Asian Research Journal of Mathematics

A Fuzzy-Hungarian Framework for Labour Allocation under Uncertain Costs in Cocoa Processing

2026-09-15T12:36:39+00:00

Efficient allocation of labour is essential for minimising operational costs and improving productivity in cocoa processing. However, conventional assignment models, such as the Hungarian Assignment Method, assume that assignment costs are known precisely; this assumption may not adequately reflect uncertainty associated with worker performance, task difficulty, fatigue, and operational conditions. This study applies fuzzy logic and the Hungarian Assignment Method to determine the most suitable allocation of workers to cocoa-processing activities at Stevefrana Farm. The study considers six processing stages: conditioning, breaking, scalping, middling, purifying, and packaging. Labour costs were initially represented as crisp values obtained from farm records and supervisors' estimates and were subsequently modelled using triangular fuzzy numbers with a symmetric uncertainty spread of 10%. The centroid method was used to defuzzify the fuzzy costs into crisp values, after which the Hungarian Assignment Method was applied to determine the optimal worker-task allocation. In addition, the robustness of the optimal assignment was evaluated using the Robustness Index (RI), Coefficient of Variation of Fuzziness (CVF), and Robustness Degree (RD), while sensitivity analysis was conducted at uncertainty levels of 5%, 10%, 15%, and 20%. The results indicate that the optimal assignment is Worker 1 to Conditioning, Worker 2 to Middling, Worker 3 to Packaging, Worker 4 to Scalping, Worker 5 to Purifying, and Worker 6 to Breaking. This allocation yields a most likely total labour cost of GH¢555 per day, with a fuzzy cost range of GH¢499.50 to GH¢610.50. The robustness analysis produced an RI of 0.20, indicating moderate robustness; a CVF of 10%, indicating uniform relative uncertainty; and an RD of 90%, indicating high reliability. The sensitivity analysis further showed that, although robustness decreases as uncertainty increases, the optimal worker-task assignment remains unchanged across the 5%-20% uncertainty levels. The study demonstrates that integrating fuzzy logic with the Hungarian Assignment Method provides a practical decision-support framework for incorporating uncertainty into labour allocation while maintaining cost efficiency. The proposed model can assist cocoa-processing managers in improving worker-task matching, budgeting for labour-cost variability, and identifying processing stages requiring greater management attention.

Graphoidal Covering Numbers of Strong and Enhanced Power Graphs of Cyclic and Dihedral Groups

2026-09-15T12:42:02+00:00

This study investigates graphoidal coverings of power graphs, enhanced power graphs, and strong power graphs associated with selected finite cyclic and dihedral groups. The analysis focuses on determining graphoidal covering numbers and on clarifying how the underlying group structure and the relevant adjacency relation affect these parameters in a systematic manner. For cyclic groups, the graphoidal covering number of the strong power graph is obtained for the cases n = 2, n = 3, n = 4, prime n ≥ 5, and composite n ≥ 6. The study also characterises the cyclic groups for which the power graph and the strong power graph are isomorphic, showing that, for n > 2, this occurs precisely when n = 2p, with p an odd prime. In addition, the graphoidal covering number of the enhanced power graph of Zn is determined for n ≥ 4. For dihedral groups, the structures of the power, enhanced power, and strong power graphs are examined, and explicit graphoidal coveringnumbers are derived for D2p, where p is an odd prime. The resulting comparison establishes a strict ordering among the three covering numbers in this case. Overall, the results show that variations in group structure and graph adjacency produce distinct graphoidal covering behaviour across the graph classes considered.

Convex-Fair Domination under Subdivision, Duplication, Total, and Switching Operations on Paths and Cycles

2026-09-16T10:14:17+00:00

This study examines the convex-fair domination parameter under four unary graph operations and structural transformations applied to path graphs P_n and cycle graphs C_n. The operations considered are subdivision, vertex duplication, total-graph construction, and vertex switching. Convex-fair domination requires a dominating set to be geodesically convex while also ensuring that every vertex outside the set has the same number of neighbours within it. Using the baseline convexfair domination values stated for paths and cycles, the manuscript derives operation-specific expressions for the resulting transformed graphs. For paths, the analysis treats the subdivision graph S(P_n), duplication graph D(P_n), total graph T (P_n), and the graph obtained by switching a pendant vertex. Corresponding results are developed for cycles through S(C_n), D(C_n), T (C_n), and single-vertex switching. The proofs use structural isomorphisms, explicit candidate dominating sets, geodesicclosure arguments, and comparisons of neighbourhood intersections for vertices outside the selected sets. Attention is also given to how transformed graph order and the placement of boundary and internal vertices affect the proposed minimum sets. The analysis therefore describes how subdivision, duplication, total-graph construction, and switching alter the interaction between domination, convexity, and fairness in linear and cyclic graph structures. These results provide a focused basis for further study of convex-fair domination under other graph operations and in broader graph classes.

Local Stability Analysis of an Infection-Age Mathematical Model for COVID-19 Disease Dynamics

2026-09-21T12:46:46+00:00

The infectiousness of individuals with an acute respiratory infection can vary with time since entry into the infectious stage, motivating epidemic models that retain infection-age structure rather than treating all infectious individuals as epidemiologically identical. This study formulates and analyses an infection-age-structured model for COVID-19 disease dynamics with susceptible, vaccinated, exposed, infection-age-structured infectious, and recovered populations. New infectious individuals enter the structured infectious class only after progression from the exposed class, so that the boundary condition is i(0,t) = gE(t). Transmission, recovery, and disease-induced mortality may depend on infection age. An explicit basic reproduction number is obtained as;

where Λ = (1-b)p, α = θ+μ, y = μ+g, and π(a) is the probability of remaining infectious to infection age a . The disease-free equilibrium is locally asymptotically stable when R₀<1 and unstable when R₀>1; a positive endemic equilibrium exists when R₀>1. For the illustrative parameter set used in the numerical analysis, the baseline value is R₀=0.4694. Multiplying the transmission kernel by three gives R₀=1.4081 and produces persistent infection in the corresponding numerical trajectory. The normalised functional sensitivity density, \(β(a)π(a)/∫_0^Tβ (s)π(s) ds,\) attains its maximum at approximately 2.49 days for the specified kernels, indicating that early infection ages make the largest model-specific contribution to . This peak is a property of the assumed mathematical functions and should not be interpreted as a direct empirical estimate of the biological peak of SARS-CoV-2 infectiousness. The results illustrate how infection-age structure can identify temporal contributions to epidemic transmission while retaining a transparent threshold criterion for local disease invasion.

A Linear Hybrid Multi-Step Approach for Efficient Numerical Integration of Third-Order Ordinary Differential Equations (IVPs)

2026-09-21T12:58:50+00:00

This study develops a fifth-order fully hybrid block method for the direct numerical solution of third-order initial value problems. A continuous approximation is constructed using interpolation and collocation, and three intra-step hybrid points are incorporated into the block formulation. The resulting discrete schemes approximate the solution together with its first and second derivatives without reducing the governing third-order equation to an equivalent system of first-order equations. The principal numerical properties of the method are examined through local truncation-error analysis, consistency, zerostability, convergence, and absolute-stability considerations. The local truncation-error analysis establishes a uniform order of five for the developed block method. The corresponding difference formulation satisfies the root condition required for zero-stability; together with consistency, this establishes convergence. The stability behaviour is further investigated through a standard third-order test equation. Four numerical initial value problems are then used to assess computational performance at a fixed step size. Across the reported test problems, the proposed method produces absolute errors smaller than those of the comparator methods presented in the manuscript, including comparisons with methods of equal or higher formal order. These results indicate that the proposed fully hybrid formulation provides accurate numerical approximations for the class of third-order initial value problems considered in this study.

Mathematical Modelling and Optimal Control of Malaria Transmission in Children under Five Years: A Scenario-based Framework for Bungoma County, Kenya

2026-09-22T13:25:05+00:00

Malaria remains an important cause of morbidity in young children in western Kenya, where intervention effectiveness depends not only on nominal coverage but also on biological efficacy, adherence, vector behaviour, insecticide susceptibility, and local transmission intensity. This study develops a deterministic human–mosquito model for children under five years of age and formulates a five-control optimal-control problem representing effective vaccine protection, long-lasting insecticidal-net protection, treatment effort, larval-source management, and adult-vector removal. The revised framework explicitly includes maturation out of the under-five population and assigns vector-control actions to biologically appropriate processes. The model is analysed for positivity, boundedness, the effective reproduction number, local and global stability of the disease-free equilibrium, and normalised control sensitivity. A weighted Lyapunov function establishes global asymptotic stability in the feasible region when the effective reproduction number is below unity. A 100-day forward–backward sweep simulation is then used as a structural scenario analysis rather than as an empirical calibration to Bungoma surveillance data. Under the no-control numerical scenario, the basic reproduction number is 1.551. A reference control profile reduces the effective reproduction number to 0.292. In the optimal-control simulation, cumulative modelled incident infections decline by 73.2% relative to the no-control trajectory while using substantially less integrated control effort than sustained high-intensity fixed-control scenarios. The results illustrate how intervention mechanisms, control effort, and terminal-time assumptions shape mathematical conclusions. They should not be interpreted as estimates of current Bungoma County incidence, programme coverage, or economic cost-effectiveness. The framework provides a transparent basis for future calibration with validated local surveillance, entomological, intervention-coverage, and cost data.