To better capture uncertainty, hesitation, and contradictory information in real-world decision-making, Smarandache[1] introduced Neutrosophic Set (NS) theory. This framework addresses the limitations of classical fuzzy sets [2], intuitionistic fuzzy sets [3], and other hybrid models [4, 5] by permitting an independent assessment of truth, indeterminacy, and falsity. To enhance practicality, Wang et al. [6] introduced Single-Valued Neutrosophic Sets (SVNS), later extended to Interval-Valued Neutrosophic Sets (IVNS) [7], which represent membership degrees as intervals for greater flexibility in handling imprecise evaluations in Multi-Criteria Decision Analysis (MCDA). Subsequent research by Chinnadurai et al. [8, 9] and Jun et al. [10, 11] explored IVNS applications in algebraic structures and parameter-based rankings. Scholars such as Ridvan [12], Liu [13], and Garg [14] further developed aggregation operators and score functions (SFs) for MCDA within neutrosophic environments. Despite these advances, traditional IVNS frameworks, constrained by their tri-partitioned structure (truth, indeterminacy, falsity), remain limited in representing complex judgment scenarios involving multidimensional uncertainty and conflicting information.

In parallel, Prospect Decision Theory (PDT) [15] has emerged as a robust psychological lens for modeling decisions under uncertainty, evaluating alternatives based on perceived gains and losses rather than absolute outcomes. The integration of PDT with fuzzy and neutrosophic sets has shown significant promise. Foundational work by [16] combined PDT with interval-valued intuitionistic fuzzy sets for stochastic MCDA, while [17] extended it to general fuzzy environments under risk. The incorporation of PDT into neutrosophic frameworks was advanced by [18] and [19], who developed algorithms for neutrosophic soft sets and single-valued neutrosophic sets, respectively, applied to group decision-making and physician selection. Further complexity was addressed by [20], who integrated cumulative prospect theory with interval neutrosophic sets using a generalized Shapley function. However, a critical limitation persists across these hybrid models: their reliance on tripartitioned structures prevents the explicit representation of contradiction where an element is simultaneously afirmed and denied. This is a common occurrence in expert judgments involving conflicting evidence or paradoxical data. To overcome this gap, we introduce a quadri-partitioned framework through Interval-Valued Intuitionistic Quadri-Partitioned Neutrosophic Soft Sets (IVIQNSS). By incorporating contradiction as a fourth independent component alongside truth, indeterminacy, and falsity, IVIQNSS provides a more nuanced and powerful ontological foundation. This structure enables a more expressive representation of decision-makers’ hesitation and conflicting assessments, allowing PDT to evaluate prospects not just under uncertainty, but under genuine cognitive conflict.

The key contributions of this work are as follows: The development of a reliable SF for IVIQNSS that efectively handles interval-based truth, indeterminacy, contradiction, and falsity. A hybrid model integrating PDT with IVIQNSS, combining objective weights (derived from data) and subjective weights (from expert opinions) to balance psychological and rational decision components. A structured algorithm for solving MCDA problems in an IVIQNSS-PDT environment, validated through a real-world case study and comparative analysis.

The remainder of this manuscript is organized as follows: Section 2 introduces essential definitions. Section 3 proposes the new SF for IVIQNSS and addresses ranking limitations. Section 4 details the integration of PDT with IVIQNSS and outlines the decision-making algorithm. Section 5 presents a practical application. Section 6 compares the proposed model with established methods to validate its performance. Section 7 concludes the paper and suggests future research directions.

This section outlines the foundational concepts required for this study, including Neutrosophic Sets (NS), Single-Valued Neutrosophic Sets (SVNS), Interval-Valued Neutrosophic Sets (IVNS), and their corresponding SFs. Throughout this section, unless explicitly mentioned otherwise, denotes a universal set.

Definition 2.1 ([1]). Let X be a universal set. A neutrosophic set (NS) on X is expressed as

,

where , and denote the truth membership, indeterminacy membership, and falsity membership functions, respectively. These mappings are defined as . The sum of the three memberships satisfies the condition

.

Definition 2.2 ([6]). A single-valued neutrosophic set (SVNS) is a constrained form of NS, given by

,

where the functions , and map each element to values within the closed interval . These are referred to as the truth, indeterminacy,and falsity membership degrees, respectively. For all , the following constraintmust hold:

.

Definition 2.3 ([7]). An interval-valued neutrosophic set (IVNS) extends the SVNS by allowing the membership values to be intervals rather than crisp numbers. It is formulated as

,

and alternatively represented as

,

where , and are interval-valued functions mapping X into the set of closed intervals within . The cumulative upper bounds of these intervals must satisfy:

.

In this section, a new score function is proposed to efectively handle the characteristics of IVIQPNSS.

We introduce a Prospect Decision Theory (PDT)-based framework designed to address Multi-Criteria Decision Analysis (MCDA) problems within a neutrosophic environment. An accompanying algorithm and flowchart are also provided to demonstrate the operational steps of the model.

The concept of PDT, initially proposed by Daniel and Atmos [15], explores how individuals make choices under uncertainty by incorporating psychological biases in decision-making processes. They defined a subjective value function, denoted by , which captures the decision-maker’s (DM’s) perception of outcomes based on the nature of the gains or losses:

Here, and represent the risk sensitivity parameters for gains and losses, respectively, where and , signifying diminishing sensitivity. The coeficient 1 \end{document} ]]> emphasizes that losses are generally perceived as more impactful than equivalent gains.

To quantify the perception of probabilities, they introduced probability weighting functions (PWFs) for both gain and loss scenarios:

In these expressions, denotes the objective probability associated with a specific attribute, while and correspond to the degrees of optimism (for gains) and pessimism (for losses), respectively. The functions and adjust the perceived importance of these probabilities based on behavioral tendencies.

The prospect value function integrates both the subjective value and its corresponding weighted probability, as shown below:

To illustrate the practical implementation of the proposed decision-making framework, we consider a realistic case study in personnel selection. The selection of four candidates and four criteria is a deliberate methodological choice for this illustrative example.

The scenario, while hypothetical, is designed to reflect a common real-world decision-making environment: a company selecting the most suitable candidate for the role of project manager. Let the set of shortlisted candidates be denoted by

The evaluation is carried out based on four key criteria, represented by the set . These parameters were selected as they represent a comprehensive yet manageable set of core competencies essential for a project manager, ensuring a balanced assessment. The criteria are defined as follows:

The data for the IVIQPNSS evaluation matrix, which forms the basis of the subsequent analysis, was constructed to simulate the kind of imprecise, uncertain, and potentially conflicting assessments that a panel of experts might provide.

Subjective weights based on expert evaluations for each criterion are given by the vector . These weights reflect the relative importance of each parameter in the context of the selection process.

Additionally, each characteristic under consideration is associated with a probability value to capture the decision-maker’s confidence or familiarity with it. The probability vector is defined as

Step 1: The decision-makers (DMs) provide the performance evaluations of each candidate in the form of an IVIQPNSM. The matrix entries represent the degree of truth, indeterminacy, and falsity associated with each candidate’s performance under diferent characteristics.

For simplification, we categorize the characteristics into four levels of risk perception:

These qualitative descriptors help in evaluating candidates’ suitability across various uncertain and complex decision contexts. The IVNSM model allows for a comprehensive and flexible analysis by integrating both subjective assessments and probabilistic reasoning.

Step 2: Determine the proposed matrices by applying Definition 3.1.

The computed score matrices , and are associated with the respective risk levels: low, moderate, elevated, and critical.

The parameter values used in this analysis are set as , , and , based on the benchmark settings suggested in the studies by Abdellaoui [21] and Gonzalez & Wu [22].

Step 3: Evaluate the positive and negative prospect scores by applying Equation (4.5).

Step 4: The normalized weighted values (NWVs) are computed using Equations 4.8 and 4.9 resulting in the vector .

Step 5: Compute the aggregated prospect scores using Equation 4.10, resulting in , and .

By comparing these merged values, the preference order is established as \mathcal { N } _ { 1 } > \mathcal { N } _ { 4 } > \mathcal { N } _ { 3 } \end{document} ]]>

Hence, based on the prospect evaluation, it can be inferred that the project manager is the most suitable candidate for managing the large-scale project.

As demonstrated, the Interval-Valued Intuitionistic Quadri-Partitioned Neutrosophic Soft Set (IVIQPNSS) provides a powerful framework for addressing Multi-Criteria Decision Analysis (MCDA) challenges characterized by uncertainty, contradiction, and vagueness. This paper has enhanced this foundation by integrating Prospect Decision Theory (PDT), thereby capturing the psychological underpinnings of decision-makers through a reference-dependent model within the IVIQP-NSS structure. The efectiveness of this hybrid IVIQPNSS-PDT framework was validated through a hypothetical case study, designed to illustrate its applicability in realistic scenarios with a manageable number of alternatives and criteria. The core contributions of this work are threefold: The introduction of a reliable and mathematically sound score function for IVIQPNSS, which efectively aggregates the four membership dimensions (truth, indeterminacy, contradiction, falsity) into a single, actionable value for ranking. A methodology that successfully balances subjective expert opinion with objective data-driven weights, ofering a flexible and robust solution for diverse decision-making contexts. A comprehensive and novel strategy for solving MCDA problems that seamlessly combines the proposed score function, parameter weighting, and the psychological insights of PDT, filling a significant gap in the existing literature. Despite its contributions, this study is not without limitations. The presented case study, while illustrative, is hypothetical; applying the framework to a large-scale, empirical problem is a necessary next step to further validate its practical utility. Furthermore, the computational complexity may increase with a larger number of criteria and alternatives. Future research will focus on addressing these limitations. Promising directions include the application of the model to large-scale, real-world empirical studies, the development of specialized software for implementation, and the integration of sophisticated aggregation operators as discussed in [23], [24], and [25]. Additionally, exploring similarity and projection-based techniques within the IVIQPNSS setting could further enhance the model’s comparative capabilities.