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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, 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
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Figure 2.  Procedure for computing the FCSTP-EQ
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Figure 3.  FCSLTP and FCSTP-EQ
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Figure 4.  Solution time of FCSLTP and FCSTP
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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

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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
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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%
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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%
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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%
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