September  2021, 17(5): 2783-2804. doi: 10.3934/jimo.2020094

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

Received  August 2019 Revised  February 2020 Published  September 2021 Early access  May 2020

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

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.

Citation: Zhimin Liu, Shaojian Qu, Hassan Raza, Zhong Wu, Deqiang Qu, Jianhui Du. Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2783-2804. doi: 10.3934/jimo.2020094
References:
[1]

M. Abbasa, Cutting plane method for multiple objective stochastic integer linear programming, European Journal of Operational Research, 168 (2006), 967-984.  doi: 10.1016/j.ejor.2002.11.006.

[2]

P. ArtznerF. DelbaenJ. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[3]

Z. Bai 2007]Bai2007The, G. Z. Bai, The transportation problem with uncertain transportation cost

[4]

V. Balachandran and S. Jain, Optimal facility location under random demand with general cost structure, Naval Research Logistics Quarterly, 23 (1976), 421-436.  doi: 10.1002/nav.3800230305.

[5]

S. Baptista, M. I. Gomes and A. P. Barbosa-Povoa, A two-stage stochastic model for the design and planning of a multi-product closed loop supply chain, Computer Aided Chemical Engineering, 30, (2012), 412–416. doi: 10.1016/B978-0-444-59519-5.50083-6.

[6]

K. S. H. Basta, Computationally efficient solution of a multiproduct, two-stage distribution-location problem, The Journal of the Operational Research Society, 45 (1994), 1316-1323. 

[7]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4614-0237-4.

[8]

K. W. Chau, A two-stage dynamic model on allocation of construction facilities with genetic algorithm, Automation in Construction, 13 (2004), 481-490.  doi: 10.1016/j.autcon.2004.02.001.

[9]

A. Chen, Shelter location-allocation model for flood evacuation planning, Journal of the Eastern Asia Society for Transportation Studies, 6 (2005), 4237-4252. 

[10]

X. ChenA. Shapiro and H. Sun, Convergence analysis of sample average approximation of two-stage stochastic generalized equations, SIAM Journal on Optimization, 29 (2019), 135-161.  doi: 10.1137/17M1162822.

[11]

X. ChenH. L. Sun and H. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Mathematical Programming, 177 (2019), 255-289.  doi: 10.1007/s10107-018-1266-4.

[12]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2002), 58-73.  doi: 10.1109/4235.985692.

[13]

L. Cooper, Location-Allocation Problems, Operations Research, 11 (1963), 331-343.  doi: 10.1287/opre.11.3.331.

[14]

L. GeldersL. Pintelon and L. N. Van Wassenhove, A location-allocation problem in a large Belgian brewery, European Journal of Operational Research, 28 (1987), 196-206.  doi: 10.1016/0377-2217(87)90218-9.

[15]

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Publishing Co. Inc., Reading, MA, 1989.

[16]

C. A. Irawan and D. Jones, Formulation and solution of a two-stage capacitated facility location problem with multilevel capacities, Annals of Operations Research, 272 (2019), 41-67.  doi: 10.1007/s10479-017-2741-7.

[17]

R. Ishfaq and C. R. Sox, Hub location-allocation in intermodal logistic networks, European Journal of Operational Research, 210 (2011), 213-230.  doi: 10.1016/j.ejor.2010.09.017.

[18]

Y. Ji, S. Qu, Z. Wu and Z. Liu, A fuzzy-robust weighted approach for multicriteria bilevel games, IEEE Transactions on Industrial Informatics, (2020). doi: 10.1109/TII.2020.2969456.

[19]

Q. Jin, X. Hui and Y. Yong, A simulated annealing methodology to multiproduct capacitated facility location with stochastic demand, The Scientific World Journal, (2015), 1–9.

[20]

J. Kennedy and R. C. Eberhart, A discrete binary version of the particle swarm algorithm, IEEE International Conference on Systems, Man, and Cybernetics, 5 (1997), 4104-4108. 

[21]

A. Klose, An lp-based heuristic for two-stage capacitated facility location problems, The Journal of the Operational Research Society, 50 (1999), 157-166.  doi: 10.1016/S0377-2217(99)00300-8.

[22]

B. Li, J. Sun, H. L. Xu and M. Zhang, A class of two-stage distributionally robust games, Journal of Industrial and Management Optimization, 15 (2019), 387–400. doi: 10.3934/jimo.2018048.

[23]

B. Li, Q. Xun, J. Sun, K. L. Teo, and C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86–97. doi: 10.1016/j.apm.2017.11.039.

[24]

I. S. LitvinchevM. Mata and L. Ozuna, Lagrangian heuristic for the two-stage capacitated facility location problem, Applied and Computational Mathematics, 11 (2012), 137-146. 

[25]

N. Loree and F. Aros-Vera, Points of distribution location and inventory management model for Post-Disaster Humanitarian Logistics, Transportation Research Part E: Logistics and Transportation Review, 116 (2018), 1-24.  doi: 10.1016/j.tre.2018.05.003.

[26]

L. R. Medsker, Hybrid Intelligent Systems, Kluwer Academic Publishers, Boston, 1995.

[27]

A. MorenoD. AlemD. Ferreira and A. Clark, An effective two-stage stochastic multi-trip location-transportation model with social concerns in relief supply chains, European Journal of Operational Research, 269 (2018), 1050-1071.  doi: 10.1016/j.ejor.2018.02.022.

[28]

S.M. MousaviR. Tavakkoli-Moghaddam and F. Jolai, A possibilistic programming approach for the location problem of multiple cross-docks and vehicle routing scheduling under uncertainty, Engineering Optimization, 45 (2013), 1223-1249.  doi: 10.1080/0305215X.2012.729053.

[29]

S. Mudchanatongsuk, F. Ordoez and J. Liu, Robust solutions for network design under transportation cost and demand uncertainty, Journal of the Operational Research Society, 59 (2008), 652–662. doi: 10.1057/palgrave.jors.2602362.

[30]

A. M. NezhadH. Manzour and S. Salhi, Lagrangian relaxation heuristics for the uncapacitated single-source multi-product facility location problem, International Journal of Production Economics, 145 (2013), 713-723.  doi: 10.1016/j.ijpe.2013.06.001.

[31]

N. Noyan, Risk-averse stochastic modeling and optimization, in Recent Advances in Optimization and Modeling of Contemporary Problems, INFORMS: PubsOnLine, 2018,221–254. doi: 10.1287/educ.2018.0183.

[32]

N. Noyan, Risk-averse two-stage stochastic programming with an application to disaster management, Computers and Operations Research, 39 (2012), 541-559.  doi: 10.1016/j.cor.2011.03.017.

[33]

N. Noyan and G. Rudolf, Optimization with multivariate conditional value-at-risk-constraints, Operations Research, 61 (2013), 990-1013.  doi: 10.1287/opre.2013.1186.

[34]

L. K. Nozick and M. A. Turnquist, A two-echelon inventory allocation and distribution center location analysis, Transportation Research Part E: Logistics and Transportation Review, 37 (2001), 425-441.  doi: 10.1016/S1366-5545(01)00007-2.

[35]

W. Ogryczak and A. Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM Journal on Optimization, 13 (2002), 60-78.  doi: 10.1137/S1052623400375075.

[36]

M. Padberg, Classical Cuts for Mixed-Integer Programming and Branch-and-Cut, Mathematical Methods of Operations Research, 53 (2001), 173-203.  doi: 10.1007/s001860100120.

[37]

N. RicciardiR. Tadei and A. Grosso, Optimal facility location with random throughput costs, Computers and Operations Research, 29 (2002), 593-607.  doi: 10.1016/S0305-0548(99)00090-8.

[38]

V. Rico-RamirezG. A. Iglesias-SilvaF. Gomez-De la Cruz and S. Hernandez-Castro, Two-stage stochastic approach to the optimal location of booster disinfection stations, Industrial and Engineering Chemistry Research, 46 (2007), 6284-6292.  doi: 10.1021/ie070141a.

[39]

R. T. Rockafellar, Coherent approaches to risk in optimization under uncertainty, in Tutorials in Operations Research, INFORMS: PubsOnline, 2007, 38–61. doi: 10.1287/educ.1073.0032.

[40]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41.  doi: 10.1007/978-1-4757-6594-6_17.

[41]

A. Ruszczyński, Decomposition methods, in Handbooks in Operations Research and Management Science, Vol. 10, Elsevier Sci. B. V., Amsterdam, 2003,141–211. doi: 10.1016/S0927-0507(03)10003-5.

[42]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory,, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898718751.

[43]

H. D. Sherali and B. M. P. Fraticelli, A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse, Journal of Global Optimization, 22 (2002), 319-342.  doi: 10.1023/A:1013827731218.

[44]

J. Shu and J. Sun, Designing the distribution network for an integrated supply chain, Journal of Industrial and Management Optimization, 2 (2006), 339-349.  doi: 10.3934/jimo.2006.2.339.

[45]

K. M. SimY. Guo and B. Shi, BLGAN: Bayesian learning and genetic algorithm for supporting negotiation with incomplete information, IEEE Transactions on Systems Man and Cybernetics Part B, 39 (2009), 198-211.  doi: 10.1109/TSMCB.2008.2004501.

[46]

H. SoleimaniM. Seyyed-Esfahani and G. Kannan, Incorporating risk measures in closed-loop supply chain network design, International Journal of Production Research, 52 (2014), 1843-1867.  doi: 10.1080/00207543.2013.849823.

[47]

T. R. StidsenK. A. Andersen and B. Dammann, A branch and bound algorithm for a class of biobjective mixed integer programs, Management Science, 60 (2014), 1009-1032.  doi: 10.1287/mnsc.2013.1802.

[48]

H. L. SunH. Xu and Y. Wang, Asymptotic analysis of sample average approximation for stochastic optimization problems with joint chance constraints via conditional value at risk and difference of convex functions, Journal of Optimization Theory and Applications, 161 (2014), 257-284.  doi: 10.1007/s10957-012-0127-1.

[49]

J. SunL. Z. Liao and B. Rodrigues, Quadratic two-stage stochastic optimization with coherent measures of risk, Mathematical Programming, 168 (2018), 599-613.  doi: 10.1007/s10107-017-1131-x.

[50]

S. A. TrusevychR. H. Kwon and A. K. S. Jardine, Optimizing critical spare parts and location based on the conditional value-at-risk criterion, The Engineering Economist, 59 (2014), 116-135.  doi: 10.1080/0013791X.2013.876795.

[51]

W. Shih, A branch and bound method for the multiconstraint zero-one knapsack problem, Journal of the Operational Research Society, 30 (1979), 369-378.  doi: 10.2307/3009639.

[52]

G. O. Wesolowsky and W. G. Truscott, The multiperiod location-allocation problem with relocation of facilities, Management Science, 22 (1975), 57-65.  doi: 10.1287/mnsc.22.1.57.

[53]

T. Westerlund and F. Pettersson, An extended cutting plane method for solving convex MINLP problems, Computers and Chemical Engineering, 19 (1995), S131–S136.

[54]

T. H. Yang, A two-stage stochastic model for airline network design with uncertain demand, Transportmetrica, 6 (2010), 187-213.  doi: 10.1080/18128600902906755.

[55]

W. ZhangK. CaoS. Liu and B. Huang, A multi-objective optimization approach for health-care facility location-allocation problems in highly developed cities such as Hong Kong, Computers Environment and Urban Systems, 59 (2016), 220-230.  doi: 10.1016/j.compenvurbsys.2016.07.001.

show all references

References:
[1]

M. Abbasa, Cutting plane method for multiple objective stochastic integer linear programming, European Journal of Operational Research, 168 (2006), 967-984.  doi: 10.1016/j.ejor.2002.11.006.

[2]

P. ArtznerF. DelbaenJ. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[3]

Z. Bai 2007]Bai2007The, G. Z. Bai, The transportation problem with uncertain transportation cost

[4]

V. Balachandran and S. Jain, Optimal facility location under random demand with general cost structure, Naval Research Logistics Quarterly, 23 (1976), 421-436.  doi: 10.1002/nav.3800230305.

[5]

S. Baptista, M. I. Gomes and A. P. Barbosa-Povoa, A two-stage stochastic model for the design and planning of a multi-product closed loop supply chain, Computer Aided Chemical Engineering, 30, (2012), 412–416. doi: 10.1016/B978-0-444-59519-5.50083-6.

[6]

K. S. H. Basta, Computationally efficient solution of a multiproduct, two-stage distribution-location problem, The Journal of the Operational Research Society, 45 (1994), 1316-1323. 

[7]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4614-0237-4.

[8]

K. W. Chau, A two-stage dynamic model on allocation of construction facilities with genetic algorithm, Automation in Construction, 13 (2004), 481-490.  doi: 10.1016/j.autcon.2004.02.001.

[9]

A. Chen, Shelter location-allocation model for flood evacuation planning, Journal of the Eastern Asia Society for Transportation Studies, 6 (2005), 4237-4252. 

[10]

X. ChenA. Shapiro and H. Sun, Convergence analysis of sample average approximation of two-stage stochastic generalized equations, SIAM Journal on Optimization, 29 (2019), 135-161.  doi: 10.1137/17M1162822.

[11]

X. ChenH. L. Sun and H. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Mathematical Programming, 177 (2019), 255-289.  doi: 10.1007/s10107-018-1266-4.

[12]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2002), 58-73.  doi: 10.1109/4235.985692.

[13]

L. Cooper, Location-Allocation Problems, Operations Research, 11 (1963), 331-343.  doi: 10.1287/opre.11.3.331.

[14]

L. GeldersL. Pintelon and L. N. Van Wassenhove, A location-allocation problem in a large Belgian brewery, European Journal of Operational Research, 28 (1987), 196-206.  doi: 10.1016/0377-2217(87)90218-9.

[15]

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Publishing Co. Inc., Reading, MA, 1989.

[16]

C. A. Irawan and D. Jones, Formulation and solution of a two-stage capacitated facility location problem with multilevel capacities, Annals of Operations Research, 272 (2019), 41-67.  doi: 10.1007/s10479-017-2741-7.

[17]

R. Ishfaq and C. R. Sox, Hub location-allocation in intermodal logistic networks, European Journal of Operational Research, 210 (2011), 213-230.  doi: 10.1016/j.ejor.2010.09.017.

[18]

Y. Ji, S. Qu, Z. Wu and Z. Liu, A fuzzy-robust weighted approach for multicriteria bilevel games, IEEE Transactions on Industrial Informatics, (2020). doi: 10.1109/TII.2020.2969456.

[19]

Q. Jin, X. Hui and Y. Yong, A simulated annealing methodology to multiproduct capacitated facility location with stochastic demand, The Scientific World Journal, (2015), 1–9.

[20]

J. Kennedy and R. C. Eberhart, A discrete binary version of the particle swarm algorithm, IEEE International Conference on Systems, Man, and Cybernetics, 5 (1997), 4104-4108. 

[21]

A. Klose, An lp-based heuristic for two-stage capacitated facility location problems, The Journal of the Operational Research Society, 50 (1999), 157-166.  doi: 10.1016/S0377-2217(99)00300-8.

[22]

B. Li, J. Sun, H. L. Xu and M. Zhang, A class of two-stage distributionally robust games, Journal of Industrial and Management Optimization, 15 (2019), 387–400. doi: 10.3934/jimo.2018048.

[23]

B. Li, Q. Xun, J. Sun, K. L. Teo, and C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86–97. doi: 10.1016/j.apm.2017.11.039.

[24]

I. S. LitvinchevM. Mata and L. Ozuna, Lagrangian heuristic for the two-stage capacitated facility location problem, Applied and Computational Mathematics, 11 (2012), 137-146. 

[25]

N. Loree and F. Aros-Vera, Points of distribution location and inventory management model for Post-Disaster Humanitarian Logistics, Transportation Research Part E: Logistics and Transportation Review, 116 (2018), 1-24.  doi: 10.1016/j.tre.2018.05.003.

[26]

L. R. Medsker, Hybrid Intelligent Systems, Kluwer Academic Publishers, Boston, 1995.

[27]

A. MorenoD. AlemD. Ferreira and A. Clark, An effective two-stage stochastic multi-trip location-transportation model with social concerns in relief supply chains, European Journal of Operational Research, 269 (2018), 1050-1071.  doi: 10.1016/j.ejor.2018.02.022.

[28]

S.M. MousaviR. Tavakkoli-Moghaddam and F. Jolai, A possibilistic programming approach for the location problem of multiple cross-docks and vehicle routing scheduling under uncertainty, Engineering Optimization, 45 (2013), 1223-1249.  doi: 10.1080/0305215X.2012.729053.

[29]

S. Mudchanatongsuk, F. Ordoez and J. Liu, Robust solutions for network design under transportation cost and demand uncertainty, Journal of the Operational Research Society, 59 (2008), 652–662. doi: 10.1057/palgrave.jors.2602362.

[30]

A. M. NezhadH. Manzour and S. Salhi, Lagrangian relaxation heuristics for the uncapacitated single-source multi-product facility location problem, International Journal of Production Economics, 145 (2013), 713-723.  doi: 10.1016/j.ijpe.2013.06.001.

[31]

N. Noyan, Risk-averse stochastic modeling and optimization, in Recent Advances in Optimization and Modeling of Contemporary Problems, INFORMS: PubsOnLine, 2018,221–254. doi: 10.1287/educ.2018.0183.

[32]

N. Noyan, Risk-averse two-stage stochastic programming with an application to disaster management, Computers and Operations Research, 39 (2012), 541-559.  doi: 10.1016/j.cor.2011.03.017.

[33]

N. Noyan and G. Rudolf, Optimization with multivariate conditional value-at-risk-constraints, Operations Research, 61 (2013), 990-1013.  doi: 10.1287/opre.2013.1186.

[34]

L. K. Nozick and M. A. Turnquist, A two-echelon inventory allocation and distribution center location analysis, Transportation Research Part E: Logistics and Transportation Review, 37 (2001), 425-441.  doi: 10.1016/S1366-5545(01)00007-2.

[35]

W. Ogryczak and A. Ruszczyński, Dual stochastic dominance and related mean-risk models, SIAM Journal on Optimization, 13 (2002), 60-78.  doi: 10.1137/S1052623400375075.

[36]

M. Padberg, Classical Cuts for Mixed-Integer Programming and Branch-and-Cut, Mathematical Methods of Operations Research, 53 (2001), 173-203.  doi: 10.1007/s001860100120.

[37]

N. RicciardiR. Tadei and A. Grosso, Optimal facility location with random throughput costs, Computers and Operations Research, 29 (2002), 593-607.  doi: 10.1016/S0305-0548(99)00090-8.

[38]

V. Rico-RamirezG. A. Iglesias-SilvaF. Gomez-De la Cruz and S. Hernandez-Castro, Two-stage stochastic approach to the optimal location of booster disinfection stations, Industrial and Engineering Chemistry Research, 46 (2007), 6284-6292.  doi: 10.1021/ie070141a.

[39]

R. T. Rockafellar, Coherent approaches to risk in optimization under uncertainty, in Tutorials in Operations Research, INFORMS: PubsOnline, 2007, 38–61. doi: 10.1287/educ.1073.0032.

[40]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41.  doi: 10.1007/978-1-4757-6594-6_17.

[41]

A. Ruszczyński, Decomposition methods, in Handbooks in Operations Research and Management Science, Vol. 10, Elsevier Sci. B. V., Amsterdam, 2003,141–211. doi: 10.1016/S0927-0507(03)10003-5.

[42]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory,, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898718751.

[43]

H. D. Sherali and B. M. P. Fraticelli, A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse, Journal of Global Optimization, 22 (2002), 319-342.  doi: 10.1023/A:1013827731218.

[44]

J. Shu and J. Sun, Designing the distribution network for an integrated supply chain, Journal of Industrial and Management Optimization, 2 (2006), 339-349.  doi: 10.3934/jimo.2006.2.339.

[45]

K. M. SimY. Guo and B. Shi, BLGAN: Bayesian learning and genetic algorithm for supporting negotiation with incomplete information, IEEE Transactions on Systems Man and Cybernetics Part B, 39 (2009), 198-211.  doi: 10.1109/TSMCB.2008.2004501.

[46]

H. SoleimaniM. Seyyed-Esfahani and G. Kannan, Incorporating risk measures in closed-loop supply chain network design, International Journal of Production Research, 52 (2014), 1843-1867.  doi: 10.1080/00207543.2013.849823.

[47]

T. R. StidsenK. A. Andersen and B. Dammann, A branch and bound algorithm for a class of biobjective mixed integer programs, Management Science, 60 (2014), 1009-1032.  doi: 10.1287/mnsc.2013.1802.

[48]

H. L. SunH. Xu and Y. Wang, Asymptotic analysis of sample average approximation for stochastic optimization problems with joint chance constraints via conditional value at risk and difference of convex functions, Journal of Optimization Theory and Applications, 161 (2014), 257-284.  doi: 10.1007/s10957-012-0127-1.

[49]

J. SunL. Z. Liao and B. Rodrigues, Quadratic two-stage stochastic optimization with coherent measures of risk, Mathematical Programming, 168 (2018), 599-613.  doi: 10.1007/s10107-017-1131-x.

[50]

S. A. TrusevychR. H. Kwon and A. K. S. Jardine, Optimizing critical spare parts and location based on the conditional value-at-risk criterion, The Engineering Economist, 59 (2014), 116-135.  doi: 10.1080/0013791X.2013.876795.

[51]

W. Shih, A branch and bound method for the multiconstraint zero-one knapsack problem, Journal of the Operational Research Society, 30 (1979), 369-378.  doi: 10.2307/3009639.

[52]

G. O. Wesolowsky and W. G. Truscott, The multiperiod location-allocation problem with relocation of facilities, Management Science, 22 (1975), 57-65.  doi: 10.1287/mnsc.22.1.57.

[53]

T. Westerlund and F. Pettersson, An extended cutting plane method for solving convex MINLP problems, Computers and Chemical Engineering, 19 (1995), S131–S136.

[54]

T. H. Yang, A two-stage stochastic model for airline network design with uncertain demand, Transportmetrica, 6 (2010), 187-213.  doi: 10.1080/18128600902906755.

[55]

W. ZhangK. CaoS. Liu and B. Huang, A multi-objective optimization approach for health-care facility location-allocation problems in highly developed cities such as Hong Kong, Computers Environment and Urban Systems, 59 (2016), 220-230.  doi: 10.1016/j.compenvurbsys.2016.07.001.

Figure 1.  Network structure of location-allocation supply chain
Figure 2.  Two-stage process of location-allocation problem
Figure 3.  Location of flour factory, food factory, Rt-mart and bread demand area in Shanghai
Figure 4.  Location-allocation supply chain network
Figure 5.  Comparisons of different algorithms
Figure 6.  Values of fitness functions and supply chain profit with different confidence
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
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
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 $Val=-1.2799e+04$
$\mathbf{e}_{best}=(1, 1, 1, 0)$ $\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 $Val=-1.2799e+04$
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 $(0, 1, 0, 1)$ $(0, 0, 1, 0, 0, 1, 0, 0, 1, 1)$ $1.2855e+04$ $-1.2660e+04$ $12035.1647$
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 $(0, 1, 0, 1)$ $(0, 0, 1, 0, 0, 1, 0, 0, 1, 1)$ $1.2855e+04$ $-1.2660e+04$ $12035.1647$
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
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
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 $(0, 1, 0, 1)$ $(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$ $1.2819e+04$
$\xi(\omega)$ $\mathbf{e}_{best}$ $\mathbf{c}_{best}$ Pro
Random $(1, 1, 1, 0)$ $(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$ $1.2952e+04$
Expected $(0, 1, 0, 1)$ $(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$ $1.2819e+04$
Table 6.  Comparison of supply chain profit of model with $ \lambda = 0.1 $ and $ \lambda = 0 $
$\lambda$ $\mathbf{e}_{best}$ $\mathbf{c}_{best}$ Pro
$\lambda=0$ $(1, 1, 1, 0)$ $(0, 0, 1, 1, 0, 1, 1, 0, 1, 0)$ $1.3007e+04$
$\lambda=0.1$ $(1, 1, 1, 0)$ $(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$ $1.2952e+04$
$\lambda$ $\mathbf{e}_{best}$ $\mathbf{c}_{best}$ Pro
$\lambda=0$ $(1, 1, 1, 0)$ $(0, 0, 1, 1, 0, 1, 1, 0, 1, 0)$ $1.3007e+04$
$\lambda=0.1$ $(1, 1, 1, 0)$ $(1, 0, 0, 0, 1, 0, 1, 0, 1, 0)$ $1.2952e+04$
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 $(0, 1, 0, 1)$ $(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$ $1.2361e+04$
$(r_{11}, r_{12})$ $h_1$ ${\mathbf e}_{best}$ ${\mathbf c}_{best}$ 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 $(0, 1, 0, 1)$ $(0, 0, 0, 1, 0, 1, 0, 0, 1, 1)$ $1.2361e+04$
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
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
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 $\mathscr{U}(0.45, 0.49)$ $\mathscr{U}(0.25, 0.28)$ $\mathscr{U}(0.11, 0.15)$ $\mathscr{U}(0.27, 0.30)$
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 $\mathscr{U}(0.45, 0.49)$ $\mathscr{U}(0.25, 0.28)$ $\mathscr{U}(0.11, 0.15)$ $\mathscr{U}(0.27, 0.30)$
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 $\mathscr{U}(0.63, 0.67)$ $\mathscr{U}(0.52, 0.55)$ $\mathscr{U}(0.54, 0.57)$ $\mathscr{U}(0.22, 0.25)$
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 $\mathscr{U}(0.63, 0.67)$ $\mathscr{U}(0.52, 0.55)$ $\mathscr{U}(0.54, 0.57)$ $\mathscr{U}(0.22, 0.25)$
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 $\mathscr{U}(1.70, 2.20)$ $\mathscr{U}(1.70, 2.20)$ $\mathscr{U}(1.50, 2.00)$ $\mathscr{U}(1.00, 1.50)$
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 $\mathscr{U}(1.70, 2.20)$ $\mathscr{U}(1.70, 2.20)$ $\mathscr{U}(1.50, 2.00)$ $\mathscr{U}(1.00, 1.50)$
Table 12.  Random demand of consumer
Demand area 1 Demand area 2 Demand area 3 Demand area 4
$\mathscr{U}(500, 525)-h_1$ $\mathscr{U}(490, 515)-h_1$ $\mathscr{U}(445, 470)-h_1$ $\mathscr{U}(455, 480)-h_1$
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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