A novel Fibonacci-inspired enhancement of the Bees Algorithm: application to robotic disassembly sequence planning

Figures & data

Figure 1. Fibonacci sequence in the family tree of a drone (adapted from Koshy, 2017).

Table 1. Metaheuristics inspired by Fibonacci.

Author(s)	Name	Fibonacci	Application	Hybrid with
Gudla and Ganguli (2005)	AHGA	Sequence	Benchmark function	GA
Yeh (2006)	MA	Sequence	Multistage supply chain network	GA, greedy
Aras and Aksen (2008)	Not named	Search	Collection centre location problem	Tabu search
Zachariadis and Kiranoudis (2010)	BLSA	Heaps	VRP
Wei et al. (2014)	AVNS	Heaps	VRP
Zhang et al. (2015)	ELS	Heaps	VRP
Da et al. (2018)	FTO	Tree	Power grid
Etminaniesfahani et al. (2018)	FIA	Retracement and time zone	Benchmark function
Sha and Pan (2018)	FSQGA	Sequence	Oil and gas (wellbore trajectory)	GA
Nematollahi et al. (2020)	GROM	Golden ratio	Benchmark function
Zhao et al. (2020)	IWPA	Sequence	Benchmark function	WPA
Hosseinian and Baradaran (2020)	MOFA	Sequence	Project scheduling problem
Abdulhameed and Rashid (2022)	CDDO	Golden ratio	Benchmark function
Khosla and Verma (2022)	Opp-PSOGWO	Sequence	Benchmark function	PSO, GWO
Tran-Ngoc et al. (2023)	FSOA	Golden ratio	Bridge structural health monitoring	SSA

Table 2. Properties of Gear Pumps A and B.

Item	Name	Disassembly point			Disassembly tool	tb(xi)
		X	Y	Z
Gear Pump A
1	Bolt A	49.4	105.5	−12.6	Spanner-I	3
2	Bolt B	74.4	81.0	−12.6	Spanner-I	3
3	Bolt C	74.4	45.0	−12.6	Spanner-I	3
4	Bolt D	49.4	20.5	−12.6	Spanner-I	3
5	Bolt E	24.4	45.0	−12.6	Spanner-I	3
6	Bolt F	24.4	81.0	−12.6	Spanner-I	3
7	Cover	49.4	63.0	−20.6	Gripper-II	4
8	Gasket	49.4	105.5	1.4	Gripper-I	3
9	Gear A	49.4	81.0	3.4	Gripper-I	6
10	Gear B	49.4	45.0	3.4	Gripper-I	6
11	Driven Shaft A	49.4	81.0	−7.6	Gripper-I	4
12	Base	49.4	81.0	49.4	Gripper-II	8
13	Driven Shaft B	49.4	45.0	152.4	Gripper-I	4
14	Packing Gland	49.4	45.0	91.4	Gripper-I	2
15	Gland Nut	49.4	45.0	96.4	Spanner-II	3
Gear Pump B
1	Bolt A	59.1	114.0	−48.4	Spanner-I	4
2	Bolt B	90.3	89.0	−48.4	Spanner-I	4
3	Bolt C	90.3	33.0	−48.4	Spanner-I	4
4	Bolt D	59.1	8.0	−48.4	Spanner-I	4
5	Bolt E	27.9	33.0	−48.4	Spanner-I	4
6	Bolt F	27.9	89.0	−48.4	Spanner-I	4
7	Cover	59.1	82.0	−64.6	Gripper-II	5
8	Gasket	59.1	114.0	−31.4	Gripper-I	4
9	Gear A	59.1	82.0	−30.9	Gripper-I	6
10	Gear B	59.1	40.0	−30.9	Gripper-I	6
11	Shaft A	59.1	40.0	−48.9	Gripper-I	4
12	Base	59.1	114.0	7.1	Gripper-II	4
13	Shaft B	59.1	82.0	136.1	Gripper-I	8
14	Gland A	59.1	94.8	34.1	Gripper-I	3
15	Gland B	59.1	94.8	41.1	Gripper-I	3
16	Gland C	59.1	94.8	48.1	Gripper-I	3
17	Gland D	59.1	94.8	55.1	Gripper-I	3
18	Gland E	59.1	82.0	79.1	Gripper-I	3
19	Bolt stud A	35.1	82.0	89.1	Spanner-II	3
20	Bolt stud B	83.1	82.0	89.1	Spanner-II	3
21	Nut A	35.1	82.0	84.1	Spanner-III	4
22	Nut B	83.1	82.0	84.1	Spanner-III	4
24	Nut C	35.1	82.0	87.1	Spanner-III	4
25	Nut D	83.1	82.0	87.1	Spanner-III	4

Table 3. Experimental design for BA_F.

		max_rv
m	maxnr	0	1	5	10	15
2	3	E1	E2	E3	E4	E5
	5	E6	E7	E8	E9	E10
	8	E11	E12	E13	E14	E15
	13	E16	E17	E18	E19	E20
	21	E21	E22	E23	E24	E25
3	3	E26	E27	E28	E29	E30
	5	E31	E32	E33	E34	E35
	8	E36	E37	E38	E39	E40
	13	E41	E42	E43	E44	E45
	21	E46	E47	E48	E49	E50
4	3	E51	E52	E53	E54	E55
	5	E56	E57	E58	E59	E60
	8	E61	E62	E63	E64	E65
	13	E66	E67	E68	E69	E70
	21	E71	E72	E73	E74	E75
5	3	E76	E77	E78	E79	E80
	5	E81	E82	E83	E84	E85
	8	E86	E87	E88	E89	E90
	13	E91	E92	E93	E94	E95
	21	E96	E97	E98	E99	E100

Table 4. Example of BA_F for a population of 51.

Experiments	n	maxnr = nr1	nr2	nr3	nr4	nr5	BAF population
E76	48	3	2	1	1	1	51
E81	44	5	3	2	1	1	51
E86	37	8	5	3	2	1	51
E91	25	13	8	5	3	2	51
E96	6	21	13	8	5	3	51

Table 5. BA_F results of the initial runs (10 independent runs).

Experiments	m	maxnr	n	NFE	Disassembly time
					max_rv = 0	max_rv = 1	max_rv = 5	max_rv = 10	max_rv = 15
E1 to E5	2	3	48	5348	173.8317	171.5567	160.185	145.065	151.3192
E6 to E10	2	5	45	5345	164.3175	165.0625	151.9225	145.7392	151.4817
E11 to E15	2	8	40	5340	160.9383	154.3392	147.3933	148.6567	143.74
E16 to E20	2	13	32	5332	150.0267	151.0992	148.7533	142.4158	142.9308
E21 to E25	2	21	19	5319	148.09	146.1292	146.4492	144.4858	144.3008
E26 to E30	3	3	48	5448	172.5958	168.9592	155.4217	146.0192	155.5033
E31 to E35	3	5	44	5444	160.6658	167.5758	149.49	143.3275	146.0117
E36 to E40	3	8	38	5438	161.3008	156.2958	146.8142	146.0458	144.9183
E41 to E45	3	13	28	5428	153.625	148.8067	143.9108	144.1133	143.245
E46 to E50	3	21	12	5412	151.1475	145.6175	141.955	142.8275	143.6292
E51 to E55	4	3	48	5548	173.1675	173.7192	153.4317	149.655	147.2775
E56 to E60	4	5	44	5544	165.3075	169.1067	148.6542	145.0658	145.8825
E61 to E65	4	8	37	5537	152.553	160.473	146.28	145.689	144.548
E66 to E70	4	13	26	5526	149.4717	150.3658	143.9317	142.383	142.0917
E71 to E75	4	21	8	5508	148.7025	147.88	144.2342	143.1633	143.0742
E76 to E80	5	3	48	5648	173.423	177.248	154.938	144.268	147.38
E81 to E85	5	5	44	5644	162.987	165.988	149.543	144.179	148.528
E86 to E90	5	8	37	5637	156.544	157.107	144.883	143.512	143.217
E91 to E95	5	13	25	5625	149.994	152.092	144.622	143.033	141.015
E96 to E100	5	21	6	5606	151.319	150.675	143.292	142.158	147.656

Bold values indicate the consistent lowest value.

Table 6. Descriptive statistics of E48, E49, E50, E68, E69 and E70 (50 runs).

	N Statistic	Minimum statistic	Maximum statistic	Mean		Std. deviation statistic
				Statistic	Std. error
E48	50	135.32	154.15	142.3295	0.54402	3.84683
E49	50	136.68	158.32	142.1598	0.61608	4.35638
E50	50	135.32	154.04	142.1537	0.54545	3.85691
E68	50	137.25	151.38	143.0255	0.51073	3.61138
E69	50	135.32	148.13	141.6850	0.39180	2.77047
E70	50	135.32	153.49	141.6220	0.52304	3.69846
Valid N (listwise)	50

Figure 2. EDBA and BA_F results (Gear Pump A).

Figure 3. EDBA and BA_F results (Gear Pump B).

Table 7. Gear Pump A best results (EDBA and BA_F).

	Best disassembly output
Sequence	2-1-6-5-4-3-7-10-9-11-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	2-1-6-5-4-3-7-10-11-9-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	3-4-5-6-1-2-7-10-11-9-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	3-4-5-6-1-2-7-10-9-11-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	5-4-3-2-1-6-7-10-11-9-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	5-4-3-2-1-6-7-10-9-11-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	6-1-2-3-4-5-7-10-11-9-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731
Sequence	6-1-2-3-4-5-7-10-9-11-8-13-15-14-12
Direction	2-2-2-2-2-2-2-2-2-2-2-2-1-1-1
Tool	1-1-1-1-1-1-4-3-3-3-3-3-2-3-4
Time	89.5731

Notes: Direction: 1 = Y+ direction, 2 = Y– direction. Tool: 1 = Spanner-I, 2 = Spanner-II, 3 = Gripper-I, 4 = Gripper-II.

Table 8. Gear Pump B best result (EDBA and BA_F).

	Best disassembly output
Sequence	24-22-20-23-21-19-1-2-3-4-5-6-7-13-10-9-8-12-14-15-16-17-18-11
Direction	1-1-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-1-1
Tool	3-3-2-3-3-2-1-1-1-1-1-1-5-4-4-4-4-5-4-4-4-4-4-4
Time	135.3167
Sequence	24-22-20-23-21-19-1-6-5-4-3-2-7-13-10-9-8-12-14-15-16-17-18-11
Direction	1-1-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-1-1
Tool	3-3-2-3-3-2-1-1-1-1-1-1-5-4-4-4-4-5-4-4-4-4-4-4
Time	135.3167

Notes: Direction: 1 = Y+ direction, 2 = Y– direction. Tool: 1 = Spanner-I, 2 = Spanner-II, 3 = Spanner-III, 4 = Gripper-I, 5 = Gripper-II.

Table 9. EDBA and BA_F average results for Gear Pump A.

Iteration_ population	EDBA				BAF
	Mean	SEM	Delta (%)	Ave_NFE	Mean	SEM	Delta (%)	Ave_NFE
100_31	90.5237	0.33	0.95	3530	90.0960	0.213	0.52	3506
200_31	90.0357	0.27	0.46	7030	89.7208	0.064	0.15	7006
300_31	89.8065	0.23	0.23	10,530	89.5731	0.000	0.00	10,506
400_31	89.6047	0.03	0.03	14,030	89.5731	0.000	0.00	14,006
500_31	89.7387	0.14	0.17	17,530	89.5731	0.000	0.00	17,506
100_41	90.6014	0.32	1.03	4540	89.7404	0.137	0.17	4516
200_41	89.8637	0.21	0.29	9040	89.5731	0.000	0.00	9016
300_41	89.6364	0.04	0.06	13,540	89.5731	0.000	0.00	13,516
400_41	89.5747	0.00	0.00	18,040	89.5731	0.000	0.00	18,016
500_41	89.5731	0.00	0.00	22,540	89.5731	0.000	0.00	22,516
100_51	89.7875	0.08	0.21	5550	89.5731	0.000	0.00	5526
200_51	89.7066	0.08	0.13	11,050	89.5731	0.000	0.00	11,026
300_51	89.6047	0.03	0.03	16,550	89.5731	0.000	0.00	16,526
400_51	89.6047	0.03	0.03	22,050	89.5731	0.000	0.00	22,026
500_51	89.6047	0.03	0.03	27,550	89.5731	0.000	0.00	27,526

Bold values indicate the consistent lowest value.

Table 10. EDBA and BA_F results for Gear Pump B.

Iteration_ population	EDBA				BAF
	Mean	SEM	Delta (%)	Ave_NFE	Mean	SEM	Delta (%)	Ave_NFE
100_31	150.9088	1.36	15.59	3530	146.6653	1.08	11.35	3506
200_31	146.918	0.90	11.60	7030	140.9998	0.36	5.68	7006
300_31	145.5225	0.91	10.21	10,530	140.5303	0.42	5.21	10,506
400_31	142.217	0.70	6.90	14,030	139.7332	0.35	4.42	14,006
500_31	142.0165	0.45	6.70	17,530	139.1772	0.27	3.86	17,506
100_41	148.5027	0.97	13.19	4540	142.6655	0.63	7.35	4516
200_41	143.5897	0.56	8.27	9040	140.5633	0.34	5.25	9016
300_41	142.4695	0.63	7.15	13,540	139.8480	0.19	4.53	13,516
400_41	142.0715	0.81	6.75	18,040	139.0933	0.23	3.78	18,016
500_41	141.9507	0.54	6.63	22,540	138.6493	0.21	3.33	22,516
100_51	148.5675	1.08	13.25	5550	141.6850	0.39	6.37	5526
200_51	143.7058	0.59	8.39	11,050	139.9833	0.29	4.67	11,026
300_51	142.9958	0.69	7.68	16,550	139.2005	0.15	3.88	16,526
400_51	142.0362	0.60	6.72	22,050	139.2242	0.15	3.91	22,026
500_51	142.0932	0.64	6.78	27,550	138.8473	0.14	3.53	27,526

Figure 4. Dunn–Sidak test results (Gear Pump A).

Group 1 = 100_31 EDBA, Group 2 = 100_31 BA_F, Group 3 = 200_31 EDBA, Group 4 = 200_31 BA_F, …, Group 29 = 500_51 EDBA, Group 30 = 500_51 BA_F.

Figure 5. Dunn–Sidak test results (Gear Pump B).

Group 1 = 100_31 EDBA, Group 2 = 100_31 BA_F, Group 3 = 200_31 EDBA, Group 4 = 200_31 BA_F, …, Group 29 = 500_51 EDBA, Group 30 = 500_51 BA_F.

Table 11. Kruskal–Wallis test results (Gear Pump A).

Kruskal–Wallis ANOVA table
Source	SS	df	MS	Chi-sq	Prob > Chi-sq
Columns	8.18798e + 06	29	282,344	263.11	1.4359e-39
Error	3.84605e + 07	1470	26,163.6
Total	4.66485e + 07	1499

Table 12. Kruskal–Wallis test results (Gear Pump B).

Kruskal–Wallis ANOVA table
Source	SS	df	MS	Chi-sq	Prob > Chi-sq
Columns	9.89515e + 07	29	3,412,119.9	533.75	3.24619e-94
Error	1.78946e + 08	1470	121,731.9
Total	2.77897e + 08	1499

Figure 6. EDBA and BA_F boxplot final results.

Table 13. Kruskal–Wallis final results (Gear Pump A).

Kruskal–Wallis ANOVA table
Source	SS	df	MS	Chi-sq	Prob > Chi-sq
Columns	103,511.1	21	4929.1	27.21	0.1641
Error	4,077,771.9	1078	3782.72
Total	4,181,283	1099

Table 14. Kruskal–Wallis final results (Gear Pump B).

Kruskal–Wallis ANOVA table
Source	SS	df	MS	Chi-sq	Prob > Chi-sq
Columns	111,867.9	6	18,644.7	11.44	0.0756
Error	3,299,830.1	343	9620.5
Total	3,411,698	349

Figure 7. PEI (histogram) and Average Disassembly Time (dot): Higher PEI and Lower Time are Better. Higher PEI represents highest performance and Lower time represents best disassembly time (desirable).

Note: The line shows the highest PEI; EDBA is represented by blue and BA_F by purple colour

Table A1. Gear Pump A normality test.

Test of normality
		Kolmogorov–Smirnov^a			Shapiro–Wilk
	Iter_pop	Statistic	df	Sig.	Statistic	df	Sig.
GearA_EDBA	100_31	0.364	50	<0.001	0.461	50	<0.001
	100_41	0.423	50	<0.001	0.522	50	<0.001
	100_51	0.482	50	<0.001	0.406	50	<0.001
	200_31	0.480	50	<0.001	0.269	50	<0.001
	200_41	0.478	50	<0.001	0.197	50	<0.001
	200_51	0.531	50	<0.001	0.248	50	<0.001
	300_31	0.517	50	<0.001	0.127	50	<0.001
	300_41	0.540	50	<0.001	0.198	50	<0.001
	300_51	0.536	50	<0.001	0.125	50	<0.001
	400_31	0.536	50	<0.001	0.125	50	<0.001
	400_41	0.536	50	<0.001	0.125	50	<0.001
	400_51	0.536	50	<0.001	0.125	50	<0.001
	500_31	0.528	50	<0.001	0.167	50	<0.001
	500_41	–	50	–	–	50	–
	500_51	0.536	50	<0.001	0.125	50	<0.001
GearA_BAF	100_31	0.456	50	<0.001	0.395	50	<0.001
	100_41	0.509	50	<0.001	0.169	50	<0.001
	100_51	–	50	–	–	50	–
	200_31	0.527	50	<0.001	0.349	50	<0.001
	200_41	-	50	–	–	50	–
	200_51	–	50	–	–	50	–
	300_31	–	50	–	–	50	–
	300_41	–	50	–	–	50	–
	300_51	–	50	–	–	50	–
	400_31	–	50	–	–	50	–
	400_41	–	50	–	–	50	–
	400_51	–	50	–	–	50	–
	500_31	–	50	–	–	50	–
	500_41	–	50	–	–	50	–
	500_51	–	50	–	–	50	–

^aLilliefors Significance Correction.

Table A2. Gear Pump B normality test.

Test of normality
		Kolmogorov–Smirnov^a			Shapiro–Wilk
	Iter_pop	Statistic	df	Sig.	Statistic	df	Sig.
GearB_EDBA	100_31	0.129	50	0.036	0.889	50	<0.001
	100_41	0.123	50	0.058	0.928	50	0.005
	100_51	0.144	50	0.011	0.915	50	0.002
	200_31	0.130	50	0.035	0.898	50	<0.001
	200_41	0.165	50	0.002	0.866	50	<0.001
	200_51	0.130	50	0.033	0.906	50	<0.001
	300_31	0.191	50	<0.001	0.825	50	<0.001
	300_41	0.159	50	0.003	0.841	50	<0.001
	300_51	0.183	50	<0.001	0.814	50	<0.001
	400_31	0.219	50	<0.001	0.735	50	<0.001
	400_41	0.278	50	<0.001	0.521	50	<0.001
	400_51	0.200	50	<0.001	0.838	50	<0.001
	500_31	0.178	50	<0.001	0.777	50	<0.001
	500_41	0.183	50	<0.001	0.850	50	<0.001
	500_51	0.229	50	<0.001	0.717	50	<0.001
GearB_BAF	100_31	0.146	50	0.009	0.900	50	<0.001
	100_41	0.171	50	<0.001	0.866	50	<0.001
	100_51	0.145	50	0.010	0.936	50	0.010
	200_31	0.110	50	0.179	0.971	50	0.255
	200_41	0.225	50	<0.001	0.838	50	<0.001
	200_51	0.240	50	<0.001	0.870	50	<0.001
	300_31	0.175	50	<0.001	0.911	50	0.001
	300_41	0.264	50	<0.001	0.865	50	<0.001
	300_51	0.335	50	<0.001	0.777	50	<0.001
	400_31	0.114	50	0.123	0.963	50	0.116
	400_41	0.268	50	<0.001	0.842	50	<0.001
	400_51	0.350	50	<0.001	0.770	50	<0.001
	500_31	0.167	50	0.001	0.948	50	0.029
	500_41	0.282	50	<0.001	0.858	50	<0.001
	500_51	0.395	50	<0.001	0.708	50	<0.001

^aLilliefors Significance Correction.

Table A3. Gear Pumps A and B homogeneity test.

Test of homogeneity of variance
		Levene statistic	df1	df2	Sig.
GearA_EDBA	Based on mean	11.653	14	735	<0.001
	Based on median	3.872	14	735	<0.001
	Based on median and with adjusted df	3.872	14	262.360	<0.001
	Based on trimmed mean	7.551	14	735	<0.001
GearA_BAF	Based on mean	17.344	14	735	<0.001
	Based on median	4.337	14	735	<0.001
	Based on median and with adjusted df	4.337	14	94.091	<0.001
	Based on trimmed mean	8.133	14	735	<0.001
GearB_EDBA	Based on mean	7.254	14	735	<0.001
	Based on median	5.442	14	735	<0.001
	Based on median and with adjusted df	5.442	14	539.180	<0.001
	Based on trimmed mean	6.670	14	735	<0.001
GearB_BAF	Based on Mean	29.396	14	735	<0.001
	Based on median	22.333	14	735	<0.001
	Based on median and with adjusted df	22.333	14	232.077	<0.001
	Based on trimmed mean	26.876	14	735	<0.001

Table B1. PEI values – Gear Pump A.

Iteration_population Algorithm	Average disassembly time	Average NFE	PEI (x 0.000001)
300_31 EDBA	89.8065	10,530	4.23
300_31 BAF	89.5731	10,506	4.25
400_31 EDBA	89.6047	14,030	3.18
400_31 BAF	89.5731	14,006	3.19
500_31 EDBA	89.7387	17,530	2.54
500_31 BAF	89.5731	17,506	2.55
100_41 BAF	89.7404	4516	9.87
200_41 BAF	89.5731	9016	4.95
300_41 EDBA	89.6364	13,540	3.30
300_41 BAF	89.5731	13,516	3.30
400_41 EDBA	89.5747	18,040	2.48
400_41 BAF	89.5731	18,016	2.48
500_41 EDBA	89.5731	22,540	1.98
500_41 BAF	89.5731	22,516	1.98
100_51 BAF	89.5731	5526	8.08
200_51 BAF	89.5731	11,026	4.05
300_51 EDBA	89.6047	16,550	2.70
300_51 BAF	89.5731	16,526	2.70
400_51 EDBA	89.6047	22,050	2.02
400_51 BAF	89.5731	22,026	2.03
500_51 EDBA	89.6047	27,550	1.62
500_51 BAF	89.5731	27,526	1.62

Table B2. PEI values – Gear pump B.

Iteration_population Algorithm	Average disassembly time	Average NFE	PEI (x 0.000001)
400_31 BAF	139.7332	14,006	2.04
500_31 BAF	139.1772	17,506	1.64
400_41 BAF	139.0933	18,016	1.60
500_41 BAF	138.6493	22,516	1.28
300_51 BAF	139.2005	16,526	1.74
400_51 BAF	139.2242	22,026	1.30
500_51 BAF	138.8473	27,526	1.05

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