Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment

Zhimin Liu; Shaojian Qu; Hassan Raza; Zhong Wu; Deqiang Qu; Jianhui Du

doi:10.3934/jimo.2020094

Article Contents

2021, Volume 17, Issue 5: 2783-2804. Doi: 10.3934/jimo.2020094

This issue Previous Article The $ F $-objective function method for differentiable interval-valued vector optimization problems Next Article Stability of ground state for the Schrödinger-Poisson equation

Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment

1.
School of Mathematics Science, Liaocheng University, Liaocheng, China
2.
Business School, University of Shanghai for Science and Technology, Shanghai, China
3.
Nanjing University of Information Science and Technology, Nanjing, China
4.
National University of Singapore, Singapore

^* Corresponding author: Shaojian Qu
^* Corresponding author: Shaojian Qu

Received: July 31, 2019

Revised: January 31, 2020

Early access: May 2020

Published: September 2021

The first author is supported by National Social Science Foundation of China (No. 17BGL083)

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Abstract

The problem of the optimal location-allocation of processing factory and distribution center for supply chain networks under uncertain transportation cost and customer demand are studied. We establish a two-stage mean-risk stochastic 0-1 mixed integer optimization model, by considering the uncertainty and the risk measure of the supply chain. Given the complexity of the model this paper proposes a modified hybrid binary particle swarm optimization algorithm (MHB-PSO) to solve the resulting model, yielding the optimal location and maximal expected return of the supply chain simultaneously. A case study of a bread supply chain in Shanghai is then presented to investigate the specific influence of uncertainties on the food factory and distribution center location. Moreover, we compare the MHB-PSO with hybrid particle swarm optimization algorithm and hybrid genetic algorithm, to validate the proposed algorithm based on the computational time and the convergence rate.

Keywords:

Two-stage mean-risk stochastic 0-1 mixed integer optimization,
conditional-value-at-risk,
location allocation,
uncertainty,
MHB-PSO.

Mathematics Subject Classification: Primary: 90C15; Secondary: 90C90.

Citation:

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Figure 1. Network structure of location-allocation supply chain

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Figure 2. Two-stage process of location-allocation problem

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Figure 3. Location of flour factory, food factory, Rt-mart and bread demand area in Shanghai

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Figure 4. Location-allocation supply chain network

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Figure 5. Comparisons of different algorithms

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Figure 6. Values of fitness functions and supply chain profit with different confidence

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Table 1. Gap analysis of various research focus of supply chain

Reference	Location type	Random parameters	Stochastic approach	Risk approach	Solving algorithm
[27]	Relief centers	Demand, supply	Two-stage	Risk-neutral	Heuristic
[Irawan and Jones2019Formulation]	Distribution center	//	Two-stage	Risk-neutral	Matheuristic
[32]	Emergency facility	Demand	Two-stage	CVaR	Bender decomposition
[6]	Warehouse	//	Two-stage	Risk-neutral	Branch-and-bound
[21]	Factory	//	Two-stage	Risk-neutral	Heuristic
[24]	Factory, warehouse	//	Two-stage	Risk-neutral	Heuristic
[37]	Facility	Throughput costs	One-stage	Risk-neutral	Heuristic
[4]	Factory	Demand	One-stage	Risk-neutral	Branch-and-bound
[5]	Facility	Demand	Two-stage	Risk-neutral	L-shaped
[30]	Distribution center	//	One-stage	Risk-neutral	Heuristic
[50]	Facility	Lead time	Two-stage	CVaR	Decomposition
[19]	Facility	Demand	One-stage	Risk-neutral	Combined simulated annealing

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Table 2. Numerical optimal solution and value of the example

$\mathbf{e}_{best}=(1, 1, 1, 0)$	$\mathbf{c}_{best}=(1, 0, 0, $	$0, 1, 0, 1, 0, 1, 0)$
$x_{111}^*=495.0740$	$x_{112}^*=935.0000$	$x^*_{113}=69.9260$	$x_{123}^*=382.7580$	$y_{111}^*=495.0740$
$y_{125}^*=424.7554$	$y_{127}^*=466.7951$	$y_{129}^*=43.4494$	$y_{135}^*=52.9868$	$y_{139}^*=399.6973$
$z_{111}^*=495.0740$	$z_{152}^*=477.7422$	$z_{171}^*=2.6723$	$z_{172}^*=7.1685$	$Pro=1.2952e+04$
$z_{174}^*=456.9543$	$z_{193}^*=443.1467$		Others=0	$Val=-1.2799e+04$

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Table 3. Comparisons of different algorithms

Algorithm	$\mathbf{e}_{best}$	$\mathbf{c}_{best}$	Pro	Val	TI
MHB-PSO	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$1.2952e+04$	$-1.2799e+04$	$9849.4900$
Hybrid PSO	$(0, 1, 1, 1)$	$(0, 0, 1, 0, 0, 1, 1, 0, 1, 0)$	$1.2861e+04$	$-1.2683e+04$	$11169.2300$
Hybrid GA	$(0, 1, 0, 1)$	$(0, 0, 1, 0, 0, 1, 0, 0, 1, 1)$	$1.2855e+04$	$-1.2660e+04$	$12035.1647$

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Table 4. Results of MHB-PSO with different parameters

System	Parameters				Results
T	N	$c_{min}$	$c_{max}$	$\mathbf{e}_{best}$	$\mathbf{c}_{best}$	Val	Error(%)
50	10	2.0	2.1	$(1, 1, 1, 0)$	$(1, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$-1.2690e+04$	0.99
100	10	2.0	2.1	$(0, 1, 1, 1)$	$(0, 0, 1, 0, 0, 1, 1, 0, 1, 0)$	$-1.2712e+04$	0.76
1000	10	2.0	2.1	$(1, 1, 1, 0)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$-1.2780e+04$	0.23
2000	10	2.0	2.1	$(1, 1, 1, 0)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$-1.2780e+04$	0.23
1000	100	2.0	2.1	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$-1.2799e+04$	0.08
1000	1000	2.0	2.1	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$-1.2799e+04$	0.08
1000	10	2.0	2.5	$(0, 1, 0, 1)$	$(0, 0, 1, 0, 1, 0, 0, 0, 1, 1)$	$-1.2801e+04$	0.06
1000	10	2.0	3.0	$(1, 1, 0, 1)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$-1.2809e+04$	0.00
1000	10	2.0	4.0	$(1, 1, 1, 0)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$-1.2780e+04$	0.23
1000	10	2.5	3.0	$(1, 1, 0, 1)$	$(0, 0, 0, 1, 1, 0, 1, 0, 1, 0)$	$-1.2721e+04$	0.69
1000	10	2.5	4.0	$(1, 1, 1, 0)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$-1.2760e+04$	0.38

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Table 5. Supply chain profit with random and expected transportation cost and demand

$\xi(\omega)$	$\mathbf{e}_{best}$	$\mathbf{c}_{best}$	Pro
Random	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$1.2952e+04$
Expected	$(0, 1, 0, 1)$	$(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$	$1.2819e+04$

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Table 6. Comparison of supply chain profit of model with $ \lambda = 0.1 $ and $ \lambda = 0 $

$\lambda$	$\mathbf{e}_{best}$	$\mathbf{c}_{best}$	Pro
$\lambda=0$	$(1, 1, 1, 0)$	$(0, 0, 1, 1, 0, 1, 1, 0, 1, 0)$	$1.3007e+04$
$\lambda=0.1$	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$1.2952e+04$

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Table 7. Effect of raw material cost and retail price on supply chain profit

$(r_{11}, r_{12})$	$h_1$	${\mathbf e}_{best}$	${\mathbf c}_{best}$	Pro
(4.6, 4.8)	14.0	$(1, 1, 1, 0)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$1.1989e+04$
(4.6, 4.8)	14.5	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$1.2952e+04$
(4.6, 4.8)	15.0	$(1, 1, 1, 0)$	$(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$	$1.3809e+04$
(4.3, 4.8)	14.5	$(0, 1, 1, 1)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$1.3233e+04$
(4.9, 4.8)	14.5	$(0, 1, 0, 1)$	$(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$	$1.2500e+04$
(4.6, 4.5)	14.5	$(0, 1, 0, 1)$	$(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$	$1.3063e+04$
(4.6, 5.0)	14.5	$(1, 1, 1, 0)$	$(0, 0, 0, 1, 0, 1, 1, 0, 1, 0)$	$1.2914e+04$
(4.3, 4.5)	14.5	$(0, 1, 1, 1)$	$(0, 0, 0, 1, 0, 0, 1, 1, 1, 0)$	$1.3424e+04$
(4.9, 5.0)	14.5	$(0, 1, 0, 1)$	$(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$	$1.2361e+04$

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Table 8. Parameters for suppliers, processing factories, and distribution centers

Index	Suppliers		Processing		plants	Distribution		centers
$s, i, j$	$a_{1s}$	$r_{1s}$	$b_{i1}$	$q_{i1}$	$f_i$	$\tau_{1j}$	$w_{1j}$	$g_j$
1	1500	4.6	850	0.80	185	500	0.25	135
2	1200	4.8	935	0.75	180	480	0.25	130
3	$/$	$/$	845	0.81	200	450	0.25	120
4	$/$	$/$	950	0.76	160	470	0.25	125
5	$/$	$/$	$/$	$/$	$/$	510	0.25	140
6	$/$	$/$	$/$	$/$	$/$	490	0.25	130
7	$/$	$/$	$/$	$/$	$/$	520	0.25	145
8	$/$	$/$	$/$	$/$	$/$	505	0.25	143
9	$/$	$/$	$/$	$/$	$/$	460	0.25	123
10	$/$	$/$	$/$	$/$	$/$	485	0.25	133

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Table 9. Random transportation cost from flour factory to food factory

Flour factory	Yingyuan food factory	Changli food factory	Ziyan food factory	Sunhong food factory
Fuxin third	$\mathscr{U}(0.18, 0.21)$	$\mathscr{U}(0.12, 0.16)$	$\mathscr{U}(0.25, 0.28)$	$\mathscr{U}(0.40, 0.43)$
Fuxin	$\mathscr{U}(0.45, 0.49)$	$\mathscr{U}(0.25, 0.28)$	$\mathscr{U}(0.11, 0.15)$	$\mathscr{U}(0.27, 0.30)$

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Table 10. Random transportation cost of food factory transporting product to Rt-mart supermarket

Rt-mart	Yingyuan food factory	Changli food factory	Ziyan food factory	Sunhong food factory
Meilanhu	$\mathscr{U}(0.26, 0.30)$	$\mathscr{U}(0.37, 0.40)$	$\mathscr{U}(0.46, 0.49)$	$\mathscr{U}(0.65, 0.68)$
Anting	$\mathscr{U}(0.46, 0.49)$	$\mathscr{U}(0.41, 0.45)$	$\mathscr{U}(0.40, 0.44)$	$\mathscr{U}(0.68, 0.72)$
Nanxiang	$\mathscr{U}(0.31, 0.34)$	$\mathscr{U}(0.27, 0.31)$	$\mathscr{U}(0.33, 0.36)$	$\mathscr{U}(0.56, 0.60)$
Yangpu	$\mathscr{U}(0.08, 0.11)$	$\mathscr{U}(0.17, 0.20)$	$\mathscr{U}(0.34, 0.37)$	$\mathscr{U}(0.41, 0.44)$
Sijing	$\mathscr{U}(0.45, 0.49)$	$\mathscr{U}(0.27, 0.31)$	$\mathscr{U}(0.18, 0.21)$	$\mathscr{U}(0.48, 0.52)$
Chunshen	$\mathscr{U}(0.36, 0.39)$	$\mathscr{U}(0.12, 0.16)$	$\mathscr{U}(0.06, 0.09)$	$\mathscr{U}(0.33, 0.37)$
Kangqiao	$\mathscr{U}(0.26, 0.29)$	$\mathscr{U}(0.11, 0.15)$	$\mathscr{U}(0.24, 0.28)$	$\mathscr{U}(0.19, 0.23)$
Songjiang	$\mathscr{U}(0.58, 0.62)$	$\mathscr{U}(0.37, 0.41)$	$\mathscr{U}(0.21, 0.25)$	$\mathscr{U}(0.51, 0.55)$
Fengxian	$\mathscr{U}(0.58, 0.62)$	$\mathscr{U}(0.36, 0.40)$	$\mathscr{U}(0.24, 0.27)$	$\mathscr{U}(0.30, 0.34)$
Nicheng	$\mathscr{U}(0.63, 0.67)$	$\mathscr{U}(0.52, 0.55)$	$\mathscr{U}(0.54, 0.57)$	$\mathscr{U}(0.22, 0.25)$

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Table 11. Random distribution cost of distribution center transporting product to consumer

Rt-mart	Demand area 1	Demand area 2	Demand area 3	Demand area 4
Meilanhu	$\mathscr{U}(1.00, 1.50)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(2.00, 2.50)$	$\mathscr{U}(1.70, 2.20)$
Anting	$\mathscr{U}(1.00, 1.50)$	$\mathscr{U}(1.40, 1.90)$	$\mathscr{U}(1.90, 2.40)$	$\mathscr{U}(1.80, 2.30)$
Nanxiang	$\mathscr{U}(1.00, 1.50)$	$\mathscr{U}(1.40, 1.90)$	$\mathscr{U}(1.90, 2.40)$	$\mathscr{U}(1.70, 2.20)$
Yangpu	$\mathscr{U}(1.10, 1.60)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(2.00, 2.50)$	$\mathscr{U}(1.20, 1.70)$
Sijing	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.00, 1.50)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.50, 2.00)$
Chunshen	$\mathscr{U}(1.40, 1.90)$	$\mathscr{U}(1.10, 1.60)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.40, 1.90)$
Kangqiao	$\mathscr{U}(1.40, 1.90)$	$\mathscr{U}(1.40, 1.90)$	$\mathscr{U}(1.60, 2.10)$	$\mathscr{U}(1.10, 1.60)$
Songjiang	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.00, 1.50)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.60, 2.10)$
Fengxian	$\mathscr{U}(1.70, 2.20)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.00, 1.50)$	$\mathscr{U}(1.60, 2.10)$
Nicheng	$\mathscr{U}(1.70, 2.20)$	$\mathscr{U}(1.70, 2.20)$	$\mathscr{U}(1.50, 2.00)$	$\mathscr{U}(1.00, 1.50)$

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Table 12. Random demand of consumer

Demand area 1	Demand area 2	Demand area 3	Demand area 4
$\mathscr{U}(500, 525)-h_1$	$\mathscr{U}(490, 515)-h_1$	$\mathscr{U}(445, 470)-h_1$	$\mathscr{U}(455, 480)-h_1$

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