A Neutrosophic set is characterized independently by a truth-membership function α(x), an indeterminacy-membership function β(x), and a falsity-membership function γ(x), and each of the functions is defined from X→[0,1].

I. Single valued NSs [1] Let X be a universe of discourse. A Single Valued Neutrosophic Set (SVNS) Bover X is an object having the form:

B = {〈y, TB(y), IB(y)), FB(y)〉: yϵX}, where TB ∶ X → [0,1], IB ∶ X → [0,1], and FB ∶ X → [0,1] with the condition

0 ≤ TB(y) + IB(y)) + FB(y) ≤ 3, for all y ∈ X.

The number TB(y), IB(y) and FB(y) denote the degree of truth - membership, indeterminacy - membership, and falsity - membership of y to X, respectively.

II. Single valued NS [2]

An SVNS is denoted and defined as A ̃ = {x, T (x), I (x), F (x)/xϵX}, where for each generic point x in X,

N A ̃ A ̃ A ̃

N N N

TA ̃ (x) called the truth membership function, IA ̃ (x) called the indeterminacy membership function and FA ̃ (x) called the falsity membership function in [0,1] and 0 ≤ TA ̃ (x) + IA ̃ (x) + FA ̃ (x) ≤ 3. For

N N N N

continuous SNVS A ̃ = ∫ 〈T (x), I (x), F (x)〉/ x , x ϵX.

N A ̃ A ̃ A ̃ i i AN N N N

III. Single valued NS An SVNS A on a universal set X is defined as a ̃ = 〈Ta ̃, Ia ̃, Fa ̃〉A = {〈TA(x), IA(x), FA(x)〉 ∶ x ∈ X}, where TA, IA, FA ∶ X → [0,1], represents the degree of membership, degree of indeterminacy, and degree of non-membership, respectively, of the element x ∈ X, such that 0 ≤ sup TA(x) + sup IA(x) + sup FA(x) ≤ 3.

IV

Let X be a nonempty set. Then an interval – valued Neutrosophic set [IVNS] A ̃ of X is defined as [2]:

IV L U L U L U

A ̃ = {〈x; [T , T ] , [I , I ] , [F , F ]〉 : xϵX},

N A ̃ A ̃ A ̃ A ̃ A ̃ A ̃

N N N N N N

where

L U L U L U L U

[T̃ , T̃ ] , [I ̃ , I ̃ ] and [F ̃ , F ̃ ] ⊂ [0,1] for all xϵX T̃ = inf(T̃ ), T̃

AN AN AN AN AN AN AN AN AN

= sup(TA ̃ ); I ̃ = inf(IA ̃ ),

N AN N U L U

I ̃ = sup(IA ̃ ); and F ̃ = inf(FA ̃ ), F ̃ = sup(FA ̃ ).

AN N AN N AN N

Let A and B be two simplified NSs, then the operations of the NSs are [1]:

A + B = {〈X, TA(x) + TB(y) − TA(x). TB(y), IA(x) + IB(y) − IA(x). IB(y), FA(x) + FB(y) − FA(x). FB(y)| x, y ∈ X〉}, A × B = {〈X, TA(x). TB(y), IA(x). IB(y) FA(x). FB(y)|x, y ∈ X〉},

( ) λ ( ) λ ( ) λ x x x

λ. A = {〈X, 1 − (1 − TA ) , (1 − TA ) , (1 − TA ) |x ∈ X〉}, λ > 0,

λ λ λ λ

A = {〈x, TA (x), IA (x), FA (x)|x ∈ X〉},λ > 0.

Let x be a generic element of a non-empty set. A Neutrosophic number A ̃ in X is defined as A ̃ = {x, 〈T̃ (x), I ̃ (x), F ̃ (x)〉/ xϵX}, for all T̃ (x), I ̃ (x) and F ̃ (x) belongs to ]0 , 1 [ where T̃ : X →

AN AN AN AN AN AN AN − + − + − +

]0 , 1 [I, ̃ : X → ]0 , 1 [ and F ̃ : X → ]0 , 1 [ are functions of truth – membership, indeterminacy

AN AN

membership, and falsity, membership in A ̃ respectively with [2]:

0 ≤ T̃ (x) + I ̃ (x) + F ̃ (x) ≤ 3 .

AN AN AN

A Single Valued Neutrosophic Number (SVNN), in the set of real numbers R with truth – membership function Ta ̃, indeterminacy-membership function Ia ̃ and falsity-membership function Fa ̃, is defined as

fa ̃(x), if a1 ≤ x < b1, 1, if b1 ≤ x < c1, Ta ̃(x) = { ga ̃(x), if c1 ≤ x < d1, 0, otherwise, la ̃(x), if a2 ≤ x < b2, 1, if b2 ≤ x < c2, Ia ̃(x) = { ma ̃(x), if c2 ≤ x < d2, 0, otherwise,

and

ha ̃(x), if a3 ≤ x < b3, 1, if b3 ≤ x < c3, Ta ̃(x) = { ka ̃(x), if c3 ≤ x < d3 0, otherwise.

Respectively, where 0 ≤ Ta ̃ + Ia ̃ + Fa ̃ ≤ 3 and ai, bi, ci, diℝ.

ai ≤ bi ≤ ci ≤ di where i = 1,2,3 and the function fa ̃, , ga ̃, la ̃, ma ̃, ha ̃, ka ̃: ℝ → [0,1]. The function fa ̃, ma ̃, ka ̃ are non-decreasing continuous functions and ga ̃, la ̃, ha ̃ are non –increasing continuous functions. SVNN is also denoted by a ̃ = 〈(a1, b1, c1, d1), (a2, b2, c2, d2), (a3, b3, c3, d3)〉.

̅ l m n p q r

A Triangular Neutrosophic Number (TNN) is denoted by V = 〈(p , p , p ), (α , α , α )〉 whose three membership functions for the truth, indeterminacy, and falsity of x can be defined as follows [3]:

(x − p )

p l m

α , p ≤ x < p , (p − p )

p p l

α (x) = α , x = p , (p − x)

p m n

α , p ≤ x < p , ( n m) p − p {0, otherwise, (p − x)

q l m

α , p ≤ x < p , ( m n) p − p q α , x = p , α (x) = (x − p )

q m n

α , p ≤ x < p (p − p ) { 1, otherwise,

(p − x)

r l m

α , p ≤ x < p , ( m n) p − p r α , x = p , α (x) = (x − p )

r m n

α , p ≤ x < p , (p − p ) { 1, otherwise,

p q r l

where 0 ≤ α (x) + α (x) + α (x) ≤ 3, x ∈ R. Accordingly, when p ≥ 0, R is called a nonnegative TNN. Similarly, when p < 0, R becomes a negative TNN.

A triangular single- valued Neutrosophic number (S ̃) is defined as (S ̃) = 〈(m , m , m ; n , n , n ; p , p , p )〉.

1 2 3 1 2 3 1 2 3

Here, πS ̃: R → [0,1] is the truth membership function, θS ̃ : R → [0,1] is the hesitation membership function, and the falsity membership function is mathematically and graphically (see Fig. 1) defined as follows [4]: x − m1 , for m1 ≤ x < m1, m2 − m1 1 , for x = m2, π̃(x) = S m − x , for m2 < x ≤ m3, m3 − m2 { 0, otherwise, x − m1 , for m1 ≤ x < m1, m2 − m1 1, for x = m2, π̃(x) = S m − x , for m2 < x ≤ m3, m3 − m2 { 0, otherwise, x − m1 , for m1 ≤ x < m1, m2 − m1 1, for x = m2, π̃(x) = S m − x , for m2 < x ≤ m3, m3 − m2 { 0, otherwise.

A single-valued trapezoidal Neutrosophic number a ̃ = 〈(a1, b1, c1, d1); wa ̃, ua ̃, ya ̃〉 is a special Neutrosophic set on the real number set R, whose truth – membership, indeterminacy–membership, and falsity–membership are given as follows [5]:

(x − a1)wa ̃ , (a1 ≤ x < b1), b1 − a1 wa ̃, (b1 ≤ x ≤ c1), μa ̃(x) = (d1 − x)wa ̃ , (c1 < x ≤ d1), d1 − c1 { 0, otherwise,

and

(b1 − x + ya ̃ (x − a1))/(b1 − a1), (a1 ≤ x < b1), ya ̃, (b1 ≤ x ≤ c1), λ (x) = x − c1 + ya ̃(d1 − x) , (c1 < x ≤ d1), (d1 − c1) {1, otherwise.

respectively.

A Single-Valued Pentagonal Neutrosophic Number (̃ M) is defined as, (Ss ̃ ) =

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3

〈[(s , t , u , v , w ); μ], [(s , t , u , v , w ); θ], [(s , t , u , v , w ); η]〉, where μ, θ, ηϵ[0,1].

The truthness function (μss ̃): ℝ → [0, μ], the indeterminacy function (θss ̃): ℝ → [θ, 1], and the falsity function (ηss ̃): ℝ → [η, 1] are given as [6]:

μ̃(x), s ≤ x < t ,

ssl1

μ̃(x), t ≤ x < u ,

ssl2

μ, x = u , μss ̃ = μsr ̃2(x), u ≤ x < v , μss ̃ r (x), v ≤ x < w , {0, otherwise, θ̃(x), s ≤ x < t ,

ssl1

θ̃(x), t ≤ x < u ,

ssl2

θ = θ, x = u ,

ss ̃

θsr ̃2(x), u ≤ x < v , θss ̃ r (x), v ≤ x < w , { 1, otherwise, η̃(x), s ≤ x < t ,

ssl1

η̃(x), t ≤ x < u ,

ssl2

η, x = u , ηss ̃ = ηsr ̃2(x), u ≤ x < v ,

3 3,

ηss ̃ r (x), v ≤ x < w {1, otherwise.

A single-valued Neutrosophic Number defined on the set of real numbers ℝ is said to be a Generalized Single Valued Neutrosophic Number (GSVNN) Ga ̃ = 〈TG , IG , FG ; ω, ρ, δ〉, with truth-membership function TG (x),

a ̃ a ̃ a ̃ a ̃

indeterminacy-membership function IG (x) and falsity-membership function FG (x) has the following

a ̃ a ̃

Characteristics:

I. TG , IG , FG ℝ → [0,1].

a ̃ a ̃ a ̃

II. TG = 0, IG = 1, FG = 1 for all x ∈ (−∞, ai] ∪ [di, ∞).

a ̃ a ̃ a ̃

III. TG (x) is strictly increasing on [a1, b1] and TG (x) is strictly decreasing on [c1, d1].

a ̃ a ̃

IG (x) is strictly decreasing on [a2, b2] and TG (x) is strictly decreasing on [c2, d2].

a ̃ a ̃

FG (x) is strictly decreasing on [a3, b3] and TG (x) is strictly increasing on [c3, d3].

a ̃ a ̃

IV. TG (x) = ω for all xϵ[b1, c1] where 0 < ω ≤ 1. IG (x) = ρ for all xϵ[b2, c2] where 0 ≤ ρ < 1. FG (x) = δ

a ̃ a ̃ a ̃

for all xϵ[b2, c2] where 0 ≤ δ < 1.

A Generalized Parabolic Single-Valued Neutrosophic Number (GPSVNN), A ̃ = 〈(T ; ω), (I ; ρ), (F ; δ) 〉,

A ̃ A ̃ A ̃

is a Neutrosophic set on the real number ℝ with truth membership function TA ̃, indeterminacy-membership function IA ̃, and falsity-membership function FA ̃, is defined as

x − a 2 ω ( ) , xϵ[a1, b1), b1 − a1 ω, xϵ[b1, c1), T̃(x) = d1 − x ω ( ) , xϵ[c1, d1), d1 − c1 { 0, otherwise, x − a 2 1 − ( ) (1 − ρ), xϵ[a2, b2), b2 − a2 ρ, xϵ[b2, c2), Ĩ(x) = d2 − x 1 − ( ) (1 − ρ), xϵ[c2, d2), d2 − c2 {1, otherwise, x − a 2 1 − ( ) (1 − δ), xϵ[a3, b3), b3 − a3 δ, xϵ[b3, c3), F̃(x) = d1 − x 1 − ( ) , xϵ[c3, d3), d1 − c1 {1, otherwise,

where, 0 ≤ TA ̃ + IA ̃ + FA ̃ ≤ 3, 0 < ω ≤ 1,0 ≤ ρ < 1, 0 ≤ δ < 1 and ai, bi, ci, di ∈ ℝ, ai ≤ bi ≤ ci ≤ di where i = 1,2,3.

A generalized TNN (N ̃) = 〈(m , m , m ; μ), (n , n , n , n ; ν), (p , p , p , p ; ζ)〉 where μ, ν, ζ ∈ [0,1]. Here,

1 2 3 1 2 3 4 1 2 3 4

πN ̃ : R → [0, μ] is the truth membership function, θN ̃ : R → [ ν, 1] is the hesitation membership function, and the falsity membership function is ηN ̃: R → [ζ, 1], where the membership functions are mathematically defined as follows [7]:

(x − m1) μ if m1 ≤ x < m2, (m2 − m1) μ if x = m2, πN ̃ (x) (m3 − x) μ if m2 < x ≤ m3, (m3 − m2) { 0 otherwise, (n2 − x) + ν(x − n1) if n1 ≤ x < n2, (n2 − n1) ν if x = n2, θN ̃ (x) (x − n2) + V(n3 − x) μ if n2 < x ≤ n3, (n3 − n2) {1 otherwise, (p2 − x) + ζ(x − p1) μ if p1 ≤ x < p2, (p2 − p1) ζ if x = p2, ηN ̃ (x) = (x − p2) + (x − p1) μ if p2 < x ≤ p3, (p3 − p2) {1 otherwise.

A Single-Valued Trapezoidal Neutrosphic Number (SVTpN) of type I, denoted as G ̃ =

TpN

(i1, i2, i3, i4; k1, k2, k3, k4; l1, l2, l3, l4) is a specific type of Neutrosophic set on the real number ℝ. It is characterized by its truth, indeterminacy, and falsity membership functions, which are defined as follows [8]: ζ − i1 , when i1 ≤ ζ < i2, i2 − i1 1, when i2 ≤ ζ ≤ i3, T̃ (ζ) =

G TpN

i4 − ζ , when i3 < ζ ≤ i4, i4 − i3 { 0, otherwise, k2 − ζ , when k1 ≤ ζ < k2, k2 − k1 0, when k2 ≤ ζ ≤ k3, I ̃ (ζ) =

G TpN

ζ − k2 , when k3 < ζ ≤ k4, k4 − k3 { 1, otherwise, l2 − ζ , when l1 ≤ ζ < l2, l2 − l1 0, when l2 ≤ ζ ≤ l3, F ̃ (ζ) =

G TpN

ζ − l3 , when i3 < ζ ≤ i4, l4 − l3 {1, otherwise.

IV IV

Let ã = 〈[{(a , b , c , d ), u IV} , {(e , f , g , h ), v IV} , {(l , m , n , p ), w IV}] 〉 and ã =

N1 1 1 1 1 a ̃ 1 1 1 1 a ̃ 1 1 1 1 a ̃ N2

N1 N1 N1

〈[{(a , b , c , d ), u IV} , {(e , f , g , h ), v IV} , {(l , m , n , p ), w IV}] 〉 be two IVTrNeNs with twelve

2 2 2 2 a ̃ 2 2 2 2 a ̃ 2 2 2 2 a ̃

N2 N2 N2

L U L U L U

components, where u IV = [u IV , u IV ] , u IV = [u IV , u IV ], v IV = [v IV , v IV ],

a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N

1 1 1 2 2 2 1 1 1

L U L U L U

v IV = [v IV , v IV ], w IV = [w IV , w IV ], w IV = [w IV , w IV ]then the following

a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N a ̃ N

2 2 2 1 1 1 2 2 2

operations hold [2].

1 1 1 1 1 2 2 2 2

Let A = 〈(s , t , u v , w ); μA, θA〉 and 〈(s , t , u v , 2); μB, θB〉 be two Single Valued Pentagonal Neutrosophic Numbers [9].

The score function concept is used to find a comparison between two IVTrNeNs. The greater the score function value, the greater the IVTrNeN. According to the base of the score and the accuracy functions of

IV

an IVTrNeN a ̃ N can be defined as follows [2]:

IV

S(ã N ) = ((8 + (a1 + b1 + c1 + d1) − (e1 + f1 + g1 + h1) − (l1 + m1 + n1

L U L U L

+ p )) × (2 + u IV + u IV − v IV − v IV − w IV

1 ã ã ã ã ã

N1 N2 N1 N1 N1

U IV

− w IV S(ã )ϵ[0,1].

ã N1

IV 1 L U

The accuracy function A(ã ) = (a + b + c + d −l − m − n − p ) × (2 + u IV + u IV −

N1 1 1 1 1 1 1 1 1 a ̃ a ̃ 4 N1 N2 L U L U

v IV − v IV − w IV − w IV ).

a ̃ N a ̃ N a ̃ N a ̃ N

1 1 1 1

Defect and imperfect handling are two of the strategies for promoting environmental sustainability. Bhavani et al. [4] framed a Neutrosophic based inventor model considering deterioration and discount on defective items. The triangular Neutrosophic representations, together with Particle Swarm Optimization (PSO), determine the optimal inventory level and costs. Bhavani and Mahapatra [7] designed an inventory model with generalized triangular Neutrosophic cost patterns considering deterioration of the items. The defective and non-defective items are classified, and the faulty items are reworked with demand as the function of quality.

Mohanta et al. [10] developed a Neutrosophic logic integrated inventory model with a two-level partial credit policy for perishable items. The proposed quality promotion and profit maximization model includes the cost parameters of preservation technology. Case studies and comparative analysis are performed to demonstrate the efficiency of Neutrosophic models. Miriam et al. [11] discussed the aspect of the rework warehouse model under a Neutrosophic environment, considering the parameters of quality conservation. The PSO technique is applied in handling quality constraints associated with the reworking of defective items. Washima et al. [12] framed a Neutrosophic inventory model for dealing with imperfect items in a two-layer supply chain system. In this work, the price and stock-dependent demand with uncertainty is modelled using Neutrosophic numbers. The proposed Neutrosophic integrated model is sensitively analyzed to demonstrate the efficiency of Neutrosophic numbers in optimizing the production costs in comparison with fuzzy and crisp representations. Supakar et al. [13] designed a Neutrosophic-based trade credit Economic Order Quantity (EOQ) inventory model for deteriorating items using various variants of PSO techniques. Linear trapezoidal Neutrosophic fuzzy numbers are used in modelling, and optimal solutions are derived using Weighted Quantum-Behaved Particle Swarm Optimization (WQPSO), Gaussian Quantum-Behaved Particle Swarm Optimization (GQPSO), and Adaptive Quantum-Behaved Particle Swarm Optimization (AQPSO). Haque et al. [8] formulated a generalized Neutrosophic inventory model with price and deterioration-dependent demand. The non-linear and intricate parameters are modelled using Laplace transforms.

Rahaman et al. [14] formulated a demand-driven inventory model with trapezoidal Neutrosophic numbers representing the deterioration and pertinent preservation technology parameters. Priskilla et al. [15] developed a manufacturing model considering defective product distribution under a Neutrosophic environment, considering the challenges of environmental degradation, resource wastage, and financial losses. An inventory model with generalized triangular Neutrosophic representations is formulated. This model includes the parameters pertaining to screening mechanisms, supplier discounts, preservation technology, advertising, shortages, and back orders. The comparative analysis of this Neutrosophic model with fuzzy and crisp representations demonstrates the competency of neutrosophy in handling multi-dimensional parameters. Surya et al. [16] formulated a Neutrosophic inventory model with price-dependent demand catering for decaying items. The trapezoidal fuzzy numbers are used in mathematical modelling of the demand parameters. Gudeta et al. [17] framed an inventory model considering a single perishable item contributing to both the economy and organizational development. The uncertainty and imprecision in fixing the price discount of defective items by both the supplier and the retailer are well handled with single-valued TNNs. The neutrosophic fuzzy optimization technique is employed to determine the maximum inventory level of the retailers. Talor’s series approximation, Mathematical, and LINGO software are used to determine optimal solutions to the Neutrosophic inventory model. The review of the above works synthesizes the efficacy of Neutrosophic integrated models in dealing with faulty items.

Rahaman et al. [18] developed an EPQ model using a Neutrosophic differential equation approach, where production rate depends on time-impacted demand and stock levels, effectively capturing uncertainty through Neutrosophic calculus and demonstrating superior cost optimization compared to traditional and crisp models. Kausar et al. [19] proposed a Neutrosophic fuzzy inventory management model using pentagonal fuzzy numbers and Graded Mean Integration Representation (GMIR) to handle uncertainty in demand, order quantity, and costs, optimizing total cost under fuzzy conditions and demonstrating robust and adaptable inventory control through case-based analysis. Alioğulları et al. [20] applied the interval-valued neutrosophic fuzzy EDAS method to evaluate and rank sustainability risk mitigation strategies in the automotive supply chain, identifying optimal approaches—such as implementing legal compliance programs—while introducing new sustainability risk dimensions and contributing a novel fuzzy EDAS application to the literature. Dubey and Kumar [21] developed a Neutrosophic inventory model using trapezoidal Neutrosophic numbers to optimize total cost without shortages, providing computational results and comparative evaluations to demonstrate the model’s effectiveness in handling both conventional and novel inventory problems. Arshi et al. [22] proposed a nonlinear programming model for multiple attribute group decision-making under interval-valued Neutrosophic numbers, using linguistic variables to capture uncertainty and cognitive hesitancy, enabling decision-makers to identify optimal alternatives even when attributes and preferences are interdependent. Dubey et al. [23] provided a comprehensive survey on applying Neutrosophic principles to inventory management, highlighting how Neutrosophic approaches address uncertain demand and imprecise data more effectively than traditional and fuzzy models, thereby improving decision-making in inventory control systems. Sindhuja et al. [24] proposed a Neutrosophic fuzzy trapezoidal number approach to inventory management that integrates environmental sustainability, optimizing energy consumption, and handling shortages while addressing uncertainties, cost variability, and regulatory challenges to promote resilient and eco-efficient supply chain operations. Das and Islam [25] developed a deterministic multi-item inventory model with quadratic demand, time-dependent holding costs, and deterioration, employing Neutrosophic and Pythagorean Hesitant Fuzzy Programming (PHFP) approaches to handle uncertain hexagonal fuzzy cost parameters, demonstrating improved total cost optimization and robustness over traditional inventory models. Pattnaik et al. [26] developed a Neutrosophic fuzzy inventory control model for pandemic conditions, integrating overage management with profit maximization, and demonstrated through numerical examples and sensitivity analysis how their approach optimizes inventory decisions under COVID-19 disruptions. Deb and Islam [27] developed a Neutrosophic interval-valued goal programming approach for a supply chain inventory model of deteriorating items with time-dependent demand, determining the optimal replenishment strategy to minimize total cost under uncertainty. Mohanta et al. [28] propose a Neutrosophic EOQ model for smart manufacturing that optimizes inventory costs by considering idle times, uncertain parameters, and deriving optimal order quantities through De-Neutrosophication and sensitivity analysis. Rajeswari et al. [29] developed a two-warehouse EOQ model for imperfect items using triangular fuzzy Neutrosophic numbers to handle uncertainty in defective proportions, computing optimal order quantity and profit while performing sensitivity analysis and De-Neutrosophication for practical insights. Surya and Mullai [30] developed a Neutrosophic multi-item inventory control model incorporating budgetary, space, and order constraints, where key parameters are represented as TNNs to more realistically handle uncertainties in determining optimal order quantities. Rajeswari et al. [31] applied Octagonal Fuzzy Neutrosophic Numbers (OFNN) to model a reusable container inventory system, accounting for container shrinkage and uncertain parameters, and demonstrated how De-Neutrosophication via the removal area method provides approximate optimal solutions while highlighting the effect of return rates through sensitivity analysis. Islam and Das [32] developed a multi-objective inventory model for multi-item systems considering time-dependent demand, holding costs, and time-varying deterioration, where shortages are partially backlogged; to reduce deterioration, a preservation condition was incorporated, and uncertainties in cost parameters were modeled using generalized trapezoidal fuzzy numbers, with solutions obtained through Neutrosophic hesitant fuzzy programming, fuzzy nonlinear programming, and fuzzy additive goal programming, validated by numerical examples and sensitivity analysis. Pal and Chakraborty [33] developed a triangular Neutrosophic EPQ model for deteriorating items with ramp-type demand and time discounting, showing that the Neutrosophic approach yields superior and more reliable results than the crisp model. Das and Islam [34] demonstrated the efficiency of their multi-item inventory model, which utilizes a Neutrosophic hesitant fuzzy programming approach to handle uncertain cost parameters under space constraints, through comprehensive numerical and sensitivity analyses.

Mohanta et al. Neutrosophic logic- Case studies and [10] based model comparative analysis

Bhavani and Neutrosophic PSO Mahapatra [7] environment Washima et al. Trapezoidal Sensitivity analysis [12] Neutrosophic numbers Supakar et al. Linear trapezoidal WQPSO, GQPSO, [13] Neutrosophic fuzzy AQPSO numbers Haque et al. [8] Generalized Laplace transform Neutrosophic numbers methods Rahaman et al. Trapezoidal Simulation [14] Neutrosophic numbers Priskilla et al. Generalized TNNs Neutrosophic fuzzy [15] optimization, LINGO, Mathematica Surya et al. [16] Trapezoidal fuzzy Analytical comparison numbers Gudeta et al. Single-valued TNNs Neutrosophic fuzzy [17] optimization Rahaman et al. TNNs Neutrosophic [18] differential equation Kausar et al. Pentagonal fuzzy GMIR [19] numbers in Neutrosophic context Alioğulları et al. Interval-valued IVN-Fuzzy EDAS [20] Neutrosophic fuzzy (MCDM) numbers Arshi et al. [22] Interval-valued Nonlinear Neutrosophic numbers programming Sindhuja et al. Neutrosophic fuzzy Python-based analysis [24] trapezoidal numbers

Das and Islam Hexagonal fuzzy and NHFP and PHFPA [25] Neutrosophic hybrid Key Features Primary Objective

Deterioration, discount, Optimal inventory defect level and cost Defective and non- Quality-based defective classification demand modeling Partial credit, perishable Profit maximization, goods, preservation quality promotion technology Quality conservation, Minimize cost and rework warehouse defects Two-layer supply chain, Cost minimization imperfect items under uncertainty Trade credit, Derive optimal EOQ deterioration

Price and deterioration- Solve non-linear dependent demand dynamics Deterioration, Demand-driven preservation tech optimization Defect distribution, Sustainability and screening, discounts, profit backorders Price-dependent Cost minimization decaying items Price discount Maximize retailer uncertainty, perishables inventory level Time-impacted Cost minimization demand, deterioration Fuzzy consumption, Cost minimization cost components

Sustainability risks, Strategy ranking automotive supply chain MADM, aggregation Decision operators optimization Shortages, energy cost, Energy optimization and environmental metrics Multi-item quadratic Total cost reduction demand

https://www.sciencedirect.com/topics/social-sciences/supply-chain-management https://www.sciencedirect.com/topics/social-sciences/carbon-t ax

Key Features Primary Objective

Overage, pandemic Profit maximization disruption

Two-warehouse EOQ, Profit maximization imperfect items Budget, space, demand Optimal order quantity Multi-item inventory, Multi-objective cost demand-dependent optimization production cost, deterioration, space constraint Ramp-type demand, Cost-effective and Weibull deterioration, reliable production reliability, time planning discounting, partial backordering Time-dependent Multi-objective demand and holding inventory cost, deterioration, and optimization preservation condition Reusable container Optimal container inventory, shrinkage, allocation and cost return rate minimization Idle time, deterioration, Cost minimization smart manufacturing, off-days Deteriorating items, EOQ optimization, time-dependent cost minimization demand, trade credit Holding, ordering, and EOQ-based cost shortage costs minimization Holding and shortage EOQ inventory costs model Demand, holding cost, To design a cost-ordering cost, effective inventory uncertainty in model incorporating parameters Neutrosophic uncertainty for better decision-making

Researchers have formulated inventory models considering green cost parameters. The costs associated with carbon emissions, including green technology, are included in the Neutrosophic models. The green costings are also discussed with other inventory management situations. Barman et al. [36] proposed a faulty handling multi-objective supply chain inventory model with single-valued trapezoidal Neutrosophic numbers. The implementation of three carbon policies in inventory management, such as carbon tax , carbon cap-and-trade, and carbon cap-and-offset are discussed in brief. Sarkar et al. [37] framed a multi-objective Neutrosophic inventory model considering the costs associated with pollution mitigation. The Neutrosophic goal programming technique is applied to minimize the costs and maximize the profit, considering warehouse management and eco-conservation. Kar et al. [38] formulated an inventory model with Neutrosophic demand considering price, marketing, service, and eco parameters. The pentagonal Neutrosophic numbers are used, and the geometric programming approach is applied in handling the non-linearity of the parameters. Numerical illustrations and comparative analysis demonstrate the efficiency of the proposed Neutrosophic model over fuzzy and crisp representations.

Resource efficiency also amounts to sustainability. In this context, Garg et al. [39] developed an inventory model based on container management organization, focusing on resource efficiency. The imbalance of the flow of containers is well modelled using trapezoidal bipolar Neutrosophic numbers. The algorithmic approach presented in this work predicts the expected inventory costs under various conditions. Sugapriya et al. [40] proposed a container inventory model with a trapezoidal bipolar Neutrosophic number. Comparison with triangular bipolar Neutrosophic numbers demonstrates the efficiency of trapezoidal representations over triangular forms. Mohanta et al. [41] developed a Neutrosophic-based inventory model using generalized trapezoidal single-valued Neutrosophic numbers, considering deteriorating items. The optimal ordering quantity, optimal opening, and closing time are determined using Neutrosophic modelling. Kar et al. [42] contributed a multi-objective perishable multi-item green inventory model considering eco-constraints with Neutrosophic representations. The model addresses the tradeoff between profit and carbon emissions with the objective of profit maximization and emission mitigation. Supakar et al. [43] proposed a green production inventory with Neutrosophic representations to handle uncertain parameters. The artificial bee colony algorithm is used in modeling the green inventory model. Loganayaki et al. [44] discussed a Neutrosophic-based inventory model with TNNs with power demand and preservation technology. The primary objective of this Neutrosophic model is cost minimization, considering shortages and linear holding costs. Sindhuja et al. [24] discussed the strategies of energizing inventory management in handling shortages using Neutrosophic trapezoidal numbers. The need of developing such models in underscoring the balance between economic efficiency and ecological responsibility is well substantiated in the illustration. Saeed et al. [45] discussed green supply chain model with refined four valued Neutrosophic optimization. The objective of this proposed Neutrosophic model is resilience maximization and minimization of Greenhouse Gases (GHG). Pal et al. [46] discussed Neutrosophic fuzzy inventory system with different demand patterns considering the impacts if carbon emissions, inflation and delayed deterioration. Non-linear triangular dense Neutrosophic numbers are used in modelling the uncertain parameters. The environmental metrics are integrated with Neutrosophic inventory models to characterize the fluctuations in sustainable cost parameters.

Mukherjee et al. [47] developed a multi-objective sustainable inventory model incorporating shortages, emissions, Radio Frequency Identification (RFID), and fuzzy-random uncertainties, where demand depends on price and marketing efforts, and demonstrated through sensitivity analysis that their approach outperforms traditional models by providing more reliable and practical decision support under real-world uncertainty. Palanivel and Venkadesh [49] proposed an integrated sustainable production inventory model that incorporates green technologies—both exponential and linear—to minimize carbon emissions, optimize resource use, and enhance supply chain resilience, offering a strategic framework that balances economic efficiency with environmental sustainability through analytical modeling and sensitivity analysis. Nafei et al. [48] proposed a sustainable inventory model with a cap-and-trade policy, incorporating price-dependent demand and inflation effects, to determine optimal inventory levels that balance economic performance with environmental responsibility, validated through numerical illustration and sensitivity analysis.

[36] Trapezoidal chain model supply chain and cap-and-offset policies integrated Sarkar et al. [37] Triangular / Neutrosophic goal Multi-objective Pollution mitigation, eco-Generalized programming inventory conservation Kar et al. Pentagonal Geometric Demand- Price, marketing, service, (model 1) [38] programming dependent and eco parameters inventory Garg et al. [39] Trapezoidal bipolar Algorithmic simulation Container Resource efficiency and inventory flow balance Sugapriya et al. Trapezoidal bipolar Comparative modelling Container Resource utilization [40] vs. triangular inventory Mohanta et al. Generalized Analytical optimization Deteriorating Sustainable ordering [28] trapezoidal items strategy Kar et al. (model Neutrosophic Multi-objective Perishable multi- Profit-emission trade-off 2) [42] (general) programming item green inventory Supakar et al. Neutrosophic Artificial bee colony Green production Green manufacturing [43] algorithm inventory Loganayaki et al. Triangular Analytical modelling Power demand Shortages and holding [44] and preservation costs tech Sindhuja et al. Trapezoidal Analytical modelling Shortage Economic–ecological [24] management balance Saeed et al. [45] Refined four-valued Neutrosophic Green supply GHG minimization, optimization chain resilience maximization Pal et al. [46] Non-linear Analytical modelling Fuzzy– Carbon emission, inflation, triangular dense Neutrosophic and delayed deterioration hybrid system Mukherjee et al. Fuzzy-random and Multi-objective Shortages, RFID, Profit and cost [47] Neutrosophic optimization with inflation, emission optimization environment sensitivity analysis Palanivel and Sustainable Sensitivity analysis and Ramp-type Carbon emission Venkadesh [49] production comparative approach demand, Weibull minimization inventory with green deterioration, technology green tech Nafei et al. [48] Neutrosophic Neutrosophic Price-dependent Sustainable inventory numbers Autocratic MCDM demand, carbon control cap, inflation

The Neutrosophic-based sustainable inventory models are computationally intensive. The three-valued logic integrated with inventory modelling and sustainability factors increases the complexity of computations. Hence, it is not easy to obtain exact analytical solutions. This is the reason for employing different optimization techniques ranging from classical mathematical programming to advanced and hybrid techniques. Rahaman et al. [18] applied Neutrosophic differential equations in modelling the Economic Production Quantity (EPQ) inventory model with production rate as a function of demand and stock. Momena et al. [50] used first-order linear Neutrosophic differential equations for handling an inventory system with insufficient data. The exploration of differential equations and numerical simulation facilitates in alleviating uncertainty in inventory systems. Haque et al. [51] applied fractional calculus for Neutrosophic - valued functions in inventory lot sizing. The choice of fractional calculus is made to capture the system’s memory and past experiences, paving the way for enhanced decision-making. Kumar and Singh [52] applied the PSO algorithm in optimizing the Neutrosophic inventory model with quality-dependent demand and payment delay. Kalaiarasi et al. [53] introduced a Neutrosophic integrated vendor -buyer model with different parameters. PSO technique, MATLAB, and Python are applied for simulation and sensitivity analysis. GJ et al. [54] used Genetic Algorithms (GAs) to optimize the Neutrosophic EOQ model with varying patterns of demand and lead time. Bhavani et al. [4] used the PSO method to devise a solution to the Neutrosophic inventory model addressing deterioration and discounts.

Kalaiarasi et al. [53] discussed inventory modelling using Neutrosophic logic. The benefits of Neutrosophic modelling in handling higher ranges of uncertainty are outlined with suitable illustrations. Bolos et al. [55] employed Neutrosophic-based nonlinear mathematical programming algorithms in framing production stock management-based inventory models. Deb and Islam [27] applied the Neutrosophic goal programming technique with the representations of triangular numbers. This model deals with the aspects of deterioration and credit risk in the supply chain. Kalaiarasi and Swathi [56] discussed the situation of quick returns. The study explored two different types of Neutrosophic numbers in modelling perfect rate, demand rate, purchase costs, and ideal order quantity. Mullai and Surya [57] discoursed on backorders to handle the shortage using Neutrosophic representations. Das and Islam [58] developed a multi-objective shortage inventory model and determined a solution using Neutrosophic non-linear programming. Different programming methods are applied under the environments of fuzzy and intuitionistic, and it is observed that Neutrosophic-based programming methods are more effective in handling complexity. Jayanthi [59] discussed overage management using Neutrosophic fuzzy geometric programming. The computational complexity of the Neutrosophic inventory models is well handled with different programming techniques.

Miriam et al. [60] proposed a Neutrosophic inventory model integrated with coordinated rework stations and distribution centers, optimized using Ant Colony Optimization (ACO), to enhance supply chain efficiency, minimize costs, and strengthen resilience against uncertainties through the combined use of Neutrosophic logic and swarm intelligence. Grützner et al. [61] developed an interlinked process-reference model for mature inventory management that supports supply chain automation through a comprehensive multi-criteria ABC analysis, integrating empirical insights from case studies, surveys, and literature to promote resilient, value-added, and sustainable inventory practices for targeted automation and decision-making. Dwivedi et al. [62] proposed a dynamic pharmaceutical inventory model for pandemic conditions that integrates environmental emission rates, preservation technology, and fuzzy learning, using advanced optimization algorithms like Ant Colony and Cuckoo Search to optimize pricing, investment, and replenishment strategies, thereby balancing profitability, sustainability, and public health objectives under uncertainty. Kar et al. [42] developed multi-objective perishable multi-item green inventory models with stock-dependent demand under uncertain finite time horizons, using a Neutrosophic optimization approach to maximize profit while minimizing wastage and carbon emissions, and demonstrated trade-offs and sensitivity through numerical examples. Anand et al. [63] applied a Neutrosophic transportation problem with a contingency technique to optimize electricity distribution, leveraging Neutrosophic logic to handle uncertainty, indeterminacy, and inconsistency, and using center-of-gravity and score functions to determine crisp values and minimize transportation costs effectively. GJ et al. [54] developed a Neutrosophic EOQ model that incorporates demand and lead-time uncertainties, using GAs and simulated annealing to optimize order quantities, demonstrating effective inventory management in highly uncertain environments through practical examples. Kousar et al. [64] developed a Multi-Objective Neutrosophic Fuzzy Linear Programming (MONFLP) model to optimize Kharif and Rabi crop production under uncertainty, maximizing net profit and output while accounting for climate, water availability, and irrigation constraints, demonstrating the model’s effectiveness in handling unpredictable agricultural parameters in Pakistan. Cakmak and Guney [65] proposed a two-stage Neutrosophic fuzzy EDAS framework integrating T2NN-LOPCOW and T2NN-ARAS methods to classify spare parts inventory and evaluate industry 4.0-based material handling technologies, demonstrating through a real-life aviation case study that Automated Guided Vehicles (AGVs) are the most effective solution and highlighting the model’s robustness and practical applicability. Adhami et al. [66] developed a Neutrosophic Compromise Programming Approach (NCPA) for multilevel supplier selection under fuzzy supply and demand, using membership, nonmembership, and indeterminacy degrees to handle uncertainty and provide satisfactory hierarchical decision-making solutions, validated through a numerical example. Cakmak and Guney [65] applied the Neutrosophic Fuzzy EDAS method to classify spare parts inventories in the aviation industry, enhancing inventory management efficiency under uncertainty while minimizing costs and ensuring high service levels. Badhotiya et al. [67] propose a Neutrosophic programming approach for integrated production-distribution planning in a two-echelon supply chain, addressing uncertainties and conflicting objectives— minimizing total cost, delivery time, and backorder level—through a tri-objective MILP model validated via a real-world automotive case study and Pareto optimality analysis. Daham [68] addresses discrete facility location problems under uncertainty by formulating Neutrosophic models that handle vague and ambiguous data for locations, distances, times, and costs, providing numerical examples to demonstrate the efficacy of these models compared to traditional deterministic approaches. Jdid et al. [69] employed Neutrosophic logic to enhance the static inventory model with deficit, allowing for more accurate handling of uncertain demand fluctuations and optimizing inventory levels while balancing deficit and storage costs. Mondal et al. [70] developed a Neutrosophic optimization-based EOQ model for seasonal products with logistic-growth demand, Weibull-distributed deterioration, partial backordering, and fully permissible delay in payment, effectively handling imprecise real-world information and providing managerial insights on cost minimization and inventory depletion timing under uncertainty. Haq et al. [71] developed a multi-criteria fuzzy Neutrosophic decision-making model for complex multi-site supply chain networks, integrating truth, indeterminacy, and falsity membership functions to optimize transportation cost and delivery time under uncertainty, and demonstrated its effectiveness through an industrial case study compared to other approaches. Piper [72] developed a Neutrosophic geometric programming-based inventory model that surpasses traditional optimization methods by effectively handling space constraints and uncertain, imprecise parameters. Kar et al. [73] extended the classical static inventory management model without deficit by incorporating Neutrosophic logic, allowing for indeterminate demand rates and uncertain data, which provides a more realistic and safe framework for determining the ideal inventory volume and associated costs, thereby enhancing operational efficiency and profitability. Islam [74] developed a multi-objective inventory model for deteriorating items under storage space constraints, incorporating time-dependent demand and holding costs, partial backlogging, and uncertain parameters modeled as generalized trapezoidal fuzzy numbers; the model is solved using Neutrosophic hesitant fuzzy programming and fuzzy nonlinear programming, with numerical examples and sensitivity analysis validating its applicability. Ahmad et al. [75] introduced a multiobjective framework using a Neutrosophic goal programming approach to address uncertainty in optimal shale gas water management. Their paper presented a Neutrosophic optimization model and algorithm for shale gas water management, which addresses conflicting constraints related to fresh water and wastewater under uncertain conditions. Islam and Kundu [76] developed a Neutrosophic optimization-based inventory model without shortages, applying geometric programming and additive operator methods to manage uncertainty effectively in production and storage parameters.

Rahman et al. Neutrosophic differential EPQ Production rate as a function of [18] equations demand and stock Momena et al. First-order linear Inventory with Uncertain demand and stock levels [50] Neutrosophic differential insufficient data equations Haque et al. [8] Fractional calculus for Inventory lot sizing Fractional derivative to capture Neutrosophic-valued system memory functions Kumar and PSO Quality-dependent Swarm intelligence for parameter Singh [52] demand with payment tuning delay Kalaiarasi et al. PSO, MATLAB, and Vendor–buyer Multi-parameter analysis [53] Python Simulation integrated model GJ et al. [54] GA EOQ with variable Evolutionary search technique demand and lead time Bhavani et al. PSO Deteriorating items Price–time deterioration factor [4] with discounts Moorthy et al. Neutrosophic logic General inventory Illustrative uncertainty handling [77] framework modelling Bolos et al. [55] Nonlinear neutrosophic Production–stock Nonlinear constraints mathematical programming management Deb and Islam Neutrosophic goal Deterioration and Multi-objective framework [58] programming (triangular credit risk numbers) Kalaiarasi and Neutrosophic numbers (two Quick returns scenario Purchase cost, demand rate, ideal Swathi [56] types) order quantity Mullai and Neutrosophic representation Shortage management Backorder and shortage rate Surya [57] for backorders Das and Islam Neutrosophic nonlinear Multi-objective Cost and shortage objectives [58] programming shortage inventory Jayanthi [59] Neutrosophic fuzzy Overage management Cost and volume optimization geometric programming Miriam et al. Neutrosophic inventory ACO Optimize the supply chain, [60] model rework, and cost Grützner et al. Multi-criteria Neutrosophic Process-reference MCABC analysis, automation, [61] evaluation modeling sustainability Dwivedi et al. Fuzzy learning–based Ant Colony and Price, infection rate, and [62] Neutrosophic hybrid Cuckoo Search environmental costs Kar et al. [42] Neutrosophic optimization Weighted-sum and Perishability, carbon emission, and GRG (LINGO) wastage Anand et al. [63] Neutrosophic fuzzy Contingency means Electricity distribution, cost numbers allocation uncertainty GJ et al. [54] Neutrosophic EOQ GA and Simulated Demand, lead time, and inventory framework Annealing cost Cakmak and Neutrosophic Fuzzy EDAS MCDM approach Aviation spare parts classification Guney [65] Badhotiya et al. Neutrosophic membership Tri-objective mixed Cost, delivery, backorder [67] functions integer programming Daham [68] Neutrosophic parameters Mathematical Facility location, cost, distance modeling Haq et al. [71] Neutrosophic set functions Compromise Multi-objective supply chain programming Mondal et al. Neutrosophic coefficients Weighted arithmetic Logistic demand, delay, [70] mean deterioration Haq et al. [71] Neutrosophic set functions Compromise Multi-objective supply chain programming Kar et al. [73] Neutrosophic set functions Geometric Setup cost, production, storage programming Islam and Neutrosophic geometric Inventory model Time-dependent holding cost, Kundu [76] programming and additive without shortages inverse relation of setup and operator production cost, storage imprecision Jdid et al. [78] Neutrosophic logic Static inventory Demand indeterminacy, ideal modeling without a deficit inventory volume Jdid et al. [78] Neutrosophic logic-based Static inventory with a Deficit cost, demand fluctuation mathematical model deficit Cakmak and Neutrosophic Fuzzy EDAS Aviation spare parts Multi-criteria classification, Guney [65] (MCDM) inventory uncertain demand Adhami et al. NCPA Supplier selection Multi-level decision structure, [66] decision model fuzzy supply and demand Kousar et al. MONFLP Crop production Canal irrigation, crop yield, and [64] optimization environmental limits Ahmad et al. Neutrosophic goal Multi-objective Freshwater allocation, wastewater [75] programming approach optimization model treatment, environmental and for shale gas water socio-economic trade-offs, and management uncertain parameters Islam [74] Neutrosophic hesitant fuzzy Multi-objective Time-dependent demand and programming and fuzzy inventory model for holding cost, partial backlogging, non-linear programming deteriorating items generalized trapezoidal fuzzy under space constraint parameters