Identification of desalination and wind power plants sites using m-polar fuzzy Aczel–Alsina aggregation information

Real-world decision-making problems often include multi-polar uncertainties dependent on multi-dimensional attributes. The m-polar fuzzy (mF) sets can efficiently handle such multi-faceted complications with T-norm based weighted aggregation techniques. The Aczel–Alsina T-norms offer comparatively flexible and accurate aggregation than the other well-known T-norm families. Consequently, this work introduced novel mF Aczel–Alsina aggregation operators (AOs), including weighted averaging (mFAAWA, mFAAOWA, mFAAHWA) and weighted geometric (mFAAWG, mFAAOWG, mFAAHWG) AOs. The fundamental properties, including boundedness, idempotency, monotonicity, and commutativity are investigated. Based on the proposed AOs, a decision-making algorithm is developed and implemented to solve two detailed multi-polar site selection problems (for desalination plant and for wind-power plant). Finally, a comparison with mF Dombi and mF Yager AOs reveals that different T-norm based AOs may yeild different solutions for the same problem.

www.nature.com/scientificreports/membership degree (MD) and non-membership degree (NMD) with their particular summation restrictions).In 1994, Zhang 9 introduced another extension of FSs, namely, bipolar FS (BFS) theory.The main idea behind this contribution is the bipolar information existing about a particular set.This bipolar information includes the positive and negative sides of an aspect (property and its counter-property) like good and bad, young and dull, fair and unfair, etc.To depict the notion of bipolarity, BFSs consider memberships from closed interval [ −1 , 1], such that 0 indicates irrelevance to the criteria, MD from [ −1 , 0) indicates satisfaction of object with the corre- sponding counter-criteria, and MD from (0, 1] depicts satisfaction with the considered criteria.Till date, several significant researches have been conducted on this theory to make more enhanced decision-making methods (see Refs. [10][11][12][13] ).Apart from the research on decision models, fuzzy theory has spread its roots in many technical fields and has been applied to many future-oriented applications.Recently, Talpur et al. 14 discussed the deep neuro-fuzzy systems, their applications, challenges, and possibilities.Deveci et al. 15 utilized an interval type-2 fuzzy set based method for saving the environment by improvising the sustainable vehicle shredding facilities.
Various daily-life scenarios are governed by multi-polar attributes affecting multi-faceted characteristics of the alternatives which leads to the emergence of multi-polar information unsolvable by conventional mathematical tools such as crisp set theory, FS theory, intuitionistic FS theory, and bipolar FS theory.Contemporary researchers are increasingly recognizing the substantial role of multi-polarity in a vast range of domains, spanning from medical sciences to engineering, management to neural fuzzy developments.For instance, consider the realm of information technology, where multi-polar technology can be employed to analyze complex information systems with varying attributes like latency, bandwidth, radio frequency, and network range.In neurobiology, the interconnectedness of neurons in the brain gather data from different other neurons, considering a multi-polar information gathering procedure.Likewise, within a social network, individuals may exhibit varying levels of effectiveness in their trading relationships, proactiveness, and sociability, all of which involve multi-polar data.To address these multi-polar scenarios, Chen et al. 16 initiated the theory of m-polar fuzzy (mF) sets, specifically formulated to tackle multi-polarity in datasets across diverse domains of modern sciences.These mF sets ( m ≥ 2 generalizing bipolar fuzzy sets) consider separate fuzzy memberships for m distinct dimensions of a particular criteria/characteristic/attribute.Some recent applications include mF algorithm applied to selection of non-tradional machining process 17 , mF networks utilized for product manufacturing problems 18 , and more.
These days, MCDM methods based on aggregation operators (AOs) are playing an increasingly significant role in various fields, including engineering, medicine, economics, environmental sciences, and more.Consequently, several decision-making techniques based on AOs have been introduced for MCDM to enhance the precision of optimal decisions, and they continue to evolve for further advancements.For example, Xu 19 explored some AOs based on intuitionistic FSs, including weighted, ordered weighted, and hybrid weighted averaging AOs (for more details on intuitionistic FS-based AOs, see Refs. 20,21).Garg et al. 22 presented Schweizer-Sklar prioritized AOs for IFSs with their decision-making applications.Peng and Yang 23 presented certain basic notions of Pythagorean FS-based AOs within an interval-valued context.Over the last decade, there have been several noteworthy studies focused on the aggregation of bipolar data using leveraging established operations.For instance, Wei et al. 24 introduced Hamacher aggregation operators tailored to bipolar data and investigated their applications in MCDM.Jana et al. 25 devised AOs based on bipolar information utilizing Dombi's operations, effectively addressing practical problems in daily-life.Moreover, Jana et al. 26 pioneered the advancement of bipolar fuzzy Dombi prioritized AOs with their innovative contributions.Subsequently, experts have investigated the aggregation of mF datasets using different established AOs.For instance, Waseem et al. 27 proposed mF Hamacher AOs and implemented them to solve MCDM problems.Khameneh and Kilicman 28 developed mF soft weighted AOs, which were effectively used to address MCDM problems.Akram et al. 29 presented mF Dombi AOs and explored their applications in MCDM.Later, Naz et al. 30 introduced innovative 2-tuple linguistic bipolar fuzzy Heronian mean AOs for group decision-making.Recently, Ali et al. 31 proposed specific arithmetic and geometric AOs for the aggregation of mF datasets with Yager's operations.For more MCDM applications of AOs, the readers may visit [32][33][34][35] .
In early 1980s, Aczel and Alsina 36 introduced the Aczel-Alsina t-norm (TN) and t-conorm (TCoN) as modified forms of the algebraic norms.Among the other TNs (and TCoNs), Aczel-Alsina TN (and TCoN) provide more accurate decisions.To demonstrate this, Farahbod and Eftekhari 37 classified nine different TN-based AOs on the basis of their aggregating accuracy in analyzing 12 different datasets.The Aczel-Alsina operators dominated in the analysis showing minimum error as compared to other operators.Consequently, recent researches have focused on Aczel-Alsina TNs and TCoNs based AOs for all the above discussed theories.Mahmood et al. 38 presented Aczel-Alsina TN and TCoN based AOs in bipolar complex fuzzy environment and explored their application in selecting the best operating system (see also, Mahmood and Ali 39 ).Akram et al. 40 used generalized orthopair fuzzy Azcel-Alsina AOs for energy resource selection.Ali et al. 41 introduced intuitionistic fuzzy soft Aczel-Alsina AOs.Wang et al. 42 utilized Aczel-Alsina based Hamy-Mean AOs for T-spherical fuzzy MCDM.Some other contributions include [43][44][45] .Considering this efficiency and applicability of Aczel-Alsina TN and TCoN, and the need for multi-polar fuzzy aggregation, this work focuses on the development of mF set-based Aczel-Alsina AOs.The motivations for the proposed work are listed below: 1. Aczel-Alsina TN/TCoN are more flexible and accurate as compared to other TNs/TCoNs in their aggregation capabilities.Existing literature 37 clearly demonstrates this dominating accuracy of the Aczel Alsina AOs. 2. Real-world decision-making problems like site selection for a massive project are often based on multifaceted information in the form of multi-agent, multi-attribute, multi-polar uncertainties.Crisp, fuzzy, and bipolar fuzzy models fail to solve such multi-faceted decision-making problems effectively.The upcoming work is structured as follows: 1. Section "Preliminaries" revisits some mF concepts and recalls the Aczel-Alsina TN and T-CoN.2. Section "mF Aczel-Alsina AOs" proposes novel mF Aczel-Alsina averaging and geometric AOs (weighted, ordered weighted, hybrid weighted).This section further analyses fundamental properties of proposed AOs.

Section "Applications to MCDM with mF information" introduces a unique MCDM algorithm based on mF
Aczel-Alsina AOs.Under the offered methodologies, detailed modeling and solutions to two multi-polar site selection problems (for a desalination facility and a wind-power plant) are presented.4. Section "Discussion" compares the proposed AOs with mF Yager AOs 31 and mF Dombi AOs 29 .The advantages and limitations of the proposed work are shortly discussed.5. Section "Conclusions and future plans" gives the conclusive remarks and future directions.

Preliminaries
The following definition recalls multi-polar fuzzy (mF) sets: Definition 2.1 16 A mF set on a universal set U is a mapping η : U → [0, 1] m .The belongingness of each alterna- tive is given by such that P i • η : [0, 1] → [0, 1] is the i th projection mapping.

mF Aczel-Alsina AOs
This section firstly gives Aczel-Alsina operations for mF numbers via Aczel-Alsina TN and TCoN, and then proceeds to development of mF Aczel-Alsina averaging and geometric AOs while discussing their basic properties.

mF Aczel-Alsina operations
Let there be three mF numbers

mF Aczel-Alsina weighted averaging AOs
This subsection establishes mFAAWA, mFAAOWA and mFAAHWA AOs and investigates some of their basic properties.In the coming developments, γ=(γ 1 , γ 2 , γ 3 , . . ., γ n ) acts as the weight vector with γ k > 0 and Proof By using induction method, for n = 1 and Hence, the result obtained by mFAAWA operator satisfies Equation (1) for n = 1 .Now, suppose that Eq. ( 1) is true for n = t , then Next it needs to be shown that the theorem holds for n = t + 1 .By putting n = t + 1 in Eq. ( 1), Hence, Eq. ( 1) verifies for n = t + 1 , which proves the theorem by induction method.
Proofs of the following two theorems are similar to the proof of Theorem 3.2, and therefore omitted.The following defines the mFAAOWA AOs.

Remark 3.1
The mFAAOWA AOs satisfy all the basic properties including idempotency, monotonicity, and boundedness as discussed in Theorems 3.2, 3.3, and 3.4, respectively.Coming definition gives the concept of mFAAHWA AOs as hybridization of the previous two AOs.
Theorem 3.12 Let η k = (P 1 • η k , . . ., P m • η k ) be the set of mF numbers with k = 1, 2, . . ., n .Then, an output value obtained after aggregation of these numbers using the mFAAOWG is also an mFNs given by  .

Applications to MCDM with mF information
This section aims to illustrate the practical implementation of newly proposed AOs.Two multi-polar site selection problems are intricately discussed and solved using newly developed algorithm based on mFAAWA and mFAAWG AOs.
Definition 4.1 Consider {X 1 , X 2 , . . ., X n } is the universal set and {T 1 , T 2 , . . ., T k } is the universe of attributes.Suppose γ = (γ 1 , γ 2 , . . ., γ k ) is the weight vector corresponding to attributes T i (1 ≤ i ≤ k) .Then the mF deci- sion-matrix representing the estimations of experts in mF environment is formulated as Algorithm 1 represents the proposed method of decision-making with mFAAWA and mFAAWG AOs. Figure 1 gives a pictorial explanation of the proposed algorithm.
Algorithm 1: Selection of best option using mFAAWA or mFAAWG operators.

Aggregation of Decision Matrix
Compute the aggregated mFNs xi corresponding to Xi s using mFAAWA or mFAAWG AOs

Score Values
Calculate the score S of each mFN xi Rank the alternatives in descending order by using score values

Ranking Decision
The alternative having max score value is the best choice.For more alternatives bearing same max value, any of these can be choosen as optimal.
Output 1. Input: a universal set X containing 'n' alternatives, a set of attributes T j with j varies from 1 to k, a weight vector γ j where k j=1 γ j = 1, and the mF decision matrix Ñ = (q it ) n×k = P 1 • η it , P 2 • η it , . . ., P m • η it n×k corresponding to the available attributes.
2. Compute the aggregated value qs corresponding to each object of the universal set using mFAAWA as follows: If mFAAWG operator is used for aggregation, then 3. Calculate the scores for each alternative X s where s varies from 1 to n.
4. Finally, rank the alternatives in descending order with respect to score values.

Output:
The alternative having the maximum score value is the best one.If more than one alternatives have the same maximum score value, then any of them could be chosen as optimal choice.

Selection of suitable site for desalination plant
The planet earth inhabiting around eight billion people has only 2.5% of its water categorized as fresh water.And unfortunately, only a fraction of this part is usable by the people on earth.The remaining water is sea water (salt water) unuseable in its current state, specifically for drinking.Many parts of the world have no or limited access to the fresh water.Such places make use of the process of desalination (removal of salts from water leaving desalinated water) in desalination plants to turn sea water into drinking and useable water.Globally around 1% of the total drinking water is provided by desalination.This seemingly low percentage accounts to more than 300 million people from around 150 countries of the world, depending partially or completely on this desalinated water for their daily requirements.The island countries and the middle east region heavily depend on desalination.
Countries like Maldives, Kuwait, Bahrain, UAE, Saudi Arabia, etc., heavily contribute to the global desalinated water production.Middle east generates around 60% desalinated water of the whole world with only Saudi Arabia producing 117 million cubic feet per day.This heavy contribution accounts for 50% of the fresh water needs of the Saudi Arabian inhabitants.Desalination is an effective solution for the drinking water requirements.However, it is a complicated process with many possible concerns including high power consumption (which may lead to more fossil fuel pollution in case of independent power sources) for pressurization, intake, and reverse osmosis; environmental concerns including possible damage to the marine life, pollution of sea water, non-uniform salinity, heavy brine (brine refers to solution of water containing a lot more salt than the sea water, returned to sea after desalination) disposal, water pre-treatment chemicals disposal, and increased erosion; social and legal concerns including the use of public water, effects to the surrounding inhabitants, effects to legal rights of correspondents, water distribution, possible pollution, employment opportunities, and finally, the financial constraints.Taking these concerns into consideration, the site for a desalination plant must be choosen wisely, as it has long-lasting impacts on the overall production, maintenance, and surrounding areas.Consider a country dependent on sea water is concerned about its increasing need of drinking water.Despite its many coastal areas, the selection of the best site for construction of a desalination plant is crucial and worthy of critical decision-making.A committee including engineers, marine biologists, desalination experts, public representatives, government representatives, approval committee members, geography and topography scientists, local non-governmental organizations, and the consulted environmental standards testing members is constructed.This committee discusses the sites with initial surveys and tests, and comes up with 15 most suitable sites for the desalination plant.For the next phase of detailed evaluation of these sites, four parameters including location, sea-water quality, technical feasibility, and environment friendliness, are choosen by the committee.These four parameters are further categorized into four sub-parameters as described below: 1. Location (i) Near a suitable brine discharge area: For minimizing the structure and energy needed to discharge brine back into the sea safely.(ii) Near a power supply source or transmission: For eliminating the need of an independent power source and minimizing the power transmission costs.(iii) Close to a water supply main conveyor: Minimal distance from the main conveyor ensures safe and cheap transport and administration of the treated water, which can be further distributed by the conveyor easily.(iv) Near a well-structured road network: Desalination plant connected to or near a structured road network allows easy transport of machinery and resources during its construction and increases accessibility for further operations.

Seawater quality
(i) Low annual silt density: Low annual silt density range represents lower fouling capacity of sea water in the RO process ensuring longer and reliable operation of the membrane surfaces.(ii) Steady and suitable temperature: Steady and suitable temperature of the water ensures health of the reverse osmosis plant.(iii) Least contamination risk: Intake away from ports or industries ensuring least contamination risk from the hazardous pollutants.(iv) Low turbidity: Low turbidity ensures lower pre-treatment processing power and costs.

Technical feasibility
(i) Intake structures invulnerability against waves and storms: Intake structures should be located and designed so that they have minimal negative affects from the possible sea-waves and storms.(ii) Open intake suitability: Suitability for open intake ensures more water production as compared to walled intake.(iii) Safe brine discharge and quick dilution: Safe discharge of brine back into the sea water with possible post-processing ensuring quick dilution of the brine is necessary and is dependant on the distance from brine disposal site, height of the plant, post-processing, and dispersal.(iv) Ease of water transportation: Topography and behaviour of the site should be helpful in structuring plant for easily transporting the water throughout the process with minimum power.

Environment freindliness
(i) Safe distance from protected areas: Conserved and protected areas like marine ecosystems, wetlands, etc., must be at a safe distance from the desalination plant to protect them from possible intake and brine discharge effects.(ii) Distant from inhabited areas: Safe distance from communities and inhabitants must be maintained to reduce the effect of possible noise and smoke pollution.(iii) Safety of marine life: The intake, structure, power-source, and byproduct discharge should pose minimum to no harm to any marine life nearby.(iv) Safety mitigation against possible erosion: The site selection ensures limitation of possible erosion in order to maintain the seabed topography, and reduce the possible adverse effects.
After analyzing the sites X i : 1 ≤ i ≤ 15 with respect to the parameters T j : 1 ≤ j ≤ 4 , the committee generates a collective report in the form of a 4F decision matrix as shown in Table 1.The committee considers Algorithm 1 (flowchart in Fig. 1) to select the best site from the available most suitable sites.Consequently, committee assigns weights to the parameters as follows: Firstly, the optimistic approach is carried out using mFAAWA aggregation to choose the best site.
Step 1 Let p = 3 .Then using the mFAAWA operator, the values qs for the desalination plant sites X s : 1 ≤ s ≤ 15 are calculated as: The score values S(q s ) of all the above computed 4F numbers qs are provided as: Step 3 Finally according to the above scores, the objects are ranked as follows: Hence, X 8 is the best site for desalination plant with mFAAWA approach.Again for a pessimistic perspective, the process is repeated with mFAAWG operator.

Location
(i) Higher grounds: High plains, grounds, round tops of hills, etc., where wind is available at high speeds without interruption.(ii) Flat terrain: Wind turbines with giant and tall structures require smooth and flat terrains to base their structures.(iii) Unrestricted land availability: Site should not be a restricted zone (like near an airport) and should not have future land uses declared in the close radius.

Power generation
(i) Near to the grid: Less distance from the electricity grid ensures low transmission costs and ease of installation.(ii) Suitability with the power flow: The newly generated power should not disturb the existing power flow and impose minimum penetration to the electrical network.(iii) Suitability for big turbines: Big turbines imply more power generation, but need bigger towers and hence the suitable site.
4. Environment-friendliness (i) Safe distance from forests: Wind turbines can be dangerous to wild life and birds, therefore a safe distance is necessary to ensure safety of these animals.(ii) Minimum noise pollution: Wind turbines cutting air cause noise which can affect the human health, therefore ideally site is distant from inhabited areas.(iii) Minimum visual impact: Site should ensure minimum shadow flickers and visual pollution.

Cost
(i) Low installation cost: Different turbines work best in different conditions.Similarly near grid sites require less transmission and transformation installations.The site should ensure best working conditions with minimum installation costs.(ii) Affordable land cost: A site away from residential and commercial areas with suitable conditions in low cost is optimal for wind power installation.(iii) Low transportation cost: Connection to structured road network and accessibility minimizing the transportation costs particularly during construction, installation, and management.
Deep analysis of the sites with respect to above parameters results in a 3F decision matrix as presented in Table 2.The following weights are assigned to the parameters T j by the committee: Using Algorithm 1, the calculations to find the best site are firstly done with mFAAWA operator.
Step 1 Let p = 3 .Then using the mFAAWA operator, the values qs for the wind power plant sites X s : 1 ≤ s ≤ 8 are calculated as:  The score values S (q s ) of all 3F numbers qs are computed as: Step 3 Finally, the sites are ranked as follows: Hence, X 1 comes to be the best site for wind power plant.Again, the process is repeated with mFAAWG aggregation.
Step 1 Let p = 3 .Then using the mFAAWG operator, the values qs for the desalination plant sites X s : 1 ≤ s ≤ 8 are determined as: Step 2 The score values S (q s ) of all 3F numbers qs are calculated as: Step 3 Finally, the sites are ranked as below: This time, X 2 comes out to be the best site for the wind power plant.

Discussion
Aggregation operators allow to accumulate and interpret a huge data set by combining the impact of multiple related information bits into a single easily understandable entity.For decision-making with uncertain information based on a number of attributes, aggregation operators offer unique approximate solutions based on their foundational structures.This reason urges decision scientists and researchers to develop varying AOs for uncertain information systems, in order to consider multiple possible solutions effected by the trade-offs due to the varying structures of these AOs.For uncertain decision-making, these AOs are often based on T-Ns/T-CoNs customized to handle specific information.In the existing literature, multiple AOs have already been defined for mF information to accumulate, interpret, and appraise complicated multi-polar uncertainties.However, despite the dominating accuracy and efficient polarity demonstrated by Aczel-Alsina TN/TCoN based AOs, no work has yet established or discussed the impact of Aczel-Alsina AOs for mF information.Consequently, this work established mF Aczel-Alsina weighted AOs and demonstrated their decision-making capability with two detailed model site-selection problems.The following subsections discuss the advantages and limitations of the proposed methods shortly.In addition, comparison with some existing AOs is presented.

Comparison
Different AOs based on different TNs and TCoNs may generate different results with the same information.In order to demonstrate this variation of outcomes, previously developed mF weighted averaging and geometric AOs based on Yager and Dombi TNs/TCoNs have been considered.Application "Selection of suitable site for desalination plant" (Site selection for desalination plant) is taken as a test case to demonstrate this comparison.Consequently, the outcomes of proposed mFAAWA and mFAAWG AOs are compared with the outcomes of pre-existing mF Yager weighted averaging (mFYWA), mF Yager weighted geometric (mFYWG), mF Dombi weighted averaging (mFDWA), and mF Dombi weighted geometric (mFDWG) AOs.Tables 3 and 4 represent the conflicting scores and corresponding rankings with the considered AOs.This comparison is graphically represented in Fig. 2. Here, the results obtained with proposed Aczel-Alsina AOs are more inclined towards those with Yager AOs.The results with mFAAWA AOs are almost consistent with those obtained with mFYWA AOs (for instance, both declare X 8 as the optimal choice), however the scores with proposed averaging aggrega- tion sandwich in between the scores with Dombi(from above) and Yager(from below) methodologies.In case of weighted geometric aggregation, mFAAWG AOs show significant variation (or comparative accuracy) from mFYWG and mFDWG AOs. Figure 2 shows that the aggregation scores obtained with proposed geometric aggregation demonstrate minimum variations from the weakest conjunction (every T-norm is bounded above by q1 = (0.6234, 0.9133, 0.8969), q2 = (0.6311, 0.7884, 0.7101), q3 = (0.6384, 0.4675, 0.6680), q4 = (0.5140, 0.9426, 0.7872), q5 = (0.3049, 0.7886, 0.4982), q6 = (0.8537, 0.8248, 0.3183), q7 = (0.6616, 0.3223, 0.7348), q8 = (0.2619, 0.6656, 0.7347).
Table 4. Rankings with different mF AOs in desalination plant site selection.

Theorem 3 . 6 (
Commutative Law) Let η k and ηk be any two families of mF numbers where k = 1, 2, 3, . . ., n .Then where ηk is a random permutation of η k .Proof It is immediately shown by Definition 3.2.

Figure 2 .
Figure 2. Comparison of mF AOs outputs in the selection of desalination plant.
1. Development of novel Aczel-Alsina AOs for mF information including weighted averaging AOs (mFAAWA, mFAAOWA, mFAAHWA) and weighted geometric AOs (mFAAWG, mFAAOWG, mFAAHWG).2. Detailed analysis of the fundamental properties of proposed AOs including idempotency, monotonicity, boundedness, and commutativity.3. Development of a working decision-making algorithm for multi-polar information based on mFAAWA and mFAAWG AOs. 4. Deep investigation of two model multi-polar site selection problems (for desalination plant and for windpower plant) with proposed techniques, and ranking of available sites against multi-polar attributes under the novel decision-making algorithm.5. Discussion on the advantageous and limiting features of proposed techniques in addition to a comparative analysis with mF Yager AOs 31 and mF Dombi AOs 29 .

Table 1 .
4F decision matrix for desalination plant sites.

Table 2 .
3F decision matrix for wind power plant sites.