• Previous Article
    Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition
  • JIMO Home
  • This Issue
  • Next Article
    A new adaptive method to nonlinear semi-infinite programming
doi: 10.3934/jimo.2020034

Solving the facility location and fixed charge solid transportation problem

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.

Citation: Gbeminiyi John Oyewole, Olufemi Adetunji. Solving the facility location and fixed charge solid transportation problem. Journal of Industrial & Management Optimization, 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. AlizadehI. MahdaviN. 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. AmiriS. J. SadjadiR. 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. CalveteC. 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. CarloV. 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. CornuéjolsR. 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. FischettiI. 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. GadegaardA. 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. GhianiL. GrandinettiF. 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. HiassatA. 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. KunduM. B. KarS. KarT. 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. LiuS. J. QuM. GohR. 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. NezhadH. 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. OguzT. 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. QuY. Y. ZhouY. L. ZhangM. WahabG. 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. RobertiE. 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. SaneiA. MahmoodiradS. NiroomandA. 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. VeenstraK. Jan RoodbergenL. 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. WolseyC. 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. WuF. ChuZ. YangZ. 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. ZhangJ. PengS. 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. AlizadehI. MahdaviN. 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. AmiriS. J. SadjadiR. 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. CalveteC. 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. CarloV. 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. CornuéjolsR. 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. FischettiI. 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. GadegaardA. 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. GhianiL. GrandinettiF. 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. HiassatA. 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. KunduM. B. KarS. KarT. 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. LiuS. J. QuM. GohR. 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. NezhadH. 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. OguzT. 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. QuY. Y. ZhouY. L. ZhangM. WahabG. 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. RobertiE. 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. SaneiA. MahmoodiradS. NiroomandA. 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. VeenstraK. Jan RoodbergenL. 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. WolseyC. 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. WuF. ChuZ. YangZ. 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. ZhangJ. PengS. 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 2.  Procedure for computing the FCSTP-EQ
Figure 3.  FCSLTP and FCSTP-EQ
Figure 4.  Solution time of FCSLTP and FCSTP
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
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{)}\ } $
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
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%
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%
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]

Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021037

[2]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[3]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[4]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[5]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[6]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

[7]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[8]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[9]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[10]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[11]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[12]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[13]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[14]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[15]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

2019 Impact Factor: 1.366

Article outline

Figures and Tables

[Back to Top]