Hybridization of Taguchi and Genetic Algorithm to minimize iteration for optimization of solution

Graphical abstract

More specific subject area: Optimization of problem using Genetic Algorithm. Method name: Selection of initial population using Taguchi for Genetic Algorithm. Name and reference of original method: This study combines two techniques, Taguchi method and Genetic Algorithm. Taguchi method is used for selecting initial population in an organized manner rather than random selection. John Holland introduced genetic algorithms in 1960 based on the concept of Darwin's theory of evolution; afterwards, his student Goldberg extended GA  Normal fillet radius at the root of a tooth HPSTC Highest point of single tooth in contact

Method details
Various methods are available to optimize a solution. The genetic algorithm is one of them and used for global optimization. GA is a work based on natural selection and fittest to survival. In GA initial population is required and it was selected by random search. GA gives solution after some iteration based on the complexity of problem [1].
Taguchi design technique gives more variables with fewer experiments using orthogonal array method. It also gives a good combination of variables within the given search area [2].
Various studies are available in the literature for hybridization of Taguchi method and Genetic Algorithm. In all hybridization, focus on a selection of initial population by Taguchi method is untouched. In this paper, instead of selecting initial population by random search, select the initial population by Taguchi design techniques. It will reduce the number of iteration to obtain a solution. This is explained with illustration.

Problem/Illustration
Gears are a very important part in power transmission. Durability and consistency in power transmission is the main focus in gear design. To improve higher durability in power transmission, current focus of in industry is on the failure occurring at a critical section of the tooth [3,4].
Failure at critical section of the tooth can be reduced by minimizing bending stress at critical section of the tooth. Higher pressure angle on the drive side is desirable to minimize bending stress at the root of a tooth [5,6]. It will change geometrical shape of standard symmetric gear tooth. Due to different pressure angle on both sides, standard normal gear becomes asymmetric gear ( Fig. 1) [5,6].

Parameters
For illustration, the parameters are given in Table 1 [7,8].

Objective function/fittness function
In this illustration, the main aim is to minimize bending stress at the critical section. Objective function/fitness function has been considered from reference [8] to calculate fitness value in the Genetic Algorithm.
Objective function/fitness function to minimize bending stress at the critical section with constraint: Nominal tooth root stress s F0 Constraints are: a d > a c ,e ! 1:1, S t ! 0.25m Optimization using GA (Initial population selected by random search) GA algorithm [1] Step 1 Generate initial value of the genes for chromosome Step 2 Calculates fitness of objective function Step 2. Repeat for all population Step 3.9

End
Step 4 Crossover (one-point crossover) Step 4. Step 4.5 generating random numbers equal to no. of chromosome select in step 4, between 1 to (length of Chromosome -1).
Step 4.6 Chromosome will be cut at a crossover point and its gens will be interchanged Step 4.7 End Step 5

Mutation
Step 5. Calculate the number of iteration Step 8

End
Initial population 9 chromosomes have been created using random search method. Parameters such as drive side pressure angle (α d ), Contact ratio (e) and tip thickness (S t )are considered as a gene for chromosomes.
As the pressure angle on the drive side (as compared to the coast side) increases, the bending stress decreases at root of the tooth [5,6]. But contact ratio (e) and tip thickness (S t ) are constraints for the same. Gear design standard procedures recommend that, the contact ratio should be higher or equal to 1.1 [3,4]. Below this contact ratio, tooth loading period increases and it is an undesirable condition for cyclic loading.
Gear design standard procedures recommend that, the tip thickness should be ! 0.4m for hardened gears. In exceptional cases, tip thickness decreases to 0.25m [3,4]. Below this tip thickness it becomes more pointed.
Tip thickness for symmetric spur gear tooth þ tana À a À tan cos À1 r p r t :cosa À cos À1 r p r t :cosa Tip thickness of asymmetric spur gear tooth S t ¼ r t :u td þ r t :u td þ tana c À a c À tan cos À1 r p r t :cosa c À cos À1 r p r t :cosa c Based on the above equations and parameter given in Table 1, the range of tip thickness is between 1.26 to 2.88, the range of contact ratio is between 1.20 to 1.61 and range of drive side pressure angle is between 20 to 38 . Created initial populations selected by random search are:

Solutions
As per reference [7], it is found that optimized or minimized bending stress for given parameters is 45.60 Mpa. The algorithm is presented to solve a problem using GA in section 3.1. Using GA, it is found that 15 iterations are required to obtain optimum bending stress when initial population is selected by random search.

Optimization using GA (Initial population selected by the Taguchi method)
The algorithm to select initial population for GA using Taguchi method [2] Step 1: Select the number of parameters Drive side pressure angle (α d ), Contact ratio (e) and tip thickness (S t ) parameters have been selected.
Step 4: Create the orthogonal array Standard L9 orthogonal array is given in Table 3, which gives the best combinations among a number of runs.
In present study, drive side pressure angle (α d ), Contact ratio (e) and tip thickness (S t ) are considered as a performance parameter. Developed L9 orthogonal array is given in Table 4.

Solutions
As per reference [7], it is found that optimized or minimized bending stress for given parameters is 45.60 Mpa. The algorithm is presented to solve a problem using GA in Section 3.1 and 4.1. Using GA along with Taguchi, it is found that 10 iterations are required to obtain optimum bending stress.

Results and discussion
From the illustration presented in the previous section, number of iterations are 15 when initial population selected by random search and number of iterations are 10 when initial population is selected using Taguchi method. Comparison of number of iterations obtained from GA, when initial population is selected by random search and initial population is selected by Taguchi method is given in Table 5.

Conclusion
It is observed that a number of iterations to solve a problem is reduced, when the Taguchi method is used to select the initial population instead of random search.