An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty

Azam Moradi; Jafar Razmi; Reza Babazadeh; Ali Sabbaghnia

doi:10.3934/jimo.2018074

Article Contents

2019, Volume 15, Issue 2: 855-879. Doi: 10.3934/jimo.2018074

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An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty

1.
School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran
2.
Faculty of Engineering, Urmia University, Urmia, West Azerbaijan Province, Iran

* Corresponding author: Tel.: +98 441 2972854; Fax: +98 441 2773591, Email address: r.babazadeh@urmia.ac.ir
* Corresponding author: Tel.: +98 441 2972854; Fax: +98 441 2773591, Email address: r.babazadeh@urmia.ac.ir

Received: December 31, 2017

Revised: December 31, 2017

Early access: June 2018

Published: January 2019

The authors are supported by INSF grant (Iran National Science Foundation).

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Abstract

The design of agile supply chain networks has attracted more attention in recent years according to the competitive business environment. Further, due to high degree of uncertainty in agile supply chains (SCs), developing robust and efficient decision making tools are of interest. In this study, an integrated approach based on principal component analysis (PCA) and multi-objective possibilistic mixed integer programming (MOPMIP) approaches is proposed to optimally design agile supply chain network under uncertainty. The PCA method is used for ranking and filtering the suppliers, constituting the first layer of the supply chain, based on agility criteria. The proposed MOPMIP model is employed to construct the agile supply chain network under uncertainty. In the proposed MOPMIP model, three objective functions including 1) total costs minimization, 2) total delivery time minimization and 3) maximization of flexibility are considered. An interactive fuzzy solution approach is used to solve the proposed MOPMILP model. Two numerical examples, is conducted to evaluate the performance and efficiency of the proposed integrated approach for agile supply chain network design under uncertainty.

Keywords:

Mathematics Subject Classification: Primary: 35Q90, 97N60, 65K05; Secondary: 90C70, 90C10, 90C29.

Citation:

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Figure 1. Here is the Caption of your figure

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Figure 2. Eigenvalue scree plot

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Figure 3. The Pareto optimal solutions for normalized values of first test problem

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Table 1. The primary defined input attributes for suppliers selection

Row	Input attribute	Row	Input attribute
1	Number of facilities	21	Working hours
2	Staff training	22	Bureaucratic
3	Education managers	23	Defective products
4	Standard simple mentation in organizations	24	Material requirements planning
5	In Stock	25	Distribution plan
6	Product price	26	The geographical location of the factory
7	Product variety	27	The geographical area covered
8	Transportation	28	The political situation in the regions covered
9	Waste	29	Infrastructure
10	Market share	30	After Sales Service
11	Career Opportunities	31	Technical Support
12	The use of new technology	32	Management
13	Production Volume	33	Response to Customer Request
14	Automation	34	E-commerce Capability
15	Communication System	35	JIT
16	Delivery	36	Packing Ability
17	Time of preparation	37	Position in the industry
18	Lot Size	38	Product appearance
19	Work in process (WIP)	39	Quality
20	Specialist operators

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Table 2. The most important agility criteria filtered through conducting brain storming meeting

Input indicators	Row
1	Specialist operators
2	The use of new technology
3	Material requirements planning
4	Distribution plan
5	Response to Customer Request
6	Technical Support
7	E-commerce Capability
8	Product variety
9	Production Volume
10	Transportation
11	After Sales Service
12	Automation
13	Communication System
14	JIT
15	Quality
16	The geographical area covered

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Table 3. The performance indicators of suppliers

Output indicators	Row
1	Product price
2	Bureaucratic
3	Delivery time
4	Work in Process (WIP)

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Table 4. Total Variance of components

Component	Initial Eigenvalue
Component	Total	Percentage of Variance	Cumulative Percentage
1	2.074	12.965	12.965
2	1.803	11.267	24.232
3	1.575	9.843	34.075
4	1.380	8.625	42.700
5	1.334	8.336	51.036
6	1.181	7.383	58.419
7	1.027	6.418	64.836
8	0.935	5.841	70.677
9	0.884	5.523	76.200
10	0.793	4.959	81.159
11	0.720	4.498	85.656
12	0.624	3.898	89.554
13	0.580	3.624	93.178
14	0.423	2.642	95.820
15	0.362	2.265	98.086
16	0.306	1.914	100.000

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Table 5. Total Variance of components

Component	Extraction Sums of Squared Loadings			Rotation Sums of Squared Loadings
Component	Total	Percentage of Variance	Cumulative Percentage	Total	Percentage of Variance	Cumulative Percentage
1	2.074	12.965	12.965	1.548	9.675	9.675
2	1.803	11.267	24.232	1.547	9.672	19.347
3	1.575	9.843	34.075	1.510	9.440	28.787
4	1.380	8.625	42.700	1.506	9.410	38.197
5	1.334	8.336	51.036	1.460	9.126	47.324
6	1.181	7.383	58.419	1.401	8.757	56.081
7	1.027	6.418	64.836	1.401	8.755	64.836

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Table 6. Rotated components

Variables	Components
Variables	1	2	3	4	5	6	7
X1	0.042	-0.192	0.615	-0.211	0.159	-0.273	-0.029
X2	-0.175	0.176	-0.015	-0.065	-0.709	-0.101	-0.061
X3	-0.083	-0.096	-0.145	0.479	-0.440	0.219	-0.067
X4	0.120	0.177	-0.220	0.609	0.221	-0.251	-0.108
X5	-0.208	-0.827	0.065	0.139	0.048	0.263	0.109
X6	-0.223	0.571	-0.207	0.099	0.256	0.182	0.418
X7	0.133	-0.134	0.205	0.753	-0.070	0.073	0.037
X8	-0.073	0.004	-0.090	-0.018	-0.016	-0.729	-0.123
X9	0.841	0.035	0.135	0.022	0.083	-0.011	0.014
X10	0.551	-0.016	-0.100	0.309	-0.035	0.370	-0.010
X11	-0.110	0.228	0.695	0.221	0.190	0.141	-0.097
X12	-0.258	0.541	0.155	0.034	-0.283	0.292	0.244
X13	0.175	-0.166	-0.147	-0.281	0.151	0.506	-0.591
X14	-0.177	0.115	-0.637	0.014	0.262	-0.176	-0.120
X15	-0.464	0.159	0.031	-0.086	0.629	-0.029	-0.048
X16	0.111	0.005	-0.030	-0.163	0.072	0.191	0.861

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Table 7. Suppliers scores driven out via PCA approach

Supplier	Score	Supplier	Score	Supplier	Score	Supplier	Score
1	-0.44143	16	-0.45093	31	-0.70673	46	0.22046
2	-0.39948	17	-0.16193	32	0.52357	47	0.646209
3	0.079517	18	0.41908	33	0.574339	48	-0.2407
4	-0.01854	19	-0.05046	34	0.054961	49	0.430716
5	0.06819	20	0.010264	35	-0.3108	50	0.175306
6	-0.16339	21	-0.29805	36	-0.29681	51	-0.12202
7	-0.47472	22	0.283236	37	0.379933	52	0.065461
8	-0.26409	23	-0.10214	38	0.083386	53	-0.80726
9	0.359434	24	-0.3874	39	-0.11785	54	-0.19505
10	-0.48728	25	0.278027	40	0.092893	55	0.258526
11	0.044161	26	0.196247	41	-0.19233	56	0.570286
12	0.50004	27	0.194767	42	0.121639	57	0.265066
13	1.064131	28	0.009783	43	-0.39538	58	0.098087
14	-0.72305	29	-0.08439	44	0.676464	59	-0.50213
15	-0.19635	30	0.266653	45	-0.02721	60	-0.39293

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Table 8. The sources of random generation of the most likely values

Parameter	Value	Parameter	Value
$\tilde fs_{st}$	$ \sim Uniform(1800000, 4300000)$	$\tilde Tr4_{idclt}$	$ \sim Uniform(80, 100)$
$\tilde fp_{pt}$	$ \sim Uniform(1700000, 4000000)$	$\tilde Cost_{mspt}$	$ \sim Uniform(20, 40)$
$\tilde fd_{dt}$	$ \sim Uniform(1600000, 3000000)$	$\tilde MI_{id}$	$ \sim Uniform(100, 120)$
$\tilde fse_{st}$	$ \sim Uniform(180000, 480000)$	$\tilde CPA1_{pft}$	$ \sim Uniform(50000, 60000)$
$\tilde fpe_{pt}$	$ \sim Uniform(170000, 400000)$	$\tilde CPA2_{djt}$	$ \sim Uniform(30000, 35000)$
$\tilde fde_{dt}$	$ \sim Uniform(160000, 300000)$	$\tilde CPA_{pdt}$	$ \sim Uniform(60000, 70000)$
$\tilde de_{ict}$	$ \sim Uniform(100, 200)$	$\tilde td_{pclt}$	$ \sim Uniform(16, 20)$
$\tilde vp_{ipt}$	$ \sim Uniform(120, 150)$	$\tilde tc_{dclt}$	$ \sim Uniform(2, 4)$
$\tilde Hd_{idt}$	$ \sim Uniform(10, 30)$	$\tilde te_{pdlt}$	$ \sim Uniform(10, 16)$
$\tilde Hp_{ipt}$	$ \sim Uniform(15, 40)$	$\tilde dis1_{pcl}$	$ \sim Uniform(200, 600)$
$\tilde Tr1_{isplt}$	$ \sim Uniform(80, 100)$	$\tilde dis2_{dcl}$	$ \sim Uniform(300, 500)$
$\tilde Tr2_{ipdlt}$	$ \sim Uniform(40, 60)$	$\tilde dis3_{pdl}$	$ \sim Uniform(100, 200)$
$\tilde Tr3_{ipclt}$	$ \sim Uniform(110, 200)$

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Table 9. The Pareto optimal solution for different values of importance coefficients

r1	r2	r3	Problem No.	Obj1	Obj2	Obj3	CPU time (Sec)
1	0	0	1	3.69E+11	1.38E+08	0.00E+00	134
1	0	0	2	3.82E+12	1.26E+09	0.00E+00	646
0	1	0	1	6.61E+12	2.44E+07	0.00E+00	164
0	1	0	2	2.50E+15	2.53E+08	0.00E+00	655
0	0	1	1	6.61E+12	1.38E+08	7.46E+07	132
0	0	1	2	2.50E+15	1.26E+09	3.74E+11	692
0.45	0.45	0.1	1	3.39E+12	7.89E+07	1.42E+07	210
0.45	0.45	0.1	2	1.13E+15	6.58E+08	8.33E+10	512
0.35	0.35	0.3	1	3.99E+12	9.30E+07	2.76E+07	167
0.35	0.35	0.3	2	1.30E+15	7.29E+08	1.46E+11	472
0.25	0.25	0.5	1	4.55E+12	1.06E+08	4.70E+07	173
0.25	0.25	0.5	2	1.58E+15	9.41E+08	2.31E+11	627
0.15	0.15	0.7	1	5.50E+12	1.16E+08	5.67E+07	159
0.15	0.15	0.7	2	1.80E+15	9.91E+08	3.07E+11	679
0.05	0.05	0.9	1	6.20E+12	1.25E+08	7.02E+07	134
0.05	0.05	0.9	2	2.25E+15	1.13E+09	3.48E+11	646
0.2	0.3	0.5	1	5.20E+12	9.83E+07	4.78E+07	164
0.2	0.3	0.5	2	1.68E+15	8.90E+08	2.25E+11	655
0.2	0.5	0.3	1	4.93E+12	7.21E+07	2.76E+07	132
0.2	0.5	0.3	2	1.70E+15	6.48E+08	1.23E+11	690
0.3	0.2	0.5	1	4.38E+12	1.13E+08	4.63E+07	210
0.3	0.2	0.5	2	1.53E+15	9.81E+08	2.31E+11	512
0.3	0.5	0.2	1	4.24E+12	6.65E+07	1.72E+07	167
0.3	0.5	0.2	2	1.53E+15	6.38E+08	8.61E+10	472
0.5	0.2	0.3	1	3.12E+12	1.08E+08	2.84E+07	173
0.5	0.2	0.3	2	9.27E+14	9.84E+08	1.12E+11	627
0.5	0.3	0.2	1	2.93E+12	9.59E+07	1.79E+07	159
0.5	0.3	0.2	2	9.47E+14	8.90E+08	9.36E+10	679

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Table 10. The studies mentioning the agility indicators in the literature

Index	Razmi et al. (2011)	Yauch (2011)	Ghodsypour and O'Brien (1998)	Min and Shin (2008)	Weber et al. (1991)	Abratt and Kleyn (2012)	Dickson (1996)	Prater et al. (2001)	Kassaee et al. (2014)	Dahmardeh et al. (2010)	Kumar et al. (2011)	Aktepe et al. (1999)	Lin (2009)	Chan and Thong (2009)	This study
1	*	*	*	*	*	*	*	*		*					*
2	*	*	*	*	*	*	*	*							*
3	*	*	*	*	*	*	*	*							*
4	*	*	*	*	*	*	*	*							*
5	*							*			*				*
6	*	*	*	*	*	*	*	*		*	*	*		*	*
7	*	*	*	*	*	*	*	*							*
8	*							*			*				*
9	*	*	*	*	*	*	*	*							*
10	*	*	*	*		*	*	*							*
11	*	*	*	*		*	*	*							*
12	*	*	*	*	*	*	*	*		*					*
13	*	*	*	*	*	*	*	*							*
14	*	*	*	*	*	*	*	*							*
15	*	*	*	*	*	*	*	*		*			*	*	*
16	*		*	*	*		*			*	*	*		*	*
17	*	*	*	*	*	*	*								*
18	*	*	*	*	*	*	*	*							*
19	*	*	*	*	*	*	*	*							*
20	*		*	*	*	*	*	*							*
21	*		*	*	*	*	*	*							*
22	*	*	*	*	*	*	*	*							*
23	*	*	*	*	*	*	*	*							*
24	*	*	*	*	*	*	*	*							*
25		*						*							*
26		*			*		*	*		*				*	*
27		*			*		*	*		*				*	*
28	*	*												*	*
29															*
30	*	*	*	*	*	*	*	*		*	*			*	*
31	*	*	*	*	*	*	*	*		*	*			*	*
32	*	*	*	*		*	*	*							*
33	*	*	*	*		*	*	*							*
34	*	*		*	*	*	*	*							*
35
36	*	*	*	*	*	*	*	*							*
37	*	*	*	*	*	*	*	*				*	*	*	*
38	*	*	*	*	*	*	*	*							*
39	*	*	*	*	*	*	*	*		*	*	*		*	*

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Table 11. Agile suppliers attributes

No.	In1	In2	In3	In4	In5	In6	In7	In8	In9	In10	In11	In12	In13	In14	In15	In16
1	0.54	0.82	0.5	0.62	0.75	0.45	0.62	0.64	0.71	0.6	0.5	0.57	0.52	0.35	0.23	0.6
2	0.5	0.04	0.63	0.63	0.95	0.46	0.6	0.76	0.4	0.77	0.5	0.6	0.8	0.53	0.38	0.47
3	0.27	0.66	0.72	0.97	0.93	0.48	0.78	0.12	0.37	0.78	0.68	0.63	0.8	0.35	0.12	0.33
4	0.43	0.34	0.71	0.75	0.92	0.56	0.72	0.44	0.53	0.68	0.64	0.68	0.72	0.37	0.32	0.3
5	0.72	0.13	0.54	0.73	0.68	0.5	0.52	0.65	0.71	0.79	0.62	0.61	0.54	0.55	0.19	0.35
6	0.71	0.45	0.81	0.52	0.9	0.52	0.55	0.02	0.31	0.72	0.7	0.65	0.7	0.34	0.1	0.34
7	0.5	0.43	0.89	0.94	0.85	0.41	0.58	0.5	0.58	0.61	0.63	0.53	0.76	0.52	0.76	0.34
8	0.34	0.77	0.74	0.55	0.74	0.42	0.7	0.06	0.8	0.76	0.4	0.63	0.79	0.44	0.24	0.49
9	0.52	0.29	0.86	0.57	0.73	0.62	0.59	0.31	0.71	0.71	0.65	0.65	0.77	0.46	0.74	0.69
10	0.39	0.58	0.65	0.76	0.67	0.66	0.62	0.4	0.21	0.61	0.42	0.6	0.52	0.56	0.94	0.36
11	0.4	0.53	0.97	0.81	0.96	0.47	0.76	0.17	0.69	0.77	0.42	0.74	0.67	0.24	0.24	0.51
12	0.58	0.07	0.75	0.54	0.65	0.6	0.53	0.1	0.86	0.72	0.63	0.74	0.7	0.3	0.06	0.55
13	0.4	0.12	0.62	0.87	0.62	0.7	0.73	0.04	0.62	0.79	0.6	0.8	0.79	0.21	0.59	0.64
14	0.34	0.81	0.87	0.86	0.63	0.67	0.52	0.94	0.2	0.73	0.49	0.52	0.77	0.5	0.22	0.32
15	0.8	0.73	0.89	0.5	0.8	0.59	0.58	0.94	0.26	0.74	0.65	0.62	0.57	0.1	0.44	0.52
16	0.62	0.27	0.57	0.89	1.00	0.44	0.7	0.67	0.2	0.62	0.44	0.56	0.59	0.41	0.86	0.65
17	0.26	0.01	0.86	0.71	0.91	0.51	0.54	0.61	0.37	0.8	0.42	0.64	0.65	0.29	0.37	0.66
18	0.33	0.63	0.96	0.93	0.61	0.65	0.73	0.23	0.44	0.71	0.7	0.64	0.54	0.52	0.17	0.63
19	0.5	0.54	0.91	0.86	0.86	0.53	0.74	0.61	0.65	0.71	0.57	0.61	0.74	0.55	0.94	0.48
21	0.33	0.89	0.68	0.62	0.81	0.59	0.51	0.38	0.51	0.75	0.51	0.77	0.79	0.41	0.86	0.35
22	0.61	0.4	0.56	0.54	0.83	0.7	0.53	0.37	0.63	0.7	0.56	0.64	0.58	0.33	0.97	0.67
23	0.68	0.84	0.98	0.9	0.88	0.44	0.69	0.03	0.6	0.74	0.43	0.59	0.72	0.14	0.25	0.61
24	0.41	0.81	0.81	0.95	0.73	0.46	0.77	0.66	0.5	0.68	0.47	0.63	0.69	0.23	0.31	0.3
25	0.27	0.44	0.63	0.55	0.94	0.69	0.72	0.18	0.28	0.74	0.61	0.71	0.58	0.21	0.00	0.59
26	0.28	0.37	0.53	0.55	0.85	0.47	0.68	0.33	0.75	0.78	0.46	0.65	0.67	0.18	0.35	0.7
27	0.64	0.79	0.63	0.7	0.89	0.45	0.57	0.15	0.69	0.77	0.66	0.71	0.65	0.25	0.55	0.52
28	0.31	0.89	0.99	0.62	0.67	0.6	0.79	0.25	0.38	0.72	0.51	0.8	0.54	0.39	0.11	0.55
29	0.6	0.02	0.76	0.73	0.64	0.7	0.5	0.67	0.53	0.64	0.42	0.79	0.62	0.45	0.99	0.36
30	0.42	0.23	0.76	0.9	0.81	0.66	0.79	0.96	0.69	0.78	0.56	0.51	0.76	0.2	0.51	0.36
31	0.8	0.48	0.8	0.51	0.83	0.43	0.66	0.84	0.3	0.78	0.48	0.58	0.69	0.48	0.12	0.31
32	0.58	0.05	0.66	0.6	0.73	0.61	0.73	0.23	0.87	0.74	0.49	0.67	0.77	0.32	0.03	0.68
33	0.57	0.21	0.86	0.71	0.64	0.54	0.78	0.45	0.55	0.74	0.68	0.77	0.56	0.31	0.99	0.41
34	0.76	0.63	0.65	0.93	0.66	0.44	0.68	0.99	0.75	0.72	0.52	0.8	0.5	0.2	0.5	0.38
35	0.28	0.04	0.71	0.67	0.72	0.43	0.57	0.44	0.72	0.71	0.45	0.5	0.75	0.41	0.29	0.47
36	0.45	0.75	0.87	0.7	0.62	0.41	0.57	0.92	0.63	0.73	0.62	0.78	0.67	0.34	0.36	0.35
37	0.73	0.1	0.75	0.98	0.73	0.55	0.56	0.34	0.75	0.69	0.62	0.52	0.73	0.23	0.61	0.57
38	0.75	0.2	0.75	0.72	0.95	0.51	0.67	0.72	0.46	0.72	0.66	0.66	0.75	0.49	0.91	0.58
39	0.8	0.98	0.52	0.86	0.86	0.68	0.5	0.31	0.69	0.65	0.63	0.64	0.76	0.17	0.27	0.43
41	0.33	0.44	0.84	0.84	0.68	0.7	0.58	0.98	0.37	0.6	0.5	0.78	0.54	0.52	0.27	0.7
42	0.66	0.97	0.59	0.7	0.78	0.59	0.77	0.72	0.44	0.68	0.67	0.71	0.66	0.16	0.34	0.54
43	0.32	0.72	0.75	0.58	0.73	0.58	0.56	0.26	0.21	0.62	0.51	0.66	0.78	0.28	0.62	0.62
44	0.52	0.17	0.56	0.99	0.76	0.65	0.77	0.59	0.46	0.73	0.67	0.67	0.5	0.41	0.87	0.56
45	0.59	0.53	0.8	0.78	0.93	0.46	0.71	0.07	0.78	0.66	0.52	0.59	0.68	0.16	0.71	0.36
46	0.46	0.01	0.77	0.77	0.99	0.5	0.71	0.76	0.9	0.78	0.59	0.6	0.79	0.48	0.46	0.54
47	0.79	0.07	0.82	0.68	0.81	0.59	0.74	0.3	0.64	0.75	0.65	0.62	0.72	0.12	0.41	0.6
48	0.41	0.18	0.64	0.59	0.79	0.46	0.73	0.89	0.76	0.64	0.62	0.54	0.58	0.53	0.22	0.41
49	0.45	0.09	0.53	0.86	0.65	0.64	0.74	0.27	0.28	0.77	0.62	0.58	0.71	0.46	0.84	0.4
50	0.4	0.49	0.91	0.62	0.88	0.58	0.75	0.39	0.35	0.71	0.65	0.78	0.64	0.13	0.39	0.44
51	0.66	0.47	0.71	0.61	0.67	0.57	0.66	0.02	0.31	0.6	0.61	0.75	0.76	0.51	0.69	0.35
52	0.62	0.42	0.81	0.54	0.93	0.54	0.52	0.11	0.22	0.69	0.68	0.79	0.62	0.35	0.55	0.69
53	0.7	0.93	0.66	0.52	0.9	0.45	0.57	0.65	0.31	0.64	0.56	0.58	0.8	0.23	0.25	0.43
54	0.25	0.91	0.82	0.99	0.63	0.65	0.58	0.88	0.36	0.76	0.46	0.78	0.64	0.52	0.17	0.62
55	0.28	0.37	0.52	0.55	0.69	0.6	0.65	0.63	0.28	0.63	0.7	0.72	0.62	0.15	0.88	0.67
56	0.64	0.42	1.00	1.00	0.91	0.69	0.71	0.11	0.89	0.77	0.51	0.62	0.56	0.42	0.21	0.63
57	0.49	0.63	0.91	0.82	0.84	0.54	0.8	0.01	0.53	0.79	0.62	0.56	0.52	0.48	0.19	0.49
58	0.49	0.97	0.56	0.7	0.6	0.66	0.58	0.26	0.82	0.8	0.42	0.61	0.77	0.46	0.59	0.56
59	0.71	0.09	0.62	0.5	0.79	0.52	0.67	0.59	0.21	0.66	0.44	0.51	0.73	0.29	0.6	0.4
60	0.27	0.06	0.97	0.68	0.98	0.59	0.58	0.25	0.2	0.69	0.4	0.75	0.68	0.54	0.72	0.49

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