Enhancing Decision Quality in Smart Manufacturing: Uncertainty-Aware Evaluation of Edge–Cloud Architectures with T-Spherical Hesitant Fuzzy Rough Sets

This study proposes a methodology that integrates edge-cloud computing architecture with the T-Spherical Hesitant Fuzzy Rough (T-SHFR) model in order to improve decision processes involving uncertainty in smart manufacturing systems. The method aims to provide more flexible and adaptable solutions in decision environments dominated by uncertainty, instability and multi-criteria structures by combining real-time data processing at the edge layer and large-scale analysis at the cloud layer. The methodology consists of system architecture design, data preprocessing, T-SHFR modeling and multi-objective optimization stages.

3.1 Model foundations and T-SHFR theory

In this section, the theoretical foundations of the proposed decision support structure are presented through the T-Spherical Hesitant Fuzzy Rough (T-SHFR) model. The mathematical definitions of the model’s core components—spherical fuzzy sets, hesitant fuzzy sets, and rough set theory—are introduced to illustrate how T-SHFR handles uncertainty, conflict, and incomplete information in decision-making environments. Furthermore, the construction of the decision matrix within the T-SHFR space is described using membership functions and approximation operators, demonstrating the model's applicability in complex multi-criteria decision-making (MCDM) scenarios, particularly in smart manufacturing contexts.

3.1.1 Spherical fuzzy sets

A spherical fuzzy set (SFS) is a recent extension of fuzzy and intuitionistic fuzzy sets, designed to represent uncertainty more flexibly in decision-making contexts. It characterizes each element by three parameters: membership degree (μ), non-membership degree (ν), and hesitancy degree (π), satisfying a spherical constraint. Formally, an SFS A in a universe X is defined as [38]:

$A=\left\{\left(x, \mu^{\mathrm{a}}(x), v^{\mathrm{a}}(x), \pi^{\mathrm{a}}(x)\right) \mid x \in X\right\}$ and spherical constraint; $\mu^{\mathrm{a}}(x)^2+v^{\mathrm{a}}(x)^2+\pi^{\mathrm{a}}(x)^2 \leq 1 \forall x \in X$.

Here, μ(x): degree of membership, ν(x): degree of non-membership, π(x): degree of hesitancy.

3.1.2 Hesitant fuzzy sets

A hesitant fuzzy set (HFS) refers to situations where the decision maker hesitates to determine a definitive membership degree for an element. More than one possible membership degree can be defined for each element. The concept was first defined by [39].

$\begin{gathered}A=\left\{\left(x, H_a(x)\right) \mid x \in X\right\} \text { and } H_a(x)=\left\{\mu_1^a(x), \mu_2^a(x), \ldots, \mu_k^a(x)\right\} \subseteq[0,1]\end{gathered}$

Here Hₐ(x) represents the set of all possible membership values for element x.

3.1.3 Rough sets

Rough set theory, developed by Pawlak [40], is a mathematical method used to deal with uncertainty, incompleteness, and ambiguity in data analysis. Under an equivalence relation R defined on the universe set U, for any subset A⊆U, elements that “certainly belong to A” and “probably belong to A” are defined separately.

The Lower Approximation: $A_{-} L=\left\{x \in U:[x]_R \subseteq A\right\}$;

The Upper Approximation: $A_{-} U=\left\{x \in U:[x]_R \cap A \neq \varnothing\right\}$,

Here, [x]_R denotes the access class of x according to the equivalence relation R.

3.1.4 T-Spherical Hesitant Fuzzy Rough Set (T-SHFR)

T-Spherical Hesitant Fuzzy Rough Set (T-SHFR) is an integrated structure used in decision-making environments with multi-layered uncertainty and ambiguity. This model combines the concepts of T-spherical fuzzy set [41], Hesitant fuzzy set [39], Rough set theory [40] and enables modeling different levels of uncertainty with three-dimensional membership, instability and lower-upper approximate sets.

The T-Spherical Hesitant Fuzzy Rough Set (T-SHFR) model aims to model the decision process more flexibly with rough set-based lower and upper approximate sets in the case of decision maker hesitancy, triple membership (T, F, U) structure of evaluation criteria, and lack of information. This structure is directly related to the T-spherical fuzzy rough aggregation operator approach proposed by Wang [42] and the weighted T-spherical fuzzy soft rough sets developed by Zhang et al. [43]. Both studies provided a strong theoretical basis for multi-criteria group decision making (MAGDM) problems by processing three uncertainty dimensions simultaneously. In a given universe, the T-SHFR for a set A is defined as:

$\tilde{A}=\left\{\left(x, \mathcal{H}(x), \mu_A^{\sim}(x)\right) \mid x \in X\right\}$

where, $\mathcal{H}(x)$ is hesitant membership values of an element x, and $\mu_A^{\sim}(x)=(T(x), F(x), U(x))$ is the T-spherical fuzzy membership function of x. For each element, the degrees of truth, falsehood and uncertainty are given, respectively. These three values must satisfy the T-spherical condition: $T(x)^t+F(x)^t+U(x)^t \leq 1$, For a universe U and a relation R, the T-SHFR is represented as: $\operatorname{T-SHFR}(A)=\left(\underset{-}{A}, \bar{A}, \mu_1, \mu_2, \mu_3\right)$, where $\mu_1, \mu_2, \mu_3$ represent the truth, uncertainty, and falsehood membership degrees of the hesitant fuzzy sets, and the approximations $\underset{-}{A}$ and $\bar{A}$ are derived using rough set theory.

3.1.5 Edge-cloud collaboration with T-SHFR in manufacturing optimization

The T-SHFR model is very suitable for modeling situations in edge-cloud based systems where the decision maker cannot make a definitive decision due to lack of data, time pressure or lack of expertise. For example: Edge Layer: Real-time unstable data. Cloud Layer: Reducing uncertainty through aggregate analysis. Thus, the T-SHFR model enables the system to make adaptive decisions at both local (edge) and global (cloud) levels. In smart manufacturing systems, edge-cloud collaboration and T-SHFR analysis are applied in optimizing decision-making under uncertain conditions. The idea is that by leveraging both local (edge) and global (cloud) information, along with the high-level fuzzy rough model, the manufacturing systems can adapt to real-time production status in a continuous manner while optimizing resource allocation, scheduling, and maintenance over the long term.

The cloud layer processes big data for predictive analytics, while the edge layer provides real-time input for the execution of immediate actions. The integration of T-SHFR enhances the decision-making process through handling multidimensional uncertainty and vagueness in the data, leading to more believable and dynamic decisions.

3.1.6 Approximations of T-SHFR

Let A⊆U be a set in the universe of discourse U, and R be the equivalence relation on U. Then, the lower and upper approximations of a T-SHFR set A are given by Zhang and Shu [44]:

Lower Approximation: $\underset{-}{A}=\left\{x \in U:[x]_R \subseteq A\right\}$

Upper Approximation: $\bar{A}=\left\{x \in U:[x]_R \cap A \neq \varnothing\right\}$

The presence of the hesitant fuzzy component modifies the approximations by including multiple possible membership degrees, enriching the decision-making process.

3.1.7 Uncertainty reduction in T-SHFR

The T-SHFR model reduces uncertainty in decision-making by providing a multi-dimensional fuzzy representation. Unlike classical fuzzy sets with a single membership value, T-SHFR incorporates multiple values reflecting various levels of uncertainty. This leads to more robust and reliable decisions under incomplete or ambiguous information.

Practical Implications of T-SHFR in Smart Manufacturing

Integrating T-SHFR into edge–cloud collaborative systems is a significant innovation for intelligent manufacturing. By processing local uncertainty at the edge and global uncertainty in the cloud, the system can optimize operations in real time while effectively handling uncertainty in scheduling, resource allocation, and maintenance. This improves overall efficiency, product quality, and cost-effectiveness.

3.1.8 Existence of optimal resource allocation using T-SHFR

Optimal resource allocation can be achieved using the T-SHFR model in multi-criteria environments involving cost, production, and uncertainty in demand and machine capabilities. By applying fuzzy decision rules and metaheuristic optimization methods (e.g., genetic algorithms, PSO), T-SHFR enables dynamic and uncertainty-aware resource distribution. Let X be the set of decisions, and let the membership function μ(x) reflect T-SHFR values. The objective is to minimize cost subject to constraints on output, resource use, and time. The result is a set of optimal solutions accounting for both efficiency and uncertainty.

3.1.9 Improved decision-making performance

The integration of T-SHFR within edge–cloud infrastructures enhance the accuracy, flexibility, and reliability of decision-making in smart manufacturing. The model’s multidimensional membership structure ensures more informed decisions under uncertainty, thereby increasing operational efficiency and system robustness.

3.2 System design and edge-cloud collaboration architecture

The proposed smart manufacturing architecture is built on a hybrid edge–cloud computing model, enabling efficient data processing and real-time decision-making via task distribution between edge devices and cloud servers. Edge devices, positioned close to production units (e.g., sensors, robots), handle local tasks such as real-time filtering, anomaly detection, and initial decisions. The cloud layer manages advanced analytics including performance monitoring, resource planning, and machine learning-based forecasting. The system architecture comprises edge, cloud, and communication layers. Preprocessing tasks such as noise reduction, normalization, and missing data imputation—are distributed across these layers. Edge-level preprocessing reduces data transmission by handling smoothing and feature extraction, while the cloud performs more intensive aggregation and trend analysis. To enhance system efficiency, adaptability, and resource use, the proposed optimization framework applies the T-SHFR model to key tasks such as production scheduling, resource allocation, and cost–energy optimization. Scheduling considers uncertainties in machine performance, demand variability, and resource availability. Real-time edge data and cloud history enable dynamic resource distribution, while predictive models support energy and cost efficiency. This iterative, feedback-driven process improves responsiveness to changing production conditions. Edge computing allows instant decisions with minimal latency, reacting swiftly to failures or demand shifts. Meanwhile, the cloud ensures large-scale optimization through predictive maintenance and scheduling. Their integration balances local agility with global intelligence, creating a dynamic and adaptive manufacturing system.

The core innovation lies in combining edge–cloud collaboration with T-SHFR analysis, which effectively captures uncertainty and hesitation in decision-making. The continuous data loop between layers enables real-time adjustments and long-term planning. The result is a robust, scalable decision support model for complex smart manufacturing environments.

3.3 Mathematical modeling with T-SHFR

The proposed T-Spherical Hesitant Fuzzy Rough (T-SHFR) model is designed to simultaneously support real-time decision making at the edge layer and long-term optimization functions at the cloud layer. This structure provides the opportunity to model the uncertainties effectively and hesitant evaluations frequently encountered in modern manufacturing systems. Optimization problems encountered in smart manufacturing environments are typically multi-objective, encompassing the following objectives: maximizing system efficiency, minimizing operating costs, reducing resource waste, and minimizing downtime. These objectives are formulated as a multi-objective optimization problem under uncertainty.

Objective 1: Maximizing Production Efficiency

Production performance is typically measured by output-oriented metrics [45]. The production efficiency E_p is defined as the ratio of the actual production output to the ideal production output. Mathematically:

$E_p=\frac{\text { Actual Output }}{\text { Ideal Output }}=\frac{\sum_{i=1}^n \text { Output }_i}{\sum_{i=1}^n \text { Ideal Output }_i}$

where, Output_i is the actual production of unit i; Ideal Output_i is the target or ideal output for unit i.

Objective 2: Minimizing Operational Costs

The operational cost C_op includes energy, labor, maintenance, and raw material costs. The total operational cost is represented as [32] (p. 106169):

$C_{\mathrm{op}}=\sum_{i=1}^n\left(c_{\text {energy}, i}+c_{\text {laboot}, i}+c_{\text {maintenance} i}+c_{\text {material}, i}\right)$

where, c_energy,i, c_laboot​,i, c_{maintenance​,i}, and c_material,i are the energy, labor, maintenance, and material costs associated with unit i, respectively.

Objective 3: Minimizing Downtime

Downtime D_down is the amount of time when machines are not in operation. The total downtime is calculated as [46]:

$D_{\text {down }}=\sum_{i=1}^n\left(t_{\text {failure}, i}+t_{\text {maintenances}, i}\right)$

where, t_failure,i is the downtime due to machine failure for unit i; t_{maintenance​,i} is the downtime due to scheduled maintenance for unit i.

3.3.1 Constraints

The optimization problem must satisfy various constraints, including production capacity, resource availability, and operational limits. The total production output must not exceed the production capacity of the system [46]:

$\sum_{i=1}^n$ Output $_i \leq$ Max Capacity

where, Output_i: Actual production of unit i, "Max Capacity" is the maximum production output that the system can handle. The amount of resources (e.g., raw materials, energy, labor) available at each point in time should be sufficient to meet production requirements. Let R_j represent the available amount of resource j, and a_ij be the amount of resource j required for unit i. Then, the resource constraint is:

$\sum_{i=1}^n a_{i j}$ Output $_i \leq R_j \quad \forall j \in\{1,2, \ldots, m\}$

where, a_ij is the amount of resource j required for production of unit i; R_j is the total available amount of resource j. The production of each unit i is dependent on the capacity of the machines available. Let M_i represent the machine capacity for unit i. The machine capacity constraint is given by:

$\sum_{i=1}^n \frac{\text { Output }_i}{M_i} \leq 1$

where, M_i is the machine's operational capacity per unit of time. The total downtime in the system must not exceed a specified threshold. Let D_max represent the maximum allowable downtime: $D_{\text {down }} \leq D_{\max }$; where, D_down is the total downtime; D_max is the allowed maximum downtime.

3.3.2 Fuzzy and rough set integration

The decision-making process in the optimization problem is affected by both uncertainty and hesitancy. To model these aspects effectively within the manufacturing system, we employ the T-Spherical Hesitant Fuzzy Rough Set (T-SHFR) approach. This model is incorporated into the objective function for resource allocation, where the fuzzy membership degrees (positive, neutral, and negative) dynamically influence the allocation process. Accordingly, the overall optimization problem is formulated as follows [47]:

$\min \sum_{i=1}^n C_{\mathrm{op}, i} \cdot \mu_1\left(x_i\right)+\lambda_1 \cdot E_p \cdot \mu_2\left(x_i\right)-\lambda_2 \cdot D_{\mathrm{down}} \cdot \mu_3\left(x_i\right)$

where,

C_op,i is the operational cost for unit i,

E_p is the production efficiency,

D_down is the downtime,

$\mu_1\left(x_i\right), \mu_2\left(x_i\right), \mu_3\left(x_i\right)$ are the membership functions corresponding to the decision xi,

$\lambda_1, \lambda_2$ are weighting factors to balance the importance of efficiency and downtime.

3.3.3 Dynamic resource allocation (edge-cloud collaboration)

The allocation process is governed by both real-time data (at the edge level) and predictive information across time intervals (at the cloud level). To capture this dynamic behavior, we define a time-dependent allocation function A(t), which evolves over time t based on both global and local information sources [18] (p.7):

$A(t)=\sum_{i=1}^n f_{\text {edge }}\left(x_i\right)+g_{\text {cloud }}\left(x_i\right)$,

where,

$f_{\text {edge }}\left(x_i\right)$ is the resource allocation function at the edge layer,

$g_{\text {cloud }}\left(x_i\right)$ is the resource allocation function at the cloud layer,

xi represents the decision variables for unit i.

3.3.4 Final optimization model

Combining the objective functions, constraints, fuzzy rough set integration, and dynamic resource allocation, the final mathematical optimization model is [46] (p. 41):

$\begin{gathered}\min \left\{\sum_{i=1}^n C_{\mathrm{op}, i} \cdot \mu_1\left(x_i\right)+\lambda_1 \cdot E_p \cdot \mu_2\left(x_i\right)-\lambda_2 \cdot D_{\mathrm{down}} \cdot\right.\left.\mu_3\left(x_i\right)\right\}\end{gathered}$

Subject to:

$\begin{gathered}\sum_{i=1}^n \text { Output }_i \leq \text { Max Capacity; } \sum_{i=1}^n a_{i j} \text { Output }_i \leq R_j \quad \forall j \in\{1,2, \ldots, m\} \\ \sum_{i=1}^n \frac{\text { output }_i}{M_i} \leq 1 ; D_{\text {down }} \leq D_{\max } ; A(t)=\sum_{i=1}^n f_{\text {edge }}\left(x_i\right)+g_{\text {cloud }}\left(x_i\right)\end{gathered}$

This model integrates real-time operations and long-term optimization in smart manufacturing systems by leveraging edge-cloud collaboration and T-SHFR analysis to effectively address uncertainty and hesitancy.

3.3.5 Proposed strategy model for T-Spherical Hesitant Fuzzy Rough (T-SHFR) set

The proposed strategy model utilizes T-Spherical Hesitant Fuzzy Rough (T-SHFR) set theory to support decision-making and optimization in environments characterized by uncertainty, imprecision, and vagueness. It is particularly designed to address complex multi-criteria decision-making (MCDM) problems such as resource allocation, system configuration, and performance evaluation.

Both possibilities are stated as a T-Spherical Hesitant Fuzzy Rough (T-SHFR) set that provides a systematic way of defining the truth (T), falsity (F), and indeterminacy (U) of each possibility regarding different criteria while considering hesitant membership.

For a negating A_i, its membership function $\mu_{A_i}^{\sim}$ in the discourse universe X is defined as [47]:

$\mu_{{A}_i}^{\sim}(x)=\left(T_{{A}_i}^{\sim}(x), F_{{A}_i}^{\sim}(x), U_{{A}_i}^{\sim}(x)\right) ;$

where, $T_{{A}_i}^{\sim}(x)$ is the truth degree of the alternative for criterion x; $F_{{A}_i}^{\sim}(x)$ is the falsehood degree; $U_{{A}_i}^{\sim}(x)$ is the uncertainty degree and the membership values satisfy the condition:

$T_{\tilde{A}_i}^{\sim}(x)^t+F_{\tilde{A}_i}^{\sim}(x)^t+U_{\tilde{A}_i}^{\sim}(x)^t \leq 1$, for some $t>0$

Hesitant fuzzy logic enables the representation of multiple possible membership degrees for each alternative with respect to a given criterion, thereby modeling the decision maker’s uncertainty or hesitation in assigning a single precise value

3.4 Evaluation and ranking

To evaluate and rank the alternatives, the T-SHFR set model integrates multi-criteria decision-making (MCDM) methods such as Fuzzy TOPSIS, VIKOR, and MOORA within the T-SHFR framework, as illustrated in Figures 1 and 2. These methods allow for robust assessment based on different perspectives—distance from ideal solutions (Fuzzy TOPSIS), compromise-based ranking (VIKOR), and ratio analysis (MOORA).

1.png

Figure 1. Decision-making process

2.png

Figure 2. T-SHFR set based MCDM model

The proposed optimization approach seeks to minimize or optimize specific objectives (e.g., cost, efficiency) in the presence of multiple conflicting criteria, vagueness, and uncertainty. The uncertainty in decision-makers’ preferences is modeled using T-Spherical Hesitant Fuzzy Rough (T-SHFR) sets. To identify the optimal solutions, optimization techniques such as genetic algorithms, particle swarm optimization, or linear programming can be applied within this framework in Figure 2.