Intelligent Automation & Soft Computing DOI:10.32604/iasc.2021.016822 | |

Article |

Solid Waste Collection System Selection Based on Sine Trigonometric Spherical Hesitant Fuzzy Aggregation Information

1Deanship of Combined First Year, Umm Al-Qura University, Makkah, Saudi Arabia

2Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Pakistan

3Department of Mathematics and Statistics, Bacha Khan University, Charsadda, 24420, Pakistan

4Department of Mathematics, Deanship of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia

*Corresponding Author: Saleem Abdullah. Email: saleemabdullah@awkum.edu.pk

Received: 10 January 2021; Accepted: 12 February 2021

Abstract: Spherical fuzzy set (SFS) as one of several non-standard fuzzy sets, it introduces a number triplet (a,b,c) that satisfies the requirement

Keywords: Spherical fuzzy set; Hesitant fuzzy set; spherical hesitant fuzzy set; Sine trigonometric aggregation information; decision making

The smart city idea is focused on the incorporation of information and communication technologies (ICTs) into city services to accumulate information for the allocation of assets and services, along with enhancing value of life and susceptibility. Difficulties and inefficient solutions in today’s cities raise a need for the smart apps to solve existing challenges effectively. In this background, many academics, architects, urban planners, and even municipal representatives have been drawn to smart city applications. In addition, a wide variety of implementations are available for the smart city concept in areas such as town planning, waste disposal, resource management and municipal services [1–3].

In particular, the problem of solid waste collection system selection is a multi-criteria decision-making problem that should also take into account environmental, social, economic and technological aspects. Due to the labor strength of the job and the use of the large number of vehicles in these processes, sample and transport are commonly considered to be the most critical and expensive phases of the process [4]. As ever, certain issues such as quick disposal of wastes from half-full bins and thus excessive fuel use of collection and transport vehicles, higher pollution levels, inefficient usage of city assets and services are raised in the predefined scheduling [5]. In this respect, the brand-new visible light communication (VLC) technique makes it possible to communicate ultra-fastly among terminals through light bulbs and to become an essential competitor for conventional radio frequency (RF) communication like Wi-Fi [6]. Without providing some other [7] communication method, VLC can supply a room’s interior lighting and data exchange at the same time.

Multiple criteria group decision making (MCGDM) method [8–14] is a significant and arising subject to depict an approach for choosing the finest alternative with group of the decision makers (DMs) and conditions. Two serious tasks are there in this procedure. The first one is to depict the atmosphere where the values of various attributes can be scrutinized successfully, while the aggregation of the depicted data is the second task. Generally, the data which depict the substances are frequently taken in the form of the deterministic or crisp in nature. Though, with the rising complications of the frameworks step by step, hardly data can be accumulated, from the records, assets and specialists, in crisp form. Therefore, to present the data more openly, a notion of fuzzy sets

In this paper, we offered the pioneering view of T- spherical HeFS (T-SHFS) by using the concept of SFS and HeFS to check the incredibility and imprecise figures in DMPs to reform the supreme alternative in conferring to list of criteria. A DMPs AOs acts the supreme role to aggregate the data. Since, for every aggregation procedure the rules of the operation perform a key role. It is necessary to construct fresh laws for the operation and aggregation of T-SHFNs. Consequently, the aim of this paper is to suggest some new laws for the operation of T-SHFSs. Therefore, by using the above stated proofs, we offered the MCGDM algorithm to grip the assessment material for T-SHFSs.

This unit contains some basic definitions of FS, InFS, PytFS, SFS, HeFS and SHFS.

Definition 1. [18] Consider the ground set

where

Definition 2. [13] Consider the ground set

where

Definition 3: Consider the ground set

Where

Definition 4. [13] A T-spherical fuzzy set (T-SFS)

where

Definition 5. Suppose

(1)

(2)

(3)

Definition 6. Let

(1)

(2)

(3)

(4)

3 New Sine Trigonometric Operational Laws for T-SHFS

We express a number of new operational laws for T-SHFSs in this part.

Definition 7. Consider a non-empty set

It is obviously understood that the

the neutral function:

and the non-membership function:

Therefore

is a T-SHFS.

Definition 8. Let

then

4 Sine Trigonometric Aggregation Operators

On the basis of STOL of T-SHFNs, we describe the below weighted averaging and geometric aggregation operators. Let

Definition 9. Let

hence the mapping T-SpHeFWA is known as sine trigonometric T-SpHeF weighted averaging operator, where

Theorem 1. Consider the set

Proof: We prove the theorem using induction method. Since for each

(1) For

As from Definition 3.1, we can see that

Assume Eq. (1) holds for

Hence, Eq. (1) also valid for

Definition 10. A sine-trigonometric T-SHFN ordered weighted average (ST-T-SpHeFOWA) operator is a mapping ST-T-SpHeFOWA

where

Theorem 2. For a collection of

.

Definition 11. A sine-trigonometric T-SHFN hybrid average (ST-T-SpHeFHA) operator is a mapping ST-T-SpHeFHA

where

Theorem 3. For a collection of

.

Definition 12. Let

then the function ST-T-SpHeFWG is called sine trigonometric T-SpHeF weighted geometric operator, where

Theorem 4. Let

Definition 13. A sine trigonometric TSpHeF ordered weighted geometric (ST-TSpHeFOWG) operator is a mapping

where

Theorem 5. Let

Definition 14. A sine trigonometric T-SpHeF hybrid weighted geometric (ST-TSpHeFHG) operator is a mapping ST-TSpHeFHG

where

Theorem 6. Let

As similar to ST-TSHFWA operator, the ST-TSpHeFOWA, ST-TSpHeFHA, ST-TSpHeFWG, ST-TSpHeFOWG and ST-TSpHeFHG operators satisfy the properties such as boundedness, monotonicity.

Fundamental Properties of the Proposed AOs

In this subsection, we scrutinized some relations between the suggested AOs and study their various major properties as given below.

Theorem 7. For two T-SHFNs

Proof: Let

and

Since for any two non-negative real numbers

which further gives that

Similarly, we can obtain

And

Hence, by using Definition (i), we get

Theorem 8. Let

(1)

Proof: We will prove part (1) only. Part (2) can be obtained in a similar way. For this, let

and

For

Similarly, we can get

Therefore, from above Eqs, we get

Here, we have settled a structure for addressing improbability in decision making (DM) under TSpHeF material. Consider a DM problem with a set of m alternatives

where

Step-1: Construct the expert evaluation matrix

where

Step-2: Construct the normalized decision matrix

Step-3: Aggregate the individual decision matrices based on the T-sphrical hesitant fuzzy aggregation operators to construct the collective matrix. Exploit the established aggregation operators to achieve the TSHFN

Step-4: Compute the score of all the values

Step-5: Rank the alternatives

We analyze the results of the established MAGDM technique with a numerical example and compare the outcomes with the one of the existing MAGDM techniques, in this area. The aim of this research is to implement T-SpHeF data methodology in a smart city area to analyze and grade alternative waste collection systems.

Case Study; Municipal Waste Collection System Selection:

Descriptions of the Problem: For human health, aesthetics, and environmental consevation, municipal solid waste (SW) management is much more needed and essential facilities. It covers all the operations and steps necessary for waste management from selection to final disposal [1]. A critical stage of an effective waste management strategy is the identification of frequently contradictory natural, social and economic requirements and the list of alternatives. The waste disposal truck drives and stops at each building in this collection system to pick up the solid waste [2,3]. Four criteria are defined in this analysis;

Innovativeness and Aesthetics

Maintenance Efficiency

Sustainability

Setup Cost Advantage

Now following are the four alternatives concept and solution of above criteria. We suggest ideas for smart city solid waste collection (SWC) in this context, in which VLC are included. In addition, we use two new SWC ideas for smart cities in which Wi-Fi connectivity and cellular connectivity are used. Smart bins fitted with sensors, microprocessor, battery packs, compaction systems as well as solar power are taken into account in all alternatives. The bins are used both to collect waste and to collect waste data.

Wi-Fi-based SWC System

Cellular Communication-based SWC System

Li-Fi-based SWC System

Waste Management Collection and Transportation with Drones

Application of Proposed MAGDM Method

Suppose, three experts

Step-1: The expert evaluation information is in the form of

Step-2: The normalized expert evaluation information in enclosed in Tab. 2:

Step-3: In this case study, we have only one expert so, we have no need to normalized

Step-4: In this step, we calculate the combined preference values of alternatives under criteria weight is

Case 1: Using WA

We apply ST-TSpHeFWA aggregation operator to the data provided in above matrix to find out the aggregated values. The combined preference values of each alternative using WA

Case 2: Using ST-TSpHeFWG aggregation operator

We apply ST-TSpHeFWG aggregation operator to above matrix to find out the aggregated values. The combined preference values of each alternative using WGST–TSHF aggregation operator is enclosed in Tab. 4:

Step-5: Score of collective overall preference values of each alternative is enclosed in Tab. 5:

Step-6: Rank the alternatives

From the above computational process, we concluded that alternative Li-Fi-based solid waste collection system

7 Reliability and Validity Test

In fact, deciding the highest suitable alternative from the decision matrices provided by the group is extremely difficult. The approach to estimate the validity and reliability of decision-making approaches was started by Ashraf et al. [21]. The steps for testing are as follows.

Test Step-1.: If we substitute the normalized element for the worse element of the alternative by presenting the appropriate alternative with no modification and also with no altering the comparable position of each decision criterion, the appropriate and effective MAGDM technique is to do so.

Test Step-2.: Through an efficient and appropriate MAGDM procedure, transitive property must be met.

Test Step-3.: When an issue with MAGDM is turned into minor issues. A combined alternative rating should be equivalent to the original rating of un-decomposed problem to ranking the alternative, we apply identical approach on minor issues used in the problem of MAGDM.

To find the best result, the MAGDM problem was transformed into a smaller one and the same proposed decision-making approach were introduced. The suitable and efficient MAGDM strategy is that the outcome would be the same as the MAGDM problem if we apply the same technique to a small problem.

Validity Test the Proposed DM Methodology

In this area [21], using the validity and reliability test mentioned above, we check the appropriation and validation of our developed methodology. The normalized spherical hesitant fuzzy information is enclosed in the Tab. 2 (given Above):

Test Step-1: We substitute the normalized element for the worse element of the alternative by presenting the appropriate alternative with no modification and also with no altering the comparable position of each decision criterion, in this step. Tab. 7 enclosed the updated decision matrix

Now, we calculate the combined preference values of each alternative under criteria weight

Case-1: Using ST-TSpHeFWA aggregation operator:

The collective overall preference values of each alternative using ST-TSpHeFWA aggregation operator is enclosed in Tab. 8:

Case-2: Using ST-TSpHeFWG aggregation operator

The combined preference values of each alternative using ST-TSpHeFWG aggregation operator is enclosed in Tab. 9:

Now, Score of collective overall preference values of each alternative is enclosed in Tab. 10:

Rank the alternatives

We get again the same alternative

We are now testing the validity test Steps-2 & 3 to demonstrate that the proposed approach is reliable and relevant. To this end, we first transformed the MAGDM problem into three smaller sub-problems such as

In this analysis, using the T-spherical hesitant fuzzy set decision process, alternative municipal SWC systems based on various ICTs are analyzed and graded. Alternative SWC concepts are built on the above four alternatives, taking into account the current situation and needs of a study area. The case study is performed in an area where municipal authorities embrace the smart city strategy and there are ongoing smart city initiatives.

Attributes which are described above are taken into account when implementing the suggested T-spherical hesitant fuzzy set (T-SHFS) methodology. The outcomes of the study indicated that the more effective solutions for the survey are are Li-Fi and visible light communication-based collection systems. The findings of this research illustrate that in the smart city area, these devices can be chosen and applied, especially in the sense of SWC. The use of fuzzy sets has helped us to effectively transform the uncertainty and complexity of local decision-makers and scientific experts’ decisions. We developed certain robust sine-trigonometric (ST) operations laws (STOLs) for T-SHFSs and concluded new aggregation operators (AOs) to calculate T-SpHeF data which are ST weighted averaging and geometric operators.

In the future studies, the suggested T-SHFS methodology proposed here can be solve by applying TOPSIS method, q-ROFS based on real emergency and supply chain.

Acknowledgement: The authors would like to thank the Deanship of Scientific Research (DRS) at Umm Al-Qura University for supporting this work by Grant Code: 19-SCI-1-01-0041.

Funding Statement: The authors would like to thank the Deanship of Scientific Research (DRS) at Umm Al-Qura University for supporting this work by Grant Code: 19-SCI-1-01-0041.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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