A Multi-Criteria Decision-Making Model with Interval-Valued Intuitionistic Fuzzy Sets for Evaluating Digital Technology Strategies in COVID-19 Pandemic Under Uncertainty.

Coronavirus diseases 2019 (COVID-19) pandemic is an essential challenge to the health and safety of people, medical members, and treatment systems worldwide. Digital technologies (DTs) have been universally introduced to improve the treatment of patients during the pandemic. Nevertheless, only a few governments have been partly successful in executing the DT strategies. In this regard, it is critical to demonstrate a suitable strategy for the governments. This problem is built based on the experts' opinions with some conflicting criteria to evaluate various types of alternatives. Hence, this research presents a new multi-criteria decision-making (MCDM) model under uncertain conditions. For this reason, interval-valued intuitionistic fuzzy sets (IVIFSs) are employed to help decision-makers (DMs) evaluate in a broader area and cope with uncertain information. Moreover, a new extended weighting method based on weighted distance-based approximation (WDBA) and a new combined ranking approach are proposed to determine the DMs' weights and rank the alternatives under IVIF conditions. The developed weighting method is constructed based on computing the DMs' weights with objective criteria weights. Furthermore, a new ranking approach is proposed by obtaining two ranking indexes separately: The first and second ranking indexes are calculated according to the positive and negative ideal solutions distances and the nature of criteria weights, respectively. Afterward, the final values of rankings are computed by considering a new aggregating procedure. The results of the proposed model represent the first alternative as the best strategy. Comparisons between the IVIF-TOPSIS and IVIF-VIKOR methods are also provided to investigate the proposed model to determine the rankings of main alternatives. Sensitivity analyses are conducted to check the reliability and the robustness of the model. For this purpose, criteria weights are analyzed to compute the dependencies' degree of the new extended weighting method. The dependencies of the ranking model are discussed on the criteria weights as well. © King Fahd University of Petroleum & Minerals 2022, Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Introduction

Coronavirus disease 2019 (COVID-19), made with a new kind of coronavirus SARS-CoV-2, was first identified in December 2019 as a case of pneumonia in Wuhan, China which has become a global pandemic, afterward [1]. World Health Organization (WHO) advertised the outbreak of a pandemic on March 11 and asked for conjunction procedures to support readiness and quick answer to the infection across the world's health parts [2]. COVID‑19 is one of the most infectious diseases in the decades for which various approaches and technologies have been introduced to support the treatment, handling, and control of the pandemic. For this reason, governments and health organizations make efforts to handle the expansion of coronavirus; they need all the help from digital technology (DT).

During the COVID-19 pandemic, digital health-based applications may significantly support associations and societies. DTs can also be used for stopping and observing measures, for example, through applications of tracing or monitoring internet searches and social media usage [3, 4]. Hence, diverse models and technologies are proposed to support the cure and control of the pandemic, including robotics and healthcare artificial intelligence (AI) applications [5, 6]. The AI technologies can be used following COVID‑19 outbreak for informing patients, disinfecting areas, as well as speeding up the search for effective treatments [7]. Furthermore, executed applications have been examined and evaluated from the viewpoint of technologies, potentially missing how policy directives contributed to the formulation of digital health landscape for COVID-19 [8]. In this way, Ting et al. [9] analyzed the DT in healthcare industry to handle the problematic position of the COVID-19 in four steps.

Meanwhile, DT selection for the healthcare industry in the COVID-19 pandemic application can be distinguished as a type of multiple criteria decision-making (MCDM) problem [10]. In addition, one of the effective methods applied in MCDM problems is the weighted distance-based approximation (WDBA) approach which has the same concept of the technique for order of preference by similarity to the ideal solution (TOPSIS) method and is categorized in the field of compromise solution approaches. As the WDBA method includes simple formulations and is straightforward, it is better than TOPSIS [11]. The WDBA technique has been successfully applied for selection processes in many MCDM problems [12]. In this regard, many researchers have begun to present new approaches to the MCDM for the DT selection of high uncertainty in pandemic and disease conditions based on uncertain linguistic terms [13]. However, the fuzzy sets (FSs) theory suggested by Zadeh in [14] has achieved great success in different fields. Meanwhile, intuitionistic fuzzy sets (IFSs) proposed by Atanassov [15] have been established to be very beneficial in dealing with vagueness. Generally, IFS examines the membership and non-membership of an element to a given set whose values are described between 0 and 1 [16]. The comparison between the mentioned approach and the initial FSs represents the IFS as the more experimental method in handling vagueness. Therefore, decision-makers (DMs) use IFSs to better demonstrate information under uncertain conditions [17, 18].

Due to the increasing complexity of the problem conditions, the lack of problem knowledge, and the limited ability of DMs relevant to the information process, it is hard to precisely describe DMs’ priorities by focusing on criteria. However, it is possible to express them as a value range. In other words, to assess a criterion, DMs may be more comfortable expressing the value of the index with satisfactory and dissatisfactory degrees using an interval value. Eventually, the IFSs under interval conditions are changed to interval-valued intuitionistic fuzzy sets (IVIFSs) [19]. The IFSs and IVIFSs have received attention in recent years, and several kinds of research have been conducted to develop and modify their theories.

Ecer and Pamucar [20] extended the MACROS technique under the IF environment to measure the performance of the insurance companies during the COVID-19 outbreak. Goker [21] proposed an integrated agile IF decision in the COVID-19 pandemic condition for a provider selection. Hezam et al. [22] presented the neutrosophic MCDM approach for the COVID-19 vaccination decision-making process. Alkan and Kahreman [23] introduced the q-rung orthopedic fuzzy TOPSIS method to select an appropriate government strategy during the COVID-19 pandemic. Narayanamoorthy et al. [24] evaluated the vaccination of Youngsters in the COVID-19 pandemic condition by PROMETHEE-II in an IF environment. Tumsekcali et al. [25] proposed an IVIF-based methodology during the COVID-19 pandemic. Zhang and Huang [26] provided the Z-IF MULTIMOORA method with the AHP approach in a real-case study of COVID-19. Boyaci and Sisman [27] suggested the selection site of the hospital in the COVID-19 pandemic condition in which Pythagorean fuzzy sets have been employed to cope with uncertain conditions. Mardani et al. [28] proposed a hesitant fuzzy MCDM technique for the selection of healthcare technology in the COVID-19 outbreak.

Saraji et al. [29] offered the MCDM problem to select suitable education techniques under hesitant fuzzy sets. Alkan and Kahraman [30] introduced IF-TOPSIS method to choose the hospital site under pandemic conditions. Chen and Lin [31] proposed an FGM decomposition-MCDM method based on FSs for selecting the smart technology tools to support mobile healthcare during and after the COVID-19 pandemic. De Andrade et al. [32] introduced an integrated MCDM method to evaluate the impact of COVID-19 on social welfare regarding the utility of food apps. Ali et al. [33] presented a hybrid MCDM approach to assess the COVID-19 pandemic on future energy systems in developing countries. Nguyen et al. [34] proposed an integrated MCDM method to evaluate financial impacts of the COVID-19 on banking and commercial policies. Toan et al. [35] presented a two-stage grey MCDM approach to select video conference software during the COVID-19 pandemic.

According to the clarification, MCDM studies on the COVID-19 pandemic respecting uncertain conditions are limited in the literature. The literature review indicates that the decision-making approaches are far from the real-world applications and do not consider them in uncertain situations. Furthermore, all the previous related works do not focus on the experts’ weights, although their opinions have an important impact on making appropriate decisions. Besides, the determination of ranking values by considering two separate ranking methods have received less attention from the studies. Accordingly, this paper proposes a new integrated IVIF-decision-making model to compute the DMs’ weights with a new extended weighting method. Also, a new hybrid ranking approach is developed by considering two ranking indexes to calculate the final ranking of the alternatives and make an appropriate decision. For this reason, the weights expressed by DMs are computed based on a new version of the WDBA method consisting of the average, negative, and positive ideal solutions under IVIF conditions. Afterward, a new combined ranking approach is proposed based on the distance between negative and positive ideal solutions. The main innovations of the paper are explained below:

Introducing a new multi-criteria decision-making model with IVIF for evaluating DT strategies under the COVID-19 pandemic.

Presenting a new extended weighting method to compute weights of DMs.

Proposing a new combined method to rank the alternatives.

Applying the empirical example in the COVID-19 pandemic environment.

The rest of the paper is organized as follows. Section 2 develops the preliminaries and the basic description; Sect. 3 presents the proposed method; Sect. 4 extends the empirical example; Sect. 5 provides the sensitivity analysis. Finally, Sect. 6 presents the conclusions and further research suggestions.

Preliminaries

This section examines the basic definitions of the IVIFS. These definitions are determined below:

Definition 1 [36]

Let be a universe. An IVIFS in U is described by Eq. (1).where , and . In these forms, is the infimum of the and is the supremum of the . This condition also applies to simultaneously.where and for . Also, if and , the IVFS changes to the IFS.

Definition 2 [37]

Let , . The mathematical operations are shown in Eqs. (4)–(7).

Definition 3 [38]

Euclidean distance is computed based on Eq. (8).

Definition 4 [39]

Normalized decision matrix is computed by Eqs. (9)–(16). Equations (9)–(12) and (13)–(16) are related to the benefit and cost criteria, respectively.

Proposed Decision Model

The proposed approach is introduced via a new version of the WDBA method. This approach is presented to obtain the weights of DMs. Moreover, a new combined method is proposed to compute the ranking of alternatives based on a new distance-based approach. This model utilizes the recent papers that have applied the weighting and ranking processes for decision-making and MCDM problems (i.e., [11, 38, 39]).

Step 1. Gathering the information from experts relevant to the criteria and the alternatives .

Furthermore, the decision-making matrix based on the sth expert opinion is shown in Eq. (17).where and determine the number of the alternatives and criteria, respectively. Also, presents the number of the DMs.

Step 2. Normalizing the decision matrix.

This process is shown in Eqs. (18) and (19) based on Definition 4.

Step 3. Constructing weighted normalized decision matrix with Eq. (20).

Step 4. Computing weights of the DMs.

In this step, the weights expressed by DMs are obtained from average, positive, and negative ideal solutions. Meanwhile, the aggregate values of the experts' opinions are needed to accomplish the DMs’ weights.

Step 4.1. Obtaining the average, positive, and negative ideal solutions based on weighting normalized decision matrix values by Eqs. (21)-(23).

Furthermore, the total of positive ideal solution and the WDBA method depicts the experts’ weights. The closer opinions of each DM to the average of the opinions earn a higher weight. For instance, suppose that a university has a teaching competition in which young teachers compete. If there are s referees as DMs for this tournament, each competitor's final score will be the average of the s scores provided by DMs. The concepts and computation method have been provided in the recent literature [e.g., 11].

Step 4.2. Calculating the distance from average, positive, and negative ideal solutions by Eqs. (24)- (26) based on Definition 3.Step 4.3. Obtaining the DMs’ values by Eqs. (27) and (28).Step 4.4. Calculating final weights of the DMs by Eq. (29).Step 5. Aggregating the decision matrix by weights of the DMs by Eq. (30).Step 6. Determining the positive and negative ideal solutions by Eqs. (31) and (32).Step 7. Computing the distances from positive and negative ideal solutions by Eqs. (33) and (34).Step 8. Obtaining initial ranking values of the collective index by Eq. (35).Step 9. Computing the secondary ranking values of the collective index by Eq. (36) based on the cost and benefit weights of the criterion .

Step 10. Calculating the final rank by Eq. (37). The collective index is provided to integrate two different ranking values that are obtained from two different ranking approaches.

Eventually, the alternatives can be ranked in descending order. The structure of the decision approach is depicted in Fig. 1.

Fig. 1

Structure of the proposed approach

Experimental Example

Here, an empirical example from the literature [28] is solved to determine the performance of the proposed approach. The decision matrix is developed via the opinions of three DMs. 24 criteria and their weights are listed in Table 1.

Table 1

Criteria description and weight evaluation [41]

Criteria	Symbol	Criteria weights
Digital treatment	C_1	0.041
Awareness and training	C_2	0.037
Health hazard prediction	C_3	0.022
Non- virtual support	C_4	0.027
Medicine development	C_5	0.037
Economic interventions	C_6	0.075
Digital divide	C_7	0.022
Lack of Digital knowledge	C_8	0.043
Cost inefficiency	C_9	0.045
Mock information	C_10	0.054
Lack of reliable data	C_11	0.034
Insecure applications	C_12	0.048
Accurate prediction system	C_13	0.035
Automatization of Healthcare	C_14	0.032
Research framework in Epidemiology	C_15	0.029
Digital health education	C_16	0.052
Digital diagnosis	C_17	0.029
Health information systems (HIS)	C_18	0.025
Non-digital interferences	C_19	0.048
Digital stratification	C_20	0.033
Privacy worries	C_21	0.057
Unaffordability	C_22	0.036
Exacerbation of paranoia	C_23	0.065
Infodemic risk	C_24	0.074

Four kinds of alternatives are applied in handling the COVID-19 pandemic condition with DTs, including strengths, weaknesses, opportunities, and threats (SWOT). The two first sections are internal elements, and the two other parts are external factors. Afterward, the linguistic variables for experts’ judgments are determined in Table 2 [40]. Furthermore, Table 3 constructs the decision matrix based on the three experts’ opinions.

Table 2

Linguistic terms to evaluate criteria and alternatives [40]

Linguistic variables	IVIF values
Extremely low (EL)	([0.00,0.05], [0.90,0.90])
Very low (VL)	([0.05,0.10], [0.80,0.90])
Low (L)	([0.10,0.20], [0.70,0.80])
Medium low (ML)	([0.30,0.40], [0.50, 0.60])
Medium (M)	([0.50,0.50], [0.50, 0.50])
Medium high (MH)	([0.50,0.60], [0.30, 0.40])
High (H)	([0.70,0.80], [0.10,0.20])
Very high (VH)	([0.80, 0.90], [0.50, 0.10])
Extremely high (EH)	([0.90,0.90], [0.00,0.05])

Table 3

Linguistic values of criteria performance

Criteria	DMs	A_1	A_2	A_3	A_4
C_1	DM_1	MH	M	ML	MH
	DM_2	M	ML	H	ML
	DM_3	H	MH	M	H
C_2	DM_1	H	MH	M	H
	DM_2	H	L	ML	H
	DM_3	M	H	M	H
C_3	DM_1	MH	ML	ML	MH
	DM_2	VH	MH	H	M
	DM_3	H	L	ML	H
C_4	DM_1	L	M	M	H
	DM_2	MH	VH	ML	M
	DM_3	M	H	M	H
C_5	DM_1	M	ML	L	M
	DM_2	M	L	H	ML
	DM_3	MH	ML	L	L
C_6	DM_1	M	M	M	MH
	DM_2	VH	M	ML	M
	DM_3	M	VL	M	VL
C_7	DM_1	MH	ML	L	L
	DM_2	M	VL	M	VL
	DM_3	H	M	VH	H
C_8	DM_1	MH	H	M	MH
	DM_2	H	MH	ML	L
	DM_3	M	M	VH	H
C_9	DM_1	ML	H	MH	M
	DM_2	MH	M	VH	VL
	DM_3	H	VH	H	M
C_10	DM_1	VH	EH	MH	H
	DM_2	M	H	H	MH
	DM_3	H	MH	H	M
C_11	DM_1	MH	EH	VH	M
	DM_2	VH	MH	MH	ML
	DM_3	H	ML	L	M
C_12	DM_1	H	VH	H	M
	DM_2	H	MH	H	M
	DM_3	H	ML	L	M
C_13	DM_1	MH	MH	H	ML
	DM_2	ML	M	M	MH
	DM_3	M	H	ML	H
C_14	DM_1	MH	H	M	VH
	DM_2	H	MH	MH	MH
	DM_3	VH	H	ML	M
C_15	DM_1	L	H	M	H
	DM_2	VH	VH	VH	M
	DM_3	M	ML	H	MH
C_16	DM_1	M	H	ML	H
	DM_2	VH	H	ML	M
	DM_3	M	ML	H	ML
C_17	DM_1	M	H	ML	ML
	DM_2	M	ML	H	H
	DM_3	H	ML	M	H
C_18	DM_1	L	MH	MH	H
	DM_2	ML	MH	ML	L
	DM_3	H	ML	L	VL
C_19	DM_1	MH	M	M	H
	DM_2	VH	MH	M	M
	DM_3	M	VH	H	M
C_20	DM_1	L	ML	ML	H
	DM_2	ML	H	MH	ML
	DM_3	ML	ML	ML	M
C_21	DM_1	MH	H	ML	ML
	DM_2	H	H	L	M
	DM_3	M	L	H	MH
C_22	DM_1	M	H	ML	ML
	DM_2	MH	ML	H	H
	DM_3	H	ML	ML	MH
C_23	DM_1	ML	H	MH	H
	DM_2	ML	MH	ML	ML
	DM_3	MH	ML	L	VL
C_24	DM_1	MH	M	MH	H
	DM_2	H	H	M	MH
	DM_3	MH	VH	H	M

Hence, the values of , and which are obtained by Eqs. (24)–(26), are determined in Table 4. Moreover, Table 4 shows the values of the DMs () and their final weights () using Eqs. (27) and (28), respectively.

Table 4

Final weights of the DMs

DMs	ϕAs	ϕPs	ϕNs	Γ^s	W^s
DM_1	0.00612	0.00825	0.08518	2.40496	0.41747
DM_2	0.00656	0.00842	0.07004	2.20183	0.38221
DM_3	0.00858	0.01318	0.00844	1.15395	0.20031

According to Table 4, the first DM has a higher priority than other experts, and DMs’ opinions may have a great influence on final decisions. In addition, Table 5 shows the positive and negative ideal solutions distances ( and ), the initial ranking value of the collective index (), the secondary ranking value of the collective index (), the final value of the collective index (), and the final ranking of alternatives that are obtained by Eqs. (33)–(37), respectively.

Table 5

Final alternatives ranking

Alternatives	Di+	Di-	ϱ_i	θ_i	C_i	Final rank
A_1	0.00290	0.00678	3.36950	0.02061	1.69505	1
A_2	0.00441	0.00566	2.29196	0.01720	1.15458	2
A_3	0.00582	0.00416	1.70054	0.01264	0.85659	3
A_4	0.00567	0.00380	1.65389	0.01155	0.83272	4

The obtained result indicates the superiority of the first alternative (i.e., strengths) over other options in the COVID-19 pandemic conditions for selecting an appropriate DT strategy. In other words, in the critical status of the COVID-19 pandemic, focusing on strengths is more important than other alternatives in the DT selection, which can help make the proper decision more effectively. Afterward, weaknesses, opportunities, and threats have the next priorities compared to each other.

Result and Discussion

In this section, the effectiveness and advantages of the proposed model are presented by comparing it with IVIF-TOPSIS and IVIF-VIKOR methods. The final ranks of the alternatives are alike, and the introduced model is valid. Table 6 determines the collective index of the proposed ranking model, IVIF-TOPSIS, and IVIF-VIKOR approaches. This table illustrates that the final ranking values result in similar outcomes.

Table 6

Comparisons among three ranking approaches

Alternatives	IVIF- TOPSIS	Ranking	IVIF-VIKOR	Ranking	Proposed approach	Ranking
A_1	0.70002	1	0.62587	1	1.69505	1
A_2	0.56171	2	0.60354	2	1.15458	2
A_3	0.41672	3	0.57412	3	0.85659	3
A_4	0.40135	4	0.563214	4	0.83272	4

Furthermore, for the proposed model, three various scenarios are carried out for each criterion. In this analysis, the weight of each index is switched to 0.1, 0.5, and 0.9, respectively, although others are proportionally kept fixed. The collective index of alternatives is reobtained with these new criteria weights. It should be noted that the sum of weights equals 1 in each scenario. The effects of the changes on alternative ranks are determined in Fig. 2.

Fig. 2

Results of sensitivity analysis for the proposed model based on the criteria weights

In Fig. 2, can be determined as the best alternative with three varying weights of the criteria for COVID-19 DT selection; it is superior to the four options in all sensitivity analyses. This denotes that the decisions made by the proposed approach are robust and effective.

Afterward, the criteria weights change among random numbers to compute the independency of DMs’ weights with these values. In 10 different cases considering various criteria weights, the first DM often has a higher degree than the second DM; Fig. 3 demonstrates this point clearly.

Fig. 3

Sensitivity analysis outcomes on dependencies among DMs weights and criteria weights

Conclusion

COVID-19 is a pandemic that has not yet been passed, appearing in China and developing in the world. Increasing the pandemic causes healthcare system organizations to face many difficulties. Hence, governments make efforts to control the behavior of the pandemic in their lifestyle. These criteria are insufficient, and so it is essential to make sure that the pandemic is managed correctly with the most suitable strategy. Most developed countries have used the digital technology (DT) to support health systems. Meanwhile, the DT selection and implementation by healthcare organizations need to be assessed and contrasted. In this regard, a multi-criteria decision-making problem (MCDM) problem arises, where many alternatives should be assessed under more than one ranking index. Nevertheless, due to uncertainties in the addressed problem and the lack of information and contradictions among a group of experts, interval-valued intuitionistic fuzzy sets (IVIFSs) that allow decision-makers (DMs) to evaluate in a broader area should be addressed. IVIFS is a powerful tool to control the vagueness condition. Using IVIFSs, especially in MCDM methods, ambiguity and uncertainty are described more strongly, and consequently, the decision-making procedure can be handled more precisely. In this respect, one of the essential issues utilized in this paper is introducing a new MCDM model considering the IVIFSs. This issue is caused by viewing the real-world uncertainty conditions and real application requirements. IVIFS is a powerful tool that presents the uncertain amounts among the interval values. Therefore, two new methods based on IVIFSs have been proposed in this paper to calculate the weights of DMs and the ranking of the alternatives. An extended weighting method is applied to determine the weights of the experts using the WDBA method. Afterward, a new ranking method is proposed by computing the positive and negative ideal distances via two various ranking approaches. Finally, a new collective index method is presented to aggregate the rankings’ values. Additionally, the mentioned methods are used to select the best DT strategy for governments. Alternatives, criteria, and information have been applied in the literature. According to the results, the first DM has high priority than other experts, and the first alternative is superior to use with the governments. This alternative is related to focusing on the strengths of the DT that helps the manager control the pandemic situation at an appropriate time. Afterward, the sensitivity analysis has been conducted to compute the independency of the proposed approach from the criteria’ weights. Firstly, the criteria weight independency is considered using the sensitivity analysis performed on the new extended DM weighting method. Secondly, the independence of the new hybrid ranking method is measured on the criteria’ weights. In these two analyses, the DMs’ weights and rankings of the alternatives are independent of criteria’ weights, and the results of the proposed new hybrid approach are robust and reliable. Likewise, the comparisons among the proposed model and two well-known IVIF-decision approaches, i.e., IVIF-technique for order of preference by similarity to ideal solution (IVIF-TOPSIS) and IVIF-VIseKriterijumska Optimizacija I Kompromisno Resenje, in Serbian (IVIF-VIKOR), have shown that the proposed approach has high performance to compute the ranking of alternatives. The final ranking values are valid by considering these two IVIF-multi-criteria decision approaches.

For future suggestions, the proposed model can be developed for other new types of fuzzy sets (FSs), e.g., interval-valued hesitant fuzzy sets and interval type-2 fuzzy sets. Various kinds of the MCDM method can be extended, in the related literature on complex decision making [42-47], to compare with this proposed approach. Moreover, the optimization technique can be used to compute the weights of the DMs in reality.