June  2018, 8(2): 157-168. doi: 10.3934/naco.2018009

A three echelon revenue oriented green supply chain network design

Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

* Corresponding author. E-mail: Ar_arshadi@khu.ac.ir

Received  April 2016 Revised  October 2017 Published  May 2018

Fund Project: The reviewing process of this paper was handled by Editors A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey. This paper was for the occasion of the 12th International Conference on Industrial Engineering (ICIE 2016), which was held in Tehran, Iran during 25-26 January, 2016.

Green supply chain network designing has been studied during last decades. As carbon emissions considered as a major index in today's activities around the world, here a three echelon-multi product network including manufacturer, distributor, retailer have been provided and tried to minimize the pollution gathered from manufacturing and distribution of products all over the chains which causes extra costs as penalty to the system.

As we faced with these penalties, the model determines selling prices of products for manufacturer and distribution center simultaneously by locating these centers in order to maximize the profits all around the network. Finally, the proposed model is solved through the numerical examples and the sensitivity analysis and important parameters are reported to find some management insights.

Citation: Ashkan Mohsenzadeh Ledari, Alireza Arshadi Khamseh, Mohammad Mohammadi. A three echelon revenue oriented green supply chain network design. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 157-168. doi: 10.3934/naco.2018009
References:
[1]

T. AbdallahA. FarhatA. Diabat and S. Kennedy, Green supply chains with carbon trading and environmental sourcing: Formulation and life cycle assessment, Applied Mathematical Modelling, 36 (2012), 4271-4285.  doi: 10.1016/j.apm.2011.11.056.  Google Scholar

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A. ChaabaneA. Ramudhin and M. Paquet, Design of sustainable supply chains under the emission trading scheme, International Journal of Production Economics, 135 (2012), 37-49.   Google Scholar

[4]

Y. J. Chen and J. B. Sheu, Environmental-regulation pricing strategies for green supply chain management, Transportation Research Part E, 45 (2009), 667-677.   Google Scholar

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A. ChoudharyS. SarkarS. Settur and M. K. Tiwari, A carbon market sensitive optimization model for integrated forward-reverse logistics, International Journal of Production Economics, 164 (2015), 433-444.   Google Scholar

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S. Elhedhli and R. Merrick, Green supply chain network design to reduce carbon emissions, Transportation Research Part D, 17 (2012), 370-379.   Google Scholar

[8]

M. El-SayedN. Afia and A. El-Kharbotly, A stochastic model for forward-reverse logistics network design under risk, Computers & Industrial Engineering, 58 (2010), 423-431.   Google Scholar

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I. HarrisM. NaimA. PalmerA. Potter and C. Mumford, Assessing the impact of cost optimization based on infrastructure modelling on CO2 emissions, International Journal of Production Economics, 131 (2011), 313-321.   Google Scholar

[10]

G. HuaT. C. E. Cheng and S. Wang, Managing carbon footprints in inventory management, International Journal of Production Economics, 132 (2011), 178-185.   Google Scholar

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B. LiM. ZhuY. Jiang and Z. Li, Pricing policies of a competitive dual-channel green supply chain, Journal of Cleaner Production, 112 (2016), 2029-2042.   Google Scholar

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S. LiuD. Kasturiratne and J. Moizer, A hub-and-spoke model for multi-dimensional integration of green marketing and sustainable supply chain management, Industrial Marketing Management, 41 (2012), 581-588.   Google Scholar

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L. Reijnders, Policies influencing cleaner production: the role of prices and regulation, Journal of Cleaner Production, 11 (2003), 333-338.   Google Scholar

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B. SarkarB. GangulyM. Sarkar and S. Pareek, Effect of variable transportation and carbon emission in a three-echelon supply chain model, Transportation Research Part E: Logistics and Transportation Review, 91 (2016), 112-128.   Google Scholar

[16]

S. C. Tseng and S. W. Hung, A strategic decision-making model considering the social costs of carbon dioxide emissions for sustainable supply chain management, Journal of Environmental Management, 133 (2014), 315-322.   Google Scholar

[17]

C. J. Vidal and M. Goetschalckx, A global supply chain model with transfer pricing and transportation cost allocation, European Journal of Operational Research, 129 (2001), 134-158.   Google Scholar

[18]

F. WangX. Lai and N. Shi, A multi-objective optimization for green supply chain network design, Decision Support Systems, 51 (2011), 262-269.   Google Scholar

[19]

C. Wu and D. Barnes, An integrated model for green partner selection and supply chain construction, Journal of Cleaner Production, 112 (2016), 2114-2132.   Google Scholar

[20]

X. XuW. ZhangP. He and X. Xu, Production and pricing problems in make-to-order supply chain with cap-and-trade regulation, Omega, 112 (2017), 248-257.   Google Scholar

[21]

B. Zhang and L. Xu, Multi-item production planning with carbon cap and trade mechanism, International Journal of Production Economics, 144 (2013), 118-127.   Google Scholar

[22]

Q. ZhangW. Tang and J. Zhang, Green supply chain performance with cost learning and operational inefficiency effects, Journal of Cleaner Production, 112 (2015), 1-18.   Google Scholar

show all references

References:
[1]

T. AbdallahA. FarhatA. Diabat and S. Kennedy, Green supply chains with carbon trading and environmental sourcing: Formulation and life cycle assessment, Applied Mathematical Modelling, 36 (2012), 4271-4285.  doi: 10.1016/j.apm.2011.11.056.  Google Scholar

[2]

K. CattaniW. GillandH. S. Heese and J. Swaminathan, Boiling frogs: pricing strategies for a manufacturer adding a direct channel that competes with the traditional channel, Production and Operations Managements, 15 (2006), 40-56.   Google Scholar

[3]

A. ChaabaneA. Ramudhin and M. Paquet, Design of sustainable supply chains under the emission trading scheme, International Journal of Production Economics, 135 (2012), 37-49.   Google Scholar

[4]

Y. J. Chen and J. B. Sheu, Environmental-regulation pricing strategies for green supply chain management, Transportation Research Part E, 45 (2009), 667-677.   Google Scholar

[5]

A. ChoudharyS. SarkarS. Settur and M. K. Tiwari, A carbon market sensitive optimization model for integrated forward-reverse logistics, International Journal of Production Economics, 164 (2015), 433-444.   Google Scholar

[6]

J. M. Cruz, Dynamics of supply chain networks with corporate social responsibility through integrated environmental decision-making, European Journal of Operational Research, 184 (2008), 1005-1031.  doi: 10.1016/j.ejor.2006.12.012.  Google Scholar

[7]

S. Elhedhli and R. Merrick, Green supply chain network design to reduce carbon emissions, Transportation Research Part D, 17 (2012), 370-379.   Google Scholar

[8]

M. El-SayedN. Afia and A. El-Kharbotly, A stochastic model for forward-reverse logistics network design under risk, Computers & Industrial Engineering, 58 (2010), 423-431.   Google Scholar

[9]

I. HarrisM. NaimA. PalmerA. Potter and C. Mumford, Assessing the impact of cost optimization based on infrastructure modelling on CO2 emissions, International Journal of Production Economics, 131 (2011), 313-321.   Google Scholar

[10]

G. HuaT. C. E. Cheng and S. Wang, Managing carbon footprints in inventory management, International Journal of Production Economics, 132 (2011), 178-185.   Google Scholar

[11]

IPCC, 2007. Climate Change. In The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (eds. S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor and H. L. iller). Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. Google Scholar

[12]

B. LiM. ZhuY. Jiang and Z. Li, Pricing policies of a competitive dual-channel green supply chain, Journal of Cleaner Production, 112 (2016), 2029-2042.   Google Scholar

[13]

S. LiuD. Kasturiratne and J. Moizer, A hub-and-spoke model for multi-dimensional integration of green marketing and sustainable supply chain management, Industrial Marketing Management, 41 (2012), 581-588.   Google Scholar

[14]

L. Reijnders, Policies influencing cleaner production: the role of prices and regulation, Journal of Cleaner Production, 11 (2003), 333-338.   Google Scholar

[15]

B. SarkarB. GangulyM. Sarkar and S. Pareek, Effect of variable transportation and carbon emission in a three-echelon supply chain model, Transportation Research Part E: Logistics and Transportation Review, 91 (2016), 112-128.   Google Scholar

[16]

S. C. Tseng and S. W. Hung, A strategic decision-making model considering the social costs of carbon dioxide emissions for sustainable supply chain management, Journal of Environmental Management, 133 (2014), 315-322.   Google Scholar

[17]

C. J. Vidal and M. Goetschalckx, A global supply chain model with transfer pricing and transportation cost allocation, European Journal of Operational Research, 129 (2001), 134-158.   Google Scholar

[18]

F. WangX. Lai and N. Shi, A multi-objective optimization for green supply chain network design, Decision Support Systems, 51 (2011), 262-269.   Google Scholar

[19]

C. Wu and D. Barnes, An integrated model for green partner selection and supply chain construction, Journal of Cleaner Production, 112 (2016), 2114-2132.   Google Scholar

[20]

X. XuW. ZhangP. He and X. Xu, Production and pricing problems in make-to-order supply chain with cap-and-trade regulation, Omega, 112 (2017), 248-257.   Google Scholar

[21]

B. Zhang and L. Xu, Multi-item production planning with carbon cap and trade mechanism, International Journal of Production Economics, 144 (2013), 118-127.   Google Scholar

[22]

Q. ZhangW. Tang and J. Zhang, Green supply chain performance with cost learning and operational inefficiency effects, Journal of Cleaner Production, 112 (2015), 1-18.   Google Scholar

Figure 1.  A three echelon supply chain network
Figure 2.  The impact of demands on selling price
Figure 3.  The impact of carbon dioxid emissions on selling price
Table 1.  Data for numerical example
$f_{j} $ Random between [200000, 300000] $a_{k} $ Random between [10000, 10100] $w_{i}^{k} $ Random between [1000, 2000]
$c_{mk} $ Random between [1000, 1100] $\gamma _{mk} $ Random between [10000, 12000] $CO_{2} $ Random between [450000000, 500000000]
$T_{mjk} $ Random between [500, 600] $v_{jk} $ Random between [200, 300] $R_{m}^{k} $ Random between [10000, 10100]
$d_{ij} $ Random between [50, 60] $T_{jik} $ Random between [500, 600] $\alpha _{ji} $ Random between [400, 450]
$d_{mj} $ Random between [50, 60] $b$ Random between [50000000, 60000000] $\alpha _{mj} $ Random between [400, 450]
$P$ Random between [6000, 7000] $q_{j} $ Random between [10000000, 10100000] $l_{k} $ Random between [1000, 2000]
$f_{j} $ Random between [200000, 300000] $a_{k} $ Random between [10000, 10100] $w_{i}^{k} $ Random between [1000, 2000]
$c_{mk} $ Random between [1000, 1100] $\gamma _{mk} $ Random between [10000, 12000] $CO_{2} $ Random between [450000000, 500000000]
$T_{mjk} $ Random between [500, 600] $v_{jk} $ Random between [200, 300] $R_{m}^{k} $ Random between [10000, 10100]
$d_{ij} $ Random between [50, 60] $T_{jik} $ Random between [500, 600] $\alpha _{ji} $ Random between [400, 450]
$d_{mj} $ Random between [50, 60] $b$ Random between [50000000, 60000000] $\alpha _{mj} $ Random between [400, 450]
$P$ Random between [6000, 7000] $q_{j} $ Random between [10000000, 10100000] $l_{k} $ Random between [1000, 2000]
Table 2.  The decision variables value
$X_{ijk}$ $\begin{array}{l} \begin{array}{ccccc} X_{111} {=\, \, \, \, \, 0\, \, \, \, }&{X_{112} =\, \, \, \, \, 0\, \, \, \, \, }&{X_{113} =1515}&{X_{114} =\, \, \, \, 0\, \, \, \, }&{X_{115} =\, \, \, \, 0\, \, \, \, } \\ {X_{121} =\, \, \, \, \, 0\, \, \, \, }&{X_{122} =1203}&{X_{123} =\, \, \, \, 0\, \, \, \, }&{X_{124} =1545}&{X_{125} =1907} \\ {X_{131} =\, 1657}&{X_{132} =\, \, \, \, 0\, \, \, \, }&{X_{133} =\, \, \, \, 0\, \, \, \, }&{X_{134} =\, \, \, \, 0\, \, \, \, }&{X_{135} =\, \, \, \, 0\, \, \, \, } \\ {X_{211} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{212} =\, \, \, \, 0\, \, \, \, }&{X_{213} =1069}&{X_{214} =\, \, \, \, 0\, \, \, \, }&{X_{215} =\, \, \, \, 0\, \, \, \, } \\ {X_{221} =1804}&{X_{222} =1983}&{X_{223} =\, \, \, \, 0\, \, \, \, }&{X_{224} =1196}&{X_{225} =1695} \\ {X_{311} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{312} =\, \, \, \, 0\, \, \, \, }&{X_{313} =1217}&{X_{314} =\, \, \, \, 0\, \, \, \, }&{X_{315} =\, \, \, \, 0\, \, \, \, } \\ {X_{321} =1639}&{X_{322} =1333}&{X_{323} =\, \, \, \, 0\, \, \, \, }&{X_{324} =1632}&{X_{325} =1012} \\ {X_{411} =1122}&{X_{412} =\, \, \, \, 0\, \, \, \, }&{X_{413} =1946}&{X_{414} =\, \, \, \, 0\, \, \, \, }&{X_{415} =\, \, \, \, 0\, \, \, \, } \\ {X_{421} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{422} =1687}&{X_{423} =\, \, \, \, 0\, \, \, \, }&{X_{424} =1709}&{X_{425} =1075} \\ {X_{511} =1510}&{X_{512} =\, \, \, \, 0\, \, \, \, }&{X_{513} =1196}&{X_{514} =\, \, \, \, 0\, \, \, \, }&{X_{515} =\, \, \, \, 0\, \, \, \, } \\ {X_{521} =\, \, \, \, 0\, \, \, \, \, \, \, }&{X_{522} =1510}&{X_{523} =\, \, \, \, 0\, \, \, \, }&{X_{524} =1273}&{X_{525} =1086}\end{array}\end{array}$ $\begin{array}{l}\\ \\ U_{mjk} \\ \\ \\ \\ \\ y_{j}\end{array}$ $\begin{array}{l} {U_{111} =9575} \\ {U_{121} =25289} \\ {U_{131} =1657} \\ \\{y_{1} =1} \\ {y_{2} =1} \\ {y_{3} =1}\end{array}$
Objective function $5.6*10^{11} $
$X_{ijk}$ $\begin{array}{l} \begin{array}{ccccc} X_{111} {=\, \, \, \, \, 0\, \, \, \, }&{X_{112} =\, \, \, \, \, 0\, \, \, \, \, }&{X_{113} =1515}&{X_{114} =\, \, \, \, 0\, \, \, \, }&{X_{115} =\, \, \, \, 0\, \, \, \, } \\ {X_{121} =\, \, \, \, \, 0\, \, \, \, }&{X_{122} =1203}&{X_{123} =\, \, \, \, 0\, \, \, \, }&{X_{124} =1545}&{X_{125} =1907} \\ {X_{131} =\, 1657}&{X_{132} =\, \, \, \, 0\, \, \, \, }&{X_{133} =\, \, \, \, 0\, \, \, \, }&{X_{134} =\, \, \, \, 0\, \, \, \, }&{X_{135} =\, \, \, \, 0\, \, \, \, } \\ {X_{211} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{212} =\, \, \, \, 0\, \, \, \, }&{X_{213} =1069}&{X_{214} =\, \, \, \, 0\, \, \, \, }&{X_{215} =\, \, \, \, 0\, \, \, \, } \\ {X_{221} =1804}&{X_{222} =1983}&{X_{223} =\, \, \, \, 0\, \, \, \, }&{X_{224} =1196}&{X_{225} =1695} \\ {X_{311} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{312} =\, \, \, \, 0\, \, \, \, }&{X_{313} =1217}&{X_{314} =\, \, \, \, 0\, \, \, \, }&{X_{315} =\, \, \, \, 0\, \, \, \, } \\ {X_{321} =1639}&{X_{322} =1333}&{X_{323} =\, \, \, \, 0\, \, \, \, }&{X_{324} =1632}&{X_{325} =1012} \\ {X_{411} =1122}&{X_{412} =\, \, \, \, 0\, \, \, \, }&{X_{413} =1946}&{X_{414} =\, \, \, \, 0\, \, \, \, }&{X_{415} =\, \, \, \, 0\, \, \, \, } \\ {X_{421} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{422} =1687}&{X_{423} =\, \, \, \, 0\, \, \, \, }&{X_{424} =1709}&{X_{425} =1075} \\ {X_{511} =1510}&{X_{512} =\, \, \, \, 0\, \, \, \, }&{X_{513} =1196}&{X_{514} =\, \, \, \, 0\, \, \, \, }&{X_{515} =\, \, \, \, 0\, \, \, \, } \\ {X_{521} =\, \, \, \, 0\, \, \, \, \, \, \, }&{X_{522} =1510}&{X_{523} =\, \, \, \, 0\, \, \, \, }&{X_{524} =1273}&{X_{525} =1086}\end{array}\end{array}$ $\begin{array}{l}\\ \\ U_{mjk} \\ \\ \\ \\ \\ y_{j}\end{array}$ $\begin{array}{l} {U_{111} =9575} \\ {U_{121} =25289} \\ {U_{131} =1657} \\ \\{y_{1} =1} \\ {y_{2} =1} \\ {y_{3} =1}\end{array}$
Objective function $5.6*10^{11} $
Table 3.  The optimal value of binary decision variable and objective function
Problem size(I*J*M*K) Optimal value of binary decision variable Objective function
5*5*5*5 $y_{j} =[11100]$ $5.6*10^{11} $
15*15*15*15 $y_{j} =[110111111001111]$ $5.1*10^{12} $
20*20*20*20 $y_{j} =[11110110101101101111]$ $9.2*10^{12} $
25*25*25*25 $y_{j} =[1111011111001111101101011]$ $1.4*10^{13} $
Problem size(I*J*M*K) Optimal value of binary decision variable Objective function
5*5*5*5 $y_{j} =[11100]$ $5.6*10^{11} $
15*15*15*15 $y_{j} =[110111111001111]$ $5.1*10^{12} $
20*20*20*20 $y_{j} =[11110110101101101111]$ $9.2*10^{12} $
25*25*25*25 $y_{j} =[1111011111001111101101011]$ $1.4*10^{13} $
Table 4.  The optimal value of binary decision variable and objective function
$S_{mk} $ $\begin{array}{ccccc} {S_{11} =9999999}&{S_{12} =1481632}&{S_{13} =1433065}&{S_{14} =1498182}&{S_{15} =1306987} \\ {S_{21} =6419183}&{S_{22} =1481632}&{S_{23} =1433066}&{S_{24} =1498182}&{S_{25} =1306987} \\ {S_{31} =6438095}&{S_{32} =1481632}&{S_{33} =1433065}&{S_{34} =1498182}&{S_{35} =1306987} \\ {S_{41} =6426776}&{S_{42} =1481633}&{S_{43} =1433065}&{S_{44} =1498182}&{S_{45} =1306988} \\ {S_{51} =6432022}&{S_{52} =1481632}&{S_{53} =1433065}&{S_{54} =1498182}&{S_{55} =1306987} \end{array}$ $U_{mjk} $ $\begin{array}{l} {U_{111} =9575} \\ {U_{121} =25289} \\ {U_{131} =1657} \end{array}$
$S_{jk}^{'} $ $\begin{array}{ccccc} {S'_{11} =9999999}&{S'_{12} =1825318}&{S'_{13} =9999999}&{S'_{14} =1904827}&{S'_{15} =1419765} \\ {S'_{21} =9999999}&{S'_{22} =9999999}&{S'_{23} =1444720}&{S'_{24} =9999999}&{S'_{25} =9999999} \\ {S'_{31} =9999999}&{S'_{32} =1493339}&{S'_{33} =1780413}&{S'_{34} =1509851}&{S'_{35} =1318645} \\ {S'_{41} =9999999}&{S'_{42} =1493334}&{S'_{43} =1444775}&{S'_{44} =1509885}&{S'_{45} =1318716} \\ {S'_{51} =9999999}&{S'_{52} =1493334}&{S'_{53} =1444775}&{S'_{54} =1509885}&{S'_{55} =1318716} \end{array}$ $y_{j} $ $\begin{array}{l} {y_{1} =1} \\ {y_{2} =1} \\ {y_{3} =1} \end{array}$
$X_{ijk} $ $\begin{array}{ccccc} {X_{111} =\, \, \, \, \, 0\, \, \, \, }&{X_{112} =\, \, \, \, \, 0\, \, \, \, \, }&{X_{113} =1515}&{X_{114} =\, \, \, \, 0\, \, \, \, }&{X_{115} =\, \, \, \, 0\, \, \, \, } \\ {X_{121} =\, \, \, \, \, 0\, \, \, \, }&{X_{122} =1203}&{X_{123} =\, \, \, \, 0\, \, \, \, }&{X_{124} =1545}&{X_{125} =1907} \\ {X_{131} =\, 1657}&{X_{132} =\, \, \, \, 0\, \, \, \, }&{X_{133} =\, \, \, \, 0\, \, \, \, }&{X_{134} =\, \, \, \, 0\, \, \, \, }&{X_{135} =\, \, \, \, 0\, \, \, \, } \\ {X_{211} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{212} =\, \, \, \, 0\, \, \, \, }&{X_{213} =1069}&{X_{214} =\, \, \, \, 0\, \, \, \, }&{X_{215} =\, \, \, \, 0\, \, \, \, } \\ {X_{221} =1804}&{X_{222} =1983}&{X_{223} =\, \, \, \, 0\, \, \, \, }&{X_{224} =1196}&{X_{225} =1695} \\ {X_{311} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{312} =\, \, \, \, 0\, \, \, \, }&{X_{313} =1217}&{X_{314} =\, \, \, \, 0\, \, \, \, }&{X_{315} =\, \, \, \, 0\, \, \, \, } \\ {X_{321} =1639}&{X_{322} =1333}&{X_{323} =\, \, \, \, 0\, \, \, \, }&{X_{324} =1632}&{X_{325} =1012} \\ {X_{411} =1122}&{X_{412} =\, \, \, \, 0\, \, \, \, }&{X_{413} =1946}&{X_{414} =\, \, \, \, 0\, \, \, \, }&{X_{415} =\, \, \, \, 0\, \, \, \, } \\ {X_{421} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{422} =1687}&{X_{423} =\, \, \, \, 0\, \, \, \, }&{X_{424} =1709}&{X_{425} =1075} \\ {X_{511} =1510}&{X_{512} =\, \, \, \, 0\, \, \, \, }&{X_{513} =1196}&{X_{514} =\, \, \, \, 0\, \, \, \, }&{X_{515} =\, \, \, \, 0\, \, \, \, } \\ {X_{521} =\, \, \, \, 0\, \, \, \, \, \, \, }&{X_{522} =1510}&{X_{523} =\, \, \, \, 0\, \, \, \, }&{X_{524} =1273}&{X_{525} =1086} \end{array}$ $g_{mjk}^{'} $ $\begin{array}{l} {g'_{111} ={\rm 95749990434}} \\ {g'_{121} ={\rm 95749990434}} \\ {g'_{131} ={\rm 16569998344}} \end{array}$
$g_{ijk} $ $\begin{array}{ccccc} {g_{111} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{112} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{113} ={\rm 15149998486}}&{g_{114} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{115} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{121} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{122} ={\rm 12029998798}}&{g_{123} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{124} ={\rm 15449998456}}&{g_{125} ={\rm 19069998094}} \\ {g_{131} =\, {\rm 16569998344}}&{g_{132} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{133} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{134} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{135} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{211} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{212} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{213} ={\rm 10689998932}}&{g_{214} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{215} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{221} ={\rm 18039998197}}&{g_{222} ={\rm 19829998018}}&{g_{223} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{224} ={\rm 11959998805}}&{g_{225} ={\rm 16949998306}} \\ {g_{311} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{312} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{313} ={\rm 12169998784}}&{g_{314} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{315} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{321} ={\rm 16389998362}}&{g_{322} ={\rm 13329998668}}&{g_{323} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{324} ={\rm 16319998369}}&{g_{325} ={\rm 10119998989}} \\ {g_{411} ={\rm 11219998879}}&{g_{412} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{413} ={\rm 19459998055}}&{g_{414} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{415} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{421} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{422} ={\rm 16869998314}}&{g_{423} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{424} ={\rm 17089998292}}&{g_{425} ={\rm 10749998926}} \\ {g_{511} ={\rm 15099998491}}&{g_{512} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{513} ={\rm 11959998805}}&{g_{514} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{515} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{521} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{522} ={\rm 15099998491}}&{\, g_{523} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{\, \, g_{524} ={\rm 12729998728}}&{g_{525} ={\rm 10859998915}} \end{array}$
Objective function $5.6*10^{11} $
$S_{mk} $ $\begin{array}{ccccc} {S_{11} =9999999}&{S_{12} =1481632}&{S_{13} =1433065}&{S_{14} =1498182}&{S_{15} =1306987} \\ {S_{21} =6419183}&{S_{22} =1481632}&{S_{23} =1433066}&{S_{24} =1498182}&{S_{25} =1306987} \\ {S_{31} =6438095}&{S_{32} =1481632}&{S_{33} =1433065}&{S_{34} =1498182}&{S_{35} =1306987} \\ {S_{41} =6426776}&{S_{42} =1481633}&{S_{43} =1433065}&{S_{44} =1498182}&{S_{45} =1306988} \\ {S_{51} =6432022}&{S_{52} =1481632}&{S_{53} =1433065}&{S_{54} =1498182}&{S_{55} =1306987} \end{array}$ $U_{mjk} $ $\begin{array}{l} {U_{111} =9575} \\ {U_{121} =25289} \\ {U_{131} =1657} \end{array}$
$S_{jk}^{'} $ $\begin{array}{ccccc} {S'_{11} =9999999}&{S'_{12} =1825318}&{S'_{13} =9999999}&{S'_{14} =1904827}&{S'_{15} =1419765} \\ {S'_{21} =9999999}&{S'_{22} =9999999}&{S'_{23} =1444720}&{S'_{24} =9999999}&{S'_{25} =9999999} \\ {S'_{31} =9999999}&{S'_{32} =1493339}&{S'_{33} =1780413}&{S'_{34} =1509851}&{S'_{35} =1318645} \\ {S'_{41} =9999999}&{S'_{42} =1493334}&{S'_{43} =1444775}&{S'_{44} =1509885}&{S'_{45} =1318716} \\ {S'_{51} =9999999}&{S'_{52} =1493334}&{S'_{53} =1444775}&{S'_{54} =1509885}&{S'_{55} =1318716} \end{array}$ $y_{j} $ $\begin{array}{l} {y_{1} =1} \\ {y_{2} =1} \\ {y_{3} =1} \end{array}$
$X_{ijk} $ $\begin{array}{ccccc} {X_{111} =\, \, \, \, \, 0\, \, \, \, }&{X_{112} =\, \, \, \, \, 0\, \, \, \, \, }&{X_{113} =1515}&{X_{114} =\, \, \, \, 0\, \, \, \, }&{X_{115} =\, \, \, \, 0\, \, \, \, } \\ {X_{121} =\, \, \, \, \, 0\, \, \, \, }&{X_{122} =1203}&{X_{123} =\, \, \, \, 0\, \, \, \, }&{X_{124} =1545}&{X_{125} =1907} \\ {X_{131} =\, 1657}&{X_{132} =\, \, \, \, 0\, \, \, \, }&{X_{133} =\, \, \, \, 0\, \, \, \, }&{X_{134} =\, \, \, \, 0\, \, \, \, }&{X_{135} =\, \, \, \, 0\, \, \, \, } \\ {X_{211} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{212} =\, \, \, \, 0\, \, \, \, }&{X_{213} =1069}&{X_{214} =\, \, \, \, 0\, \, \, \, }&{X_{215} =\, \, \, \, 0\, \, \, \, } \\ {X_{221} =1804}&{X_{222} =1983}&{X_{223} =\, \, \, \, 0\, \, \, \, }&{X_{224} =1196}&{X_{225} =1695} \\ {X_{311} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{312} =\, \, \, \, 0\, \, \, \, }&{X_{313} =1217}&{X_{314} =\, \, \, \, 0\, \, \, \, }&{X_{315} =\, \, \, \, 0\, \, \, \, } \\ {X_{321} =1639}&{X_{322} =1333}&{X_{323} =\, \, \, \, 0\, \, \, \, }&{X_{324} =1632}&{X_{325} =1012} \\ {X_{411} =1122}&{X_{412} =\, \, \, \, 0\, \, \, \, }&{X_{413} =1946}&{X_{414} =\, \, \, \, 0\, \, \, \, }&{X_{415} =\, \, \, \, 0\, \, \, \, } \\ {X_{421} =\, \, \, \, \, 0\, \, \, \, \, \, }&{X_{422} =1687}&{X_{423} =\, \, \, \, 0\, \, \, \, }&{X_{424} =1709}&{X_{425} =1075} \\ {X_{511} =1510}&{X_{512} =\, \, \, \, 0\, \, \, \, }&{X_{513} =1196}&{X_{514} =\, \, \, \, 0\, \, \, \, }&{X_{515} =\, \, \, \, 0\, \, \, \, } \\ {X_{521} =\, \, \, \, 0\, \, \, \, \, \, \, }&{X_{522} =1510}&{X_{523} =\, \, \, \, 0\, \, \, \, }&{X_{524} =1273}&{X_{525} =1086} \end{array}$ $g_{mjk}^{'} $ $\begin{array}{l} {g'_{111} ={\rm 95749990434}} \\ {g'_{121} ={\rm 95749990434}} \\ {g'_{131} ={\rm 16569998344}} \end{array}$
$g_{ijk} $ $\begin{array}{ccccc} {g_{111} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{112} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{113} ={\rm 15149998486}}&{g_{114} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{115} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{121} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{122} ={\rm 12029998798}}&{g_{123} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{124} ={\rm 15449998456}}&{g_{125} ={\rm 19069998094}} \\ {g_{131} =\, {\rm 16569998344}}&{g_{132} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{133} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{134} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{135} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{211} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{212} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{213} ={\rm 10689998932}}&{g_{214} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{215} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{221} ={\rm 18039998197}}&{g_{222} ={\rm 19829998018}}&{g_{223} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{224} ={\rm 11959998805}}&{g_{225} ={\rm 16949998306}} \\ {g_{311} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{312} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{313} ={\rm 12169998784}}&{g_{314} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{315} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{321} ={\rm 16389998362}}&{g_{322} ={\rm 13329998668}}&{g_{323} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{324} ={\rm 16319998369}}&{g_{325} ={\rm 10119998989}} \\ {g_{411} ={\rm 11219998879}}&{g_{412} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{413} ={\rm 19459998055}}&{g_{414} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{415} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{421} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{422} ={\rm 16869998314}}&{g_{423} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{424} ={\rm 17089998292}}&{g_{425} ={\rm 10749998926}} \\ {g_{511} ={\rm 15099998491}}&{g_{512} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{513} ={\rm 11959998805}}&{g_{514} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{515} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {g_{521} =\, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{g_{522} ={\rm 15099998491}}&{\, g_{523} =\, \, \, \, \, \, \, \, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, }&{\, \, g_{524} ={\rm 12729998728}}&{g_{525} ={\rm 10859998915}} \end{array}$
Objective function $5.6*10^{11} $
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