Designing a new sustainable Test Kit supply chain network utilizing Internet of Things

3.1. Problem statement

In this study, a sustainable SC of coronavirus test kits network (STKSCN) is designed. An MO, multi-period, and multi-resource mathematical model is formulated to implement this network. The proposed network has four levels, suppliers, Distribution Centers (DCs), hospitals and WM centers, which are interconnected respectively. To better understand the network, Fig. 1 is depicted. According to the health of patients is a priority, and we may see a sudden increase in hospital referrals and the need for more test kits, and also the lack of a large number of kits in the country, in the network provided, both internal manufacturer of coronavirus test kits and external supplier of these kits are used. There are two types of test kits in this network: PCR and Anti-body tests. Patients with suspected COVID symptoms should have a PCR test, and patients who have been through the disease and want to make sure they have antibodies in their body want to have Anti-body test. The kits are first transported to DCs that located and constructed by the government. After packing, they are sent to hospitals according to the declared demand. After use, the test kits are immediately placed by the operator in a special box (safety box) that is placed in each hospital room and are collected at the end of the day and sent directly to the waste disposal unit to be incinerated and disinfected by the incinerator. Of course, not all hospitals have incinerators. Due to the weakness of our country’s health system and the critical situation in this chain, we considered that hospitals that do not have incinerators, take their medical waste to appropriate locations for landfilling to be buried there following health protocols.

An IoT-based application that can be installed on a mobile phone or computer is designed. Data is collected by this application and after analysis, decision and allocation, the necessary information is sent to hospitals and patients. The data gathering process is shown in Fig. 2. The way this application works is that the patient has to answer a few questions after entering the application and registering their first and last name and address. Questions about their clinical symptoms or previous tests. For the PCR test, if at least 2 out of 5 questions are answered in the affirmative, the patient is allowed to take the test, and the nearest hospital is assigned to their based on their geographical location. The hospital address is also sent to the patient in the same application. For the Anti-body test, if the patient has two necessary conditions, they are allowed to take this test, and the same steps are performed for them. The steps that the patient must go through in the application are shown in Fig. 3, Fig. 4.

Cost management in the health logistics network is essential during the pandemic condition. On the other hand, the amount of plastic waste and infectious waste has increased significantly. In the long run, the increase of plastic waste will cause irreparable damage to the environment, and improper management of infectious waste will be dangerous for public health. The purposes of the proposed model in this research are to minimize network costs, minimize negative environmental impacts and increase the level of social indicators so that it includes all three aspects of sustainability. The assumptions of current problem are explained as follow:

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  There are two types of test kits. The first category is PCR test kits that indicate the presence or absence of the virus, and the second category is Antibody test kits that are used to assess the patient’s immune system.

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  The DCs and landfills must be located.

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  The hospitals, manufacturers, and foreign suppliers are pre-determined.

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  Some hospitals are not equipped with incinerators. These hospitals transfer their waste to the landfill at the end of each period.

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  DCs and landfills have a fixed establishment cost.

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  The DCs, manufacturers, and foreign suppliers have limited capacity.

3.2. Notations

The notations, parameters, and decision variables are stated.

3.3. Mathematical modeling

A MO, multi-period, multi-resource, distribution-location–allocation-inventory-WM model is presented in subsection aims to minimizing total network costs, negative environmental effects and negative social effects.

The objective function (1) seeks to minimize total costs of network. Respectively, two first equations are Purchasing cost, transportation cost between manufacturer and DCs, fixed establishing cost of DC, transportation cost between suppliers and DCs, transportation cost between DCs and hospitals, incinerator cost, establishing cost of landfill centers, transportation cost between hospitals and landfill centers, landfilling cost, and two last equations consider inventory holding cost and penalty shortage cost.

This objective function (2) aims to reduce adverse environmental impact on SC. The parentheses contain amount of carbon emission during transportation from manufacturer to DC, supplier to DC, DC to hospital, and hospital to landfill center. The second equation calculate amount of carbon emission in burning Used kits with incinerator.

This Eq. (3) purpose to minimize the adverse social index with considering population density rate in routes between manufacturer to DC, supplier to DC, DC to hospital, and hospital to landfill center.

Constraints (4)–(6) ensure that DC, manufacturer and supplier cannot receive, produce and supply kits exceed of their limited capacity at each period.

Constraint (7) is demand constraint, and refer to amount of shortage in each hospital per period.

Constraints (8)–(11) ensure that kits are transported to or from DC, and transported to landfill center if they are opened.

Constraints (12)–(15) calculate the number of vehicles used between manufacturers to DCs, suppliers to DCs, DCs to hospitals and, hospitals to landfill centers at each period.

This Eq. (16) indicates balancing for inventory level of each DC during each period.

Constraint (17) Specifies an upper limit for amount of kit type $i$ transported to DC $r$ at period $t$.

Constraints (18), (19) determine number of used kits that must be burn and number of used kits that must be transported to landfill, respectively.

Constraints (20)–(23) ensure that, at most, each DC is allocated to one manufacturer, each DC is allocated to one supplier, each landfill center is allocated to one hospital, and each hospital is allocated to one DC.

Constraints (24), (25) define domain of decision variables.

4.1. Augmented $\mathit{ϵ}$-constraint2 (AUGMECON2)

AUGMECON2 improves the AUGMECON which uses slack variables in each iteration. Additional iterations are eliminated and calculation times are reduced (Mavrotas and Florios, 2013). The formulation of this method according to our presented model is defined as follows:

where $e_2,e_3$ are parameters for RHS that are plotted for specific iterations of the network points of Objective 2 and 3. The parameters $r_2$ and $r_3$ are the domains of the objective functions. $\left\{S_2,S_3\right\}$ are the slack variables of the constraints and eps $\in$ $[10^{-6},10^{-3}]$.

4.2. Non-dominated sorting genetic algorithm (NSGA-II)

NSGA-II is one of the most successful and widely used algorithms introduced by (Deb et al., 2002). The advantages of this algorithm include less computational complexity and compatibility with constraints. The NSGA-II process consists of three main phases: generating the population, computing crowding distance for members, and selecting the solution which satisfies all objective functions. Crowding distance is a factor for better selection of solutions in terms of dispersion. More information about this method can be in some other studies (Taleizadeh et al., 2008, Taleizadeh et al., 2011, Farrokhi-Asl et al., 2020, Ghasemi et al., 2019, Ghasemi et al., 2020, Zhang et al., 2017). The main steps and the pseudo code of the NSGA-II are presented in Fig. 5, Fig. 6.

4.3. Pareto envelope-based selection algorithm (PESA-II)

PESA-II was introduced by (Corne et al., 2001). This algorithm is an evolutionary optimization algorithm that solves MO problems based on Pareto selection and according to the concept of genetic algorithm. This algorithm actually improves the PESA algorithm. Region-based selection is considered an individual selection feature. Selective fitness is to be assigned to the region of the objective space. This selection enhances the Pareto frontier. For more information about PESA-II, scholars can read these studies (Arjmand et al., 2020, Omidi Brojeni et al., 2021). The main steps and the pseudo code of the PESA-II are depicted in Fig. 7, Fig. 8.

5.1. Numerical example

The price of foreign and domestic kits is estimated according to the information announced by the Ministry of Health. Distance between manufacturer and supplier to DC and, DC to hospital and, hospital to landfill center are generated randomly by uniform distribution. Other parameters are estimated based on Expert opinion. Several numerical tests problem in small, medium, and large scale are presented in Table 3. The examples contain the number of test kits, manufacturer, supplier, DCs, landfills, hospitals, and periods. The range of parameters is shown in Table 4.

The results of solving the model by exact methods are shown in Fig. 9 and Table 5. Small and medium size of problem are solved by AUGMECON2.

5.2. Performance metrics

For measure the quality of the solution obtained by proposed algorithms, several performance metrics are presented. Convergence, diversity and number of solutions are aspects that metrics considered (Riquelme et al., 2015). In order to take these features into account and compare meta-heuristic algorithms, a number of performance metrics namely SM, CM, SNS, IGD and MID are purposed.

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  MID (Goodarzian et al., 2021b)

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  MS (Farrokhi-Asl et al., 2020)

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  IGD (Zhang et al., 2009)

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  SNS (Goodarzian et al., 2020)

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  NPS (Zahiri et al., 2018, Govindan et al., 2015)

5.3. Parameter tuning

Adjusting the parameters can raise the performance of the proposed methods, and makes the conditions for comparing them fairer. In this study, Taguchi method is utilized to tuning parameters. Taguchi was introduced by Taguchi (1986). Taguchi method is different from common engineering methods. Taguchi emphasizes on quality control during product and process design, while other common methods are based on inspection during the production process or after product production. Taguchi uses common statistical tools in his quality improvement methods and has simplified this method by identifying a set of powerful solutions in designing experiments and analyzing the results. Parameters are divided into controllable and uncontrollable factor. Taguchi design uses orthogonal arrays. Two different orthogonal designs are used for these two sets of parameters. Internal array for controllable variables and outer array for intrusive variables. The combination of two internal and external arrays provides a crossed array that provides information about the interactions between controllable and intrusive variables. By facilitating the experimental design process, orthogonal arrays make it possible to examine main and interaction effects by performing the least number of experiments in a reasonable amount of time. The factors and levels of algorithms are shown in Table 6.

In this approach, after adjusting the parameters of the algorithms by using a number of orthogonal arrays, the total number of experiments is reduced (Goodarzian et al., 2021a). The Taguchi method is used for the NSGA-II and PESA-II, the Orthogonal Array L9 and L27 with three levels, which specified in Table 7, Table 8 respectively. To attain the optimum level between the proposed levels for the proposed methods, Relative Percent Deviation (RPD) and S/N indices have been calculated.

In Taguchi method, a loss function ($L_i\text{)}$ is used to calculate the changes between the results and the desired value. The formulation of the S/N, is shown in Eq. (27).

Since the model’s objective functions are aimed to minimized, the formula for the loss function is calculated as follows in Eq. (28) and S/N ratio is provided in Eq. (29).

$n$ indicates the orthogonal array and $y_i$ illustrates the value of the solution for the orthogonal array. Fig. 10, Fig. 11 show the S/N ratio for the proposed algorithms, respectively.

5.4. Comparison of the proposed methods

In this section, the solution of meta-heuristic algorithms is compared together by their CPU time and purposed assessment metrics. The objective function values in each algorithm are provided in Table 9. Also, the behavior of algorithms is compared by their CPU time in Fig. 12, and based on figure, it is obvious that PESA-II is swifter than NSGA-II for each test problem. As mentioned earlier, five performance metrics were proposed for better and fairer comparison of algorithms. The results are provided in Table 10, Table 11. In Fig. 13, Fig. 14, some examples are run and the obtained Pareto front of each algorithm is shown. Furthermore, according to the Pareto optimal analyses, a set of statistical comparison is presented to find the best method. Relative Deviation Index (RDI) that is common metric is calculated.

The formulation of RDI is as follow (Mehdizadeh et al., 2015):

$Al{g}_{sol}$ shows the value obtained with a specific measurement scale of the method. $Bes{t}_{sol}$ indicates the best solution between all algorithm, and ${\mathrm{Max}}_{sol}\mathrm{and}{\mathrm{Min}}_{sol}$ are maximum and minimum values between all resulted. The results are indicated in Fig. 15. The lower value of RDI is better. So PESA-II is More efficient.

5.5. Comparison of results

In this article, to compare and evaluate the presented meta-heuristic algorithms, five different assessment metrics named MID, IGD, SNS, NPS, and MS are presented. According to the results of Table 10, Table 11, we come to the conclusion that the PESA-II algorithm is more efficient and better than the NSGA-II algorithm. In order, the (higher the MS performance metrics, higher the NPS, higher the SNS, lower the IGD, and lower the MID), the more efficient the algorithm is. Also, based on Fig. 12, the amount of time needed to solve the PESA-II algorithm is less than NSGA-II, as a result, this algorithm is faster and smoother. Also, for additional analysis, we proposed the RDI index, which is calculated according to Eq. (30), and its results are shown in Fig. 15. It is worth pointing out that the lower RDI value indicates better quality and efficiency. Therefore PESA-II is more efficient.

5.6. Case study

In order to validate the proposed model and come conditions closer to real world, a case study in Tehran is investigated. Tehran is the capital of Iran. Fig. 16 indicates its location. With the outbreak of the corona pandemic, the distribution and allocation of COVID test kits has become big challenges for decision makers due to the shortage of test kits and the sudden increase in cases suspected of having COVID19. Eight hospitals, six DCs, four landfills, four suppliers, and three manufacturers are considered in case study. Their locations are shown in Fig. 17. Other information is provided in Table 12, Table 13, Table 14. Also, two types of test kits, and fifteen units of time period are determined. The dataset of case study is obtained from Application of IoT, google map, ministry of health and treatment, and Municipality of Tehran city. The amount of distance between network components are calculated by google map. The amount of demand for each hospital in each period is obtained from IoT application, the ministry of health and municipality of Tehran provided other data.

5.7. Case study results

The outcomes of case study are investigated by PESA-II algorithm. Table 15, Table 16, Table 17, Table 18, Table 19, Table 20 shows case study solutions. It should be noted that Nikan, Milad and Erfan hospitals are equipped with incinerators and do not need to transfer their infectious waste to landfills.

As you can see in Table 15, Saadat Abad, Nobonyad, Piroozi and Rah Ahan are the locations that obtained for DC, but Haft-e-Tir and Jomhoury are rejected locations. Also, according to Table 16 Ariashahr, Khaksefid and Darband are selected for landfill locations, whereas Yaftabad is rejected. Table 17 indicates the amount of case study solution for each objective function. Table 18, Table 19, Table 20 shows the allocation relationship between manufacturers and DCs, suppliers and DCs, hospitals and DCs, and hospitals and landfills. For example, the manufacturer that stated in Sedeghie location is allocated to Saadat Abad DC, or Kasra hospital is allocated to Piroozi DC.

5.8. Sensitivity analysis

A set of sensitivity analysis is performed to evaluate and identify purposed model. The proposed case study (two kinds of COVID test kits, three manufacturers, four suppliers, six DCs, four landfills, eight hospitals and fifteen periods of time) is selected for this section. Objective functions, several important parameters like demand ($D_{iHt}$), capacity of vehicle (CC) and Capacity of DCs (CI) are considered to sensitivity analysis. The variations of objective functions relative to parameters are measured relative. In Fig. 18, 3D plot of three objective function is shown. For instance, the behavior of the objective functions and other parameters are measured in variation of a $\pm \text{\%}30$ fluctuation in demand. Fig. 19, Fig. 20 illustrate the results of analysis.

According to Fig. 19, As the capacity of vehicles increases, the first objective function decreases because the number of vehicles used in a period decreases dramatically and the cost of transportation decreases dramatically, which ultimately reduces the final cost. For example, when vehicle capacity increase 10%, total cost decreases 3653. Objective functions 2 and 3 are also reduced because environmental pollution and social harms are reduced by reducing vehicle traffic. Then, increasing the capacity of vehicles can have better results.

As you can see in Fig. 20, as the capacity of DCs increases, the total shortfall for a hospital decreases dramatically. Here Nikan hospital has been selected as a sample. When the capacity of DC increases 20%, the total shortage of Nikan hospital decreases to 45.

5.9. Managerial insight

As mentioned in the literature section, there is a lack of mathematical modeling in research investigating COVID 19, and most of them are conceptual. In this study, a MILP model is presented with three objective functions, and several mathematical constraints. Also, it solves by exact and meta-heuristic algorithms, and the results are compared. Healthcare decision makers can utilize the methods used in this paper to handle their plans for pandemic and reduce its harmful effects.

Nowadays, cities are going to smartness in every aspect. Internet of things (IoT) is one of the smartness elements that play a key role in decision makers’ plans. In recent years, IoT has made a major contribution in medical progression in developed countries. Fig. 21 indicates influences that IoT can be in healthcare management, but still in developing countries has not remarkable role in this aspect.

Using a platform based on the IoT for a health supply chain during a pandemic primarily increases the speed of the process, which is the most important point in crisis control and management. In addition, due to the reduction of direct human intervention and human errors, it also improves the accuracy and security of the network. It reduces people’s visit to get their turn and crowding in this condition, and as a result, costs and environmental damages are also reduced. Therefore, it can be concluded that in all aspects, the use of such a network can be effective and profitable for supply chain managers. This research which has also been investigated in Iran can help decision makers to develop and use IoT systems to prioritize Suspicious patients to take the COVID test. IoT leads to appropriate allocation, location and distribution in healthcare systems. Also, in these fatal situations where the number of test kits is low in developing countries, more critical patients will have access to the kit sooner with proper planning and prioritization.

Also, if the capacity of vehicles that carry test kits or other medical products related to the pandemic is increased, it can reduce the total costs. Due to the decrease in the number of required cars, the amount of emission of polluting gases is also reduced and the environmental indicators are improved, of course, on the condition that managers pay attention to the emission rate of polluting gases of the cars that are going to be replaced.