Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences

Shoufeng Ji; Jinhuan Tang; Minghe Sun; Rongjuan Luo

doi:10.3934/jimo.2021051

Article Contents

2022, Volume 18, Issue 3: 1949-1977. Doi: 10.3934/jimo.2021051

This issue Previous Article Multi-aircraft cooperative path planning for maneuvering target detection Next Article Fresh produce price-setting newsvendor with bidirectional option contracts

Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences

1.
School of Business Administration, Northeastern University, Shenyang 110167, China
2.
School of Economics and Management, Shenyang Aerospace University, Shenyang 110136, China
3.
Department of Management Science and Statistics, University of Texas at San Antonio, San Antonio, TX 78249-0632, USA

^* Corresponding author: Dr Rongjuan Luo
^* Corresponding author: Dr Rongjuan Luo

Received: June 30, 2020

Revised: December 31, 2020

Published: May 2022

This research was supported by National Science Foundation of China [grant numbers 71572031, 71971049 and 71702112]

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Abstract

A combined location-routing-inventory system (CLRIS) in a three-echelon supply chain network is studied with environmental considerations. Specifically, a bi-objective mixed integer programming model is formulated for the CLRIS to deal with the trade-offs between the total cost and the carbon-capped difference (CCD). A multi-objective particle swarm optimization (MOPSO) heuristic solution procedure is developed and implemented to solve the bi-objective mixed integer programming problem. The bi-objective mixed integer programming model and the MOPSO heuristic procedure are applied to a real-life problem as an illustrative example. The approximate nondominated frontier formed by solutions not dominated by others can be used for the decision makers to make trade-offs between the total cost and the CCD. Sensitivity analyses are conducted, and the relationship among the carbon cap, CCD, the total cost and the carbon prices are examined, and relevant managerial insights are provided. Comparisons with other existing algorithms show that the MSPSO heuristic procedure has very good performance.

Keywords:

Mathematics Subject Classification: Primary: 65K05; Secondary: 90C08.

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Figure 1. A three-echelon supply chain network

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Figure 2. Locations of the Plants, the potential DCs and the Retailers

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Figure 3. The nondominated frontier of the CLRIS considering CCD

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Figure 4. The nondominated frontier of the CLRIS considering carbon emissions

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Figure 5. Nondominated frontiers of the model minimizing CCD by varying the carbon cap

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Figure 6. Box plots for the three evaluation indicators

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Figure 7. The obtained nondominated frontiers of the all three procedures

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Table 1. Summary of relevant literature for low carbon supply chains

Literature category	Issues considered	Related publications
Translating carbon emissions into cost	Carbon cost; carbon price; pollution cost; cap-and-trade; emissions trading scheme	Tseng and Hung [44]; Treitl et al. [43]; Alhaj et al. [1]; Bai et al. [5]; Wang, Tao and Shi [50].
Treating carbon emissions as a constraint in supply chain models	Carbon footprints; carbon-constrained	Benjaafar et al. [8]; Xu et al. [53]; Lam and Gu [23]; Martí et al. [26].
Policies for carbon emission reduction	Carbon trading; carbon tax	Bazan et al. [7]; Wang et al. [47]; Wang, Zhao and Herty [49]; Saxena et al. [34]; Xin et al. [51]; Huang et al. [18]; Samuel et al. [33].
Multiple objectives with carbon emissions	Trade-offs between cost and environment	Paksoy et al. [30]; Musavi and Bozorgi-Amiri [29]; Daryanto et al. [10].
Carbon emission reduction with coordination mechanisms	Green production; green purchasing; coordination mechanism; life cycle assessment	Palmer [31]; Jaber et al. [19]; Du et al. [13]; Xu et al. [52]; Zhang et al. [56].
Supply chain network design	Location; routing; inventory	Min et al. [27]; Wang, Tao and Shi [50]; Javid and Azad [20]; Farahani et al. [15]; Tang et al. [42]; Zhalechian et al. [55].

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Table 2. Distances from the plants to the potential DCs (km)

	Shenbei	Yuhong	Sujiatun	Dongling
Fushun	77.5	71.1	64.6	36.5
Liaoyang	112.8	65.5	44.7	35.8

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Table 3. Parameters of the potential DCs

	Shenbei	Yuhong	Sujiatun	Dongling
Lead time (days)	3	2	3	4
Service level	95%	95%	95%	95%

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Table 4. Areas and fixed costs of the potential DCs

	Capacity (ton)	Fixed cost per period ()	Holding cost (/ton/day)	Areas (m2)
Shenbei	340	20000	0.20	1100
Yuhong	500	18000	0.25	1600
Sujiatun	310	11200	0.20	1000
Dongling	380	15600	0.25	1200

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Table 5. Details of some typical solutions on the nondominated frontier

Solutions	Plants	DC Status				Total cost	CCD
Solutions	Plants	Shenbei	Yuhong	Sujiatun	Dongling	Total cost	CCD
1	Fushun	1	0	1	1	80153.5	4451.7
1	Liaoyang	1	0	0	0	80153.5	4451.7
2	Fushun	1	1	0	1	81875.5	3677.2
2	Liaoyang	1	1	0	0	81875.5	3677.2
3	Fushun	0	1	0	1	88125.9	3025.7
3	Liaoyang	1	1	0	0	88125.9	3025.7
4	Fushun	1	1	0	1	93750.3	2418.4
4	Liaoyang	1	1	0	1	93750.3	2418.4
5	Fushun	1	1	0	1	100104.9	1975.1
5	Liaoyang	1	1	1	0	100104.9	1975.1
6	Fushun	1	1	1	0	110139.3	1575.7
6	Liaoyang	1	0	1	0	110139.3	1575.7
7	Fushun	1	1	1	0	117697.4	1273.5
7	Liaoyang	1	1	1	0	117697.4	1273.5
8	Fushun	0	1	1	1	122385.8	968.1
8	Liaoyang	0	1	1	1	122385.8	968.1
9	Fushun	1	1	1	1	134982.2	613.9
9	Liaoyang	0	1	1	1	134982.2	613.9

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Table 6. Values of the evaluation indicators for MOPSO, NSGAII and MOEA/D

	Convergence			Diversity			Dominance
	MOPSO	NSGAII	MOEA/D	MOPSO	NSGAII	MOEA/D	MOPSO	NSGAII	MOEA/D
1	0.258	0.179	2.320	0.507	0.518	0.691	0.909	0.476	0.000
2	0.126	0.251	2.261	0.520	0.494	0.775	0.878	0.466	0.000
3	0.193	0.142	1.208	0.508	0.540	0.694	0.814	0.485	0.000
4	0.170	0.254	1.273	0.516	0.482	0.841	0.828	0.441	0.000
5	0.160	0.244	1.679	0.468	0.471	0.738	0.913	0.428	0.000
6	0.158	0.260	1.741	0.559	0.479	0.808	0.820	0.463	0.000
7	0.137	0.241	1.375	0.493	0.497	0.698	0.826	0.535	0.000
8	0.177	0.106	2.207	0.480	0.627	0.817	0.775	0.517	0.000
9	0.233	0.092	1.471	0.516	0.545	0.728	0.789	0.371	0.000
10	0.250	0.103	1.050	0.455	0.493	0.713	0.834	0.526	0.000
11	0.239	0.171	1.607	0.521	0.598	0.789	0.817	0.330	0.000
12	0.106	0.179	1.800	0.485	0.557	0.804	0.892	0.326	0.000
13	0.146	0.188	1.952	0.406	0.533	0.724	0.884	0.478	0.000
14	0.131	0.181	1.667	0.490	0.493	0.704	0.871	0.448	0.000
15	0.115	0.158	2.385	0.433	0.449	0.779	0.905	0.476	0.000
16	0.158	0.195	2.158	0.530	0.482	0.662	0.868	0.557	0.000
17	0.122	0.268	1.276	0.480	0.485	0.710	0.841	0.486	0.000
18	0.193	0.169	1.727	0.489	0.517	0.663	0.892	0.546	0.000
19	0.168	0.220	1.700	0.550	0.576	0.637	0.926	0.403	0.000
20	0.172	1.295	1.333	0.478	0.567	0.787	0.833	0.353	0.000
21	0.143	1.724	1.696	0.502	0.517	0.708	0.866	0.303	0.000

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Table A1. Distances from the potential DCs to the retailers (km)

	Shenbei	Yuhong	Dongling
1	82.6	47.7	55.6
2	80.1	47.0	37.3
3	80.5	27.9	54.2
4	64.5	35.7	42.3
5	62.3	40.0	47.3
6	64.6	49.8	15.7
7	61.2	55.6	7.0
8	58.2	45.0	27.3
9	64.2	36.1	49.1
10	66.1	27.2	49.5
11	66.0	12.0	50.0
12	66.2	48.2	60.1
13	63.8	60.6	12.0
14	56.6	40.2	58.9
15	55.2	35.2	67.1
16	58.1	7.0	55.3
17	62.1	54.2	42.0
18	56.9	42.1	41.1
19	50.3	38.4	47.9
20	45.7	39.0	52.1
21	50.1	32.2	50.0
22	50.1	12.0	57.1
23	40.1	36.1	37.0
24	50.1	48.9	12.2
25	30.2	45.9	52.1
26	25.8	40.1	67.1
27	32.1	40.2	55.3
28	23.2	38.4	55.3
29	25.8	42.6	65.1
30	22.4	48.9	54.2
31	20.5	52.6	62.1
32	23.7	46.0	70.1
33	7.0	55.6	66.2
34	15.0	61.3	72.1

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Table A2. Distances between the retailers (km)

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34
1	0.0	17.1	15.9	18.7	50.2	52.5	55.6	52.5	48.9	50.2	49.5	48.9	58.8	52.5	47.9	51.1	60.0	50.1	52.5	49.1	50.2	50.2	55.3	55.3	56.3	52.1	56.3	62.3	67.3	69.2	70.1	75.2	77.1
2	17.1	0.0	18.7	7.0	10.1	18.8	50.2	20.1	15.9	15.9	24.1	52.5	55.1	18.8	19.8	45.1	53.2	47.1	20.1	50.2	50.2	51.2	53.2	54.7	55.7	53.2	54.9	66.1	66.1	67.1	67.1	69.2	70.1
3	15.9	18.7	0.0	17.3	47.5	49.9	52.1	48.5	45.2	17.3	16.6	16.8	60.0	45.2	47.5	45.7	60.0	49.9	45.1	46.2	44.1	50.3	60.3	60.0	57.3	49.8	55.1	57.3	57.3	60.1	61.3	67.1	68.9
4	18.7	7.0	17.3	0.0	7.0	55.1	40.1	30.2	25.1	7.0	15.2	32.1	41.9	30.2	27.1	25.7	41.9	56.2	30.2	29.5	35.1	26.7	36.9	36.3	41.9	39.1	36.9	50.1	53.4	57.3	59.9	64.7	66.3
5	50.2	10.1	47.5	7.0	0.0	7.0	49.6	7.2	5.9	28.1	50.5	25.8	32.6	27.1	13.8	45.1	32.6	44.3	40.1	46.2	49.9	43.2	32.6	36.6	39.8	39.8	37.6	55.3	55.3	59.9	62.0	65.0	66.0
6	52.5	18.8	49.9	55.1	7.0	0.0	18.8	8.7	20.4	24.1	29.4	48.9	55.1	55.1	57.8	35.9	54.1	55.1	54.1	32.4	48.9	58.8	57.8	48.9	53.9	54.8	55.5	58.4	58.7	62.4	67.1	68.6	68.1
7	55.6	50.2	52.1	40.1	49.6	18.8	0.0	18.8	49.6	50.2	47.1	52.1	5.1	49.6	50.2	54.1	10.0	49.8	47.1	50.1	52.1	48.6	19.8	19.6	23.8	50.1	50.2	49.2	46.3	54.1	59.2	59.2	60.1
8	52.5	20.1	48.5	30.2	7.2	8.7	18.8	0.0	6.4	15.6	20.6	28.7	19.6	14.2	20.6	35.9	19.6	24.2	22.1	20.6	28.7	19.6	20.6	28.7	30.7	32.5	34.6	50.3	45.3	50.3	54.2	55.2	66.4
9	48.9	15.9	45.2	25.1	5.9	20.4	49.6	6.4	0.0	15.2	18.9	25.8	26.8	6.4	15.2	35.6	26.8	27.8	10.1	35.6	38.9	30.2	38.9	39.7	40.1	37.8	41.1	45.6	45.8	47.6	50.2	50.6	50.9
10	50.2	15.9	17.3	7.0	28.1	24.1	50.2	15.6	15.2	0.0	5.0	17.3	55.1	15.2	5.0	15.9	45.2	36.8	15.2	36.8	38.8	39.6	45.2	43.2	43.2	40.1	43.2	45.4	47.9	47.4	45.2	52.1	52.0
11	50.2	24.1	16.6	15.2	50.5	29.4	47.1	20.6	18.9	5.0	0.0	10.0	45.9	27.1	9.0	7.1	27.1	20.1	16.9	16.9	17.5	18.9	27.1	26.7	27.1	25.9	32.2	36.5	45.9	47.2	46.2	53.1	53.0
12	48.9	52.5	16.8	32.1	25.8	48.9	52.1	28.7	25.8	17.3	10.0	0.0	60.0	25.8	9.2	10.0	50.0	40.3	25.8	11.2	12.3	40.3	49.0	45.1	44.3	40.3	44.5	40.3	50.1	53.7	56.5	58.2	59.9
13	58.8	55.1	60.0	41.9	32.6	55.1	5.1	19.6	26.8	55.1	45.9	60.0	0.0	35.6	41.9	45.9	12.1	20.3	32.6	50.1	45.9	35.6	17.0	43.1	45.1	55.1	58.1	59.8	45.1	62.8	69.1	59.8	60.2
14	52.5	18.8	45.2	30.2	27.1	55.1	49.6	14.2	6.4	15.2	27.1	25.8	35.6	0.0	6.2	15.2	45.1	16.1	5.9	23.3	29.1	20.1	40.1	38.7	38.5	29.1	32.1	37.7	39.8	42.3	45.2	46.6	47.9
15	47.9	19.8	47.5	27.1	13.8	57.8	50.2	20.6	15.2	5.0	9.0	9.2	41.9	6.2	0.0	15.2	45.7	48.9	10.2	15.6	20.1	21.2	40.7	41.2	40.2	38.0	40.2	43.5	45.1	47.9	47.1	48.2	48.9
16	51.1	45.1	45.7	25.7	45.1	35.9	54.1	35.9	35.6	15.9	7.1	10.0	45.9	15.2	15.2	0.0	37.1	29.1	25.7	10.1	11.1	23.4	37.1	32.7	33.3	23.4	30.0	31.2	36.1	46.1	45.2	47.7	49.2
17	60.0	53.2	60.0	41.9	32.6	54.1	10.0	19.6	26.8	45.2	27.1	50.0	12.1	45.1	45.7	37.1	0.0	18.7	21.1	45.2	44.5	25.6	8.8	18.7	25.5	45.2	46.6	55.8	45.1	50.1	68.1	63.1	60.0
18	52.1	50.2	45.2	39.1	49.8	18.8	10.0	22.1	25.9	40.1	23.1	48.9	15.0	25.9	57.8	31.2	11.9	7.7	18.8	40.1	44.4	21.8	7.5	44.4	24.5	43.1	46.8	48.7	40.1	45.1	55.1	58.1	58.7
19	50.1	47.1	49.9	56.2	44.3	55.1	49.8	24.2	27.8	36.8	20.1	40.3	20.3	16.1	48.9	29.1	18.7	0.0	7.4	35.9	40.6	10.0	22.7	20.0	24.7	27.8	35.7	40.7	49.7	50.7	52.3	57.9	58.7
20	52.5	20.1	45.1	30.2	40.1	54.1	47.1	22.1	10.1	15.2	16.9	25.8	32.6	5.9	10.2	25.7	21.1	7.4	0.0	20.1	25.5	5.6	27.6	28.7	32.5	30.7	35.7	45.1	45.0	43.1	48.7	49.2	50.1
21	49.1	50.2	46.2	29.5	46.2	32.4	50.1	20.6	35.6	36.8	16.9	11.2	50.1	23.3	15.6	10.1	45.2	35.9	20.1	0.0	8.5	22.2	40.7	34.5	32.1	18.7	24.5	38.0	42.7	43.0	43.7	45.6	46.5
22	50.2	50.2	44.1	35.1	49.9	48.9	52.1	28.7	38.9	38.8	17.5	12.3	45.9	29.1	20.1	11.1	44.5	40.6	25.5	8.5	0.0	27.8	45.2	40.1	38.1	18.9	26.0	30.1	44.1	44.6	45.0	46.7	48.0
23	50.2	51.2	50.3	26.7	43.2	58.8	48.6	19.6	30.2	39.6	18.9	40.3	35.6	20.1	21.2	23.4	25.6	10.0	5.6	22.2	27.8	0.0	27.1	14.2	15.8	20.7	24.2	33.3	38.9	40.1	43.3	45.1	45.2
24	55.3	53.2	60.3	36.9	32.6	57.8	19.8	20.6	38.9	45.2	27.1	49.0	17.0	40.1	40.7	37.1	8.8	22.7	27.6	40.7	45.2	27.1	0.0	13.6	19.5	33.2	34.8	45.2	33.8	38.5	46.1	55.1	50.1
25	55.3	54.7	60.0	36.3	36.6	48.9	19.6	28.7	39.7	43.2	26.7	45.1	43.1	38.7	41.2	32.7	18.7	20.0	28.7	34.5	40.1	14.2	13.6	0.0	6.1	18.9	17.0	41.8	20.0	25.0	40.0	45.2	46.1
26	56.3	55.7	57.3	41.9	39.8	53.9	23.8	30.7	40.1	43.2	27.1	44.3	45.1	38.5	40.2	33.3	25.5	24.7	32.5	32.1	38.1	15.8	19.5	6.1	0.0	20.0	17.9	36.1	30.1	36.1	36.2	37.1	38.2
27	52.1	53.2	49.8	39.1	39.8	54.8	50.1	32.5	37.8	40.1	25.9	40.3	55.1	29.1	38.0	23.4	45.2	27.8	30.7	18.7	18.9	20.7	33.2	18.9	20.0	0.0	7.3	18.2	36.2	38.1	38.1	40.2	44.2
28	56.3	54.9	55.1	36.9	37.6	55.5	50.2	34.6	41.1	43.2	32.2	44.5	58.1	32.1	40.2	30.0	46.6	35.7	35.7	24.5	26.0	24.2	34.8	17.0	17.9	7.3	0.0	13.5	30.1	36.9	40.2	42.1	43.5
29	62.3	66.1	57.3	50.1	55.3	58.4	49.2	50.3	45.6	45.4	36.5	40.3	59.8	37.7	43.5	31.2	55.8	40.7	45.1	38.0	30.1	33.3	45.2	41.8	36.1	18.2	13.5	0.0	35.6	37.3	20.0	45.2	46.2
30	67.3	66.1	57.3	53.4	55.3	58.7	46.3	45.3	45.8	47.9	45.9	50.1	45.1	39.8	45.1	36.1	45.1	49.7	45.0	42.7	44.1	38.9	33.8	20.0	30.1	36.2	30.1	35.6	0.0	7.2	37.3	30.2	32.1
31	69.2	67.1	60.1	57.3	59.9	62.4	54.1	50.3	47.6	47.4	47.2	53.7	62.8	42.3	47.9	46.1	50.1	50.7	43.1	43.0	44.6	40.1	38.5	25.0	36.1	38.1	36.9	37.3	7.2	0.0	25.6	20.1	26.1
32	70.1	67.1	61.3	59.9	62.0	67.1	59.2	54.2	50.2	45.2	46.2	56.5	69.1	45.2	47.1	45.2	68.1	52.3	48.7	43.7	45.0	43.3	46.1	40.0	36.2	38.1	40.2	20.0	37.3	25.6	0.0	38.2	45.6
33	75.2	69.2	67.1	64.7	65.0	68.6	59.2	55.2	50.6	52.1	53.1	58.2	59.8	46.6	48.2	47.7	63.1	57.9	49.2	45.6	46.7	45.1	55.1	45.2	37.1	40.2	42.1	45.2	30.2	20.1	38.2	0.0	8.2
34	77.1	70.1	68.9	66.3	66.0	68.1	60.1	66.4	50.9	52.0	53.0	59.9	60.2	47.9	48.9	49.2	60.0	58.7	50.1	46.5	48.0	45.2	50.1	46.1	38.2	44.2	43.5	46.2	32.1	26.1	45.6	8.2	0.0

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