Medical drug shortages can have significant effects not only on patient
health but also on a government. Demand uncertainty, distribution, and inventory
policy are regarded as the main factors causing such shortages, and to avoid them
medical institutes tend to order or store more medical drugs than they need,
resulting in medical wastes that need to be properly addressed in order to mitigate
any financial burden and impact on the environment.
The inventory routing problem (IRP) appears to be a good approach to
model and solve the problem. This study thus extends IRP to propose a multiobjective
two-stage model that considers demand uncertainty and environmental
issues concerning the distribution of medical drugs. The first objective focuses on
the basic costs of inventory and distribution. Since medical wastes affect the
environment and should be recycled, we also include the concept of reverse
logistics. The second objective minimizes vehicle emissions resulting from
distribution and recycling processes. To deal with uncertainty, we apply two
different approaches based on the demand pattern probability to confirm the
robustness of the proposed model. To solve the model, the study employs the Latin
hypercube sequence to conducted the scenarios and develops a metaheuristic based
on the simulated annealing algorithm.
Finally, we report the results of numerical studies, including the
performance of the proposed models and the effect on the economy and
environment. The use of the Latin Hypercube and Annealing Simulation algorithm
is capable of solving the proposed model of IRP and can provide competitive results
compared to other algorithms. Furthermore, the model can show the resulting
emissions based on the process of distribution and waste management.

Alinaghian, M., Zamani, M., 2019. A bi-objective fleet size and mix green inventory routing problem, model and solution method. Soft Comput. 23, 1375–1391.
Alvarez, A., Munari, P., Morabito, R., 2018. Iterated local search and simulated annealing algorithms for the inventory routing problem. Int. Trans. Oper. Res. 25, 1785–1809.
Andersson, H., Hoff, A., Christiansen, M., Hasle, G., Løkketangen, A., 2010. Industrial aspects and literature survey: Combined inventory management and routing. Comput. Oper. Res. 37, 1515–1536.
Apsaliamova, S.O., Alekseenko, S.N., Khashir, B.O., Khuazhev, O.Z., Drozdov, A.N., 2019. Medical waste management: Technologies ANS innovations. Int. J. Eng. Adv. Technol. 9, 5546–5551.
Azadeh, A., Elahi, S., Farahani, M.H., Nasirian, B., 2017. A genetic algorithm-Taguchi based approach to inventory routing problem of a single perishable product with transshipment. Comput. Ind. Eng. 104, 124–133.
Beck, M., Buckley, J., O’Reilly, S., 2019. Managing pharmaceutical shortages: an overview and classification of policy responses in Europe and the USA. Int. Rev. Adm. Sci.
Bell, W.J., Dalberto, L.M., Fisher, M.L., Greenfield, A.J., Jaikumar, R., Kedia, P., Mack, R.G., Prutzman, P.J., 1983. Improving the Distribution of Industrial Gases With an on-Line Computerized Routing and Scheduling Optimizer. Interfaces (Providence, Rhode Island) 13, 4–23.
Bertazzi, L., Coelho, L.C., De Maio, A., Laganà, D., 2019. A matheuristic algorithm for the multi-depot inventory routing problem. Transp. Res. Part E Logist. Transp. Rev. 122, 524–544.
Černý, V., 1985. Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. J. Optim. Theory Appl. 45, 41–51.
Coelho, L.C., Cordeau, J.-F., Laporte, G., 2013. Thirty Years of Inventory Routing. Transp. Sci. 48, 1–19.
Crama, Y., Rezaei, M., Savelsbergh, M., Woensel, T. Van, 2018. Stochastic Inventory Routing for Perishable Products. Transp. Sci. 52, 526–546.
Eglese, R.W., 1990. Simulated annealing: A tool for operational research. Eur. J. Oper. Res. 46, 271–281.
Eskandari-Khanghahi, M., Tavakkoli-Moghaddam, R., Taleizadeh, A.A., Amin, S.H., 2018. Designing and optimizing a sustainable supply chain network for a blood platelet bank under uncertainty. Eng. Appl. Artif. Intell. 71, 236–250.
Freimer, M.B., Linderoth, J.T., Thomas, D.J., 2012. The impact of sampling methods on bias and variance in stochastic linear programs. Comput. Optim. Appl. 51, 51–75.
Gallien, J., Rashkova, I., Atun, R., Yadav, P., 2016. National Drug Stockout Risks and the Global Fund Disbursement Process for Procurement. POMS 12662.
Gils, T., Bossard, C., Verdonck, K., Owiti, P., Casteels, I., Mashako, M., Van Cutsem, G., Ellman, T., 2018. Stockouts of HIV commodities in public health facilities in Kinshasa: Barriers to end HIV. PLoS One 13, 1–12.
Jafarkhan, F., Yaghoubi, S., 2018. An efficient solution method for the flexible and robust inventory-routing of red blood cells. Comput. Ind. Eng. 117, 191–206.
Jia, T., Li, X., Wang, N., Li, R., 2014. Integrated Inventory Routing Problem with Quality Time Windows and Loading Cost for Deteriorating Items under Discrete Time. Math. Probl. Eng. 2014, 1–14.
Jovanović, V., Manojlović, J., Jovanović, D., Matic, B., Đonović, N., 2016. Management of pharmaceutical waste in hospitals in Serbia – Challenges and the potential for improvement. Indian J. Pharm. Educ. Res. 50, 695–702.
Kazemi, S.M., Rabbani, M., Tavakkoli-Moghaddam, R., Shahreza, F.A., 2017. Blood inventory-routing problem under uncertainty. J. Intell. Fuzzy Syst. 32, 467–481.
Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1987. Optimization by Simulated Annealing, Readings in Computer Vision. Morgan Kaufmann Publishers, Inc.
Kleywegt, A.J., Nori, V.S., Savelsbergh, M.W.P., 2003. The Stochastic Inventory Routing Problem with Direct Deliveries. Transp. Sci. 36, 94–118.
Li, M., Wang, Z., Chan, F.T.S., 2016. A robust inventory routing policy under inventory inaccuracy and replenishment lead-time. Transp. Res. Part E Logist. Transp. Rev. 91, 290–305.
Li, Y., Guo, H., Wang, L., Fu, J., 2013. A hybrid genetic-simulated annealing algorithm for the location-inventory- routing problem considering returns under E-supply chain environment. Sci. World J. 2013.
Marczak, H., 2016. Logistics of waste management in healthcare institutions. J. Ecol. Eng. 17, 113–118.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E., 1953. Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092.
Micheli, G.J.L., Mantella, F., 2018. Modelling an environmentally-extended inventory routing problem with demand uncertainty and a heterogeneous fleet under carbon control policies. Int. J. Prod. Econ. 204, 316–327.
Niakan, F., Rahimi, M., 2015. A multi-objective healthcare inventory routing problem; a fuzzy possibilistic approach. Transp. Res. Part E Logist. Transp. Rev. 80, 74–94.
Nikzad, E., Bashiri, M., Oliveira, F., 2019. Two-stage stochastic programming approach for the medical drug inventory routing problem under uncertainty. Comput. Ind. Eng. 128, 358–370.
Nonzee, N.J., Luu, T.H., 2019. Cancer Policy: Pharmaceutical Safety, Cancer Treatment and Research. Springer International Publishing.
Olsson, A.M.J., Sandberg, G.E., 2002. Latin Hypercube Sampling for Stochastic Finite Element Analysis. J. Eng. Mech. 128, 121–125.
Pasquet, A., Messou, E., Gabillard, D., Minga, A., Depoulosky, A., Deuffic-Burban, S., Losina, E., Freedberg, K.A., Danel, C., Anglaret, X., Yazdanpanah, Y., 2010. Impact of drug stock-outs on death and retention to care among HIV-infected patients on combination antiretroviral therapy in Abidjan, Côte d’ivoire. PLoS One 5.
Pholdee, N., Bureerat, S., 2015. An efficient optimum Latin hypercube sampling technique based on sequencing optimisation using simulated annealing. Int. J. Syst. Sci. 46, 1780–1789.
Rahbari, M., Naderi, B., Mohammadi, M., 2018. Modelling and Solving the Inventory Routing Problem with CO2 Emissions Consideration and Transshipment Option. Environ. Process. 5, 649–665.
Rahimi, M., Baboli, A., Rekik, Y., 2016. Sustainable Inventory Routing Problem for Perishable Products by Considering Reverse Logistic. IFAC-PapersOnLine 49, 949–954.
Rahimi, M., Baboli, A., Rekik, Y., 2017. Multi-objective inventory routing problem: A stochastic model to consider profit, service level and green criteria. Transp. Res. Part E Logist. Transp. Rev. 101, 59–83.
Schmidt, R., Voigt, M., Mailach, R., 2019. Uncertainty Management for Robust Industrial Design in Aeronautics. Springer International Publishing.
Shaabani, H., Kamalabadi, I.N., 2016. An efficient population-based simulated annealing algorithm for the multi-product multi-retailer perishable inventory routing problem. Comput. Ind. Eng. 99, 189–201.
Soysal, M., 2015. Decision Support Modeling for Sustainable Food Logistics Management.
Soysal, M., 2016. Closed-loop Inventory Routing Problem for returnable transport items. Transp. Res. Part D Transp. Environ. 48, 31–45.
Soysal, M., Çimen, M., Belbağ, S., Toğrul, E., 2019. A review on sustainable inventory routing. Comput. Ind. Eng. 132, 395–411.
Sun, Q., Chien, S., Hu, D.-W., Ma, B.-S., 2018. Optimizing the Location-Inventory-Routing Problem for Perishable Products Considering Food Waste and Fuel Consumption. CICTP 2018 1545–1560.
Tarpani, R.R.Z., Alfonsín, C., Hospido, A., Azapagic, A., 2020. Life cycle environmental impacts of sewage sludge treatment methods for resource recovery considering ecotoxicity of heavy metals and pharmaceutical and personal care products. J. Environ. Manage. 260, 109643.
Teodor Gabriel Crainic, G.L., 1998. Fleet Management and Logistics, Fleet Management and Logistics. Kluwer Academic Publishers.
Videau, M., Lebel, D., Bussières, J.F., 2019. Drug shortages in Canada: Data for 2016–2017 and perspectives on the problem. Ann. Pharm. Fr. 77, 205–211.
Wang, X.P., Wang, M., Ruan, J.H., Li, Y., 2018. Multi-objective optimization for delivering perishable products with mixed time windows. Adv. Prod. Eng. Manag. 13, 321–332.
Yadollahi, E., Aghezzaf, E.H., Walraevens, J., Raa, B., Claeys, D., 2019. Evaluating approximate solution models for the stochastic periodic inventory routing problem. J. Manuf. Syst. 50, 25–35.
Yu, Y., Chu, C., Chen, H., Chu, F., 2012. Large scale stochastic inventory routing problems with split delivery and service level constraints. Ann. Oper. Res. 197, 135–158.
Zhao, Y., Guo, Z., Niu, F., Yu, Y., Wang, S., 2019. Global sensitivity analysis of passive safety systems of FHR by using meta-modeling and sampling methods. Prog. Nucl. Energy 115, 30–41.