Bi-Objective Redundancy Allocation Problem

• Bi-Objective RAP is a framework that optimizes the trade-off between system reliability and cost under explicit resource constraints using advanced uncertainty modeling.
• It incorporates interval Type-2 fuzzy sets to robustly handle deep uncertainties in component performance and cost parameters.
• Advanced metaheuristic methods like PSO and GA variants efficiently approximate Pareto-optimal fronts for complex, repairable systems.

The Bi-Objective Redundancy Allocation Problem (RAP) addresses the joint design of system architectures to simultaneously optimize two conflicting goals, typically maximizing system reliability or availability and minimizing total cost, under explicit resource constraints and complex component-level uncertainties. RAP arises in the design of multi-stage or multi-component systems that demand high operational reliability or availability—critical in transportation, telecommunications, process plants, and repairable infrastructure. Formulations span both active and standby redundancy, k-out-of-n and arbitrary topologies, incorporate repair and dynamic maintenance, and increasingly utilize advanced uncertainty modeling—especially interval Type-2 Fuzzy Sets (IT2 FSs). RAP is non-convex, non-linear, and mixed-integer by nature; state-of-the-art solution methodologies predominantly rely on metaheuristic optimization, multi-objective techniques, and stochastic modeling frameworks.

1. Mathematical Formulation and System Structures

Bi-objective RAP for multi-stage systems is classically defined by selecting, at each stage $i$, the number $n_i$ of redundant components and their reliability parameters $r_{ij}$, to optimize:

• System reliability $R_s$ (or availability $A_s$)
• System cost $C_s$

subject to constraints on aggregated weight, volume, and bounds on $n_i$, $r_{ij}$ (Ashraf et al., 2020, Kundu, 2020, Yeh, 2020, Oszczypała et al., 20 Dec 2025). The system topology is fundamental; common models include series-parallel, parallel-series, k-out-of-n, and hybrid networks. Reliability and cost aggregation depend on the configuration:

• Series-Parallel: $$R_s^{SP} = \prod_{i=1}^m \left[1-\prod_{j=1}^{n_i}(1 - r_{ij})\right]$$
• Parallel-Series: $$R_s^{PS} = 1 - \prod_{i=1}^m \left[\prod_{j=1}^{n_i}(1 - r_{ij})\right]$$

Cost functions incorporate both intrinsic component parameters (e.g., scaling factors $a_i$, $B_i$) and system-level penalties (e.g., interconnection cost via $e^{n_i/4}$). Repairable system formulations add Markov modeling for dynamic failure/recovery cycles (see §4) and introduce sequential decision variables for maintenance.

2. Uncertainty Modeling: Interval Type-2 Fuzzy Sets

Precision computation of component reliability is frequently impossible due to deep uncertainties: small-sample statistical variability, environmental effects, manufacturing tolerances, and divergent expert judgments. RAP thus benefits from modeling parameters (especially reliability and cost) as Interval Type-2 Fuzzy Numbers (IT2 FNs), which capture the “footprint of uncertainty” (FOU) bounded by upper (UMF) and lower (LMF) membership functions (Ashraf et al., 2020, Kundu, 2020). A typical triangular IT2 FN for $R_i$ is parameterized so that $$\widetilde{R}_i = \{(x, \mu) \;|\; x \in [0, 1], \; \mu \in [\underline{\mu}(x), \overline{\mu}(x)]\}$$.

Fuzzy aggregation leverages the extension principle: series-topology reliability is the intersection of fuzzy reliabilities, with minimum UMF/LMF at each $x$, while cost is unified via maximum UMF/LMF over subsystems. The Enhanced Karnik–Mendel (EKM) method is applied for type-reduction, yielding centroid intervals, which are then defuzzified to representative crisp values for optimization.

3. Multi-Objective Optimization Techniques

Resolving the bi-objective trade-off in RAP requires tracing Pareto-optimal fronts (POF) subject to the imposed constraints and uncertainties. Traditional approaches include:

• Weighted-sum scalarization: $$F(\mathbf{x}) = w_1(1 - R_s^*) + w_2\, C_s^*/C^{\max}$$
• Global Lp-norm criterion, desirability functions, fuzzy-programming (Zimmermann’s max–min), and interactive classification (NIMBUS) (Kundu, 2020).

Metaheuristics, especially Particle Swarm Optimization (PSO), Genetic Algorithms (GA), Multi-Objective PSO (MOPSO), and recently, large-scale benchmarking of 65 algorithms—including NSGA-II, NNIA, CMOPSO, CMODEFTR, and DSPCMDE—are the standard solution vehicles (Ashraf et al., 2020, Yeh, 2020, Oszczypała et al., 20 Dec 2025). These techniques approximate the nondominated front efficiently for highly nonlinear, nonconvex, and high-dimensional RAP instances. Scaled Binomial Initialization (SBI) notably accelerates convergence and improves hypervolume for binary-encoded metaheuristics in large repairable-system benchmarks.

Table: Sample Pareto Points (Series-Parallel RAP under IT2 Fuzzy, PSO solution) (Ashraf et al., 2020)

weights	$R_s$	$C_s$
(1,1)	0.8676	437.08
(1,0.5)	0.9111	484.01
(0.8,0.2)	0.8730	457.86
(0.2,0.8)	0.6854	544.59
(0.5,1.0)	0.9166	485.97

4. Modeling, Repairability, and Dynamic Maintenance

Classical RAP formulations are static and do not incorporate repair dynamics. Advanced models utilize Continuous-Time Markov Decision Processes (CTMDPs) to capture the stochastic evolution of repairable systems and optimize not only the redundancy design (first-stage, binary install decisions $x_{ij}$) but also the operational maintenance policy (second-stage, action selection per system state $s$) (Fairley et al., 17 Apr 2025, Oszczypała et al., 20 Dec 2025). Objective functions are long-run averages:

• Operating cost rate $g^o(\mu)$
• Failure rate $g^f(\mu)$

The two-stage “Bi-Objective Integrated Design and Dynamic Maintenance Problem” (BO-IDDMP) formalizes the linkage between initial architecture and subsequent repair actions. Solution methods span exact Mixed Integer Linear Programs (MILP) for small systems and the bespoke Approximate Pareto Population (APP) heuristic for practical scalability. Results show that integrating dynamic maintenance yields strictly richer Pareto fronts and enables more precise tuning of reliability vs. cost trade-offs.

5. Benchmarking, Redundancy Strategies, and System Complexity

Recent large-sample studies benchmark multi-objective metaheuristics for RAP in repairable, k-out-of-n systems with selectable redundancy strategies: cold, warm, hot, and mixed standby (Oszczypała et al., 20 Dec 2025). CTMC modeling is essential for quantifying subsystem availability under different strategies and repair kinetics. Core findings include:

• Hot standby and mixed redundancy consistently dominate Pareto fronts; cold and warm standby are rarely optimal.
• Under strict weight constraints, hot standby is preferred; allowing more spares shifts the front toward mixed strategies.
• System topology (series, parallel, bridge, hybrid) alters the shape of trade-off curves, but the strategic trend persists.

Solver effectiveness depends on computational budget and system size: SBI-initialized genetic and swarm-based algorithms (e.g., NSGAII-ARSBX-SBI, CMOPSO-SBI) excel at different evaluation budgets. Complex systems (dimension ≥20) require much larger computational budgets (≥10⁵ evaluations) for adequate Pareto front coverage.

6. Statistical Performance Analysis and Comparative Insights

Algorithmic comparisons utilize statistical metrics: generational distance (GD) to simulated fronts, hypervolume (HV) indicators, spacing (SP) for solution diversity, and multivariate analysis of variance (M-ANOVA). Key trends across studies (Ashraf et al., 2020, Yeh, 2020):

• PSO-based approaches yield Pareto fronts with superior adherence to expert weight preferences, lower solution variance, and smaller GD than GAs.
• Simplified Swarm Optimization with adaptive update mechanisms and compulsory pBest replacement produces uniformly-spaced fronts and outperforms NSGA-II in diversity (SP).
• Heuristic methods (APP) in CTMDP-based RAPs recover virtually all supported Pareto points with orders-of-magnitude faster runtime than exact solvers.

7. Guidelines and Practical Recommendations

RAP modelers are advised to:

• Adopt interval type-2 fuzzy modeling for deep reliability uncertainty.
• Evaluate sensitivity to defuzzification method (KM, UB, N-T, geometric centroid).
• Employ several multi-objective solution techniques to expose the full spectrum of feasible trade-offs.
• Use CTMC/CTMDP frameworks when repairability or dynamic maintenance is integral.
• Scale optimization budgets to system size and complexity; leverage advanced initialization (SBI) in metaheuristics.
• Visualize and analyze Pareto fronts to guide design choices.

A plausible implication is that further integration of uncertainty quantification, dynamic decision models, and advanced multi-objective metaheuristics will continue to expand the design flexibility and operational performance of complex engineered systems subject to both economic and reliability constraints.