/10.5772/intechopen.83747*


### **Table 1.**

which are associated with the MRs. The Transmission Power (Pt) needed to associ-

**Figure 1** shows the fuzzy DE control flow model for optimizing the transmission cost (TC) metric. In general the DE key operators are fixed. In order to overcome the premature convergence which usually occurs in DE, a fuzzy DE approach is implemented wherein the scaling factor (S) and cross over constants (CR) are

As the network size increases the load of the network, transmission power and the distance between the router and client varies in different instances. The proposed FDE is a knowledge based system which dynamically selects the best search parameters from the fuzzy set. The Mamdani's fuzzy inference method is used to map the output function. The output of each rule is a fuzzy set and the output fuzzy set is the aggregation of all the sets. The sample rule set is constructed and given as follows for the fuzzy inputs CR, S and output F(x) using IF-THEN structure.

The categorical value for CR includes {low, average, high}, the categorical values for S include {low, average, high}. The decision attribute F(x) i.e. output has four categorical values that include {very low, low, medium, high, very high}. A total number of possible fuzzy inference rules will be 9(3\*3), hence there are two linguistic states. The fuzzy rules are given in **Table 1**. An example membership plot of the input variables CR, S and output variable F(x) using triangular membership

Defuzzification is used to get the crisp values from the fuzzy inference rules. The input fuzzy set 'μ' is defuzzified into crisp value 'c' using centroid technique.

ate a MC with MR also varies between low and high.

*Wireless Mesh Networks - Security, Architectures and Protocols*

adapted using the fuzzy inference engine (FIS) [12].

IF (CR is low) AND (S is low) THEN F(x) is low.

*Optimization control flow model using fuzzy differential evolution.*

function is shown in **Figure 2**.

**Figure 1.**

**138**

IF (CR is average) AND (S is low) THEN F(x) is low.

IF (CR is high) AND (S is average) THEN F(x) is medium.

**4.1 Proposed optimization control flow model**

*Fuzzy rules-DE control inputs F(x).*

### **Figure 2.**

*Membership plot of DE inputs CR, S and output F(x) with surface view plot.*

For example the linguistic values of CR (low = 0.3, average = 0.6, high = 1) and S (low = 0.2, average = 0.6, high = 1) the crisp output can be calculated by

$$\mathbf{C}(out) = \sum \mathbf{x}i\,\mu(\mathbf{x}i)/\sum \mu(\mathbf{x}i)\tag{10}$$

where *xi* is the CR or S and *μ*(*xi*) is the linguistic value.

As the network control parameters like the minimum distance (Dmin), Traffic Load (TL) and Transmission power (Pt) are uncertain parameters, a fuzzy inference method is used to map the input parameters with output cost metric function TC which refers only the energy consumption of mesh nodes. A sample rule set is given by:


The categorical value for Dmin includes {short, medium, long, very long}, the categorical values for *TL* include {low, medium, high} and the categorical value for *Pt* includes {low, medium, high}.The decision attribute i.e. output cost has 3 categorical values that include {low, acceptable, high}. A total number of possible fuzzy inference rules will be 36(4\*3\*3), hence there are three linguistic states. The fuzzy rules for TC are tabulated in **Table 2**. with respect to four categorical values of minimum distance (a) short (b) medium (c) long and (d) very long. For example the linguistic values of *D*min ranges from zero to 1 km and categorized as (short = 0.2; medium = 0.4; long = 0.6, very long = 1), the values of *Pt* ranges from 1 to 15 dBm and categorized as (low = 5; medium = 10; high = 15). The values of *TL* is the number of clients associated to a MR which ranges from 1 to 45 and categorized for fuzzy rule as (low = 15; medium = 30; high = 45).

Input: TL➔Traffic Load; Pt➔Transmission power; Distance.

Output: TC➔Transmission Cost.

Output variables: L➔Low; A ➔Acceptable; H➔High.

The membership plots of each input variable and the output variable shown in **Figure 3(a)–(d)**. The input variables are minimum distance, traffic load and transmission power. The fuzzy rule viewer and surface view plot are shown in **Figures 4** and **5**.


**Figure 3.**

**141**

*Membership plot of network inputs and output transmission cost metric.*

*Energy Aware Router Placements Using Fuzzy Differential Evolution*

*DOI: http://dx.doi.org/10.5772/intechopen.83747*

**Table 2.** *Fuzzy rules\_Transmission cost.* 3. IF (Dmin is long) AND (TL is high) AND (Pt is low) THEN TC is high.

high.

**Figures 4** and **5**.

(a) Distance: short TC

(b) Distance: medium TC

(c) Distance: long TC

(d) Distance: very long TC

*Fuzzy rules\_Transmission cost.*

**Table 2.**

**140**

4.IF (Dmin is very long) AND (TL is high) AND (Pt is high) THEN TC is very

The categorical value for Dmin includes {short, medium, long, very long}, the categorical values for *TL* include {low, medium, high} and the categorical value for *Pt* includes {low, medium, high}.The decision attribute i.e. output cost has 3 categorical values that include {low, acceptable, high}. A total number of possible fuzzy inference rules will be 36(4\*3\*3), hence there are three linguistic states. The fuzzy rules for TC are tabulated in **Table 2**. with respect to four categorical values of minimum distance (a) short (b) medium (c) long and (d) very long. For example

(short = 0.2; medium = 0.4; long = 0.6, very long = 1), the values of *Pt* ranges from 1 to 15 dBm and categorized as (low = 5; medium = 10; high = 15). The values of *TL* is the number of clients associated to a MR which ranges from 1 to 45 and categorized

The membership plots of each input variable and the output variable shown in

**Low Medium High**

**Figure 3(a)–(d)**. The input variables are minimum distance, traffic load and transmission power. The fuzzy rule viewer and surface view plot are shown in

Low L L A Medium L A H High A A H

Low L L H Medium L A H High A H H

Low A H H Medium A H H High L A A

Low H H H Medium A H H High A A H

the linguistic values of *D*min ranges from zero to 1 km and categorized as

Input: TL➔Traffic Load; Pt➔Transmission power; Distance.

Output variables: L➔Low; A ➔Acceptable; H➔High.

for fuzzy rule as (low = 15; medium = 30; high = 45).

*Wireless Mesh Networks - Security, Architectures and Protocols*

Output: TC➔Transmission Cost.

**Pt TL**

**Figure 3.** *Membership plot of network inputs and output transmission cost metric.*

greater than the consumed energy and the power consumed by the both the mesh nodes must be less than the maximum power allocated. Even if any MR has no client association and it is in idle state, still there is some minimum amount of energy consumption. In order to overcome this, the proposed energy aware scheme turns the inactive mesh routers to sleep mode thus minimizing the energy consumption [13]. The energy consumption level of a node at any time of the simulation can be determined by finding the difference between the current energy value and initial energy

Step 1: The network input parameters like the number of mesh routers and

Step 2: Compute the distance of clients with respect to each router and store in a matrix and calculate the minimum distance at each generation. Step 3: The DE control parameters CR,S are applied to the fuzzy inference

Step 4: The important parameters like distance between MR and MC, Traffic Load(TL) for each router and transmission power of router are fuzzy

The performance of the proposed scheme is analyzed based on three important

*Throughput*. Network throughput is the average rate of successful message

the destination node to the number of data packets transmitted.

*Packet delivery rate*. The ratio of the average number of data packets received by

*Failure rate (FR)*. In a time slot there is a possibility for a MC not to be assigned to a MR, hence they get disconnected which is referred as connection failure. The network performance is evaluated through a metric failure rate and it is defined as the number of failures to the attempts to make the connection.

The proposed approach is evaluated in NS2 simulator and compared with the existing algorithms. The MRs is deployed in a large terrain area of 1000 m 1000 m

ruled and adaptively tuned for optimal setting.

If Pt = f(medium, high) then MR ∈ active mode.

Step 5: If TL = f(medium, high, very high) and If Dmin = f(short, medium) and.

> If TL = f(low, very low) and. If Dmin = f(long, very long) and. If Pt = f(low) then MR ∈ sleep mode.

Step 6: Check the QOS constraints

Compute the fitness value.

delivery over a communication channel.

**6. Simulation results and discussion**

Mc(j) ∈ MR(i) Er(new) = Er-1

Mc(j) ∉ Mr.(i)

metrics PDR, throughput and FR.

Step 7: If true then

Else if.

Then.

**5.1 Performance metrics**

Step 9: End

**143**

engine and adaptively controlled to select the best optimum inputs for

The methodology is illustrated as follows:

*Energy Aware Router Placements Using Fuzzy Differential Evolution*

clients are specified.

*DOI: http://dx.doi.org/10.5772/intechopen.83747*

best result.

value.

**Figure 4.**

*Fuzzy rule viewer.*
