Fuzzy critical path method based on ranking methods using hexagonal fuzzy numbers for decision making

Abstract

In this paper a ranking procedure based on Hexagonal Fuzzy numbers is applied to find a critical path in a fuzzy project network. Here we introduce hexagonal fuzzy numbers for indicating duration of each activity time. Modified arithmetic operations are applied to find fuzzy latest finish time which gives precise result as there will be no negative time. Our proposed method provides better way of finding fuzzy critical path and aid in decision making in a complex project network. A numerical example is conferred to illustrate the potential use of the proposed approach.

Keywords

Fcpm fuzzy project network hexagonal fuzzy number ranking methods decision making

1 Introduction

In todays competitive world competing project on time is essential to be successful in business. The investor wants to optimise resource utilisation to complete the project on time with reduced cost to meet the challenge in business scenario. Critical path method employed by project managers to compute minimal project duration and to identify critical path.The unknown problem that could occur in practical situation can be very well managed using this fuzzy Critical path method. In real life problems, decision makers generally face lot of confusions, ambiguity due to the involvement of uncertanity. To deal with such kind of vagueness the fuzzy set theory proffers the possibility to construct decision models with vague data. The data values need to be any fuzzy numbers. Fuzzy set was first proposed by Zadeh [21] as a mean of handling uncertanity about the time duration of activities in the network planning. The backward pass is performed to calculate the fuzzy latest-start and latest-finish times were presented by [11 , 17]. Critical path method has been applied in business management, factory production, etc., and analysed by [4, 12]. Slyptsov and Tyshchuk [18] pointed out that the critical path method is a useful tool for planning and managing of complicated projects in real world applications.

Chen and Huang [6] introduced a graphical display of project activities (tasks), an estimate of how long the project will take. Abdullah et al. [2] developed a general algorithm to solve decision making problems accomplished by bipolar fuzzy soft set. Elizabeth and Sujatha [9] extended the methods of fuzzy critical path and the fuzzy critical length are presented in the nature of fuzzy membership function. Mehdi et al. [13] a new method for ranking fuzzy numbers based on the left and right using distance method and a-cut has been presented. Amit Kumar et al. [5] asserted that ranking of fuzzy numbers play an vitalt role in decision making, optimization, forecasting etc. Eslamipoor et al. [10] indicated a new revised method for ranking fuzzy numbers which can avoid problem for ranking fuzzy numbers. Chen and Cheng [7] proposed a metric distance method to rank fuzzy numbers. Tzeu-Chen Han et al. [19] originated the fuzzy critical path method to expose airport’s ground critical operation processes. Zhong and Xu [20] examined the multiple attribute decision making (MADM) problems in which attribute values take the form of hesitant triangular fuzzy information. Ravi Shankar et al. [15] manipulated on trapezoidal fuzzy numbers to rank the set of fuzzy numbers in a fuzzy project network. Abdullah and Amin [3] observed a generalized fuzzy soft expert set criterion is accomadating to decide the suitability of an S-box to image encryption applications. Stefan and Zielinski [8] anticipated that the concept of fuzzy criticality. Abalfazl et al. [1] applied two linear programming models in order to calculate earliest and latest events time in project scheduling problem. Ravi Shankar et al. [16] exploited trapezoidal fuzzy numbers using metric distance ranking method to find a critical path for the fuzzy project network.

In this paper the authors extended to determine a fuzzy critical path using hexagonal fuzzy number in metric distance ranking and sanguansat and chen ranking methods. We use individual fuzzy number for metric distance and set of fuzzy number is used for sanguansat and chen ranking methods. The arithmetic operations are modified to find backward pass calculation, this gives precise result as there will be no negative time in fuzzy subtraction operation. Since in physical activity negative time doesn’t exist and has no relevance. So the proposed method is very useful to find fuzzy critical path for decision makers.

The rest of this paper is organized as follows. In Section 2, we briefly review some basic definitions. Formulation of hexagonal fuzzy number and modified arithmetic operations are proposed in Section 3. In Section 4, the algorithm and procedure for finding critical path in a network are presented. In Section 5, we apply the proposed fuzzy ranking methods and numerical problem is solved. Finally, the conclusions of the work are presented in Section 6.

2 Preliminaries

In this section, we briefly review some basic definitions of fuzzy sets, metric distance and critical path method.

Definition 2.1. Critical Path Method Network analysis technique used in complex project plans with a large number of activities. Critical path method diagram all activities, time required for their completion and how each activity is related to the previous and next activity. A sequence of activities is called a ‘path,’ and the longestpath in the diagram is the critical path. It is ‘critical’ because all activities on it must be completed in the designated time, otherwise the whole project will be delayed. Critical path method is commonly used with all forms of projects, including construction, aerospace and defense, software development, research projects, product development, engineering, and plant maintenance etc.

Definition 2.2. [21] Let X be a set. A fuzzy set A on X is defined to be a function A : X → [0, 1] or μ_A : X → [0, 1] Equivalently, A fuzzy set A is defined to be the class of objects having the following representation: A ={ (x, μ_A (x) , x ∈ X) } where μ_A : X → [0, 1], is a function called the membership function of A.

Definition 2.3. A fuzzy number is a fuzzy subset in the universe of discourse X that is both convex and normal, and satisfying the following conditions:

μ_A (x) is piecewise continuous

μ_A (x) is a convex, i.e., μ_A (λx + (1 - λ) y) ≥ min (μ_A (x) , μ_A (y)), ∀x, y ∈ R and λ ∈ [0, 1].

A is normal, i.e., there exist an element x₀ ∈ R such that μ (x₀) =1.

Definition 2.4. A Fuzzy number A interms of mean and standard deviation is denoted by $\begin{matrix} A & = & (g_{A}^{L} (y), g_{A}^{R} (y)) \\ = & ((μ - σ) + σ y, (μ + σ) - σ y) \end{matrix}$ y ∈ [0, 1] , it is obtained as follows, $\begin{matrix} y = \frac{g_{A}^{L} (y) - (μ - σ)}{σ} \Rightarrow g_{A}^{L} (y) = (μ - σ) + σ y \\ y = \frac{(μ + σ) - g_{A}^{R} (y)}{σ} \Rightarrow g_{A}^{R} (y) = (μ + σ) - σ y \\ \int_{0}^{1} d ((g_{A}^{L} (y), g_{A}^{R} (y)), 0) dy = \int_{0}^{1} \frac{g_{A}^{L} (y) + g_{A}^{R} (y)}{2} dy \end{matrix}$ $mean μ = \frac{g_{A}^{L} (y) + g_{A}^{R} (y)}{2}$ (1)

Definition 2.5. The distane between two points P (x₁, y₁) and Q (x₂, y₂) is $\begin{matrix} d (P (x_{1}, y_{1}), Q (x_{2}, y_{2})) \\ = \sqrt{(Δ x)^{2} + (Δ y)^{2}} \\ = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} \end{matrix}$

The metric distance between $A_{α}^{} = (A_{L} (α), A_{R} (α))$ and 0 is calculated as follows

$D (A_{α}^{}, 0) = {[\int_{0}^{1} (A_{L} (α)^{2}) + \int_{0}^{1} (A_{R} (α)^{2})]}^{\frac{1}{2}}$ (2)

Definition 2.6. (Properties of Metric Distance) Let X be a non-empty set and d : X × X → R be a function that satisfies the following conditions.

(d1) : d (x, y) ≥0 ∀ x, y ∈ X and d (x, y) =0 iff x = y

(d2) : d (x, y) = d (y, x) for x, y ∈ X

(d3) : d (x, y) ≤ d (x, z) + d (z, y) for any x, y, z ∈ X

Since d is a metric by the condition (d3), we have

$\begin{matrix} d (x, y) \leq d (x, z) + d (z, y) \\ d (x, y) - d (z, y) \leq d (x, z) \end{matrix}$ (3)

Again interchanging x and z, we obtain from (1)

$\begin{matrix} d (z, y) - d (x, y) \leq d (z, x) = d (x, y) \\ - (d (x, y) - d (z, y)) \leq d (x, z) \end{matrix}$ (4) combining 1 and 2, it follows that $d (x, y) \leq d (x, z) + d (z, y)$ (5)

The function d (. , .) satisfying (d1) , (d2) and (d3) is called a metric and the structure (X, d) is called a metric space.

Definition 2.7. (Axioms) Given $r, \tilde{r}, \hat{r} \in R$ , one says that $\tilde{r}$ is between r and $\hat{r}$ and one writes $[r, \tilde{r}, \hat{r}]$ if $\forall s, t \in A : \tilde{r_{s}} < \hat{r_{t}} \Rightarrow r_{s} < r_{t}$ or $\hat{r_{s}} < \hat{r_{t}}$ and $\tilde{r_{s}} = \tilde{r_{t}} \Rightarrow (r_{s} - r_{t}) (\hat{r_{s}} < \hat{r_{t}}) \leq 0 .$

Given $r, \tilde{r}, \hat{r} \in R, d (r, \hat{r}) = d (r, \tilde{r}) + d (\tilde{r}, \hat{r})$ whenever [ $r, \tilde{r}, \hat{r}$ ]

$d (σ (r), σ (\hat{r})) = d (r, \hat{r})$ for every permutation σ of the elements of r and $\hat{r}$ .

if an alternative is added to A and if this alternative is ranked first or last relative to both r and $\hat{r}$ , so that r and $\hat{r}$ become r^* and $\hat{r^{*}}$ respectively, then $d (r^{*} {and \hat{r^{*}}) = d (r, \hat{r}) .$

$min_{r, \hat{r} \in R} d (r, \hat{r}) \in {0, 1}$ that is, the minimum positive distance is 1.

3 Formulation of hexagonal fuzzy number for ranking

In current section, we present a new method for ranking fuzzy numbers. Ravi Shankar et al. [15, 16] have studied on ranking methods using triangular and trapezoidal fuzzy numbers. Here we have extended ranking methods and modified arithmetic operations using hexagonal fuzzy number to find fuzzy critical path based on Sanguansat and Chen and Metric distance ranking methods which is not researched so far. Motivation behind this study is this method can be employed where a particular problem cannot be solved by triangular or trapezoidal fuzzy numbers.

Hexagonal fuzzy number and modified arithmetic operations are presented in the following subsections 3.1 and 3.2.

3.1 Hexagonal fuzzy number

Fuzzy numbers (A, B) is a hexagonal fuzzy number denoted by (A, B) = (a, b, c, d, e, f) where (a, b, c, d, e, f) are real numbers and its membership functions μ_A (x) and μ_B (x) are given below. $μ_{A} (x) = {\begin{matrix} \frac{x - a}{c - a}, a \leq x \leq c \\ 1, c \leq x \leq d \\ \frac{x - f}{d - f}, d \leq x \leq f \\ 0, otherwise \end{matrix}$ (6) $μ_{B} (x) = {\begin{matrix} \frac{x - a}{b - a}, a \leq x < b \\ 0.8, b \leq x \leq e \\ \frac{f - x}{f - e}, e < x \leq f \\ 0, otherwise \end{matrix}$ (7) is called a hexagonal fuzzy number (A, B) = (a, b, c, d, e, f)

Under universe of discourse [0,1] as described in Fig. 1 has relationship μ_A (x) and μ_B (x) defined in Equations (6) and (7) where a ≤ b ≤ c ≤ d ≤ e ≤ f, $\begin{matrix} a = f = 0, \\ b = e = 0.8, \\ c = d = 1 . \end{matrix}$ (Instead of membership value 0.8, we can choose any fuzzy membership value between 0 and 1).

3.2 Modified arithmetic operations

Let A₁ and A₂ be two hexagonal fuzzy numbers parameterized by (a₁, a₂, a₃, a₄, a₅, a₆) and (b₁, b₂, b₃, b₄, b₅, b₆) respectively. The simplified fuzzy number arithmetic operations between the fuzzy numbers A₁ and A₂ are as follows: Fuzzy numbers addition ⊕ : (a₁, a₂, a₃, a₄, a₅, a₆) ⊕ (b₁, b₂, b₃, b₄, b₅, b₆) = (a₁ + b₁, a₂ + b₂, a₃ + b₃, a₄ + b₄, a₅ + b₅, a₆ + b₆) .

Fuzzy numbers subtraction ⊖ : (a₁, a₂, a₃, a₄, a₅, a₆) ⊖ (b₁, b₂, b₃, b₄, b₅, b₆) = (max(0, a₁ - b₆) , max(0, a₂ - b₅) , max(0, a₃ - b₄) , max(0, a₄ - b₃) , max(0, a₅ + b₂) , max(0, a₆ + b₁)) .

4 Proposed fuzzy critical path method based on ranking of fuzzy numbers

Theorem 4.1. Assume that the fuzzy activity times of all activities in a fuzzy project network are hexagonal fuzzy numbers, then there exists fuzzy critical path in the project network.

A fuzzy project network is an acyclic digraph, where the vertices represent events and the direct edges represent the activities, to be performed in a project. Formally, A fuzzy project network is represented by M = (V, A, T). Let V ={ v₁, v₂, . . . , v_n } be a set of vertices, where v₁ and v_n are the start and final events of the project, and each v_i belongs to some path from v₁ to v_n. Let A ⊂ V × V be the set of a directed edge a_ij = (v_i, v_j), that represents the activities to be performed in the project. Activity a_ij is then represented by one, and only one, arrow with a tail event v_i, and a head event v_j. For each activity a_ij, a fuzzy number t_ij ∈ T is defined, where t_ij is the fuzzy time required for the completion of a_ij. A critical path is a longest path from v₁ to v_n, and an activity a_ij on a critical path is called a critical activity. Let FE_i and FL_i be the earliest fuzzy event time and the latest fuzzy event time for event i, respectively. Let FE_j and FL_j be the earliest event time and the latest event time for event j, respectively. Let D_j ={ i/i ∈ V and a_ij ∈ A } be a set of events obtained from event j ∈ V and i < j. We then obtain E_j using the following equations

$\begin{matrix} {FE}_{j} & = & max_{i \in D_{j}} [{FE}_{i} \oplus t_{ij}] and \\ {FE}_{1} & = & {FL}_{1} = 0 \end{matrix}$ (8)

Similarly, let H_i ={ j/j ∈ V and aij ∈ A } be a set of events obtained from event i ∈ V and i < j. We obtain FL_i using the following equations

${FL}_{i} = min_{j \in H_{i}} [{FL}_{j} ⊖ t_{ij}] and {FE}_{n} = {FL}_{n}$ (9)

The interval [FE_i, FL_j] is the time during which a_ij must be completed. When the earliest fuzzy event time and latest fuzzy event time have been obtained, we can calculate the total float of each activity. For activity i-j in a fuzzy project network, the total float FT_ij of the activity i - j can be computed as follows:

${FT}_{ij} = [{FL}_{j} ⊖ {FE}_{i} ⊖ t_{ij}]$ (10)

Hence we can obtain the earliest fuzzy event time, latest fuzzy event time, and the total float of every activity by using Equations (8–10).

Algorithm and Applications

One of the main aims of this paper is to extend the fuzzy critical path algorithm using hexagonal fuzzy number and construct its algorithmic form allowing us to determine fuzzy critical path by Sanguansat and Chen ranking and Metric distance ranking methods of hexagonal fuzzy number. Consider the fuzzy project network, where the duration time of each activity in a fuzzy project network is represented by hexagonal fuzzy number.

Step 1: Calculate FE_i’s using Equation 8.

Step 2: Calculate FL_j’s using Equation 9.

Step 3: Calculate FT_ij’s for each activity (i, j) using Equation 10.

Step 4: Find all the possible paths and calculate the total slack fuzzy time of each path.

Step 5: Rank the total slack fuzzy time of each path using Sanguansat and Chen and Metric distance ranking Methods.

Step 6: The path having minimum rank in step 5 is the critical path.

In the following subsection we demonstrated the ranking procedure for Sanguansat and Chen and Metric distance ranking methods using hexagonal fuzzy number.

4.1 Method-I

Assume that there are n fuzzy numbers A ={ A₁, A₂, A₃ . . . , A_n }. Here we use hexagonal fuzzy number to rank set of fuzzy number.

Step 1: Transform each hexagonal fuzzy numbers

into a standardized fuzzy number $A_{i}^{*}$ ,

$\begin{matrix} A_{i}^{*} = (\frac{a_{i 1}}{k}, \frac{a_{i 2}}{k}, \frac{a_{i 3}}{k}, \frac{a_{i 4}}{k}, \frac{a_{i 5}}{k}, \frac{a_{i 6}}{k}) \\ = (a_{i 1}^{*}, a_{i 2}^{*}, a_{i 3}^{*}, a_{i 4}^{*}, a_{i 5}^{*}, a_{i 6}^{*}) \end{matrix}$ (11)

Where k = max ∥ a_ij ∥ , |a_ij| denotes the absolute value of a_ij and ∥a_ij∥ denotes the upper bound of |a_ij|, 1 ≤ i ≤ n and 1 ≤ j ≤ 6 .

Step 2: Calculate standard deviation of each standardized fuzzy number $A_{i}^{*}$

$σ_{A_{i}^{*}} = \sqrt{\frac{\sum_{j = 1}^{6} (a_{ij}^{*} - \bar{a_{i}})^{2}}{6 - 1}}$ (12) where $\bar{a_{i}}$ denotes the mean of the values $a_{i 1}^{*}, a_{i 2}^{*}, a_{i 3}^{*}, a_{i 4}^{*}, a_{i 5}^{*}, a_{i 6}^{*}$ and 1 ≤ i ≤ n.

Step 3: Calculate the areas ${Area}_{iL}^{-}$ and ${Area}_{iR}^{-}$ of each standardized fuzzy number $A_{i}^{*}$ where

$\begin{matrix} {Area}_{iL}^{-} = \frac{(a_{i 1}^{*} + 1) + (a_{i 2}^{*} + 1) + (a_{i 3}^{*} + 1)}{3}, \\ {Area}_{iR}^{-} = \frac{(a_{i 4}^{*} + 1) + (a_{i 5}^{*} + 1) + (a_{i 6}^{*} + 1)}{3} . \end{matrix}$ (13) then calculate the areas ${Area}_{iL}^{+}$ and ${Area}_{iR}^{+}$ of each standardized fuzzy number $A_{i}^{*}$ where $\begin{matrix} {Area}_{iL}^{+} = \frac{(1 - a_{i 1}^{*}) + (1 - a_{i 2}^{*}) + (1 - a_{i 3}^{*})}{3}, \\ {Area}_{iR}^{+} = \frac{(1 - a_{i 4}^{*}) + (1 - a_{i 5}^{*}) + (1 - a_{i 6}^{*})}{3} \end{matrix}$ (14)

Step 4: Calculate the values ${XI}_{A_{i}^{*}}$ and ${XD}_{A_{i}^{*}}$ of each standardized fuzzy number $A_{i}^{*}$

$\begin{matrix} {XI}_{A_{i}^{*}} = {Area}_{iL}^{-} + {Area}_{iR}^{-}, \\ {XD}_{A_{i}^{*}} = {Area}_{iL}^{+} + {Area}_{iR}^{+} + σ_{A_{i}^{*}} \end{matrix}$ (15)

Step 5: Calculate the ranking score $A_{i}^{*}$ of each standardized fuzzy number $A_{i}^{*}$

$Score (A_{i}^{*}) = \frac{{XI}_{A_{i}^{*}} - {XD}_{A_{i}^{*}}}{{XI}_{A_{i}^{*}} + {XD}_{A_{i}^{*}} + k}$ (16) where k = 1 if ${XI}_{A_{i}^{*}} - {XD}_{A_{i}^{*}} \geq 0$

k = -1 otherwise and 1 ≤ i ≤ n

4.2 Method-II

Here we make use of hexagonal fuzzy number to rank individual fuzzy number. Let A and B be two fuzzy numbers defined as follows: $\begin{matrix} f_{A} (x) & = & {\begin{matrix} f_{A}^{L} (x), x \leq m_{A} \\ f_{A}^{R} (x), x \geq m_{A} \end{matrix} \\ f_{B} (x) & = & {\begin{matrix} f_{B}^{L} (x), x \leq m_{B} \\ f_{B}^{R} (x), x \geq m_{B} \end{matrix} \end{matrix}$

Where m_A and m_B are the mean of A and B. The metric distance between A and B can be calculated as follows:

$\begin{matrix} D (A, B) & = & [\int_{0}^{1} (g_{A}^{L} (y) - g_{B}^{L} (y))^{2} dy \\ {+ \int_{0}^{1} (g_{A}^{R} (y) - g_{B}^{R} (y))^{2} dy]}^{\frac{1}{2}} \end{matrix}$ (17) where $g_{A}^{L}, g_{A}^{R}, g_{B}^{L}$ and $g_{B}^{R}$ are the inverse functions of $f_{A}^{L}, f_{A}^{R}, f_{B}^{L}$ and $f_{B}^{R}$ respectively. In order to rank fuzzy numbers, let the fuzzy number B = 0 then the metric distance between A and 0 is calculated as follows

$D (A, 0) = {[\int_{0}^{1} (g_{A}^{L} (y))^{2} dy + \int_{0}^{1} (g_{A}^{R} (y))^{2} dy]}^{\frac{1}{2}}$ (18)

The large value of D (A, 0), is the better ranking of A. A hexagonal fuzzy number A = (a₁, a₂, a₃, a₄, a₅, a₆) can be approximated as a symmetry fuzzy number S [μ, σ] , μ denotes the mean of A, σ denotes the standard deviation of A in terms of Mean and Standard Deviation is defined as follows:

$f_{A} (x) = {\begin{matrix} \frac{x - (μ - σ)}{σ}, if μ - σ \leq x \leq μ \\ \frac{(μ + σ) - x}{σ}, if μ \leq x \leq μ + σ \end{matrix}$ (19) where μ and σ are calculated as follows $σ = \frac{2 (a_{6} - a_{1}) + (a_{5} - a_{2}) + (a_{4} - a_{3})}{6}$ (20) $μ = \frac{a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6}}{6}$ (21)

The inverse functions $g_{A}^{L}$ and $g_{A}^{R}$ of $f_{A}^{L}$ and $f_{A}^{R}$ respectively, are shown as follows: $g_{A}^{L} = (μ - σ) + σ \times y$ (22) $g_{A}^{R} = (μ + σ) - σ \times y$ (23)

5 Numerical example

To illustrate the proposed method, consider the following example intended for fuzzy critical path method using hexagonal fuzzy number for decision making. Consider the project network, Node 1 - Getting rid of the old furniture. Node 2 - Painting the walls. Node 3 - Fixing the ceiling. Node 4 - Installing the new furniture. Node 5 - Choosing the new curtains. Node 6 - Hanging the new curtains.

5.1 Example for proposed method-I

The project given in Fig. 2 the fuzzy activity times are represented by hexagonal fuzzy number. Each fuzzy activity, activity time and total float are specified in the Table 1.

The possible paths, total float and total fuzzy slack times are calculated and given in the Table 2.

by using the Equations (11–(16) we have calculated the following values $\begin{matrix} A_{1}^{*} & = & (0.8076, 0.0961, 0.1923, 0.3846, \\ 0.5769, 0.5769), \bar{a_{1}} = 0.4390 \\ A_{2}^{*} & = & (0.9615, 0, 0.0384, 0.2884, \\ 0.7692, 0.3269), \bar{a_{2}} = 0.3974 \\ A_{3}^{*} & = & (1, 0, 0, 0.2884, 0.5769, \\ 0.9330), \bar{a_{3}} = 0.4647 \\ A_{4}^{*} & = & (0.9230, 0, 0, 0.1730, \\ 0.4615, 0.1730), \bar{a_{4}} = 0.2884 \\ A_{5}^{*} & = & (0.7307, 0.0576, 0.1346, 0.2884, \\ 0.4615, 0.4230), \bar{a_{5}} = 0.3493 . \\ σ_{A_{1}^{*}} & = & 0.3428, σ_{A_{2}^{*}} = 0.3896, \\ σ_{A_{3}^{*}} & = & 0.4408, σ_{A_{4}^{*}} = 0.3536, σ_{A_{5}^{*}} = 0.2441 \\ {Area}_{1 L}^{-} & = & 1.3653, {Area}_{2 L}^{-} = 1.3333, \\ {Area}_{3 L}^{-} & = & 1.3333, {Area}_{4 L}^{-} = 1.3076, \\ {Area}_{5 L}^{-} & = & 1.3076, {Area}_{1 R}^{-} = 1.5128, \\ {Area}_{2 R}^{-} & = & 1.4615, {Area}_{3 R}^{-} = 1.5961, \\ {Area}_{4 R}^{-} & = & 1.2691, {Area}_{5 R}^{-} = 1.3909, \\ {Area}_{1 L}^{+} & = & 0.6346, {Area}_{2 L}^{+} = 0.5705, \\ {Area}_{3 L}^{+} & = & 0.6666, {Area}_{4 L}^{+} = 0.6923, \\ {Area}_{5 L}^{+} & = & 0.6923, {Area}_{1 R}^{+} = 0.4872, \\ {Area}_{2 R}^{+} & = & 0.5385, {Area}_{3 R}^{+} = 0.4039, \\ {Area}_{4 R}^{+} & = & 0.7308, {Area}_{5 R}^{+} = 0.6090 \end{matrix}$

By using ${Area}_{iL}^{-}$ , ${Area}_{iR}^{-}$ , ${Area}_{iL}^{+}$ , ${Area}_{iR}^{+}$ and $σ_{A_{i}^{*}}$ from this we can get the values of ${XI}_{A_{i}}^{*}, {XD}_{A_{i}}^{*}$ and Score $(A_{i}^{*})$ values. The values are determined and tabulated in the Table 3.

From the calculated scores, the path which is having minimum score is 1-4-5-6. This is the critical path for the considered fuzzy project network.

5.2 Example for proposed method-II

The project given in Fig. 2 the fuzzy activity times are represented by hexagonal fuzzy number. Each fuzzy activity, activity time and total float are specified in the Table 4.

The possible paths, total float and total fuzzy slack times are calculated and given in the Table 5.

by using the Equations (17)– (23) we have calculated the following values of standard deviation, mean, inverse functions and metric distance values of each path.

(i) Total fuzzy slack time for the path 1 - 2 -6 is (0, 10, 20, 30, 25, - 22)

σ = -3.166, μ = 10.5

$g_{A}^{L} (y) = 13.666 - 3.166 y, g_{A}^{R} (y) = 7.334 + 3.166 y$

D (A, 0) =15.072

(ii) Total fuzzy slack time for the path 1 - 3 -5 - 6 is (-7, - 1, 15, 40, 27, - 35)

σ = -0.5, μ = 6.5

$g_{A}^{L} (y) = 7 - 0.5 y, g_{A}^{R} (y) = 6 + 0.5 y$

D (A, 0) =9.021

(iii) Total fuzzy slack time for the path 1 - 2 -5 - 6 is (-18, - 3, 15, 30, 39, - 37)

σ = 3.166, μ = 4.333

$g_{A}^{L} (y) = 1.167 + 3.166 y, g_{A}^{R} (y) = 7.499 - 3.166 y$

D (A, 0) =6.650

(iv) Total fuzzy slack time for the path 1 - 4 -5 - 6 is (-24, - 9, 9, 24, 39, - 39)

σ = 2.5, μ = 0

$g_{A}^{L} (y) = - 2.5 + 2.5 y, g_{A}^{R} (y) = 2.5 - 2.5 y$

D (A, 0) =1.290

(v) Total fuzzy slack time for the path 1 - 4 -6 is (-5, 4, 15, 24, 24, - 23)

σ = -1.166, μ = 6.5

$g_{A}^{L} (y) = 7.666 - 1.166 y, g_{A}^{R} (y) = 5.334 + 1.166 y$

D (A, 0) =9.241

By using μ, σ, $g_{A}^{L}$ and $g_{A}^{R}$ from this we can get the metric distance rank D (A, 0) of each path in fuzzy project network is calculated and tabularized in the Table 6.

From the calculated values the path having minimum rank is 1 - 4 -5 - 6. This is the critical path for the considered fuzzy project network.

6 Conclusion

In this paper, we have presented a new ranking approach for decision making based on fuzzy critical path method using hexagonal fuzzy number. We applied modified arithmetic operations, since it eliminates negative time in fuzzy backward calculations. This new approach is highly suitable for decision making in complex project in reallife situations. This method can be used where a particular problem cannot be solved by triangular and trapezoidal fuzzy numbers. This paper solved with six fuzzy number system. In future we expect to use more fuzzy numbers and to study the comparison among them. The fuzzy critical path method is very hot and important area of research, so it will be very interesting to use metric distance ranking and sanguansat and chen ranking by using hexagonal fuzzy number to determine the fuzzy critical path.

Footnotes

Acknowledgments

The authors would like to thank the Associate Editor Prof. Jian Wu and anonymous referees for their suggestions and comments which have led to an improvement in both the quality and clarity of the paper.

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