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Accelerated cuckoo optimization algorithm for the multi-objective welding process

  • Mohamed Arezki MellalEmail author
  • Abdellah Salhi
  • Edward J. Williams
Open Access
Technical Paper
  • 191 Downloads

Abstract

Welding is a well-known process in manufacturing industries due to its importance. Several process parameters should be tuned in order to perform a high-quality welding. Usually, the problem is described as an optimization one and the challenge is to reconcile conflicting objectives. This paper deals with a multi-objective welding process namely the submerged arc welding process, involving five objectives. The weighted sum approach is used to handle it. An accelerated cuckoo optimization algorithm is implemented for this process model and applied to a practical instance of it. On this practical example, the superiority of the proposed optimization technique has been demonstrated in terms of better solutions and fewer required generations of the cuckoos relative to the basic COA and four other optimization algorithms.

Keywords

Welding process Process parameters Accelerated cuckoo optimization algorithm (ACCOA) 

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

Flowchart of the accelerated cuckoo optimization algorithm

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)

Objective

Method

I (Ampere)

V

(Volt)

S

(cm/min)

F

(cm/min)

Optimum result

Required no. of iterations

BW*

ACCOA

416.357

29.337

20.000

226.852

17.062

8

COA

416.200

29.328

20.000

227.037

17.062

20

PPA

415.5685

29.1592

20

229.3993

17.0748

179

TLBO [11]

412.000

29.000

20.000

228.000

17.110

Jaya [16]

416.200

29.327

20.000

227.043

17.062

25

Q-O Jaya [16]

416.500

29.342

20.000

226.940

17.062

17

R*

ACCOA

426.0490

31.9591

9.3009

191.0474

1.3312E − 05

6

COA

450.0000

32.0000

4.0000

202.6015

0.0011

14

PPA

450

32

5.7628

190

9.3467E − 04

23

TLBO [11]

378.0000

31.0000

18.0000

214.0000

0.0086

Jaya [16]

375.8213

30.9250

7.1382

233.7626

0.00355

15

Q-O Jaya [16]

350.0000

30.8981

4.9221

236.2409

0.0027

12

P*

ACCOA

449.9999

30.0266

12.8725

242.3283

13.1402

7

COA

450.0000

30.0303

12.9001

242.7443

13.1401

19

PPA

450.0000

29.8164

12.1109

226.7450

13.0652

15

TLBO [11]

444.0000

29.0000

5.0000

241.0000

11.16

Jaya [16]

450.0000

30.1887

4.0000

277.1496

11.50

24

Q-O Jaya [16]

368.4300

30.3395

11.9418

241.3833

11.50

15

TS*

ACCOA

450.0000

32.0000

11.7662

259.2621

944.0975

5

COA

450.0000

32.0000

11.7946

259.3121

944.0971

18

PPA

450.0000

32.0000

4.0000

259.1989

881.7714

95

TLBO [11]

448.0000

32.0000

11.0000

253.0000

940.90

Jaya [16]

450.0000

32.0000

11.7660

259.2569

944.12

20

Q-O Jaya [16]

450.0000

32.0000

11.7660

259.2569

944.12

9

H*

ACCOA

350.0000

28.0000

4.0000

310.0000

36.66

2

COA

350.0000

28.0000

4.0000

310.0000

36.66

3

PPA

350.0000

28.0000

4.0000

310.0000

36.66

1

TLBO [11]

350.0000

28.0000

4.0000

307.0000

36.65

Jaya [16]

350.0000

28.0000

4.0000

310.0000

36.66

3

Q-O Jaya [16]

350.0000

28.0000

4.0000

310.0000

36.66

2

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)

Method

I

(Ampere)

V

(Volt)

S

(cm/min)

F

(cm/min)

BW

R

P

TS

H

Min Z

Required no. of iterations

ACCOA

382.5168

32.0000

19.6968

214.9693

20.9075

7.5969E − 06

8.1865

812.3678

30.9921

− 0.1065

6

COA

404.5336

28.0591

17.6312

206.8119

21.8616

0.0010

9.0596

770.4275

30.0361

− 0.0108

15

PPA

450.0000

32.0000

5.7625

190.0000

26.2462

6.9561E − 04

9.0434

784.4098

34.2587

− 0.0469

78

TLBO [11]

445.0000

32.0000

7.0000

193.0000

27.05

0.826

9.32

846.6

33.45

19.00

Jaya [16]

423.1719

29.8221

4.0000

267.0907

20.89

0.0152

11.19

856.75

29.69

0.5644

13

Q-O Jaya [16]

382.41

29.416

20.0000

190.0000

19.47

0.0062

10.36

717.99

29.02

0.1933

18

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)

Method

I

(Ampere)

V

(Volt)

S

(cm/min)

F

(cm/min)

BW

R

P

TS

H

Min Z

Required no. of iterations

ACCOA

449.6237

30.6568

4.0000

251.0311

22.4001

0.0011

12.0065

900.1521

32.9043

− 0.3215

3

COA

450.0000

30.7105

4.0000

248.9756

22.3942

7.1693E − 04

11.9645

900.6669

32.9908

− 0.2907

15

PPA

450.0000

32.0000

5.0089

190.0000

25.9241

3.9623E − 04

9.4577

789.8573

34.3685

− 0.0720

138

Jaya [16]

350.0000

28.0000

4.0000

190.0000

26.865

5.636

6.9176

672.855

36.077

− 0.0064

11

Q-O Jaya [16]

350.0000

28.0000

4.0000

190.0000

26.865

5.636

6.9176

672.855

36.077

− 0.0064

7

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

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

Fig. 3

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

Fig. 4

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

Fig. 5

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

Fig. 6

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

Fig. 7

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

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.

Notes

Acknowledgements

We are grateful to ESRC for funding this research under grant number ES/L011859/1.

Compliance with ethical standards

Conflict of interest

The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.LMSS, Faculty of Technology (FT)M’Hamed Bougara UniversityBoumerdesAlgeria
  2. 2.Department of Mathematical SciencesUniversity of EssexColchesterUK
  3. 3.Industrial and Manufacturing Systems Engineering Department, College of Engineering and Computer ScienceUniversity of MichiganDearbornUSA
  4. 4.Decision Sciences, College of BusinessUniversity of MichiganDearbornUSA

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