Scientific Programming

Volume 2016, Article ID 6158208, 8 pages

http://dx.doi.org/10.1155/2016/6158208

## The Intelligence of Octagonal Fuzzy Number to Determine the Fuzzy Critical Path: A New Ranking Method

Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

Received 8 December 2015; Accepted 23 February 2016

Academic Editor: Piotr Luszczek

Copyright © 2016 S. Narayanamoorthy and S. Maheswari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This research paper proposes a modified ranking approach to determine the critical path in fuzzy project network, where the duration of each activity time is represented by an octagonal fuzzy number. In this method, a modified subtraction formula is carried out on fuzzy numbers. This modified method works well on fuzzy backward pass calculations as there will be no negative time. The analysis is expected to show that the fuzzy number which is used in this paper is more effective in determining the critical path in a fuzzy project network and possibility of meeting the project time. A numerical example is given and compared with trapezoidal, triangular fuzzy numbers through proposed ranking method.

#### 1. Introduction

The range of project management application has significantly expanded. The project management concerning the scheduling and monitoring progress within the cost, time, and duration is becoming important to obtain competitive priorities such as time delivery. The activity duration time in project is known and deterministic; critical path method (CPM) has been demonstrated to be a useful tool in the planning and control of complicated projects in management and engineering applications. The critical path method worked out at the beginning of the 1960s. In reality, it is often difficult to obtain estimates of activity time, due to the uncertainty of information. In this situation, Zadeh proposed an alternative way of this kind of decision-making environment.

#### 2. Related Work

Critical path method (CPM) has been demonstrated to be a useful tool in managing projects in an efficient manner to meet the challenges presented by Hillier and Lieberman [1]. In business management, Ahuja et al. [2] discussed factory production using critical path method. Mon et al. [3] and Dubois et al. [4] presented a project network defined as a set of activities that must be performed according to precedence constraints stating which must start after the completion of other specified activities. Natarajan et al. [5] and Sivarenthinamohan [6] presented the backward pass which is performed to calculate the fuzzy latest-start and latest finish times. Khalaf [7] applied fuzzy theory to find fuzzy total float, free float, and fuzzy independent float for a project network with activity duration modeled as triangular fuzzy numbers. Oladeinde and Itsisor [8] obtained the fuzzy early start, early finish, fuzzy latest start, and latest finish as well as the critical activities with triangular activity duration. Shahsavari Poura et al. [9] presented another fuzzy critical path method where the processing times of all activities follow trapezoidal fuzzy numbers. Soundararajan et al. [10] ascertained that Artificial Neural Network is an alternative method for predicting the mechanical properties and appropriate results can be estimated rather than measured, thereby reducing the testing time and cost.

Zadeh [11] introduced an alternative way to deal with imprecise data to employ the concept of fuzziness. Liang and Han [12] developed an algorithm which is presented to perform critical path analysis in a fuzzy environment. Jain [13] proposed the concept of fuzzy numbers. Chanas and Radosinski [14] analyzed the use of fuzzy numbers in the network planning. M. H. Oladeinde and C. A. Oladeinde [15] considered the effectiveness of the decision-makers’ risk attitude index for fuzzy critical path analysis. Han et al. [16] presented the fuzzy critical path method to find out airport’s ground critical operation processes. Lin [17] presented practical support to the management of organizations in order to make a formation program of human resources. Ravi Shankar et al. [18] used trapezoidal fuzzy numbers to rank the set of fuzzy numbers in a fuzzy project network. Karimirad et al. [19] anticipated a novel method for ranking of fuzzy numbers is proposed that is based on the real numbers. Rotarescu [20] offered practical support to the management of organizations in order to make a formation program of human resources. Kazemi et al. [21] used the fuzzy Delphi-analytical hierarchy process method and also the ranking method would help product designers to decide on appropriate materials in a consistent method.

This paper presents another approach, which has not been proposed in the literature so far, to analyze the critical path in a project network with fuzzy activity times. A modified arithmetic operation is used on octagonal, trapezoidal, and triangular fuzzy numbers and applied to the float time for each activity in the fuzzy project network. We compared trapezoidal and triangular with octagonal fuzzy numbers and which one is more effective to determining the activity criticalities to find the critical path in a network.

#### 3. Preliminaries

In this section, we present the most basic notations and definitions, which we used throughout this work. We start with defining a fuzzy set.

##### 3.1. Fuzzy Set

Let be a set. A fuzzy set on is defined to be a function or . Equivalently, A fuzzy set is defined to be the class of objects having the following representation: where is a function called the membership function of .

##### 3.2. Fuzzy Number

The fuzzy number is a fuzzy set whose membership function satisfies the following conditions:(1) is piecewise continuous.(2)A fuzzy set of the universe of discourse is convex.(3)A fuzzy set of the universe of discourse is called a normal fuzzy set if exists.

##### 3.3. Trapezoidal Fuzzy Number

A fuzzy number with membership function in the form is called a trapezoidal fuzzy number .

##### 3.4. Triangular Fuzzy Number

A fuzzy number with membership function in the form is called a triangular fuzzy number .

#### 4. Formulation of Octagonal Fuzzy Number for Ranking

In the current section, we present a new method for ranking fuzzy numbers. Ravi Shankar et al. and Narayanamoorthy and Maheswari [18, 22] have studied ranking methods using triangular and trapezoidal fuzzy numbers. Here we have extended ranking methods and modified arithmetic operations using octagonal fuzzy number to find fuzzy critical path. Octagonal fuzzy number and modified arithmetic operations are presented in the following subsections.

##### 4.1. Octagonal Fuzzy Number

Here, we present a method for ranking fuzzy numbers. This method calculates the ranking scores of the octagonal fuzzy number. Fuzzy numbers are an octagonal fuzzy number denoted by where are real numbers and its membership functions , , and are given below: Figure 1 depicts that the graphical representation of an octagonal fuzzy number .