Multi-objective optimization of abrasive waterjet machining process using Jaya algorithm and PROMETHEE Method

Abstract

In this work, the process parameters optimization problems of abrasive waterjet machining process are solved using a recently proposed metaheuristic optimization algorithm named as Jaya algorithm and its posteriori version named as multi-objective Jaya (MO-Jaya) algorithm. The results of Jaya and MO-Jaya algorithms are compared with the results obtained by other well-known optimization algorithms such as simulated annealing, particle swam optimization, firefly algorithm, cuckoo search algorithm, blackhole algorithm and bio-geography based optimization. A hypervolume performance metric is used to compare the results of MO-Jaya algorithm with the results of non-dominated sorting genetic algorithm and non-dominated sorting teaching–learning-based optimization algorithm. The results of Jaya and MO-Jaya algorithms are found to be better as compared to the other optimization algorithms. In addition, a multi-objective decision making method named PROMETHEE method is applied in this work in order to select a particular solution out-of the multiple Pareto-optimal solutions provided by MO-Jaya algorithm which best suits the requirements of the process planer.

Appendices

Appendix A

Now the steps of the improved PROMETHEE method for selection of an optimal solution from the set of Pareto-optimal solution based on the order preference of the decision maker are demonstrated. For the purpose of demonstration the multi-objective optimization problem of AWJM process described in “Optimization methodology” section is considered.

Step 1 :

A multi-objective decision making problem for AWJM process is formulated by considering the 50 Pareto-optimal solutions provided by MO-Jaya algorithm as alternatives and the multiple objectives such as Kerf and $R_{a}$ are considered as selection criteria. Both the criteria are considered as non-beneficial criteria. Table 11 shows the objective data for multi-objective decision problem of AWJM process. In Table 11, the fifty solutions are considered as alternatives while Kerf and $R_{a}$ are considered as two attributes.

Step 2 :

The next step is to assign the weights of the criteria based on the order preference of decision maker. For the purpose of demonstration equal importance is assigned to all the criteria (i.e. $w_{1}=w_{2} = 0.5$).However, if a decision maker wants to assign unequal weights of relative importance to the criteria then he/she may use the analytical hierarchy process (AHP) method for determining weights of relative importance, explained by Rao and Patel (2010).

Table 11 Objective data for multi-objective decision making problem of AWJM process (case study 1)

Step 3 :

After assigning the weights of relative importance to the criteria next step is to select a preference function. Rao and Patel (2010) had suggested six preference functions namely, usual function, U-shape function, V-shape function, level function, linear function and Gaussian function. In this work the V-shape function is used as the preference function. For more details about calculating the values of preference indices using V-shape preference function the readers may refer to Rao and Patel (2010).

Step 4 :

Now each alternative is to be compared with every other alternative and the preference indices are to be calculated considering each criterion separately. For the purpose of demonstration, the preference indices resulting from pairwise comparison of 50 alternative solutions with respect to the criteria Kerf and $R_{a}$ are shown in Tables 12 and 13, respectively.

Step 5 :

Once the preference values resulting from pairwise comparison of each alternative with every other alternative is calculated for each criteria separately, the next step is to determine the weighted average of the preference functions according to Eq. (5). The weighted average of preference function values are shown in Table 14.

Step 6 :

The values of leaving flow, entering flow and net flow are calculated for each alternative according to Eqs. (6), (7) and (8). Considering, the values of net flow rank is assigned to each alternative following the higher-the-better approach.The values of leaving flow, entering flow, net flow and rank of each alternative is shown in Table 14.

Table 12 The preference values resulting from pairwise comparison of 50 alternative solutions with respect to the criterion Kerf

Table 13 The preference values resulting from pairwise comparison of 50 alternative solutions with respect to the criterion $R_{\mathrm{a}}$

Table 14 Leaving, entering, net flow values and ranking of fifty alternatives

Table 14 shows that for equal importance to both the criteria (i.e. $w_{1}=w_{2}= 0.5$) the improved PROMETHEE method has suggested alternative solution no. 40 as the first choice. Mainly for the purpose of demonstration equal weights are considered for all the criteria. However, if a decision maker wants to assign unequal weights to the criteria then he/she may use the analytical hierarchy process (AHP) method for determining weights of relative importance, explained by Rao and Patel (2010).

Appendix B

In order to demonstrate the working of MO-Jaya algorithm, the multi-objective optimization problem formulated in “Optimization methodology” section is considered in this work. The objective functions are expressed by Eqs. (10) and (11). The input process parameters and their respective upper and lower bounds are expressed by Eqs. (12) to (15). For the purpose of demonstration, a population size of five is considered and two iterations of MO-Jaya algorithm are shown now. The randomly generated initial population is shown in Table 15. The ranks are assigned to each solution based on the ranking methodology discussed in “Ranking methodology” section and crowding distance (CD) is computed as discussed in “Computing the crowding distance” section. The solution with the highest rank and highest CD value is the best solution and the solution with the lowest rank is selected as the worst solution.

Now the new values of the variables are calculated according to Eq. (1). For the purpose of demonstration, the random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable A are considered as 0.91 and 0.15, respectively. The random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable B are considered as 0.67 and 0.5, respectively. The random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable C are considered as 0.25 and 0.65, respectively. The random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable D are considered as 0.36 and 0.75, respectively. The new values of variables A, B, C and D are calculated and are shown in Table 16 along with the corresponding values of Kerf and $R_{a}$.

Now the new solutions are combined with the initial solutions and a combined population is formed. The combined population is shown in Table 17. The ranks and CD value are determined for all the solutions in the combined population. Now based on the ranks and CD value five good solutions are selected from the combined population. These five solutions which are shown in Table 18 will act as initial population for the next iteration.

Iteration 2

The solutions selected at the end of the first iteration, shown in Table 18 are used as initial population for the second iteration. Now in the second iteration, the initial solutions (Table 18) are modified according to Eq. (1) and are shown in Table 19. For the purpose of demonstration the random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable A are considered as 0.54 and 0.29, respectively. The random number $\alpha _{1}$ and $\alpha _{2}$ for variable B are considered as 0.48 and 0.37, respectively. The random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable C are considered as 0.62 and 0.14, respectively. The random numbers $\alpha _{1}$ and $\alpha _{2}$ for variable D are considered as 0.73 and 0.55, respectively.

The combined population is shown in Table 20. Five good solutions are selected from the combined population based on the ranks and CD which are shown in Table 21. These five solutions will act as initial solutions for the third iteration of MO-Jaya algorithm.

Table 15 Initial population

Table 16 New values of the variables and objective functions

Table 17 Combined population

Table 18 Selection of candidate solutions based on non-dominance rank and crowding distance

Table 19 New values of the variables and objective functions

Table 20 Combined population

Table 21 Selection of candidate solutions based on non-dominance rank and crowding distance