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# Enhancing Critical Path Problem in Neutrosophic Environment Using Python

VIT-AP University, Inavolu, Besides AP Secretariat, Amaravati AP, India

* Corresponding Author: Ranjan Kumar. Email:

(This article belongs to the Special Issue: Advances in Ambient Intelligence and Social Computing under uncertainty and indeterminacy: From Theory to Applications)

*Computer Modeling in Engineering & Sciences* **2024**, *140*(3), 2957-2976. https://doi.org/10.32604/cmes.2024.051581

**Received** 09 March 2024; **Accepted** 14 May 2024; **Issue published** 08 July 2024

## Abstract

In the real world, one of the most common problems in project management is the unpredictability of resources and timelines. An efficient way to resolve uncertainty problems and overcome such obstacles is through an extended fuzzy approach, often known as neutrosophic logic. Our rigorous proposed model has led to the creation of an advanced technique for computing the triangular single-valued neutrosophic number. This innovative approach evaluates the inherent uncertainty in project durations of the planning phase, which enhances the potential significance of the decision-making process in the project. Our proposed method, for the first time in the neutrosophic set literature, not only solves existing problems but also introduces a new set of problems not yet explored in previous research. A comparative study using Python programming was conducted to examine the effectiveness of responsive and adaptive planning, as well as their differences from other existing models such as the classical critical path problem and the fuzzy critical path problem. The study highlights the use of neutrosophic logic in handling complex projects by illustrating an innovative dynamic programming framework that is robust and flexible, according to the derived results, and sets the stage for future discussions on its scalability and application across different industries.## Keywords

Planning multiple tasks to develop and execute the project within the allotted time frame is an essential part of project management [1]. Time restrictions are a common source of failure while working manually in many industries. Project managers use scheduling tools like Gantt charts and network planning to address such issues. Previous researchers were involved in the study of Gantt charts due to their less complex nature. Furthermore, large-scale and more complicated project execution has emerged since 1950, leading to the development of project management models, i.e., network analysis.

Network analysis deals with the coordination of project scheduling and the identification of task interdependencies to design and analyze [2]. This analytical framework employs two primary methodologies: the CPM (Critical Path Method) and the PERT (Program Evaluation and Review Technique). Kelly and Walker created CPM [3], which provides definitive project execution schedules in a chronological pattern based on deterministic time estimation [4]. PERT was created by Malcolm et al. [5] and is also called the backward research method [6] because it has a three-time estimation that takes the account of uncertainty [7]. Our study examines the critical path problem (CPP) using the CPM approach, which is a common method in project management, to distinguish between critical and non-critical tasks. This makes it easier to solve problems and avoid delays. CPP improves the project’s efficiency by helping to determine the minimum feasible time for task completion [8]. Utilizing CPP involves a multitude of operational metrics, including the calculation of the maximum time allowance, the earliest and latest initiation, and the corresponding completion time [9]. Traditional CPP practices dictate that fixed-time estimations represent this project activity. However, predicting future events in the real world is difficult due to the inherent unpredictability of dynamic project environments [10].

An anthology of researchers demonstrated the classical critical path problem (CCPP) study in project scheduling across a variety of domains. However, improving project management and control in the CCPP faces challenges in anticipating and estimating parameters involving uncertainty to calculate time deviations. In such a scenario, Zadeh introduced the concept of fuzzy logic to address the limitations of classical set theory while dealing with the study of vagueness and uncertainty in real-world situations [11]. Following up on Zadeh’s theory, Atanassov [12] introduced the legerdemain concept of intuitionistic fuzzy sets (IFS) in 1986, involving both membership and non-membership functions. To advance the study of uncertainty, researchers developed triangular fuzzy numbers (TrFN) [13] and trapezoidal fuzzy numbers (TFN) [14] to represent uncertainty. More fuzzy numbers have been made, like octagonal [15], heptagonal [16], and hendecagonal [17]. This shows that the research has gone beyond the initial forms. Researchers further conducted the study to extend zadeh’s pioneering work in fuzzy logic to a variety of practical applications. In one of these efforts, Mehlawat et al. [18] used IFS to study multi-criteria decision-making (MCDM) for critical path selection. An advanced methodology to implement fuzzy methods to handle the challenges of project management. Another study by Revathi et al. [17] showed how flexible and useful fuzzy critical path problems (FCPP) in managing agricultural projects. It uses a variety of fuzzy parameters, such as trapezoidal, heptagonal, and hendecagonal fuzzy numbers, to accurately handle the complexity of agricultural data. Senussi et al. [19] also used the TFN parametric form to account for uncertainty in planning projects that will make important contributions to the field. Ganesan et al. [20] analyzed another notable study of inter-valued parameters in operational networks. Further, many studies have implemented fuzzy environments in different optimization techniques such as supply chain [21], transportation [22] and so on. Although fuzzy logic has improved, it cannot fully capture real-world uncertainty. There are still unresolved issues while solving CCPP and FCPP under uncertain circumstances. This shift is being driven to address the research gaps leading to neutrosophic logic.

To adopt such parameters, Smarandache [23] introduced the neutrosophic set (NS) in 1998 with three integrands: truthiness, indeterminacy, and falsity, unlike the general fuzzy and IFS. With the daily progress of research, Wang et al. [24] proposed a single-valued neutrosophic set (SVNS) that solves complex problems by involving the study of uncertain parameters. Investigations by Chakraborty et al. [25,26] examined different categories of trapezoidal and triangular neutrosophic numbers. Fernandez et al. [27] and Abdel-Basset et al. [28] looked into the method using a single-valued trapezoidal neutrosophic number (SVTNN); this technique employs neutrosophic PERT (NPERT) to effectively navigate unpredictable settings via network analysis. Based on these findings, Nagalakshmi et al. [29] compared the study of NS and FCPP. This comparative research shows that neutrosophic sets can provide greater versatility in risk assessment. Another work by Priyadharsini et al. [30] evaluated triangular NPERT analysis for estimating project time and costs. A lot of researchers are studying NS under different optimization methods, such as the shortest path [31], minimum spanning tree [32], MCDM [33], linear programming problem [34], and so on. In addition to their theoretical and practical uses, ongoing research also uses computer implementations of these advanced ideas. These have greatly improved tools, such as the NCMPy package for managing neutrosophic cognitive maps [35] and the open-source python neutrosophic package [36], which are based on this theoretical base. The main study is about how to solve CPP in a neutrosophic environment using the python programming language. This will enhance the effectiveness and efficiency of computing environments by enabling the effective application of neutrosophic understanding.

The following is a list of the key research contributions to the development of the CPP objective:

• According to recent literature, CPP solves complex problems by including uncertainty study.

• To represent uncertainty, our proposed model uses CPP in a neutrosophic environment (NCPP). It uses a single-valued triangular neutrosophic number (TrSVNN) to represent the uncertainty and ambiguity that come with project timelines. Furthermore, the initiation also addresses a new set of problems with varying uncertainty parameters.

• The task involves implementing a score function that aims to quantify the accuracy of project analysis.

• Developed the proposed methodology in python, a computational programming language, to elucidate the nuances of neutrosophic logic.

• Implementing a comparative analysis that reveals neutrosophic logic’s superior capabilities over classical and fuzzy in demonstrating its enhanced effectiveness while dealing with uncertainty and complexity.

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 literature utilizes a scoring approach under neutrosophic study as a further extension of FCPP.

• A comparative study analysis is conducted on our proposed model to that of previous existing FCPP and CCPP models.

• An innovation to this study is the use of python for computational execution, which aids in better quality decision-making.

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.

The paper includes a background on the fundamental concepts of FS, NS, TrSVNN is as follows:

Definition 2.1. Fuzzy Set [11]: A set

Definition 2.2. Neutrosophic Set (NS) [23]: A set

Definition 2.3. Triangular Single-Valued Neutrosophic Number (TrSVNN) [25]: TrSVNN is defined as

In special case, when

Definition 2.4. Comparison between two SVTNN [37]: Consider two SVTNN as

1.

2. If

where

1. Score function is defined as:

2. Accuracy function is defined as:

3. Certainty function is defined as:

Note: If the SVTNN

Definition 2.5. Arithmetic operations between two (TrSVNN) [38]: Let

• Addition:

• Subtraction:

Definition 2.6. Binary operations between two (TrSVNN): Let

2.1 List of Abbreviation Used throughout This Paper

• CPP represents “critical path problem”.

• TrFN represents “triangular fuzzy number”.

• TFN represents “trapezoidal fuzzy numbers”.

• TrIFS represents “triangular intuitionistic fuzzy sets”.

• CCPP represents “classical critical path problem”.

• NS represents “neutrosophic set”.

• TrSVNN represents “single-valued triangular neutrosophic number”.

• SVTNN represents “single-valued trapezoidal neutrosophic number”.

• CP represents “critical path”.

• NPP represents “neutrosophic non-critical possible paths”.

• NCPP represents “neutrosophic critical path problem”.

• NCPL represents “neutrosophic critical path length”.

• NCP represents “neutrosophic critical path”.

• CCPL represents “critical crisp path length”.

• FCPP represents “fuzzy critical path problems”.

• FCP represents “fuzzy critical path”.

• FCPL represents “fuzzy critical path length”.

The exploration to delve the study of existing CCPP and the proposed NCPP is discussed above to enhance the decision-making framework in addressing the uncertainty in project scheduling.

3.1 Existing Critical Path Network Problem Formulation under Classical Environment

The CCPP implements a dynamic programming in the network, a cyclic-directed graph

The weight

3.2 Proposed Critical Path Network Problem Formulation under Neutrosophic Environment

The NCPP and edge weights

The interval within edge weights

Refining this expression yields:

If

where equality holds at least one pathway since

Transitioning neutrosophic on both sides of the equation, the modified Eq. (3.6) evolves to:

Before proceeding with the further step, the implementation of the score function

Let

The reformulation of Eq. (3.1) is as follows: for any vertex

Updating and applying neutrosophic on both sides of the Eq. (3.10) allows us to compare the aggregated neutrosophic weights as follows:

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

Consequently, from the Eqs. (3.5), (3.10), and (3.12), the decision maker (DM) choose appropriate bounds:

The dynamic programming problem recursion for the neutrosophic critical path problem from the Eqs. (3.9), and (3.12) is thus formalized as:

Encapsulate Eq. (3.14), where

3.3 Proposed Algorithm for Solving TrSVNN NCPP

The numerical analysis shows a network structure defined in Fig. 2 from [39], where nodes 1 to 5 are project activities. Initial activity durations are based on the existing CCPP. The proposed algorithmic method transmits the initial activity durations to the TrSVNN context. Decision-makers utilize the interval for each activity and choose appropriate values corresponding to truth, indeterminacy, and falsity

4.1 Existing Classical Critical Path Problem (CCPP)

Example 4.1. In CCPP [39], assuming the edge weights of the network (ref Fig. 2) are as follows:

Solution: The working model from Eq. (3.1) is calculated, and the CP of the classical case from node 1 to node 5 is

4.2 Comparing with the Existing Method

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,

Example 4.2.1. Case-I: If the DM chooses the condition as

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:

From Steps 6 to 8, the case-I

Finally, the NCPL is

Example 4.2.2. Case-II: If the DM chooses the condition as

Solution: Similary, by implementing our proposed algorithmn in Subsection 3.3, the bounds are structured as:

In contrast, case-II

Based on the condition

Example 4.2.3. Case-III: If the DM chooses the hybrid approach condition as

Solution: Similarly, implementing the algorithmic approach from steps 1 to 9, the obtained lower and upper bounds as:

Upon the analysis using three hybrid case-III

For case-III, the NCPL is

Synthesizing the neutrosophic decision-making parameter involving three distinct cases to illustrate the complexity of project uncertainty and equivalence. The condition

4.3 Analysis of Neutrosophic Path Lengths Under Varied Degrees

Example 4.3. Within this framework, the study now advances by exploring neutrosophic with varied degrees of uncertainty. From the previous Example 4.2, the study outlines TrSVN with

Solution: From Table 6, the proposed algorithm from steps 1 to 9 evaluates the result outcomes of NCPP. It illustrates different neutrosophic conditions by varying uncertainty parameters; the obtained SCCPL is 17.36 days, when illustrated the condition

The comparison between Example 4.2 and Example 4.3 involving the study results having the same NCP as

A comparison of NCPP with both conventional CCPP and FCCPP was the primary emphasis of Examples 4.2 and 4.3. More conventional systems tend to simplify or ignore the inherent uncertainties in project management activities; the main goal was to evaluate NCPP’s ability to accommodate and dynamically adapt to these uncertainties.

Our study proposes a structured NCPP model that integrates into project management networks. This dynamic method’s adaptive algorithm updates project uncertainty dynamically. By leveraging the capabilities of TrSVNN, the NCPP facilitates a refined measurement of uncertainty that plays a crucial role in the field of complex projects. Using NCPP in three different situations gives a more varied result while keeping the same NCP as

Acknowledgement: The authors express their sincere gratitude to the reviewers and the chief-editor for their valuable insights and suggestions, which significantly enhanced the quality and depth of our paper.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Navya Pratyusha M., Kumar R.; analysis and interpretation of results: Navya Pratyusha M., Kumar R.; draft manuscript preparation: Navya Pratyusha M. The authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: All the data contained in this study from the necessary sources can potentially access or by getting in touch with the paper’s corresponding author.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

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*Computer Modeling in Engineering & Sciences*,

*140*

*(3)*, 2957-2976. https://doi.org/10.32604/cmes.2024.051581

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*Comput. Model. Eng. Sci.*, vol. 140, no. 3, pp. 2957-2976. 2024. https://doi.org/10.32604/cmes.2024.051581

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