1 Introduction

Manufacturing processes are the baselines for any industrial firm to design and make a product. There is a wide range of processes involved such as turning, grinding, milling, ultrasonic machining, abrasive jet machining, and water jet machining, depending on the target product of the manufacturer. Manufacturers strive to optimize these processes individually or overall as the manufacturing circumstance may dictate. Consequently, in the literature, most of the processes are formulated as optimization problems. These often improve the process performances by providing the best values for the process parameters. Unfortunately, these problems are often intractable meaning that classical approaches are not effective.

In the last decade, it has been observed that soft computing methods (computational intelligence) are powerful enough to solve this kind of problems. In [1], a genetic algorithm has been applied to optimize the fiber-reinforced composite injection molding process. The heat-treatment process of an alloy of titanium has been optimized in [2] by using the Taguchi method, while the turning of the same alloy has been optimized in [3] by integrating the gray relational analysis with the Taguchi method. The production time of the multi-pass milling process has been optimized by using the artificial bee colony (ABC) approach, the particle swarm optimization (PSO), and simulated annealing (SA) in [4], whereas the cuckoo optimization algorithm (COA) appears in [5]. The unit production cost of the multi-pass turning process has been minimized by using the teaching–learning-based optimization algorithm (TLBO) [6] and the COA [7]. The machining parameters of other traditional and non-traditional processes have been investigated in [8, 9]. In engineering optimization, the objectives may vary and conflict at the same time, in which case, the problem becomes multi-objective. It can be converted into a single objective by combining the objectives or solving by a Pareto approach.

Some authors focused on the optimization of the important manufacturing process of welding [10,11,12,13,14,15,16]. The aim of the current work is to deal with the multi-objective optimization problem of the submerged arc welding process (SAW) [11, 16]. Our approach is to convert it into a single objective using the weighted sum method. This allows handling the different objectives by resorting to weights assigned to each objective function [17]. In [11], a regression model has been established by experimental means and the optimization problem has been solved using the teaching–learning-based optimization algorithm, whereas in [16], the Jaya algorithm has been improved for this purpose. An accelerated cuckoo optimization algorithm (ACCOA) is implemented for solving the problem in the current work.

The remainder of the paper is organized as follows: Section 2 defines the multi-objective problem of the SAW process expressed by the weighted sum method. Section 3 describes the steps of the implemented ACCOA. A discussion of the obtained results is given in Sect. 4. Finally, the conclusion summarizes the paper and outlines some further likely developments.

2 Multi-objective model representation of the submerged arc welding

The submerged arc welding process is defined by an arc maintained between a continuously fed bare wire electrode and the workpiece and a blanket of powdered flux which generates a protective gas shield. It is an economical method of metal joining [18].

The multi-objective optimization problem of the submerged arc welding of Cr–Mo–V steel investigated here is based on the empirical formulation developed by Rao and Kalyankar [11, 16]. The problem involves two minimization objectives: bead width (BW) in mm and weld reinforcement (R) in mm, and three maximization objectives: weld penetration (P) in mm, tensile strength (TS) in MPa, and weld hardness (H) in Rc. The control parameters of the considered SAW process are the welding current (I) in Amp, voltage (V) in volts, welding speed (S) in cm/min, and wire feed (F) in cm/min. It should be noted that the weld reinforcement must be greater than zero.

The regression models of the objectives are given as follows:

Bead width (BW)

$$\begin{aligned} {\text{Minimize}}\;{\text{BW}}& = 475.425 - 0.9814I - 15.0015V + 2.4805S\\ & \quad - 0.351F + 0.001179I^{2} + 0.25575V^{2} \\ & \quad- 0.109781S^{2} + 0.000773F^{2} + \varepsilon_{\text{BW}} \\ \end{aligned}$$
(1)

where εBW is the error term which takes value 0.656092.

Weld reinforcement (R)

$$\begin{aligned} {\text{Minimize}}\;R &= 931.851 - 2.45118I - 30.4892V - 2.44028S \\ & \quad+ 0.111489F { + }0.0778514IV + 0.00841464IS \\ & \quad- 0.0171696VS + \varepsilon_{R} \, \\ \end{aligned}$$
(2)

where εR is the error term which takes value 0.60023.

Weld penetration (P)

$$\begin{aligned} {\text{Maximize}}\;P &= - 668.516 + 0.094333I + 43.0883V \\ &\quad+ 0.47667S + 0.064944F - 0.000092I^{2} \\ & \quad- 0.7175V^{2} - 0.018515S^{2} - 0.000134F^{2} + \varepsilon_{P} \\ \end{aligned}$$
(3)

where εP is the error term equal to 0.623273.

Tensile strength (TS)

$$\begin{aligned} {\text{Maximize}}\;{\text{TS}} &= - 1148.73 - 0.1934I + 20.1667V + 9.5S\\ & \quad + 9.774F + 0.001467I^{2} - 0.0834V^{2} \\ & \quad - 0.4037S^{2} - 0.01885F^{2} + \varepsilon_{\text{TS}} \\ \end{aligned}$$
(4)

where εTS is the error term equal to 2.170478.

Weld hardness (H)

$$\begin{aligned} & {\text{Maximize}}\;H = 772.444 - 1.45667I - 30V - 0.04167S + 0.00556F \\ & \quad + 0.0018I^{2} + 0.5V^{2} + \varepsilon_{H} \\ \end{aligned}$$
(5)

where εH is the error term equal to 0.078337.

The upper and lower bounds of each parameter are given as follows:

$$350\;({\text{Ampere}}) \le I \le 450\;({\text{Ampere}})$$
(6)
$$28\;({\text{Volt}}) \le V \le 32\;({\text{Volt}})$$
(7)
$$4\;({\text{cm/min}}) \le S \le 20\;({\text{cm/min}})$$
(8)
$$190\;({\text{cm/min}}) \le F \le 310\;({\text{cm/min}})$$
(9)

In the literature, two scenarios are considered for the above objectives, i.e., with and without error terms.

The result of combining objectives using the weighted sum method can be written as follows:

$$\begin{aligned} {\text{Minimize }}Z &= w_{1} \left( {\frac{BW}{{BW^{*} }}} \right) + w_{2} \left( {\frac{R}{{R^{*} }}} \right) - w_{3} \left( {\frac{P}{{P^{*} }}} \right)\\ &\quad - w_{4} \left( {\frac{TS}{{TS^{*} }}} \right) - w_{5} \left( {\frac{H}{{H^{*} }}} \right) \end{aligned}$$
(10)

where BW*, R*, P*, TS*, and H* are the optimal values of the objectives when the problem is solved as a single-objective problem. Here, the values of the weights used in [16] are maintained, i.e., wi= 0.2 for i = 1,…,5.

3 Accelerated cuckoo optimization algorithm

The cuckoo optimization algorithm (COA) has been introduced by Rajabioun in [19]. It is a soft computing method inspired by the special lifestyle of the cuckoo. This bird has the trait of laying its eggs in other birds’ nests of different species. The patterns of invaded birds eggshells are mimicked to evade recognition which may result in the destruction of the eggs. However, this is not always successful and some dissimilar eggs are indeed destroyed. It is also the case that, some cuckoo chicks will starve after hatching, as they eat more than the chicks of the invaded species. The algorithm is based on an egg laying radius (ELR) and the migration of mature cuckoos. Its effectiveness has been proved and it has been implemented for solving various engineering optimization problems, such as the PID controller [19, 20], pattern recognition [21], replacement of obsolete components [22, 23], data mining and clustering [24, 25], combined heat and economic power dispatch [26], and machining parameters [5, 7, 9, 27]. The procedure which determines ELR and that which sets the run parameters form crucial steps of COA. They may be the aspects of the algorithm which contribute to the loss of the best solution when dealing with combined objective functions.

In the current work, the ELR is replaced by a binary procedure to improve COA when solving the problem with the combined objective function of the SAW process. This led to the so-called accelerated cuckoo optimization algorithm (ACCOA). It is implemented as follows:

3.1 ACCOA: the accelerated cuckoo optimization algorithm

Begin ACCOA

  • Step 1: Generate a random number of solutions which represents a set of candidate habitats.

    $$\begin{array}{*{20}l} {{\text{Habitat}}_{1} = [I,V,S,F]} \hfill \\ {{\text{Habitat}}_{2} = [I,V,S,F]} \hfill \\ \vdots \hfill \\ {{\text{Habitat}}_{N} = [I,V,S,F]} \hfill \\ \end{array}$$
    (11)

    where N is the number of total habitats.

  • Step 2: Dedicate some eggs to each cuckoo.

  • Step 3: Binary egg laying.

    Some of the dedicated eggs hatch and those remaining are detected and destroyed by the invaded birds. A binary value is randomly generated for each egg.

    $$Egg = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{if}}\;{\text{the}}\;{\text{egg}}\;{\text{is}}\;{\text{not}}\;{\text{recognized}}} \hfill \\ {\text{else}} \hfill & {} \hfill \\ 1 \hfill & {} \hfill \\ \end{array} } \right.$$
    (12)

    Equation (12) is used for the intensification of the algorithm.

  • Step 4: Limit the total number of surviving cuckoos.

  • Step 5: Evaluate fitness.

  • Step 6: Find the best habitat.

  • Step 7: Migrate the cuckoo to the best habitat.

  • Step 8: If the number of cuckoo iterations is reached, stop; otherwise, go to Step 2.

End ACCOA

It should be noted that the number of habitats is constant at each iteration, and the best habitat is introduced in the next iteration. Figure 1 shows the flowchart.

Fig. 1
figure 1

Flowchart of the accelerated cuckoo optimization algorithm

Full size image

4 Results and discussion

ACCOA has been coded in MATLAB 2015 and run on a personal computer with a processor G620 (2.60 GHz, Sandy Bridge, 4 GB Memory, Windows 7, 32 bits). The algorithm has been applied to the five objectives as single-objective problems [see Eq. (1)–(5)]. It has then been applied to the problem involving the combination of all five objective functions in a single objective [see Eq. (10)].

Table 1 summarizes the optimal results obtained by the teaching–learning-based algorithm (TLBO) [11], Jaya algorithm (Jaya) [16], quasi-oppositional-based Jaya algorithm (Q-O Jaya) [16], the plant propagation algorithm (PPA) [28,29,30], the simple cuckoo optimization algorithm (COA) [19], and the accelerated cuckoo optimization algorithm (ACCOA) for the single regression models without the error terms, where the bold type represents the best value. The number of habitats fixed for the COA and the ACCOA is 20 in order to compare the different performances. As reported in [11, 16], the population size was fixed to 20 for Jaya and Q-O Jaya in order to be able to compare the required number of generations to reach the optimal solutions. The optimal BW is similar for ACCOA, COA, Jaya algorithm and Q-O Jaya, i.e., 17.062. However, the ACCOA required the smallest number of iterations (eight iterations). PPA provided better results (17.0748) than TLBO (17.110). For the R objective, ACCOA decreased its value to 1.3312E − 05 and required only six iterations, compared to COA (0.0011 with 14 iterations), Jaya algorithm (0.00355 with 15 iterations), PPA (9.3467E − 04 with 23 iterations), and Q-O Jaya (0.0027 with 12 iterations). For the P objective, ACCOA (equal to 13.1402) has also outperformed the other algorithms: COA (13.1402), TLBO (11.16), Jaya algorithm (11.50), PPA (13.0652), and Q-O Jaya (11.50). The values provided by Jaya and Q-O Jaya for the TS objective are better than those of COA and ACCOA. However, the value achieved by ACCOA is better than that of COA and required fewer generations. The H objective value is similar for ACCOA, ACO, Jaya algorithm, PPA, and Q-O Jaya (36.66). PPA required only one generation. It seems that this objective has reached the maximum possible value.

Table 1 Optimal results for a single-objective problem (without error terms)
Full size table

Table 2 summarizes the optimal results obtained for the combined objective. It clearly shows that the objective value provided by ACCOA (− 0.1065) is better than those of COA (− 0.0108), PPA (− 0.0469), TLBO (19.00), Jaya algorithm (0.5644), and Q-O Jaya (0.1933). Moreover, ACCOA required fewer iterations (six iterations) than the rest of algorithms. PPA came second, overall.

Table 2 Optimal results for combined objective (without error terms)
Full size table

The optimal results for the combined objective, considering the error terms, are reported in Table 3. Here again, ACCOA outperforms the other methods. The optimal value of ACCOA is (− 0.3215), whereas that of COA is (− 0.2907), PPA (− 0.0720), and those of the Jaya algorithm and the Q-O Jaya are similar (− 0.0064). Furthermore, ACCOA required fewer generations (three iterations) compared to COA (15 iterations), PPA (78 iterations), Jaya algorithm (11 iterations), and Q-O Jaya (seven iterations).

Table 3 Optimal results for combined objective (with error terms)
Full size table

It should be noted that the required CPU times in the combined objective are as follows: without errors (ACOA: 17.25 s; ACCOA: 3.55 s) and with errors (ACOA: 17.67 s; ACCOA: 1.89 s). Figures 2, 3, 4, 5, 6, and 7 summarize the performance of each algorithm for the combined objective. The optimal value by TLBO in the case without errors has not been illustrated.

Fig. 2
figure 2

Optimal value of z for combined objective (without error terms)

Full size image
Fig. 3
figure 3

Required number of iterations for combined objective (without error terms)

Full size image
Fig. 4
figure 4

Required CPU time (s) for combined objective (without error terms)

Full size image
Fig. 5
figure 5

Optimal value of z for combined objective (with error terms)

Full size image
Fig. 6
figure 6

Required number of iterations for combined objective (with error terms)

Full size image
Fig. 7
figure 7

Required CPU time (s) for combined objective (with error terms)

Full size image

5 Conclusions

The goal of this paper was to evaluate the efficiency and robustness of a number of relatively new heuristics on a well-known multi-objective problem that arises in manufacturing. One of these algorithms, namely ACCOA which we introduce here for the first time, is a modification (acceleration) of the well-known cuckoo optimization algorithm. The test problem is the multi-objective optimization model of the submerged arc welding process expressed with the weighed sum method. In the literature, the problem has five objectives: the bead width, the weld reinforcement, the weld penetration, the tensile strength, and the weld hardness. ACCOA implements a binary decision to avoid the disadvantage due to the egg laying radius of the original cuckoo optimization algorithm. The results reveal the effectiveness of the current approach in terms of better results (robustness) and lower numbers of required iterations (efficiency) for reaching the optimum results. The disadvantage of the current work is related to the decision on the number of eggs. Further work on this issue is underway. Moreover, work on an application to a welding process involving more than five objectives and four decision variables will be reported in the future. On the other hand, availability of adequate equipment will experimentally investigate the results.