Hybrid Quantum–Classical Framework for Fuzzy 0–1 Integer Linear Programming: α-Level Trade-off Analysis via Quadratic Unconstrained Binary Optimization Reformulation

Integer Linear Programming (ILP) represents a fundamental class of optimization problems with wide-ranging applications in logistics, finance, telecommunications, and resource allocation. The 0–1 ILP variant, where variables are restricted to binary values $\{$0, 1$\}$, is particularly significant as it models numerous combinatorial optimization problems with substantial practical importance. However, traditional 0–1 ILP formulations assume precise knowledge of all parameters, which rarely aligns with real-world conditions where costs, demands for resources, and constraints are often imprecise or vaguely defined [1].

Fuzzy set theory, introduced by Zadeh [2], provides a mathematical framework for modeling this uncertainty through membership functions and fuzzy numbers. While fuzzy optimization methods have been developed over decades, they face substantial computational challenges when applied to large-scale combinatorial problems due to their exponential worst-case complexity. Concurrently, quantum computing has emerged as a promising paradigm for tackling combinatorial optimization, with approaches like quantum annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) showing potential for speedup on certain problem classes [3].

Modern industrial systems feature growing complexity, dynamism and uncertainty. Decision-making under such conditions often relies on imprecise information stemming from measurement errors, subjective judgement, prediction inaccuracies, and intrinsic process variability. Such uncertainty appears widely in industrial intelligence applications.

Production Resource Allocation: Manufacturing systems should allocate limited resources (machines, labor, and materials) across competing production activities. Demand forecasts, processing time, and resource availability are rarely known with certainty, thus necessitating optimization under fuzzy parameters. Production managers often express constraints such as “approximately 100 units of raw material are available” or “the processing time is about 5 hours”, which naturally fit the fuzzy paradigm.

Manufacturing Scheduling: Problems of job shop scheduling and flow shop scheduling involve fuzzy processing time, fuzzy due dates, and fuzzy machine availability. The objective often involves minimizing fuzzy makespan or maximizing fuzzy customer satisfaction, thus turning fuzzy ILP into a natural modeling framework.

Supply Chain Optimization: Supply chain networks face uncertainties in demand, supplier reliability, transportation costs, and lead time. Fuzzy optimization enables robust decision-making that accounts for these uncertainties while maintaining operational efficiency.

Energy Management Systems: Power systems optimization involves fuzzy load forecasts, fuzzy generation capacities, and fuzzy transmission constraints. The integration of renewable energy sources introduces additional uncertainty, rendering fuzzy optimization essential for reliable energy management.

Healthcare Resource Management: Hospital scheduling, patient flow optimization, and medical resource allocation involve imprecise parameters such as uncertain patient arrival time, variable durations of treatment, and unpredictable resource requirements.

Portfolio Optimization: Financial decision-making under uncertainty involves fuzzy returns, fuzzy risks, and fuzzy investment constraints, requiring sophisticated optimization tools that could handle imprecision.

The proposed Fuzzy-Hybrid Quantum-Classical Optimization Algorithm (F-HQCOA) framework dealt with these industrial challenges by providing a scalable and quantum-assisted solution methodology that could handle the inherent uncertainty of real-world industrial problems, while generating a spectrum of solutions to support risk-aware decision-making.

Despite these advancements, a significant gap exists in the literature regarding the integration of fuzzy optimization methodologies with quantum computing approaches. The main contributions of this work are summarized as follows:

1. A formal $\alpha$-cut-based transformation framework for converting fuzzy 0–1 ILP problems into a family of deterministic Quadratic Unconstrained Binary Optimization (QUBO) models;

2. A F-HQCOA that solves each $\alpha$-level problem efficiently using quantum optimization techniques;

3. A systematic parametric analysis framework that generates a spectrum of solutions across $\alpha$-levels, enabling evaluation of trade-offs between feasibility satisfaction and objective performance;

4. An empirical evaluation demonstrating scalability and solution diversity compared to classical fuzzy optimization methods;

5. A discussion of the limitations of current quantum approaches and the potential for extending the framework toward multi-objective fuzzy optimization.

Notably, the proposed framework differs from classical Pareto-based multi-objective optimization. It employs an \(\alpha\)-parametric method generating deterministic problems for each \(\alpha\), and solutions over varying \(\alpha\) illustrate the trade-off between feasibility satisfaction and objective value. This distinction prevents misinterpretation against conventional multi-criteria optimization.

The paper proceeds as follows. Section 2 reviews relevant studies on fuzzy optimization and quantum computing. Section 3 outlines the theoretical framework and Section 4 specifies F-HQCOA. Section 5 describes experimental configurations, whose outcomes are analyzed in Section 6. Section 7 discusses implications and limitations, and Section 8 concludes with future research.

Recent advances in fuzzy integer programming have explored multi-objective formulations, interactive decision-making approaches, and metaheuristic solution techniques [4], [5]. In particular, multi-objective fuzzy optimization has been extensively studied using Pareto-based methods, weighted aggregation, and fuzzy goal programming.

In parallel, significant progress has been made in quantum optimization, particularly in QUBO formulations for combinatorial problems [6]. Hybrid quantum–classical algorithms such as QAOA and QA have been applied to many Nondeterministic Polynomial (NP)-hard problems, including scheduling, routing, and portfolio optimization [7], [8].

The fusion of fuzzy optimization and quantum computing remains underexplored. Existing deterministic approaches neglect uncertainty in quantum optimization. This work bridges the gap by integrating fuzzy modeling into QUBO-based quantum optimization.

Fuzzy optimization approaches have evolved significantly since Zadeh’s pioneering work on fuzzy sets [2]. As an early contribution, Zimmermann [9] introduced fuzzy linear programming with multiple objective functions, while Verdegay [10] developed fundamental duality results. Traditional fuzzy optimization methods could be broadly categorized into possibilistic programming, fuzzy stochastic programming, and robust fuzzy programming approaches.

A fundamental challenge in fuzzy optimization involves handling imprecise parameters without dramatically increasing computational complexity. Recent advancements focused on granular differentiability (gr-differentiability) concepts to address limitations of earlier Hukuhara-based differentiation approaches. González-Rodelas et al. [11] proposed gr-differentiability to overcome issues such as increasing support closure length and unnatural behavior in modeling phenomena. This approach maintains a single fuzzy differential equation without multiple solution forms, to provide significant advantages for optimization under uncertainty.

Specifically for fuzzy integer programming, researchers have developed specialized branch-and-bound algorithms, cutting plane methods, and metaheuristics tailored to handle fuzzy constraints and objectives [12]. However, these approaches continue to face scalability challenges with problem sizes encountered in practical applications, necessitating new computational paradigms.

Quantum optimization has emerged as a promising approach for combinatorial problems, with two primary paradigms: QA and gate-based quantum algorithms like QAOA [3]. Quantum annealers, such as those produced by D-Wave Systems, are specialized quantum computers designed to find low-energy states of Ising Hamiltonians, which could be mapped to QUBO problems [6].

Recent research has demonstrated the potential of quantum computing for various combinatorial optimization problems, though applications to fuzzy optimization remain largely unexplored. Current literature primarily focused on deterministic problems, with limited attention to uncertain or fuzzy environments. This gap is particularly notable given the potential for quantum approaches to tackle the exponential complexity of fuzzy combinatorial optimization.

Our review identified a significant research gap at the intersection of fuzzy optimization and quantum computing. While both fields have advanced independently, few works have attempted to integrate fuzzy set theory with quantum optimization approaches. The existing literature on fuzzy integer programming primarily focused on classical solution methods, which are facing scalability limitations. Meanwhile, quantum optimization research has predominantly handled deterministic problems without considering the imprecision and uncertainty inherent in real-world optimization scenarios.

This paper bridged these distinct research streams by developing a unified framework that leveraged the strengths of both fuzzy optimization theory and quantum computing capabilities. Our approach enabled the solution of larger-scale fuzzy ILP problems than previously possible with classical methods alone, while providing decision-makers with a spectrum of solutions representing different trade-offs between optimality and feasibility satisfaction.

We considered a fuzzy 0–1 ILP problem with imprecise parameters in both the objective function and constraints. Formally, the problem is defined as:

\[ \text{Maximize } \tilde{c}^T x \quad \]

subject to,

\[ A x \precsim \tilde{b}\text{,} \quad x\in\{0,1\}^n \]

where, $\tilde{c}$ = ($\tilde{c}_1$, $\tilde{c}_2$, $\dots$, $\tilde{c}_n$) and $\tilde{b}$ = ($\tilde{b}_1$, $\tilde{b}_2$, $\dots$, $\tilde{b}_m$) denote vectors of triangular fuzzy numbers (TFNs). The symbol $\precsim$ stands for “approximately less than or equal to”.

Each TFN is expressed as ($a^p$, $a^m$, $a^o$), where $a^p$ denotes the pessimistic value, $a^m$ denotes the most likely value, and $a^o$ denotes the optimistic value [13].

The membership function of a TFN $\tilde{a}$ = ($a^p$, $a^m$, $a^o$) was given by:

\[ \mu_{\tilde{A}}(x)= \begin{cases} 0 & \text{if } x < a^p\\[3pt] \dfrac{x - a^p}{a^m - a^p} & \text{if } a^p \leq x < a^m\\[3pt] 1 & \text{if } x = a^m\\[3pt] \dfrac{a^o - x}{a^o - a^m} & \text{if } a^m < x \leq a^o\\[3pt] 0 & \text{if } x > a^o \end{cases} \]

Figure 1 shows a TFN $\tilde{A}$ = ($a^p$, $a^m$, $a^o$) with its membership function $\mu_{\tilde{A}}(x)$. Three $\alpha$-cuts ($\alpha$ = 0.0, $\alpha$ = 0.5, $\alpha$ = 1.0) are shown, demonstrating how each cut yielded a specific crisp interval [$a_\alpha^L$,$a_\alpha^R$].

To convert the fuzzy problem into a form amenable to quantum optimization, we employed the concept of $\alpha$-cuts. For a fuzzy number $\tilde{a}$ and a specified $\alpha \in$ [0,1], the $\alpha$-cut was defined as the crisp set:

\[ [\tilde{a}]_\alpha = \bigl\{ x \in X \,\big|\, \mu_{\{\tilde{a}\}}(x) \geq \alpha \bigr\} \]

For a TFN $\tilde{a}$ = ($a^p$, $a^m$, $a^o$), the $\alpha$-cut yielded the interval:

\[ [\tilde{a}]_\alpha = \bigl[ a^p + \alpha\bigl(a^m - a^p\bigr),\; a^o - \alpha\bigl(a^o - a^m\bigr) \bigr] \]

Rationale for Transformation Strategies: The transformation of fuzzy parameters to crisp values requires careful consideration of the optimization objective and constraint orientation.

In maximization problems, using the pessimistic value of the objective coefficients $c_j(\alpha) = c_j^p + \alpha\bigl(c_j^m - c_j^p\bigr)$ ensures that the resulting objective value is a conservative estimate of the achievable profit. This is appropriate for risk-averse decision-makers who prefer guaranteed lower bounds on profit. The parameter $\alpha$ controls the degree of optimism: at $\alpha$ = 0, the decision-maker accepts the most optimistic estimate $c_j^p$; at $\alpha$ = 1, the decision-maker accepts the most likely estimate $c_j^m$.

In maximization problems with resource constraints $({Ax} \precsim \tilde{b})$, using the optimistic value of the capacity $b_i(\alpha) = b_i^o - \alpha(b_i^o - b_i^m)$ creates a stricter constraint. At $\alpha$ = 0, the constraint is most relaxed (using the optimistic bound $b_i^o$); at $\alpha$ = 1, the constraint is most restrictive (using the most likely bound $b_i^m$). This conservative transformation ensures that solutions satisfying the crisp constraints will satisfy the original fuzzy constraints to at least degree $\alpha$.

Lemma 1 (Monotonicity of Feasible Region). For $\alpha_1 \le \alpha_2$, the feasible region of the transformed crisp problem at $\alpha_2$ is a subset of the feasible region at $\alpha_1$.

Proof: For $\alpha_1 \le \alpha_2$, we have $b_i(\alpha_2) \le b_i(\alpha_1)$ since $b_i(\alpha) = b_i^o - \alpha(b_i^o - b_i^m)$ is decreasing in $\alpha$. Therefore, the constraint $\displaystyle\sum_{j} a_{ij}x_j \le b_i(\alpha_2)$ implies $\displaystyle\sum_{j} a_{ij}x_j \le b_i(\alpha_1)$. Hence, the feasible region is monotonically decreasing with $\alpha$.

This monotonicity property is critical for the parametric analysis. As $\alpha$ increases, the feasible region shrinks, leading to potentially lower objective values but higher confidence in feasibility.

To illustrate the $\alpha$-cut transformation process, consider a small example with two variables and one constraint:

Fuzzy Problem:

\[\text{Maximize}\ \tilde{c}_1 x_1+\tilde{c}_2 x_2\]

subject to,

\[ \tilde{a}_1 x_1+\tilde{a}_2 x_2 \precsim \tilde{b}\]

\[x_1, x_2 \in \{0,1\}\]

where,

\[ \tilde{c}_1 = (8,10,12),\quad \tilde{c}_2 = (12,15,18) \]

\[ \tilde{a}_1 = (1.5,2.0,2.5),\quad \tilde{a}_2 = (2.0,2.5,3.0) \]

\[ \tilde{b} = (4.0,4.5,5.0) \]

Transformation at $\alpha$ = 0.2

Objective coefficients:

\[ c_1(0.2) = 8 + 0.2(10 - 8) = 8.4, c_2(0.2) = 12 + 0.2(15 - 12) = 12.6 \]

Constraint Right-Hand Side (RHS):

\[ b(0.2) = 5.0 - 0.2(5.0 - 4.5) = 4.9 \]

Crisp problem:

\[ \text{Maximize}\ 8.4x_1 + 12.6x_2 \]

subject to,

\[ \bigl(1.5 + 0.2(2.0 - 1.5)\bigr)x_1 + \bigl(2.0 + 0.2(2.5 - 2.0)\bigr)x_2 \le 4.9 \]

LHS coefficients:

\[ a_1(0.2) = 1.6, a_2(0.2) = 2.1 \]

Feasible binary solutions:

\[ (0,0): 0 \le 4.9; (1,0): 1.6 \le 4.9; (0,1): 2.1 \le 4.9; (1,1): 3.7 \le 4.9 \]

Optimal solution:

\[ (1,1)\ \text{with objective value}\ 8.4 + 12.6 = 21.0 \]

Transformation at $\alpha$ = 0.8

Objective coefficients:

\[ c_1(0.8) = 8 + 0.8(10 - 8) = 9.6, c_2(0.8) = 12 + 0.8(15 - 12) = 14.4 \]

Constraint RHS:

\[ b(0.8) = 5.0 - 0.8(5.0 - 4.5) = 4.6 \]

LHS coefficients:

\[ a_1(0.8) = 1.5 + 0.8(2.0 - 1.5) = 1.9, a_2(0.8) = 2.0 + 0.8(2.5 - 2.0) = 2.4 \]

Feasible binary solutions:

\[ (0,0): 0 \le 4.6; (1,0): 1.9 \le 4.6; (0,1): 2.4 \le 4.6; (1,1): 4.3 \le 4.6 \]

Optimal solution:

\[ (1,1)\ \text{with objective value}\ 9.6 + 14.4 = 24.0 \]

Transformation at $\alpha$ = 1.0

Objective coefficients:

\[ c_1(1.0) = 10, c_2(1.0) = 15 \]

Constraint RHS:

\[ b(1.0) = 4.5 \]

LHS coefficients:

\[ a_1(1.0) = 2.0, a_2(1.0) = 2.5 \]

Feasible binary solutions:

\[ (0,0): 0 \le 4.5; (1,0): 2.0 \le 4.5; (0,1): 2.5 \le 4.5; (1,1): 4.5 \le 4.5 \]

Optimal solution:

\[ (1,1)\ \text{with objective value}\ 10 + 15 = 25.0 \]

Analysis of the Example: As $\alpha$ increases from 0.0 to 1.0:

The objective coefficients increase (from pessimistic to most likely values), resulting in higher objective values. The constraint RHS decreases (from optimistic to most likely values), rendering stricter constraint. However, in this example, the optimal solution (1,1) remains feasible for all $\alpha$-levels.

The satisfaction degree $\mu$ for each solution was computed as the minimum membership grade:

For $\alpha$ = 0.2:

\[ \mu = \text{min}\Bigl(\mu_{\tilde{c}_1}(8.4),\mu_{\tilde{c}_2}(12.6),\mu_{\tilde{b}}(4.9)\Bigr) = \text{min} (0.2,0.2,0.8) = 0.2 \]

For $\alpha$ = 0.8:

\[ \mu = \text{min} \Bigl(\mu_{\tilde{c}_1}(9.6),\mu_{\tilde{c}_2}(14.4),\mu_{\tilde{b}}(4.6)\Bigr) = \text{min} (0.8,0.8,0.2) = 0.2 \]

The satisfaction degree $\mu$ is the minimum membership grade across all fuzzy constraints and the objective function for a given solution. Different solutions may have different $\mu$ values, and the solution's $\mu$ is not necessarily equal to the $\alpha$-level used in the transformation.

This example demonstrated how different $\alpha$-levels transformed the same fuzzy problem into different crisp problems, leading to a spectrum of solutions with varying objective values and feasibility satisfaction levels.

Each crisp $\alpha$-ILP problem was converted to a QUBO problem by incorporating constraints as penalty terms. The general QUBO form is:

\[ \text{minimize:}\ x^{T}Qx \]

where, $x$ is a vector of binary decision variables and $Q$ is an $n \times n$ real-valued matrix.

Constraint $i$ was incorporated using a quadratic penalty term:

\[ P_i \left(\sum_{j} a_{ij}x_j - b_i(\alpha)\right)^2 \]

The complete QUBO formulation for a given $\alpha$ became:

\[ H(x,\alpha) = -\sum_{j} c_{j,\alpha}x_j + \sum_{i} P_i \left(\max\Bigl(0,\sum_{j} a_{ij}x_j - b_{i,\alpha}\Bigr)\right)^2 \]

where, $P_i$ are sufficiently large penalty weights that ensure constraint satisfaction in optimal solutions [14]. The $\alpha$-Cut transformation rules are summarized in Table 1.

The F-HQCOA integrated fuzzy set theory with quantum computation through a structured four-stage process.

$\alpha$-Level Selection: Determine a set of $\alpha$ values $\{\alpha_1$, $\alpha_2$, ..., $\alpha_k\} \in$ [0,1] to explore different feasibility satisfaction levels.

Defuzzification: Transform the fuzzy problem into crisp QUBO problems for each $\alpha$-level using the $\alpha$-cut method.

Quantum Processing: Solve each QUBO problem using quantum algorithms (QA or QAOA).

Classical Refinement: Filter and refine solutions to ensure feasibility and evaluate satisfaction degrees.

The end-to-end workflow of the F-HQCOA, as shown in Figure 2, illustrates the four-stage process from fuzzy input to solutions across $\alpha$-levels.

The steps of the proposed algorithm are detailed below. See Appendix B for further details of the implementation.

The F-HQCOA framework supported two primary quantum processing approaches:

QA: Utilizes quantum fluctuations to find low-energy states of the QUBO problem. This approach is implemented on specialized quantum annealers like D-Wave systems.

QAOA: A hybrid quantum-classical algorithm that uses a parameterized quantum circuit to prepare a state that encodes the solution, with classical optimization of the parameters.

For both approaches, we employed warm-starting techniques that used classical solutions to initialize quantum algorithms in promising regions of the solution space [7]. The strengths and limitations of the aforementioned two approaches are summarized in Table 2.

Remarks on the Interpretation of Solutions: Solutions at different $\alpha$-levels are not a Pareto front; each $\alpha$-level defines an independent deterministic problem, and the solutions form a parametric trade-off curve reflecting the effect of constraint satisfaction on the objective value.

The satisfaction degree ($\mu$) was computed after optimization from the fuzzy membership functions and used only to evaluate solution quality at each $\alpha$-level, not as a QUBO objective.

It is important to clarify the fundamental difference between the proposed $\alpha$-level parametric approach and traditional multi-objective optimization.

$\alpha$-Level Parametric Approach (Proposed): Each $\alpha$-level defines an independent deterministic optimization problem. Solutions across $\alpha$-levels form an $\alpha$-level trade-off curve showing how constraint satisfaction affects the achievable objective value; thus, this is a parametric analysis rather than true multi-objective optimization.

Multi-Objective Optimization (Classical): Classical multi-objective optimization simultaneously optimizes conflicting objectives, such as profit and satisfaction degree, producing a Pareto front of non-dominated trade-off solutions.

The degree of satisfaction ($\mu$) was computed as a post-solution evaluation metric based on the membership functions of the fuzzy parameters. It was not directly optimized within the QUBO formulation but was used to assess the quality of solutions obtained at each $\alpha$-level.

We evaluated F-HQCOA on two well-established NP-hard problem classes adapted to fuzzy environments.

Fuzzy Multi-Dimensional Knapsack Problem (F-MDKP): Items have fuzzy values $\tilde{c}_j$ and fuzzy weights $\tilde{w}_{ij}$. The goal is to maximize total value without exceeding fuzzy capacity constraints for each dimension.

Fuzzy Set Covering Problem: Elements have fuzzy costs $\tilde{c}_j$ and fuzzy coverage requirements $\tilde{a}_{ij}$. The objective is to minimize total cost while satisfying all fuzzy coverage constraints.

For each problem class, we generated multiple instances with varying sizes ($n$ = 20, 40, 60, 100 variables) and complexity parameters. Fuzzy parameters were modeled as TFNs with perturbations of around 15–25% crisp values. More details for complexity analysis are referenced to Appendix C.

The computational environment specifications are detailed in Table 3.

This paper compared F-HQCOA against three classical fuzzy optimization approaches:

Classical Fuzzy Branch-and-Bound: An exact method that incorporates $\alpha$-cuts in the branching process;

Fuzzy Genetic Algorithm: A metaheuristic approach using evolutionary computation with fuzzy fitness evaluation;

Fuzzy Simulated Annealing: A trajectory-based metaheuristic adapted for fuzzy environments.

For F-HQCOA, we tested both QA (on D-Wave Advantage quantum annealer) and QAOA (on Quantum Information Software Kit (Qiskit) simulator with noise models) implementations. See Appendix A for details of the experimental results.

We evaluated algorithm performance using multiple metrics under a fair and reproducible experimental protocol.

Satisfaction Degree ($\mu$):

The minimum membership grade across all satisfied fuzzy constraints and the objective function. For a solution $x^{*}$, we computed:

\[ \mu(x^{*}) = \text{min}\big(\mu_{\tilde{c}}(c^{T}x^{*}), \text{min}_{i} \mu_{\tilde{b}_{i}}(a_{i}^{T}x^{*})\big) \]

Optimality Gap: Percentage difference from the optimal solution of the deterministic ($\alpha$ =1) case, computed as:

\[ \text{Gap} = \frac{\text{Optimal}_{\alpha=1} - \text{Objective}_{x^{*}}}{\big|\text{Optimal}_{\alpha=1}\big|} \times 100\% \]

Time-to-Solution (TTS): Wall-clock time required to find the best solution. For fair comparison, TTS was measured as:

Classical Solvers: Total execution time from problem input to final solution output.

Quantum Approaches: Total execution time including QUBO construction, embedding, quantum sampling (including all anneals/QAOA iterations), and classical post-processing.

Diversity: Average pairwise Hamming distance between solutions in the solution set, computed as:

\[ D = \frac{1}{k(k-1)}\sum_{i=1}^{k}\sum_{j=i+1}^{k}\sum_{l=1}^{n}\big|x_{i}^{(l)} - x_{j}^{(l)}\big| \]

where, $k$ is the number of solutions in the set.

Experimental Protocol: For each benchmark instance, we performed 20 independent runs to account for the probabilistic nature of quantum algorithms. The reported results were the median values across these runs, with the interquartile range (IQR) provided for key metrics.

Classical Solver (Gurobi): Time limit of 3,600 seconds. The optimality tolerance was set to $10^{-6}$. No warm-start was used to ensure a fair comparison with the quantum approaches.

QA (D-Wave): The QUBO matrix was embedded using the D-Wave's minorminer heuristic. The annealing time was set to 20 microseconds with 1,000 anneals per run. The chain strength was tuned using the D-Wave's default scaling method. The reported TTS included embedding time.

QAOA (Qiskit): The circuit was executed on the Qiskit Aer simulator with a noise model approximating current quantum hardware (IBM's “fake\_montreal" backend). The Constrained Optimization BY Linear Approximations (COBYLA) was run with a maximum of 200 iterations. The reported TTS included classical optimization time.

QUBO Construction and Post-Processing: For all quantum approaches, the reported TTS included the time for QUBO construction, penalty matrix computation, and classical refinement steps.

All experiments were conducted on a high-performance computing cluster, with quantum simulations leveraging the Qiskit Aer noise-aware simulator to mimic real hardware constraints. The experimental parameters are given in Table 4.

The F-HQCOA algorithm successfully generated a spectrum of solutions representing different trade-offs between optimality and feasibility satisfaction. Figure 3 shows the characteristic trade-off curve obtained for a representative F-MDKP instance with $n$ = 60 variables.

As shown in Table 5, both classical and quantum approaches [15],[16] found higher-quality solutions at lower $\alpha$-levels (greater constraint relaxation). However, the quantum approach (using QA) achieved significantly faster TTS across all $\alpha$-levels, with speedups of $4.3\times$ to $16.2\times$ compared to the Classical Fuzzy Branch-and-Bound Algorithm.

Figure 3 shows trade-off curve between objective value and satisfaction degree across $\alpha$-levels for a fuzzy MDKP instance ($n$ = 60). Each point corresponds to the optimal solution obtained for a specific $\alpha$-level, to illustrate the impact of feasibility requirements on objective performance.

The trade-off curve obtained by F-HQCOA for a Fuzzy Multi-Dimensional Knapsack Problem instance ($n$ = 60) shows the trade-off between the Solution Objective Value and the overall Satisfaction Degree $\mu$. Each point represents a solution found for a different $\alpha$-level.

We evaluated the scalability of F-HQCOA by testing problems of increasing size. Figure 4 shows how TTS scales with problem size for the different algorithms.

Scaling of median TTS with problem size (number of variables $n$) for the Classical Fuzzy Branch-and-Bound solver and the F-HQCOA (QA) approach. The $y$-axis is on a logarithmic scale. The shaded regions represent the IQR over 20 problem instances. For larger problem instances ($n$ = 100), classical solvers frequently failed to find feasible solutions within a one-hour time limit for higher $\alpha$-levels, while F-HQCOA remained effective ( Table 6).

The quantum approach maintained solution quality within 5–8% of the best-known solutions while achieving feasible computation times.

Figure 5 is a conceptual illustration of the distinction between the proposed $\alpha$-level parametric analysis (left) and true multi-objective Pareto front (right). In the proposed approach, each $\alpha$-level yielded a single solution, and the trade-off curve represented how the objective value changed as feasibility requirements varied. In contrast, a true Pareto front consisted of multiple non-dominated solutions from a single multi-objective optimization problem.

A key advantage of F-HQCOA was its ability to generate diverse solutions across the $\alpha$-level spectrum. We measured diversity using average pairwise Hamming distance between solutions [17]. The quantum approach generated more diverse solution sets (average Hamming distance 28.7) compared to classical approaches (average Hamming distance 19.3), thus providing decision-makers with a broader range of alternatives. Table 7 compares $\alpha$-level parametric analysis and multi-objective optimization.

Table 8 compares F-HQCOA with classical fuzzy optimization methods.

It should be noted that in comparison studies, effective factors differentiated the performance of different quantum approaches as given in Table 9.

Figure 6 shows a parametric $\alpha$-level trade-off curve obtained by F-HQCOA.

Each point corresponds to a solution derived from a specific $\alpha$-level, to illustrate how increasing feasibility requirements (lower $\mu$) impact the objective value. The curve represents a parametric analysis rather than a true Pareto front. It is emphasized that $\alpha$-levels define independent optimization problems rather than simultaneous objective trade-offs.

To demonstrate the practical applicability of the F-HQCOA framework, we developed an industrial-inspired production planning problem. A manufacturing facility produced three products using two types of resources (machining hours and labor hours). The problem parameters were modeled as TFNs to reflect real-world uncertainties:

Product values:

Product 1: $\tilde{c}_1$ = (75, 100, 125)

Product 2: $\tilde{c}_2$ = (120, 150, 180)

Product 3: $\tilde{c}_3$ = (90, 110, 130)

Resource requirements (machining hours):

Product 1: $\tilde{a}_{11}$ = (2.0, 2.5, 3.0)

Product 2: $\tilde{a}_{12}$ = (3.0, 3.5, 4.0)

Product 3: $\tilde{a}_{13}$ = (1.5, 2.0, 2.5)

Resource requirements (labor hours):

Product 1: $\tilde{a}_{21}$ = (1.5, 2.0, 2.5)

Product 2: $\tilde{a}_{22}$ = (2.0, 2.5, 3.0)

Product 3: $\tilde{a}_{23}$ = (1.0, 1.5, 2.0)

Resource capacities:

Machining hours: $\tilde{b}_1$ = (90, 100, 110)

Labor hours: $\tilde{b}_2$ = (80, 90, 100)

The objective is to maximize total production value while respecting the fuzzy resource capacities.

We applied the F-HQCOA framework with $\alpha$-levels {0.0, 0.2, 0.5, 0.8, 1.0}. The results are presented in Table 10.

The results demonstrated that the F-HQCOA framework generated a spectrum of production plans that trade off between resource utilization (or constraint satisfaction) and production value. A manager with a conservative risk profile might select the $\alpha$ = 0.8 solution, which utilized 95.8% of machining capacity and 91.2% of labor capacity, to achieve a value of 2,135 with a satisfaction degree of 0.78. Conversely, a manager seeking to maximize production value might select the $\alpha$ = 0.0 solution, which achieved a value of 2,450 with full (1.00) satisfaction but represented a less certain plan.

This industrial-oriented example illustrated the practical utility of the F-HQCOA framework for real-world decision-making under uncertainty.

It is important to emphasize that the observed computational advantages of the proposed approach should be interpreted with caution. The comparison involves fundamentally different computational paradigms, including specialized quantum hardware and classical optimization solvers. Therefore, the reported speedups do not represent a direct one-to-one comparison but rather indicate the potential efficiency of hybrid quantum approaches under specific conditions.

Our results demonstrated that the F-HQCOA framework effectively confronted the challenge of solving fuzzy 0-1 ILP problems by leveraging quantum computing capabilities. However, several important caveats should be considered when interpreting the computational performance comparison.

Fundamental Paradigm Differences: The classical and quantum approaches operate under fundamentally different computational paradigms. Classical solvers like Gurobi are mature, deterministic systems that run on general-purpose central processing units with decades of optimization. Quantum approaches, in contrast, are probabilistic, run on specialized hardware (or simulators), and are in an early stage of development. As such, the reported speedups should not be interpreted as a direct one-to-one comparison but rather as an indication of the potential efficiency of hybrid quantum approaches under specific conditions.

Components of TTS: The reported TTS for quantum approaches includes all phases of the computation: QUBO construction, minor embedding (for QA), quantum execution (including multiple anneals or QAOA iterations), and classical post-processing. This comprehensive measurement ensures a fair comparison with classical solvers, which similarly include all computational phases.

Hardware-Specific Considerations: The performance of quantum approaches is highly dependent on the specific hardware used. Our experiments used a D-Wave Advantage quantum annealer and a Qiskit simulator with noise models. Results may vary in different quantum hardware configurations.

Parameter Sensitivity: The performance of quantum approaches is sensitive to various parameters, including annealing time, number of anneals, QAOA depth, and penalty weights. Our parameter choices were made based on preliminary tuning experiments, but optimal parameters may vary across problem instances.

Scalability and Future Hardware: Current quantum processors have limited qubit counts and coherence times, restricting the problem sizes that can be effectively addressed. As quantum hardware continues to improve, the computational advantages of hybrid approaches like F-HQCOA are expected to grow. However, the results reported in this paper should be interpreted as indicative of the potential rather than as definitive evidence of quantum supremacy for fuzzy optimization.

The experimental details are shown in Table 11.

While the F-HQCOA framework demonstrates strong performance on standard benchmark problems and the industrial-inspired production planning example, a comprehensive industrial case study with proprietary data and complex operational constraints is beyond the current scope of this methodological paper. Such applications would require engagement with industrial partners and are the subject of ongoing research. However, the framework's flexibility in modeling TFNs and generating solution spectra makes it directly applicable to a wide range of industrial decision-making scenarios, including those with complex constraints and uncertain operating conditions. The main advantages of F-HQCOA are as follows.

Scalability: Ability to handle larger problem instances than classical fuzzy optimizers.

Solution Diversity: Generation of multiple solutions representing different risk-reward profiles.

Flexibility: Adaptability to various quantum computing platforms and fuzzy problem types.

However, several limitations should be noted [18], [19]. A key limitation of the current framework is that it does not explicitly formulate a multi-objective optimization problem incorporating both objective value and satisfaction degree. Instead, it relies on parametric variation of $\alpha$-levels. Future work should investigate formulations where satisfaction degree is directly integrated into the objective function, enabling true multi-objective optimization. Other limitations include the following aspects.

Hardware Constraints: Current quantum processors have limited qubit counts and coherence times, restricting problem sizes.

Noise Sensitivity: Quantum algorithms are susceptible to various errors that can affect solution quality.

Parameter Tuning: Performance depends on careful selection of penalty weights and algorithm parameters.

The F-HQCOA framework offers practical value across multiple industrial domains where uncertainty and computational complexity intersect, as illustrated by the following applications.

Production and Operations Management: In manufacturing environments, production planners frequently encounter imprecise parameters such as “approximately 100 units of raw material are available” or “the processing time is about 5 hours”. The F-HQCOA framework enables planners to systematically explore how different levels of constraint satisfaction ($\alpha$-levels) affect production targets. For instance, a conservative ($\alpha$ = 1.0) solution ensures feasibility with high certainty but may sacrifice production volume, while a more aggressive ($\alpha$ = 0.5) solution increases output but with lower feasibility confidence. The solution spectrum generated by F-HQCOA provides operations managers with actionable insights for risk-aware decision-making.

Supply Chain Design and Logistics: Supply chain networks face uncertainties in demand, transportation costs, and supplier reliability. The F-HQCOA framework can model these uncertainties using TFNs and generate solutions that balance cost minimization with feasibility satisfaction. For example, a logistics company could use F-HQCOA to determine optimal warehouse locations and inventory levels while accounting for fuzzy demand forecasts and fuzzy transportation costs. The ability to obtain a spectrum of solutions allows supply chain managers to select configurations that align with their organization's risk tolerance.

Energy Systems Optimization: Power system operators must make decisions under uncertain load forecasts and renewable energy generation. The F-HQCOA framework can address unit commitment problems, economic dispatch, and energy storage optimization with fuzzy parameters. The parametric analysis over $\alpha$-levels enables energy managers to evaluate trade-offs between cost reduction and reliability assurance.

Healthcare Resource Allocation: Hospital administrators face challenges in allocating limited resources (beds, staff, equipment) under uncertain patient demand and treatment durations. The F-HQCOA framework provides a decision-support tool that generates resource allocation plans with varying levels of feasibility satisfaction, enabling administrators to balance service quality with resource constraints.

The ability to obtain a spectrum of solutions rather than a single optimum allows decision-makers to select alternatives based on additional criteria not formally represented in the optimization model, such as managerial preferences, strategic priorities, or qualitative factors. Table 12 maps industrial application of the proposed method.

This paper presented a novel F-HQCOA for solving 0–1 ILP problems with fuzzy parameters. Our key contributions include:

A formal framework for transforming fuzzy ILP problems into families of crisp QUBO problems using $\alpha$-cuts;,The F-HQCOA algorithm that integrates fuzzy set theory with quantum optimization;,Extensive empirical evaluation demonstrating the effectiveness of our approach on standard benchmark problems;,Analysis showing quantum advantage for larger problem instances and higher feasibility requirements.

A potential extension of this work involves reformulating the problem as a bi-objective optimization model:

\[ \text{Maximize } \big(f(x), \mu(x)\big) \]

where, $f(x)$ represents the objective function value and $\mu(x)$ denotes the degree of satisfaction. Such a formulation would enable the generation of true Pareto-optimal solutions and provide a more rigorous treatment of trade-offs under uncertainty.

Several promising directions for future research emerged from this work:

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As quantum computing technology continues to advance, we anticipate that the Hybrid Quantum-Classical approaches like F-HQCOA would gradually take up a significant position in decision-making under uncertainty.