A Multi-objective Particle Swarm Optimization Based on Pareto Archive for Integrated Production and Distribution Planning in A Green Supply Chain

Figures & data

Figure 1. An overview of the studied supply chain

Figure 2. Pseudo-code of the proposed pa-PSO

Figure 3. Pseudo-code of the VNS procedure

Table 1. Two problem instances for model validation

Problem number	u	l	TC	TD	TS	TK	p	t	c	j	i	h	s
1	2	2	4	5	2	2	2	2	2	2	2	2	2
2	2	2	4	5	2	2	2	4	5	2	2	2	2

Table 2. Solutions of ɛ-constraint with GAMS for the first sample

Solutions	f1	f2
1	4682.351	2735.078
2	4663.351	2602.621
3	4647.351	2485.076
4	4628.351	2352.619
5	4608.351	2214.46
6	4589.351	2082.002
7	4570.351	1949.545
8	4528.351	1860.11
9	4509.351	1727.653
10	4009.727	1596.596
11	3308.38	1518.006

Table 3. Solution of the proposed pa-PSO for the first sample

	f1	f2
Values	4507.107	2589.304

Table 4. Small size problems

Problem number	u	l	TC	TD	TS	TK	p	t	c	j	i	h	s
3	5	5	5	5	5	5	1	7	7	7	3	2	2
4	5	5	5	5	5	5	2	7	7	7	3	2	2
5	5	5	5	5	5	5	3	7	7	7	3	2	2
6	5	5	5	5	5	5	1	8	10	8	6	2	2
7	5	5	5	5	5	5	2	8	10	8	6	2	2
8	5	5	5	5	5	5	3	8	10	8	6	2	2
9	5	5	5	5	5	5	1	9	15	9	7	2	2
10	5	5	5	5	5	5	2	9	15	9	7	2	2
11	5	5	5	5	5	5	3	9	15	9	7	2	2

Table 5. Large size problems

Problem no.	u	l	TC	TD	TS	TK	p	t	c	j	i	h	s
12	10	10	10	10	10	10	1	10	30	20	10	5	2
13	10	10	10	10	10	10	2	10	30	20	10	5	2
14	10	10	10	10	10	10	3	10	30	20	10	5	2
15	10	10	10	10	10	10	1	11	70	40	15	5	2
16	10	10	10	10	10	10	2	11	70	40	15	5	2
17	10	10	10	10	10	10	3	11	70	40	15	5	2
18	10	10	10	10	10	10	1	12	90	45	15	5	2
19	10	10	10	10	10	10	2	12	90	45	15	5	2
20	10	10	10	10	10	10	3	12	90	45	15	5	2

Table 6. The results of the proposed pa-PSO and NSGA-II algorithm

Problem Number		NSGA-II		pa- PSO
	Diversity metric	Spacing metric	Quality metric	Diversity metric	Spacing metric	Quality metric
3	333.01	0.6601	14.629	633.2	0.8694	85.371
4	415.5	0.8649	0.989	790.6	1.003	99.011
5	777.1	0.99	0	919.5	0.7634	100
6	879.3	0.4562	0	1092.3	0.9911	100
7	906.6	0.7941	20.48	1213.7	1.3482	79.52
8	992.4	0.7054	10.215	1609.4	0.889	89.785
9	1232.9	0.90480	2194.6	1.22	100
10	1176.2	0.44	0	2427.3	0.99	100
11	1720.6	0.59	0	3568.4	1.09	100
12	1302.6	0.73	30.87	1599.5	0.88	69.13
13	1399.4	0.45	22.35	1694.8	0.69	77.65
14	1549.2	0.93	0	2834.7	1.23	100
15	1666.5	0.74	5.24	11009	0.98	94.76
16	2709.3	0.44	15.51	11517	0.99	84.49
17	2858.6	0.91	0	11900	1.65	100
18	11029	0.58	0	22210	0.76	100
19	110424	0.77	16.97	26342	0.94	83.03
20	110950	0.64	0	30077	1.34	100

Figure 4. Computational times of two algorithms for 10 test problems

Table 7. General information for the case study problem

Indices	u	l	TC	TD	TS	TK	p	t	c	j	i	h	s
Values	3	14	4	5	2	3	41	12	21	2	2	10	2

Table 8. Objective functions values of non-dominated solutions from two algorithms

pa-PSO		NSGA-II
f1	f2	f1	f2
1174525	624633.3	1164585	792463.3
1174562.3	624662.4	1164829.8	792534.4
1174603.6	624673.5	1165388.2	792596.7
1174644.7	624690.3	1165987.6	792702
1174684.2	624719.9	1166271.7	792811.1
1174715.3	624736.9	1166647.8	792883.5
1174754.1	624763.1	1167240.6	792940.4
1174786.2	624781.9	1167775.8	793009.9
1174828	624808	1168363	793128.5
1174868.9	624826.2	1168705.4	793170.1
1174908.5	624838.6	1169107.8	793248.9
1174947.3	624856.2	1169552.6	793328.6
1174988.9	624876	-	-
1175024	624888	-	-
1175052.6	624906.7	-	-

Table 9. The metric values for the proposed PSO and the NSGA-II algorithms

	Pa-PSO			NSGA-II
Diversity metric	Spacing metric	Quality metric	Diversity metric	Spacing metric	Quality metric
				metric
100	0.715	1490.3	0	0.307	1186.2

Figure 5. Pareto front of case study with two algorithms