A Novel Hybrid Whale Optimization Algorithm to Solve a Production-Distribution Network Problem Considering Carbon Emissions

Document Type : Original Article


1 Department of Industrial Engineering, Shomal University, Amol, Iran

2 Department of Industrial Engineering, University of Science and Technology of Mazandaran, Behshahr, Iran

3 Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran


Nowadays, there is a great deal of attention for regulations of carbon emissions to enforce the decision-makers of production and distribution networks to redesign their systems satisfactorily. The literature has seen a rapid interest in developing novel metaheuristics to solve this problem as a complicated optimization problem. Such difficulties motivate us to address a production-distribution network design problem considering carbon emissions policies among the first studies in this area by a novel hybrid whale optimization algorithm. Accordingly, a mixed integer non-linear programming model has been developed. To tackle the proposed problem, a new hybrid metaheuristic based on whale optimization algorithm and simulated annealing as a successful optimizer is employed to solve the proposed problem. The calibration of the algorithms has been designed by Taguchi method, comprehensively. Finally, an extensive analysis has been evaluated through a comparative study along with some assessment metrics of Pareto solutions.


1. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M. and Mirjalili, S.,
"A set of efficient heuristics for a home healthcare problem",
Neural Computing and Applications,  (2019), 1-21. 
2. Hajiaghaei-Keshteli, M. and Fathollahi-Fard, A.M., "A set of
efficient heuristics and metaheuristics to solve a two-stage
stochastic bi-level decision-making model for the distribution
network problem", Computers & Industrial Engineering,  Vol.
123, (2018), 378-395. 
3. Absi, N., Dauzère-Pérès, S., Kedad-Sidhoum, S., Penz, B. and
Rapine, C., "Lot sizing with carbon emission constraints",
European Journal of Operational Research,  Vol. 227, No. 1,
(2013), 55-61. 
4. Ghosh, A., Jha, J. and Sarmah, S., "Optimizing a two-echelon
serial supply chain with different carbon policies", International
Journal of Sustainable Engineering,  Vol. 9, No. 6, (2016), 363377.
5. Beamon, B.M., "Supply chain design and analysis:: Models and
methods", International Journal of Production Economics, 
Vol. 55, No. 3, (1998), 281-294. 
6. Benjaafar, S., Li, Y. and Daskin, M., "Carbon footprint and the
management of supply chains: Insights from simple models",
IEEE Transactions on Automation Science and Engineering, 
Vol. 10, No. 1, (2012), 99-116. 
7. Golmohamadi, S., Tavakkoli-Moghaddam, R. and HajiaghaeiKeshteli, M., "Solving a fuzzy fixed charge solid transportation problem
using batch transferring by new approaches in metaheuristic",
Electronic Notes in Discrete Mathematics, Vol. 58, (2017), 143-150.
8. Fu, Y., Tian, G., Fathollahi-Fard, A.M., Ahmadi, A. and Zhang,
C., "Stochastic multi-objective modelling and optimization of an
energy-conscious distributed permutation flow shop scheduling
problem with the total tardiness constraint", Journal of Cleaner
Production,  Vol. 226, (2019), 515-525. 
9. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M. and TavakkoliMoghaddam,
R., "The social engineering optimizer (SEO)",
Engineering Applications of Artificial Intelligence,  Vol. 72,
(2018), 267-293. 
10. Zhang, B. and Xu, L., "Multi-item production planning with
carbon cap and trade mechanism", International Journal of
Production Economics,  Vol. 144, No. 1, (2013), 118-127. 
11. Fathollahi-Fard, A.M., Govindan, K., Hajiaghaei-Keshteli, M.
and Ahmadi, A., "A green home health care supply chain: New
modified simulated annealing algorithms", Journal of Cleaner
Production,  Vol. 240, (2019), 118200. 
12. Bouchery, Y., Ghaffari, A., Jemai, Z. and Tan, T., "Impact of
coordination on costs and carbon emissions for a two-echelon
serial economic order quantity problem", European Journal of
Operational Research,  Vol. 260, No. 2, (2017), 520-533. 
13. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M., Tian, G. and Li,
Z., "An adaptive lagrangian relaxation-based algorithm for a
coordinated water supply and wastewater collection network
design problem", Information Sciences,  (2019).
14. Chen, C.-L. and Lee, W.-C., "Multi-objective optimization of
multi-echelon supply chain networks with uncertain product
demands and prices", Computers & Chemical Engineering,  Vol.
28, No. 6-7, (2004), 1131-1144. 
15. Chen, X., Benjaafar, S. and Elomri, A., "The carbon-constrained
eoq", Operations Research Letters,  Vol. 41, No. 2, (2013), 172179.
16. Darvish, M., Larrain, H. and Coelho, L.C., "A dynamic multi-
plant lot-sizing and distribution problem", International Journal 
of Production Research,  Vol. 54, No. 22, (2016), 6707-6717.
17. Daskin, M.S., Coullard, C.R. and Shen, Z.-J.M., "An inventory-
location model: Formulation, solution algorithm and 
computational results", Annals of Operations Research,  Vol.
110, No. 1-4, (2002), 83-106. 
18. Dobos, I., "Tradable permits and production-inventory strategies
of the firm", International Journal of Production Economics, 
Vol. 108, No. 1-2, (2007), 329-333. 
19. Sabri, E.H. and Beamon, B.M., "A multi-objective approach to
simultaneous strategic and operational planning in supply chain
design", Omega,  Vol. 28, No. 5, (2000), 581-598. 
20. Chan, F.T., Chung, S. and Wadhwa, S., "A hybrid genetic
algorithm for production and distribution", Omega,  Vol. 33, No.
4, (2005), 345-355. 
21. Hua, G., Cheng, T. and Wang, S., "Managing carbon footprints in
inventory management", International Journal of Production
Economics,  Vol. 132, No. 2, (2011), 178-185. 
22. Jaber, M.Y., Glock, C.H. and El Saadany, A.M., "Supply chain
coordination with emissions reduction incentives", International
Journal of Production Research,  Vol. 51, No. 1, (2013), 69-82. 
23. Li, J., Su, Q. and Ma, L., "Production and transportation
outsourcing decisions in the supply chain under single and
multiple carbon policies", Journal of Cleaner Production,  Vol.
141, (2017), 1109-1122. 
24. Shu, J., Teo, C.-P. and Shen, Z.-J.M., "Stochastic transportationinventory
network design problem", Operations Research, Vol. 53, No. 1, (2005), 48-60.
25. Hajiaghaei-Keshteli, M. and Fard, A.M.F., "Sustainable closedloop
supply chain network design with discount supposition",
Neural Computing and Applications,  Vol. 31, No. 9, (2019),
26. Fathollahi-Fard, A.M. and Hajiaghaei-Keshteli, M., "A stochastic
multi-objective model for a closed-loop supply chain with
environmental considerations", Applied Soft Computing,  Vol.
69, (2018), 232-249. 
27. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M. and Mirjalili, S.,
"Hybrid optimizers to solve a tri-level programming model for a
tire closed-loop supply chain network design problem", Applied
Soft Computing,  Vol. 70, (2018), 701-722. 
28. Sahebjamnia, N., Fathollahi-Fard, A.M. and Hajiaghaei-Keshteli,
M., "Sustainable tire closed-loop supply chain network design:
Hybrid metaheuristic algorithms for large-scale networks",
Journal of Cleaner Production,  Vol. 196, (2018), 273-296. 
29. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M. and Mirjalili, S.,
"Multi-objective stochastic closed-loop supply chain network
design with social considerations", Applied Soft Computing, 
Vol. 71, (2018), 505-525. 
30. Xu, X., He, P., Xu, H. and Zhang, Q., "Supply chain coordination
with green technology under cap-and-trade regulation",
International Journal of Production Economics,  Vol. 183,
(2017), 433-442. 
31. Toptal, A., Özlü, H. and Konur, D., "Joint decisions on inventory
replenishment and emission reduction investment under different
emission regulations", International Journal of Production
Research,  Vol. 52, No. 1, (2014), 243-269. 
32. Fathollahi-Fard, A.M., Hajiaghaei-Keshteli, M. and TavakkoliMoghaddam,
R., "A bi-objective green home health care routing problem",
Journal of Cleaner Production,  Vol. 200, (2018),
33. Samadi, A., Mehranfar, N., Fathollahi Fard, A. and HajiaghaeiKeshteli,
M., "Heuristic-based metaheuristics to address a
sustainable supply chain network design problem", Journal of
Industrial and Production Engineering,  Vol. 35, No. 2, (2018),
34. Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P., "Optimization by
simulated annealing", Science,  Vol. 220, No. 4598, (1983), 671680.
35. Mirjalili, S. and Lewis, A., "The whale optimization algorithm",
Advances in Engineering Software,  Vol. 95, (2016), 51-67.