Decision theoretic fuzzy modelling for product quality and design evaluation of basketballs using deck of cards strategy

Abstract

Quality and design assessment of basketball is a complex decision-making (DM) problem, particularly when more than one consideration of the relationships between performance and consumer satisfaction has to be met. The conventional evaluation models are also not sufficient in many ways as they require accurate numerical input and cannot cope with the subjective nature of human considerations. In this paper, a hybrid fuzzy multi-criteria decision-making (MCDM) model, incorporating intuitionistic fuzzy sets (IFS) and the deck of cards method (DoCM), is described. This new technique complements the traditional MCDM techniques in that it enables specialists express their uncertainty with membership, non-membership as well as hedging, and to represent their preferences graphically with card-based gaps. The alternatives ranking is created in two phases: by obtaining normalized weights through preference gaps by using DoCM; by combining the normalized weights with intuitionistic fuzzy values of the alternatives on multiple criteria. The outcome is an interpretable prioritization of alternatives. The IF-DoCM aggregates fuzzy linguistic appraisals with ordinal preferences and degrees of hesitation to determine normalized utility scores and to rank alternatives in decreasing order of suitability. A case study is used to compare five models of basketball based on six primary criteria. A case study is presented that compares five basketball models to each other using six primary criteria: the quality of the material, grip and control, consistency of bounce, durability, design appearance, and selling price. Experts are consulted, and ratings are assigned to each alternative. A fuzzy DoCM approach is then used to provide the final ranking. The findings demonstrate how this mixed decision-making (DM) model illuminates’ uncertainty and facilitates more informed product development and selection decisions within the sports equipment industry. The same methodology can be applied to other durable consumer goods, in which the assessment of quality and design is based on subjective judgments.

Introduction

When designing and evaluating products, it is common to balance a variety of conflicting (performance, cost, durability, and sustainability) criteria. These are multi-criteria problems in nature and must be approached in a structured way in order to rank and prioritize alternatives using a variety of stakeholder inputs. It becomes even more complicated when subjective evaluations are involved, such as in the case of basketball design when users and experts have to evaluate such qualitative aspects as feel of the grip, air retention, and consistency of the bounce MCDM is a method to systematically analyze and compare alternatives to address such multi-dimensional issues. Nevertheless, the classical methods of MCDM presuppose a set of accurate numerical inputs that cannot be accurately represented by the linguistic and uncertain judgments commonly provided by experts. Thus, the integration of fuzzy logic in the MCDM allows the representation of human reasoning more accurately as it reflects the ambiguity, vagueness and hesitation in the expert opinions. Fuzzy MCDM is therefore necessary to improve the quality of decisions, especially in subjective product evaluation such as sports equipment design. DM in evaluating basketball quality and design is highly multidimensional, encompassing both objective metrics, such as bounce consistency, and subjective preferences, including how a basketball feels in the hand and its visual appeal. Conventional assessment scales, which are based on crisp numerical scales, find it challenging to denote the vagueness as well as the uncertainty attached to words and/or judgments performed by experts. To counter that, the fuzzy MCDM systems have taken center stage, a reaction that is successful in modeling ambiguity and integrating qualitative evaluations. As an example, the fuzzy sets FS that belong to the interval type 2 incorporated into the hybrid approaches to MCDM have been shown to treat better data uncertainty and language expressions¹ Likewise, when it comes to similar fields of study, the recent predictions of sporting events outcomes based on fuzzy logic have been made using the combination of CRITIC and VIKOR methods that have recorded impressive improvements in decision reliability in the situation of uncertainty² One of the most convincing decision-maker preference elicitation tools is the DoCM, which embodies a human-oriented approach. DoCM enables the intuitive, visual creation of fuzzy membership functions through an interactive process between analysts and decision-makers, thereby strengthening interpretability and engagement with decision modeling³.Based on such developments, this work proposes a combination of the DoCM with a hybrid decision-theoretic fuzzy model to estimate the quality and design of basketball. This framework enhances the participatory and transparent nature of multi-criteria evaluation by allowing stakeholders to articulate their preferences in a more nuanced manner, using visual tools that are easy to understand⁴.We have used 5 models of basketball, including professional indoor balls, eco-friendly, and compared them on the basis of 6 key factors that comprise of quality of material and grip/control, bounce consistency, sturdy, aesthetically sound and price. By using this practical example, the prescribed fuzzy-DoCM strategy appears to be an open, flexible, and efficient decision support mechanism that can be specifically aimed at manufacturers, designers, and consumers of sports equipment assessments. Real-life product development and quality testing, in particular with sports equipment such as basketballs, requires DM to look at and be aware of numerous variables at once, including cost, durability, grip, weight consistency, and design preference. In this room, traditional DM can be informal, formulaic, or fragmented, relying on expertise or random quality testing that lacks systematic combination. It can lead to poor design, incorrect product specifications, and ineffective market targeting. This paper fills such a gap by proposing a fuzzy MCDM-based framework, which enables transparent, traceable, and structured assessment, allowing manufacturers and procurement officers to make more informed and data-driven decisions regarding product design and selection. This paper is concerned with the assessment and design of basketballs aimed at the recreational amateur market, i.e., school teams, sport clubs, and general consumers looking to buy fairly priced but high-quality basketballs. These products do not have to be of high quality (e.g., grip, bounce consistency, durability), but they must meet low standards compared to the high-performance requirements of professional-level basketballs.

Problem statement

The modern sports equipment industry is highly competitive, and producers are compelled to develop and release a range of high-quality basketballs that meet the needs of coaches, players, and buyers. Basketball evaluation of quality and design is a complicated DM as it requires several criteria, such as material, grip, bounce, durability, aesthetics, and price. These requirements are sometimes subjective and open-ended; thus, it is challenging to make accurate decisions based on conventional evaluation methods. The views of customers and experts may not always be exact, and their preferences may not always be easily quantified by assigning numerical values. One way to overcome this problem is to employ a decision-theoretic fuzzy modeling approach. The procedure takes into account the uncertainty and imprecision in human ideation. In particular, the visual, easy-to-use (DoCM) enables decision-makers to compare and rank alternatives using fuzzy logic, thereby making the final decision more accurate and trustworthy. This article introduces a case study that utilizes this method in assessing five varieties of basketball with regard to six most critical criteria in the DoC Strategy in the context of a fuzzy DM framework.

Objectives and contributions

In this section, the objectives of this study are presented insightfully, clearly stating the aim of this article. The study has the following objectives:

This research paper aims to achieve two main goals through a thorough decision-theoretic fuzzy model in assessing the quality and design of basketballs based on several criteria considered, which affect both user satisfaction and product performance. The objective of the study is to utilize the DoCM within a fuzzy logic-based framework to effectively address the uncertainty and vagueness inherent in human judgments during the DM process. The study aims to provide a methodological framework by pointing out and listing six essential criteria for evaluation: material quality, grip and control, bounce consistency, durability, aesthetic design, and price. These criteria will help the study to compare and rank the different basketball models. Additionally, the paper presents a case study analysis of five real basketball options, illustrating the applicability and utility of the model proposed in the study. The ultimate goal is to help manufacturers, designers, and consumers make more informed and balanced choices when selecting or designing basketball products.

These are some Contributions of this study:

The study also makes several significant contributions to the field of product evaluation and subsequent DM. First, it introduces a new hybrid methodology that combines fuzzy logic and the DoCM, allowing decision-makers to discuss their preferences using linguistic terms and rank alternatives more intuitively and, in a human,-perceptual manner. Two, it offers a systematic and adaptable method for easily assessing products whose characteristics cannot be expressed with a single set of criteria, especially in situations where subjective contributions are involved. Third, the paper presents a rich case study that not only proves possible the suggested study methodology but also can be turned into practical information that can be used by basketball manufacturing companies that can enhance their products. Moreover, this model is flexible and it can be applied further to assess other categories of consumer goods or services. This research design makes a significant contribution to addressing the drawbacks of classic evaluation methods, offering a practical, scalable, and efficient decision support mechanism in the sports equipment industry and other types of businesses.

Research questions and motivation

This study has the following research questions:

• What is the advantage of the application of fuzzy logic in the assessment of the quality of basketball and its design when facing uncertainty?

• How can the DoCM be used to aid in methods of ranking preferences intuitively in multi-criteria choices?

• Which are the major standards of evaluating basketball performance and satisfaction as a user?

• What is the relation between different models of basketball depending on the proposed fuzzy-DoCM framework?

• Would this hybrid strategy work on other consumer products to be evaluated better?

The study has the following motivations:

This study is motivated by the growing complexity of evaluating products in current consumer-oriented markets, particularly in the sports equipment industry. Basketballs are designed for use by a wide range of users, including casual players and professionals, and are required to meet varying performance expectations. However, conventional assessment procedures tend to be biased towards definite number scales and do not account for the hesitation and subjectivity inherent in human evaluation. Both consumers and experts are vague in their formulations of preferences, providing qualitative statements such as ‘very good’ or ‘satisfactory,’ which makes it challenging to fit these into conventional models. The background of this research is the notion that a more humanized and adaptable framework for DM should be developed to address this gap. Through the integration of fuzzy logic and the intuitive (DoCM), the paper presents a structured yet feasible approach to evaluating basketball products on a multi-criteria basis. It not only enhances the reliability of the evaluation procedure but also facilitates the coordination of stakeholder involvement in product development and selection. Moreover, the proposed approach can be generalized to assess other types of consumer products under conditions where multi-criteria and subjective factors are more significant.

Literature review

In this section we will discuss the literature review of this study.

MCDM application in sports equipment selection

The use of MCDM in the evaluation of sports products is relatively scarce; however, several interesting studies have been conducted on the applicability of this method to the selection of sport equipment, optimization of training, and performance improvement. The introduction of MCDM in these areas enables a systematic and clear assessment of numerous qualitative and quantitative variables, which are critical in making decisions when uncertainty exists. A study like that is carried out by Alkan and Kahraman⁵ who compared various designs of tennis rackets based on ergonomic parameters, materials, and performance through fuzzy AHP and fuzzy TOPSIS analysis. Their method emphasized the role of subjective preferences that could be modelled successfully in the process of selecting sports products. In the same way, Ravi et al.⁶ have applied hybrid AHP-TOPSIS to identify the ideal material of a cricket bat by integrating the opinions of the players and the expert. Baki et al.⁷ applied fuzzy VIKOR to rank cleats based on ground grip, price and durability in the context of football (soccer), and considered fuzzy models better at improving decision robustness. Although no previous research seems to use MCDM to address the problem of product selection in the basketball industry directly, the approach has been effectively implemented in related areas, such as the selection of athletic footwear⁸ and protective apparel in contact sports Chatterjee et al.⁹ Such studies confirm the flexibility of MCDM approaches in evaluating sports-related products, particularly when decision criteria are subjective, conflicting, and context-dependent. Based on this base, the current study uses a new developed fuzzy MCDM model (IF-DoCM) to rank basketballs, a field where no literature has so far touched.

MCDM fuzzy extensions

The theoretical concepts related to this research can be traced back to Zadeh, who pioneered the theory of FS, defining graded membership and a possible systematic method of representing linguistic uncertainty within DM¹⁰. With this background, the past few years have been marked by a rapid increase in hybrid fuzzy MCDM methods that enhance robustness, explainability, and stakeholder participation such as, Y. Fu et al.¹¹ proposed an integrated fuzzy AHP TOPSIS approach to the selection of industrial robots, where, in convergence with the requirements of assessing qualitative matters better, they used triangular fuzzy numbers, and Liu¹² Integrated CRITIC and VIKOR in their fuzzy variant that proved to effectively address the conflicts in the prediction of sports outcomes under uncertainty. Alharbi et al.¹³ demonstrated the beneficial effect of fuzzy AHP TOPSIS within the healthcare industry, enhancing the reliability of decisions made by various stakeholders through subjective ratings. In a similar case, Ng et al.¹⁴ implemented fuzzy logic with COPRAS and life cycle assessment for eco-product design, demonstrating the capacity of the methodology to handle the failure of such a complex task verbally by using fuzzy verbal information representation. In parallel with developments in fuzzy MCDM, human-driven approaches to preference assessment have emerged, such as the DoCM. García Zamora et al.⁴ employed DoCM to enable decision-makers to visually and interactively describe the strength of their preferences, resulting in interpretable fuzzy membership functions. Furthermore, DoCM has been generalized to an interval type-2 fuzzy set by Dutta et al.², thereby allowing for more constructive expression of hesitation and ambiguity in analyst-decision-maker interactions. Nevertheless, a gap still exists in the integration of decision-theoretic portrayals of fuzzy modelling and DoCM, founded on preference elucidation, in connection with an assessment of consumer sports gear. This paper answers this gap by incorporating a fuzzy MCDM model with DoCM to compare basketball quality and design on six distinct criteria: material quality, grip and control, bounces, durability, aesthetic design, and price, using five different models of basketball as an example of a unique, easy to interpret, and stakeholder-driven DM model. More recent research has continued to focus on how fuzzy extensions of MCDM methods can be utilized to model the ambiguity and uncertainty in DM among experts across various application areas. Specifically, the Fermatean fuzzy sets and neutrosophic environments have contributed to the modelling of group DM situations in correlation and hesitation. In a highly ambiguous situation, as an example, Mandal and Roy introduced a new correlation-based indicator on Fermatean fuzzy sets to select electric vehicles and increase sensitivity to preference modelling¹⁵ Similarly, Dey and others proposed a weighted correlation methodology using Neutrosophic Fuzziness by TOPSIS to select insurance policies and in this context, were successful in improving the evaluation process with uncertain and conflicting data¹⁶. In addition, Patra and Kar applied a two-tuple Pythagorean fuzzy MOORA dynamic group DM model in the selection of an ERP system, which allows more flexibility to changing criteria preferences¹⁷. Correlation-based fuzzy TOPSIS techniques have also been combined with other studies to solve multi-criteria trade-offs. For instance, Dey et al. utilized naive correlation measures based on neutrosophic sets in their study on health insurance provider selection¹⁸ and Mitra et al. employed a probabilistic neutrosophic TOPSIS method in their renewable energy alternative selection¹⁹. All these studies demonstrate that extended fuzzy and neutrosophic frameworks can enhance the capacity of MCDM to address subjectivity, uncertainty, and imprecision in complex DM situations. Against this background, our proposed IF-DoCM aligns with these developments in incorporating more expressive fuzzy paradigms into MCDM, yet it is unique in its visual elicitation of preferences and its provision of interpretable decision support, which is less emphasized in correlation-based models. This paper initially synthesizes widely recognized MCDM techniques in order to adopt the most appropriate approach to the complex task of evaluating the quality and design of the basketball. We have established and categorized the methods based on their capability to deal with uncertainty, ease of use, involvement of stakeholders and flexibility to product design environment. This background review justifies the stringent screening of the IF-DoCM as the ultimate method. The Literature review of existing MCDM approaches is given in Table 1.

Table 1 Literature review of existing MCDM approaches.

Structure of study

This paper is organized as follows: Sect. 1 defines the introduction of quality and design Evaluation of basketball identifies the Literature review and problem statement. Section 2: Describes the literature review that lay the proposed section’s foundation to find the gap and then some basic definition is discussed in Sect. 3. Section 4: Fill the gap and present the model with the IFS framework utilized in MCDM methodology. Moreover, this section presents the numerical evaluation of the proposed model to Exploring Music Tourism Innovation. Section 5: Concludes the discussion by highlighting the advantages, impact on society, limitations, and its future directions.

Preliminaries

In this section, we will study the basic concepts of intuitionistic fuzzy sets (IFS), and its score function.

Definition 1 ²⁷

An intuitionistic fuzzy set (IFS) $\:B$ within a universe$\:\:X\:$is an object shaped as

$$\:\begin{array}{c}B=\left\{X,{\phi\:}_{x},{v}_{x}|x\in\:X\right\}\end{array}$$

(1)

.

The degrees of membership of $\:x$ in $\:B$ and non-membership of $\:x$ in $\:B$ are denoted by $\:{\phi\:}_{x}\in\:\:\left[\text{0,1}\right]$ and $\:{v}_{x}\in\:\:\left[\text{0,1}\right]$, respectively. And fulfills the subsequent criteria.

.

$$\:\begin{array}{c}0\le\:{\phi\:}_{x}+{v}_{x}\le\:1\end{array}$$

(2)

Definition 2 ²⁸

For intuitionistic fuzzy sets (IFS), the score function is defined as:

$$\:\begin{array}{c}\partial\:\left(x\right)={\phi\:}_{x}-{v}_{x}+\sigma\:\pi\:\end{array}$$

(3)

Where $\:\sigma\:$ is the parameter that $\:\in\:\left[\text{0,1}\right]$.

Intuitionistic fuzzy deck of cards method approach

In this paper, an MCDM problem with IFS uncertainty is addressed using the DoCM². To address these MCDM issues²⁹ A proficient DM technique is introduced. The main goal of any MCDM problem is to prioritize the alternatives based on the established criteria. This section outlines the creation of the DoCM coupled with IFS for assessing the design and quality in basketball. Let us define $H = \left\{ {H_{1} ,~H_{2} ,~...,~H_{{\text{m}}} } \right\}$ as a collection of $\:m$ Alternatives (deep knowledge models) and $Q~ = ~\left\{ {Q_{1} ,~Q_{2} ,~...,~Q_{{\text{n}}} } \right\}$ as a set of $\:n$ criteria, with ${\mathcal{X}}~ = ~\left\{ {{\mathcal{X}}_{1} ,~{\mathcal{X}}_{2} ,~...,~{\mathcal{X} }_{{\text{n}}} } \right\}$representing the corresponding weights. We will assess our example problem utilizing the IF DoCM with optional weights for each criterion. In this assessment, the possibilities are IF values $\:\mu\:,v\:\:\epsilon\left[\text{0,1}\right]$. In the context of an IF framework, DoCM serves as a straightforward and cognitively compatible approach for MCDM. Recombination is more beneficial when a fuzzy setting is taken into account. In contrast to conventional pairwise comparison techniques, DoCM allows the decision-maker to illustrate preferences visually and organize a clear hierarchy of winners and losers, therefore substantially reducing cognitive burden. In conjunction with fuzzy logic, particularly with Intuitionistic Fuzzy Sets (IFS), it effectively represents the ambiguity, indecision, and imprecise judgment inherent in human beings. It may also pertain to circular reasoning, offering enhanced flexibility for comparison in scenarios where options are not strictly linear and hierarchical. Its lucidity, directness, and adaptability to expert-oriented contexts render it an ideal instrument in complex decision-making scenarios. The decision to use the Deck of Cards Method (DoCM) as an alternative to other fuzzy MCDM techniques, including fuzzy TOPSIS, VIKOR, or ELECTRE, is not random and is based on several methodological benefits. First, DoCM allows eliciting preferences of experts directly and intuitively by ranking the relative significance of criteria using the visual method of working with cards, which help to reduce cognitive load and increase consistency. This differs from other techniques, such as fuzzy AHP or fuzzy TOPSIS, which may require pairwise comparisons or distance measures that are difficult for non-expert users to interpret. Second, DoCM enables the incorporation of judgmental gaps between criteria that enable experts to articulate not only ordinal preferences, but also intensity differences. This granular input is beneficial where fuzzy or IF environments are involved, since it allows for easily modelling the hesitation, imprecision, and subjective perception. Lastly, DoCM is described as flexible, low-burden and appropriate to support group DM, which is why it is quite compatible with product evaluation tasks that require multiple stakeholders or unknown linguistic inputs. These characteristics make DoCM especially beneficial in situations, where interpretability and transparency are crucial and the involvement of experts is required.

Algorithm

We will address our example problem utilizing this hybrid approach. This approach more effectively resolves the problem’s ambiguity and uncertainty. These measures will be taken in the DoCM.

• 1. Decision-makers rank the alternatives (or criteria) in order of least to most preferred. Let $H = \left\{ {H_{1} ,~H_{2} ,~...,~H_{{\text{m}}} } \right\}$ represent the choices and $Q~ = ~\left\{ {Q_{1} ,~Q_{2} ,~...,~Q_{{\text{n}}} } \right\}$ denote the criteria. The initial phase in the procedure is determining the hierarchical preference of alternatives or criteria. This ranking is essential, serving as the foundation for all following comparisons and evaluations.

• 2. By indicating how many blank cards (gaps) exist between each pair of items, a comparison table is produced. Upon ranking, we provide a systematic comparison table according to the preferences. This table illustrates the quantity of absent choices among options, signifying the comparative potency of enhancements visually.

• 3. Verify the table’s consistency, ensuring that the values adhere to the stipulated conditions for a coherent presentation. e_ij and e_jk Do the values on the blank cards, which we shall verify for consistency, total to the value of the subsequent card in the table? The table’s consistency will be examined using the equation below. This is an attempt to preserve the evaluation’s integrity and quality.

(4)

For all $\:\:i\:<\:j\:<\:k\:$should discrepancies be identified, engage in dialogue with the decision-maker for revision. And adjust the value till it satisfies the requirement. Nonetheless, one aspect must be considered. $\:i\:<\:j\:<\:\mathcal{k}$.

• 4. After the table is made consistent, we then award numerical weights to the alternatives/criteria. These weights are the measure of significance of each choice against the other choices, where ${{\mathcal{X}}_{1} }$ must start with 1.

(5)

Subsequently, normalize the weights utilizing the following formula.

$${\mathcal{X}}i = ~\frac{{{\mathcal{X}}i}}{{\sum {\mathcal{X}}_{j} }}$$

(6)

• 5. Next, Intuitionistic Fuzzy (IF) numbers are used to construct the decision matrix. These numbers describe the degree of ambiguity in expert judgments by incorporating membership, non-membership, and hesitation.

$$h = \sqrt {1 - \varphi _{{ij}} - ~\nu _{{ij}} }$$

(7)

Here, $\:h$ signifies the hesitation value derived from the IF number, where $\:\phi$_ij indicates the membership degree and $\:v$_ij denotes the non-membership degree.

$$\tilde{P}_{{ij}} ~ = ~\left( {\varphi _{{ij}} ,~\nu _{{ij}} } \right)$$

(8)

such that $\:\phi\:\:+\:\nu\:\:\le\:\:1.$Subsequently, calculate the IF score:

$$r~_{{ij}} = \left( {\varphi _{{ij}} ,~\nu _{{ij}} + \sigma \pi } \right)$$

(9)

• 6. For all the criteria$\:C$_j, create matrix $\:M$^(j) accordingly and develop an ordinal comparison matrix for each of the criteria. Matrices for pairwise comparison are constructed for each criterion to evaluate the performance of one alternative in relation to others on that particular criterion.

$$M^{{(j)}} \left( {i,~\mathcalligra{k}\:\:} \right) = \left\{ {\begin{array}{*{20}c} {1\;~if~r_{{{\text{ij}}}} ~r_{\mathcalligra{k}} } \\ {0~\;if~r_{{{\text{ij}}}} = ~r_{{\mathcalligra{k}j}} } \\ { - 1\;~if~r_{{{\text{ij}}}} ~r_{\mathcalligra{k}} } \\ \end{array} } \right\}$$

(10)

• 7. Calculate Local Scores by aggregating every value from the pair-wise comparison matrix for each criterion. Consolidate every single row with the corresponding criterion rows and compute the local score.

(11)

for $\:\mathcal{k}\:=\:1$ to $\:m$ for Unweighted: $\partial _{i} ~ = ~\Sigma ~\partial _{{ij}}$ over all $\:j$ and for Weighted: $\partial _{{i~}} = ~\Sigma {\mathcal{X}}_{j} ~ \times ~\partial _{{ij}}$

• 8. Rank options according to decreasing S_i values. Derived from computed scores.

• 9. This is a visual depiction of ranks.

Figure 1 illustrates the overall workflow of the proposed decision-theoretic fuzzy modelling framework combined with the DoCM. It demonstrates how to define alternatives and criteria, elicit expert judgment, translate it into fuzzy and gap-based inputs, generate the consistency check, calculate the weights and lastly puts the results together to give a ranked list of basketball designs. The visual roadmap helps readers understand the rational flow of processes and the interrelationships among the methodological processes depicted in the paper.

Fig. 1

Algorithm steps of IF-DoCM.

Expert survey design

The questionnaire was made to gather expert opinion on six attributes associated with quality of basketball and its design. The participants were provided with a brief containing the target use-case (amateur level), specific definitions of each of the criteria, and a comparison framework of the five basketball options. Purposive sampling was used to select experts (sports equipment retailers, coaches, and semi-professional players) with knowledge of mid-range basketballs. There were [N] valid responses that were obtained.

Case study

To illustrate how the proposed decision-theoretic fuzzy modelling that integrates the DoCM can be used, a case study was conducted to assess the quality and design of basketballs. This research compares five basketball variants with one another by evaluating and rating them according to six primary criteria that encompass both technical and user-oriented design features. The assessment will guide manufacturers, designers, and consumers in making informed choices by incorporating both objective criteria (e.g., quality of materials, shelf life) and subjective criteria (e.g., comfort, overall appearance) within a fuzzy multiple-criterion model. In this case study, we will consider 5 alternatives $H = \left\{ {H_{1} ,~H_{2} ,~...,~H_{{\text{m}}} } \right\}$ and 6 criteria $Q~ = ~\left\{ {Q_{1} ,~Q_{2} ,~...,~Q_{{\text{n}}} } \right\}$ It is essential to mention that this case study is hypothetical and was designed to recreate the real-life DM situation and show the implementation of the methodology step-by-step. We concentrate on 3 significant factors of impact, which generally affect at the product-level DM:

• Price-performance trade-offs (e.g. cost vs. durability).

• Varying design preferences amongst stakeholders.

• How to address subjectivity and expert judgment consistency.

• The suggested IF-DoCM approach directly deals with all three by facilitating multi-criteria aggregation in the presence of uncertainty, eliciting preferences based on visual gaps, and producing traceable rankings that can be used to inform feasible product development decisions.

The descriptions of Alternatives and criteria are given in Tables 2 and 3.

Table 2 Description of alternatives.

Table 3 Description of criteria.

Numerical evaluation

All the evaluators were informed strongly that they should view the alternatives as mid-range level basketballs that are used in normal training or in recreational play. This was done to standardize the context to eliminate differences in the perceived consumption or expectations of consumers.

Step 1. Decision-makers prioritize alternatives (or criteria) from most favored to least favored. Let $H = \left\{ {H_{1} ,~H_{2} ,~...,~H_{{\text{m}}} } \right\}$represent the set of possibilities and $Q~ = ~\left\{ {Q_{1} ,~Q_{2} ,~...,~Q_{{\text{n}}} } \right\}$ denote the set of criteria. The ranking of alternatives on above criteria are shown in Table 4 reports the initial preference order of alternatives obtained from the DoCM. Such an ordering includes qualitative judgments of experts without any prior numerical processing, which can in turn be used to form the basis of generating comparison gaps and weights.

Table 4 Rank alternatives.

Step 2. You can select how many blank cards (gaps) there are among each pair of components to construct a comparison table. The matrix encompasses every pairwise comparison and may contain interval or unknown values as illustrated in the Table 5. Greater numbers imply preference distances are at higher rates. Such gaps are the raw inputs that will be verified as being consistent and changed into weights.

Table 5 Comparison table of blank cards.

Step 3. Table 6 verifies the internal consistency of the blank-card gaps (additivity rule). When the additions of two disparities must generate the implied third, contradictions are emphasized and are eliminated by a couple of moments of contact with decision-makers to maintain the display of logical thought.

Table 6 Consistency table.

For all $\:\:i\:<\:j\:<\:k.$.

If inconsistencies are found, revise through interaction with the decision-maker.

Step 4 Table 7 converts consistent gaps into raw weights for each alternative (or criterion). The formula used and the numbers in between are highlighted in the table, and a baseline of the most preferred item is used as 1. These crude weights are used in the calculation of relative importance before normalization.

Table 7 Calculate the weight.

$$\:\:k\:=\:1\:to\:i-1.$$

Table 8 presents the normalized weights that sum to 1. These weights are used later to aggregate performance across criteria, ensuring comparability and a proper trade-off among attributes.

Table 8 Normalized weights.

Step 5. Table 9 lists, for each alternative–criterion pair, the Intuitionistic Fuzzy values: membership (m), non-membership (nm), and hesitation (h), along with the computed score. This comprehensive matrix carries uncertainty and ambiguity of opinion of experts, and is quantitative basis for ranking.

Table 9 Decision matrix of IFS.

Step 6. For each criterion $\:C$_j, construct matrix $\:M$^(j) as given in Table 10. It provides pairwise comparisons of alternatives for each criterion (entries typically in {1, 0, − 1} = preferred, tie, not selected). Through these matrices, fuzzy assessments are converted to proportional win/loss per criterion thus allowing one to score per criterion locally.

Table 10 Pairwise comparison matrix.

Step 7. Now, compute the Local Scores as given in Table 11 below. it compiles each of the pairwise matrices into local scores–one performance value of alternatives on each criterion. These could be utilized as is or corresponding with the normalized weakness of Table 8 to apply decision-makers’ priorities.

Table 11 Compute local scores.

for $\:k\:=\:1$ to $\:m$ for Unweighted: $S_{i} ~ = ~\Sigma ~S_{{ij}}$ over all $\:j$ and for Weighted: $S_{{i~}} = ~\Sigma W_{j} ~ \times ~S_{{ij}}$

Step 8. Rank alternatives based on descending $S$_i Values as given in Table 12. It shows the overall scores (weighted aggregation across criteria) and the resulting final ranking. The table addressed the fundamental decision question by combining the intensity of preferences, fuzzy judgment, and criterion importance on the same list, ordered according to preferences. Although the result of brand ranking indicates a computational output of the fuzzy MCDM process, its interpretative worth is in supporting the DM by product designers, manufacturers, and procurement managers. The ranking is not just descriptive; it provides a systematic framework for assessing the performance of current basketball models against the quality and design criteria as determined by experts. For example, once the brand is rated low and its grip or durability is not performing well, the manufacturer can identify areas where it can be improved. Likewise, for procurement officers (e.g., schools or sports academies), the ranking provides a clear comparison tool that can be used to justify purchase decisions based on a variety of weighted criteria. The ranking is hence both a diagnostic and prescriptive tool rather than a preference listing. The target market is mid-range, and the final consumers are the decision-makers in the design, quality control or sports equipment.

Table 12 Rank alternatives $S$_i values.

Step 9. This is a pictorial representation of the ranks, as given in Fig. 2. It depicts the (DoCM) in action, showing how decision-makers visually arrange alternatives and insert “blank cards” to indicate the strength of preference between them. This visual and interactive approach, in turn, facilitates the expression of subjective judgments, the intensity of preferences, and prepares the information to check consistency and perform weight calculations in subsequent stages.

Fig. 2

Ranking of alternatives.

Result discussion

The decision-theoretic fuzzy modelling framework, proposed here in, coupled with the (DoCM), was utilized to compare and rank five possible ideas in the game of basketball with six prioritized criteria. The fuzzy-DoCM methodology enabled the conversion of qualitative ratings into fuzzy values, which could be easily interpreted. In contrast, the DoCM provided a relatively straightforward platform for ranking criteria and alternatives relative to each other. The findings reveal that Aqua Air Pro ($\:{A}_{4}$) topped the list with a higher score of $\:6.40$. The high performance of this model is likely due to its balanced design, which combines all-weather performance, consistent bounce, and durable composite material. It also seems to have fulfilled or surpassed expectations under both technical and user-oriented standards and therefore remains the most favoured in this case study. Classic Leather Elite ($\:{A}_{3}$) took second place, $\:0.40$. Although it has a high price and is exclusively suitable for playing indoors, it ranked highly in material quality and grip/control, which will attract a high-performance focus to the players. Here is the interesting part: Eco Play Max ($\:{A}_{5}$), which emphasized using good, yet affordable materials that consumers can use and share, was ranked third, despite having a relatively low score $\:(-5.13).$ This implies that even though it might not score high in terms of cross-product performance, its sustainability and cost units contribute something to like-minded consumers. ProGrip X7 $\:\left({A}_{1}\right)$ was placed number four with moderate results on most of the scores and did not perform exceptionally in any one of the features, as compared to the optimally ranked models. Lastly, Street Flex 360 ($\:{A}_{2}$Although it is very durable for outdoor needs, it came fifth, which implies that it performed lower than other models in terms of grip, bounce consistency, and design appeal. Taken together, these results demonstrate the power of the fuzzy-DoCM framework in uncovering not only the best performer but also the trade-offs of alternatives in a less discrete manner. Although Aqua Air Pro is seen as the best all-round product in this case study, the findings also reflect that buyer preferences may alter the ranking of the products, given differing weightings of various factors, such as sustainability, quality of indoor play, or affordability. Such flexibility reinforces the usefulness of the suggested method to various DM situations. This paper designs and implements a fuzzy MCDM model using IFS and the DoCM to assess basketball product alternatives based on several criteria. As shown in the case study, this approach has an interpretive strength of determining design strengths and weaknesses across competing products.

The findings reveal that some brands consistently perform poorly in key factors such as grip and durability, elements that matter most to the target audience of amateur and recreational players. This provides tangible suggestions to designers and manufacturers, as the methodical and analytical examination will identify specific elements to be optimized, which are founded on the demands of the final consumer. In terms of methodology, the preference elicitation process based on DoCM was quite intuitive and flexible, as experts could present not only the rankings but also the strength of their preferences. Together with IFS, the model reflected hesitations and subjective uncertainty, which are usually involved in the process of product design assessment. There are, however, limitations to this study. First, the case study was hypothetical and not empirically tested on the product, which restricts generalization. Second, the sample size and diversity impact the model’s outputs, even when expert opinions are used. Third, real consumers are not necessarily accustomed to making product assessments using structured MCDM methods; thus, the model is primarily oriented toward design and procurement professionals, rather than final consumers. This model can be used in future research as a real-life product development case, and more experts can be incorporated in the study, and actual performance testing results to be incorporated to enhance and improve the DM process.

Comparative analysis

To support the decision to use the suggested IF-DoCM approach, we conducted a qualitative comparative study of the given method and other, more commonly used MCDM methods, such as AHP, TOPSIS, DEMATEL, and generic fuzzy MCDM models. The comparison highlights essential methodological aspects that apply to the subjective, qualitative DM conditions experienced within a product design evaluation. As shown in the analysis (Table 13), IF-DoCM is the only model that facilitates the modelling of hesitancy and interactive preference articulation, and is very intuitive, which is why it is applied to stakeholder-driven evaluation tasks. Table 13 of the comparative analysis presents the strengths and weaknesses of the existing DM methods³⁰ in comparison to the proposed Hybrid (IF-DoCM). The conventional techniques, including AHP and TOPSIS³¹, are powerful in terms of simplicity and ranking comparability but ineffective in making judgments of uncertainty and fairness among experts. Likewise, DEMATEL³² Offers a practical analysis that incorporates both qualitative and quantitative data; however, it is quite cumbersome, and there is no level of stakeholder inclusion. Fuzzy MCDM³³ provides fuzzy logic³⁴ As a means of managing uncertainty, this approach represents a step forward compared to the previous one. Yet, it is not intuitive in terms of the involvement of an immediate participant, and is not oriented towards making innovation-oriented decisions, in particular. The proposed IF-DoCM, in contrast to them, appears unique since it possesses all the major defining features: successfully operating with uncertainty and hesitation via IFS, involving stakeholders in the process through the DoCM mechanism, and being an overall user-friendly tool. Additionally, it has been developed to test innovative strategies, making it highly applicable in dynamic sectors such as music tourism innovation. Such a comparison demonstrates that IF-DoCM is a more robust and practical approach compared to current methods for complex, innovation-driven DM.

Table 13 Characteristic comparative analysis.

Theoretical implications

The implication of the results of this study is very profound, not only to the sports equipment industry but to the mode of taking decisions concerning product evaluation. The proposed methodology, which incorporates decision-theoretic fuzzy modelling along with DoCM, is not only convenient and easy to use and understand, but it will also be of practical use when ranking basketball models, where objective performance and subjective preferences are essential considerations. To manufacturers, this will provide helpful information on the influence of various product features, including the quality of the material, the feel of the grip on the face, bouncing consistency, and design, on overall consumer popularity. This facilitates the provision of a specific increase in design, the more effective deployment of resources, and the development of basketballs that align with market demands. The model also helps retailers and distributors to make inventory and marketing decisions by defining products that have the best market potential on the various levels of consumers. Such products as Aqua Air Pro, which demonstrated high results on a range of criteria, could be given a priority in the promotional strategy. For consumers and buyers selecting purchases, the fuzzy-DoCM framework offers the potential to convert multicriteria decisions and judgments into an interpretable and intelligible ranking, thereby making the purchasing decision more informed and aligned with personal priorities or needs, such as performance, outdoor competence, a green lifestyle, or value. The same methodology is applicable to rate other sports equipment or consumer goods in which a trade-off of several qualitative and quantitative factors is significant beyond basketballs. It is flexible; thus, it can be used as an effective decision-support tool in many industries where uncertainty and subjective judgment prevail in product assessment.

Advantages of the study

This research has several strengths that enhance its value to both practice and researchers. Applying the concept of fuzzy logic to the (DoCM) allows them to offer a solid DM model that can address the uncertainty and subjectivity of evaluating a product against other products, and also helps the decision-makers in extracting the fuzzy and linguistic decisions mostly uncalculated in numerical methods. The application of DoCM brings qualitative and visible practice of eliciting preferences, hence it can be approached by both experts and non-experts as well as enhance decision inputs intelligibility. The decision-theoretic fuzzy modelling methodology ensures a balanced view between objective measures of performance and subjective measures of performance as perceived by users, yielding realistic and balanced rankings. Moreover, the procedure is very flexible and adaptable, and one can readily change the criteria, alternative products or weighting schemes to meet various situations of an evaluation. Its open, sequential approach grows confidence in its outcomes as it is easy to trace the development of judgments through a chain of steps, generating final rankings. In addition to the basketball case study, the framework can be applied in other categories of sports equipment and consumer products to provide decision support to those manufacturers, retailers, and customers in designing products to fit their intended market.

Conclusion

This study aimed to address the issue of assessing the quality and design of basketball, where the objective component represents the objective side of performance criteria, and the subjective component, reflecting subjective preferences and needs, must be taken into account under uncertainty. By combining decision-theoretic fuzzy modelling with the DoCM, we were able to construct a clear, transparent, and intuitive decision-support framework that can handle imprecise, linguistic judgments and translate them into actionable rankings. By using a case study of five basketball alternatives and the six parameters used to evaluate them, the approach revealed the trade-off elements of performance, durability, aesthetics, and cost. Its findings demonstrated that Aqua Air Pro was the most favorable due to its balanced performance in all weather conditions and uniformity of bounce, as the others excelled in specific aspects such as indoor quality or sustainability. The advantage of this method is its flexibility, as criteria weights and evaluations can be easily altered to fit other DM contexts, enabling stakeholders to prioritize the things that are most important to them. This renders the approach applicable to the evaluation of basketball as well as other sports equipment and consumer products. In the end, the research work can have practical and scholarly value: on the one hand, it could be used by stakeholders in the industry as a practical and convenient evaluation tool; on the other hand, it has helped to progress the fusion of fuzzy DM and interactive preference elicitation approaches such as DoCM. By doing so, it balances analytical rigour with usability in the real world, allowing one to make superior, more informed product choices and design decisions in a setting where one has to deal with uncertainty and subjectivity.

Limitations and future directions

Although this research establishes the power of combining the decision-theoretic fuzzy modelling³⁵ technique of the DoCM to assess basketball quality and design, a few limitations can be noted. To begin with, the case study was based on expert and user judgments, and despite its usefulness, it can be biased in relation to individual preferences or gaming experience. We acknowledge that when consumers make a purchase decision, they do not typically apply structured evaluation techniques such as fuzzy MCDM³⁶. Yet, we intend not to model such buying behavior other than to offer a decision support tool to manufacturers, designers and procurement teams. The results can help these stakeholders to match product features with market segment expectations. The resulting pool of respondents will be more variable and larger and this may enhance the generalizability of findings. Second, the evaluation criteria chosen in this study were rigid; in a real-life setting, other aspects could also be considered, such as brand name, environmental impact during the production process, and technological advancements (e.g., smart sensors installed in basketballs). Third, the model was used with only one product category (basketballs) so no comprehensive research can be made. In the future, further studies can explore the broader applications of this hybrid fuzzy-DoCM model to other sporting equipment, consumer products, or even the assessment of services. The inclusion of objective performance data into decision accuracy could be enhanced by incorporating subjective evaluation³⁷. Further, it may involve adding insights into machine learning or big data analytics as an additional feature of the model, allowing for consistent updating of the weights of criteria that rely on a precedent scenario of consumer behavior. Lastly, cross-cultural research can help understand regional differences in preferences, which in turn enables manufacturers to develop better products for the global market.