A robust algorithmic cum integrated approach of interval-valued fuzzy hypersoft set and OOPCS for real estate pursuit

Motivation of proposed study

Many researchers made rich contributions regarding the utilization of fuzzy TOPSIS for handling various DM situations. Eraslan (2015) presented a DM system structured on TOPSIS and soft set theory. Eraslan & Karaaslan (2015) introduced a DM TOPSIS technique based on a fuzzy soft environment. Ashtiani et al. (2009) and Mokhtarian (2015) extended TOPSIS method to interval-valued fuzzy sets. Tripathy, Sooraj & Mohanty (2017) employed a novel approach to IVFS-set for discussing DM situation. The HS-set provides a more simplistic modified version of the S-set that settles the barriers of the FS-set by making use of the multi-argument approximation operator (MAAO) instead of the single-argument approximation operator (SAAO). This tool maps sub-parametric entities into a power set of the universal set. It focuses on attribute segmentation into a non-overlapping attribute value set. These features make it a completely new mathematical tool for dealing with situations involving uncertainty and risk. This enhances the effectiveness and efficiency of DM. In some cases, DM requires a more strategic approach than just selecting the best available goods or services. In such situations, what is paramount may rely on numerous factors. One such case may occur when the experts are hesitant and they give their opinion in terms of linguistic values that are required to be transformed to interval-valued fuzzy values, i.e., membership degree for approximating an alternative based on opted attributes to deal with roughness-based imprecision.

As the TOPSIS approach heavily relies on multi-parameter DM, a sophisticated MADM has been employed to rate potential residential construction options. In order to do this, a brand-new approach called “OOPCS (optimal order preference correlation strategy)” is created on the foundation of FHS-sets with interval settings. By using this technique, a collection of characteristics can be divided into several sub-attributed valued sets, where each attribute correlates to a different valued set. Data might be in the range between the lower bond and the higher bond due to the intervals employed in this approach. Decision makers assign weight in two phases. In the first phase, decision-makers rank each component of the HS-set without taking into account the alternatives. In order to avoid bias, each k-tuple element of the hypersoft model is assigned a weight based on the preferences expressed by each decision-maker. A weighted vector is then created that maintains the relevance of each tuple member at a constant level. In the second phase, each decision-maker ranks each alternative based on its associated tuple value. This dual ranking minimizes the bias-ness of decision-makers towards any specific alternative. After observing the above literature, it is quite transparent that there is a need to initiate a mathematical framework that may tackle the following concerns and issues collectively:

1. How can the limitations of IVFS-set like structures be managed regarding the partitioning of attributes into their related non-intersecting subclasses consisting of subattributive values?

2. How can traditional TOPSIS be modified for a multi-argument approximate operator?

The proposed mathematical structure, i.e.,  IVFHS-set, can easily tackle the above issues through their integration and characterization.

The novelty of the proposed framework is that the adopted mathematical structure, i.e.,  IVFHS-set, is more flexible as compared to relevant existing literature because it generalizes most of the pre-developed fuzzy and soft set like structures. It not only copes with a large amount of data by introducing lower and upper bounds but also assists the decision-makers in making decisive comments by considering multiple arguments simultaneously. The proposed framework is actually an integration of IVFHS-set theory and its modified TOPSIS which has not been addressed by anyone so far in the literature.

The significant contributions of the study are outlined below:

1. The well-known DM technique TOPSIS is modified for IVFHS-sets by considering multi-argument approximate mapping with fuzzy graded approximations.

2. An innovative DM strategy called OOPCS is established, which employs a modified TOPSIS method and weight vectors of parametric tuples-based matrices and alternative-based matrices for the evaluation process by considering related decision makers’ approximations.

3. A robust algorithmic approach is utilized to evaluate appropriate residential buildings by integrating modified TOPSIS, OOPCS, and aggregation operations of the IVFHS-set. Moreover, the flexibility and reliability of the proposed strategy are assessed through structural comparison.

This research work is structured as follows: Section 2 outlines the fundamentals of IVFHS-sets, F-sets, and S-sets, along with their interval-valued hybrids. Section 3 describes the main process for the standard HS in a sequence of steps. In Section 4, a new strategy TOPSIS is developed based on OOPCS-set theory for group IVFHS mechanisms. Section 5 demonstrates the efficiency of the proposed method through a real-world application along with a comparative study and sensitivity analysis. Section 6 concludes the research work with future directions.

Modification of TOPSIS

The TOPSIS is a helpful and pragmatic strategy for evaluating and choosing a variety of alternatives employing distance measures. TOPSIS operations include decision matrix normalization, distance measurements, and aggregation operations (Shih, Shyur & Lee, 2007). A literature review of research (Hwang & Yoon, 1981; Yoon, 1987) on the TOPSIS technique is recommended for a better understanding of the concept. The TOPSIS technique adopted by Eraslan & Karaaslan (2015) has been described in this section, and its modified form for the interval-valued hypersoft set is described in the next section. Step by step TOPSIS technique is described hereafter and demonstrated in Fig. 2.

Let ${\aleph }_n=\left\{1,2,...,n\right\}\forall n\in \mathfrak{N}$ (the set of natural numbers)

1. Let M_D be the decision matrix defined in Eq. (1) demonstrated in the form of Table 1, where A_i, i ∈ ℵ represent alternative and ζ_j, j ∈ ℵ represent criteria. (1)${\displaystyle {\mathbf{M}}_{\mathbf{D}}={\left[{\phi }_{ij}\right]}_{m\times n}}$

2. Construction of normalized decision matrix D_N displayed in Table 2. Where each entry of normalized decision matrix D_N can be calculated using Eq. (2). (2)${\displaystyle {\gamma }_{ij}=\frac{{\phi }_{ij}}{\sqrt{\sum _{k=1}^m{{\phi }^2}_{kj}}},\forall {\phi }_{ij}\ne 0}$

3. Formation of weighted normalized decision matrix D_WN = [ζ_ij]_m×n = [ϖ_jγ_ij]_m×n, i ∈ ℵ_m, displayed in the form of Table 3, where ${\varpi }_j=\frac{{\mathbf{W}}_j}{{\sum }_{k=1}^m{\mathbf{W}}_j},j\in \left\{1,2,...,n\right\}$, such that ${\sum }_{k=1}^m{\varpi }_j=1$, and W_j are weights for criteria ζ_j, j ∈ ℵ_n. ${\mathbf{D}}_{\mathbf{WN}}={\left[{\zeta }_{ij}\right]}_{m\times n}$

4. Calculation of A^+ive, the +ive ideal solution (PIS) and A^−ive, the −ive ideal solution (NIS). (3)${\displaystyle {\mathbf{A}}^{+ive}=\left\{{\zeta }_1^{+},{\zeta }_2^{+},...,{\zeta }_j^{+},...,{\zeta }_n^{+}\right\}=\left\{\left(\underset{i}{max}{\zeta }_{ij}|j\in {{\rm Y}}_1\right),\left(\underset{i}{min}{\zeta }_{ij}|j\in {{\rm Y}}_2\right),i\in {\aleph }_m\right\}}$ (4)${\displaystyle {\mathbf{A}}^{-ive}=\left\{{\zeta }_1^{-},{\zeta }_2^{-},...,{\zeta }_j^{-},...,{\zeta }_n^{-}\right\}=\left\{\left(\underset{i}{min}{\zeta }_{ij}|j\in {{\rm Y}}_1\right),\left(\underset{i}{max}{\zeta }_{ij}|j\in {{\rm Y}}_2\right),i\in {\aleph }_m\right\}}$ where Υ₁ is benefit attribute set and Υ₂ is cost attribute set.

5. Calculation of separation measure of +ive ideal ${\mathbf{SM}}_i^{+}$ solution and separation measure of _ive ideal ${\mathbf{SM}}_i^{-}$ solution. (5)${\displaystyle {\mathbf{SM}}_i^{+}=\sqrt{\sum _{j=1}^n{\left({\zeta }_{ij}-{{\zeta }^{+}}_j\right)}^2},\forall i\in {\aleph }_m}$ and (6)${\displaystyle {\mathbf{SM}}_i^{-}=\sqrt{\sum _{j=1}^n{\left({\zeta }_{ij}-{{\zeta }^{-}}_j\right)}^2},\forall i\in {\aleph }_m}$

6. Calculation of relative nearness of alternatives to the optimal solution (7)${\displaystyle {\mathbf{C}}_i^{+}=\frac{{\mathbf{SM}}_i^{-}}{\left({\mathbf{SM}}_i^{-}+{\mathbf{SM}}_i^{+}\right)},0\le C_i^{+}\le 1,\forall i\in {\aleph }_m}$

7. Ranking the preference order.

Strategy for selected parameters

The choice of residence is influenced by a number of parameters. Criteria (parameters) can be of decisive importance for the DM process. Therefore, careful selection and calculation are required. It is a long-standing practice that residence is often selected solely on the basis of location. The criteria (parameters) discussed in this paper are found to be much more relevant and significant to residence selection than the criteria previously discussed by many researchers in different studies. The suggested study is concerned with the interval-valued fuzzy hypersoft environment, which means that sub-parameters are also taken into consideration and may fruitfully fulfill the requirements of the user environment.

Operational role of selected parameters

1. Location: Location has a tremendous influence on the housing market. Housing units vary depending on their surroundings, the type of neighborhood they are located in, and how close they are to centers of work and retail. This is because their locations are set, making them immobile. The location-specific area additionally indicates that a home’s environment may have a serious influence on how much it is worth. Residences in town committees and housing schemes have more value than others.

2. Price: The cost of a residence is directly affected by a number of factors. Location and other physical facilities boost the price of the residence. Approximating the appropriate manufacturing cost is also necessary as it indicates the proposed worth of the residence.

3. Plot size: The shape and size of the plot are important factors as they directly affect the physical appearance and beauty of a residential building. Square-shaped plots with a size of about 300 square meters are preferred over rectangular plots for symmetric designs.

4. Adaptation to weather: The basic concept is to make sure that the design of the house is in accordance with the climate. For example, being extremely cold or hot during the winter or summer months, respectively, can affect the air conditioning and heating systems. As well as heat preservation systems like insulation, both are very important.

5. Covered area: The ratio of the constructed portion of land versus the unconstructed portion varies with location and need. Normally, in metropolitan areas, the constructed portion dominates the unconstructed portion of residences. For a healthy environment, this ratio should be 1:1.

6. Number of bedrooms: The bedroom serves as the most significant room in the house in many respects. The importance of getting decent sleep is becoming increasingly clear given the pressures that the digital age throws on our time. In addition to enhancing focus and productivity, enough sleep has been linked to improved immune system performance, mental wellness, and even weight loss. One or two bedrooms are enough for a small family, but demand increases with the increase in the number of individuals in the family.

7. Access to the main road: The worth of a residence varies with distance from the main road/highway. Residences near main roads and highways have more value than others.

8. Number of bathrooms: A pleasant, warm bath or a hot shower in the bathroom may help you relax after a long day of work and provide that little moment of isolation. This transforms the bathroom into a safe zone rather than simply an additional room in the house. The number of bathrooms varies with the density of the population in the building. Normally, at least one bathroom is necessary for each living room (bedroom).

9. Number of floors: A building with several stories above the ground is referred to as a “multi-story building” in this context. Typically, a structure is considered a multi-story if it comprises more than two levels. Living in a building like that has both benefits and drawbacks.

10. Green surrounding: Numerous health advantages, particularly reduced illness and premature death, increased average lifespan, fewer mental health issues, decreased heart disease, improved mental abilities in adolescents and elders, and healthier infants, are linked to green space. A residential building with green surroundings is preferable to a building in or near an industrial zone.

11. Recreation facility: Zoos and cinemas are the places where people go for recreation. Zoos are especially popular with children. Zoos also put a great deal of attention on scientific study and wildlife management in addition to entertaining and instructing the general public. It is becoming increasingly popular to give animals more room and recreate their natural environments. The availability of playgrounds and public parks also has certain positive impacts on society. A residential building near such places is indeed a blessing.

12. Educational Institution: Students who live near to a school can commute by bike or on foot, breathing in the fresh air while doing their part to preserve the environment. The burden on parents to plan for the school run on a hectic morning is further reduced by easy access to the school. In order to avoid having to go too far in quest of these essential facilities, people hunt for residences close to important amenities like public transit, stores, and schools. The school catchment regions are becoming increasingly important to many home purchasers and renters.

13. Health Facility: You can maintain your lifestyle quality by having access to high-quality health-care close by. The ease of routine check-ups, peace of mind during emergencies, and reduced travel time can all be attributed to living close to a hospital.

14. Security and safety system: Protecting individuals and their personal property against dangers, such as theft, violence, vandalism, or indeed any hazard that might endanger resident safety, is the goal of residential security. In a nutshell, their responsibility is to identify risks, look into them, and quickly take appropriate action. To buy a residence, the security plans and safety systems of that locality should be taken into consideration.

15. Market: The availability of fresh, regionally produced food and frequently cheap organic food at public markets benefits the overall health of the neighborhood. They also serve as the focal point of the area, promoting pride. Neighborhood markets contain the freshest and healthiest products and normally include locals or farmers selling their wares.

16. Materials used in architecture: Building materials signify structural existence. It proves the prevalence of a sense of beauty in a design and, as a result, confirms the stainability of the construction.

Application

Finding the precise image of society in terms of the indicators needed to choose a residence from the possible choices was the initial stage of this exploratory work. The data was obtained through conversations among residents of the region, comprising estate advertising or marketing personnel or representatives, professionals and personalized specialists, respondents with previous experience in selecting or acquiring a home, and individuals with no previous experience in residence choice. A discussion was undertaken, and the participants were given a survey. They had to grade the relevance of the parameters in house choosing on a scale ranging from one to 5. Participants were asked to specify.

The following example (problem) can now be solved step-by-step by employing this group DM algorithm:

1. Let us say a real estate agent “A” has a variety of homes $\ddot{\theta }=\left\{{\ddot{\theta }}_1,{\ddot{\theta }}_2,{\ddot{\theta }}_3,{\ddot{\theta }}_4,{\ddot{\theta }}_5\right\}$, each of which may be described by a set of parameters $\mathbf{X}=\left\{{\mho }_1,{\mho }_2,{\mho }_3,...,{\mho }_{16}\right\}$ along with the description and measuring units, as elaborated in Table 8. The parameters ℧_j stand for attributes location, price, plot size, adaptation to weather, covered area, number of bedrooms, access to the main road, number of bathrooms, number of floors, green surrounding, recreation facility nearby, educational institution nearby, health facility nearby, security and safety system, market nearby, and material used for construction, respectively, for j = 1, 2, 3, ..., 16. The attribute valued sets corresponding to each parameter along with the prescribed value and fuzzified interval value given below and elaborated in Table 9. $X_1=\left\{\right.$non-ideal, ideal$\left.\right\}=\left\{{\mho }_{11},{\mho }_{12}\right\}$ $X_2=\left\{\right.$Less than or equal to 0.1 M$, greater than 0.1M$ up to 0.15M$, greater than 0.15M$$\left.\right\}=\left\{{\mho }_{21},{\mho }_{22},{\mho }_{23}\right\}$ $X_3=\left\{\right.$less than or equal to 300, greater than 300$\left.\right\}=\left\{{\mho }_{31},{\mho }_{32}\right\}$ $X_4=\left\{\right.$extremely hot climate (EH), extremely cold climate (EC), EH&EC$\left.\right\}=\left\{{\mho }_{41},{\mho }_{42},{\mho }_{43}\right\}$ $X_5=\left\{\right.$less than or equal to 50%, greater than 50%$\left.\right\}=\left\{{\mho }_{51},{\mho }_{52}\right\}$ $X_6=\left\{\right.$less than 3, 3 to 5, greater than 5$\left.\right\}=\left\{{\mho }_{61},{\mho }_{62},{\mho }_{63}\right\}$ $X_7=\left\{\right.$main road, less than 2km, greater than 2km$\left.\right\}=\left\{{\mho }_{71},{\mho }_{72},\mho 73\right\}$ $X_8=\left\{\right.$less than 3, 3 to 4, more than 4$\left.\right\}=\left\{{\mho }_{81},{\mho }_{82},{\mho }_{83}\right\}$ $X_9=\left\{\right.$1, 2, 3, more than 3$\left.\right\}=\left\{{\mho }_{91},{\mho }_{92},{\mho }_{93},{\mho }_{94}\right\}$ $X_{10}=\left\{\right.$available, not available$\left.\right\}=\left\{{\mho }_{101},{\mho }_{102}\right\}$ $X_{11}=\left\{\right.$available, not available$\left.\right\}=\left\{{\mho }_{111},{\mho }_{112}\right\}$ $X_{12}=\left\{\right.$less than or equal to 300 m, more than 300 m but less than 1 km, more than 1 km$\left.\right\}=\left\{{\mho }_{121},{\mho }_{122},{\mho }_{123}\right\}$ $X_{13}=\left\{\right.$less than or equal to 500 m, more than 500 m$\left.\right\}=\left\{{\mho }_{131},{\mho }_{132}\right\}$ $X_{14}=\left\{\right.$less than or equal to 5 min, greater than 5 min$\left.\right\}=\left\{{\mho }_{141},{\mho }_{142}\right\}$ $X_{15}=\left\{\right.$less than or equal to 500 m, greater than 500 m$\left.\right\}=\left\{{\mho }_{151},{\mho }_{152}\right\}$ $X_{16}=\left\{\right.$concrete, wood$\left.\right\}=\left\{{\mho }_{161},{\mho }_{162}\right\}$ By consulting experts some sub-attributes are preferred over others. In X₁, ℧₁₂ is preferred over others. In X₂, ℧₂₂ is preferred, in X₃, ℧₃₂ is preferred, in X₄, ℧₄₃ is preferred, in X₅, ℧₅₁ and ℧₅₂ are given equal preference. In X₆, ℧₆₂ is preferred, in X₇, ℧₇₂ is preferred, in X₈, ℧₈₂ is preferred, in X₉, ℧₉₁, ℧₉₂ are given equal preference. In X₁₀, ℧₁₀₁ is preferred, in X₁₁, ℧₁₁₁ is preferred, in X₁₂, ℧₁₂₂ is preferred, in X₁₃, ℧₁₃₁ is preferred, in X₁₄, ℧₁₄₁ is preferred, in X₁₅, ℧₁₅₁ is preferred, and in X₁₆, ℧₁₆₁ and ℧₁₆₂ are given equal preference. Now ${\mathfrak{X}}^{\star }=X_1\times X_2\times ...X_{16}=\left\{{\mho }_1^{\star },{\mho }_2^{\star },...,{\mho }_8^{\star }\right\}$ where each ${\mho }_i^{\star },i=1,2,...,8$ is a sixteen tuple element. Then consider the following examples: Consider a situation where a real estate agent is approached by three decision-makers Δ₁, Δ₂ and Δ₃ to purchase a residential building. Prior to making a decision, each decision-maker must take into account his/her own set of criteria. Following that, they can build their interval-valued fuzzy hypersoft sets. Next, we choose a residential building based on the specifications of the sets of decision-makers by using OOPCS as a fuzzy set theory DM approach. Assume that Δ₁, Δ₂, Δ₃ create their respective interval-valued fuzzy hypersoft sets ${\mathbf{D}}_1^{\star },{\mathbf{D}}_2^{\star },{\mathbf{D}}_3^{\star }$ respectively that are displayed in the form of matrices as follow; ${\mathbf{D}}_1^{\star }=$ $\begin{array}{cc}\hfill \begin{array}{c}\hfill \hfill \\ \hfill {\ddot{\theta }}_1\hfill \\ \hfill {\ddot{\theta }}_2\hfill \\ \hfill {\ddot{\theta }}_3\hfill \\ \hfill {\ddot{\theta }}_4\hfill \\ \hfill {\ddot{\theta }}_5\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill \begin{array}{cccccccc}\hfill {\mho }_1^{\ast }\hfill & \hfill {\mho }_2^{\ast }\hfill & \hfill {\mho }_3^{\ast }\hfill & \hfill {\mho }_4^{\ast }\hfill & \hfill {\mho }_5^{\ast }\hfill & \hfill {\mho }_6^{\ast }\hfill & \hfill {\mho }_7^{\ast }\hfill & \hfill {\mho }_8^{\ast }\hfill \end{array}\hfill \\ \hfill \left(\begin{array}{cccccccc}\hfill \left[0.1,0.3\right]\hfill & \hfill \left[0.2,0.5\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.2,0.6\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.2,0.7\right]\hfill & \hfill \left[0.5,0.7\right]\hfill & \hfill \left[0.4,0.5\right]\hfill \\ \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.7,0.8\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.4,0.7\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.1,0.4\right]\hfill \\ \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.2,0.6\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.4,0.7\right]\hfill & \hfill \left[0.3,0.6\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.2,0.3\right]\hfill & \hfill \left[0.4,0.6\right]\hfill \\ \hfill \left[0.5,0.7\right]\hfill & \hfill \left[0.2,0.6\right]\hfill & \hfill \left[0.2,0.4\right]\hfill & \hfill \left[0.2,0.6\right]\hfill & \hfill \left[0.3,0.4\right]\hfill & \hfill \left[0.1,0.3\right]\hfill & \hfill \left[0.6,0.7\right]\hfill & \hfill \left[0.5,0.6\right]\hfill \\ \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.5,0.8\right]\hfill & \hfill \left[0.3,0.7\right]\hfill & \hfill \left[0.6,0.8\right]\hfill & \hfill \left[0.5,0.7\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.3,0.5\right]\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}$${\mathbf{D}}_2^{\star }=$ $\begin{array}{cc}\hfill \begin{array}{c}\hfill \hfill \\ \hfill {\ddot{\theta }}_1\hfill \\ \hfill {\ddot{\theta }}_2\hfill \\ \hfill {\ddot{\theta }}_3\hfill \\ \hfill {\ddot{\theta }}_4\hfill \\ \hfill {\ddot{\theta }}_5\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill \begin{array}{cccccccc}\hfill {\mho }_1^{\ast }\hfill & \hfill {\mho }_2^{\ast }\hfill & \hfill {\mho }_3^{\ast }\hfill & \hfill {\mho }_4^{\ast }\hfill & \hfill {\mho }_5^{\ast }\hfill & \hfill {\mho }_6^{\ast }\hfill & \hfill {\mho }_7^{\ast }\hfill & \hfill {\mho }_8^{\ast }\hfill \end{array}\hfill \\ \hfill \left(\begin{array}{cccccccc}\hfill \left[0.2,0.3\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.6,0.8\right]\hfill & \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.3,0.7\right]\hfill & \hfill \left[0.6,0.8\right]\hfill & \hfill \left[0.3,0.4\right]\hfill & \hfill \left[0.5,0.8\right]\hfill \\ \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.2,0.4\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.6,0.8\right]\hfill & \hfill \left[0.3,0.7\right]\hfill & \hfill \left[0.5,0.7\right]\hfill & \hfill \left[0.5,0.7\right]\hfill & \hfill \left[0.3,0.5\right]\hfill \\ \hfill \left[0.5,0.7\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.2,0.3\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.2,0.4\right]\hfill & \hfill \left[0.2,0.6\right]\hfill & \hfill \left[0.4,0.7\right]\hfill & \hfill \left[0.5,0.8\right]\hfill \\ \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.7,0.9\right]\hfill & \hfill \left[0.2,0.4\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.3,0.7\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.2,0.4\right]\hfill \\ \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.4,0.7\right]\hfill & \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.6,0.7\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.2,0.3\right]\hfill & \hfill \left[0.6,0.7\right]\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}$${\mathbf{D}}_3^{\star }=$ $\begin{array}{cc}\hfill \begin{array}{c}\hfill \hfill \\ \hfill {\ddot{\theta }}_1\hfill \\ \hfill {\ddot{\theta }}_2\hfill \\ \hfill {\ddot{\theta }}_3\hfill \\ \hfill {\ddot{\theta }}_4\hfill \\ \hfill {\ddot{\theta }}_5\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill \begin{array}{cccccccc}\hfill {\mho }_1^{\ast }\hfill & \hfill {\mho }_2^{\ast }\hfill & \hfill {\mho }_3^{\ast }\hfill & \hfill {\mho }_4^{\ast }\hfill & \hfill {\mho }_5^{\ast }\hfill & \hfill {\mho }_6^{\ast }\hfill & \hfill {\mho }_7^{\ast }\hfill & \hfill {\mho }_8^{\ast }\hfill \end{array}\hfill \\ \hfill \left(\begin{array}{cccccccc}\hfill \left[0.1,0.2\right]\hfill & \hfill \left[0.3,0.4\right]\hfill & \hfill \left[0.2,0.3\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.2,0.5\right]\hfill & \hfill \left[0.3,0.7\right]\hfill \\ \hfill \left[0.3,0.4\right]\hfill & \hfill \left[0.3,0.6\right]\hfill & \hfill \left[0.2,0.5\right]\hfill & \hfill \left[0.1,0.3\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.2,0.5\right]\hfill & \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.3,0.6\right]\hfill \\ \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.3,0.4\right]\hfill & \hfill \left[0.5,0.8\right]\hfill & \hfill \left[0.5,0.6\right]\hfill & \hfill \left[0.5,0.8\right]\hfill & \hfill \left[0.2,0.5\right]\hfill & \hfill \left[0.4,0.7\right]\hfill & \hfill \left[0.2,0.4\right]\hfill \\ \hfill \left[0.2,0.6\right]\hfill & \hfill \left[0.4,0.5\right]\hfill & \hfill \left[0.3,0.6\right]\hfill & \hfill \left[0.6,0.8\right]\hfill & \hfill \left[0.7,0.8\right]\hfill & \hfill \left[0.3,0.4\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.2,0.5\right]\hfill \\ \hfill \left[0.7,0.9\right]\hfill & \hfill \left[0.2,0.3\right]\hfill & \hfill \left[0.4,0.7\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.6,0.7\right]\hfill & \hfill \left[0.4,0.6\right]\hfill & \hfill \left[0.3,0.5\right]\hfill & \hfill \left[0.4,0.8\right]\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}$

2. Construction of weighed interval-valued fuzzy parameter matrix MD displayed in Table 10:

3. Mean difference of weighted interval-valued fuzzy parameter matrix M by $x=\frac{b-a}{2}$ obtained for interval (a, b) is displayed in the Table 11

4. Construction of weighted vector W by utilizing Eq. (9), as follow ${\varpi }_1=\frac{0.1+0.1+0.1}{3}=0.1$, ${\varpi }_2=\frac{0.15+0.05+0.2}{3}=0.133$, ${\varpi }_3=\frac{0.1+0.05+0.05}{3}=0.066$, ${\varpi }_4=\frac{0.2+0.1+0.15}{3}=0.15$, ${\varpi }_5=\frac{0.05+0.05+0.15}{3}=0.083$, ${\varpi }_6=\frac{0.25+0.15+0.1}{3}=0.166$, ${\varpi }_7=\frac{0.1+0.1+0.05}{3}=0.083$, ${\varpi }_8=\frac{0.05+0.15+0.1}{3}=0.1$ So ${\sum }_{i=1}^8{\varpi }_i={\varpi }_1+{\varpi }_2+{\varpi }_3+{\varpi }_4+{\varpi }_5+{\varpi }_6+{\varpi }_7+{\varpi }_8=0.881$. Therefore $W_1=\frac{0.1}{0.881}=0.11351$, ${\mathbf{W}}_2=\frac{0.133}{0.881}=0.15096$, ${\mathbf{W}}_3=\frac{0.066}{0.881}=0.07491$, ${\mathbf{W}}_4=\frac{0.15}{0.881}=0.17026$, ${\mathbf{W}}_5=\frac{0.083}{0.881}=0.09421$, ${\mathbf{W}}_6=\frac{0.166}{0.881}=0.18842$, ${\mathbf{W}}_7=\frac{0.083}{0.881}=0.09421$, ${\mathbf{W}}_8=\frac{0.1}{0.881}=0.11351$. Hence $\mathbf{W}=\left(0.11351,0.15096,0.07491,0.17026,0.09421,0.18842,0.09421,0.11351\right)$

5. Assume that decision makers Δ₁, Δ₂, Δ₃ create their respective interval-valued fuzzy hypersoft sets ${\mathbf{D}}_1^{\star },{\mathbf{D}}_2^{\star }$ and ${\mathbf{D}}_3^{\star }$ that are displayed in Tables 12, 13 and 14 respectively;

6. Construction of average interval-valued fuzzy hypersoft decision matrix using Eq. (10) displayed in Table 15

7. Construction of Mean difference of average interval-valued fuzzy parameter matrix V^⋆ displayed in Table 16 by $x=\frac{b-a}{2}$ for interval (a, b)

8. Weighted fuzzy decision matrix V^⋆ displayed in Table 17 is constructed utilizing Eq. (11) as follow:

9. Fuzzy valued +ive ideal solution (⁺IS) and −ive ideal solution (⁻IS) can be obtained using Eqs. (12) and (13) as follow A⁺=⁺IS = $\left\{\begin{array}{c}{\breve{\zeta }}_1^{+}=0.0132432117,{\breve{\zeta }}_2^{+}=0.0176125032,{\breve{\zeta }}_3^{+}=0.0124852497,\hfill \\ {\breve{\zeta }}_4^{+}=0.0227007658,{\breve{\zeta }}_5^{+}=0.0125619614,{\breve{\zeta }}_6^{+}=0.0282630000,\hfill \\ {\breve{\zeta }}_7^{+}=0.0109914807,{\breve{\zeta }}_8^{+}=0.0151354234\hfill \end{array}\right\}$A⁻=⁻IS = $\left\{\begin{array}{c}{\breve{\zeta }}_1^{-}=0.0075677117,{\breve{\zeta }}_2^{-}=0.0150960000,{\breve{\zeta }}_3^{-}=0.0062422503,\hfill \\ {\breve{\zeta }}_4^{-}=0.0141877658,{\breve{\zeta }}_5^{-}=0.0047105000,{\breve{\zeta }}_6^{-}=0.0188420000,\hfill \\ {\breve{\zeta }}_7^{-}=0.0078514614,{\breve{\zeta }}_8^{-}=0.0113510000\hfill \end{array}\right\}$

10. Calculating ${\mathbf{SM}}_i^{+}$ and ${\mathbf{SM}}_i^{-}$ for i = 1, 2, 3 from Eqs. (5) and (6) ${\mathbf{SM}}_1^{+}=$ $\sqrt{\left\{\begin{array}{c}{\left(0.0132432117-0.0075677117\right)}^2+{\left(0.0176125032-0.0150960000\right)}^2+\hfill \\ {\left(0.0124852497-0.0062422503\right)}^2+{\left(0.0227007658-0.0170260000\right)}^2+\hfill \\ {\left(0.0125619614-0.0109914807\right)}^2+{\left(0.0282630000-0.0251239228\right)}^2+\hfill \\ {\left(0.0109914807-0.0094210000\right)}^2+{\left(0.0151354234-0.0151354234\right)}^2\hfill \end{array}\right\}}$ $=\sqrt{\left\{\begin{array}{c}{\left(0.0056755000\right)}^2+{\left(0.0025165032\right)}^2+\hfill \\ {\left(0.0062429994\right)}^2+{\left(0.0056747658\right)}^2+\hfill \\ {\left(0.0015704807\right)}^2+{\left(0.0031390772\right)}^2+\hfill \\ {\left(0.0015704807\right)}^2+{\left(0\right)}^2\hfill \end{array}\right\}}$$=\sqrt{\left\{\begin{array}{c}0.00003221130+0.00000633279+\hfill \\ 0.00003897504+0.00003220297+\hfill \\ 0.00000246641+0.00000985381+\hfill \\ 0.00000246641+0\hfill \end{array}\right\}}$$=\sqrt{0.00012450873}=0.011158348$ Similarly ${\mathbf{SM}}_1^{-},{\mathbf{SM}}_2^{+},{\mathbf{SM}}_2^{-},{\mathbf{SM}}_3^{+},{\mathbf{SM}}_3^{-},{\mathbf{SM}}_4^{+},{\mathbf{SM}}_4^{-},{\mathbf{SM}}_5^{+},{\mathbf{SM}}_5^{-}$ can be calculated: ${\mathbf{SM}}_1^{-}=0.01018597849$, ${\mathbf{SM}}_2^{+}=0.01101842605$, ${\mathbf{SM}}_2^{-}=0.00935871757$, ${\mathbf{SM}}_3^{+}=0.00943778278$, ${\mathbf{SM}}_3^{-}=0.01348368274$, ${\mathbf{SM}}_4^{+}=0.01180231684$, ${\mathbf{SM}}_4^{-}=0.01127467957$, ${\mathbf{SM}}_5^{+}=0.01532198987$, ${\mathbf{SM}}_5^{-}=0.00697154082$

11. Relative closeness of alternatives to the ideal solution can be calculated as follow: ${\mathbf{RC}}_1^{+}=\frac{{\mathbf{SM}}_1^{-}}{{\mathbf{SM}}_1^{-}+{\mathbf{SM}}_1^{+}}=\frac{0.01018597849}{0.01018597849+0.011158348}=\frac{0.01018597849}{0.02134432649}=0.4772218273$ Hence ${\mathbf{RC}}_1^{+}=0.4772218273$ Similarly ${\mathbf{RC}}_2^{+}=0.45927524213$, ${\mathbf{RC}}_3^{+}=0.5882556998$, ${\mathbf{RC}}_4^{+}=0.48856789548$, ${\mathbf{RC}}_5^{+}=0.31271586887$

12. Ranking the preference order is ${\ddot{\theta }}_5\le {\ddot{\theta }}_2\le {\ddot{\theta }}_1\le {\ddot{\theta }}_4\le {\ddot{\theta }}_3$ As benefit parameters are preferred over cost parameters, so ${\ddot{\theta }}_3$ is selected. The pictorial form of ranking is shown in Fig. 4.