An intuitionistic dense fuzzy AHP-TOPSIS method for military robot selection

Abstract

Land, marine and airborne are the three types of military robots used in the war-field. Land robots are the most crucially considered robots. Selecting a military land robot for a specific purpose is one of the challenging problems for a decision-maker to find the most preferred alternative when it involves fuzziness and uncertainty. Intangible factors are used while selecting the appropriate robotic system as it effectively deals with fuzziness. Intuitionistic dense fuzzy set, which is the combination of intuitionistic fuzzy set and dense fuzzy set, is capable of dealing with intangible factors. This study aims to design the integrated model on intuitionistic dense fuzzy AHP-TOPSIS to choose the most preferable military land robots under various circumstances. Robots for different types of situations, namely bomb disposal, search and rescue, surveillance and reconnaissance and war-fighter are considered. Moreover, the intuitionistic dense fuzzy AHP is utilized to calculate the subjective weights of the criteria and intuitionistic dense fuzzy TOPSIS is used to rank the alternatives. Further, a sensitivity analysis is examined to demonstrate the quality of the outcome and the results are compared with the fuzzy set, intuitionistic fuzzy set, and dense fuzzy set to show the efficiency of the proposed methodology.

Keywords

Robot selection intuitionistic dense fuzzy set intuitionistic trapezoidal dense fuzzy AHP intuitionistic trapezoidal dense fuzzy TOPSIS

Abbreviations

AHP

Analytic Hierarchy Process

BEAR

Battlefield Extraction-Assist Robot

Bomb Disposal robots

Battery

Cam

Camera Cap Capability

Consistency index

Consistency ratio

Criteria weights

DFS

Dense fuzzy set

DOF

Degree of freedom

Equally Important

ExI

Extremely Important

F-AHP

Fuzzy AHP

F-TOPSIS

Fuzzy TOPSIS

High Importance

IDFS

Intuitionistic Dense Fuzzy Set

IFS

Intuitionistic Fuzzy Set

ITpDF-AHP

Intuitionistic Trapezoidal Dense Fuzzy AHP

ITpDF-AHP-TOPSIS

Intuitionistic Trapezoidal Dense Fuzzy AHP-TOPSIS

ITpDFN

Intuitionistic Trapezoidal Dense Fuzzy Number

ITpDF-NIS

Intuitionistic Trapezoidal Dense Fuzzy Negative Ideal Solution

ITpDF-PIS

Intuitionistic Trapezoidal Dense Fuzzy Positive Ideal Solution

ITpDFS

Intuitionistic Trapezoidal Dense Fuzzy Set

ITpDF-TOPSIS

Intuitionistic Trapezoidal Dense Fuzzy TOPSIS

Intermediate Importance

MAARS

Modular Advanced Armed Robotic System

MARCbot

Multi-function Agile Remote-Controlled Robot

MATILDA

Mesa Associates Tactical Integrated Light-Force Deployment Assembly

MCDM

Multi-Criteria Decision Making

Moderate Important

Payload

QoS

Quality of Service

Random Index

S&R

Search and Rescue robots

Surveillance and Reconnaissance robots

THeMIS

Tracked Hybrid Modular Infantry System

TOPSIS

Technique for Order of Preference by Similarity to Ideal Solution

VHI

Very High Importance

War-fighter robots

1 Introduction

A massive war between Russia and Ukraine has recently engulfed the world, which is still invading. Many people have died as a result of this. According to the latest estimates (25-08-2022) from US officials quoted by “The New York Times”, around 25,000 Russian soldiers have been killed and more than 20,000 have been injured in Ukraine. Meanwhile, Russia’s defense ministry claims that over 9,000 Ukrainian soldiers and paramilitary fighters have been killed, with more than 8,000 wounded. This count could rise as the war is still invading. Military robots are used to prevent the loss of human life. By encouraging military robots, life-threatening tasks can be performed without losing human lives. Military robots are remote-controlled or autonomous robots that were designed for various military applications. Bomb disposal, search and rescue, surveillance and reconnaissance, and war-fighter are considered to be the four most vital applications in the military field. Land, marine and airborne are the three types of military robots used in the war field where land robots are the most crucially considered robots. The primary platforms for the land robots are wheeled, tracked, legged and wearable. The military robots were used in World War II and Cold War in the form of German Goliath-tracked mines and Soviet teletanks. In July 2002, during a war in Afghanistan, robots were used in ground combat for the first time. Since then, robots have been developed and trained for various purposes in the military field. The selection of robots for a specific application is just as crucial as using robots in military. When robots replace soldiers in military, the robot should wield more power than human. Additionally, it should take the position of the fieldworker’s ability, precision, and flexibility. Every military application demands for a unique characterization of robots that can mimic humans. Considering the four vital applications namely bomb disposal, search and rescue, surveillance and reconnaissance, and war-fighter the characterization of the robots for each situation is different. Robots used for bomb disposal are considered to have good camera with high degrees of freedom and the range to be high for the sensor of the remote which serves as a basic need. For search and rescue operations the basic characterization considered for the robot is its speed, camera and carrying capacity. In case of surveillance and reconnaissance, the basic criteria considered for the robot is its camera, range and battery. However, in case of war-fighter, number of weapons, camera and payload for the robots are given more weightage than the others. Therefore, the importance of the criteria varies with respect to their desired applications. Humans are capable of adapting their behaviour in accordance with their circumstances. However, it is different when a robot is involved. Hence, selecting a military land robot to replace humans for the desired application is considered to be the most challenging problem for a decision-maker in recognizing the most preferred alternative.

The approaches to robot selection are divided into five main groups. They are,

Multi-criteria decision-making (MCDM) models

Production system performance optimization models

Computer-assisted models

Statistical models

Other approaches

Among these, MCDM is one of the most widely used decision-making methodologies involving quantitative and qualitative factors in various fields and applications [1-8]. In reality, assigning accurate numerical values to these criteria is problematic. Thus, to address such types of problems, Zadeh (1965) [9] pioneered the fuzzy set theory where the elements and its memberships are given in ordered pairs. The selection of robots is made easy when fuzzy set is adopted in the MCDM methods. By including the element’s belongingness and non-belongingness of the set, fuzzy set was extended into intuitionistic fuzzy set (IFS). By integrating learning experience in the fuzzy set, the elements are considered to be a converging functions that converge to an element in the set and hence De (2016) [10] extended the fuzzy set into a dense fuzzy set (DFS). To overcome the drawbacks of DFS, De (2018) [11] combined the IFS and DFS and introduced an intuitionistic dense fuzzy set (IDFS). IDFS is found to be more general and flexible when all sets are compared. Thus, the purpose of this research is to propose MCDM in an intuitionistic dense fuzzy environment.

1.1 Review on dense fuzzy sets

A fuzzy set is a set that will have elements with their corresponding degree of membership. This idea was proposed by L.A. Zadeh in 1965 [9]] when the set with vague boundaries. For the proper use of linguistic variables, values are to be assigned. Thus, fuzzy numbers were introduced by L.A. Zadeh [12]. When the non-belongingness of the elements of the set was also to be denoted, IFS was initiated by Atanassov in 1986, which sometimes includes its hesitancy [13]. Every vague moments changes by incoperating learning experience to the fuzzy sets. Considering this idea in the fuzzy environment, De and Beg [10, 11] introduced a triangular dense fuzzy set. A DFS is a set whose elements are a sequence of functions that converges to an element in the set as n→ ∞. Swethaa and Felix [14] introduced DFS with respect to trapezoidal fuzzy number for various defuzzification methods. IDFS was introduced by Maity, De and Mondal [15] as it provides a hesitancy index that states the requirement of information for the element whether it belongs to the set or not. This paves an uncomplicated way for the decision-maker to rank his/her preferences. Thus, in this paper, the MCDM method is used in this IDFS on the trapezoidal fuzzy number for ranking the most preferred alternative. The problem of robot selection for the MCDM model in the field of Intuitionistic trapezoidal dense fuzzy number (ITpDFN) is examined which serves as the novelty for the paper.

1.2 Review on MCDM techniques

Several MCDM techniques were used in choosing the optimal probable options in recent years. Since 1960s, many theoretical and applied articles and books were written based on MCDM which has been an active research area. Technique for order performance by similarity to ideal solution (TOPSIS), Vlse kriterijumska optimizacija kompromisno resenje (VIKOR), Analytical hierarchy process (AHP), preference ranking organization method for enrichment evaluation (PROMETHEE), Weighted aggregated sum product assessment (WASPAS), etc., are some popular MCDM techniques that were developed over the last few decades. AHP and TOPSIS are the most commonly used MCDM methods. AHP is used in finding the criteria weights and TOPSIS is used in finding the preferred ranks for the alternatives. AHP was developed by Thomas L. Saaty in the 1970s and has been refined since then [16]. A paper in the journal of mathematical psychology [17] precisely described the AHP method. TOPSIS was coined in 1981 by Yoon and Hwang [18]. TOPSIS under fuzzy environment was used in the spillway selection [19]. Even though both numerical and linguistic terms can be dealt with AHP and TOPSIS, however, it cannot handle the vagueness and uncertainties that exist in a robot selection process [20]. The vagueness in the human preferences can be represented by incorporating fuzzy logic with the pair-wise relation comparison which was represented in [21]. Thus, Fuzzy AHP (F-AHP) model was introduced. The earliest reference of F-AHP was found to be dated from 1983 [22]. Then in 1992, TOPSIS was introduced in the fuzzy environment by S. J. Chen and C. L. Hwang [23].]. Since its inception, F-AHP and fuzzy TOPSIS (F-TOPSIS) has been widely used in many fields like supplier selection [24, 25], financial risk management [26], robot selection [27, 28] and so on. F-AHP is a scientific approach that assists to take a decision under fuzzy conditions. Patil [29] explained F-AHP in various distinctive applications in combination with the other methods. A review paper was conducted by Singh on AHP and F-AHP applications from the year 2010 to 2015 in various fields [30]. A dynamic sustainable management index in leisure agriculture was done with the help of F-AHP [31].

F-TOPSIS is one of the most effective methods for finding an optimal solution among a variety of options [18]. In 2016 [32], the equipment selection was made using the F-TOPSIS method and the result concluded that a F-TOPSIS method can assist the engineers to evaluate the alternatives in mining engineering effectively. A simplified description of F-TOPSIS method for MCDM was done in the year 2017 [33]. In 2020 [34], AHP-TOPSIS methods were used to find the best inspired shopping mall site selection. In the same year [35], optimal material selection for the different components of the cashew juice extractor was made using F-TOPSIS.

Extending the F-AHP and F-TOPSIS methods for the robot selection problem, numerous articles are present in various fields like industrial robots [36], medical robots [37, 38] and so on. In this article, military robots are considered and robot selections on the basis of four major applications namely bomb disposal, search and rescue, surveillance and reconnaissance and war-fighter are studied [39]. Considering these applications, a list of robots and their possible criteria are taken and robot selection is performed by combining intuitionistic trapezoidal dense fuzzy (ITpDF)-AHP (which gives the weightage) and ITpDF-TOPSIS (which lists the preferred ranking) in the field of intuitionistic dense fuzzy set.

1.3 Motivation

The uncertainty and vagueness of the fuzzy parameter reduces with the learning experience of the decision maker/ observer with respect to the frequency of obervation/ turn-over/ cycle time/ number of decision maker, etc. With this inkling, De and beg [10, 11] in 2016 introduced the DFS where they considered the elements to be a sequence of functions that converge to an element in the set. Taking up this idea to the intuitionistic fuzzy context, Maity, De and Mondal [15] introduced intuitionistic dense fuzzy set as it provides a hesitancy of the elements. The idea of knowing the elements belongingness and non-belongingness of the set including the hesitency index paves an uncomplicated way for the decision-maker to rank his/her preferences.

The literature has revealed some limitations stating that the formally proposed AHP and TOPSIS methods cannot be used directly for Intuitionistic trapezoidal dense fuzzy set (ITpDFS) and hence the motivation for the study are listed below.

With the help of ITpDFS, the fuzziness in the belongignness and non-belongingness of the elements becomes shorter and the excution is made with more approximation. Thus, introducing MCDM with ITpDF makes more sense.

A novel method is needed to propose any MCDM method under ITpDF field.

Moreover, the literature reveals that no study has documented the use of the AHP-TOPSIS approach in an ITpDF environment and so they are to be innovated. This idea provokes and motivates to indroduce an MCDM model under ITpDF.

Military robots can perform life-threatening tasks without the lose of human lives. But, recognizing the most preferred military robot to replace humans with respect to their application is a challinging task. Therefore, the idea of military robot selection using MCDM in ITpDF is found to be lagging and thus, they are incorporated in this article.

1.4 Contribution and novelty

The following are the paper’s major contributions

The concept of ITpDFN is defined and explained through graphical visualizations.

A novel AHP and TOPSIS methods are discussed under the ITpDF context which is named to be intuitionistic dense fuzzy AHP-TOPSIS (ITpDF-AHP-TOPSIS) methods.

In the context of ITpDF, traditional AHP and TOPSIS techniques are extended to deal with complex MCDM problems involving decision experts and criteria weights.

This study also identifies a new linguistic scale under ITpDF as this covers the scope of uncertain information when compared with the other extensions of fuzzy sets.

To demonstrate the utility and flexibility of ITpDF, four different types of situations namely bomb disposal, search and rescue, surveillance and reconnaissance and war-fighter robots has been considered.

To certify the obtained results, sensitivity investigation and comparative analysis are taken up.

The novelty of the present study are listed below.

The new concept of ITpDFN are introduced in this study.

The AHP-TOPSIS methods are incorporated with the defined ITpDFN and the novel model of ITpDF-AHP-TOPSIS are utilited to rank the most preferred alternatives.

The military robot selection is taken as the real time application for four different situation. Thus, robot selection on ITpDF-AHP-TOPSIS method in the field of ITpDF are constructed.

The paper is structured as follows: Section-1 covers the introduction part. Section-2 discussess some basic preliminaries of ITpDFS. Section-3 proposes a novel ITpDF-AHP-TOPSIS model. Section-4 dictates the data availability statement of the proposed problem and illustrates the application of the proposed model which includes the theory background of the robots used for robot selection and confers a case study with four applications. Futher, the sensitivity and comparative analysis of robot selection are executed. Finally, section-5 concludes the work with some suggestions for future research.

2 Theoretical background

This section contains some basic definitions on the concept of ITpDFS.

Definition 1. [40] An intuitionistic fuzzy set (IFS) $\tilde{A} = {(x, μ_{\tilde{A}} (x), γ_{\tilde{A}} (x)) : x \in ℝ}$ , where $μ_{\tilde{A}} \in [0, 1]$ and $γ_{\tilde{A}} \in [0, 1]$ determines the degree of belongingness and non-belongingness which satisfies the condition $0 ⩽ μ_{\tilde{A}} (x) + γ_{\tilde{A}} (x) ⩽ 1$ as $π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}} (x) - γ_{\tilde{A}} (x)$ called the Intuitionistic fuzzy index or hesitancy index of x in $\tilde{A}$ as $0 ⩽ π_{\tilde{A}} (x) ⩽ 1$ .

Definition 2. [10, 11] Let $\tilde{A}$ be a fuzzy set whose elements are sequence of functions that maps natural numbers $n \in ℕ$ with a crisp number x as n→ ∞, then the set considered is known as dense fuzzy set (DFS) where $μ_{\tilde{A}} : X \times ℕ \to [0, 1]$ .

It can be articulated as follows to represent them as a set: $\tilde{A} = {\begin{matrix} ((x, 1), μ_{\tilde{A}} (x, 1)), ((x, 2), μ_{\tilde{A}} (x, 2)), \\ ((x, 3), μ_{\tilde{A}} (x, 3)), \dots, ((x, n), μ_{\tilde{A}} (x, n)) \end{matrix}}$

The dense fuzzy set is graphically shown in Fig. 1.

Fig. 1

Diagrammatical depiction of DFS.

Definition 3. An intuitionistic trapezoidal dense fuzzy set (ITpDFS) $\tilde{A} = {b f_{n}, b, c, c g_{n}; b k_{n}, b, c, c l_{n}}$ with a′ = bk_n < a = bf_n < b = b < c = c < d = g_nc < d′ = l_nc where f_n, g_n, k_n and l_n are functions that converges to 1 as n→ ∞ then $\tilde{A} \to [b, c]$ . Thus $\tilde{A}$ is known as ITpDFS.

Example 4. An example is formed from the definition (3) for the ITpDFS as follows:

Membership function, $\tilde{A} = {b (1 - \frac{ρ}{1 + n}), b, c, c (1 + \frac{σ}{1 + n})},$ (1) for 0 < ρ, σ < 1

The degree of membership is represented by this function for 0 ⩽ n are as follows, $μ (x, n) = {\begin{matrix} 0, & if x < b (1 - \frac{ρ}{1 + n}) \\ {\frac{x - b (1 - \frac{ρ}{1 + n})}{\frac{ρ b}{1 + n}}}, & if b (1 - \frac{ρ}{1 + n}) ⩽ x ⩽ b \\ 1, & if b ⩽ x ⩽ c \\ {\frac{c (1 + \frac{σ}{1 + n}) - x}{\frac{σ c}{1 + n}}}, & if c ⩽ x ⩽ c (1 + \frac{σ}{1 + n}) \\ 0, & if x > c (1 + \frac{σ}{1 + n}) \end{matrix}$ (2)

Non-membership function, $\begin{matrix} \tilde{A} = {b (1 - (1 - ρ) e^{- n}), b, c, c (1 + (1 + σ) e^{- n})}, \\ for 0 < ρ, σ < 1 \end{matrix}$ (3)

The non-membership function for 0 ≤ n is defined as follows: $γ (x, n) = {\begin{matrix} 0, & if x < b (k_{n}) \\ {\frac{b - x}{b - b (k_{n})}}, & if b (k_{n}) ⩽ x ⩽ b \\ 1, & if b ⩽ x ⩽ c \\ {\frac{x - c}{c (l_{n}) - c}}, & if c ⩽ x ⩽ c (l_{n}) \\ 0 & if x > c (l_{n}) \end{matrix}$ (4)

The graphical representation of ITpDFS are shown in Figs. 2 3.

Fig. 2

Membership functions of ITpDFS.

Fig. 3

Non-Membership functions of ITpDFS.

Definition 5. The three types of arithmetic operations for ITpDFN are listed as follows.

Arithmetic operations based on (α, β)-cut method

Arithmetic operations based on fuzzy number

Arithmetic operations based on extension principle

(i) Operations based on (α, β)-cut method

If $\tilde{A}$ is an ITpDFN, then the (α, β)-cut is known by $\begin{matrix} {\tilde{A}}_{α, β} = {\begin{matrix} [A_{1} (α), A_{2} (α)] \\ [{A_{1}}^{'} (β), {A_{2}}^{'} (β)] \end{matrix} \begin{matrix} , α \in [0, 1] \\ , β \in [0, 1] \end{matrix} \\ with α + β ⩽ 1 \end{matrix}$

For instance,

Let $\tilde{A} = (a = b f_{n}, b, c, d = c g_{n}; a^{'} = b k_{n}, b, c, d^{'} = c l_{n})$ be a ITpDFN where f_n, g_n, k_n, l_n are the converging sequence respectively. Then the (α, β)–cut is represented as, $\begin{matrix} {\tilde{A}}_{α, β} = {[A_{1} (α), A_{2} (α)]; [{A_{1}}^{'} (β), {A_{2}}^{'} (β)]}, \\ α + β ⩽ 1, α, β \in [0, 1] \end{matrix}$ where, $A_{1} (α) = α (b - b f_{n}) + b f_{n};$ $A_{2} (α) = c g_{n} - α (c g_{n} - c)$ $A_{1^{'}} (β) = b - β (b - b k_{n});$ $A_{2^{'}} (β) = c + β (c l_{n} - c))$ and, $(α (b - b f_{n}) + b f_{n}) + (β (x - x k_{n}) + x k_{n})$

The interval value of the ITpDFN for both membership and non-membership is defined as [α (b - bf_n) + bf_n, cg_n - α (cg_n - c)] & [b - β (b - bk_n) , β (cl_n - c) + c] respectively.

Property-1: Let $\tilde{A} = (a = b f_{n}, b, c, d = c g_{n},; a^{'} = b k_{n}, b, c, d^{'} = c l_{n})$ and $\tilde{B} = (w = x f_{n}, x, y, z = y g_{n}; w^{'} = x k_{n}, x, y, z^{'} = y l_{n})$ be two ITpDFN. The arithmetic operations (+ , - , × , ÷) defined for (α, β)-cuts when n→ ∞ can be determined as below.

$\tilde{A} + \tilde{B} = (b + x, c + y; b + x, c + y)$

$\tilde{A} - \tilde{B} = (b - y, c - x; b - y, c - x)$

$\tilde{A} \times \tilde{B} = (bx, cy; bx, cy)$

$\tilde{A} \div \tilde{B} = (\frac{b}{y}, \frac{c}{x}; \frac{b}{y}, \frac{c}{x})$

(ii) Operations based on intuitionistic dense fuzzy numbers

(i) $\tilde{A} + \tilde{B} =$

${\begin{matrix} (b + x) f_{n}, b + x, c + y, (c + y) g_{n}; \\ (b + c) k_{n}, b + x, c + y, (c + y) l_{n} \end{matrix}}$

when n→ ∞,

$\tilde{A} + \tilde{B} = {\begin{matrix} 0, b + x, c + y, 0; \\ 0, b + x, c + y, 0 \end{matrix}}$

(ii) $\tilde{A} - \tilde{B} =$

${\begin{matrix} (b - y) f_{n}, b - y, c - x, (c - x) g_{n}; \\ (b - y) k_{n}, b - y, c - x, (c - x) l_{n} \end{matrix}}$

when n→ ∞,

$\tilde{A} - \tilde{B} = {\begin{matrix} 0, b - y, c - x, 0; \\ 0, b - y, c - x, 0 \end{matrix}}$

(iii) $\tilde{A} \times \tilde{B} =$

${\begin{matrix} min (bx, by, cx, cy), bx, cy, max (bx, by, cx, cy); \\ max (bx, by, cx, cy), bx, cy, \min (bx, by, cx, cy) \end{matrix}}$

(iv) $\tilde{A} \div \tilde{B} =$

${\begin{matrix} min (\frac{b}{y}, \frac{c}{x}; \frac{b}{y}, \frac{c}{x}), \frac{b}{x}, \frac{c}{y}, max (\frac{b}{y}, \frac{c}{x}; \frac{b}{y}, \frac{c}{x}); \\ max (\frac{b}{y}, \frac{c}{x}; \frac{b}{y}, \frac{c}{x}), \frac{b}{x}, \frac{c}{y}, min (\frac{b}{y}, \frac{c}{x}; \frac{b}{y}, \frac{c}{x}) \end{matrix}}$

(iii) Operations based on extension principle

If $\tilde{A} = (a = b f_{n}, b, c, d = c g_{n}; a^{'} = b k_{n}, b, c, d^{'} = c l_{n})$ and $\tilde{B} = (w = x f_{n}, x, y, z = y g_{n}; w^{'} = x k_{n}, x, y, z^{'} = y l_{n})$ are two ITpDFNs, then the arithmetic operations (* = + , - , × , ÷) depending on the extension principle for both membership and non-membership is determined with the formula. $μ_{\tilde{A} * \tilde{B}} (y) = \begin{matrix} \sup \\ \tilde{A} * \tilde{B} \end{matrix} [\min (μ_{\tilde{A}} (x, n), μ_{\tilde{B}} (y, n))]$ $γ_{\tilde{A} * \tilde{B}} (y) = \begin{matrix} \inf \\ \tilde{A} * \tilde{B} \end{matrix} [\min (γ_{\tilde{A}} (x, n), γ_{\tilde{B}} (y, n))]$

3 Proposed methodology

In this section, a novel method is designed by extending the AHP and TOPSIS method into the field of ITpDF and an algorithm for ITpDF-AHP-TOPSIS method is given. It is divided into three stages: first, the standard AHP approach is used to determine the consistency of the pair-wise comparison matrix; second, the ITpDF-AHP method is used to determine the weights of the criteria; and third, the ITpDF-TOPSIS method is used to rank the alternatives.

3.1 Algorithm of intuitionistic dense fuzzy AHP-TOPSIS

Stage-1 AHP process

Step-1: Develop a hierarchical structure.

A hierarchical structure can be developed by considering the goal at the top, which is the purpose of the problem, the criteria in the middle, which is the possible and important keywords/characteristics of the problem and the alternatives at the bottom level which is basically the types of possible choices depending upon the applications of the problem.

Step-2: Create a pair-wise comparison matrix (P).

A pair wise comparison matrix can be created with criteria C ={ C₁, C₂, …, C_n } using scale of relative importance which are listed in Table 1 and a general pair wise comparison matrix is obtained in Equation (5).

Table 1
Scale of relative importance table

Score Degree of Importance

1 Equally Important (EI)

3 Moderate Important (MI)

5 High Importance (HI)

7 Very High Importance (VHI)

9 Extremely Important (ExI)

2,4,6,8 Intermediate Importance (IV)

1/3, 1/5, 1/7, 1/9 Values of inverse comparison

Score	Degree of Importance
1	Equally Important (EI)
3	Moderate Important (MI)
5	High Importance (HI)
7	Very High Importance (VHI)
9	Extremely Important (ExI)
2,4,6,8	Intermediate Importance (IV)
1/3, 1/5, 1/7, 1/9	Values of inverse comparison

$P = [\begin{matrix} \begin{matrix} p_{11} & p_{12} & \begin{matrix} \dots & p_{1 n} \end{matrix} \end{matrix} \\ \begin{matrix} p_{21} & p_{22} & \begin{matrix} \dots & p_{2 n} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ \\ p_{n 1} \end{matrix} & \begin{matrix} ⋮ \\ p_{n 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ p_{nn} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]$ (5) where P = [p_ij] , p_ij > 0, where, i, j = 1, 2, …, n, then $p_{ij} = {\begin{matrix} 1, if i = j \\ \frac{1}{p_{ji}}, if i \neq j \end{matrix}$

Step-3: Compute normalized pair-wise comparison matrix ( $\tilde{P}$ ).

$\tilde{P}$ can be gained by dividing every element of the column with its sum value. The normalized matrix is given in Equation (6) $\tilde{P} = [\begin{matrix} {\tilde{p}}_{11} & {\tilde{p}}_{12} & \dots & {\tilde{p}}_{1 n} \\ {\tilde{p}}_{21} & {\tilde{p}}_{22} & \dots & {\tilde{p}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{p}}_{n 1} & {\tilde{p}}_{n 2} & \dots & {\tilde{p}}_{nn} \end{matrix}]$ (6) where ${\tilde{P}}_{i j} = \frac{p_{i j}}{\sum_{i = 1}^{n} p_{i j}}, i, j = 1, 2, \dots, n$ .

Step-4: Evaluate the criteria weight (W^*).

The importance level of each criterion can be determined by aggregating each row of the normalized pair-wise comparison matrix. $W^{*} = \frac{\sum_{i = 1}^{n} {\tilde{p}}_{in}}{n}$ (7)

Step-5: Evaluate the consistency for the criteria weights.

The consistency for the criteria weights can be calculated by the following steps,

The pair-wise comparison matrix (P) can be taken from Equation (5) and multiply each element of the column with respective criteria weights.

The sum of each value in the row can be used to calculate the weighted sum value.

For each row, calculate the ratio of the weighted sum values with the criteria weights and by computing the average of these ratio values the λ_max can be attained.

Consistency index (CI) can be calculated by Equation (8). $CI = \frac{λ_{\max} - n}{n - 1}$ (8) where n is the number of criteria taken.

Consistency ratio (CR) can be found by the formula given in Equation (9) where CR should be less than 0.1.

CR = \frac{CI}{RI} < 0.1

(9) where RI is the random index, which is the consistency index of randomly generated pair-wise comparison matrix. The random index table for 1 to 10 criteria is given in Table 2.

Table 2

Random index table

n	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

Thus, the criteria weights are found to be consistence if the CR is less than 0.1.

Stage-2 ITpDF-AHP process

Step-6: Obtain the scale of relative importance table for ITpDFS. When the consistency can be found to be less than 0.1, the numeric values from Table 1 are converted into ITpDF values by the definition (3), where the functions f_n, g_n, k_n and l_n are the converging sequence. They are represented in Table 3.

Table 3

Scale of relative preference for ITpDFS

Score	Linguistic terms	ITpDFS
1	EI	(1f_n, 1, 1, 1g_n ; 1k_n, 1, 1, 1l_n)
3	MI	(2.5f_n, 2.5, 3.5, 3.5g_n ; 2.5k_n, 2.5, 3.5, 3.5l_n)
5	HI	(4.5f_n, 4.5, 5.5, 5.5g_n ; 4.5k_n, 4.5, 5.5, 5.5l_n)
7	VHI	(6.5f_n, 6.5, 7.5, 7.5g_n ; 6.5k_n, 6.5, 7.5, 7.5l_n)
9	ExI	(8.5f_n, 8.5, 9, 9g_n ; 8.5k_n, 8.5, 9, , 9l_n)
2,4,6,8 - Intermediate Importance (IV)
1/3, 1/5, 1/7, 1/9 - Values of inverse comparison

The intermediate importance of the values are taken by the formula $(\frac{b f_{n} + b}{2}, b, c, \frac{c g_{n} + c}{2})$ . Replacing the crisp numeric values to the ITpDFN in the pair wise comparison matrix (P) which is represented in Equation (5), a fuzzified pair-wise comparison matrix (X) is obtained in (10). $X = [\begin{matrix} \begin{matrix} x_{11} & x_{12} & \begin{matrix} \dots & x_{1 n} \end{matrix} \end{matrix} \\ \begin{matrix} x_{21} & x_{22} & \begin{matrix} \dots & x_{2 n} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ \\ x_{n 1} \end{matrix} & \begin{matrix} ⋮ \\ x_{n 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ x_{nn} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]$ (10) where $x_{ij} = (a_{ij}, b_{ij}, c_{ij}, d_{ij}; a_{ij}^{'}, b_{ij}, c_{ij}, d_{ij}^{'}),$ i, j = 1, 2, …, n

Step-7: Attain the geometric mean of the criteria.

ITpDF geometric mean value can be calculated using Equation (11). $\begin{matrix} r_{i} = [{(a_{i 1} * a_{i 2} * \dots * a_{in})}^{\frac{1}{n}}, {(b_{i 1} * b_{i 2} * \dots * b_{in})}^{\frac{1}{n}}, \\ {(c_{i 1} * c_{i 2} * \dots * c_{in})}^{\frac{1}{n}}, {(d_{i 1} * d_{i 2} * \dots * d_{in})}^{\frac{1}{n}}; \\ (a_{i 1}^{'} * a_{i 2}^{'} * \dots * a_{in}^{'}) \frac{1}{n}, {(b_{i 1} * b_{i 2} * \dots * b_{in})}^{\frac{1}{n}}, \\ {(c_{i 1} * c_{i 2} * \dots * c_{in})}^{\frac{1}{n}}, (d_{i 1}^{'} * d_{i 2}^{'} * \dots * d_{in}^{'}) \frac{1}{n}] \end{matrix}$ (11) where i = 1, 2, … n

Step-8: Calculate the criteria weight (w_j).

ITpDF weights for all the criteria can be calculated using Equation (12) $w_{j} = ({w_{j}}^{mem}; {w_{j}}^{non - mem}) = r_{i} \otimes (r_{1} + r_{2} + \dots + r_{n})^{- 1}$ (12) where i, j = 1, 2, … n

Step-9: Normalize the criteria weights $({\tilde{w}}_{j})$ .

The ITpDF weights can be normalized using Equation (13). ${\tilde{w}}_{j} = ({\tilde{w}}_{j}^{mem}; {\tilde{w}}_{j}^{non - mem})$ (13) where, ${\tilde{w}}_{j}^{mem} = \frac{w_{j}^{mem}}{\sum_{j = 1}^{n} max {d_{j}^{mem}}}$ $\begin{matrix} {\tilde{w}}_{j}^{non - mem} = \frac{w_{j}^{non - mem}}{\sum_{j = 1}^{n} max {d_{j}^{non - mem}}}, \\ j = 1, 2, \dots n \end{matrix}$

Stage-3 ITpDF-TOPSIS process

Step-10: Frame the decision matrix (D).

Decision matrix can be framed with the set of alternatives A ={ A₁, A₂, …, A_m } and criteria C ={ C₁, C₂, …, C_n } in ITpDF matrix obtained in (14) from the Equations (3) using Table 4. Here, the choices are taken to be the alternatives and the keywords/characteristics of the choices are taken to be the criteria.

Table 4

Linguistic variables and their values

Linguistic	Intuitionistic trapezoidal fuzzy number
preferences
VL	(0.1f_n,0.1,0.2,0.2g_n; 0.1k_n,0.1,0.2,0.2l_n)
L	(0.2f_n,0.2,0.3,0.3g_n; 0.2k_n,0.2,0.3,0.3l_n)
M	(0.4f_n,0.4,0.5,0.5g_n; 0.4k_n,0.4,0.5,0.5l_n)
H	(0.6f_n,0.6,0.7,0.7g_n; 0.6k_n,0.6,0.7,0.7l_n)
VH	(0.8f_n,0.8,0.9,0.9g_n; 0.8k_n,0.8,0.9,0.9l_n)

$D = {[r_{ij}]}_{m \times n} = [\begin{matrix} r_{11} & r_{12} & \begin{matrix} \dots & r_{1 n} \end{matrix} \\ r_{21} & r_{22} & \begin{matrix} \dots & r_{2 n} \end{matrix} \\ \begin{matrix} ⋮ \\ r_{m 1} \end{matrix} & \begin{matrix} ⋮ \\ r_{m 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ r_{mn} \end{matrix} \end{matrix} \end{matrix}]$ (14) where, $\begin{matrix} r_{ij} = (a_{ij}, b_{ij}, c_{ij}, d_{ij}; a_{ij}^{'}, b_{ij}, c_{ij}, d_{ij}^{'}), \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$

Step-11: Compute the normalized ITpDF decision matrix (N).

The criteria are divided into beneficial and non-beneficial/cost criteria and the normalized ITpDF decision matrix can be computed using Equations (16) respectively. $N = [\begin{matrix} {\tilde{r}}_{11} & {\tilde{r}}_{12} & \begin{matrix} \dots & {\tilde{r}}_{1 j} \end{matrix} \\ {\tilde{r}}_{21} & {\tilde{r}}_{22} & \begin{matrix} \dots & {\tilde{r}}_{2 j} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{r}}_{i 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{r}}_{i 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{r}}_{ij} \end{matrix} \end{matrix} \end{matrix}]$ where i = 1, 2, …, m , j = 1, 2, …, n .

•For beneficial criteria, ${\tilde{r}}_{ij} = [\frac{a_{ij}}{d_{j}^{+}}, \frac{b_{ij}}{d_{j}^{+}}, \frac{c_{ij}}{d_{j}^{+}}, \frac{d_{ij}}{d_{j}^{+}}; \frac{a ij'}{d_{j}^{' +}}, \frac{b_{ij}}{d_{j}^{' +}}, \frac{c_{ij}}{d_{j}^{' +}}, \frac{d_{ij}^{'}}{d_{j}^{' +}}]$ (15) where $d_{j}^{+} = max (d_{ij})$ and $d_{j}^{' +} = max (d_{ij}^{'})$

For cost criteria, ${\tilde{r}}_{ij} = [\frac{{a_{j}}^{-}}{a_{ij}}, \frac{{a_{j}}^{-}}{b_{ij}}, \frac{{a_{j}}^{-}}{c_{ij}}, \frac{{a_{j}}^{-}}{d_{ij}}; \frac{{a_{j}}^{' -}}{a ij'}, \frac{{a_{j}}^{' -}}{b_{ij}}, \frac{{a_{j}}^{' -}}{c_{ij}}, \frac{{a_{j}}^{' -}}{d_{ij}^{'}}]$ (16) where a_j^- = min(a_ij) and $a_{j}^{' -} = min (a_{ij}^{'})$

Step-12: Calculate the weighted normalized ITpDF decision matrix (W).

The weighted normalized ITpDF decision matrix can be obtained by multiplying the respective weights of the criteria found in stage-2 and are shown in Equation (17) $W = {[{\tilde{z}}_{ij}]}_{m \times n} = [\begin{matrix} {\tilde{z}}_{11} & {\tilde{z}}_{12} & \begin{matrix} \dots & {\tilde{z}}_{1 n} \end{matrix} \\ {\tilde{z}}_{21} & {\tilde{z}}_{22} & \begin{matrix} \dots & {\tilde{z}}_{2 n} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{z}}_{m 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{z}}_{m 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{z}}_{mn} \end{matrix} \end{matrix} \end{matrix}]$ (17) where, ${\tilde{z}}_{ij} = {\tilde{r}}_{ij} \times {\tilde{w}}_{j}, i = 1, 2, \dots, m and j = 1, 2, \dots, n$

Step-13: Determine the ITpDF positive ideal solution (ITpDF-PIS) and ITpDF negative ideal solution (ITpDF -NIS)

From the weighted normalized ITpDF decision matrix, the ITpDF-PIS and ITpDF-NIS can be calculated using Equations (19) given below. $\begin{matrix} A^{+} = ({\tilde{z}}_{1}^{+}, {\tilde{z}}_{2}^{+}, \dots, {\tilde{z}}_{n}^{+}), \\ where {\tilde{z}}_{j}^{+} = max_{i} {\tilde{z}}_{ij} \end{matrix}$ (18) $\begin{matrix} A^{-} = ({\tilde{z}}_{1}^{-}, {\tilde{z}}_{2}^{-}, \dots, {\tilde{z}}_{n}^{-}), \\ where {\tilde{z}}_{j}^{-} = min_{i} {\tilde{z}}_{ij} \end{matrix}$ (19)

Step-14: Compute the Euclidean distance.

Fig. 4

Overall architecture of the integrated ITpDF-AHP-TOPSIS method.

The Euclidean distance (d) between alternative and the ITpDF-PIS and ITpDF-NIS are calculated to obtain their positive (d_i⁺) and negative (d_i^-) distance by Equation (20), ${d_{i}}^{+} = \sum_{j = 1}^{n} d ({\tilde{z}}_{ij}, {\tilde{z}}_{j}^{+}), {d_{i}}^{-} = \sum_{j = 1}^{n} d ({\tilde{z}}_{ij}, {\tilde{z}}_{j}^{-})$ (20) where ${\tilde{z}}_{ij}$ is the element of the i^th alternative with respect to j^th criteria and ${\tilde{z}}_{j}^{+}$ , ${\tilde{z}}_{j}^{-}$ are the ITpDF-PIS and ITpDF-NIS, respectively.

Step-15: Determine the final ranking for the alternatives.

Closeness coefficient (CC_i) for each alternative can be computed using Equation (21) and are ranked in decending order. $C C_{i} = \frac{{d_{i}}^{-}}{{d_{i}}^{-} + {d_{i}}^{+}}$ (21)

4 Application of ITpDF -AHP-TOPSIS: Selecting the appropriate military robot

A war is a fierce armed struggle between two or more nations, governments, societies, or paramilitary organisations. A Mesolithic cemetery in Jebel Sahaba, which is estimated to be roughly 14,000 years old, is the earliest trace of prehistoric warfare. From then different types of war namely, asymmetric war, biological war, nuclear war, coldwar, etc., had conquered. A war yields nothing but the death of civilians. Recently, the Russo-Ukrainian war is a conflict between Russia and Ukraine that is still invading. The most recent figures from US authorities, cited by “The New York Times”, state that more than 25,000 Russian soldiers have died and more than 20,000 have been injured in Ukraine. According to the Russian Defense Ministry, approximately 9,000 Ukrainian army and paramilitary fighters have died, and another 8,000 have been wounded. As the war continues, this count could emerge. Soldiers who serve their country are often expected to carry out dangerous tasks that could slay them. As a result, the death rate of humans increases. By engaging the services of military robots, life-threatening work can be performed without the loss of human lives. One of the most difficult problem for a decision maker is to select a military land robot for a specific purpose. Selecting a robot for a particular action is at ease when the problem is analysed by MCDM model in an ITpDF environment. In this study, the ITpDF-AHP method is used to determine the weights for the criteria, while the ITpDF-TOPSIS method is used to determine the robots’ preference ranking.

5 I. PackBot

PackBot [41, 42] is a type of military robot by Endeavor Robotics, an international robotics company founded in 2016, created by iRobot. This robot operates in the mode of a joystick. It weights 14.3 kg with its length 68.6 cm to 88.9 cm, its width 52.1 cm and its height of 88.9 cm. It has a payload of 2.7 kg with the capability to climb a 60-degree incline. It drives through mud and operates in all weather conditions. Its battery capacity is about 4 to 8 hours and it can be away from the operator up to the range of 1 km. Its speed is about 9.3 km/hr and it has a pulling capacity of about 20 kg. It has 4 cameras and it cost $240,000. Its main purpose is for bomb disposal, hazmat search, reconnaissance and other dangerous missions. The image of the PackBot is shown in Fig. 5.

Fig. 5

PackBot.

6 II. TALON

TALON [42, 43] was built and designed by Foster-Miller, a company owned by Qinetiq North America. It’s a tracked, lightweight, military UGV robot. It was urbanized to protect troops and to treat explosive threats. It is also used for search and rescue. It is a remote-controlled robot weighting about 27 to 45 kg, its length about 86.4 cm, its weight 57.2 kg and its height 27.9 cm. Its payload is about 45 kg and can work through sand, water, snow and can climb stairs. The degree of freedom for its arm is 7 and can turn to a 360-degree angle. Its battery holds for about 8 hours and has an extensive battery of 5 hours. Its range is about 1 km. It is the fastest robot considered and has about 7-speed settings. It can pull 77.11 kg and can lift 9.07 kg weighted objects. It is built with 1 gun in it. It costs $230,000 and its price drops to $150,000 when it is considered in mass production. The image of the robot is given in Fig. 6

Fig. 6

TALON.

7 III. MAARS

The operator can be 800 to 1000 meters away from the robot and operate them. Its speed is about 12 km/hr. It has a pulling capacity of about 140 kg and can lift up to 54 kg. It is mounted with guns and 4 grenades. Its cost is about $300,000 and it has a wide range of purposes namely Reconnaissance, surveillance and target acquisition (RSTA), hostage rescue, assaults ambushes, forced entry, body trapped areas, site security, improvised explosive device, detection and defeat. Figure 7 is the representation of the MAARS robot.

Fig. 7

MAARS.

8 IV. MATILDA

Mesa Associates Tactical Integrated Light-Force Deployment Assembly which is also known as MATILDA [44] was created and designed for surveillance and reconnaissance by the Mesa Robotics Corporation. It is a remote-controlled robot, which weights 40 to 52 lbs with the size of 26l*20w*12h. It has a payload of 56.7 kg and the capability to climb inclined steps of 50 degrees. It works till 6-10 hours and has an operational extension of 2 hrs. It has a range of operation of 600 meters. It has a speed of 3.22 km/hr. It can pull to 226.8 kg weighting object. As it has a very high pulling capacity it can be used for rescue purpose also. It has 2 cameras and costs $25,000. It also serves as a reason for explosive device neutralization, target surveillance, transport, material pickup, firing and law enforcement and weapon transport. Figure 8 represents the MATILDA robot.

Fig. 8

MATILDA.

9 V. Milos / Little Milosh

Miloš, also called Little Milosh [45], is a remote-controlled unmanned ground vehicle (UGV) urbanized by the Military Technical Institute Belgrade. Its size is 1.73l*0.77w*0.475h. It has a payload to carry a sick variant. It has a capability of climbing steps which is inclined of about 45 degrees. It has an electric engine powered on a Li-ion battery and a battery saver of about 90 minutes. Its range is 1 km and its speed is about 12.5 km/hr. It has 2 cameras and a laser ranging up to 2000 meters. It has one or two weapons in it. It can be used for reconnaissance of the Battlefield and it can also be used as an anti-tank weapon. The picture of the robot is represented in Fig. 9.

Fig. 9

Milos/Little Milosh.

10 VI. Ripsaw

The Ripsaw [46] is a series of developmental UGV designed by Howe & Howe Technologies for evaluation by the US army. It is a remote-controlled robot weighting 4100 kg. It has a payload of 910 kg, the capability to climb stairs and ranges up to 1 km. Its speed is about 150 km/hr. It has a day and night video camera, a thermal camera and a laser range finder. It can be mounted with 2 types of guns and its main area is for fighting purpose. It is also used for several purposes including convey protection perimeter defense, surveillance, rescue, border patrol, crowd control and explosive ordnance disposal. The robot is pictured in Fig. 10.

Fig. 10

Ripsaw.

11 VII. Dragon Runner

Dragon Runner [47] is a military UGV developed for urban war. They have three modes of operations namely drive mode, sentry mode and watch mode. This robot weights about 9 kg and so this robot can be carried and thrown. Its size is 23l*20w*7.5h. It has a payload of sensors, manipulator arms, cameras, two-way audio, etc. It can climb stairs and has a 4 degree of freedom. It works for 6 hours with a maximum speed of 70 km/hr. It ranges 650 meters away from the operator and can pull objects weighting approximately 10 kg. It has 6 cameras and is capable of taking videos. It cost around 140,869 dollars and this robot is mainly used for reconnaissance/surveillance, situational awareness, sentry or to be thrown. The image of the robot is given in Fig. 11.

Fig. 11

Dragon runner.

12 VIII. DRDO Daksh ROV

Daksh [48] is a remote-controlled robot that is used to locate, handle and destroy hazardous objects safely without human loss. This robot is electrically powered and was urbanized by Defence Research and Development Organisation. The robot is owned by the Indian army. It is a wheeled robot that can negotiate steep slopes, navigate staircases, tow transport mediums and find the way through narrow paths to reach hazardous materials. This robot can scan cars for explosive devices and has an inbuilt shotgun, which helps to break open locked doors. It has 6 degrees of freedom and can operate for 3 hours. Its range is about 500 meters and pulls about 20 kg. It has various uses like defusing bombs, recovers bombs and also in locating and destroying hazardous objects safely. It is represented in Fig. 12.

Fig. 12

DRDO Daksh ROV.

13 IX. BEAR

Vecna Robotics developed a remotely controlled robot which was named Battlefield Extraction-Assist Robot (BEAR) [49] used in pulling out the wounded soldiers from the combat zone without human loss. This bear-faced UGV uses powerful hydraulics which helps the robot to move over long distances and travel through rough terrain like stairs by carrying humans and other heavy objects. It is 6 feet tall and has a payload of 227 kg. It has a range of up to 500 meters and can pull or lift 230 kg to 240 kg. It has 3 cameras and can be loaded with an M-4 rifle. It can be used for many situations like search & rescue, transporting supplies, lifting heavy objects, clearing obstacles, handling hazardous material, reconnaissance and inspecting for miles. The image of the robot is given in Fig. 13.

Fig. 13

BEAR.

14 X. MARCbot

Exponent Inc. for the United States Army Rapid Equipping Force created a UGV named Multi-function Agile Remote-Controlled Robot (MARCbot) [50]. It is a remotely operated robot. It weights 15 kg and has a size of 24l*19.5w*13.5h. It has a payload of 25 lbs and a battery working for 6 hours. It ranges 300 meters away from the controller and has a camera in it. It costs 19,000 dollars and usage of Demolition vehicle, reconnaissance robot, and war-fighter. It is made known in Fig. 14.

Fig. 14

MARCbot.

15 XI. THeMIS

Tracked Hybrid Modular Infantry System (THeMIS) [51], a tracked UGV, built for war-fighter and various military applications by Milrem Robotics in Estonia. It weights about 1630 kg and has a payload of 750 to 1200 kg. It has the capability of climbing stairs with 31-degree inclines. It is a diesel and electric engine with 12-15 hrs of working time, with its maximum speed of 25 km/hr and can pull 150 kg force. It has 3 cameras and has a machine gun which is light or heavy. The image for the robot is given in Fig. 15.

Fig. 15

THeMIS.

4.1 The datasets of the military robots discussion

4.1.1 Robots in military field

Robots are a type of automated machine that can do specialised jobs with little or no help from humans. As a result, this form of innovation is required in all areas. Soldiers’ lives are often put in danger in the military field. By encouraging military robots, life-threatening missions can be done without the loss of human lives. Military robots are remotely controlled or autonomous robots that have been built for a number of military uses, including transportation, surveillance, bomb disposal, search and rescue or attack, among many others. (https://en.wikipedia.org/wiki/Unmanned_ground_vehicle). The three categories of military robots utilized on the combat are land, marine, and airborne. The most essential robots in combat are the land robots. Land robots are most generally constructed on wheeled, tracked, legged, and wearable platforms [52, 53]. Recently, the world has been engulfed by a massive war between Russia and Ukraine, which is still occurring. According to “The New York Times”, more than 40,000 people are estimated to die or been injured as a result of this war. Military robots were found to be more important in order to avoid human loss, as they can reduce the death rate of human lives.

4.1.2 A description of the dataset

The data set contains different types of military land robots taken as alternatives and their specialized characteristics which is known as criteria. The data of these alternatives and criteria are collected from the websites and from the literature review. With the help of field experts, data sets were framed for four kinds of most dangerous situations namely bomb disposal, search and rescue, surveillance and reconnaissance and war-fighter.

4.1.3 The dataset of the alternatives

Alternatives are found to be the kind of robots taken for the specified situations like bomb disposal, search and rescue, surveillance and reconnaissance and war-fighter. The data for these alternatives like PackBot [41, 42], TALON [42, 43], Dragon Runner [47], DRDO Daksh DOV [48], BEAR [49], MATILDA [44], MAARS [42], Milos [45], MARCbot [50], THeMIS [51], and Ripsaw [46] are collected and grouped according to the situations mentioned which are listed below.

Bomb disposal: Robots like DRDO Daksh DOV, PackBot, TALON and Dragon Runner are found to be the bomb disposal robots.

Search and rescue: BEAR, MATILDA and TALON are the search and rescue robots considered.

Surveillance and reconnaissance: PackBot, Dragon Runner, MATILDA, MAARS, Milos, MARCbot and BEAR are considered for the surveillance and reconnaissance robots.

War-fighter: BEAR, MAARS, THeMIS and Ripsaw are the considered war-fighter robots.

4.1.4 The dataset of the criteria

Criteria are the various characteristic of the robots, which changes according to the considered situation. Thus, the selection of the criteria is important as it is the key point to find the best robot (alternative) among others. In this article, the criteria are considered according to the experts’ opinion and according to the review of literature [39].

4.2 Adaptation of the proposed method for selecting military robots

In this section, robots are selected for four main military applications namely bomb disposal, search and rescue, surveillance and reconnaissance, and war-fighter, with preference ranking for robots depending on the necessities. PackBot, TALON, Dragon Runner, DRDO Daksh DOV, BEAR, MATILDA, MAARS, Milos, MARCbot, THeMIS, and Ripsaw are some of the robots considered for the problems. Robots are selected depending on their capabilities. The speculations and their key characteristics for each application are tabulated in Table 5, which were derived from a literature review [39] and on the expert’s opinion.

Table 5
Criteria presumptions for each application

Criteria

ROBOTS Cam DOF Cap Range Bt Cost Speed Pulls PD Weapon

Bomb disposal 2 5 20’ 152.4m 4hrs $20,000 – – – –

Search and rescue 2 – 45’ 500m 6hrs $100,000 20km 60–70kg 40 kg –

War-fighter 2 – 30’ stairs 500m 10 hrs $250,000 20–30 km/hr – 500 kg 1 Heavy Gun

Surveillance and 3 – 45’ stairs 400m 5-6 hrs $20,000 10 km/hr – – –

reconnaissance

Criteria
Bomb disposal	2	5	20’	152.4m	4hrs	$20,000	–	–	–	–
Search and rescue	2	–	45’	500m	6hrs	$100,000	20km	60–70kg	40 kg	–
War-fighter	2	–	30’ stairs	500m	10 hrs	$250,000	20–30 km/hr	–	500 kg	1 Heavy Gun
Surveillance and	3	–	45’ stairs	400m	5-6 hrs	$20,000	10 km/hr	–	–	–
reconnaissance

Some abbreviations of the considered parameters are Cam-camera, DOF-degree of freedom, Cap-capability, Bt-battery, PD-payload.

The initial analysis relies on bomb disposal. Here robots are used to keep the human bomb expert out of harm. Considering the bomb disposal application, the most important characteristic of the bomb disposal robots are sorted according to the survey taken with the help of members of the Los Angeles Sheriff Department’s Special Enforcement Bureau (LASD-SEB) [39]. Here, 65 responses were received and the main characteristics/key factors for a bomb disposal operation considered are camera, degree of freedom, capability, range, battery and cost. Thus, the robots that can perform the bomb disposal operation are PackBot, TALON, Dragon Runner, and DRDO Daksh DOV, which were taken from the list of robots given earlier in section-2. The role of robot in replacing human for a bomb disposal application can only be done by its characteristics/key factors. Thus, they are listed below.

Camera: Acts as a human eye for the robot. Thus, for this application, the basic number of camera needed is 2.

Degree of freedom: This enhances the motion capabilities of robots. Thus, according to the survey, the need of DOF is found to be 5 for the good movements of the robots.

Capability: It refers to how much the loads can be carried by the robot. Here the survey demands 18.4 kg of caring capability for the robot.

Range: The range for the remote control is found to be 152.4 m as it helps to avoid any damage or lose in lives in case of any explosion.

Battery: The battery capacity determines the capability of working hour for the robots and thus, the survey demands 4 hours of battery capacity.

Cost: Cost is always considered to be one of the non-beneficial criteria for the robot selection problem and thus, the expected cost from the survey is found to be $20,000.

Hence, with all these characterisation, the specialized bomb disposal robots are selected from the list of robots given in section-2 and a linguistic matrix is framed on expert’s opinion which is examined in Table 14.

Table 14

Linguistic matrix on expert’s opinion

Robots	Cam	DOF	Cap	Range	Bt	Ct
Packbot	H	VH	M	VH	VH	VH
TALON	VH	H	VH	VH	H	VH
Dragon Runner	VH	L	VL	VH	M	VH
DRDO Daksh	L	H	M	VH	L	H
DOV
Presumption	2	5	18.14kg	152.4m	4hr	$20,000

Evaluation of the robot selection problem for bomb disposal by the proposed method.

Stage-1 AHP process:

Step-1: A hierarchical structure is developed on robot selection for the bomb disposal and the structure is diagrammatically represented in Fig. 16.

Fig. 16

Flowchart of the hierarchical structure.

Step-2: A pair-wise comparison matrix is created for six criteria of the robots, represented in Table 6, which is formed with the help of scale of relative importance from the Table 1.

Table 6

Pair-wise comparison matrix on the basis of criteria of the problem

Criteria	Camera	DOF	Capability	Range	Battery	Cost
Camera	1	2	3	4	5	9
DOF	0.5	1	3	4	4	8
capability	0.333	0.333	1	3	4	6
Range	0.25	0.25	0.333	1	2	6
Battery	0.2	0.25	0.25	0.5	1	4
Cost	0.111	0.125	0.167	0.1667	0.25	1

Step-3: The normalized pair-wise comparison matrix is obtained using equation (6) by dividing each element of the matrix with its summation of the respective column elements which are given in Table 7.

Table 7

Normalized pair-wise comparison matrix

Criteria	Camera	DOF	Capability	Range	Battery	Cost
Camera	0.4176	0.5053	0.3871	0.3158	0.3077	0.2647
DOF	0.2088	0.2526	0.3871	0.3158	0.2462	0.2353
Capability	0.1392	0.0842	0.1290	0.2368	0.2462	0.1765
Range	0.1044	0.0631	0.0430	0.0790	0.1231	0.1765
Battery	0.0835	0.0632	0.0323	0.0395	0.0615	0.1177
Cost	0.0464	0.0316	0.0215	0.0132	0.0153	0.0294

Step-4: The importance level of each criterion can be determined by finding the average of each row with the number of criteria of the normalized pair-wise comparison matrix which is represented in equation (7). The calculated criteria weights are given in Table 8.

Table 8

Criteria weights

	Criteria weight
Cam	0.3664
DOF	0.2743
Cap	0.1687
Range	0.0982
Bt	0.0663
Ct	0.0262

Step-5: Using (i) –(v) from step-5, the consistency ratio is checked by calculating the λ_max values. The λ_max values are calculated by taking the average between the weighted sum values to the criteria weights. The framed pair-wise comparison matrix is found to be consistent if it is less than 0.10. The values are shown in Table 9.

Table 9

Examining the consistency ratio

Criteria	Weighted sum value	Criteria weight	Weight sum value/criteria weight	λ_max	Consistency index
Cam	2.3811	0.3664	6.499		=0.0712
DOF	1.8311	0.2743	6.6758	(6.49 + 6.67 +
cap	1.0993	0.1687	6.5178	6.52 + 6.16 +
Range	0.6045	0.0982	6.1575	6.10 + 6.18)/	Consistency ratio
Bt	0.4043	0.0663	6.1015	6=6.3560	=0.05742<0.1
Ct	0.1623	0.0262	6.1841	where RI=1.24

Thus, the consistency is less than 0.10.

Stage-2 ITpDF-AHP

Step-6: The crisp numeric values are converted to ITpDF numeric values by substituting them from Table 3 and are represented in Table 10.

Table 10

Matrix for the ITpDF numeric values

Criteria	Cam	DOF	Cap
Cam	(1,1,1,1; 1,1,1,1)	(1.5, 1.5, 2.5, 2.5; 1.5, 1.5, 2.5, 2.5)	(2.5, 2.5, 3.5, 3.5; 2.5, 2.5, 3.5, 3.5)
DOF	(0.66, 0.66, 0.4, 0.4; 0.66, 0.66, 0.4, 0.4)	(1,1,1,1;1,1,1,1)	(2.5, 2.5, 3.5, 3.5; 2.5, 2.5, 3.5, 3.5)
Cap	(0.4, 0.4, 0.285, 0.285; 0.4, 0.4, 0.285, 0.285)	(0.4, 0.4, 0.29, 0.29; 0.4, 0.4, 0.29, 0.29)	(1,1,1,1; 1,1,1,1)
Range	(0.29, 0.29, 0.22, 0.22; 0.28,	(0.28, 0.28, 0.22, 0.22; 0.28,	(0.4, 0.4, 0.28, 0.29; 0.4,
	0.28, 0.22, 0.22)	0.28, 0.22, 0.22)	0.4, 0.29, 0.29)
Bt	(0.22, 0.22, 0.18, 0.18; 0.22,	(0.28, 0.28, 0.22, 0.22; 0.28,	(0.28, 0.28, 0.22, 0.22; 0.28,
	0.22, 0.18, 0.18)	0.28, 0.22, 0.22)	0.28, 0.22, 0.22)
Ct	(0.12, 0.12, 0.11, 0.11; 0.12,	(0.13, 0.13, 0.12, 0.12; 0.13,	(0.18, 0.18, 0.16, 0.16; 0.18,
	0.12, 0.11, 0.11)	0.13, 0.12, 0.12)	0.18, 0.16, 0.16)
Criteria	Range	Battery	Cost
Cam	(3.5, 3.5, 4.5, 4.5; 3.5, 3.5, 4.5, 4.5)	(4.5, 4.5, 5.5, 5.5; 4.5, 4.5, 5.5, 5.5)	(8.5, 8.5, 9, 9; 8.5, 8.5, 9, 9)
DOF	(3.5, 3.5, 4.5, 4.5; 3.5, 3.5, 4.5, 4.5)	(3.5, 3.5, 4.5, 4.5; 3.5, 3.5, 4.5, 4.5)	(7.5, 7.5, 8.5, 8.5; 7.5, 7.5, 8.5, 8.5)
Cap	(2.5, 2.5, 3.5, 3.5; 2.5, 2.5, 3.5, 3.5)	(3.5, 3.5, 4.5, 4.5; 3.5, 3.5, 4.5, 4.5)	(5.5, 5.5, 6.5, 6.5; 5.5, 5.5,6.5, 6.5)
Range	(1,1,1,1;1,1,1,1)	(1.5, 1.5, 2.5, 2.5; 1.5, 1.5, 2.5, 2.5)	(5.5, 5.5, 6.5, 6.5; 5.5, 5.5, 6.5, 6.5)
Bt	(0.66, 0.66, 0.4, 0.4; 0.66, 0.66, 0.4,0.4)	(1,1,1,1;1,1,1,1)	(3.5, 3.5, 4.5, 4.5; 3.5, 3.5, 4.5, 4.5)
Ct	(0.18, 0.18, 0.15, 0.15; 0.18,	(0.29, 0.29, 0.22, 0.22; 0.29,	(1,1,1,1;1,1,1,1)
	0.18, 0.15, 0.15)	0.29, 0.22, 0.22)

Step-7: The ITpDF-AHP method using geometric mean from Equation (11) and the calculated values are shown in Table 11.

Table 11

Geometric mean

	Geometric mean
Cam	(83.672, 83.672, 324.84, 324.84; 83.67, 83.67,
	324.84, 324.84)
DOF	(25.52, 25.520, 40.16, 40.16; 25.52, 25.52, 40.16, 40.16)
Cap	(1.28,1.28, 1.39, 1.39; 1.28, 1.28, 1.39, 1.39)
Range	(0.05, 0.05, 0.04, 0.04; 0.04, 0.04, 0.038, 0.038)
Bt	(0.01, 0.01, 0.0026, 0.0026; 0.007, 0.007,
	0.0027, 0.0027)
Ct	(2.46×10^-5, 2.4×10^-5, 1.2×10^-5, 1.2×10^-5;
	2.×10^-5, 2.5×10^-5, 1.15×10^-5, 1.15×10^-5)

Step-8: ITpDF weights for all the criteria can be calculated using Equation (12) and are given in Table 12.

Table 12

ITpDF weights

Criteria	ITpDF weight
Cam	(0.228, 0.228, 2.939, 2.939; 0.228, 0.228, 2.939, 2.939)
DOF	(0.069, 0.069, 0.363, 0.363; 0.069, 0.069, 0.363, 0.363)
Cap	(0.004, 0.004, 0.013, 0.0126; 0.004, 0.004,
	0.013, 0.0127)
Range	(0.0001, 0.0001, 0.0004, 0.0004; 0.0001,
	0.0001, 0.0004, 0.0004)
Bt	(1.93×10^-5, 1.93×10^-5, 2.44×10^-5, 2.44×10^-5;
	1.93×10^-5, 1.93×10^-5, 2.44×10^-5, 2.44×10^-5)
Ct	(6.74×10^-7, 6.74×10^-7, 1.04×10^-7, 1.04×10^-7;
	6.74×10^-8, 6.74×10^-8, 1.04×10^-7, 1.04×10^-7)

Step-9: The ITpDF weights are normalized using Equation (13) and are represented in Table 13.

Table 13

Normalized ITpDF weights

Criteria	Normalized ITpDF weights
Cam	(0.068, 0.068, 0.886, 0.886; 0.068, 0.068, 0.886, 0.886)
DOF	(0.021, 0.021, 0.109, 0.109; 0.021, 0.021, 0.109, 0.109)
Cap	(0.001, 0.001, 0.003, 0.003; 0.001, 0.001, 0.003, 0.003)
Range	(3.69×10^-5, 3.69×10^-5, 0.0001, 0.0001; 3.69×10^-5
	3.69×10^-5, 0.0001, 0.0001)
Bt	(5.81×10^-6, 5.81×10^-6, 7.35×10^-6, 7.35×10^-6;
	5.80 × 10^-6, 5.81 × 10^-6, 7.35 × 10^-6, 7.35 × 10^-6)
Ct	(2.03×10^-8, 2.03×10^-8, 3.12×10^-8, 3.13×10^-8;
	2.033×10^-82.03×10^-8, 3.12×10^-8, 3.127×10^-8)

Stage-3 ITpDF-TOPSIS:

Step-10: The decision matrix is framed with the help of the decision makers and the presumptions are taken from one of the survey report of the Robotics for Law Enforcement: Beyond Explosive Ordnance Disposal [39]. Thus, the decision matrice is given in Table 14 which represents VL = Very Low, L = Low, M = Medium, H = High and VH = Very High and their ITpDFN values are tabulated in Table 15.

Table 15

Relative preference values when n→ ∞

Linguistic terms	ITpDFN
VL	(0.1,0.1,0.2,0.2; 0.1,0.1,0.2,0.2)
L	(0.2,0.2,0.3,0.3; 0.2,0.2,0.3,0.3)
M	(0.4,0.4,0.5,0.5; 0.4,0.4,0.5,0.5)
H	(0.6,0.6,0.7,0.7; 0.6,0.6,0.7,0.7)
VH	(0.8,0.8,0.9,0.9; 0.8,0.8,0.9,0.9)

Step-11: The criteria are divided into beneficial and non-beneficial/cost criteria and the normalized ITpDF decision matrix are computed using Equations (16) respectively. The normalized ITpDF decision matrix is shown in Table 16.

Table 16

Normalized ITpDF decision matrix

Robots	Camera	DOF	Capability
Packbot	(0.67, 0.67, 0.78, 0.78;	(0.89, 0.89, 1, 1;	(0.44, 0.44, 0.56, 0.56;
	0.67, 0.67, 0.78, 0.78)	0.89, 0.89, 1, 1)	0.44, 0.44, 0.56, 0.56)
TALON	(0.89, 0.89, 1, 1;	(0.67,0.67, 0.78, 0.78;	(0.89, 0.89, 1, 1;
	0.89, 0.89, 1, 1)	0.67, 0.67, 0.78, 0.78)	0.89, 0.89, 1, 1)
Dragon	(0.89, 0.89, 1, 1;	(0.22, 0.22, 0.33, 0.33;	(0.11, 0.11, 0.22, 0.22;
Runner	0.89, 0.89, 1, 1)	0.22,0.22,0. 33, 0.33)	0.11, 0.11, 0.22, 0.22)
DRDO	(0.22, 0.22, 0.33, 0.33;	(0.67, 0.67, 0.78, 0.78;	(0.44, 0.44, 0.56, 0.56;
Daksh DOV	0.22, 0.22, 0.33, 0.33)	0.67, 0.67, 0.78, 0.78)	0.44, 0.44, 0.56, 0.56)
Robots	Range	Battery	Cost
Packbot	(0.89, 0.89, 1, 1;	(0.89,0.89,1,1;	(0.75, 0.75, 0.67, 0.67;
	0.89, 0.8, 1,1)	0.89, 0.89, 1, 1)	0.75, 0.75, 0.67, 0.67)
TALON	(0.89, 0.89, 1, 1;	(0.67, 0.67, 0.78, 0. 78;	(0.75, 0.75, 0.67, 0.67;
	0.89, 0.89, 1, 1)	0.67, 0.67, 0.78, 0.78)	0.75, 0.75, 0.67, 0.67)
Dragon	(0. 89, 0.89, 1, 1;	(0.44, 0.44, 0.56, 0.56;	(0.75, 0.75, 0.67, 0.67;
Runner	0.89, 0.89, 1, 1)	0.44,0.44, 0.56, 0.56)	0.75,0.75, 0.67, 0.67)
DRDO	(0.89, 0.89, 1, 1;	(0.22, 0.22, 0.33, 0.33;	(1, 1, 0.86, 0.86;
Daksh DOV	0.89, 0.89, 1, 1)	0.22, 0.22, 0.33, 0.33)	1, 1, 0.86, 0.86)

Step-12: The weighted normalized ITpDF decision matrix is calculated by multiplying the normalized ITpDF decision matrix with their respective criteria weights which are shown in Table 17.

Table 17

Weighted normalized ITpDF decision matrix

Weighted normalized ITpDF DM
Robots	Camera	DOF	Capability
Packbot	(0.05, 0.05, 0.69, 0.69;	(0.02, 0.02, 0.11, 0.11;	(0.001, 0.001, 0.002, 0.002;
	0.05, 0.05, 0.69, 0.69	0.02, 0.02, 0.11, 0.11)	0.0005, 0.0005, 0.002, 0.002)
TALON	(0.06, 0.06, 0.89, 0.89;	(0.01, 0.01, 0.09, 0.09;	(0.001, 0.001, 0.004, 0.004;
	0.06, 0.06, 0.89, 0.89)	0.01, 0.01, 0.09, 0.09)	0.001, 0.001, 0.004, 0.004)
Dragon	(0.06, 0.06, 0.89, 0.89;	(0.005,0.005, 0.037, 0.037;	(0.0001, 0.0001, 0.001, 0.001;
Runner	0.06, 0.06, 0.89, 0.89)	0.005, 0.005, 0.04, 0.04)	0.0001, 0.0001, 0.001, 0.001)
DRDO	(0.02, 0.02, 0.29, 0.29;	(0.01, 0.01, 0.09, 0.09;	(0.0005, 0.0005, 0.002, 0.002;
Daksh DOV	0.02, 0.02, 0.29, 0.29)	0.01, 0.01, 0.09, 0.09)	0.001, 0.001, 0.002, 0.002)
A+	(0.06, 0.06, 0.89, 0.89;	(0.02, 0.02, 0.11, 0.11;	(0.001, 0.001, 0.004, 0.004;
	0.06, 0.06, 0.89, 0.89)	0.02, 0.02, 0.11, 0.11)	0.001, 0.001, 0.004, 0.004)
A-	(0.02, 0.02, 0.3, 0.3;	(0.01, 0.01, 0.04, 0.04;	(0.0001, 0.0001, 0.001, 0.001;
	0.02, 0.02, 0.3, 0.3)	0.04, 0.01, 0.04, 0.04)	0.0001, 0.0001, 0.001, 0.001)
Robots	Range	Battery	Cost
Packbot	(3.3×10^-5, 3.3×10^-5, 0.0001,	(5.2 ×10^-6,5.2×10^-6, 7.4 ×10^-6,	(1.5×10^-8, 1.5×10^-8, 2.1×10^-8,
	0.0001; 3.3×10^-5, 3.3×10^-5,	7.4×10^-6, 5.2×10^-6; 5.2 ×10^-6,	2.1×10^-8; 1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001)	7.4×10^-6, 7.4×10^-6)	2.1×10^-8, 2.1×10^-8)
TALON	(3.3×10^-5, 3.3×10^-5,	(3.9×10^-6, 3.9×10^-6,	(1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001;	5.7×10^-6, 5.7×10^-6;	2.1×10^-8, 2.1×10^-8;
	3.23×10^-5, 3.3×10^-5,	3.9×10^-6, 3.9×10^-6,	1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001)	5.7×10^-6, 5.7×10^-6)	2.1×10^-8, 2.1×10^-8)
Dragon Runner	(3.3×10^-5, 3.3×10^-5,	(2.6×10^-6, 2.6×10^-6,	(1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001;	4.1×10^-6, 4.1×10^-6;	2.1×10^-8, 2.1×10^-8;
	3.3×10^-5, 3.3×10^-5,	2.6×10^-6, 2.6×10^-6,	1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001)	4.1×10^-6, 4.1×10^-6)	2.1×10^-8, 2.1×10^-8)
DRDO	(3.3×10^-5, 3.3E-05,	(1.3×10^-6, 1.3×10^-6,	(2.03×10^-8, 2.03×10^-8,
Daksh	0.0001, 0.0001;	2.5×10^-6, 2.5×10^-6;	2.7×10^-8, 2.7×10^-8;
DOV	3.3×10^-5, 3.3×10^-5,	1.3×10^-6, 1.3×10^-6,	2.03×10^-8, 2.03×10^-8,
	0.0001, 0.0001)	2.5×10^-6, 2.5×10^-6)	2.7×10^-8, 2.7×10^-8)
A+	(3.3×10^-5, 3.3×10^-5,	(5.2×10^-6, 5.2×10^-6,	(2.03×10^-8, 2.03×10^-8,
	0.0001, 0.0001;	7.4×10^-6, 7.4×10^-6;	2.7×10^-8, 2.7×10^-8;
	3.3×10^-5, 3.3×10^-5,	5.2×10^-6, 5.2×10^-6,	2.03×10^-8, 2.03×10^-8,
	0.0001, 0.0001)	7.4×10^-6, 7.4×10^-6)	2.7×10^-8, 2.7×10^-8)
A-	(3.3×10^-5, 3.3×10^-5,	(1.3×10^-6, 1.3×10^-6,	(1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001;	2.5×10^-6, 2.5×10^-6;	2.1×10^-8, 2.1×10^-8;
	3.3×10^-5, 3.3×10^-5,	1.3×10^-6, 1.3×10^-6,	1.5×10^-8, 1.5×10^-8,
	0.0001, 0.0001)	2.5×10^-6, 2.5×10^-6)	2.1×10^-8, 2.1×10^-8)

Step-13: The ITpDF–PIS and ITpDF–NIS are calculated by finding the maximum and minimum values of each criterion respectively. Thus, the values are shown in Table 17.

Step-14: The Euclidean distance from each alternative to the ITpDF-PIS and ITpDF-FNIS are calculated using Equation (20) and the calculated values are shown in Tables 18 19.

Table 19

Distance from each alternative to the ITpDF-NIS

A -	Cam	DOF	Cap	Range	Bt	Ct	d-
Packbot	0.559	0.105	0.002	0	8.832×10^-6	0	0.67
TALON	0.838	0.0701	0.004	0	5.89×10^-6	0	0.913
Dragon Runner	0.838	0	0	0	2.95×10^-6	0	0.838
DRDO Daksh DOV	0	0.0701	0.002	0	0	1.107×10^-8	0.072

Table 18

Distance from each alternative to the ITpDF-PIS

A +	Cam	DOF	Cap	Range	Bt	Ct	d+
Packbot	0.28	0	0.002	0	0	1.11×10^-8	0.282
TALON	0	0.035	0	0	2.9E-06	1.11×10^-8	0.0351
Dragon Runner	0	0.105	0.004	0	5.895E-06	1.11×10^-8	0.1095
DRDO Daksh DOV	0.84	0.035071499	0.002	0	8.83E-06	0	0.876

Step-15: The CC_i for each robots are calculated by dividing the ITpDF-NIS with the summation of ITpDF-PIS and ITpDF-NIS and the alternatives are ranked relating to the CC_i which is given in Table 20.

Table 20

Closeness coefficient and ranking

Robots	CC _i	RANK
Packbot	0.7026	3
TALON	0.9630	1
Dragon Runner	0.8844	2
DRDO Daksh DOV	0.0760	4

4.3 Observations

In the section, the calculations are worked for all the four applications by selecting all possible robots tailored to the needs and purposes, and the results are tabulated and pictured below.

4.3.1 Bomb disposal robots

ITpDF-AHP -TOPSIS methods are worked on robot selection and the preferred ranking for the robots are made for the bomb disposal robots. The presumptions are taken from a survey report which was created to establish law-enforcement needs for non-explosive ordnance disposal robots. With the help of members of the Los Angeles Sheriff Department’s Special Enforcement Bureau (LASD-SEB), a questionnaire was created, and more than 65 responses were received. The decision matrixes were framed by the decision-makers. It is observed that the robots are prioritized as TALON, Dragon Runner, and DRDO Daksh DOV as first, second, third and forth respectively. Hence, the preferred robot is found to be TALON. TALON is graded first as it was urbanized to protect troops and to treat explosive threats, and it can work through sand, water, snow, and stairs. It has a battery life of about 13 hours and is the fastest robot in the competition, with about 7 speed settings. Figure 17 depicts the ranking of bomb disposal robots in diagrammatic form.

Fig. 17

Level of bomb disposal robots.

4.3.2 War-fighter robots

The war-fighter robots are found to be MAARS, THeMIS, Ripsaw and BEAR. The MCDM method is used to rank the robots, and Ripsaw is ranked first as it is equipped with two types of guns and has a main fighting area. The results are collected and tabulated in Table 21, besides being represented graphically in the form of a pie chart in Fig. 18.

Table 21
War-fighter robots

Robots CC _i RANK

MAARS 0.5910 3

THeMIS 0.5913 2

Ripsaw 0.6138 1

BEAR 0.1378 4

Robots	CC _i	RANK
MAARS	0.5910	3
THeMIS	0.5913	2
Ripsaw	0.6138	1
BEAR	0.1378	4

Fig. 18

Level of War-fighter robots.

Fig. 23

Sensitivity analysis for Surveillance and reconnaissance robots.

4.3.3 Surveillance and reconnaissance robots

The robots for a surveillance and reconnaissance are found to be MAARS, MATILDA, Packbot, Dragon Runner, Milos, Mackbot and BEAR. The MCDM method is used to select robots, and MAARS is ranked first due to its broad purpose in reconnaissance and surveillance. The results are tabulated in Table 22, and are also represented graphically in the form of a pie chart in Fig. 19.

Table 22
Surveillance and reconnaissance robots

Robots CC _i RANK

MAARS 0.9957 1

MATILDA 0.3378 5

Packbot 0.3614 4

BEAR 0.0031 6

Dragon Runner 0.6623 2

Milos 0.3635 3

Mackbot 0.0011 7

Robots	CC _i	RANK
MAARS	0.9957	1
MATILDA	0.3378	5
Packbot	0.3614	4
BEAR	0.0031	6
Dragon Runner	0.6623	2
Milos	0.3635	3
Mackbot	0.0011	7

Fig. 19

Level of surveillance and reconnaissance robots.

Fig. 21

Sensitivity analysis for Bomb disposal robot.

Fig. 22

Sensitivity analysis for War-fighter robots.

4.3.4 Searcsh and rescue robots

The robots for search and rescue are found to be TALON, MATILDA and BEAR. The ITpDF-AHP-TOPSIS method is used to prioritize the robots, and MATILDA is preferred first as it has a high pulling capacity and can pull up to 226.8 kg of weighting object. It can also be used for rescue. The results are tabulated in Table 23, and they are also represented graphically Fig. 20.

Table 23
Search and rescue robots

Robots CC _i RANK

TALON 0.6921 1

MATILDA 0.0378 3

BEAR 0.3766 2

Robots	CC _i	RANK
TALON	0.6921	1
MATILDA	0.0378	3
BEAR	0.3766	2

Fig. 20

Levels of search and rescue robots.

Fig. 24

Sensitivity analysis for Search and rescue robots.

Fig. 25

Comparative analysis for Bomb disposal robot.

Fig. 26

Comparative analysis for War-fighter robots.

Fig. 27

Comparative analysis for Surveillance and reconnaissance robots.

Fig. 28

Comparative analysis for Search and rescue robots.

As a result, the selection of preferred robots for each application is done according to the ranking obtained from the calculations.

4.4 Sensitivity analysis

In this current section, the impact on increasing the n values of the elements in the sets are investigated using sensitivity analysis. The values are taken to be n = 1, 5, 10, 100, 1000, 1000 and analyzed by calculating them for all the four applications and the ranking of the robots are done and analysed and are tabulated in Table 24.

Table 24
Sensitive analysis on the n values of the sets

Application The value of n

n=1 n=5 n=10 n=100 n=1000 n=10000

Bomb Disposal BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4

War-fighter robots WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4

Surveillance and reconnaissance robots SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7

Search and rescue robots S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2

Application	The value of n
Bomb Disposal	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4
War-fighter robots	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4
Surveillance and reconnaissance robots	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7
Search and rescue robots	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2

4.5 Comparative analysis

The comparative analysis is performed by ranking the alternatives by using fuzzy AHP-TOPSIS, intuitionistic fuzzy AHP-TOPSIS, dense fuzzy AHP-TOPSIS models with ITpDF-AHP-TOPSIS model and the rank outcomes are compared and tabulated in Table 25. From the comparative analysis, the following results are witnessed.

BOMB DISPOSAL: BR represents the bomb disposal robots which includes Packbot (BR1), TALON (BR2), Dragon Runner (BR3) and DRDO Daksh DOV (BR4). It is observed that, for bomb disposal robots the TALON ranks first, as it has its specilizied characteristic of seven cameras and 7 DOF, whereas the dragon runner ranks second with 6 cameras and 4 DOF. Accordingly, PackBot ranks third and DRDO Daksh DOV ranks forth. Therefore, the preferred robot is found to be TALON.

WAR-FIGHTER: WF represents the war-fighter robots which includes MAARS (WF1), THeMIS (WF2), Ripsaw (WF3) and BEAR (WF4) and found that Ripsaw (WF3) ranks first as it has three weapons inbuilt with the payload of 3600 kgs in addition. Hence, the ranking is ordered as WF3 >WF1 >WF2 >WF4.

SURVEILLANCE AND RECONNAISSANCE: SR represents the surveillance and reconnaissance robots, which includes MAARS (SR1), MATILDA (SR2), Packbot (SR3) and BEAR (SR4), Dragon Runner (SR5), Milos (SR6) and Mackbot (SR7). The predominent characterizations are included and thus it ranks in the order where SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7.

SEARCH AND RESCUE: S&R represents the search and rescue robots where TALON (S&R1), MATILDA (S&R2) and BEAR (S&R3) are the robots considered and are ordered as S&R1 >S&R3 >S&R2.

Table 25
Comparative analysis table on the values of the sets

Application Fuzzy AHP-TOPSIS Intuitionistic fuzzy AHP-TOPSIS Dense fuzzy AHP-TOPSIS Intuitionistic dense fuzzy AHP-TOPSIS

Bomb Disposal BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4 BR2 >BR3 >BR1 >BR4

War-fighter robots WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4 WF3 >WF1 >WF2 >WF4

Surveillance and reconnaissance robots SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7 SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7

Search and rescue robots S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2 S&R1 >S&R3 >S&R2

Application	Fuzzy AHP-TOPSIS	Intuitionistic fuzzy AHP-TOPSIS	Dense fuzzy AHP-TOPSIS	Intuitionistic dense fuzzy AHP-TOPSIS
Bomb Disposal	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4
War-fighter robots	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4
Surveillance and reconnaissance robots	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7
Search and rescue robots	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2

Consequently, from the Table 25 it is found that the ranking produced by all the sets are the same but it is claimed that the calculation is framed by including the learning experience of the decision maker and hence it is concluded that the approximation is higher in ITpDFS than the others.

4.6 Advantages and limitations

The main advantages of this study are listed below.

As the DFS are taken in the intuitionistic environment, both the belongingness and non-belongingness of each elements are obtained with respect to the n values.

ITpDFS are used in the MCDM models to include the learning experience of a decision maker with respect to the frequency in their observation.

By the use of ITpDFS in the MCDM models, the fuzziness in the fuzzy parameters reduces when the elements converge to an interval.

Robot selection under MCDM methodologies are made simpler and sensible while prioritizing the robots in the ITpDF environments.

Some limitations of this proposed methods are listed below.

Using the ITpDF-AHP, the weights for the criteria were determined (subjective weights). The result may differ if the weights are determined in accordance with the objective weights.

In this study, the data set is collected from the literature and online sources. However, primary data collections can also be done and therefore, the ranking may change with respect to the data set.

The data collection for the military robots seemed to be a bit challenging.

The comparison of the ranking orders with the real-time results is still lagging as they are not assured.

5 Conclusion

Robots are often considered to be cheaper to use over humans as they make replacements over human lives in handling dangerous situations. Though they might not replace human’s spontaneous thinking, they are smarter and more efficient than humans. The majority of robots today are used to perform repetitive tasks or jobs that are too dangerous for humans to perform. The military is one of such fields and robots introduced in the military field are more practical. One of the most difficult problems for a decision maker is choosing the most preferred military land robot for a specific purpose.

In this study, list of most widely used robots in the military field were taken for four major applications and the robot selection for those applications were done on ITpDF-AHP-TOPSIS methods. The problem was considered in the field of intuitionistic dense fuzzy environment which provides results with higher approximation than the others. Four applications in the military field namely bomb disposal, search and rescue, surveillance and reconnaissance and war-fighter were taken. The robots were selected depending on their construction purpose for various applications and the preference ranking for each application were calculated. Comparative analysis is worked on for fuzzy set, intuitionistic fuzzy set, dense fuzzy set and intuitionistic dense fuzzy set to describe the effectiveness of ITpDFS. Sensitivity analysis is done for n=1, 5, 10, 100, 1000, 1000 and observed that the ranking is identical for all the values of n.

Ranking methods and aggregation operators can also be introduced with respect to ITpDFS. Not only in the field of land, robot selection can be done for the fields like marine and airborne also, which serves as a future scope for this study. Robot selection with other MCDM models can be incorporated. Furthermore, the aforementioned limitations can be solved in future research.

Funding

The authors have not received any funding.

Data availability

On reasonable request, the data sets used in the investigation are available from the corresponding author.

Declarations

Conflict of interest

The authors have no conflict interest.

Authorship contributions

The authors are equally contributed to the present study. The manuscript preparation and data collection were employed by Swethaa S and Felix A.

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Application	The value of n
	n=1	n=5	n=10	n=100	n=1000	n=10000
Bomb Disposal	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4	BR2 >BR3 >BR1 >BR4
War-fighter robots	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4	WF3 >WF1 >WF2 >WF4
Surveillance and reconnaissance robots	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7	SR1 >SR5 >SR6 >SR3 >SR2 >SR4 >SR7
Search and rescue robots	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2	S&R1 >S&R3 >S&R2

An intuitionistic dense fuzzy AHP-TOPSIS method for military robot selection

Abstract

Keywords

Abbreviations

1 Introduction

1.1 Review on dense fuzzy sets

1.2 Review on MCDM techniques

1.3 Motivation

1.4 Contribution and novelty

2 Theoretical background

3.1 Algorithm of intuitionistic dense fuzzy AHP-TOPSIS

Table 1 Scale of relative importance table Score Degree of Importance 1 Equally Important (EI) 3 Moderate Important (MI) 5 High Importance (HI) 7 Very High Importance (VHI) 9 Extremely Important (ExI) 2,4,6,8 Intermediate Importance (IV) 1/3, 1/5, 1/7, 1/9 Values of inverse comparison

5 I. PackBot

4.1.1 Robots in military field

4.1.2 A description of the dataset

4.1.3 The dataset of the alternatives

4.1.4 The dataset of the criteria

4.2 Adaptation of the proposed method for selecting military robots

4.3.1 Bomb disposal robots

Table 21 War-fighter robots Robots CC i RANK MAARS 0.5910 3 THeMIS 0.5913 2 Ripsaw 0.6138 1 BEAR 0.1378 4

Table 22 Surveillance and reconnaissance robots Robots CC i RANK MAARS 0.9957 1 MATILDA 0.3378 5 Packbot 0.3614 4 BEAR 0.0031 6 Dragon Runner 0.6623 2 Milos 0.3635 3 Mackbot 0.0011 7

Table 23 Search and rescue robots Robots CC i RANK TALON 0.6921 1 MATILDA 0.0378 3 BEAR 0.3766 2

5 Conclusion

Funding

Data availability

Declarations

Conflict of interest

Authorship contributions

References

Table 1
Scale of relative importance table

Score Degree of Importance

1 Equally Important (EI)

3 Moderate Important (MI)

5 High Importance (HI)

7 Very High Importance (VHI)

9 Extremely Important (ExI)

2,4,6,8 Intermediate Importance (IV)

1/3, 1/5, 1/7, 1/9 Values of inverse comparison

Table 21
War-fighter robots

Robots CC _i RANK

MAARS 0.5910 3

THeMIS 0.5913 2

Ripsaw 0.6138 1

BEAR 0.1378 4

Table 22
Surveillance and reconnaissance robots

Robots CC _i RANK

MAARS 0.9957 1

MATILDA 0.3378 5

Packbot 0.3614 4

BEAR 0.0031 6

Dragon Runner 0.6623 2

Milos 0.3635 3

Mackbot 0.0011 7

Table 23
Search and rescue robots

Robots CC _i RANK

TALON 0.6921 1

MATILDA 0.0378 3

BEAR 0.3766 2