Implementing the proposed NSAHP framework in a real world case study in Tonekabon Electricity Distribution Company (TEDC), demonstrating its practical applicability and effectiveness [15], [32–34]. Comparative analysis: conducting a comprehensive comparison of NSAHP with conventional FAHP and Technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS) methods to evaluate performance and robustness under different conditions of data completeness and consistency [9], [12], [35], [36]. Management insights, providing practical guidance for decision makers in service and technical organizations, highlighting the benefits of adopting advanced MCDM techniques for employee performance evaluation [13], [37–40]. By addressing the limitations of traditional AHP and FAHP, the NSAHP framework provides a more reliable and comprehensive methodology for employee performance evaluation and advances decision making techniques in complex organizational environments [26]. This research fills a gap in the practical application of distributed Neutrosophic MCDM methods in technical service organizations. However, most of the quasi Rang Kutta schemes reported in the literature are limited to fixed step formulations without adaptive memory or dynamic weighting. This work fills this gap by developing a family of quasi Rang Kutta with adaptive memory (AMERKLP) that is able to achieve higher convergence rates with derivative free formulations.

A NS is defined as [16], [39]:

A = {(x, FA(x), IA(x), TA(x)) ∥ x ∈ X}, FA(x), IA(x), TA(x) ∈ [0,1], (1)

where TA(x) is truth membership and IA(x) is indeterminacy membership and FA(x) is falsity membership. Remark 1. If IA(x) = 0 and TA(x) + FA(x) ≤ 1, the NS reduces to an intuitionistic fuzzy set [17], [18].

A SVNN is defined as

a ̃ = (F, I, T), F, I, T ∈ [0,1], 0 ≤ F + I + T ≤ 3 (3)

For a ̃ = (Fa, Ia, Ta) and b ̃ = (Fb, Ib, Tb), Algebraic operations are defined consistently as

a ̃ ⊕ b ̃ = (Fa + Fb − FaFb, IaIb, TaTb). (4)

a ̃⨂ b ̃ = (FaFb, Ia + Ib − IaIb, Ta + Tb − TaTb). (5)

Theorem 1 (commutativity).

a ̃ ⊕ b ̃ = b ̃ ⊕ a ̃, a ̃ ⨂ b ̃ = b ̃ ⨂ a ̃. (6)

Remark 1. This operation will be applied later to pairwise comparison matrices in the NAHP framework. The SVNN representation allows for the management of correctness, incorrectness, and partial uncertainty in expert judgments, which is crucial for reliable evaluation of employee performance.

Theorem 2 (associativity).

a ̃ ⊕ (b ̃ ⊕ c ̃) = (a ̃ ⊕ b ̃) ⊕ c ̃.

a ̃ ⨂ (b ̃⨂ c ̃) = ( a ̃ ⨂ b ̃) ⨂ c ̃.

For criteria C = {Cn, … , C2, C1}, the periwise comparison matrix:

1 m12 ⋯ m1n M = [ ⋮ ⋱ ⋮ ], mij > 0, (9) mn1 mn2 ⋯ 1

and the weight vector ω satisfies:

Mω = λmaxω. (10)

Consistency Ratio (CR) is defined as λmax − n CR = . (11) n − 1

Remark 1, in Neutrosophic and fuzzy extensions (FAHP, NAHP), consistency checks are still necessary, but the Neutrosophic representation allows the model to tolerate minor inconsistency while stil producing relaible weights.

Pairwise comparisons are modeled as triangular fuzzy members:

m ̃ = (uij,mij, lij). (12)

With membership function:

x − uij , uij ≤ x ≤ mij, mij − uij μm ̃ ij(x) = lij − x (13) , mij ≤ x ≤ lij, lij − mij {0 otherwise.

Remark 3. FAHP takes into account the ambiguity in human judgments, but unlike NAHP, it does not model uncertainty directly, unlike NAHP.

IN NAHP, pairwise comparisions are Neutrosophic numbers:

M ̃ = [m ̃ ]n×n, m ̃ = (Fij,Iij, Tij). (14)

Inverse symmetry property:

m ̃ = (Fij−1, Iij, Tij). (15)

Definision 1 (Neutrosophic score). For alternative Ak:

S(Ak) = ∑ ωi(Fik − Tik). (16)

For incomplete or unreliable data, sparse optimization is applied:

ω ‖Mω − λmaxω‖1 + α‖ω‖1. (17)

Extended with additional constraints if needed:

ω ‖Mω − λmaxω‖1 + α‖ω‖1 + β ∑ ωi. (18)

Remark 4. Sparse MCDM ensures reliable weight extraction even with incomplete evaluations, which is a common situation in human resource datasets.

Dynamic uses the Neutrosophic fractional integral:

α 1 t μ−1 −λ(t−τ)γ ′ (0NDt )f(t) = ∫ (t − τ) e f (τ)dτ, α > 0. (19)

Integral and operator forms:

1 T μ−1 −λτγ MT = ∫ τ e dτ. (20)

1 t μ−1 −λ(t−τ)γ (Tv)(t) = u0 + ∫ (t − τ) e F(τ, v(τ))dτ. (21)

General equation:

u(t) = u0 + ∫ K(t − τ) G(τ, u(τ))dτ. (22)

Lemma 1 (boundednees). Let f(t) be continuous and bounded on [0, T], |f(t)| ≤ M. Then the Neutrosophic fractional integral

α 1 t μ−1 −λ(t−τ)γ ′ (0NDt )f(t) = ∫ (t − τ) e f (τ)dτ. (23)

Is convergent.

Theorem 2 (existence and uniqueness). Let G: [0, T] → ℝ × ℝ be Lipschitz in the second variable with constant L > 0. Then

u(t) = u0 + ∫ K(t − τ) G(τ, u(τ))dτ, (24)

where K(t − τ) = (t − τ)μ−1e−λ(t−τ) , has a unique solution on [0, T].

Corollary 1 (stability/ continuous dependece).

Let the assumptions of Theorem 2 hold. Suppose u(t) and v(t) are solutions of

u(t) = u0 + ∫ K(t − τ) G(τ, u(τ))dτ. (25)

v(t) = v0 + ∫ K(t − τ) G(τ, u(τ))dτ. (26)

With γ ≥ 0, λ ≥ 0, γ ≥ 0. Then

‖u − v‖∞,[0,T] ≤ CT|u0− v0|, (27)

where CT is finite under standard Lipschitz conditions.

L sup t Γ(L μ) ∫τt K(t−s)ds CT = 1 + t∈[0,T] ∫ K(t − τ) (e ) dτ < ∞. (28)

If

L sup t t∈[0,T] ∫ K(t − τ) dτ < 1. (29)

Then,

L sup t Γ(L μ) ∫τt K(t−s)ds CT = 1 + t∈[0,T] ∫ K(t − τ) (e ) dτ < ∞. (30)

|u0− v0| ‖u − v‖∞,[o,T] ≤ .

L sup t (31) 1 − t∈[0,T] ∫0 K(t − τ) dτ Γ(μ)

Remark 5. Small changes in initial conditions produce relatively small changes in Neutrosophic scores, ensuring stable assessment of employee performance under conditions of uncertainty [4], [17].

Remark 6. These formulas form the basis of the proposed Neutrosophic sparse AHP method with fractional dynamics. In the following sections, the model will be applied to real world employee performance data to demonstrate its robustness to uncertainty, missing information, and dynamic evolution.

We consider the fractional integral equation

1 t μ−1 −λ(t−τ)γ St(Ak) = S0(Ak) + ∫ (t − τ) e G(τ, S(τ))dτ, (39)

where s(τ) ≡ s(τ, Ak).

Assumptions:

I. Regularity of G.

The function G: [0, T] × R → R is continuous in τ and Lipschitz continuous in its second argument i.e.

|G(τ, x) − G(τ, y)| ≤ L|x − y|, ∀τ ∈ [0, T], x, y ∈ R, (40)

Where L > 0 is the Lipschiz constant.

II. Kernel properties Define the Kernel

(t − τ)μ−1e−λ(t−τ)γ K(t − τ) = . (41) Γ(μ)

For t ≥ τ, K(t − τ) is nonnegative and integrable over [0, T]. We introduce

sup

QT = τ∈[0,T] ∫ K(t − τ)d τ (42)

III. Contraction condition Let q = LQT < 1. This smallness requirement can be ensured either by shortening the time horizon T, reducing the effective Lipschitz constant L, or tuning the fractional parameters γ, μ, λ to decrease the Kernel mass QT.

Theorem 3 (existence and uniqueness). Under Assumptions (19)-(22), Eq. (21) admits a unique continuous solution S(∙) ∈ C([0, T]).

Proof: define the operator T: C([0, T]) → C([0, T]) by

(Tu) = S0 + ∫ K(t − τ)G(τ, u(τ))dτ. (43)

For any v, u ∈ C([0, T]), applying the Lipschize property of G yields.

‖(Tu − Tv) ‖∞ ≤ L‖u − v ‖∞ τ∈[0 sup ,T] ∫ K(t − τ)d τ = q‖u − v ‖∞. (44) Since q < 1, the operator T is a contraction on the complete metric space (C([0, T]), ‖∙‖∞). By the Banach fixed point theorem, T admits a unique fixed points S, which is precisely the unique Solution of (22).

Theorem 4 (stability with respect to initial data). Let S(∙) and S ̃(∙) be Solutions of (22) corresponding to initial conditions S0 and S ̃0. Then, for all t ∈ [0, T]

‖St − S0) ‖∞ ≤ |S0 − S ̃0|. (45) 1 − q

Subtracting the equations satisfied by S and S ̃, we obtain

S(t) − S ̃(t) = (S0 − S ̃0) + ∫ K(t − τ)[G(τ, s(τ)) − G(τ, s ̃(τ))]dτ. (46)

Taking the supremum norm and applying (3.9) gives

‖St − S0) ‖∞ ≤ |S0 − S ̃0| + q‖St − S0) ‖∞. (47)

Rearranging yields

(1 − q)‖S − S ̃) ‖ ≤ |S0 − S ̃0|, ∞ (48)

Which directly implies (3.14).

Corollary 2. (existence-type bound). By iterating (3.15) and applying a Gronwall-type argument, one may obtain

‖S − S ̃) ‖ ≤ |S0 − S ̃0|exp(L sup ∫ K(t − τ)d τ)|S0 − S ̃0|eLQT. (49)

∞ τ∈[0,T]

This bound is weaker than Eq. (45) when q is small, but it provides an alternative. Continuous dependence estimate that is useful in resolvent Kerner and series expansion argument. Remarks and implications for NS-AHP

TEDC, located in Mazandaran Province, Iran, is responsible for providing electricity services to a mix of urban and rural customers. The company employs engineers, field technicians, administrative staff, and customer service personnel. Since workforce performance directly affects system reliability, power outage recovery, and customer satisfaction, TEDC is a suitable domain to test a sophisticated evaluation method under uncertainty.

We selected m = 55 employees in three departments (operations, maintenance, customer service) for the pilot evaluation. We elicited expert judgments from p = 25 senior managers on n = 6 criteria as follows:

C₁: technical skill, C₂: safety compliance, C₃: response time, C₄: customer orientation C₅: initiative and innovation and C₆: reliability and presence.

To extract pairwise comparisons under uncertainty, we used linguistic scales augmented with Neutrosophic elements. For example, a manager might say “criterion 1 is more important among 2 criteria, but with uncertainty” and assign (T, I, F) = (0.7, 0.2, 0.1). Some comparisons were left blank when the manager was unsure, creating scatter. After collecting these judgments, we constructed a Neutrosophic sparse comparison matrix for each expert. We then aggregated them (e.g., via Neutrosophic incremental aggregation) to form a composite AM̃̃.

We implemented NSAHP with sparse optimization for weight extraction, which is solved via a convex solver (e.g. CVX in MATLAB). The hyperparameter α controlling the scatter was tuned by cross validation: we tested α at {0.001, 0.01, 0.05, 0.1}. The best balanced scatter of α and the fitting error. As baseline methods, we applied the following:

FAHP using triangular fuzzy numbers and normalization. TOPSIS (classical and fuzzy versions) on the values of the categorical or fuzzy criteria extracted from expert rating scales. For each method, employees were ranked and we compared the consistency, sensitivity, and robustness to missing values.

We conducted sensitivity tests by varying the dispersion regularizer α, the fractional parameters μ, λ, and γ, and the growth factor κ. We observed: For α < 0.005, the weight vector became too dense and lost its robustness; For α > 0.05, the excessive dispersion led to the neglect of the relevant criteria. Varying μ in [0.6,1.0] had a moderate effect on the dynamic ranking; the adjustments of λ and γ affected the long term convergence rate but did not change the top rankings for stable employees. If κ is larger (e. g. 0.05), the rapid growth exaggerates small initial differences NSAHP remains stable under moderate changes. These analyses show that the method is not overly sensitive to parameter tuning a desirable property for real world applications.

In Fig. 1, the weights obtained from NS-AHP are less volatile compared to FAHP and reduce the weight of less important criteria in a soft dispersion manner. This behavior provides more stability in the decision-making process and limits the effect of "dominant criteria dominance.

In Fig. 2, the weights obtained from NSAHP are less volatile compared to FAHP and reduce the weight of less important criteria in a soft dispersion manner. This behavior provides more stability in the decision making process and limits the effect of dominant criteria dominance. In Fig. 3, even with about 20% of the data removed, the NSAHP method maintains more than 80% stability of the top rankings; while FAHP and TOPSIS have significant losses. This indicates the high power of NSAHP in handling incomplete and uncertain data, which is very valuable for real world decision-making.

In Fig. 4, the NS-AHP method has the highest correlation with expert consensus (ρ ≈ 0.9, τ ≈ 0.8). This result shows that adding the dispersion term and using the neotrosophic structure has increased the model's compatibility with human understanding of the importance of criteria. In Fig. 5, the top employees (e.g., E₁ and E₂) have a steady upward trend and maintain their position over time. This indicates that the fractional model reflects the long term stability of performance better than static models (FAHP or TOPSIS) and is more suitable for evaluating continuous performance.