In recent years, there has been an increasing focus within the academic community on developing the study of the neutrosophic field to discover innovative applications in varied domains. Despite the progress in understanding and applying TrSVNN, a multitude of unresolved theories and challenges continue to persist. The primary objective of this research article is to shed light on the concepts of the neutrosophic domain and offer a novel viewpoint on its possible applications. Our novelty includes:

Developed a novel approach while employing TrSVNN, an effective and simple model for handling uncertain information.

The structure of the article unfolds as follows: Section 2 defines the prelims useful for the development of the document. Section 3 gives the methodology about the existing classical and neutrosophic environment, where the discussion of classical critical path is derived in Subsection 3.1 and introduces the proposed devlopment of neutrosophic formulation on the working principle of CPP mentioned in Subsection 3.2 and the proposed algorithmn is breifly discussed in Subsection 3.3. Further, Section 4 solves numerical example study that provides existing CCPP in Section 4.1 and solves existing literature in Section 4.2, further the new set of problem of NCPP using three different cases was implemented in Subsection 4.3 and lastly conclusion.

CPP represents “critical path problem”.

The CCPP implements a dynamic programming in the network, a cyclic-directed graph G=(V,A). In this graph, V is the set of vertices with numbers from 1 (source) to m (destination), and A is the set of directed edges. The project initialization starts with zero, indicating that there is no accumulated duration at the beginning. (3.1)f(m)=0f(k)=maxk<δ{Ckδ+f(δ)|<k,δ>∈A}

The weight Ckδ of the directed edge from one vertex k to another vertex δ is represented by the duration. f(k) quantifies the length of the longest critical path commencing at vertex k and concluding at vertex m.

The NCPP and edge weights Ckδ in the network are treated as indeterminate, reflecting the inherent ambiguity of the data. Each edge weight Ckδ is defined within the acceptable range forming an interval, with lower (F_kδ1) and upper bound (F¯kδ2), by maintaining the indeterminate nature of project activity durations by satisfying the conditions 0≤F_kδ1<Ckδ and, 0<F¯kδ2. The suitable edge weight within the range [Ckδ−F_kδ1,Ckδ+F¯kδ2] for Ckδ leads to the construction of such selected intervals form a specific type of neutrosophic number, termed to be TrSVNN (C kδN∗), which is defined by the triplet corresponding to its truth (dpN), indeterminacy (epN), and falsity (fpN) been depicted in Fig. 1 as follows:

(3.2)CkδN∗=(Ckδ−F_kδ1,Ckδ,Ckδ+F¯kδ2;dpN,epN,fpN);where0<F_kδ1<Ckδ,0<F¯kδ2

The interval within edge weights [Ckδ−F_kδ1,Ckδ+F¯kδ2] is considered as neutrosophic framework, where Fkδ=F¯kδ2−F_kδ1 derives its variation of the upper and lower bound. This alignment concept is derived from the Definition 2.4. The neutrosophic edge weights C kδN∗ can be calculated using the formula: (3.3)E′(CkδN∗)=Ckδ−F_kδ1+4Ckδ+Ckδ+F¯kδ26×(2+dpN−epN−fpN3)

Refining this expression yields: E′(CkδN∗)=6Ckδ+(F¯kδ2−F_kδ1)6×(2+dpN−epN−fpN3) (3.4)E′(CkδN∗)=Ckδ+(F¯kδ2−F_kδ16)×(2+dpN−epN−fpN3) (3.5)E′(CkδN∗)=(Ckδ+Fkδ6)×(2+dpN−epN−fpN3)>0

If F¯kδ1=F_kδ2 from Eq. (3.4), it simplifies the neutrosophic edge weight E′(C kδN∗)=Ckδ is obtained to a classical sense. Where E′(CkδN∗)=(Ckδ+Fkδ6)×(2+dpN−epN−fpN3) be the neutrosophic estimate of the edge weight, reflecting the transition from the potential range of values to a classical measure. Considering the finite number of paths between the two nodes within the network can be deducted from the existence of numerous neutrosophic paths from any node k to node m. Thus, for a path Q=⟨k,k1,k2..,kr(k),m⟩ leading to the sequence of the node pairs ⟨k,k1⟩,⟨k1,k2⟩,..,⟨kr(k),m⟩∈A for, f(k)=Ckk1+Ck1k2+.....+Ckr(k)m. Accordingly, the aggregate critical path length f(k) from vertex k to vertex m is expressed as follows: (3.6)f(k)=Ckk1+Ck1k2+⋯+Ckr(k)m≥Ckw1+Cw1w2+⋯+CwQ(w)mwhere equality holds at least one pathway since f(k) represents the maximum path length for all possible routes Q=⟨k,w1,w2,..,wQ(w),m⟩ from vertex k to vertex m. Thus, f(k)=max{Ckw1+Cw1w2+..+CwQ(w),m/Q=⟨k,w1,w2,...,wQ(w),m⟩}

Transitioning neutrosophic on both sides of the equation, the modified Eq. (3.6) evolves to: (3.7)C kk1N∗⊕C k1k2N∗⊕..⊕C kr(k)mN∗≥C kw1N∗⊕C w1w2N∗⊕...⊕C wQ(w)mN∗

Before proceeding with the further step, the implementation of the score function (E′) defined from Definition 2.4 for the path length Ckw1,Cw1w2,CwQ(w)m into crisp is as follows: E′(C kk1N∗⊕C k1k2N∗⊕..⊕C kr(k)mN∗)≥E′(C kw1N∗⊕C w1w2N∗⊕...⊕C wQ(w)mN∗) (3.8)C kk1N+C k1k2N+..+C kr(k)mN≥C kw1N+C w1w2N+...+C wQ(w)mN

Let [f(k)]N be the length of the neutrosophic critical path, where at least one equality holds from the possible paths from vertex k to vertex m in the network G=(V,A) with {C kδN∗/<k,δ>∈A}. From Eq. (3.8), where f(k)=Ckk1+Ck1k2+.....+Ckr(k)m calculates from vertex k and vertex δ within the network m. [f(k)]N=Ckk1N+Ck1k2N+...+Ckr(k)mN (3.9)[f(δ)]N=Cδδ1N+Cδ1δ2N+...+Cδr(δ)mN

The reformulation of Eq. (3.1) is as follows: for any vertex k in the graph, the length of any path from k to m implies; f(k)≥Ckδ+f(δ),∀k<δ,(k,δ)∈A represents the critical path where at least one equal sign holds for all possible paths, which can be mathematically represented as: (3.10)Ckk1+Ck1k2+...+C kr(k)m≥Ckδ+Cδδ1+Cδ1δ2+...+C δr(δ)m,∀k<δ,(k,δ)∈A

Updating and applying neutrosophic on both sides of the Eq. (3.10) allows us to compare the aggregated neutrosophic weights as follows: (3.11)C kk1N∗⊕C k1k2N∗⊕...⊕C kr(k)mN∗≥CkδN∗⊕Cδδ1N∗⊕Cδ1δ2N∗⊕...⊕Cδr(δ)mN∗∀k<δ,(k,δ)∈A

Ensuring the preservation of at least one instance of equality from Definition 2.4 and Eq. (3.5) refined within the edge connections from vertex k to m is as follows: (3.12)C kk1N+C k1k2N+...+C kr(k)mN≥CkδN+Cδδ1N+Cδ1δ2N+..+Cδr(δ)mN∀k<δ,(k,δ)∈A

Consequently, from the Eqs. (3.5), (3.10), and (3.12), the decision maker (DM) choose appropriate bounds: F¯kk1,F¯k1k2,..,F¯kr(k)m,F_kδ,F_δδ1,..,F_δr(δ)m to satisfy: (3.13)F¯kk1+F¯k1k2+..+F¯kr(k)m≥F_kδ+F_δδ1+..+F_δr(δ)m∀k<δ/<k,δ>∈A

The dynamic programming problem recursion for the neutrosophic critical path problem from the Eqs. (3.9), and (3.12) is thus formalized as: (3.14)[f(k)]N=maxk<δ{(C kδN∗+[f(δ)]N)/⟨k,δ⟩∈A}f(m)N=⟨(0,0,0);1,0,0⟩

Encapsulate Eq. (3.14), where [f(k)]N designates the length of the critical path from node k to node m in the neutrosophic sense. In this scenario, if the edges are equivalent to F¯kδ2=F_kδ1, then obtained C kδN∗=Ckδ effectively transforms the neutrosophic to classical sense.

Example 4.1. In CCPP [39], assuming the edge weights of the network (ref Fig. 2) are as follows: C12=5, C13=10, C14=3, C24=4, C34=2, C45=8. Our aim is to implement the CCPP model to find the critical path for the given network (Fig. 2).

Solution: The working model from Eq. (3.1) is calculated, and the CP of the classical case from node 1 to node 5 is 1→3→4→5, within the total duration of CCPL is 20 days. The estimation utilizes a deterministic CCPP approach that ignores uncertainty about activity duration. The neutrosophic technique, which integrates uncertainty and variability into the critical path analysis, is introduced in the next subsection.

In the upcoming study, the representation of TrSVNN is considered to encapsulate the uncertainty and imprecision of project activity’s time durations using the same network diagram of Fig. 2. In Example 4.2, (dpN,epN,fpN) are constantly employed a uniform approach of (1,0,0) in evaluating the possible duration of activities with three distinct cases, i.e., Case-I: (F_kδ1<F¯kδ2), Case-II: (F_kδ1>F¯kδ2) and hybrid approach of Case-III: (F_kδ1<F¯kδ2),(F_kδ1>F¯kδ2),(F_kδ1=F¯kδ2) based on the relationship between F_kδ1 and F¯kδ2, which are critical parameters in our neutrosophic model.

Example 4.2.1. Case-I: If the DM chooses the condition as F_kδ1<F¯kδ2.

Solution: Step 1 defines the project network (ref Fig. 2). Using the algorithmic steps from 2 to 5, the DM chooses the appropriate lower and upper bounds as follows: F_121=1, F¯122=2, F131=3, F¯132=4, F_141=2, F¯142=4, F_241=1, F¯242=2, F_341=1, F¯342=2, F_451=3, F¯452=4 to satisfy the condition from Eq. (3.2) is determined in Table 1 as follows:

From Steps 6 to 8, the case-I (F_kδ1<F¯kδ2) calculates arithmetic operations from terminal node m by intializing with f(m)N=⟨(0,0,0);1,0,0⟩ that defines the optimal assumption. Further, the designed computation of the project network perform neutrosophic arithmetic operation and results are generated from python programming depicted in Figs. 3a–3c.

Finally, the NCPL is ⟨(13,20,30);1,0,0⟩ being the maximum time taken from node 1 to node 5 having its NCP is 1→3→4→5 as mentioned in Fig. 3a. Using Definition 2.4, the completion of the project calculates the Suggested CCPL (SCCPL) is 20.50 days and other NPP are being identified in Fig. 3b with 11.50 days having the NCP of 1→4→5 and NCPL ⟨(6,11,19);1,0,0⟩ and Fig. 3c with 17.50 days by having the NCP as 1→2→4→5 and respective NCPL of ⟨(12,17,25);1,0,0⟩ unveils alternate project activity sequencing that could mitigate risks and improve project responsiveness to changing situations.

Example 4.2.2. Case-II: If the DM chooses the condition as F_kδ1>F¯kδ2.

Solution: Similary, by implementing our proposed algorithmn in Subsection 3.3, the bounds are structured as: F_121=4, F¯122=1, F_131=6, F¯132=5, F_141=2, F¯142=1, F_241=2, F¯242=1, F_341=2, F¯342=1, F_451=4, F¯452=2 to satisfy the condition from Eq. (3.2) determined in Table 2 as follows:

In contrast, case-II (F_kδ1>F¯kδ2) is considered to incorporate a more conservative and risk-averse approach, which means incorporating a higher degree of uncertainty yields to shorter duration, and the results are depicted in Figs. 4a–4c.

Based on the condition (F_kδ1>F¯kδ2). The obtained NCPL is <(8,20,28);1,0,0> and the NCP of 1-3-4-5 yields a project completion time of SCCPL with 19.33 days, as observed in Fig. 4a. Alternative NPP-1 and NPP-2 have been explored in Figs. 4b and 4c, respectively, similar to case I of Example 4.2.1. However, there is a change in the NCCPL, having 10.50 days and 16.00 days, and their respective NCPLs are <(5,11,14);1,0,0> and <(7,17,21);1,0,0)>.

Example 4.2.3. Case-III: If the DM chooses the hybrid approach condition as (F_kδ1>F¯kδ2), (F_kδ1<F¯kδ2), (F_kδ1=F¯kδ2)

Solution: Similarly, implementing the algorithmic approach from steps 1 to 9, the obtained lower and upper bounds as: F_121=1, F¯122=2, F_131=4, F¯132=3, F_141=1, F¯142=1, F_241=3, F¯242=2, F_341=1, F¯342=2, F_451=5, F¯452=3 to satisfy the Eq. (3.2) is determined in Table 3 as follows: