Autonomous Mobile Robot Selection in Smart Warehouses Considering Cybersecurity and Integration Requirements

Abstract

Autonomous mobile robots (AMRs) are increasingly used in warehouse intralogistics to improve material flow, flexibility, productivity, and operational continuity. However, selecting an appropriate AMR is no longer limited to mechanical performance or acquisition cost, since modern warehouse robots operate as networked cyber-physical systems that must interact with enterprise software, fleet management platforms, communication infrastructures, and cybersecurity mechanisms. This study proposes an integrated Pythagorean fuzzy multi-criteria decision-making (MCDM) framework for evaluating AMR alternatives in warehouse operations by jointly considering economic, technical, physical, software-related, integration-oriented, and security-related criteria. Expert judgments obtained from eight specialists, including four academics and four private-sector professionals, were modeled using Pythagorean fuzzy numbers to capture uncertainty and hesitation in linguistic assessments. The Pythagorean Fuzzy Indifference Threshold-Based Attribute Ratio Analysis (PF-ITARA) method was employed to determine criterion weights based on threshold-sensitive discrimination among alternatives, while Pythagorean Fuzzy VIšekriterijumsko KOmpromisno Rangiranje (PF-VIKOR) was used to rank four AMR alternatives according to a compromise solution logic. The results show that investment cost, maneuverability, total cost of ownership, integration and validation requirements, and ease of programming and commissioning are the most influential criteria. Cybersecurity-related criteria, particularly data confidentiality, system integrity, monitoring and incident response readiness, authentication control, and role-based access control, also received notable importance levels. Among the evaluated alternatives, MiR250 achieved the best overall performance and emerged as the most suitable compromise solution, followed by OMRON LD-250, HIKROBOT Forklift AGV, and KUKA KMP 600-S diffDrive. The proposed framework provides a transparent and practically applicable decision-support tool for AMR procurement by integrating operational performance, digital interoperability, and cybersecurity readiness into a unified evaluation structure.

1. Introduction

Investments in robotic automation have become strategically important in manufacturing and intralogistics systems due to objectives such as increased productivity, consistent quality, improved occupational safety, traceability and operational flexibility. However, the robot market is not limited to industrial manipulators; the proliferation of platforms such as collaborative robots and autonomous mobile robots has expanded the problem of selecting the right robot to a much broader technological and integration space.

Robot selection is by its nature a multi-criteria decision problem. Technical criteria such as payload capacity, reach, accuracy, speed and cycle time are considered alongside economic criteria such as purchase and integration costs, energy consumption and lifecycle costs. In addition, qualitative criteria such as supplier support, maintainability, compliance with safety standards, software ecosystem and compatibility with existing line or warehouse management systems directly affect the quality of the decision. Since a significant number of these criteria conflict with each other, approaches that optimize a single metric are insufficient in most applications. Therefore, robot selection in the literature is mostly addressed within the framework of Multi-Criteria Decision-Making (MCDM). MCDM methods bring transparency to the decision-making process in terms of weighting criteria, consistently comparing alternatives and modeling uncertain expert judgments. In particular, linguistic evaluations, which are common in practice, can be quantified using fuzzy set theory and its derivatives, allowing the methods to be applied under realistic conditions.

The shared studies primarily address robot selection from an MCDM perspective, framing it as a criterion-based ranking or selection problem in different application contexts. While early studies mostly focused on core criteria such as performance and cost, newer studies include system-level dimensions such as integration, safety, energy management, standardization, cloud architectures and cybersecurity in the selection set.

In robot selection problems, criteria can generally be grouped into six main categories. This set is expanded with sub-criteria such as navigation maturity, fleet management, charging strategies, and wireless communication performance. [1,2,3,4]. In the latter half of the 1990s and the beginning of the 2000s, scoring and hierarchical approaches, which could be considered more deterministic, came to the forefront for robot selection. Models that consider both objective and subjective criteria with revised forms of weighted sums and structured weighting and ranking based on the analytical hierarchy process (AHP) are typical examples of this trend [1]. During the same period, studies emphasizing the investment evaluation dimension [2] and studies relating it to performance measurement models [3] contributed to integrating the selection problem with financial and operational goals.

Since 2010, the literature has expanded significantly in terms of both methodological diversity and hybridization. Studies comparing consensus ranking and outranking approaches [4], distance-based selection approaches and extensions of the vIšeKriterijumska optimizacija I kompromisno rešenje (VIKOR) methodology in an intuitionistic fuzzy environment have shown that different decision logics can be adapted to robot selection. New methodological proposals that consider both objective and subjective criteria and solution proposals based on the multi-objective optimization by ratio analysis (MULTIMOORA) that model uncertainty with grey numbers support this diversification.

During this period, decision support systems [5] and systematic literature reviews [6,7] both facilitated method selection for practitioners and clarified the conceptual boundaries of the research field. Studies such as interval-valued fuzzy TOPSIS [8] and interval valued fuzzy complex proportional assessment (fuzzy COPRAS) [4] are examples of how uncertainty is addressed with richer representations.

The 2015 to 2016 period saw a rise in the use of integrated fuzzy MCDM frameworks and outranking methods. Examples include integrated fuzzy approaches that consider both objective and subjective criteria [9] and selection applications based on the preference ranking organization method for enrichment evaluation II (PROMETHEE II) [10]. Adapted MCDM proposals under interval type-2 fuzzy sets [11] and the extension of PROMETHEE for robot selection [10] further advanced the logic of uncertainty modeling and outranking.

Uncertainty representation is central to the literature because robot selection decisions are often based on limited data, expert opinions and linguistic evaluations. Classical fuzzy numbers and intuitionistic fuzzy sets have expanded into more expressive structures such as interval-valued, type-2, pythagorean and q-rung orthopair fuzzy models [10]. This expansion aims to incorporate the decision-maker’s levels of certainty or uncertainty into the model in a more flexible way.

There is a growing trend in studies using 2-tuple representations and q-rung orthopair fuzzy structures, particularly for more precise encoding of linguistic information. The application of the CODAS-based 2-tuple linguistic q-rung orthopair fuzzy approach to robot selection [12] and the integration of the elimination et choix traduisant la realité (ELECTRE) based multi-attribute group decision-making (MAGDM) framework with linguistic q-rung orthopair information [13] are noteworthy in terms of group decision-making and the formalization of linguistic scales. Recently, new operator proposals such as trigonometric Pythagorean fuzzy normal numbers and aggregation operators [14] and fuzzy neural network-based analyses [15] demonstrate the intersection of this field with data-driven or learning methods.

Compared with recent fuzzy MAGDM studies, including the Einstein aggregation operators for Pythagorean fuzzy soft sets proposed by Zulqarnain et al. [16] and the interval-valued probabilistic linguistic T-spherical fuzzy TOPSIS-based cloud storage provider selection model developed by Gurmani et al. [17], the present study differs in both methodological structure and application scope. These studies mainly focus on developing advanced fuzzy information aggregation or ranking mechanisms for general MAGDM problems. In contrast, this study develops an application-oriented PF-ITARA–PF-VIKOR framework for AMR selection in warehouse intralogistics. The proposed model emphasizes threshold-sensitive criterion weighting, compromise-based ranking, and the explicit inclusion of integration and cybersecurity requirements in AMR procurement decisions.

The focus of application in the literature has expanded over time from being solely for industrial manipulator selection to include newer robot classes such as cobots and AMRs. For cobot selection, a hybrid AHP-TOPSIS-based MCDM application [18] emphasizes the need to naturally integrate human–robot interaction and safety constraints into the selection process. On the industrial robot side, fuzzy AHP and TOPSIS integrated frameworks [19] and hybrid approaches combining objective weighting with criteria importance through intercriteria correlation (CRITIC) and consensus ranking with VIKOR [20] demonstrate the maturity of methodological hybridization.

In mobile robot and AMR selection, the criteria set and system architecture become more distinct. Selecting mobile robots using fuzzy extended VIKOR in specific applications, such as hospital pharmacies [21] is an example of context-oriented criteria formulation. Studies classifying the current state and research gaps for AMRs in intralogistics [22] and studies examining the transferability of AMR functions to the cloud using AHP [23] transform the selection problem into a robot and digital infrastructure collaborative design.

This expansion also incorporates issues such as energy storage and standardization into the selection problem. Studies discussing the power consumption, pack characteristics and future perspective of Li-ion batteries used in AMRs [24] and the standard interface requirements in AGV/FTS systems [25] show that the selection criteria have shifted from the product level to the ecosystem level. A study evaluating autonomous robot alternatives in warehouse optimization using AHP [26] also demonstrates that the selection of AMR or autonomous robots can be directly linked to operational design decisions.

Finally, the risk and safety dimension of the selection problem is not limited solely to physical safety. Studies linking cybersecurity requirements in industrial machine control systems with functional safety [27] emphasize that robot selection and integration decisions should be considered in conjunction with cyber-physical risk management. An integrated framework combining fuzzy TOPSIS with picture fuzzy combined compromise solution (CoCoSo) for AMR selection [28] is a current example reflecting both methodological hybridization and the diversification of criteria within the context of AMR.

Despite these developments, the existing literature still presents several gaps regarding AMR selection for contemporary warehouse environments. First, many robot selection studies mainly emphasize conventional technical, operational, and economic criteria, whereas integration readiness and cybersecurity requirements are often treated as secondary or post-selection issues. Second, AMRs used in warehouse intralogistics increasingly operate as networked cyber-physical systems that interact with WMS, MES, ERP, fleet management platforms, communication infrastructures, and remote software services. However, the selection literature has not sufficiently incorporated these system-level and security-oriented requirements into a unified evaluation structure. Third, although fuzzy MCDM methods have been widely applied to robot selection problems, limited attention has been given to combining threshold-sensitive criterion weighting with compromise-based ranking under Pythagorean fuzzy uncertainty for cybersecurity- and integration-aware AMR selection.

To address these gaps, this study develops an integrated Pythagorean fuzzy MCDM framework for selecting autonomous mobile robots in warehouse operations. The proposed model evaluates AMR alternatives through a comprehensive criterion structure covering economic, technical, physical, software-related, integration-oriented, and security-related dimensions. Expert judgments obtained through linguistic assessments are modeled using Pythagorean fuzzy numbers to capture uncertainty and hesitation in the decision-making process. The Pythagorean Fuzzy Indifference Threshold-Based Attribute Ratio Analysis (PF-ITARA) method is employed to determine criterion weights because it considers the discriminative power of criteria through indifference thresholds and reduces exclusive dependence on subjective weighting judgments. Subsequently, the Pythagorean Fuzzy VIKOR (PF-VIKOR) method is used to rank the candidate AMR solutions because it provides a compromise-based ordering by simultaneously considering overall group utility and individual regret. This integrated structure enables the selection process to reflect both the relative importance of criteria and the need for a balanced AMR alternative under conflicting evaluation dimensions.

The main contributions of this study are threefold. First, it proposes a cybersecurity- and integration-aware AMR selection framework that extends conventional robot evaluation criteria by explicitly incorporating digital interoperability, access control, data protection, monitoring readiness, and vulnerability management requirements. Second, it introduces a hybrid PF-ITARA and PF-VIKOR decision model that combines Pythagorean fuzzy uncertainty modeling, semi-objective criterion weighting, and compromise-based alternative ranking. Third, it provides a practical decision-support structure for warehouse managers, automation engineers, IT/OT integration teams, and cybersecurity stakeholders by demonstrating how AMR procurement can be evaluated from a system-level perspective rather than a purely hardware-centered perspective. Taken together, the novelty of the study lies not in the isolated use of a single fuzzy MCDM method, but in the joint design of (i) a cybersecurity- and integration-aware AMR criterion architecture for warehouse intralogistics and (ii) a PF-ITARA–PF-VIKOR workflow that links threshold-sensitive weighting with compromise ranking under Pythagorean fuzzy uncertainty [22,25].

Accordingly, the problem addressed in this study can be stated as follows: given four candidate AMR alternatives for warehouse intralogistics and a set of 36 evaluation criteria grouped into economic, technical, physical, software, integration, and security dimensions, determine the most suitable AMR under uncertain and partly subjective expert judgments. The decision problem is inherently multi-criteria because the alternatives must be assessed simultaneously with respect to operational performance, implementation effort, digital interoperability, and cybersecurity readiness in a warehouse environment increasingly characterized by networked cyber-physical interactions [22,25]. To address this problem, the study employs a Pythagorean fuzzy PF-ITARA–PF-VIKOR framework in which criterion importance is derived through threshold-sensitive weighting and the alternatives are ranked according to a compromise solution logic [28].

To clarify the positioning of the present study,

Table 1. Comparison of representative fuzzy MCDM-based robot/AMR selection studies.

Study	Application Focus	Methodology	Main Evaluation Scope	Main Difference
[29]	Industrial robot selection	Fuzzy MCDM	Technical and economic robot selection criteria	Foundational fuzzy robot selection study; does not address AMRs, enterprise integration, or cybersecurity.
[8]	Industrial robot selection	IVFF TOPSIS	Conventional robot performance and selection criteria	Focuses on uncertainty modeling in robot selection, but remains hardware/performance-oriented.
[4]	Industrial robot selection	IVFF COPRAS	Technical and operational robot evaluation criteria	Uses interval-valued fuzzy decision modeling, but does not include digital integration or security readiness.
[9]	Industrial robot selection	Integrated fuzzy MCDM	Objective and subjective robot selection criteria	Provides a hybrid fuzzy selection model, but does not treat robots as networked cyber-physical systems.
[11]	Industrial robot selection	Interval type-2 fuzzy MCDM	Uncertain expert-based robot evaluation	Improves uncertainty representation, but does not address AMR-specific integration or cybersecurity issues.
[10]	Industrial robot selection	Extended PROMETHEE	Robot ranking using outranking logic	Emphasizes outranking-based robot selection rather than system-level AMR deployment requirements.
[21]	Mobile robot selection for hospital pharmacy	Fuzzy extended VIKOR	Context-specific mobile robot selection criteria	Addresses mobile robot selection, but cybersecurity and enterprise-level integration are not central criteria.
[20]	Arc welding robot selection	2-tuple linguistic q-rung orthopair fuzzy CODAS	Linguistic and uncertainty-based robot evaluation	Advances fuzzy linguistic modeling, but focuses on welding robots rather than warehouse AMRs.
[13]	Robot selection	Linguistic q-rung orthopair fuzzy ELECTRE	Group decision-making and robot ranking	Uses advanced fuzzy outranking logic, but does not include integration or cybersecurity as decision dimensions.
[30]	Robot cybersecurity	Comprehensive survey	Cybersecurity risks, threats, and protection mechanisms in robotics	Provides cybersecurity background, but does not propose a robot/AMR selection model.
[22]	AMRs in intralogistics	Literature review	AMR state of the art, limitations, and research gaps	Highlights AMR-related challenges, but does not develop a fuzzy MCDM-based AMR selection framework.
[28]	AMR selection	Fuzzy TOPSIS and picture fuzzy CoCoSo	Subjective and objective AMR selection measures	Provides a recent fuzzy AMR selection framework, but cybersecurity and integration readiness are not jointly structured as core selection dimensions.
This Study	Warehouse AMR selection	PF-ITARA and PF-VIKOR	Economic, technical, physical, software, integration, and security criteria	Develops a cybersecurity- and integration-aware AMR selection framework under Pythagorean fuzzy uncertainty.

compares representative fuzzy MCDM-based robot/AMR selection studies with the proposed framework in terms of application focus, methodological approach, evaluation scope, and the treatment of integration and cybersecurity dimensions.

The remainder of this paper is organized as follows. Section 2 presents the methodological background and the proposed PF-ITARA and PF-VIKOR framework. Section 3 describes the case study, including the AMR alternatives, evaluation criteria, expert panel, criterion weighting results, and alternative rankings. Section 4 discusses the theoretical and managerial implications of the findings. Finally, Section 5 concludes the study and outlines future research directions.

2. Materials and Methods

The methods of this study consist of candidate AMR alternatives, evaluation criteria derived from the literature and expert knowledge, and linguistic assessment data gathered from multiple stakeholders. Beyond conventional technical and cost-related factors, the material set also includes enterprise integration and cybersecurity-oriented criteria, reflecting the practical requirements of contemporary warehouse intralogistics. Expert judgments are modeled using fuzzy sets to address uncertainty and subjectivity, thereby forming the basis for criterion weighting, alternative ranking, and sensitivity analysis within the proposed fuzzy MCDM framework (Figure 1).

2.1. Preliminaries of Pythagorean Fuzzy Sets

Fuzzy set theory, introduced by Zadeh [31] provides an effective framework for representing vague and imprecise information through a membership function. Atanassov [32] extended this concept by proposing intuitionistic fuzzy sets, in which both membership and non-membership degrees are defined. Yager [33] further generalized this structure through Pythagorean fuzzy sets (PFSs), in which the squared sum of the membership and non-membership degrees is constrained to be less than or equal to one. Owing to this property, PFSs provide greater flexibility for representing uncertainty and hesitation in expert judgments. This suppleness allows membership and non-membership degrees to distance a wider range, thus reducing hesitation more exactly in expert assessments.

Definition 1. 

A PFS $\widetilde{PF}$ in the universe of discourse X is defined as in Equation (1), where ${\mu }_{pf}\left(x\right)$ and $v_{pf}\left(x\right)$ denote the membership and non-membership degrees, respectively. The basic condition of a Pythagorean fuzzy set is given in Equation (2), and the hesitancy degree is calculated according to Equation (3).

$$\widetilde{\mathrm{P}\mathrm{F}}=\left\{\mathrm{x},{\mathsf{\mu }}_{\mathrm{p}\mathrm{f}}\left(\mathrm{x}\right),{\mathrm{v}}_{\mathrm{p}\mathrm{f}}\left(\mathrm{x}\right)|\mathrm{x}\in \mathrm{X}\right\}$$

(1)

For each membership and non-membership function  $x\in X$, the following equations create:

$$0\le {{\mathsf{\mu }}_{\mathrm{p}\mathrm{f}}\left(\mathrm{x}\right)}^2+{{\mathrm{v}}_{\mathrm{p}\mathrm{f}}\left(\mathrm{x}\right)}^2\le 1$$

(2)

$${\mathsf{\pi }}_{\mathrm{p}\mathrm{f}}\left(\mathrm{x}\right)=\sqrt{1-{{\mathsf{\mu }}_{\mathrm{p}\mathrm{f}}\left(\mathrm{x}\right)}^2-{{\mathrm{v}}_{\mathrm{p}\mathrm{f}}(\mathrm{x})}^2}$$

(3)

Definition 2. 

Let $\lambda >0$and $\tilde{A}=\left({\mu }_A,v_A\right)$ and $\tilde{B}=\left({\mu }_B,v_B\right)$ be two PF numbers (PFNs). The algebraic operations defined on PFNs are given in Equations (4)–(7) [34,35]:

$$\tilde{\mathrm{A}}\oplus \tilde{\mathrm{B}}=\left(\sqrt{{\mathsf{\mu }}_{\mathrm{A}}+{\mathsf{\mu }}_{\mathrm{B}}-{\mathsf{\mu }}_{\mathrm{A}}{\mathsf{\mu }}_{\mathrm{B}}},{\mathrm{v}}_{\mathrm{A}}{\mathrm{v}}_{\mathrm{B}}\right)$$

(4)

$$\tilde{\mathrm{A}}\otimes \tilde{\mathrm{B}}=\left({\mathsf{\mu }}_{\mathrm{A}}{\mathsf{\mu }}_{\mathrm{B}},\sqrt{{{\mathrm{v}}_{\mathrm{A}}}^2+{{\mathrm{v}}_{\mathrm{B}}}^2-{{\mathrm{v}}_{\mathrm{A}}}^2{{\mathrm{v}}_{\mathrm{B}}}^2}\right)$$

(5)

$${\tilde{\mathrm{A}}}^{\mathsf{\lambda }}=\left({{\mathsf{\mu }}_{\mathrm{A}}}^{\mathsf{\lambda }},\sqrt{1-{\left(1-{{\mathrm{v}}_{\mathrm{A}}}^2\right)}^{\mathsf{\lambda }}}\right)$$

(6)

$$\mathsf{\lambda }\tilde{\mathrm{A}}=\left(\sqrt{1-{\left(1-{{\mathsf{\mu }}_{\mathrm{A}}}^2\right)}^{\mathsf{\lambda }}},{{\mathrm{v}}_{\mathrm{A}}}^{\mathsf{\lambda }}\right)$$

(7)

Definition 3. 

The score function and the normalized score function of PFN $\tilde{A}$ are presented in Equations (8) and (9), respectively [34,36]. These functions are used to transform PF evaluations into comparable scalar values.

$$\mathrm{F}\left(\tilde{\mathrm{A}}\right)={\left({\mathsf{\mu }}_{\mathrm{A}}\right)}^2-{\left({\mathrm{v}}_{\mathrm{A}}\right)}^2$$

(8)

$${\mathrm{F}}^{\mathrm{*}}\left(\tilde{\mathrm{A}}\right)={\displaystyle \frac{\left(\mathrm{F}\left(\tilde{\mathrm{A}}\right)+1\right)}{2}}$$

(9)

Definition 4. 

Let ${\tilde{A}}_j=\left({{\mu }_A}_j,{v_A}_j\right)$, $j=(1,\dots \dots n)$ be a collection of PFNs, and let $w_j$ denote the associated weight vector satisfying ${\displaystyle {\sum }_{j=1}^n}{w}_j=1$. The Pythagorean Fuzzy Weighted Average (PFWA) operator is defined in Equation (10) [37]. This operator is used to aggregate the assessments of multiple experts into a single PF evaluation.

$${\mathrm{P}\mathrm{F}\mathrm{W}\mathrm{A}}_{{\mathrm{w}}_{\mathrm{j}}}\left({\tilde{\mathrm{A}}}_1,\dots ,{\tilde{\mathrm{A}}}_{\mathrm{j}}\right)={\mathrm{w}}_1{\tilde{\mathrm{A}}}_1⨁{\mathrm{w}}_2{\tilde{\mathrm{A}}}_2\cdots \cdots ⨁{\mathrm{w}}_{\mathrm{n}}{\tilde{\mathrm{A}}}_{\mathrm{n}}=\left(\sqrt{1-{\displaystyle \prod _{\mathrm{j}=1}^{\mathrm{n}}}{\left({1-\left({\mathsf{\mu }}_{\mathrm{j}}\right)}^2\right)}^{{\mathrm{w}}_{\mathrm{j}}}},{\displaystyle \prod _{\mathrm{j}=1}^{\mathrm{n}}}{{\mathrm{v}}_{\mathrm{j}}}^{{\mathrm{w}}_{\mathrm{j}}}\right)$$

(10)

Definition 5. 

The Euclidean and Taxicab distance measures between two PFNs are defined in Equations (11) and (12), respectively [38]. These measures are employed in the subsequent ranking stage of the proposed framework.

$$\mathrm{u}(\tilde{\mathrm{A}},\tilde{\mathrm{B}})=\sqrt{{\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\mu }}_{\mathrm{A}}^2-{\mathsf{\mu }}_{\mathrm{B}}^2\right|+\left|{\mathrm{v}}_{\mathrm{A}}^2-{\mathrm{v}}_{\mathrm{B}}^2\right|+\left|{\mathsf{\pi }}_{\mathrm{A}}^2-{\mathsf{\pi }}_{\mathrm{B}}^2\right|\right)}$$

(11)

$$\mathrm{ı}(\tilde{\mathrm{A}},\tilde{\mathrm{B}})={\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\mu }}_{\mathrm{A}}-{\mathsf{\mu }}_{\mathrm{B}}\right|+\left|{\mathrm{v}}_{\mathrm{A}}-{\mathrm{v}}_{\mathrm{B}}\right|+\left|{\mathsf{\pi }}_{\mathrm{A}}-{\mathsf{\pi }}_{\mathrm{B}}\right|\right)$$

(12)

2.2. PF-ITARA-VIKOR Integration

In this study, the ITARA and VIKOR methods are integrated within a Pythagorean fuzzy environment in order to determine criterion weights and rank the alternatives under uncertainty. ITARA is adopted as a semi-objective weighting method that exploits the dispersion of alternative performances together with indifference thresholds to identify criterion importance [39,40]. VIKOR is then used to obtain a compromise ranking by jointly considering group utility and individual regret [41]. By extending this hybrid structure to the Pythagorean fuzzy setting, linguistic expert judgments can be incorporated while preserving the uncertainty inherent in the evaluation process [42].

The proposed hybrid framework offers several methodological advantages that make it more suitable than traditional MCDM techniques. The specific reasons for selecting this integrated model are summarized below:

PF-ITARA uses an explicit indifference threshold to eliminate minor data gaps, ensuring that weights reflect the true discriminating power of the criteria [43].

The PF-ITARA approach demonstrates greater stability than entropy-based methods and reduces the cognitive burden on experts compared to traditional pairwise comparison techniques like AHP [44].

PF-VIKOR determines a compromise solution by balancing group utility and individual regret, which is essential for handling the conflicting criteria typical in mobile robot selection [45].

The PF-VIKOR method includes rigorous acceptance tests for advantage and stability to ensure the final ranking is robust and reliable [44].

The construction of the PF decision matrix and the aggregation of expert judgments are carried out as follows.

Step 1: A decision matrix is constructed based on the evaluations of the experts. In this process, the experts assess each alternative with respect to each criterion by using the linguistic terms presented in

Table 2. Linguistic terms [43].

Linguistic Term	$\mathbf{P}\mathbf{F}\mathbf{N}\mathbf{s} (\mathit{\mu },\mathit{v})$
Very Good (VG)	(0.8, 0.3)
Good (G)	(0.7, 0.35)
Medium Good (MG)	(0.6, 0.5)
Fair (F)	(0.45, 0.65)
Medium Poor (MP)	(0.35, 0.75)
Poor (P)	(0.3, 0.85)
Very Poor (VP)	(0.25, 0.9)

. Subsequently, these linguistic assessments are transformed into their corresponding PFNs according to the scale provided in Table 2.

Step 2: The individual expert assessments are aggregated into a single collective PF evaluation by using the PFWA operator.

Step 3: The normalized score values of the aggregated PF assessments are then computed.

Step 4: The normalized decision matrix is obtained according to Equation (13), where ${\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{j}}$ denotes the normalized assessment of the alternative ${\mathrm{A}}_{\mathrm{i}}$ under criterion ${\mathrm{C}}_{\mathrm{j}}$.

$${\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{{\mathrm{F}}^{\mathrm{*}}\left({\mathsf{\xi }}_{\mathrm{l}\mathrm{j}}\right)}{{\displaystyle {\sum }_{\mathrm{l}=1}^{\mathrm{m}}} \left({\mathrm{F}}^{\mathrm{*}}\left({\mathsf{\xi }}_{\mathrm{l}\mathrm{j}}\right)\right)}},\mathrm{i}=1,\dots ,\mathrm{m};\mathrm{j}=1,\dots ,\mathrm{n}$$

(13)

Step 5: For each criterion, the normalized assessments are ordered from the smallest to the largest value, as shown in Equation (14).

$${\mathrm{L}}_{(1)\mathrm{j}}=\underset{1\le \mathrm{i}\le \mathrm{m}}{\mathrm{m}\mathrm{i}\mathrm{n}} \mathrm{N}{\mathrm{R}}_{\mathrm{i}\mathrm{j}}<\cdots <{\mathrm{L}}_{(\mathrm{m})\mathrm{j}}=\underset{1\le \mathrm{i}\le \mathrm{m}}{\mathrm{m}\mathrm{a}\mathrm{x}} {\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{j}},\mathrm{j}=1,\dots ,\mathrm{n}$$

(14)

${\mathrm{L}}_{(1)\mathrm{j}}$ and ${\mathrm{L}}_{(\mathrm{m})\mathrm{j}}$ represent the lowest (order 1) and highest (order m) normalized assessments under the criterion ${\mathrm{C}}_{\mathrm{j}}$.

Step 6: The ordered distances between two consecutive normalized assessments are computed by using Equation (15). These distances indicate the degree of separation among alternatives under each criterion.

$${\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}={\mathrm{L}}_{(\mathrm{t}+1)\mathrm{j}}-{\mathrm{L}}_{(\mathrm{t})\mathrm{j}},\mathrm{t}=1,\dots ,\mathrm{m}-1;\mathrm{j}=1,\dots ,\mathrm{n}$$

(15)

${\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}$ is the variance between successive normalized assessments ${\mathrm{L}}_{(\mathrm{t}+1)\mathrm{j}}$ and ${\mathrm{L}}_{(\mathrm{t})\mathrm{j}}$.

Step 7: For each criterion ${\mathrm{C}}_{\mathrm{j}}$ an indifference threshold ($\mathrm{I}{\mathrm{T}}_{\mathrm{j}}$) is specified by the experts. The normalized indifference threshold $(\mathrm{n}\mathrm{I}{\mathrm{T}}_{\mathrm{j}}$) is calculated by Equation (16):

$$\mathrm{n}\mathrm{I}{\mathrm{T}}_{\mathrm{j}}={\displaystyle \frac{\mathrm{I}{\mathrm{T}}_{\mathrm{j}}}{{\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{m}}} {\mathsf{\xi }}_{\mathrm{l}\mathrm{j}}}}$$

(16)

where $\mathrm{n}\mathrm{I}{\mathrm{T}}_{\mathrm{j}}$ serves as a foundation for determining whether or not a distance ${\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}$ is important.

Step 8: The significant ordered distances are determined according to Equation (17). If the distance between two consecutive ordered values exceeds the corresponding normalized indifference threshold, that distance is treated as meaningful; otherwise, it is set to zero.

$${\mathrm{O}\mathrm{R}}_{\mathrm{t}\mathrm{j}}=\left\{\begin{array}{c}{\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}-\mathrm{n}\mathrm{I}{\mathrm{T}}_{\mathrm{j}}\mid {\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}>\mathrm{n}\mathrm{I}{\mathrm{T}}_{\mathrm{j}}\\ 0\mid {\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}\le \mathrm{n}\mathrm{I}{\mathrm{T}}_{\mathrm{j}}\end{array},\mathrm{t}=1,\dots ,\mathrm{m}-1;\mathrm{j}=1,\dots ,\mathrm{n};\mathsf{\xi }>0\right.$$

(17)

If ${\mathrm{\varnothing }}_{\mathrm{t}\mathrm{j}}$ surpasses ${\mathrm{T}}_{\mathrm{j}}$, ${\mathrm{O}\mathrm{R}}_{\mathrm{t}\mathrm{j}}$ is considered important, attracting the importance of ${\mathrm{C}}_{\mathrm{j}}$. Otherwise, ${\mathrm{O}\mathrm{R}}_{\mathrm{t}\mathrm{j}}$ is equal to zero.

Step 9: The criterion weights are calculated using Equation (18). In this step, the parameter λ determines the distance metric employed in the weighting process, such as Manhattan (λ = 1), Euclidean (λ = 2), or Tchebycheff (=∞)

$${\mathsf{\omega }}_{\mathrm{j}}={\displaystyle \frac{{\left({\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{m}-1}} {\mathrm{O}\mathrm{R}}_{\mathrm{t}\mathrm{j}}^{\mathsf{\lambda }}\right)}^{1/\mathsf{\lambda }}}{{\displaystyle {\sum }_{\mathrm{l}=1}^{\mathrm{n}}} {\left({\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{m}-1}} {\mathrm{O}\mathrm{R}}_{\mathrm{t}\mathrm{l}}^{\mathsf{\lambda }}\right)}^{1/\mathsf{\lambda }}}},\mathrm{j}=1,\dots ,\mathrm{n};\mathsf{\lambda }\in \{1,\dots ,\mathrm{\infty }\}$$

(18)

where $\mathsf{\omega }={\left({\mathsf{\omega }}_1,\dots ,{\mathsf{\omega }}_{\mathrm{j}},\dots ,{\mathsf{\omega }}_{\mathrm{n}}\right)}^{\mathrm{T}}$ represent the vector of importance.

Step 10: For each criterion, the positive ideal PF value (${\mathrm{P}\mathrm{S}}_{\mathrm{j}}^{+}$) and negative ideal PF value (${\mathrm{P}\mathrm{S}}_{\mathrm{j}}^{-}$) are identified according to Equations (19) and (20). These ideal values are determined by considering whether the criterion is of a benefit or cost type.

$${\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{+}=\left\{\begin{array}{c}\arg \underset{\mathrm{i}}{\max }\{\mathrm{S}({\tilde{\mathrm{e}}}_{\mathrm{i}\mathrm{j}})\}, \mathrm{if} \mathrm{criterion} \mathrm{j} \mathrm{is} \mathrm{benefit}\\ \arg \underset{\mathrm{i}}{\min }\{\mathrm{S}({\tilde{\mathrm{e}}}_{\mathrm{i}\mathrm{j}})\}, \mathrm{if} \mathrm{criterion} \mathrm{j} \mathrm{is} \mathrm{cost}\end{array}\right.$$

(19)

$${\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{-}=\left\{\begin{array}{c}\arg \underset{\mathrm{i}}{\min }\{\mathrm{S}({\tilde{\mathrm{e}}}_{\mathrm{i}\mathrm{j}})\}, \mathrm{if} \mathrm{criterion} \mathrm{j} \mathrm{is} \mathrm{benefit}\\ \arg \underset{\mathrm{i}}{\max }\{\mathrm{S}({\tilde{\mathrm{e}}}_{\mathrm{i}\mathrm{j}})\}, \mathrm{if} \mathrm{criterion} \mathrm{j} \mathrm{is} \mathrm{cost}\end{array}\right.$$

(20)

Step 11: The group utility value $({\mathrm{T}}_{\mathrm{i}})$ and individual regret value $({\mathrm{I}}_{\mathrm{i}})$ of each alternative are computed by Equations (21) and (22), respectively. Here, $({\mathrm{T}}_{\mathrm{i}})$ reflects the overall performance of an alternative across all criteria, whereas $({\mathrm{I}}_{\mathrm{i}})$ represents its worst-case performance with respect to the most unfavorable criterion.

$${\mathrm{T}}_{\mathrm{i}}={\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{n}}} {\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{j}}\mathrm{E}\left({\tilde{\mathrm{e}}}_{\mathrm{i}\mathrm{j}},{\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{+}\right)}{\mathrm{E}\left({\tilde{\mathrm{P}}}_{\mathrm{j}}^{+},{\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{-}\right)}}$$

(21)

$${\mathrm{I}}_{\mathrm{i}}=\underset{\mathrm{j}}{\max }\left\{{\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{j}}\mathrm{E}\left({\tilde{\mathrm{e}}}_{\mathrm{i}\mathrm{j}},{\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{+}\right)}{\mathrm{E}\left({\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{+},{\widetilde{\mathrm{P}\mathrm{S}}}_{\mathrm{j}}^{-}\right)}}\right\}$$

(22)

Step 12: The compromise index $({\mathrm{Q}}_{\mathrm{i}})$ is calculated using Equation (23). This index combines the group utility and individual regret measures, where the parameter z represents the weight assigned to the majority utility strategy [46].

$${\mathrm{Q}}_{\mathrm{i}}=\mathrm{z}{\displaystyle \frac{\left({\mathrm{T}}_{\mathrm{i}}-{\mathrm{T}}^{+}\right)}{{(\mathrm{T}}^{-}-{\mathrm{T}}^{+})}}+\left(1-\mathrm{z}\right){\displaystyle \frac{\left({\mathrm{I}}_{\mathrm{i}}-{\mathrm{I}}^{+}\right)}{{(\mathrm{I}}^{-}-{\mathrm{I}}^{+})}}$$

(23)

where ${\mathrm{T}}^{+}=\underset{\mathrm{i}}{\min }{\mathrm{T}}_{\mathrm{i}};{\mathrm{T}}^{-}=\underset{\mathrm{i}}{\max }{\mathrm{T}}_{\mathrm{i}}$ and ${\mathrm{I}}^{+}=\underset{\mathrm{i}}{\min }{\mathrm{I}}_{\mathrm{i}};{\mathrm{I}}^{-}=\underset{\mathrm{i}}{\max }{\mathrm{I}}_{\mathrm{i}}$.

Step 13: Finally, the alternatives are ranked according to their ${\mathrm{T}}_{\mathrm{i}}$, ${\mathrm{I}}_{\mathrm{i}}$, and ${\mathrm{Q}}_{\mathrm{i}}$ values. Let ${\mathsf{\delta }}^{\prime }$ denote the alternative with the lowest ${\mathrm{Q}}_{\mathrm{i}}$, and ${\mathsf{\delta }}^{″}$ denote the second-ranked alternative according to ${\mathrm{Q}}_{\mathrm{i}}$. The final compromise solution is then determined on the basis of the VIKOR ranking conditions.

3. Case Study

This case study aims to systematically evaluate autonomous mobile robot (AMR) alternatives for warehouse operations by considering not only conventional technical and economic requirements but also software, integration, and cybersecurity-related concerns. As warehouses increasingly rely on digitalized and interconnected intralogistics systems, selecting an appropriate AMR has become a complex decision problem shaped by operational efficiency, system compatibility, safety, and data security considerations. In this context, four AMR alternatives were identified and comparatively assessed.

In order to reflect the diversity of warehouse AMR applications, the alternative set was deliberately composed of different solution types. MiR250 (A1) and OMRON LD-250 (A2) represent AMR solutions for flexible point-to-point material transport in dynamic intralogistics environments, where infrastructure-light navigation, adaptability to layout changes, and battery-supported continuous operation are critical [22,24]. HIKROBOT Forklift AGV (A3) represents a forklift-oriented autonomous vehicle class for pallet-oriented material handling; this class is associated with additional perception, pallet localization, and obstacle-detection challenges in warehouse settings [22]. KUKA KMP 600-S diffDrive (A4) represents an intralogistics transport alternative in which interoperability, standardized communication, and integration with heterogeneous fleet-control structures become increasingly important [25]. All alternatives were furthermore considered as networked industrial assets that must satisfy cybersecurity-related requirements such as secure access, update management, and protection against cyber-physical threats [27]. Thus, the four alternatives were selected to provide heterogeneity in transport function, implementation burden, and digital integration exposure, allowing the proposed framework to be tested on a realistic warehouse decision set.

The evaluation framework was organized under six main criteria groups, namely economic, technical, physical and precision-related, software and functional flexibility, integration capability, and security dimensions, encompassing a comprehensive set of sub-criteria. The complete list of criteria considered in the study is presented in

Table 3. Criteria used in the AMR evaluation framework.

	Sub-Criteria	Definitions	Refs.
Economic	Investment Cost	Refers to the initial purchase and installation costs of the AMR system.	[37,38]
	Total Cost of Ownership	Represents the total life-cycle cost, including energy, maintenance, service, and operating expenses.	[39]
	Availability and Durability	Refers to the system’s ability to operate continuously and withstand failures.	[40]
	Service Contract Coverage	Refers to the adequacy of the maintenance, service, and support coverage provided by the vendor.	[21]
	Vendor and Technical Support Performance	Represents the effectiveness of the supplier and technical support network.	[37]
	Ease of Maintenance	Refers to the extent to which maintenance and repair operations can be carried out easily.	[39]
Technical	Load Carrying Capacity	Refers to the maximum load that the robot can safely transport.	[21]
	Speed and Throughput Capacity	Refers to the robot’s travel speed and the amount of transport work it can perform within a given time.	[37,39,40]
	Maneuverability	Refers to the ability to move in narrow spaces and complex environments.	[47]
	Platform Stability	Refers to the robot’s ability to maintain balance during movement.	[37]
	Power Capacity	Refers to the extent to which the robot’s battery and power system can meet operational requirements.	[48]
	Operating Area Coverage	Refers to the limits of the robot’s ability to perform tasks under different working conditions.	[38,41,48]
Physical	Physical Constraints	Refers to the robot’s physical limitations, such as size, weight, and space requirements.	[47,48]
	Repeatability	Refers to the robot’s ability to perform the same task repeatedly with similar results.	[37,41]
	Positioning Accuracy	Refers to the precision with which the robot can reach the target position correctly.	[21,42]
	Repeatability Error	Refers to the magnitude of positional deviations occurring in repeated tasks.	[38]
	Operational Safety	Refers to the level at which the robot ensures the safety of humans and equipment during operation.	[48]
	Compliance with Standards and Regulations	Refers to the robot’s level of compliance with relevant technical standards and legal regulations.	[8,37]
Software	Human–Machine Interface	Refers to the extent to which users can control the robot system easily and effectively.	[14,37]
	Programming Flexibility	Refers to the robot’s ability to be programmed for different tasks.	[37,40]
	Ease of Programming and Commissioning	Refers to the ease of the robot’s setup and programming processes.	[48]
	Functional Flexibility and Reconfigurability	Refers to the robot’s ability to adapt to changing task requirements.	[48]
	Control Unit Resources	Refers to the processing power and memory capacity of the control system.	[38,41]
	Infrastructure Compatibility	Refers to the level at which the robot can operate compatibly with the existing warehouse infrastructure.	[37,40]
Integration	WMS/MES/ERP Integration	Refers to the robot’s ability to exchange data with warehouse and production management systems.	[46,49]
	Fleet Management System Compatibility	Refers to the level at which the robot can operate compatibly with centralized fleet management systems.	[50]
	API/SDK Quality and Data Exchange	Refers to the adequacy of the robot’s software interfaces for data integration.	[46,51]
	Integration and Validation Requirements	Refers to the time and technical effort required for integrating the robot system.	[51,52]
	Network Segmentation Compatibility	Refers to the level at which the robot can comply with the corporate network security architecture.	[53,54]
	Monitoring and Incident Response Readiness	Refers to the infrastructure required for monitoring system events and responding to incidents.	[55]
Security	Authentication Control	Refers to the reliability of the authentication mechanisms for users accessing the system.	[55,56]
	Authorization and Role-Based Access Control	Refers to the definition of access rights according to user authorizations.	[30]
	System Integrity	Refers to the level of protection of system software and data against unauthorized modifications.	[30]
	Data Confidentiality	Refers to the level of protection of system data against unauthorized access.	[30,57]
	Update and Vulnerability Management	Refers to the extent to which security updates can be applied regularly within the system.	[30]
	Evidence of Security Management	Refers to whether the vendor possesses security policies and certifications.	[58]

.

To capture the uncertainty and hesitation inherent in expert judgments, the analysis was carried out within a Pythagorean fuzzy environment. Based on the evaluations of eight experts, the criterion weights were determined using the PF-ITARA method. These weights were then incorporated into the PF-VIKOR approach to rank the AMR alternatives. This hybrid framework was designed to address the multidimensional nature of the AMR selection problem and to provide decision-makers with a structured and practically relevant basis for prioritizing the most suitable alternative.

Within the scope of the study, an expert panel consisting of eight members was formed to strengthen the reliability and practical relevance of the evaluation process. The panel was deliberately balanced to include four experts from academia and four experts from the private sector in order to combine methodological knowledge with field-based operational experience. The selected experts had backgrounds related to warehouse automation, autonomous mobile robots, industrial engineering, intralogistics, enterprise systems integration, maintenance planning, and industrial cybersecurity. They contributed to the identification and refinement of the evaluation criteria, the assessment of their relative importance, and the appraisal of AMR alternatives under the proposed decision framework. Detailed information about the experts is presented in

Table 4. Information about experts.

ID	Experience	Education	Job Title	Research Area/Expertise
E1	17 years	PhD in Industrial Engineering	Professor	MCDM, warehouse systems, logistics optimization
E2	13 years	PhD in Mechatronics Engineering	Associate Professor	Autonomous systems, mobile robotics, robot control
E3	11 years	PhD in Computer Engineering	Associate Professor	Industrial software systems, data integration, cyber-physical systems
E4	9 years	PhD in Industrial Engineering	Assistant Professor	Decision support systems, smart manufacturing, uncertainty modeling
E5	16 years	MSc in Industrial Engineering	Warehouse Automation Manager	Intralogistics operations, AMR deployment, process improvement
E6	14 years	MSc in Electrical and Electronics Engineering	Robotics Integration Lead	AMR integration, control architecture, commissioning and validation
E7	12 years	MSc in Computer Engineering	Industrial Cybersecurity Specialist	Network security, access control, industrial cybersecurity governance
E8	10 years	BSc in Mechanical Engineering	Maintenance and Technical Services Manager	Maintenance planning, system reliability, serviceability and lifecycle performance

.

The experts were selected using a purposive expert sampling approach, since the evaluation problem requires specialized knowledge rather than random stakeholder representation. Three eligibility criteria were considered in forming the panel: (i) direct academic or professional experience related to warehouse automation, AMR systems, robotics, intralogistics, system integration, maintenance, or cybersecurity; (ii) sufficient professional seniority to evaluate both technical and managerial aspects of AMR adoption; and (iii) familiarity with industrial decision-making processes involving technology selection or system implementation. The panel was intentionally structured to include both academic and private-sector perspectives. Academic experts contributed methodological knowledge on MCDM, robotics, cyber-physical systems, and uncertainty modeling, while private-sector experts provided practical insights into AMR deployment, commissioning, integration, maintenance, and cybersecurity operations. Therefore, the selected panel was considered appropriate for evaluating the multidimensional AMR selection problem addressed in this study.

To determine the relative importance of the criteria, the evaluations provided by the eight experts were first transformed into Pythagorean fuzzy numbers (PFNs). Using the linguistic assessment scale defined in the study, the experts evaluated the criteria listed in Table 3 according to their perceived importance (see Table S1 in Supplementary Materials). The individual judgments were then aggregated using the PFWA operator. In this study, equal expert weights were assigned because all panel members satisfied the predefined eligibility requirements, including relevant professional or academic experience, domain knowledge in AMR selection, warehouse automation, system integration, or cybersecurity, and familiarity with industrial decision-making processes. In addition, the expert panel was deliberately balanced by including four academic experts and four private-sector professionals, thereby combining methodological and practical perspectives. Therefore, equal weighting was considered appropriate to avoid introducing additional subjective bias into the aggregation process and to reflect a consensus-based group decision-making structure. Following this aggregation process, integrated Pythagorean fuzzy values and their corresponding score values were obtained for each criterion. The resulting score values, derived from the consolidated expert assessments, are reported in Tables S2 and S3 (Supplementary Materials). These results constituted the main input for the subsequent computation of criterion weights within the PF-ITARA procedure.

Based on the aggregated Pythagorean fuzzy values, score values were obtained for each criterion. These scores were then used to normalize the decision matrix so that the criteria could be evaluated on a comparable basis. After the normalization step, the criterion values were arranged in ascending order, and the distances between successive values were calculated. These ordered distances formed the basis of the threshold analysis. In accordance with expert assessments, an indifference threshold was assigned to each criterion and then normalized. Subsequently, the total importance scores of the criteria were computed, and the results are reported in

Table 5. Indifference threshold and the total score for each criterion.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
$I{T}_j$	0.05	0.05	0.03	0.1	0.1	0.07	0.1	0.05	0.05	0.1	0.1	0.1
Sum	1.846	2.088	2.619	2.649	2.615	2.428	2.513	2.460	2.337	2.752	2.412	2.558
$NI{T}_j$	0.027	0.024	0.011	0.038	0.038	0.029	0.040	0.020	0.021	0.036	0.041	0.039
	C13	C14	C15	C16	C17	C18	C19	C20	C21	C22	C23	C24
$I{T}_j$	0.1	0.1	0.1	0.1	0.07	0.05	0.07	0.03	0.03	0.07	0.1	0.03
Sum	2.105	2.689	2.689	2.689	2.853	2.752	2.527	2.593	2.270	2.448	2.610	2.611
$NI{T}_j$	0.048	0.037	0.037	0.037	0.025	0.018	0.028	0.012	0.013	0.029	0.038	0.011
	C25	C26	C27	C28	C29	C30	C31	C32	C33	C34	C35	C36
$I{T}_j$	0.07	0.1	0.1	0.05	0.1	0.07	0.1	0.1	0.1	0.1	0.1	0.1
Sum	2.640	2.622	2.542	2.056	2.417	2.417	2.269	2.269	2.276	2.261	2.429	2.410
$NI{T}_j$	0.027	0.038	0.039	0.024	0.041	0.029	0.044	0.044	0.044	0.044	0.041	0.041

.

Using the consecutive differences, discriminative distances were derived and employed to identify the relative importance levels of the criteria. In the final step of the PF-ITARA procedure, the normalized weight coefficients of all criteria were calculated, and the resulting weights are given in

Table 6. Criteria weights.

	Weight	R		Weight	R
Investment Cost	0.0648	1	Human–Machine Interface	0.0253	20
Total Cost of Ownership	0.0525	3	Programming Flexibility	0.0301	16
Availability and Durability	0.0292	17	Ease of Programming and Commissioning	0.0483	5
Service Contract Coverage	0.0121	34	Functional Flexibility and Reconfigurability	0.0352	8
Vendor and Technical Support Performance	0.0121	33	Control Unit Resources	0.0112	35
Ease of Maintenance	0.0328	14	Infrastructure Compatibility	0.0347	10
Load Carrying Capacity	0.0126	31	WMS/MES/ERP Integration	0.0178	24
Speed and Throughput Capacity	0.0315	15	Fleet Management System Compatibility	0.0122	32
Maneuverability	0.0538	2	API/SDK Quality and Data Exchange	0.0136	27
Platform Stability	0.0109	36	Integration and Validation Requirements	0.0510	4
Power Capacity	0.0148	26	Network Segmentation Compatibility	0.0260	19
Operating Area Coverage	0.0151	25	Monitoring and Incident Response Readiness	0.0350	9
Physical Constraints	0.0333	13	Authentication Control	0.0342	11
Repeatability	0.0134	28	Authorization and Role-Based Access Control	0.0342	11
Positioning Accuracy	0.0132	29	System Integrity	0.0391	7
Repeatability Error	0.0132	29	Data Confidentiality	0.0392	6
Operational Safety	0.0197	23	Update and Vulnerability Management	0.0246	22
Compliance with Standards and Regulations	0.0281	18	Evidence of Security Management	0.0252	21

.

The obtained results show that Investment Cost received the highest weight (0.0648), indicating that the initial financial burden of AMR adoption is the most influential consideration in the evaluation process. This criterion is followed by Maneuverability (0.0538), Total Cost of Ownership (0.0525), Integration and Validation Requirements (0.0510), and Ease of Programming and Commissioning (0.0483). These findings suggest that decision makers attach particular importance not only to economic feasibility, but also to the practical deployability of AMR systems in real warehouse settings. In other words, alternatives that are cost-efficient, easy to integrate, and adaptable to operational conditions are evaluated more favorably.

The results also indicate that several security- and software-related criteria occupy relatively prominent positions. In particular, Data Confidentiality (0.0392), System Integrity (0.0391), Functional Flexibility and Reconfigurability (0.0352), Monitoring and Incident Response Readiness (0.0350), Infrastructure Compatibility (0.0347), Authentication Control (0.0342), and Authorization and Role-Based Access Control (0.0342) received notable weights. This pattern highlights that AMR selection is not shaped solely by mechanical or operational performance, but also by digital compatibility and cybersecurity readiness. By contrast, criteria such as Platform Stability (0.0109), Control Unit Resources (0.0112), Service Contract Coverage (0.0121), and Fleet Management System Compatibility (0.0122) were assigned comparatively lower weights. Overall, the weighting results demonstrate that the AMR selection problem is primarily driven by a combination of economic considerations, operational suitability, integration effort, and secure system deployment requirements.

Subsequently, the decision matrix was established by transforming the linguistic evaluations into Pythagorean fuzzy numbers. Since the assessment framework was built on a linguistic scale reflecting more favorable judgments through higher preference levels, all criteria were treated as benefit-type criteria in the PF-VIKOR analysis. After determining the aggregated evaluations and incorporating the final criterion weights obtained from the PF-ITARA procedure, the $S_i$, $R_i$, and $Q_i$ values were computed for each AMR alternative. In this stage, the weighted decision structure was used, meaning that the performance values of the alternatives were evaluated together with the corresponding criterion importance coefficients. In addition, the parameter $v$ was set to 0.5 to ensure a balanced consideration of group utility and individual regret. The resulting $S_i$, $R_i$, and $Q_i$ values for the alternatives are reported in

Table 7. $S_i$, $R_i$ and $Q_i$ values and Rankings.

	Score			Rank
	${\mathit{S}}_{\mathit{i}}$	${\mathit{R}}_{\mathit{i}}$	${\mathit{Q}}_{\mathit{i}}$	${\mathit{S}}_{\mathit{i}}$	${\mathit{R}}_{\mathit{i}}$	${\mathit{Q}}_{\mathit{i}}$
A1. MiR250	0.118	0.019	0.000	1	1	1
A2. OMRON LD-250	0.182	0.031	0.169	2	2	2
A3. HIKROBOT Forklift AGV	0.936	0.054	0.879	4	3	4
A4. KUKA KMP 600-S diffDrive	0.370	0.065	0.654	3	4	3

, while the final ranking of the AMR alternatives, based on ascending $Q_i$ values, is presented in Table 7.

The PF-VIKOR results reveal a clear separation among the AMR alternatives. A1 (MiR250) achieved the best overall performance and ranked first, with the lowest $S_i$, $R_i$, and $Q_i$ values, indicating that it provides the most balanced compromise solution across the evaluated criteria. This result suggests that MiR250 offers a strong combination of economic feasibility, operational suitability, integration capability, and security-related performance. A2 (OMRON LD-250) ranked second, showing relatively strong performance, although it remained behind A1 in terms of both overall utility and regret measures. This indicates that, while A2 represents a competitive option, it does not achieve the same level of balance across the decision criteria as the top-ranked alternative. A4 (KUKA KMP 600-S diffDrive) ranked third, implying that its performance is less consistent within the overall evaluation framework, despite showing some acceptable criterion-specific strengths. A3 (HIKROBOT Forklift AGV) obtained the lowest rank, mainly due to its comparatively unfavorable compromise index and the highest regret value, which points to weaker performance under at least one critical criterion. Overall, the findings demonstrate that A1 constitutes the most suitable AMR alternative for the warehouse context considered in this study, while the remaining options exhibit varying degrees of limitation in achieving a similarly balanced performance profile.

4. Sensitivity Analysis

To examine the effect of changes in the indifference threshold $I{T}_j$, eight threshold scenarios were generated. S0 represents the baseline setting. In S1 and S2, all $I{T}_j$ values were decreased and increased by 25%, respectively, while S3 and S4 represent greater global changes with 50% decrease and 50% increase. Since Investment Cost was identified as the most influential criterion, S5, S6, and S7 were designed as criterion-specific stress scenarios by changing only the threshold of Investment Cost while keeping all other $I{T}_j$ values unchanged. The resulting PF-VIKOR compromise scores are presented in

Table 8. Threshold sensitivity results based on PF-VIKOR $Q_i$ values.

Alternative	S0	S1	S2	S3	S4	S5	S6	S7
A1. MiR250	0	0	0	0	0	0	0	0
A2. OMRON LD-250	0.169	0.171	0.166	0.141	0.162	0.139	0.204	0.209
A3. HIKROBOT Forklift AGV	0.879	0.888	0.86	0.866	0.862	0.786	0.983	1
A4. KUKA KMP 600-S diffDrive	0.654	0.645	0.667	0.638	0.685	0.662	0.649	0.518

.

The results indicate that the final ranking is highly stable under alternative threshold configurations. A1 remains the best-performing alternative in all scenarios, with a ${ Q}_i$ value of 0.000, confirming that its leading position is not affected by changes in the $I{T}_j$ values. A2 also consistently remains the second-best alternative across all scenarios. Although the scores of A3 and A4 fluctuate under different threshold assumptions, these changes do not affect the dominance of A1. Therefore, the sensitivity analysis shows that the proposed PF-ITARA and PF-VIKOR framework provides robust results against reasonable changes in the indifference thresholds, including the stress scenarios applied to Investment Cost.

In addition to the threshold sensitivity analysis, the robustness of the PF-VIKOR results was examined by varying the majority utility weight $z$ from 0 to 1 with an interval of 0.1. This analysis was conducted to observe how different emphases on group utility and individual regret affect the final compromise scores of the AMR alternatives. The obtained $Q_i$ values are presented in

Table 9. Sensitivity analysis results for different $z$ values.

Alternative	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
A1. MiR250 (Mobile Industrial Robots)	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
A2. OMRON LD-250	0.260	0.242	0.224	0.205	0.187	0.169	0.151	0.133	0.115	0.097	0.078
A3. HIKROBOT Forklift AGV (FMR-FA)	0.757	0.781	0.806	0.830	0.854	0.879	0.903	0.927	0.951	0.976	1.000
A4. KUKA KMP 600-S diffDrive	1.000	0.931	0.862	0.793	0.723	0.654	0.585	0.516	0.447	0.378	0.309

.

The results show that the leading position of A1 remains unchanged for all $z$ values, with a $Q_i$ value of 0.000 in every scenario. This confirms that MiR250 provides the most stable compromise solution regardless of the relative emphasis placed on group utility or individual regret. A2 also preserves its second position across all scenarios, indicating that its performance is robust under different VIKOR preference settings. In the lower ranks, a rank reversal is observed between A3 and A4. A3 performs better when lower $z$ values are considered, whereas A4 becomes more favorable as the weight of group utility increases. However, this change only affects the third and fourth positions and does not alter the main conclusion of the study. Overall, the analysis confirms that the top-ranked alternatives are highly stable, and the PF-VIKOR results are robust against changes in the majority utility weight.

5. Comparative Analysis

To further examine the robustness of the obtained ranking results, a comparative analysis was conducted using the PF-TOPSIS method. PF-TOPSIS was selected as a benchmark method because it ranks alternatives according to their relative closeness to the positive ideal solution and distance from the negative ideal solution [59]. The same aggregated Pythagorean fuzzy decision matrix and the final criteria weights obtained from the PF-ITARA procedure were used in this comparative analysis. The PF-TOPSIS scores and corresponding rankings are presented in

Table 10. Comparative ranking results based on PF-TOPSIS.

Alternative	PF-TOPSIS Score	Rank
A1. MiR250	0.915	1
A2. OMRON LD-250	0.835	2
A3. HIKROBOT Forklift AGV	0.073	4
A4. KUKA KMP 600-S diffDrive	0.624	3

.

As shown in Table 10, the PF-TOPSIS results fully confirm the ranking obtained from the PF-VIKOR analysis. A1 remains the best-performing alternative, followed by A2, A4, and A3. This consistency indicates that the final ranking is not dependent on a single ranking logic. While PF-VIKOR identifies the most suitable compromise solution by considering group utility and individual regret, PF-TOPSIS evaluates alternatives according to their closeness to the ideal solution. Therefore, the identical ranking obtained from both methods strengthens the reliability of the proposed framework and supports the conclusion that MiR250 is the most suitable AMR alternative in the considered warehouse selection problem.

6. Discussion

The results support the central premise of this study that AMR selection in digitally connected warehouse environments should not be evaluated solely through conventional technical and cost criteria, but also through integration feasibility and cybersecurity readiness. This interpretation is consistent with recent literature in which AMRs are increasingly considered components of broader intralogistics and cyber-physical systems rather than stand-alone transport devices [22]. In this respect, the present findings reinforce the view that warehouse automation decisions are becoming progressively more system-oriented and digitally conditioned.

The weighting results show that investment cost, maneuverability, total cost of ownership, integration and validation requirements, and ease of programming and commissioning are the most influential criteria in the evaluated decision context. The prominence of investment- and lifecycle-related criteria is in line with earlier robot selection studies emphasizing economic feasibility and operational efficiency as primary drivers of technology adoption [1,2,3,5]. At the same time, the strong role of maneuverability and deployability is broadly consistent with recent warehouse and AMR studies, where navigation capability, implementation practicality, and operational adaptability are treated as essential considerations [22,26]. More importantly, the high weight assigned to integration and validation requirements suggests that implementation burden should be treated as part of the selection decision itself rather than as a purely post-selection engineering issue. This observation complements studies emphasizing standardized interfaces, interoperability, and system-level compatibility in AMR ecosystems [25]. By contrast, the relatively low weights assigned to criteria such as platform stability and control unit resources may indicate that some technical characteristics are increasingly perceived as baseline expectations rather than major differentiators among current AMR alternatives.

A particularly notable result is the visible prominence of cybersecurity-related criteria in the final weighting structure. This is not merely symbolic within the model. Data confidentiality and system integrity ranked sixth and seventh, monitoring and incident response readiness ranked ninth, and authentication control together with role-based access control shared the eleventh position. Unlike much of the earlier robot selection literature, which primarily focuses on economic, technical, and operational dimensions [5,11], the present study demonstrates that cybersecurity has become a meaningful decision dimension in its own right. This result is compatible with recent work arguing that cybersecurity and functional safety are increasingly intertwined in industrial automation and control environments [27]. However, the present study goes beyond that perspective by embedding cybersecurity directly into the AMR selection framework rather than treating it only as a deployment or governance concern. In this sense, the study also extends recent AMR-oriented fuzzy decision models by showing that cyber resilience should be evaluated explicitly at the procurement stage [28].

From a methodological perspective, the PF-ITARA-VIKOR framework contributes to the robot selection literature by combining Pythagorean fuzzy modeling, semi-objective weighting, and compromise ranking within a single decision architecture. Previous robot selection studies have successfully employed TOPSIS, PROMETHEE, VIKOR, AHP-based hybrids, and other fuzzy MCDM extensions. Compared with many earlier approaches, the ITARA stage derives criterion importance through threshold-sensitive discrimination among alternatives, thereby reducing exclusive dependence on purely subjective weighting schemes [11,47]. The VIKOR stage, in turn, identifies a compromise solution by jointly considering group utility and individual regret [41]. This combination is particularly valuable in AMR selection, where cost, operational performance, implementation effort, and cybersecurity requirements may conflict with one another and where expert judgments are inherently uncertain.

The alternative ranking results also merit closer discussion. MiR250 emerged as the leading alternative because it achieved the lowest S, R, and Q values, indicating the most balanced compromise across the evaluated criteria. OMRON LD-250 followed as a competitive option, yet its higher utility and regret values suggest a weaker overall balance than the top-ranked alternative. A more nuanced interpretation arises in the comparison between HIKROBOT Forklift AGV and KUKA KMP 600-S diffDrive. Although KUKA exhibited a better group utility value than HIKROBOT, its higher regret value and the worst compromise index indicate that it performs less favorably under at least one critical criterion. This explains why HIKROBOT ranked above KUKA in the final VIKOR ordering. This interpretation is fully consistent with the compromise logic of VIKOR, where the preferred alternative is the one closest to the ideal compromise rather than the one with the best-isolated performance [41]. Overall, the ranking results suggest that AMR procurement decisions should prioritize balanced performance across economic, operational, integration, and security dimensions rather than isolated strengths in a limited number of criteria.

The findings also carry managerial implications. In the broader context of Industry 4.0, smart warehousing, and cyber-physical production systems, AMR selection increasingly shapes not only material-handling performance but also interoperability, digital resilience, and secure operational continuity. Accordingly, procurement decisions should not be made only by operations or industrial engineering teams. They should also involve IT/OT integration specialists, cybersecurity stakeholders, and maintenance personnel from the outset. Selection processes that overlook issues such as API/SDK quality, network segmentation compatibility, monitoring readiness, or update and vulnerability management may underestimate downstream integration effort, lifecycle cost, and cyber exposure [25,27]. Therefore, the present results suggest that AMR selection should be embedded within broader digital transformation and risk-governance practices.

Despite these contributions, several limitations should be acknowledged. The study is based on four AMR alternatives and expert judgments obtained from a limited panel, which may restrict the generalizability of the findings across different warehouse scales, industries, and operational conditions. In addition, although the unified linguistic structure enabled a coherent Pythagorean fuzzy analysis, treating all criteria as benefit-oriented simplifies the native structure of some variables, especially economic ones. Another limitation of the present study is that the proposed framework treats the evaluation criteria as independent. However, in real warehouse AMR selection problems, some criteria may influence one another. For instance, integration requirements may affect the total cost of ownership, cybersecurity readiness may be related to monitoring capability and system integrity, and infrastructure compatibility may shape commissioning efforts. Therefore, future studies may extend the proposed framework by incorporating interdependence analysis through DEMATEL, ANP, or correlation-based screening methods. Such extensions would help identify causal relationships, feedback effects, or redundant criteria and provide a more network-oriented decision model for AMR selection. Future research may therefore test the proposed framework on larger sets of AMR alternatives, apply it in different industrial domains, combine expert judgments with objective operational data, and compare its ranking stability with other advanced fuzzy and hybrid decision-making approaches. Further extensions may also explore adaptive, learning-supported, or digital twin-enabled decision structures for AMR selection in rapidly evolving intralogistics environments.

7. Conclusions

This study presented a cybersecurity- and integration-aware framework for AMR selection in warehouse environments by integrating PF-ITARA and PF-VIKOR within a Pythagorean fuzzy setting. By extending conventional evaluation structures with integration readiness and security-related requirements, the proposed model reflects the increasing complexity of AMR procurement in digitally connected warehouse systems.

The findings show that AMR selection is shaped not only by financial and operational considerations, but also by implementation effort, system compatibility, and cybersecurity performance. Within the evaluated decision setting, MiR250 emerged as the most suitable compromise solution. More broadly, the results indicate that effective AMR procurement should move beyond hardware-centered comparison and adopt a system-level perspective that captures both deployability and secure digital interoperability.

Overall, the study offers a transparent and practically applicable decision-support framework for uncertain and multidimensional AMR selection problems. Future research may strengthen this framework by testing larger alternative sets, applying it in different industrial contexts, and combining expert judgments with objective operational data.