Multi-criteria spherical fuzzy regret based evaluation of healthcare equipment stocks

1Introduction

Epidemic outbreaks are special conditions that may create supply chain interruptions. They create an unusual type of risk in the supply chain systems. Outbreaks cause long-lasting and unpredicted scaling disruption in the supply chains, and the infrastructures between the customers and products are damaged. The significant difference in the epidemic outbreaks is that they have a long term impact both on the production and demand side. Providing the necessary equipment is critical for coping with such outbreaks. Unless an ample amount of healthcare equipment is provided, more people are infected, and controlling the epidemic becomes harder. An efficient method to make optimally utilize the healthcare equipment/tools is to manage the inventory levels in the supply chain by classifying the equipment based on its different attributes.

Face masks, hand sanitary products, and medical gloves are good examples of healthcare products that are easy to produce and provide under normal circumstances. These products are essential for decreasing the contamination rate under outbreaks [10]. Under COVID 19 outbreak, not only the production of the masks but also the supply chain has been heavily disrupted [17]. In many countries finding these products become very hard [10]. Healthcare workers face significant problems for acquiring these products, which are vital for their health. The healthcare organizations realize the importance of defining the inventory levels for these products.

After the outbreak, the usual globalized, multifaceted value chains with the international delivery of millions of products and probable quick formation of new relations and supply chains may change. This outbreak exposure the weakness of just-in-time based lean supply chain systems. In the future, the logistics systems can be improved by increasing reverse logistics and controlling inventories with the new technologies. In order to satisfy the need, supply, and production of healthcare equipment may become more local. Keeping stocks of products that come through long distances with midway storing and depots can become more beneficial [13]. In order to be prepared for the outbreaks while keeping the expenses at the optimal levels, companies should define their level of inventories. Under more defined conditions, the inventory levels can be optimized by using deterministic or stochastic optimization methods. However, the situations such as outbreaks make the decision making process more complicated. Before defining the exact amount of the stock, the stock policy should be determined. In the stock policy, the decision-makers classify the healthcare product inventory levels as low, medium, or high based on different factors. The factors such as the criticality of the product, the easiness of producing the product, and necessary inventory conditions should be considered for defining the inventory levels in the long term. The proposed approach can easily classify the products based on their risk levels, and the appropriate inventory levels can be defined after this classification.

Defining the inventory keeping policy for the long term and selecting the inventory level of healthcare products is a multi-criteria decision-making problem. The severity of keeping stock of the healthcare products can be ranked, and the inventory levels of the products can be defined based on these scores. The imprecision and vagueness of the epidemic outbreaks make this decision hard to evaluate. Whether there will be a severe outbreak, local outbreak, or no outbreaks within three years’ frame depends on many complex factors that cannot be predicted in advance. A decision model should consider these different scenarios and also should be able to consider multiple factors.

Moreover, the decision-makers may have hesitancies while evaluating the attributes of the healthcare equipment. Fuzzy sets are useful tools for dealing with decision making problems [2, 4, 8, 9], Spherical fuzzy sets are one of the recent extensions of fuzzy sets. The benefit of Spherical fuzzy sets is that they better represent the hesitancy of the decision-makers. In the Spherical fuzzy sets, the degree of membership, the degree of non-membership, and the degree of hesitancy can be defined separately, but the total sum of the squares should be equal or less than one. In this study, we propose a novel regret based Spherical fuzzy multi-criteria evaluation method where the predefined factors are evaluated under different scenarios, and the hesitancy of the decision-makers are modeled with the Spherical fuzzy sets. In order to show the applicability, the proposed model is applied to facial mask inventory level selection in a hospital.

The rest of the paper is organized as follows, Section 2 summarizes the recent literature for healthcare equipment inventory levels under outbreaks. Section 3 gives the preliminaries of the spherical fuzzy sets. Section 4 gives the details of the proposed multi-criteria spherical fuzzy regret based approach. In Section 5, the proposed approach is applied to defining the importance of stock-keeping in different healthcare equipment. The robustness of the rankings is checked via sensitivity analysis in Section 6. Section 7 concludes and gives further studies.

2Healthcare equipment management under outbreak conditions

We conducted a literature survey to understand the current literature of outbreaks and supply chain problems. The mathematical models have been widely used in improving healthcare systems [12]. Figure 1 shows the studies that focus on outbreaks and their impact on the supply chains in the Scopus database.

Fig. 1

Studies that focus on outbreaks and supply chains (Scopus database).

The concept of outbreaks is mainly considered for food safety, food supply chains, and the traceability of the products. The supply chain problems under outbreaks have become a popular concept with the COVID 19 crises. Since, in order to be better prepared for the following outbreaks, we have to learn lessons from COVID 19 [13]. Notably, the shortage of critical but straightforward healthcare equipment increases the focus on this area.

Several studies tried to model the supply chains under COVID 19. Many researchers highlighted the vulnerability of the supply chains under outbreaks [6, 7]. Ivanov [7] developed a simulation-based approach to examine and predict the impacts of epidemic outbreaks on the supply chain performance. Govindan et al. [5] focus on patient levels in healthcare. They propose a decision support system based on knowledge and a fuzzy inference system for dealing with demand management in the healthcare supply chain to reduce the stress in the community. Their risk levels classify the patients. Fond et al. [14] utilize composite Monte-Carlo simulation for modeling future events such as patient numbers and understanding the composite data relationships of the outbreaks. The proposed model uses the deep learning network and fuzzy rule induction in the simulation model for obtaining better stochastic intuitions about the outbreak.

Skipworth et al. [15] highlighted the importance of outsourcing in supply chains catastrophic costs in a healthcare system. Two supply chain outsourcing decisions, public-to-private, and public-to-public are evaluated. Healthcare systems of different European countries are evaluated with this new conceptual framework.

After COVID 19, they are facing the problems associated with the shortage of face masks and protective equipment, overcoming the shortage become of the most critical research problem [10]. Wu et al. [17] highlighted the impact of face mask shortage under COVID 19 conditions. Similarly, Boskoski et al. [1] emphasized the usage of protective equipment such as masks under the COVID-19 outbreak. The appropriate reuse methods are revealed for dealing with the shortage of this equipment. Dargaville et al. [3] also focused on the facemask and protective equipment shortage. The researchers claim that the researchers should focus on multidisciplinary approaches for overcoming the shortages under outbreaks. Musazzi et al. [16] focus on medicine shortages in Europe. They criticize the current European regulatory framework on medicine shortages.

In the literature, being prepared for the outbreaks and keeping the necessary amount of stocks has been highlighted by many researchers. However, keeping stock is costly. The methods that solely depend on the objective criteria such as order cost, stock keeping, and demand quantities may not be realistic since some subjective criteria such as criticalness of the healthcare equipment should also be taken into account. Besides, in a usual healthcare organization, the number of equipment utilized is very high, and under different conditions, usage rates, costs, and availability may change. Using the expert judgments for prioritizing the equipment storage can be beneficial for overcoming outbreak crisis. Five criteria can be selected for evaluating the total regret of not keeping stock of the healthcare equipment. Distance to the potential producers stands for both the availability of local producers and the geographical closeness to production facilities. The usage quantity of healthcare equipment shows the demand level for the necessary product. Order costs of the healthcare equipment stand for the cost of providing the equipment. The criticalness of using healthcare equipment shows how critical it is to use the healthcare items under normal conditions or outbreak conditions. The last criteria, cost of keeping healthcare equipment inventory stands for the stock keeping costs.

3Preliminaries of spherical fuzzy sets

The basics of the spherical fuzzy sets ( S˜ ) are defined in this section [11].

The score of a spherical fuzzy set can be utilized to rank or defuzzify.

The distance between two spherical fuzzy sets D(S˜1, SF˜2) can be obtained as follows:

4Multi-criteria spherical fuzzy regret based evaluation method

The proposed multi-criteria spherical fuzzy regret approach tries to minimize the total opportunity cost under different conditions. In the selection process, the objective is to minimize the total maximum regret. In this study, first, the weights of criteria are determined by using the Spherical fuzzy AHP approach defined by Gundogdu and Kahraman [11]. Then, the maximum regrets under each criterion are defined by using the novel approach based on Spherical fuzzy evaluations. The steps of the proposed model can be given as follows.

The first step is defining the decision model; the criteria and evaluation alternatives are determined.

The second step is the pairwise comparison of the evaluation criteria. The scale given in Table 1 is utilized for evaluating pairwise comparisons. The linguistic terms in the pairwise comparison matrix are converted to their corresponding score indices by using Table 1.

In the third step, we aggregate the evaluations of the decision-makers by using SWAM method given in Equation 8.

In the fourth step, we obtain the weights w_j of criteria by using the Score function defined in Equation (9).

In the fifth step, we evaluate the alternatives with respect to each criterion under different conditions. The scale given in Table 2 is utilized for evaluating the alternatives.

In the sixth step, we aggregate the evaluations of the different decision-makers by SWAM operator and obtain an aggregated spherical fuzzy decision matrix given in Equation (10).

where A˜ijs denotes the spherical fuzzy evaluation of i^th alternative with respect to j^th criterion under the scenario s

In the seventh step, we define the best alternative with respect to each criterion (j = 1, 2, …, m) under every scenario s, (s = 1, …, l) by using the score function defined in Equation (6).

In the eighth step, we calculate the distances between A˜js* and A˜ijs using Equation (12) and obtain the regret value RV_ij for i^th alternative with respect to j^th criterion under s^th scenario.

The next step is calculating the maximum regret value of each alternative with respect to each criterion by using Equation (13).

In the tenth step, the total weighted sum of maximum regret values, TR_i are calculated by using Equation (14).

In the last step, the alternatives are ranked reversal based on TR_i where smaller TR_i values indicate a better alternative.

5An application: Defining stock-keeping necessity for different healthcare equipment

The proposed regret based spherical fuzzy multi-criteria evaluation methodology is implemented to defining the importance of stock-keeping for different healthcare equipment in a healthcare institution. Five criteria are defined after a literature review and meetings with decision-makers. These criteria are as follows;

Distance to the potential producers (C1) is the combination of availability of local production and also the geographical closeness to production facilities.

The usage quantity of the healthcare equipment (C2) is the quantity of the product that will be required under different conditions.

Order costs of the healthcare equipment (C3) is the cost of providing the equipment.

The criticalness of using the healthcare equipment (C4) shows how important it is to use the healthcare items under different conditions.

The cost of keeping healthcare equipment inventory (C5) refers to the stock-keeping costs.

In the evaluation process, a decision-maker who is an experienced physician and also responsible for the supply chain of the medical products in a healthcare institution evaluates the criteria and alternatives. Table 3 shows the pairwise linguistic comparison of criteria.

The linguistic scale given in Table 1 is utilized to convert the linguistic evaluations into the spherical fuzzy sets. Table 4 shows the spherical fuzzy representations of the pairwise linguistic comparisons.

The spherical fuzzy AHP methodology is utilized to obtain the weights of the criteria. Table 5 shows the spherical and normalized crisp weights of the criteria.

After defining the weights of the criteria, the necessity of keeping stock for different healthcare equipment with respect to the criteria are evaluated by the decision-makers. The decision-maker considered the factors under three scenarios. The aspects of different healthcare equipment are evaluated under a global epidemic outbreak, a local epidemic outbreak, or under normal conditions. Table 6 summarizes the linguistic evaluation of the decision-maker.

For instance, the first equipment has an unfavorable characteristic with respect to the distance to the potential producer’s criteria under normal conditions. This means that the availability of local produce and also the geographical closeness to production facilities is limited; thus, we have to provide this product from a long-distance supplier. Moreover, when there is a local or a global outbreak, the procurement of this equipment will be very hard and therefore considered very unfavorable. Table 7 shows the spherical fuzzy sets that are converted from linguistic evaluations.

In Table 7, the usage quantity of the healthcare equipment 1 is absolutely unfavorable that is represented by SPH (0.1,0.9,0.1) under any scenario. The usage quantity of this equipment is very high. Thus, it requires an ample amount of stock to keep. We applied the proposed approach and obtained the regret values RV_ijs and maximum regret values Max RV_ij given in Table 8.

After following the steps of the proposed methodology maximum regret values Max RV_ij. Moreover, the total regret of not keeping stock is obtained. The healthcare equipment is ranked based on the importance of keeping the stocks of these items. The results are summarized in Table 9.

According to these results, healthcare equipment 9 has the maximum total regret, which means that it is very vulnerable to crises such as outbreaks. It will be beneficial to keep stock for this item, whereas healthcare equipment 5 has a deficient total regret value, which means we can apply JIT type minimum stock approaches for managing the inventories of HE 9.

6Sensitivity analysis

The effects of the possible changes in the weights of the criteria are observed with a sensitivity analysis. Figures 2–6 show the results of a one-at-a-time sensitivity analysis. In these figures, X-axis represents the selected criterion weight, while Y-axis represents the scores of alternatives. In the one-at-a-time sensitivity analysis, we change the value of a specific criterion’s weight as the other criteria weights are changed proportionally to the original weights, and total weights of the criteria kept as one. The dark red line represents the current weight of the related criterion.

Fig. 2

Sensitivity Analysis-Distance to the potential production markets.

Fig. 3

Sensitivity Analysis-Usage quantity of the healthcare equipment.

Fig. 4

Sensitivity Analysis-Order cost.

Fig. 5

Sensitivity Analysis-criticalness of using the healthcare equipment.

Fig. 6

Sensitivity Analysis-cost of keeping healthcare equipment inventory.

The sensitivity analysis shows that the stock-keeping decision is highly dependent on the changes in the criteria weights. Thus, decision-makers should be very cautious while assigning weights to the criteria. The proposed pairwise comparison based criteria weight definition approach enhances the process.

Figure 3 shows that although the overall score of the healthcare units 4 and 9 are very close, the small changes in the weight of the usage quantity do not change the ranking of the first two equipment.

Figure 4 shows that the ranking of the healthcare equipment alternatives is highly sensitive to the changes in the weight of the order cost. The decision-makers should carefully consider the changes in the rankings.

Figure 5 shows that the ranking of the healthcare equipment alternatives is partially sensitive to the changes in the weight of the criticalness.

The ranking of the alternatives changes based on the changes in the weights of the criteria “cost of keeping inventory.” Healthcare equipment 8 becomes vulnerable when the weight of this alternative is increased.

7Conclusion and further suggestions

In this paper, a practical multi-criteria decision-making method was proposed to define healthcare equipment inventory levels under outbreaks. The different healthcare equipment is classified and ranked based on the criticalness of keeping stock. In order to define the criticalness of stock-keeping, five criteria are selected, and the healthcare equipment is evaluated with respect to these criteria. Defining the weights of the criteria is crucial. In order to enhance the process, the weights of the criteria are defined by using spherical fuzzy AHP. The different healthcare products are evaluated with a spherical fuzzy, regret based approach where different scenarios such as local and global epidemics are considered. The proposed model is applied to a real case, and the robustness of the results is checked with a one-at-a-time analysis.

This study has some limitations, and some areas can be further developed. While the study takes a comprehensive perspective, the decision criteria can be enhanced and modified to the needs of the different healthcare intuitions. Future research could analyze the results of the prioritization and stock levels. The results of the proposed method can be compared with other methods, or another type of fuzzy sets can be utilized for the comparison such as neutrosophic sets [18]. A SWOT analysis with fuzzy AHP integration can be also used for further research [19].