American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy
  • JIMO Home
  • This Issue
  • Next Article
    Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets
July  2021, 17(4): 1557-1575. doi: 10.3934/jimo.2020034

Solving the facility location and fixed charge solid transportation problem

Gbeminiyi John Oyewole , and  Olufemi Adetunji

Department of Industrial and Systems Engineering, University of Pretoria, Pretoria 0002, South Africa

* Corresponding author: Gbeminiyi John Oyewole

Received  May 2019 Revised  August 2019 Published  February 2020

In this paper, a new variant of the Solid Transportation Problem (STP) that incorporates both facility location and Fixed Charge Solid Transportation Problem (FCSTP) is presented with significant applications in logistics. It integrates decisions of diverse planning horizons: operational, tactical and strategic. The problem is termed Fixed Charge Solid Location and Transportation Problem (FCSLTP). Benchmark data obtained from the literature was extended for experimentation purposes. Solution to the FCSLTP was obtained using CPLEX commercial optimization solver. A Lagrange Relaxation Heuristic (LRH) was developed as an alternative solution for users not possibly having access to CPLEX. We further defined an equivalent FCSLTP in the main paper and termed this as FCSTP-EQ. The FCSTP-EQ was compared to our FCSLTP to investigate possible cost savings with both formulations. Results obtained showed CPLEX outperforming the Lagrange relaxation heuristic developed both in the upper bound and lower bound generation for the problem sizes considered. Additionally, the cost savings obtained using the FCSLTP was consistently better than the FCSTP-EQ. The upper bound generation capability of Lagrange relaxation could possibly be improved by using better search methods such as metaheuristics. Under certain conditions, the FCSTP could feasibly be used as a starting solution to solve the FCSLTP.

Keywords: Solid transportation problem, facility location, fixed charge problem, Lagrange relaxation heuristic, CPLEX.
Mathematics Subject Classification: Primary: 90B06, 90B10; Secondary: 90C11.
Citation: Gbeminiyi John Oyewole, Olufemi Adetunji. Solving the facility location and fixed charge solid transportation problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1557-1575. doi: 10.3934/jimo.2020034
References:
[1]

M. Agar and S. Salhi, Lagrangean heuristics applied to a variety of large capacitated plant location problems, J. Oper. Res. Soc., 49 (1998), 1072-1084.  doi: 10.1057/palgrave.jors.2600621.  Google Scholar

[2]

U. Akinc and B. M. Khumawala, An efficient branch and bound algorithm for the capacitated warehouse location problem, Management Sci., 23 (1977), 585-594.  doi: 10.1287/mnsc.23.6.585.  Google Scholar

[3]

M. Alizadeh, I. Mahdavi, N. Mahdavi-Amiri and S. Shiripour, A capacitated location-allocation problem with stochastic demands using sub-sources: An empirical study, Applied Soft Computing, 34 (2015), 551-571.  doi: 10.1016/j.asoc.2015.05.020.  Google Scholar

[4]

M. Amiri, S. J. Sadjadi, R. Tavakkoli-Moghaddam and A. Jabbarzadeh, An integrated approach for facility location and supply vessel planning with time windows, J. Optim. Industrial Engineering, 12 (2018), 151-165.  doi: 10.22094/JOIE.2018.544109.1517.  Google Scholar

[5]

H. I. Calvete, C. Galé and J. A. Iranzo, An improved evolutionary algorithm for the two-stage transportation problem with fixed charge at depots, OR Spectrum, 38 (2016), 189-206.  doi: 10.1007/s00291-015-0416-9.  Google Scholar

[6]

D. Canca and E. Barrena, The integrated rolling stock circulation and depot location problem in railway rapid transit systems, Transportation Res. Part E: Logistics Transportation Rev., 109 (2018), 115–138. doi: 10.1016/j.tre.2017.10.018.  Google Scholar

[7]

H. J. Carlo, V. David and G. Salvat, Transportation-location problem with unknown number of facilities, Comput. Industrial Engineering, 112 (2017), 212-220.  doi: 10.1016/j.cie.2017.08.003.  Google Scholar

[8]

T. Christensen, Network Design Problems with Piecewise Linear Cost Functions, Ph.D thesis, Institut for Økonomi in Aarhus Universitet, 2013. Google Scholar

[9]

G. Cornuéjols, R. Sridharan and J. M. Thizy, A comparison of heuristics and relaxations for the capacitated plant location problem, European J. Oper. Res., 50 (1991), 280-297.  doi: 10.1016/0377-2217(91)90261-S.  Google Scholar

[10]

M. Fischetti, I. Ljubić and M. Sinnl, Benders decomposition without separability: A computational study for capacitated facility location problems, European J. Oper. Res., 253 (2016), 557-569.  doi: 10.1016/j.ejor.2016.03.002.  Google Scholar

[11]

M. L. Fisher, The Lagrangian relaxation method for solving integer programming problems, Management Sci., 27 (1981), 1-18.  doi: 10.1287/mnsc.27.1.1.  Google Scholar

[12]

S. L. Gadegaard, A. Klose and L. R. Nielsen, An improved cut-and-solve algorithm for the single-source capacitated facility location problem, EURO J. Comput. Optim., 6 (2018), 1-27.  doi: 10.1007/s13675-017-0084-4.  Google Scholar

[13]

G. Ghiani, L. Grandinetti, F. Guerriero and R. Musmanno, A Lagrangean heuristic for the plant location problem with multiple facilities in the same site, Optim. Methods Softw., 17 (2002), 1059-1076.  doi: 10.1080/1055678021000039184.  Google Scholar

[14]

G. Guastaroba and M. G. Speranza, A heuristic for BILP problems: The single source capacitated facility location problem, European J. Oper. Res., 238 (2014), 438-450.  doi: 10.1016/j.ejor.2014.04.007.  Google Scholar

[15]

K. B. Haley, New methods in mathematical programming - The solid transportation problem, Oper. Res., 10 (1962), 448-463.  doi: 10.1287/opre.10.4.448.  Google Scholar

[16]

A. Hiassat, A. Diabat and I. Rahwan, A genetic algorithm approach for location-inventory-routing problem with perishable products, J. Manufacturing Systems, 42 (2017), 93-103.  doi: 10.1016/j.jmsy.2016.10.004.  Google Scholar

[17]

K. Hindi and K. Pieńkosz, Efficient solution of large scale, single-source, capacitated plant location problems, J. Oper. Res. Soc., 50 (1999), 268-274.  doi: 10.1057/palgrave.jors.2600698.  Google Scholar

[18]

K. Holmberg and J. Ling, A Lagrangean heuristic for the facility location problem with staircase costs, in Operations Research Proceedings, Operations Research Proceedings, 1995, Springer, Berlin, Heidelberg, 1996, 66–71. doi: 10.1007/978-3-642-80117-4_12.  Google Scholar

[19]

IBM ILOG CPLEX Optimization Studio Cplex User'S Manual, IBM Corp., 2016. Available from: https://www.ibm.com/support/knowledgecenter/SSSA5P_12.7.0/ilog.odms.studio.help/pdf/opl_languser.pdf. Google Scholar

[20]

A. Klose and S. Görtz, A branch-and-price algorithm for the capacitated facility location problem, European J. Oper. Res., 179 (2007), 1109-1125.  doi: 10.1016/j.ejor.2005.03.078.  Google Scholar

[21]

P. Kundu, M. B. Kar, S. Kar, T. Pal and M. Maiti, A solid transportation model with product blending and parameters as rough variables, Soft Computing, 21 (2017), 2297-2306.  doi: 10.1007/s00500-015-1941-9.  Google Scholar

[22]

R. Lima, IBM ILOG CPLEX - What is Inside of the Box?, Proc. 2010 EWO Seminar, 2010. Available from: http://egon.cheme.cmu.edu/ewo/docs/rlima_cplex_ewo_dec2010.pdf. Google Scholar

[23]

Z. M. Liu, S. J. Qu, M. Goh, R. P. Huang and S. L. Wang, Optimization of fuzzy demand distribution supply chain using modified sequence quadratic programming approach, J. Intell. Fuzzy Systems, 36 (2019), 6167-6180.  doi: 10.3233/JIFS-181997.  Google Scholar

[24]

I. Ljubić and E. Moreno, Outer approximation and submodular cuts for maximum capture facility location problems with random utilities, European J. Oper. Res., 266 (2018), 46-56.  doi: 10.1016/j.ejor.2017.09.023.  Google Scholar

[25]

S. M. Mousavi and S. T. A. Niaki, Capacitated location allocation problem with stochastic location and fuzzy demand: A hybrid algorithm, Appl. Math. Model., 37 (2013), 5109-5119.  doi: 10.1016/j.apm.2012.10.038.  Google Scholar

[26]

A. M. Nezhad, H. Manzour and S. Salhi, Lagrangian relaxation heuristics for the uncapacitated single-source multi-product facility location problem, Internat. J. Prod. Econ., 145 (2013), 713-723.  doi: 10.1016/j.ijpe.2013.06.001.  Google Scholar

[27]

M. Oguz, T. Bektas and J. A. Bennell, Multicommodity flows and Benders decomposition for restricted continuous location problems, European J. Oper. Res., 266 (2018), 851-863.  doi: 10.1016/j.ejor.2017.11.033.  Google Scholar

[28]

C. Ou-Yang and R. Ansari, Applying a hybrid particle swarm optimization Tabu search algorithm to a facility location case in Jakarta, J. Industrial Prod. Engineering, 34 (2017), 199-212.   Google Scholar

[29]

G. J. Oyewole and O. Adetunji, On the capacitated step-fixed charge and facility location problem: A row perturbation heuristic, Appl. Math, 12 (2018), 1033-1045.  doi: 10.18576/amis/120516.  Google Scholar

[30]

M. S. Puga and J. S. Tancrez, A heuristic algorithm for solving large location–inventory problems with demand uncertainty, European J. Oper. Res., 259 (2017), 413-423.  doi: 10.1016/j.ejor.2016.10.037.  Google Scholar

[31]

S. J. Qu, Y. Y. Zhou, Y. L. Zhang, M. Wahab, G. Zhang and Y. Y. Ye, Optimal strategy for a green supply chain considering shipping policy and default risk, Comput. Industrial Engineering, 131 (2019), 172-186.  doi: 10.1016/j.cie.2019.03.042.  Google Scholar

[32]

A. Rahmani and M. Yousefikhoshbakht, Capacitated facility location problem in random fuzzy environment: Using ($\alpha$, $\beta$)-cost minimization model under the Hurwicz criterion, J. Intell. Fuzzy Systems, 25 (2013), 953-964.  doi: 10.3233/IFS-120697.  Google Scholar

[33]

R. Roberti, E. Bartolini and A. Mingozzi, The fixed charge transportation problem: An exact algorithm based on a new integer programming formulation, Management Sci., 61 (2014), 1275-1291.  doi: 10.1287/mnsc.2014.1947.  Google Scholar

[34]

G. Sá, Branch-and-bound and approximate solutions to the capacitated plant-location problem, Oper. Res., 17 (1969), 1005-1016.  doi: 10.1287/opre.17.6.1005.  Google Scholar

[35]

M. Sanei, A. Mahmoodirad, S. Niroomand, A. Jamalian and S. Gelareh, Step fixed-charge solid transportation problem: A Lagrangian relaxation heuristic approach, Comput. Appl. Math., 36 (2017), 1217-1237.  doi: 10.1007/s40314-015-0293-5.  Google Scholar

[36]

M. Veenstra, K. Jan Roodbergen, L. C. Coelho and S. X. Zhu, A simultaneous facility location and vehicle routing problem arising in health care logistics in the Netherlands, European J. Oper. Res., 268 (2018), 703-715.  doi: 10.1016/j.ejor.2018.01.043.  Google Scholar

[37]

L. A. Wolsey, C. Cordier and H. Marchand, Cutting planes for integer programs with general integer variables, Math. Programming, 81 (1998), 201-214.  doi: 10.1007/BF01581105.  Google Scholar

[38]

T. Wu, F. Chu, Z. Yang, Z. Zhou and W. Zhou, Lagrangean relaxation and hybrid simulated annealing tabu search procedure for a two-echelon capacitated facility location problem with plant size selection, Internat. J. Prod. Res., 55 (2017), 2540-2555.  doi: 10.1080/00207543.2016.1240381.  Google Scholar

[39]

B. Zhang, J. Peng, S. Li and L. Chen, Fixed charge solid transportation problem in uncertain environment and its algorithm, Comput. Industrial Engineering, 102 (2016), 186-197.  doi: 10.1016/j.cie.2016.10.030.  Google Scholar

show all references

References:
[1]

M. Agar and S. Salhi, Lagrangean heuristics applied to a variety of large capacitated plant location problems, J. Oper. Res. Soc., 49 (1998), 1072-1084.  doi: 10.1057/palgrave.jors.2600621.  Google Scholar

[2]

U. Akinc and B. M. Khumawala, An efficient branch and bound algorithm for the capacitated warehouse location problem, Management Sci., 23 (1977), 585-594.  doi: 10.1287/mnsc.23.6.585.  Google Scholar

[3]

M. Alizadeh, I. Mahdavi, N. Mahdavi-Amiri and S. Shiripour, A capacitated location-allocation problem with stochastic demands using sub-sources: An empirical study, Applied Soft Computing, 34 (2015), 551-571.  doi: 10.1016/j.asoc.2015.05.020.  Google Scholar

[4]

M. Amiri, S. J. Sadjadi, R. Tavakkoli-Moghaddam and A. Jabbarzadeh, An integrated approach for facility location and supply vessel planning with time windows, J. Optim. Industrial Engineering, 12 (2018), 151-165.  doi: 10.22094/JOIE.2018.544109.1517.  Google Scholar

[5]

H. I. Calvete, C. Galé and J. A. Iranzo, An improved evolutionary algorithm for the two-stage transportation problem with fixed charge at depots, OR Spectrum, 38 (2016), 189-206.  doi: 10.1007/s00291-015-0416-9.  Google Scholar

[6]

D. Canca and E. Barrena, The integrated rolling stock circulation and depot location problem in railway rapid transit systems, Transportation Res. Part E: Logistics Transportation Rev., 109 (2018), 115–138. doi: 10.1016/j.tre.2017.10.018.  Google Scholar

[7]

H. J. Carlo, V. David and G. Salvat, Transportation-location problem with unknown number of facilities, Comput. Industrial Engineering, 112 (2017), 212-220.  doi: 10.1016/j.cie.2017.08.003.  Google Scholar

[8]

T. Christensen, Network Design Problems with Piecewise Linear Cost Functions, Ph.D thesis, Institut for Økonomi in Aarhus Universitet, 2013. Google Scholar

[9]

G. Cornuéjols, R. Sridharan and J. M. Thizy, A comparison of heuristics and relaxations for the capacitated plant location problem, European J. Oper. Res., 50 (1991), 280-297.  doi: 10.1016/0377-2217(91)90261-S.  Google Scholar

[10]

M. Fischetti, I. Ljubić and M. Sinnl, Benders decomposition without separability: A computational study for capacitated facility location problems, European J. Oper. Res., 253 (2016), 557-569.  doi: 10.1016/j.ejor.2016.03.002.  Google Scholar

[11]

M. L. Fisher, The Lagrangian relaxation method for solving integer programming problems, Management Sci., 27 (1981), 1-18.  doi: 10.1287/mnsc.27.1.1.  Google Scholar

[12]

S. L. Gadegaard, A. Klose and L. R. Nielsen, An improved cut-and-solve algorithm for the single-source capacitated facility location problem, EURO J. Comput. Optim., 6 (2018), 1-27.  doi: 10.1007/s13675-017-0084-4.  Google Scholar

[13]

G. Ghiani, L. Grandinetti, F. Guerriero and R. Musmanno, A Lagrangean heuristic for the plant location problem with multiple facilities in the same site, Optim. Methods Softw., 17 (2002), 1059-1076.  doi: 10.1080/1055678021000039184.  Google Scholar

[14]

G. Guastaroba and M. G. Speranza, A heuristic for BILP problems: The single source capacitated facility location problem, European J. Oper. Res., 238 (2014), 438-450.  doi: 10.1016/j.ejor.2014.04.007.  Google Scholar

[15]

K. B. Haley, New methods in mathematical programming - The solid transportation problem, Oper. Res., 10 (1962), 448-463.  doi: 10.1287/opre.10.4.448.  Google Scholar

[16]

A. Hiassat, A. Diabat and I. Rahwan, A genetic algorithm approach for location-inventory-routing problem with perishable products, J. Manufacturing Systems, 42 (2017), 93-103.  doi: 10.1016/j.jmsy.2016.10.004.  Google Scholar

[17]

K. Hindi and K. Pieńkosz, Efficient solution of large scale, single-source, capacitated plant location problems, J. Oper. Res. Soc., 50 (1999), 268-274.  doi: 10.1057/palgrave.jors.2600698.  Google Scholar

[18]

K. Holmberg and J. Ling, A Lagrangean heuristic for the facility location problem with staircase costs, in Operations Research Proceedings, Operations Research Proceedings, 1995, Springer, Berlin, Heidelberg, 1996, 66–71. doi: 10.1007/978-3-642-80117-4_12.  Google Scholar

[19]

IBM ILOG CPLEX Optimization Studio Cplex User'S Manual, IBM Corp., 2016. Available from: https://www.ibm.com/support/knowledgecenter/SSSA5P_12.7.0/ilog.odms.studio.help/pdf/opl_languser.pdf. Google Scholar

[20]

A. Klose and S. Görtz, A branch-and-price algorithm for the capacitated facility location problem, European J. Oper. Res., 179 (2007), 1109-1125.  doi: 10.1016/j.ejor.2005.03.078.  Google Scholar

[21]

P. Kundu, M. B. Kar, S. Kar, T. Pal and M. Maiti, A solid transportation model with product blending and parameters as rough variables, Soft Computing, 21 (2017), 2297-2306.  doi: 10.1007/s00500-015-1941-9.  Google Scholar

[22]

R. Lima, IBM ILOG CPLEX - What is Inside of the Box?, Proc. 2010 EWO Seminar, 2010. Available from: http://egon.cheme.cmu.edu/ewo/docs/rlima_cplex_ewo_dec2010.pdf. Google Scholar

[23]

Z. M. Liu, S. J. Qu, M. Goh, R. P. Huang and S. L. Wang, Optimization of fuzzy demand distribution supply chain using modified sequence quadratic programming approach, J. Intell. Fuzzy Systems, 36 (2019), 6167-6180.  doi: 10.3233/JIFS-181997.  Google Scholar

[24]

I. Ljubić and E. Moreno, Outer approximation and submodular cuts for maximum capture facility location problems with random utilities, European J. Oper. Res., 266 (2018), 46-56.  doi: 10.1016/j.ejor.2017.09.023.  Google Scholar

[25]

S. M. Mousavi and S. T. A. Niaki, Capacitated location allocation problem with stochastic location and fuzzy demand: A hybrid algorithm, Appl. Math. Model., 37 (2013), 5109-5119.  doi: 10.1016/j.apm.2012.10.038.  Google Scholar

[26]

A. M. Nezhad, H. Manzour and S. Salhi, Lagrangian relaxation heuristics for the uncapacitated single-source multi-product facility location problem, Internat. J. Prod. Econ., 145 (2013), 713-723.  doi: 10.1016/j.ijpe.2013.06.001.  Google Scholar

[27]

M. Oguz, T. Bektas and J. A. Bennell, Multicommodity flows and Benders decomposition for restricted continuous location problems, European J. Oper. Res., 266 (2018), 851-863.  doi: 10.1016/j.ejor.2017.11.033.  Google Scholar

[28]

C. Ou-Yang and R. Ansari, Applying a hybrid particle swarm optimization Tabu search algorithm to a facility location case in Jakarta, J. Industrial Prod. Engineering, 34 (2017), 199-212.   Google Scholar

[29]

G. J. Oyewole and O. Adetunji, On the capacitated step-fixed charge and facility location problem: A row perturbation heuristic, Appl. Math, 12 (2018), 1033-1045.  doi: 10.18576/amis/120516.  Google Scholar

[30]

M. S. Puga and J. S. Tancrez, A heuristic algorithm for solving large location–inventory problems with demand uncertainty, European J. Oper. Res., 259 (2017), 413-423.  doi: 10.1016/j.ejor.2016.10.037.  Google Scholar

[31]

S. J. Qu, Y. Y. Zhou, Y. L. Zhang, M. Wahab, G. Zhang and Y. Y. Ye, Optimal strategy for a green supply chain considering shipping policy and default risk, Comput. Industrial Engineering, 131 (2019), 172-186.  doi: 10.1016/j.cie.2019.03.042.  Google Scholar

[32]

A. Rahmani and M. Yousefikhoshbakht, Capacitated facility location problem in random fuzzy environment: Using ($\alpha$, $\beta$)-cost minimization model under the Hurwicz criterion, J. Intell. Fuzzy Systems, 25 (2013), 953-964.  doi: 10.3233/IFS-120697.  Google Scholar

[33]

R. Roberti, E. Bartolini and A. Mingozzi, The fixed charge transportation problem: An exact algorithm based on a new integer programming formulation, Management Sci., 61 (2014), 1275-1291.  doi: 10.1287/mnsc.2014.1947.  Google Scholar

[34]

G. Sá, Branch-and-bound and approximate solutions to the capacitated plant-location problem, Oper. Res., 17 (1969), 1005-1016.  doi: 10.1287/opre.17.6.1005.  Google Scholar

[35]

M. Sanei, A. Mahmoodirad, S. Niroomand, A. Jamalian and S. Gelareh, Step fixed-charge solid transportation problem: A Lagrangian relaxation heuristic approach, Comput. Appl. Math., 36 (2017), 1217-1237.  doi: 10.1007/s40314-015-0293-5.  Google Scholar

[36]

M. Veenstra, K. Jan Roodbergen, L. C. Coelho and S. X. Zhu, A simultaneous facility location and vehicle routing problem arising in health care logistics in the Netherlands, European J. Oper. Res., 268 (2018), 703-715.  doi: 10.1016/j.ejor.2018.01.043.  Google Scholar

[37]

L. A. Wolsey, C. Cordier and H. Marchand, Cutting planes for integer programs with general integer variables, Math. Programming, 81 (1998), 201-214.  doi: 10.1007/BF01581105.  Google Scholar

[38]

T. Wu, F. Chu, Z. Yang, Z. Zhou and W. Zhou, Lagrangean relaxation and hybrid simulated annealing tabu search procedure for a two-echelon capacitated facility location problem with plant size selection, Internat. J. Prod. Res., 55 (2017), 2540-2555.  doi: 10.1080/00207543.2016.1240381.  Google Scholar

[39]

B. Zhang, J. Peng, S. Li and L. Chen, Fixed charge solid transportation problem in uncertain environment and its algorithm, Comput. Industrial Engineering, 102 (2016), 186-197.  doi: 10.1016/j.cie.2016.10.030.  Google Scholar

Figure 1.  Schematic representation of FCSLTP
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Procedure for computing the FCSTP-EQ
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  FCSLTP and FCSTP-EQ
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Solution time of FCSLTP and FCSTP
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Problem sizes and number of instances used for experimentation
Problem Size No. Problem Size
$ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $		 No of instances
1
2
3
4
5
6
7
8
9
10		 5$ \mathrm{\times} $5$ \mathrm{\times} $2
5$ \mathrm{\times} $8$ \mathrm{\times} $2
7$ \mathrm{\times} $10$ \mathrm{\times} $2
8$ \mathrm{\times} $8$ \mathrm{\times} $2
10$ \mathrm{\times} $10$ \mathrm{\times} $3
10$ \mathrm{\times} $20$ \mathrm{\times} $3
15$ \mathrm{\times} $30$ \mathrm{\times} $4
20$ \mathrm{\times} $20$ \mathrm{\times} $5
25$ \mathrm{\times} $38$ \mathrm{\times} $8
35$ \mathrm{\times} $42$ \mathrm{\times} $9		 5
5
5
5
5
5
5
5
5
5
Table Options

Download as excel

Problem Size No. Problem Size
$ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $		 No of instances
1
2
3
4
5
6
7
8
9
10		 5$ \mathrm{\times} $5$ \mathrm{\times} $2
5$ \mathrm{\times} $8$ \mathrm{\times} $2
7$ \mathrm{\times} $10$ \mathrm{\times} $2
8$ \mathrm{\times} $8$ \mathrm{\times} $2
10$ \mathrm{\times} $10$ \mathrm{\times} $3
10$ \mathrm{\times} $20$ \mathrm{\times} $3
15$ \mathrm{\times} $30$ \mathrm{\times} $4
20$ \mathrm{\times} $20$ \mathrm{\times} $5
25$ \mathrm{\times} $38$ \mathrm{\times} $8
35$ \mathrm{\times} $42$ \mathrm{\times} $9		 5
5
5
5
5
5
5
5
5
5
Table 2.  Parameter distribution used for experimentation
Parameter Distribution
$ {\boldsymbol{S}}_{\boldsymbol{i}} $ U(200,400)
$ {\boldsymbol{D}}_{\boldsymbol{j}} $ U(50,100)
$ {\boldsymbol{T}}_{\boldsymbol{r}} $ U(800, 1800)
$ {\boldsymbol{c}}_{\boldsymbol{ijr}} $ U(20,150)
$ {\boldsymbol{H}}_{\boldsymbol{ijr}} $ U(200,600)
$ {\boldsymbol{F}}_{\boldsymbol{i}}\boldsymbol{=}\boldsymbol{U}\left(\boldsymbol{0},\boldsymbol{90}\right)\boldsymbol{+\ }\sqrt{{\boldsymbol{S}}_{\boldsymbol{i}}}\boldsymbol{\ }\boldsymbol{U}\boldsymbol{(}\boldsymbol{100},\boldsymbol{110}\boldsymbol{)} $
$ {\boldsymbol{M}}_{\boldsymbol{ijr}}\boldsymbol{=}{\boldsymbol{\mathrm{min}} \boldsymbol{(}{\boldsymbol{S}}_{\boldsymbol{i}}\boldsymbol{,\ }{\boldsymbol{D}}_{\boldsymbol{j}},{\boldsymbol{T}}_{\boldsymbol{r}}\boldsymbol{)}\ } $
Table Options

Download as excel

Parameter Distribution
$ {\boldsymbol{S}}_{\boldsymbol{i}} $ U(200,400)
$ {\boldsymbol{D}}_{\boldsymbol{j}} $ U(50,100)
$ {\boldsymbol{T}}_{\boldsymbol{r}} $ U(800, 1800)
$ {\boldsymbol{c}}_{\boldsymbol{ijr}} $ U(20,150)
$ {\boldsymbol{H}}_{\boldsymbol{ijr}} $ U(200,600)
$ {\boldsymbol{F}}_{\boldsymbol{i}}\boldsymbol{=}\boldsymbol{U}\left(\boldsymbol{0},\boldsymbol{90}\right)\boldsymbol{+\ }\sqrt{{\boldsymbol{S}}_{\boldsymbol{i}}}\boldsymbol{\ }\boldsymbol{U}\boldsymbol{(}\boldsymbol{100},\boldsymbol{110}\boldsymbol{)} $
$ {\boldsymbol{M}}_{\boldsymbol{ijr}}\boldsymbol{=}{\boldsymbol{\mathrm{min}} \boldsymbol{(}{\boldsymbol{S}}_{\boldsymbol{i}}\boldsymbol{,\ }{\boldsymbol{D}}_{\boldsymbol{j}},{\boldsymbol{T}}_{\boldsymbol{r}}\boldsymbol{)}\ } $
Table 3.  Mean values for best lower bound and upper bound computation per solution method
Problem Size No. Problem Size $ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $ Total Problem Instances mean $ {\boldsymbol{LB}}_{\boldsymbol{LRH}} $ (best) mean $ {\boldsymbol{UB}}_{\boldsymbol{LRH}} $ (best) mean $ {\boldsymbol{LB}}_{\boldsymbol{CPLEX}} $ (best) mean $ {\boldsymbol{UB}}_{\boldsymbol{CPLEX}} $ (best)
1 5$ \mathrm{\times} $5$ \mathrm{\times} $2 5 10879.80 18505.79 17859.01 17860.29
2 5$ \mathrm{\times} $8$ \mathrm{\times} $2 5 17322.20 28333.22 25534.45 26333.42
3 8$ \mathrm{\times} $8$ \mathrm{\times} $2 5 15736.80 29614.83 25534.45 25667.43
4 7$ \mathrm{\times} $10$ \mathrm{\times} $2 5 22063.60 35494.88 33925.34 33925.34
5 10$ \mathrm{\times} $10$ \mathrm{\times} $3 5 18764.20 34423.95 29758.67 29758.67
6 10$ \mathrm{\times} $20$ \mathrm{\times} $3 5 39061.60 62065.47 58664.97 58813.86
Table Options

Download as excel

Problem Size No. Problem Size $ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $ Total Problem Instances mean $ {\boldsymbol{LB}}_{\boldsymbol{LRH}} $ (best) mean $ {\boldsymbol{UB}}_{\boldsymbol{LRH}} $ (best) mean $ {\boldsymbol{LB}}_{\boldsymbol{CPLEX}} $ (best) mean $ {\boldsymbol{UB}}_{\boldsymbol{CPLEX}} $ (best)
1 5$ \mathrm{\times} $5$ \mathrm{\times} $2 5 10879.80 18505.79 17859.01 17860.29
2 5$ \mathrm{\times} $8$ \mathrm{\times} $2 5 17322.20 28333.22 25534.45 26333.42
3 8$ \mathrm{\times} $8$ \mathrm{\times} $2 5 15736.80 29614.83 25534.45 25667.43
4 7$ \mathrm{\times} $10$ \mathrm{\times} $2 5 22063.60 35494.88 33925.34 33925.34
5 10$ \mathrm{\times} $10$ \mathrm{\times} $3 5 18764.20 34423.95 29758.67 29758.67
6 10$ \mathrm{\times} $20$ \mathrm{\times} $3 5 39061.60 62065.47 58664.97 58813.86
Table 4.  Mean Gap% of each solution method using the best mean lower bound (CPLEX)
Problem Size No. Problem Size $ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $ mean $ {\boldsymbol{LB}}_{\boldsymbol{CPLEX}} $ (best) Gap% LRH Gap % CPLEX
1 5$ \mathrm{\times} $5$ \mathrm{\times} $2 17859.01 3.62% 0.007%
2 5$ \mathrm{\times} $8$ \mathrm{\times} $2 25534.45 7.72% 0.1%
3 8$ \mathrm{\times} $8$ \mathrm{\times} $2 25534.45 15.98% 0.05%
4 7$ \mathrm{\times} $10$ \mathrm{\times} $2 33925.34 4.63% 0.00%
5 10$ \mathrm{\times} $10$ \mathrm{\times} $3 29758.67 15.68% 0.00%
6 10$ \mathrm{\times} $20$ \mathrm{\times} $3 58664.97 5.8% 0.03%
Table Options

Download as excel

Problem Size No. Problem Size $ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $ mean $ {\boldsymbol{LB}}_{\boldsymbol{CPLEX}} $ (best) Gap% LRH Gap % CPLEX
1 5$ \mathrm{\times} $5$ \mathrm{\times} $2 17859.01 3.62% 0.007%
2 5$ \mathrm{\times} $8$ \mathrm{\times} $2 25534.45 7.72% 0.1%
3 8$ \mathrm{\times} $8$ \mathrm{\times} $2 25534.45 15.98% 0.05%
4 7$ \mathrm{\times} $10$ \mathrm{\times} $2 33925.34 4.63% 0.00%
5 10$ \mathrm{\times} $10$ \mathrm{\times} $3 29758.67 15.68% 0.00%
6 10$ \mathrm{\times} $20$ \mathrm{\times} $3 58664.97 5.8% 0.03%
Table 5.  Comparison between the FCSTP EQ and FCSLTP using CPLEX under 9000secs computation time
Problem No Problem Size $ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $ Total no. of Instances $ \boldsymbol{FCSTP}\boldsymbol{\ } $ EQ mean $ \boldsymbol{FCSLTP} $ mean Cost Difference % Cost Difference
1 5$ \mathrm{\times} $5$ \mathrm{\times} $2 5 18480.39 17860.29 620.10 3%
2 5$ \mathrm{\times} $8$ \mathrm{\times} $2 5 28333.22 26333.42 1999.80 8%
3 8$ \mathrm{\times} $8$ \mathrm{\times} $2 5 29064.07 25667.43 3396.64 13%
4 7$ \mathrm{\times} $10$ \mathrm{\times} $2 5 35807.10 33925.34 1881.76 6%
5 10$ \mathrm{\times} $10$ \mathrm{\times} $3 5 32147.15 29758.67 2388.48 8%
6
7
8
9
10		 10$ \mathrm{\times} $20$ \mathrm{\times} $3
15$ \mathrm{\times} $30$ \mathrm{\times} $4
20$ \mathrm{\times} $20$ \mathrm{\times} $5
25$ \mathrm{\times} $38$ \mathrm{\times} $8
35$ \mathrm{\times} $42$ \mathrm{\times} $9		 5
5
5
5
5		 62797.43
85778.98
61498.56
106532.31
120932.51
 58813.86
77653.71
50054.12
89098.73
96508.73		 3983.57
8125.27
11444.44
17433.58
24,423.78		 7%
10%
23%
20%
25%
Table Options

Download as excel

Problem No Problem Size $ \boldsymbol{m}\boldsymbol{\times }\boldsymbol{n}\boldsymbol{\times }\boldsymbol{a} $ Total no. of Instances $ \boldsymbol{FCSTP}\boldsymbol{\ } $ EQ mean $ \boldsymbol{FCSLTP} $ mean Cost Difference % Cost Difference
1 5$ \mathrm{\times} $5$ \mathrm{\times} $2 5 18480.39 17860.29 620.10 3%
2 5$ \mathrm{\times} $8$ \mathrm{\times} $2 5 28333.22 26333.42 1999.80 8%
3 8$ \mathrm{\times} $8$ \mathrm{\times} $2 5 29064.07 25667.43 3396.64 13%
4 7$ \mathrm{\times} $10$ \mathrm{\times} $2 5 35807.10 33925.34 1881.76 6%
5 10$ \mathrm{\times} $10$ \mathrm{\times} $3 5 32147.15 29758.67 2388.48 8%
6
7
8
9
10		 10$ \mathrm{\times} $20$ \mathrm{\times} $3
15$ \mathrm{\times} $30$ \mathrm{\times} $4
20$ \mathrm{\times} $20$ \mathrm{\times} $5
25$ \mathrm{\times} $38$ \mathrm{\times} $8
35$ \mathrm{\times} $42$ \mathrm{\times} $9		 5
5
5
5
5		 62797.43
85778.98
61498.56
106532.31
120932.51
 58813.86
77653.71
50054.12
89098.73
96508.73		 3983.57
8125.27
11444.44
17433.58
24,423.78		 7%
10%
23%
20%
25%
[1]

Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017
[2]

Gaidi Li, Zhen Wang, Dachuan Xu. An approximation algorithm for the $k$-level facility location problem with submodular penalties. Journal of Industrial & Management Optimization, 2012, 8 (3) : 521-529. doi: 10.3934/jimo.2012.8.521
[3]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056
[4]

Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021002
[5]

Bariş Keçeci, Fulya Altıparmak, İmdat Kara. A mathematical formulation and heuristic approach for the heterogeneous fixed fleet vehicle routing problem with simultaneous pickup and delivery. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1069-1100. doi: 10.3934/jimo.2020012
[6]

G.S. Liu, J.Z. Zhang. Decision making of transportation plan, a bilevel transportation problem approach. Journal of Industrial & Management Optimization, 2005, 1 (3) : 305-314. doi: 10.3934/jimo.2005.1.305
[7]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171
[8]

Johannes Elschner, George C. Hsiao, Andreas Rathsfeld. An inverse problem for fluid-solid interaction. Inverse Problems & Imaging, 2008, 2 (1) : 83-120. doi: 10.3934/ipi.2008.2.83
[9]

Peter Monk, Virginia Selgas. An inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2009, 3 (2) : 173-198. doi: 10.3934/ipi.2009.3.173
[10]

Guowei Hua, Shouyang Wang, Chi Kin Chan, S. H. Hou. A fractional programming model for international facility location. Journal of Industrial & Management Optimization, 2009, 5 (3) : 629-649. doi: 10.3934/jimo.2009.5.629
[11]

Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[12]

Marco Castrillón López, Pedro Luis García Pérez. The problem of Lagrange on principal bundles under a subgroup of symmetries. Journal of Geometric Mechanics, 2019, 11 (4) : 539-552. doi: 10.3934/jgm.2019026
[13]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[14]

Paulina Ávila-Torres, Fernando López-Irarragorri, Rafael Caballero, Yasmín Ríos-Solís. The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty. Journal of Industrial & Management Optimization, 2018, 14 (2) : 447-472. doi: 10.3934/jimo.2017055
[15]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[16]

Liping Zhang, Soon-Yi Wu. Robust solutions to Euclidean facility location problems with uncertain data. Journal of Industrial & Management Optimization, 2010, 6 (4) : 751-760. doi: 10.3934/jimo.2010.6.751
[17]

Behrad Erfani, Sadoullah Ebrahimnejad, Amirhossein Moosavi. An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1801-1834. doi: 10.3934/jimo.2019030
[18]

Anh Son Ta, Le Thi Hoai An, Djamel Khadraoui, Pham Dinh Tao. Solving Partitioning-Hub Location-Routing Problem using DCA. Journal of Industrial & Management Optimization, 2012, 8 (1) : 87-102. doi: 10.3934/jimo.2012.8.87
[19]

Xuefeng Wang. The heterogeneous fleet location routing problem with simultaneous pickup and delivery and overloads. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1147-1166. doi: 10.3934/dcdss.2019079
[20]

Feliz Minhós, T. Gyulov, A. I. Santos. Existence and location result for a fourth order boundary value problem. Conference Publications, 2005, 2005 (Special) : 662-671. doi: 10.3934/proc.2005.2005.662

2019 Impact Factor: 1.366

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences