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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  July 2021 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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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{)}\ } $
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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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