Mathematical modeling for optimizing the blood supply chain network

Dehaghani, Amir Rahimzadeh; Nawaz, Muhammad; Sultanie, Rohullah; Quartey- Papafio, Tawiah Kwatekwei Article Mathematical modeling for optimizing the blood supply chain network Modern Supply Chain Research and Applications Provided in Cooperation with: Emerald Publishing Limited Suggested Citation: Dehaghani, Amir Rahimzadeh; Nawaz, Muhammad; Sultanie, Rohullah; Quartey- Papafio, Tawiah Kwatekwei (2021) : Mathematical modeling for optimizing the blood supply chain network, Modern Supply Chain Research and Applications, ISSN 2631-3871, Emerald, Bingley, Vol. 3, Iss. 3, pp. 174-190, https://doi.org/10.1108/MSCRA-09-2020-0024 This Version is available at: https://hdl.handle.net/10419/314888 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Mathematical modeling for optimizing the blood supply chain network Amir Rahimzadeh Dehaghani Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran Muhammad Nawaz Department of Business Administration, National College of Business Administration and Economics, Lahore, Pakistan Rohullah Sultanie Department of Construction and Real Estate, School of Civil Engineering, Southeast University, Nanjing, China, and Tawiah Kwatekwei Quartey-Papafio College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, China Abstract Purpose –This research studies a location-allocation problem considering the m/m/m/k queue model in the blood supply chain network. This supply chain includes three levels of suppliers or donors, main blood centers (laboratories for separation, storage and distribution centers) and demand centers (hospitals and private clinics). Moreover, the proposed model is a multi-objective model including minimizing the total cost of the blood supply chain (the cost of unmet demand and inventory spoilage, the cost of transport between collection centers and the main centers of blood), minimizing the waiting time of donors in blood donating mobile centers, and minimizing the establishment of mobile centers in potential places. Design/methodology/approach –Since the problem is multi-objective and NP-Hard, the heuristic algorithm NSGA-II is proposed for Pareto solutions and then the estimation of the parameters of the algorithm is described using the design of experiments. According to the review of the previous research, there are a few pieces of research in the blood supply chain in the field of design queue models and there were few works that tried to use these concepts for designing the blood supply chain networks. Also, in former research, the uncertainty in the number of donors, and also the importance of blood donors has not been considered. Findings –A novel mathematical model guided by the theory of linear programming has been proposed that can help health-care administrators in optimizing the blood supply chain networks. Originality/value –By building upon solid literature and theory, the current study proposes a novel model for improving the supply chain of blood. Keywords Blood supply chain, Location-allocation problem, Blood products, NSGA-II, Queuing theory Paper type Research paper MSCRA 3,3 174 © Amir Rahimzadeh Dehaghani, Muhammad Nawaz, Rohullah Sultanie and Tawiah Kwatekwei Quartey-Papafio. Published in Modern Supply Chain Research and Applications. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/ legalcode Funding: The study did not receive any funding from any agency. Conflict of Interest: The authors declare that they have no conflict of interest. The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/2631-3871.htm Received 17 September 2020 Revised 4 March 2021 3 May 2021 Accepted 4 May 2021 Modern Supply Chain Research and Applications Vol. 3 No. 3, 2021 pp. 174-190 Emerald Publishing Limited 2631-3871 DOI 10.1108/MSCRA-09-2020-0024 1. Introduction The management of blood product consumption considers as one of the most complex issues in health-care systems due to some issues such as limitation of blood resources, perishability, special preservation conditions of blood products, and high wastage and shortage costs of blood products (Ramezanian and Behboodi, 2017). Blood supply chain management is one of the major challenges of health systems. Human blood is a scarce resource that is produced only by humans themselves, and there is currently no other chemical product or process that can be used as a substitute and there is currently no other chemical product or process that can be used as a substitute Zahiri et al. (2014). Many patients need blood transfusions every day for various reasons (Ramezanian and Behboodi, 2017). Some people, such as dialysis patients and long-term patients, need blood products to survive permanently, and some other patients need blood products due to surgical resection. In cases such as open surgery or neonatal surgery, according to some medical requirements, only fresh blood products should be used (Ghasemi, 2019). The amount of blood products in hospitals is a function of the number of daily accidents and this amount follows an uncertain trend (Hosseini-Motlagh et al., 2020). For this reason, hospitals prefer to order more blood products to make sure the blood required is not low. However, excessive ordering of blood products by hospitals is not possible due to some factors and limitations in the blood supply chain (Zahiri et al., 2015). The first and most important factor is the limited amount of blood in the regional blood center. The regional blood center is responsible for the distribution of blood between different hospitals and devotes the limited amount of blood available (Hosseini-Motlagh et al., 2020;Ghasemi et al., 2019;Zahiri et al., 2014,2015;Alfonso et al., 2012). Another reason is that blood is a scarce and perishable product, and storing large quantities of it can lead to spoilage. One of the most important limitations for discussing the issue of location-allocation in the blood supply chain is the issue of time and price (Ghasemi et al., 2019;Zahiri et al., 2014). Many previous studies have been devoted to the blood supply chain. In former research, the uncertainty in the number of donors, and also the importance of blood donors has not been considered. For example, Ramezanian and Behboodi (2017) designed a blood supply chain network under uncertainties in supply and demand considering social aspects. But the number of donors under uncertainty does not consider. Alfonso et al. (2012) studied the processes of collecting blood concerning establishing the cost of temporary and permanent blood facilities in France, but at this research, the number of donors and also the importance of blood donors has not been considered under uncertainty. Zahiri et al. (2014) presented a robust possibilistic programming approach to multi-period location-allocation of organ transplant centers under uncertainty. Ghasemi (2019) presented a model for Location Allocation Problem After Disaster Blood Supply Chain. But in this research, the uncertainty in the number of donors has not been considered. In the current study, a new model for minimizing time and cost with the queue theory considering the blood donor’s roles has been applied. It aims to minimize the total cost of the blood supply chain and the waiting time for donors in mobile centers using a three-level nonlinear mathematical model. In the present study and at all levels, blood products have a specific life cycle and bypassing their time, they will be considered as waste products. While recognizing the fact that the importance of the donor position in the blood supply chain is missed in previous works, the current study fills the gap. The rest of the study is organized as follows. In section 2, we briefly summarized the previous studies in the field of placement-assignment and blood supply chain and we pointed to the lack of these studies. In section 3 with considering queuing model theory, we minimized the waiting time of donors in the mobile blood centers and the time of donating. Also in section 3 by proposing a nonlinear and multi-objective model, we reduced the cost of wastes, transportation, and lack of products, and then with a realistic example, we solved this mode Optimizing the blood supply chain network 175 using the NSGA-II heuristic algorithm. In section 4, we analyzed the solutions obtained in section 3 and we also described the methods to estimate the parameters of the algorithm. Finally considering the research problem, research process and obtained results, we proposed a solution for future works. 2. Literature review In this research, we focus on two different topics including the blood supply chain and the problem of placement-assignment. This section explains related works that have been done for both focuses. 2.1 Blood supply chain Fazli-Khalaf et al. (2019) discussed the problem of controlling available blood and optimized the available amount by proposing a model. S ¸ahinyazan et al. (2015) used a spreadsheet for calculating data envelopment analysis (DEA); their conclusion was among the scale of the studied blood centers, expanding the level of operation more than one point causes the reduction in efficiency concerning the scale. Alfonso et al. (2014) established the benchmarking goals in hospitals for blood red cells. Their logical regression results indicated various pair groups. Between these groups, benchmarking could be done. They also used bubble sheets for indicating the pairs. Williams et al. (2020) used a simulation model using FORTRAN to simulate the daily deliveries of the simple blood bank. They used this simulation as proof for their theory model. Surveys illustrate that the researchers do not tend to design the blood supply chain using queue models, and few works tried to use these concepts for designing the blood supply chain networks. Hosseini-Motlagh et al. (2016) combined numerical analysis with Markov chains. They developed a model in which it was able to convert blood demand to access possibility and blood usage to efficiency as a function of the blood distribution policy of the local blood center and blood storage policy of the hospital blood bank. 2.2 Placement-assignment Osorio et al. (2015) used integer linear programming to model the blood product shipments to Australian hospitals using the effective cost method. They studied changes from customerbased to seller-based to see which model is better. They also compared their method with an alternative solution based on neighborhood search. As the result, they realized they could reduce travel costs considerably by having a more complex delivery policy by 30 percent. Heidari-Fathian and Pasandideh (2018) considered a solution for the two objectives model for the distribution of perishable materials in the blood supply chain in their studies. These goals were minimizing the total cost of the system and balancing the distances passed by vehicles. Their model was solved by epsilon limitation and NSGA-II methods and then by some indexes, and the results were evaluated. Finally, Govindan et al. (2016) proposed a simulation of the supply chain of very perishable products. They presented a delivery chain composed of a distribution center and few platelet using centers. Their objective function minimized the maximum possible amount of shortage. They also proposed a new strategy of ordering products based on remained product’s lifetime. Later, comparison was made with strategies proposed by Fortsch and Khapalova (2016) called “EWA”and “Order-Up-To”under centralized and decentralized conditions. The result showed that the rate of wastes in the centralized condition was reduced significantly from 19.6 percent to 4.04 percent. As the related works suggests, the scholars considered different sections of the blood supply chain, but it is obvious that the insignificant amount of studies optimized the complete blood supply chain (Rahmani, 2019;Dutta et al., 2018;Ramezanian and Behboodi, 2017; Chaiwuttisak et al., 2016;Khalili-Damghani et al., 2015;Zahiri et al., 2015). Queuing theory is MSCRA 3,3 176 the mathematical study of the congestion and delays of waiting in line. Queuing theory (or queue theory) examines every component of waiting in line to be served, including the arrival process, service process, number of servers, number of system places and the number of customers –which might be people, data packets, cars, etc. Also, queue theory is was not the attention of researchers, and the researches on queue theory were not related to the blood supply chain and were more related to the stock of blood banks. The importance of ordering and delivering very perishable blood products while considering the minimization of the total cost of the supply chain is an important topic that was not considered a lot by researchers while it is very critical. Blood donation is critical to all transfusion therapy, as it provides the starting product. In the United States and many other economically developed nations, all of the blood is given by volunteer and nonremunerated donors. Donated whole blood is then made into transfusable components, which include, but are not limited to, packed red blood cells (RBCs), platelets and frozen plasma or cryoprecipitate (Zeger et al., 2007). Therefore, in this paper, we focused on the queue theory to minimize the total cost of the chain and we achieve two other goals that are minimizing the waiting time for customers and minimizing the establishment time for mobile blood centers. 3. Modeling In this section, we first describe the problem, its assumptions, limitations, and we propose three objective mathematical models. Then, we describe the three objectives solution method for the NSGA-II algorithm. 3.1 Problem statement Respecting the goals of this research which are minimizing the total cost of the blood supply chain and minimizing the waiting time for donors in mobile centers, a three-level nonlinear mathematical model is presented. In this research and at all levels, blood products have a specific life cycle, and bypassing their time, they will be considered as waste products. It is worth noting that blood demand in this model follows the uniform distribution function, and entering donors to the mobile blood centers follows the exponential distribution function. Also, we took advantage of queue theory for potential places for collecting blood such as reducing the waiting time of donors in blood centers and optimize the time for the establishment of mobile blood centers. In other words, the proposed model is a combination of placement-assignment problem and queue theory problems that aims to achieve three goals include first, minimizing the transportation cost, waste and lack of demand; second, minimizing the establishment of mobile blood centers and third, reducing the waiting time of donors in blood centers. Also considering the input and output strategy of blood products, we decided to use FIFO for queuing model. 3.2 Assumptions The proposed model has been built on the following assumptions: (1) In each period, blood collecting from each potential place is done only by one mobile blood center. (2) Mobile blood centers bring collected blood only to their original main blood center. (3) Transportation cost between main blood centers and mobile blood centers would be considered. (4) Respecting different blood products, the blood demands of each hospital are only assigned to one main blood center. Optimizing the blood supply chain network 177 (5) Main blood centers cover potential places only if they are in their specific coverage distance range. (6) A potential place can be blood collected only once during certain periods. (7) Each red blood cell and platelet would be considered as waste after 25 and 3 days, respectively. (8) Each demand should be responded to in the same period. (9) Initially, stock is 0 for all three levels. (10) Processing one blood unit produces one platelet unit and one red blood cell unit. (11) The cost of wastes and perishing blood products would be considered. (12) Responding to the demands is based on FIFO policy. (13) Main blood centers can transfer blood to each other. (14) Input and service distribution functions follow Poisson and exponential distribution functions, respectively. (15) The collected blood should be sent to the lab before 6 h. (16) Blood collecting centers have a specific capacity. 3.3 Mathematical model of the problem 3.3.1 Indexes. I: potential places for collecting blood J: blood main centers (test and refinement) L: mobile blood centers R: blood products T: planning periods A: products’remained life cycle H: demand centers (hospitals) 3.3.2 Parameters. mi: Service rate to donors by potential center i λi: Donor see rate in potential center i B: maximum establishment time for a mobile blood center at a potential blood center H: maximum capacity of a mobile center h: minimum capacity of a mobile center M: maximum number of servers in mobile centers ttij: transfer time for collected blood from the potential center Ito main center j e: maximum time for keeping the blood before refinement dij: distance matrix between potential center iand main center j MSCRA 3,3 178 COr: cost of each unit of demanded blood product r Cr: the cost of each unit of not fulfilled demand in center j CC: the fixed cost of transferring blood between main blood centers E: possible distance for assigning potential centers to main blood centers N: possible distance for transferring blood between main blood centers SS: possible distance for assigning demand centers to main blood centers tdjh: distance matrix between demand center hand main center j ddj1j2: distance matrix between main centers j 1 and j 2 csij: the cost of transferring blood between potential center iand main center j RLr: maximum allowed time for keeping product r Det hr: demanding amount of product rin period tby hospital h U: a very large number ε :a very small number 3.3.3 Variables. π i: probability of presence of kdonors in the potential center i π 0i: probability of presence of no donors in the potential center i λi: entering rate of donors to mobile blood centers of potential center i Li: average number of customers in potential center i Tt ilj: the establishment time for mobile blood center I of the main center jin the potential center iin the period t mi: number of beds in potential center i ki: capacity of mobile centers in potential center i Int jar: stock of product rin main center jwith remained life cycle ain the period t Ft jarh: the amount of shipped product rwith remained life cycle a from main center jto hospital h Gj1j2: if main center j 1 can send its extra blood to main center j 2 then 1 else 0 Ot jr: waste amount of product rin main center jin period t Xt ilj: if potential center iin period tis assigned to mobile center I from the main center jthen 1 else 0 wij: if main center jcovers the potential center ithen 1 else 0 Qgt raj2j1: shipped amount of blood from blood center j 1 to j 2 Qst raj1j2: shipped amount of blood from main blood center j 1 to j 2 Ylj: if mobile center iis assigned to main center jthen 1 else 0 Zjh: if the demand of hospital his assigned to main center jthen 1 else 0 Optimizing the blood supply chain network 179 Pt jr: not responded demands (lack of blood products) st ijl: the amount of shipped blood from potential center iby mobile center jwhich is assigned to the main center I 3.3.4 Proposed model. In the model, the first function, or Function (1), aims to minimize the cost of not fulfilled demand, cost of perishing existing stock, cost of shipment from collecting centers to main centers and transferring blood between main centers. MINX tX jX r Pt jrCrþX tX jX r CorOt jr þX tX jX lX i csijXt ijl bTt ilj ettijc þX j1X j2 CC ddj1j2Gj1j2 Function (3) aims to minimize the waiting time for donors in mobile centers. MinX tX lX jX i Li λi Xt ijl Function (3) aims to minimize the establishment time of mobile centers in potential centers. MinX tX lX jX i Tt ijl Xt ijl Function (4) calculates the input rate of donors in mobile blood centers. λi¼λið1 π iÞ∀i Function (5) calculates the probability of the presence of kdonors in a mobile blood center. π i¼λi miki mðki−miÞ i ki!!,"X mi−1 n¼0λi miki1 ki!þλi mimi1 mi!X ki n¼miλi mimiki−mi#∀i Function (6) calculates the probability of the presence of no donors in a mobile blood center. π 0i¼1,"X mi−1 n¼0λi min1 n!þλi mimi1 mi!X ki n¼miλi mimin−mi#∀i Function (7) calculates the average of customers in a potential center waiting for service. Lqi¼ π 0i miλi mimiλi mimi 1λi mimi2 *1λi mimiki−miþ1 1λi mimiðkimiþ1Þλi mimiki−mi∀i Function (8) calculates the average of customers in a potential center. Li¼Lqiþmi π 0iX mi−1 n¼0λi minmin n!∀i MSCRA 3,3 180 Function (9) guarantees that the capacity of the mobile center is more than servers. ki≥mi∀i Function (10) guaranteed that the efficiency of a system is maximum. λi mimi ≤1 ε ∀i Function (11) shows the maximum number of servers. mi≤M∀i Function (12) shows the upper bound and lower bound of the capacity of each mobile center. hXt ilj ≤ki≤HXt ilj ∀i;l;j;t Function (13) shows the maximum time that a mobile center can be established in a potential center. Tt ijl ≤BXt ilj ∀i;l;j;t Function (14) shows the input blood to main centers. Tt ijl λiXt ilj ¼st ijl ∀i;l;j;t Function (15) guarantees that each mobile center in each period can be assigned to a maximum of one potential center. X i Xt ilj ≤1∀l;j;t Function (16) guarantees that each area is covered by only one mobile center. X l Xt ilj ≤1∀i;j;t Function (17) guarantees that in each planning period collect blood from each area is done at most once. X t Xt ilj ≤1∀i;j;l Function (18) indicates an area is covered by the main center only if it is in the distance limitation. dijwij ≤E∀i;j Function (19) guarantees collecting blood from each area is done by a mobile center assigned to the main center that covers that area. Xt ilj ≤Ylj ∀i;j;l;t Function (20) indicates blood collection is done by a mobile center only if the mobile center is assigned to the main center that covers that area. Xt ilj ≤wij ∀i;j;l;t Optimizing the blood supply chain network 181 Scenario Average Scenario 1 0.52734 Scenario 2 0.484011 Scenario 3 0.504859 Scenario 4 0.514896 Scenario 5 0.431364 Scenario 6 0.454843 Scenario 7 0.506213 Scenario 8 0.499034 Scenario 9 0.495434 Table 9. RPD values Figure 3. The objective function Figure 4. The S/N diagram Figure 5. The RPD diagram MSCRA 3,3 188 For our example, we choose level 1 from RPD. Other parameters are set based on S/N. Finally, the crossover from level 1, a mutation from level 3, the population from level 3 and iteration from level 3. 5. Conclusion and recommendations Perishable product supply chain problems include various types in the health sector such as blood product issues. Nowadays due to the lack of viable alternatives to blood and blood products, the only blood that is donated by benevolent that can save other lives. The lack of viable alternatives, the limited shelf life and the constant need for blood and blood products from one side and random demand and irregular supply of this product from the donor, and the complexity of matching demand with supply effectively from the other side doubled the importance of this issue. In this research, we present a new model for minimizing time and cost with the queue theory considering blood donor’s roles. We aim to minimize the total cost of the blood supply chain and minimize the waiting time for donors in mobile centers; a three-level nonlinear mathematical model is presented. In this research and at all levels, blood products have a specific life cycle, and bypassing their time, they will be considered as waste products. Indeed, we presented a three-objective model and solved it with the NSGA-II algorithm. This model was based on queue theory and aimed at the placement-assignment of blood products supply chain networks. For modeling the problem, we considered real conditions such as time limitations, waiting times in blood collecting stations, the life cycle of blood, etc. combination of queue theory, and placement-assignment problem was one of the innovations of this work. Finally, we solve this model by NSGA-II and we saw that this algorithm has a good performance and in proper time can answer the problem. Considering the importance of uncertainty and resilience in supply chain management (Mahmoudi et al., 2021), in the future, the issue of uncertainty in the number of donors, as well as the importance of blood donors can also be considered to develop robust and resilient frameworks. The focus of this study is on blood donation and the benefits for health-care sectors. The study help reduces stress in communities and improves physical health, and helps to distance oneself from negative emotions. Also, the model presented in this study can alleviate cases and restrictions related to waiting for people in need of blood, as well as improve planning in the healthcare, technology and health information sectors. Before generalizing the findings of the study across the board, future studies can apply the proposed model in different environments, so its significant strengths and limitations are known. References Alfonso, E., Xie, X., Augusto, V. and Garraud, O. 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Corresponding author Muhammad Nawaz can be contacted at: [email protected]om For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: [email protected] MSCRA 3,3 190