An analyst may optimise an alternative objective while judging its policy by a benchmark cost. We give a sufficient certificate that the resulting expected loss lies within a stated allowance. For finite-horizon additive rewards on a common feasible system, the Preference-Class Gap is a uniform distance between canonical representatives modulo common positive scale and component constants. The Pathwise Distortion Index is benchmark regret normalised by the sum of component ranges. A component-oscillation bound implies PDI≤min{1,2NPPG} , and a certified optimisation error enters additively. These certificates do not require the benchmark optimum; an inconclusive certificate calls for a separate audit. We distinguish net regret from coupling-dependent gross diagnostics. An inventory study includes a hand-derived certificate for a nearby objective and a fixed-cost-free surrogate with a proved base-stock recursion. In a synthetic inventory study, a declared benchmark loss allowance distinguishes certification before benchmark optimisation from adoption or rejection after a value audit. A surrogate without fixed ordering costs admits a base-stock recursion on the specified finite grid, reducing the work required to solve the alternative model.
We investigate the problem of finding a common zero of a finite family of maximal monotone operators in a real Hilbert space. To address this problem, we introduce two new two-step inertial proximal point algorithms and establish their weak convergence. We also derive a best-iterate rate for the squared successive increments. Furthermore, we apply the main results to signal recovery and image restoration problems and compare the proposed schemes with one-step, non-inertial, cyclic, and product-space Douglas–Rachford methods.
This study examines the impacts of black swan events on markets using Heaviside geometric Brownian motion with implications for economic and trade disruption management. We prove existence and uniqueness of solution paths to identify critical tipping points that may signal possible market instability points. Numerical simulations presented in this work suggest that markets are sensitive to changes in the tail index of black swans and are non-linear with asset prices, leading to market shocks. Findings could inform contingency plans and resource optimization procedures for decision-making in managing economic disruptions in markets. Further research could explore event dynamics and predictive capabilities to enhance an understanding of black swan events in markets.
Multi-objective optimization requires balancing conflicting objectives while maintaining convergence and diversity. This paper proposes NDBCO, a multi-objective extension of Bézier Curve-based Optimization (BCO) that retains the mathematical forms of the original linear, quadratic, and cubic Bézier search operators and their inherited parameter schedules. NDBCO introduces four structural changes: NSGA-II environmental selection, rank-1 guide selection through a crowding-distance tournament, a bounded accumulation archive used only for retention and reporting, and reuse of rank/crowding-distance state from environmental selection. All algorithms are evaluated on a common terminal-population basis. The study covers 18 ZDT, DTLZ, and UF benchmark problems against six established competitors and 12 constrained engineering design problems against seven competitors, including Two-Arch2. Six indicators (IGD, IGD ^+ , GD, HV, SP, and NR) are analysed using Friedman ranking with Holm-corrected post-hoc comparisons. On the benchmark suite, NDBCO ranks third overall, with no statistically significant difference from NSGA-II or NSGA-III on IGD, IGD ^+ , HV, or SP. On the engineering suite, NDBCO attains the best overall Friedman rank, with no significant difference from NSGA-II, NSGA-III, or Two-Arch2. Cumulative ablation identifies environmental selection as the dominant measurable incremental contributor to terminal quality; the accumulation archive is retention-only, while rank/crowding-distance reuse changes some trajectories without a significant aggregate quality effect. Archive-capacity sensitivity shows identical terminal populations for A_max∈{N,2N,4N} under matched seeds. These results support NDBCO as a competitive multi-objective optimizer while clarifying the roles of its components.
Evaluating Sustainable Development Goals (SDGs) requires integrating heterogeneous expert judgments across qualitative sustainability dimensions—a task for which traditional numerical frameworks cause information loss and representational bias. We propose a heterogeneous fuzzy linguistic model (HFLM) based on the 2-tuple linguistic representation model and the Computing with Words (CWW) paradigm and present a decision support tool implementing it end-to-end. The framework supports three aggregation levels: expert assessments are consolidated per criterion via 2-tuple weighted average or ordered weighted averaging (OWA) operators; criterion results are combined into a global evaluation via a Choquet integral governed by a 2-additive capacity, capturing non-additive SDG interactions. The capacity is built from fuzzy densities obtained through a structured Deck-of-Cards (Simos–Roy–Figueira) protocol together with pairwise interaction terms from targeted specialist judgement. The framework is first verified computationally against analytical baselines, then subjected to a controlled simulation experiment under a factorial design, and finally illustrated through a fully synthetic case study—a simulated 50-expert multidisciplinary panel evaluating Internet of Things (IoT)-enabled healthcare for ageing populations (SDG 3)—designed to test its mathematical robustness, heterogeneity handling, and discriminative capacity under controlled conditions. The results show that the Choquet aggregation identifies Rural Connectivity as the structural bottleneck of the synthetic scenario and quantifies its amplification, displacing the global score below every weighted-average baseline without requiring the blanket pessimism of an OWA operator, while the tool is designed to make this discriminative capacity accessible to practitioners without specialist knowledge of the formalism.
