Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization

Xia Zhao; Jianping Dou

doi:10.3934/jimo.2018095

Article Contents

2019, Volume 15, Issue 3: 1263-1288. Doi: 10.3934/jimo.2018095

This issue Previous Article A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems Next Article Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers

Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization

Xia Zhao^a,b, , and
Jianping Dou^c,

a.
Center for Food Security and Strategic Studies, Nanjing University of Finance and Economics, 128 Tielubeijie St., Nanjing 210003, China
b.
Collaborative Innovation Center of Modern Grain Circulation and Safety, Nanjing University of Finance and Economics, 128 Tielubeijie St., Nanjing 210003, China
c.
School of Mechanical Engineering, Southeast University, Nanjing 211189, China

* Corresponding author: Xia Zhao
* Corresponding author: Xia Zhao

Received: July 2017

Revised: January 2018

Early access: June 2018

Published: April 2019

This work is supported by the Programs of National Natural Science Foundation of China(No.51575108 and No.71403114), China Special Fund for Grain-scientific Research in the Public Interest(No.201513004), the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD) and Qinglan Project.

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Abstract

Motivated by observing the importance of logistics cost in the cost structure of some products, this paper aims at multi-objective optimization of integrating supply chain network design with the selection of transportation modes (TMs) for a single-product four-echelon supply chain composed of suppliers, production plants, distribution centers (DCs) and customer zones. The key design decisions are the number, capacity and location of plants and DCs, the flow of products through the network, and the selection of TMs for each flow path. A bi-objective mixed integer linear programming model is first formulated. The two incompatible objectives are minimizing the total cost and maximizing the demand fill rate. The model is validated by applying to the case of the design of fresh apple supply chain. Then, a new metaheuristic, called multi-objective modified particle swarm optimization (MMPSO), is presented to find non-dominated solutions. A new modified binary PSO for updating binary variables along with the adaptive mutation is incorporated into the MMPSO. The MMPSO is compared with a multi-objective basic PSO (MBPSO) and the NSGA-Ⅱ against three small cases and six randomly generated medium and large size problems. The comparative results indicate that the MMPSO is better than the NSGA-Ⅱ and the MBPSO with respect to solution quality and computation efficiency for the problem.

Keywords:

Mathematics Subject Classification: 90B80.

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Figure 1. The studied single product four-echelon supply chain network

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Figure 2. The solution representation of binary and continuous variables by a particle

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Figure 3. Solution procedure of the MILP model by PSO and LINGO

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Figure 8. Location map of an apple supply chain in Shanxi Province of China

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Figure 4. Solution report of case 1 by LINGO given DFR as one

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Figure 5. Non-dominated solutions of case 1 obtained by three algorithms

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Figure 6. Non-dominated solutions of problem 4 obtained by three algorithms

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Figure 7. Average and best values of ERs and ∆ versus nine problems

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Table 1. Key decision variables of five typical non-dominated solutions for case 1

#	Plant locations						DC locations
	YA	TC	BO	XY	WN	XA	BJ	SH	SZ	WH	ZZ	CD	XA
1	1	0	0	1	1	0	0	0	0	1	1	0	1
2	1	1	0	0	1	0	0	0	0	1	1	0	1
3	1	0	0	1	0	1	0	0	0	1	1	0	1
4	1	0	0	1	1	1	0	0	0	1	1	1	1
5	1	1	1	1	1	0	0	0	0	1	1	1	1
Demand	BJ	SH	SZ	WH	CS	ZZ	CD	CQ	XA
1	24000	32000	16000	32000	40000	24000	32000	24000	40000
2	24000	32000	16287.6	33855.8	41188	27999.6	36140.1	24000	40000
3	24000	32000	19765.1	35230.7	48442	25584.1	33739.3	27232.1	50000
4	30000	32687.3	20000	40000	46619.4	30000	39736.1	24000	50000
5	30000	40000	20000	40000	50000	30000	40000	30000	50000
TM	Supplier-to-Plant			Plant-to-DC			DC-to-CZ
1	truck			rail			rail+truck
2	truck			rail			rail+truck
3	truck			rail+truck			rail+truck
4	truck			rail+truck			rail+truck
5	truck			rail+truck			rail+truck
#	1	2	3	4	5
Cost/1E8	4.7783	5.0033	5.3648	5.6808	5.9819
DFR	0.8	0.835	0.897	0.949	1.0

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Table 6. Expected maximal sale amounts of fresh apples for nine CZs(K tons)

CZs	BJ	SH	SZ	WH	CS	ZZ	CD	CQ	XA
Apples	30	40	20	40	50	30	40	30	50

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Table 7. Capacities and procumbent costs for five suppliers

Locations	YA	TC	BO	XY	WN
Capacity (K tons)	120	60	60	160	100
cost (RMB/ton)	610	620	630	620	630

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Table 8. Variable costs of producing fresh apples for potential plants (RMB/ton)

Sites	YA	TC	BO	XY	WN	XA
Cost	500	520	540	530	530	520

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Table 9. Variable costs of warehousing for nine sites (RMB/ton)

DCs	BJ	SH	SZ	WH	CS	ZZ	CD	CQ	XA
Cost	420	430	440	410	400	400	400	400	400

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Table 10. Unit transportation costs for supplier-to-plant (RMB/ton)

cost	YA	TC	BO	XY	WN	XA
YA	0	103.2	212.4	156	139.8	157.2
TC	103.2	0	135	53.4	48	53.4
BO	212.4	135	0	99.6	153	120
XY	156	53.4	99.6	0	53.4	21
WN	139.8	48	153	53.4	0	35.4

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Table 11. Unit transportation costs of truck and rail (RMB/ton)

cost	BJ	SH	SZ	WH	CS	ZZ	CD	CQ	XA
YA	127.6	150.8	199.6	116.1	143.3	75.9	100.8	93.9	37.6
YA	516	877.2	984.6	517.8	607.8	298.2	533.4	508.8	157.2
TC	110.2	133.4	182.2	98.7	125.9	58.5	83.4	78.5	20.2
TC	588	866.4	899.4	467.4	529.2	300.6	444	406.2	53.4
BO	114.2	137.4	186.2	102.7	129.9	62.5	61.4	82.5	24.2
BO	720.6	976.8	912.6	543	566.4	427.2	322.2	324	120
XY	102.9	126.1	174.9	91.5	118.6	51.3	72.6	67.8	12.9
XY	636.6	879.6	866.4	454.8	503.4	327.6	393.6	352.2	21
WN	97	120.2	187.7	85.5	112.7	45.3	78.6	73.7	15.4
WN	589.8	829.2	852	416.4	481.2	273.6	440.4	383.4	35.4
XA	101.2	124.4	173.2	90	116.9	49.5	74.4	69.5	11.2
XA	624	858.6	852.6	435	486.6	308.4	405.6	355.2	0
BJ	0	120.3	189.1	102.7	130.7	62.9	164.4	167	101.2
BJ	0	670.2	1167	645	816	391.8	1030.2	958.2	624
SH	120.3	0	137.6	72.8	99.6	86.1	172.5	173.8	124.4
SH	670.2	0	757.2	481.8	600.6	578.4	1159.8	1005.6	858.6
SZ	189.1	137.6	0	103.9	75.3	153.5	187.9	164.3	173.2
SZ	1167	757.2	0	528	381	816	858.6	689.4	852.6
WH	102.7	72.8	103.9	0	38.3	51.4	113.1	102.7	90
WH	645	481.8	528	0	176.4	291	679.8	523.2	435
CS	130.7	99.6	75.3	38.3	0	78.6	112.9	93.3	116.9
CS	816	600.6	381	176.4	0	442.8	616.2	442.8	486.6
ZZ	62.9	86.1	153.5	51.4	78.6	0	112.7	115.5	49.5
ZZ	391.8	578.4	816	291	442.8	0	690	586.8	308.4
CD	164.4	172.5	187.9	113.1	112.9	112.7	0	34.7	74.4
CD	1030.2	1159.8	858.6	679.8	616.2	690	0	175.8	405.6
CQ	167	173.8	164.3	102.7	93.3	115.5	34.7	0	69.5
CQ	958.2	1005.6	689.4	523.2	442.8	586.8	175.8	0	355.2
Note: the i numbers denote unit cost of rail, and non-i numbers denote unit cost of truck.

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Table 5. Three metrics for MMPSO, MBPSO and NSGA-Ⅱ against three medium and three large randomly generated problems

Pro.	Item	ER			CT/seconds			$\Delta$
Pro.	Item	MMPSO	MBPSO	NSGA-Ⅱ	MMPSO	MBPSO	NSGA-Ⅱ	MMPSO	MBPSO	NSGA-Ⅱ
Pro 4	avg.	0.367	0.96	0.52	767.1	784.9	798.3	0.303	0.455	0.436
	best	0.3	0.933	0.13	737.6	762.3	773.1	0.199	0.394	0.373
	std.	0.0782	0.0279	0.288	35.8	20	29.3	0.049	0.047	0.051
Pro 5	avg.	0.513	0.9	0.547	1147.9	1193.8	1174.6	0.318	0.714	0.453
	best	0.433	0.8	0.433	1091.6	1138.7	1131	0.266	0.695	0.408
	std.	0.0506	0.085	0.0989	36.7	41.2	41.6	0.039	0.014	0.043
Pro 6	avg.	0.487	0.953	0.52	1487.6	1472.1	1491.9	0.315	0.502	0.461
	best	0.433	0.9	0.3	1424.8	1421.8	1404.4	0.232	0.358	0.412
	std.	0.0581	0.04	0.1609	46.4	38.69	100.8	0.062	0.084	0.041
Pro 7	avg.	0.488	0.958	0.519	1511.4	1529.3	1538.9	0.304	0.436	0.423
	best	0.4	0.85	0.35	1484.8	1494.3	1526.6	0.239	0.356	0.378
	std.	0.0598	0.054	0.165	16.29	38.03	12.52	0.059	0.042	0.037
Pro 8	avg.	0.48	0.946	0.48	1947.2	1953.6	1951.6	0.346	0.506	0.435
	best	0.375	0.9	0.4	1933.2	1932.8	1944.6	0.292	0.421	0.302
	std.	0.087	0.025	0.043	13.7	14.5	6.43	0.041	0.046	0.057
Pro 9	avg.	0.472	0.958	0.544	2351.2	2383	2355.8	0.323	0.509	0.413
	best	0.35	0.95	0.325	2343	2316.3	2348.1	0.239	0.432	0.303
	std.	0.078	0.013	0.188	7.62	9.21	6.74	0.079	0.053	0.069

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Table 2. Characteristics of all randomly generated SCND problems

Problem	#	I	J	K	S	P	M
Medium size	4	8	10	12	19	3	3
	5	10	12	14	23	3	3
	6	12	14	15	25	3	3
Large size	7	14	15	16	29	5	4
	8	15	18	18	33	5	4
	9	17	19	20	35	5	4

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Table 3. Comparison of location decisions for case 3 with and without TM selection

Plant locations					DC locations
YA	TC	BO	XY	WN	BJ	SH	SZ	ZZ	WH	XA
1	1	0	1	1	1	1	1	1	1	1
1	1	0	1	1	0	1	0	1	1	1
TM Selection
Optional TMs		Supplier-to-Plant				Plant-to-DC			DC-to-CZ
Truck only		Truck				Truck			Truck
Truck and Rail		Truck				Rail			Rail
Note: the first row with i numbers is binary decision without TM selection, and the second row is with TM selection

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Table 4. Three metrics for MMPSO, MBPSO and NSGA-Ⅱ against three small cases

Case	Metric	Algorithm	Run 1	Run 2	Run 3	Run 4	Run 5	Average	Best
Case1	ER	MMPSO	15/20	12/20	8/20	3/20	7/20	$45\%$	$15\%$
		MBPSO	19/20	11/20	12/20	14/20	16/20	72%	55%
		NSGA-Ⅱ	11/20	8/20	10/20	12/20	10/20	51%	40%
	CT/seconds	MMPSO	46.550	49.901	45.615	41.948	48.298	46.462	41.948
		MBPSO	53.029	54.990	51.532	54.416	52.358	53.265	51.532
		NSGA-Ⅱ	56.441	53.788	52.416	52.791	54.148	53.917	52.416
	$\Delta$	MMPSO	0.478	0.467	0.313	0.322	0.462	0.408	0.313
		MBPSO	0.290	0.326	0.357	0.339	0.315	0.326	0.290
		NSGA-Ⅱ	0.580	0.470	0.626	0.426	0.645	0.549	0.426
Case2	ER	MMPSO	11/20	8/20	9/20	8/20	11/20	47%	40%
		MBPSO	20/20	15/20	18/20	16/20	16/20	85%	75%
		NSGA-Ⅱ	11/20	12/20	12/20	10/20	9/20	54%	50%
	CT/seconds	MMPSO	54.413	52.953	54.179	59.585	59.405	56.107	52.953
		MBPSO	50.104	53.325	55.099	60.025	54.380	54.587	50.104
		NSGA-Ⅱ	56.471	58.846	60.847	55.968	57.705	57.967	55.968
	$\Delta$	MMPSO	0.283	0.369	0.326	0.275	0.341	0.319	0.275
		MBPSO	0.329	0.353	0.269	0.391	0.357	0.339	0.269
		NSGA-Ⅱ	0.631	0.578	0.455	0.517	0.503	0.537	0.455
Case3	ER	MMPSO	8/20	9/20	8/20	14/20	8/20	47%	40%
		MBPSO	17/20	14/20	17/20	13/20	16/20	77%	65%
		NSGA-Ⅱ	7/20	10/20	12/20	12/20	10/20	51%	35%
	CT/seconds	MMPSO	43.607	43.898	37.106	39.952	42.460	41.405	37.106
		MBPSO	43.612	42.369	43.306	42.728	48.301	44.063	42.369
		NSGA-Ⅱ	45.038	48.282	44.382	45.383	46.448	45.907	44.382
	$\Delta$	MMPSO	0.365	0.257	0.373	0.338	0.286	0.324	0.257
		MBPSO	0.412	0.379	0.408	0.292	0.238	0.346	0.238
		NSGA-Ⅱ	0.536	0.544	0.562	0.498	0.571	0.542	0.498

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