Uncertain project network analysis with fuzzy–PERT and Interval Type–2 fuzzy activity durations

Abstract

Project network analysis is often subject to different uncertainties that can occur in real world scenarios. These uncertainties are not always measurable which makes its statistical estimation hard, so the use of third party information coming from experts is useful to provide an estimation of activity durations of the project. Projects usually have multiple experts who provide different estimates of their perceptions about activity durations which can be represented as Type-2 fuzzy sets in cases where experts do not agree on their estimates. Such disagreement/ambiguitites can be either expressed with Interval Type-2 fuzzy numbers or the proposed fuzzy-PERT set which is an extension of the classical PERT distribution to a fuzzy environment. To do so, a mathematical formulation of the fuzzy problem is presented, two solution methods are described and two illustrative examples are solved.

1. Introduction y motivation

The classical approach to find solutions of a deterministic project network is the CPM (critical path method) which finds the shortest path to finish the project and the most popular method to involve probabilistic uncertainty is the PERT (Program Evaluation and Review Technique) method which assumes each activity duration to behave as a Beta–distributed random variable described by three parameters: pessimistic, expected and optimist values. The use of Beta distribution and the PERT method has been first described by Malcolm et al. [24] as a nonlinear counterpart of the triangular distribution whose main goal is to address probabilistic uncertainty in the behavior of activity durations adding flexibility to the analysis. However, every project is singular and activity durations often lack of historical data to be estimated which is a common issue to be addressed, so the use of alternative information sources (comparative analysis, expert based estimates, etc.) have become a popular approach in many projects and fuzzy sets are appropriate for involving information coming from experts into project network analysis.

Fuzzy sets theory is useful in cases where no data to perform a proper statistical analysis is available, so third-party information coming from experts can be used to define activities duration. To do so, Dubois, Fortin & Ziełiński [10] and McCahon [25] firstly proposed the use of fuzzy uncertain activity durations in CPM and network flow problems then Fargier et al. [11] proposed the use of fuzzy sets and PERT techniques in series parallel graphs and Wang & Hao [38] have proposed the use of fuzzy linguistic variables in network analysis; Chen [8] and Chen & Wang [9] have included fuzzy imprecision coming from non-probabilistic uncertainty associated with the perception of an expert about the project and used α-cuts representation of a fuzzy set to obtain the fuzzy set of optimal durations of a project and Glisovic et al. [18] applied those results to forecasting of a project duration as a decision support system; Lootsma [23] analyzed the relationship between stochastic and fuzzy PERT techniques with triangular membership functions, Chen & Huang [7] used fuzzy PERT to a supply chain seen as a project network using triangular fuzzy sets; Buckley & Jowers [3] combined Monte Carlo simulation techniques with fuzzy PERT methods to involve randomness to the analysis and Kazemi, Fakhori & Shakourloo [20] used fuzzy logic systems to represent activity durations using fuzzy linguistic variables; Rambaud & Pérez [33] first addressed several experts estimates using a bet value equivalent to its mean. A first approach of Interval Type–2 Fuzzy Numbers (IT2FN) to project network analysis was proposed by Mohagheghi et al. [31] who use trapezoidal non–normal IT2FS activity durations/costs in project cash flow problems; Aramesh, Mousavi & Mohagheghi [2] formalized a mathematical programming model for project networks with resource constraints considered as non–normal IT2FSs and activity durations are constrained to agreement α and increasing indexes λ; Akan & Bayar [1] proposed an IT2FS-based PERT method (forward/backward pass and slack calculations) and compared the obtained results to a defuzzified classical PERT and simulation based PERT approaches.

The main drawback of those previous PERT works involving IT2FNs is that they are focused to find crisp values that satisfy certain crisp conditions/constraints so the obtained solution is a single value which implies a loss of fuzzy information. It is important to remark that fuzzy activity durations must lead to a fuzzy set of optimal durations of the project and a set of possible optimal paths/solutions and those previous works did not clearly state the role of fuzzy sets in the optimal duration of the project and how different solutions can appear. This way, we propose a mathematical framework to solve project networks under Type–2 fuzzy uncertainty which mainly comes from disagreement between different experts about activity durations, so the union of all experts perceptions can be represented with IT2FNs and/or the proposed Interval Type–2 fuzzy–PERT set to then obtain the set of optimal project durations associated to a set of possible routes/solutions as a sequel of the fuzzy model proposed by Chen [8]. To do so, two main methods are introduced: an α–cuts which computes the set of fuzzy project optimal durations and a method based on the expectations/centroids of all interval Type–2 fuzzy activity durations regardless its membership function.

The paper is organized as follows: Section 1 presents the introduction and motivation; Section 2 presents basic concepts on fuzzy sets; Section 3 introduces the fuzzy network problem and the fuzzy–PERT sets; Section 4 presents two examples: an Interval Type–2 fuzzy sets example and an Interval Type–2 fuzzy–PERT example and some concluding remarks are presented in Section 5.

2. Fuzzy sets concepts and definitions

Fuzzy sets are denoted by capital letters A with membership function ${\mu }_A(x)$ and Type–2 fuzzy sets are denoted by emphasized capital letters $\tilde{A}$ with membership function ${\mu }_{\tilde{A}}(x)$ where $x\in X$ is an element of a subset X of the reals $X\subseteq \mathbb{R}$. ${\mu }_A(x)$ measures the affinity degree of an element $x\in X$ to the concept/word/label A and $\tilde{A}$ measures uncertainty over A. $\mathcal{P}(X)$ is the class of all crisp sets, ${\mathcal{F}}_1(X)$ is the class of all fuzzy sets, ${\mathcal{F}}_1(\mathbb{R})$ is the class of all fuzzy numbers, ${\mathcal{F}}_2(X)$ is the class of all Type–2 fuzzy sets, ${\mathcal{F}}_2(\mathbb{R})$ is the class of all Type–2 fuzzy numbers and ∫ denotes the Lebesgue integral. Thus, a fuzzy set A is a set of ordered pairs of an element x and its membership degree, ${\mu }_A(x)$, i.e.

$$A=\{(x,{\mu }_A(x))|x\in X\}.$$ (1)

The support of $A\in {\mathcal{F}}_1(X)$ namely $S(A)$ is the set of all values $x\in X$ with positive membership i.e.

$$S(A)=\{x|{\mu }_A(x)>0\}=[\check{x},\hat{x}].$$ (2)

The core of $A\in {\mathcal{F}}_1(X)$ namely $K(A)$ is the set of all values $x\in X$ with maximum membership i.e.

$$K(A)=\{x|\underset{\forall x}{\sup }{\mu }_A(x)\}=[{\check{x}}_c,{\hat{x}}_c]$$ (3)

The cardinality of $A\in {\mathcal{F}}_1(X)$ namely $|A|$ is the total area of A:

$$|A|=\underset{X}{\int }{\mu }_A(x)dx$$ (4)

The centroid of $A\in {\mathcal{F}}_1(X)$ of $C(A)$ is the mathematical center of gravity of A over X i.e.

$$C(A)=\frac{1}{|A|}\underset{X}{\int }x{\mu }_A(x)dx$$ (5)

A fuzzy set A is said to be normal if $\exists (x|{\mu }_A(x)=1)$ i.e. ${\max }_{\forall x}{\mu }_A(x)=1$. Normal fuzzy sets, in particular the subclass of fuzzy numbers is popular in practical applications. The α–cut of $A\in {\mathcal{F}}_1(X)$ namely ${}^{\alpha }A$ is the set of all values $x\in X$ such that $\alpha \in [0,1]$, i.e.:

$${}^{\alpha }A=\{x|{\mu }_A(x)⩾\alpha \}\forall x\in X$$ (6)

$${}^{\alpha }A=[\underset{x}{\inf }{}^{\alpha }\mu _A(x),\underset{x}{\sup }{}^{\alpha }\mu _A(x)]=[\check{A}(\alpha ),\hat{A}(\alpha )]$$ (7)

which satisfies the following properties:

$$\alpha =0\Rightarrow {}^{\alpha }A=S(A)\text{(support)}$$ (8)

$$\underset{x}{\sup }{\mu }_A(x)\Rightarrow {}^{\alpha }A=x_c\text{(core)}$$ (9)

$${\alpha }_1<{\alpha }_2\Rightarrow {}^{{\alpha }_1}A\subseteq {}^{{\alpha }_2}A\text{(monotonicity)}$$ (10)

and allow to define the following special fuzzy set ${}_{\alpha }A$ (also known as an interval–fuzzy set of level α):

$${}_{\alpha }A=\{({}^{\alpha }A,\alpha )|\alpha \in [0,1]\}.$$ (11)

A fuzzy set A can also be represented as the fuzzy standard union namely ∪ of all its α–cuts (see Klir & Yuan [22]) which is known as the h–representation of A (a.k.a horizontal representation or decomposition of A). The h–slice representation of A given a set of $L\subseteq [0,1]$ α–cuts, is:

$$A\equiv \bigcup _{\alpha \in L}{}_{\alpha }A.$$ (12)

It is important to note that convexity is not a requirement for the h–representation. A particular and interesting subclass of normal fuzzy sets defined over the reals is the class of fuzzy numbers which is popular in theory, applications and real analysis.

Definition 1

Let $A:\mathbb{R}\to [0,1]$ be a fuzzy subset of the reals, then $A\in {\mathcal{F}}_1(\mathbb{R})$ is called a fuzzy number if there exists a closed interval $[{\check{x}}_c,{\hat{x}}_c]\ne \varnothing$ with membership degree ${\mu }_A(x)$ such that:

$${\mu }_A(x)=\{\begin{array}{cc}\hfill 1\hfill & \mathit{\text{for}}x\in [{\check{x}}_c,{\hat{x}}_c],\hfill \\ \hfill l(x)\hfill & \mathit{\text{for}}x\in [-\infty ,{\check{x}}_c],\hfill \\ \hfill r(x)\hfill & \mathit{\text{for}}x\in [{\hat{x}}_c,\infty ],\hfill \end{array}$$ (13)

where ${\mu }_A(x)=1$ for the subset $x\in [{\check{x}}_c,{\hat{x}}_c]$, $l:(-\infty ,{\check{x}}_c)\to [0,1]$ is monotonic increasing, continuous from the right i.e. $l(x)=0$ for $x<\check{x}$; $r:({\hat{x}}_c,\infty )\to [0,1]$ is monotonic decreasing continuous from the left i.e. $r(x)=0$ for $x>\hat{x}$.

Fig. 1 shows a triangular fuzzy set/number A, its support $S(A)$, core $x_c$ and ${}^{\alpha }A=[\check{x}(\alpha ),\hat{x}(\alpha )]$. Nonlinear IT2FNs (Gaussian, exponential, logistic membership functions, etc.) are among the most popular shapes in analysis, but they are unbounded i.e. $S(\tilde{A})\in [-\infty ,\infty ]$ which leads to have non-compact fuzzy sets leading to unbounded/unfeasible solutions. To overcome this issue we propose to truncate unbounded fuzzy sets as follows.

Definition 2

Let $A\in {\mathcal{F}}_1(\mathbb{R})$ be a unbounded fuzzy number, then a β-truncated fuzzy number $A^{\beta }$ is an ordered pair $\{(A,{\mu }_A^{\beta }(x))|x\in X\}$ such that:

$${\mu }_A^{\beta }(x)=\{\begin{array}{cc}\frac{{\mu }_A(x)-\beta }{1-\beta },\hfill & x\in [\inf {\mu }_A^{-1}(\beta ),\sup {\mu }_A^{-1}(\beta )]\hfill \\ 0,\hfill & x\notin [\inf {\mu }_A^{-1}(\beta ),\sup {\mu }_A^{-1}(\beta )]\hfill \end{array}$$ (14)

where $\beta \in [0,1)$ is a scale factor.

Figure 1.

Triangular fuzzy set A.

2.1. Interval Type–2 fuzzy sets/numbers

An Interval Type-2 Fuzzy Set (IT2FS) is denoted by emphasized capital letters $\tilde{A}$ with a membership function ${\mu }_{\tilde{A}}(x)$ where $x\in X$ is an element of a subset X of the reals $X\subseteq \mathbb{R}$. ${\mu }_{\tilde{A}}(x)$ measures the uncertainty degree of a value $x\in X$ regarding the concept/word/label A (see Mendel [27], Mendel [28], Bustince [5], Bustince et al. [4], and Türksen [35]). Let $\mathcal{L}([0,1])$ be the set of all closed subintervals over $[0,1]$ i.e.

$$\mathcal{L}([0,1])=\{\mathbf{x}=[\underset{\_}{x},\bar{x}]|(\underset{\_}{x},\bar{x})\in {[0,1]}^2,\underset{\_}{x}⩽\bar{x}\},$$

then an IT2FS $\tilde{A}$ is an ordered pair $\{(x,{\mu }_{\tilde{A}}(x))|x\in X\}$ such that:

$${\mu }_{\tilde{A}}(x)=\{[{\underset{\_}{\mu }}_{\tilde{A}}(x),{\bar{\mu }}_{\tilde{A}}(x)]|({\underset{\_}{\mu }}_{\tilde{A}}(x),{\bar{\mu }}_{\tilde{A}}(x))\in {[0,1]}^2,{\underset{\_}{\mu }}_{\tilde{A}}(x)⩽{\bar{\mu }}_{\tilde{A}}(x)\}$$

where ${\mu }_{\tilde{A}}(x)$ is the membership function of $\tilde{A}$, the set of primary memberships or primary variable is X and its domain of uncertainty associated to an element $x\in X$ is the subset $J_x=[{\underset{\_}{\mu }}_{\tilde{A}}(x),{\bar{\mu }}_{\tilde{A}}(x)]\subseteq [0,1]$. $\tilde{A}$ is completely characterized by a upper membership function UMF$(\tilde{A})={\bar{\mu }}_{\tilde{A}}=\bar{A}$ and a lower membership function LMF$(\tilde{A})={\underset{\_}{\mu }}_{\tilde{A}}=\underset{\_}{A}$. An IT2FS can also be seen as the union of infinite type-1 fuzzy sets A into FOU$(\tilde{A})$ namely embedded fuzzy sets $A_e$ (see dashed line in Fig. 2).

Figure 2.

Interval Type-2 fuzzy set $\tilde{A}$.

An Interval Type-2 fuzzy number (IT2FN) is the extension of a fuzzy number (see Figueroa-García et al. [16]) which means that $\tilde{A}$ is an IT2FS whose UMF$(\tilde{A})$ and LMF$(\tilde{A})$ are fuzzy numbers i.e. convex normal fuzzy sets whose ${}^{\alpha }A$ is a closed interval for all $\alpha \in [0,1]$ (see Figueroa-García et al. [16]).

Definition 3 Interval Type-2 Fuzzy Number (IT2FN) —

Let ${\mathcal{F}}_1(\mathbb{R})$ be the class of all fuzzy numbers and ${\mathcal{F}}_2(\mathbb{R})$ be the class of all IT2FNs, then $\tilde{A}\in {\mathcal{F}}_2(\mathbb{R})$ iff there exists a closed interval $[{\check{x}}_c^u,{\hat{x}}_c^u]\ne \varnothing$ for each UMF$(\tilde{A})$ and LMF$(\tilde{A})$ such that

$${\mu }_{\tilde{A}}(x)=\{\begin{array}{cc}\hfill 1\hfill & \mathit{\text{for}}x\in [{\check{x}}_c^u,{\hat{x}}_c^u],\hfill \\ \hfill \tilde{l}(x)\hfill & \mathit{\text{for}}x\in [-\infty ,{\check{x}}_c^l],\hfill \\ \hfill \tilde{r}(x)\hfill & \mathit{\text{for}}x\in [{\hat{x}}_c^l,\infty ],\hfill \end{array}$$ (15)

where $\tilde{l}:(-\infty ,{\check{x}}_c^l)\to \mathcal{L}([0,1]),J_x\subseteq [0,1]$ is monotonic increasing, continuous from the right, and $\tilde{l}(x)=0$ for $x<{\check{x}}^u$, and $\tilde{r}:({\hat{x}}_c^l,\infty )\to \mathcal{L}([0,1]),J_x\subseteq [0,1]$ is monotonic decreasing, continuous from the left, and $\tilde{r}(x)=0$ for $x>{\hat{x}}^u$.

Now, if both UMF$(\tilde{A})$ and LMF$(\tilde{A})$ are fuzzy numbers, then $\tilde{A}$ is an IT2FN (see Fig. 2). Now, the core of an IT2FN namely $K(\tilde{A})$ is:

$$K(\tilde{A})=\{x|{\bar{\mu }}_{\tilde{A}}(x)=1\}=[{\check{x}}_c^u,{\hat{x}}_c^u]$$ (16)

where ${\check{x}}_c^u={\hat{x}}_c^u=x_c$ for single–core IT2FNs. Now, the support of an IT2FN $\tilde{A}\in {\mathcal{F}}_2(\mathbb{R})$ is:

$$S(\tilde{A})=\{x|{\bar{\mu }}_{\tilde{A}}(x)>0\}=[{\check{x}}^u,{\hat{x}}^u].$$ (17)

The α-cut of $A\in {\mathcal{F}}_1(X)$ is defined as ${}^{\alpha }A=\{x|{\mu }_A(x)⩾\alpha \}$. This simplifies the computation of any fuzzy continuous function $f(x)$ (see Dubois & Prade [32] and Nguyen [15]). In our case $z^{⁎}=c^{\prime }{x}^{⁎}$ comes from $f({\tilde{A}}_i)(z^{⁎})$, so the computation of $f({\tilde{A}}_i)(z^{⁎})$ through α-cuts instead of mapping x is an easy way to map ${\mu }_{\tilde{A}}(x)$. Thus, the idea is to compute $f({\mu }_{\tilde{A}}^{-1})\to x$ instead. First, the α-cut of an IT2FN $\tilde{A}$ namely ${}^{\alpha }\tilde{A}$ (see Figueroa [12]) is the union of all α-cuts of all fuzzy sets embedded into $J_x=[{\underset{\_}{\mu }}_{\tilde{A}}(x),{\bar{\mu }}_{\tilde{A}}(x)]$:

$${}^{\alpha }\tilde{A}=\{x|{\mu }_{\tilde{A}}(x)⩾\alpha \};\alpha \in [0,1]$$ (18)

$${}^{\alpha }\tilde{A}=\{x|\exists {\mu }_{A_e}(x)\in J_x\};\alpha \in [0,1],J_x\subseteq [0,1]$$ (19)

The boundaries of each α-cut are defined as follows.

Definition 4 Boundaries of ${}^{\alpha }\tilde{A}$ —

Let the bounds of an IT2FN $\tilde{A}\in {\mathcal{F}}_2(\mathbb{R})$, ${}^{\alpha }\tilde{A}$ be defined as the α-cuts of its UMF$(\tilde{A})=\bar{A}$ and LMF$(\tilde{A})=\underset{\_}{A}$ as follows:

$${}^{\alpha }\underset{\_}{A}=[\underset{x}{\inf }{}^{\alpha }\underset{\_}{A}(x);\underset{x}{\sup }{}^{\alpha }\underset{\_}{A}(x)]=[{\check{A}}^l(\alpha ),{\hat{A}}^l(\alpha )]$$ (20)

$${}^{\alpha }\bar{A}=[\underset{x}{\inf }{}^{\alpha }\bar{A}(x);\underset{x}{\sup }{}^{\alpha }\bar{A}(x)]=[{\check{A}}^u(\alpha ),{\hat{A}}^u(\alpha )]$$ (21)

The boundaries of ${}^{\alpha }\tilde{A}$ also allow us to define the following interval Type–2 fuzzy set of level α

$${}_{\alpha }\tilde{A}=\{({}^{\alpha }\tilde{A},\alpha )|\alpha \in [0,1]\},$$ (22)

and the h–slice representation of $\tilde{A}$ given a set of $L\subseteq [0,1]$ α–cuts, is

$$\tilde{A}\equiv \bigcup _{\alpha \in L}{}_{\alpha }\tilde{A}.$$

A graphical representation of ${}^{\alpha }\tilde{A}$ is provided in Fig. 2.

The centroid of $\tilde{A}$ namely $C(\tilde{A})$ is the optimization problem of finding the bounds $C_l(\tilde{A}),C_r(\tilde{A})$ of the centroids of all possible embedded sets $A_e$ enclosed into FOU$(\tilde{A})$.

Definition 5 Wu & Mendel [39] —

Let $A_e\in {\mathcal{F}}_1(X)$ be an embedded type–1 fuzzy set $A_e$ enclosed into FOU$(\tilde{A})$. The centroid $C(\tilde{A})$ of an IT2FS is the union of the centroids of all $A_e$, i.e.,

$$C(\tilde{A})=\bigcup _{A_e}C(A_e)=[C_l(\tilde{A}),C_r(\tilde{A})]C_l(\tilde{A})=\underset{A_e}{\min }C(A_e),C_r(\tilde{A})=\underset{A_e}{\max }C(A_e)$$

The relative variance of $\tilde{A}$ is the following optimization problem of finding two bounds $v_l(\tilde{A}),v_r(\tilde{A})$.

Definition 6 Wu & Mendel [39] —

Let $A_e\in {\mathcal{F}}_1(X)$ be an embedded type–1 fuzzy set $A_e$ enclosed into FOU$(\tilde{A})$. The relative variance $V_r(\tilde{A})$ regarding $C_c(\tilde{A})$ is the union of the variances of all $A_e$, i.e.,

$$V_r(\tilde{A})=\bigcup _{A_e}{v}_r(A_e)=[v_l(\tilde{A}),v_r(\tilde{A})]v_l(\tilde{A})=\underset{A_e}{\min }v(A_e),v_r(\tilde{A})=\underset{A_e}{\max }v(A_e),v(A_e)=\frac{{\sum }_{i=1}^N{(x_i-C_c(A_e))}^2{\mu }_{A_e}(x_i)}{{\sum }_{i=1}^N{\mu }_{A_e}(x_i)}$$

The most efficient iterative methods to obtain $C_l(\tilde{A}),C_r(\tilde{A}),V_l(\tilde{A})$ and $V_r(\tilde{A})$ are the KM and EKM algorithms (see Mendel & Liu [29], Wu & Mendel [40] and Mendel [41]) since no exact methods are available to compute them. Alike unbounded type-1 fuzzy sets, we can also define a β-truncated IT2FN.

Definition 7

Let $\tilde{A}\in {\mathcal{F}}_2(\mathbb{R})$ be a unbounded IT2FN, then a β-truncated fuzzy number ${\tilde{A}}^{\beta }$ is characterized by a β-truncated UMF$(\tilde{A})$ ${\bar{\mu }}_{\tilde{A}}^{\beta }(x)$ and a β-truncated LMF$(\tilde{A})$ ${\underset{\_}{\mu }}_{\tilde{A}}^{\beta }(x)$ where $\beta \in [0,1)$ is a scale factor.

Unbounded fuzzy sets are not suitable in project network analysis since in practice an unbounded support allows unrealistic activity durations i.e. $-\infty ⩽t_{ij}⩽\infty$. This way, β-truncated fuzzy numbers allow to include nonlinear experts perceptions about $t_{ij}$ which must be nonnegative and finite-measurable.

3. The fuzzy project network problem

A deterministic project network P is a triplet of n vertexes V, a set of feasible arcs/activities $(i,j)\in A$ and its activities duration $t_{ij}\in R^{+}$ i.e. $P=\langle V,A,t_{ij}\rangle$ whose mathematical programming formulation is

$$\begin{gathered} z=\mathrm{Max}\sum _i\sum _j{t}_{ij}{x}_{ij} \\ s.t\sum _j{x}_{1j}=1, \end{gathered}$$ (23)

$$\sum _j{x}_{ij}-\sum _k{x}_{ki}=0,i\notin \{1,n\}$$ (24)

$$\sum _k{x}_{kn}=1,x_{ij}\in \{0,1\},(i,j)\in A$$ (25)

where $x_{ij}$ is a decision variable of activating flow from node i to node j. Constraints (23) and (25) guarantee to activate flow from node one (source) to the terminal node (sink) and constraint (24) defines feasible arcs.

In projects where no historical data to define $t_{ij}$ are available, information coming from experts can be used as a proper source information to represent uncertainties that occur in real world. An expert can represent his/her individual knowledge about an activity duration $t_{ij}$ by using a fuzzy set $T_{ij}$ in order to measure ambiguity around it, but some projects involve different experts who do not necessarily agree with each other, so the use of Type-2 fuzzy sets are useful to represent uncertainty around $t_{ij}$ and to enclose all experts perceptions into a single measure bounded by two fuzzy sets $\underset{\_}{T}$ and $\bar{T}$.

Fig. 3 shows the idea behind a single expert associated to a single fuzzy set and a group of experts associated to an IT2FS. Multiple experts can define different embedded fuzzy sets $A_e$ which can be comprised into the FOU of an IT2FS (see Fig. 2) while a single expert can just have an ambiguous perception of $t_{ij}$ which can be represented by interval (or ambiguous) memberships $[{\underset{\_}{T}}_{ij},{\bar{T}}_{ij}]$.

Figure 3.

Single/group experts project.

Zadeh [42] and Klir & Folger [21] refer to fuzzy sets to measure imprecision or gradualness about a label/word/concept with well defined membership values/function and a type-2 fuzzy set is associated to ambiguity associated to the lack of certainty about the membership degrees of a fuzzy set (see Mendel [26]). Comprehensive works about how to setup type-2 membership functions/degrees have been proposed by Mendel [27] and Mendel & Wu [30].

In practice, an IT2FS can be composed as either the union of multiple experts perceptions given as embedded fuzzy sets $A_e$ or an expert providing ambiguous memberships as interval-valued degrees. In both cases, the lower/upper bounds of activity durations are $[{\underset{\_}{T}}_{ij},{\bar{T}}_{ij}]$ obtained by the $\min /\max$ values of membership for every value $t\in \mathbb{R}$ which are defined by project analysts who know the limits of activity durations under ambiguity or uncertainty.

Projects usually have experts for every activity each one having a different perception about every activity duration $t_{ij}$, so their perceptions can be represented by different shapes such as triangular, Gaussian, exponential, trapezoidal, etc. However, it is also possible to define a fuzzy version of the PERT distribution and a Type–2 fuzzy PERT set which are useful to represent two different but related uncertainty levels: imprecision via classical fuzzy sets or uncertainty (disagreement/ambiguity) via Type–2 fuzzy sets.

3.1. Fuzzy and Interval Type–2 fuzzy-PERT approaches

One of the most popular approaches to represent nonlinear activity times is the PERT distribution (see Malcolm et al. [24]) which is also used in risk analysis and decision making (see Vose [36]). The fuzzy–PERT set can be defined as follows.

Definition 8

The fuzzy–PERT set T of an activity duration $t⩾0$ is defined as follows:

$$T(t)=\frac{{(t-a)}^{\gamma -1}{(c-t)}^{\theta -1}}{{(b-a)}^{\gamma -1}{(c-b)}^{\theta -1}};t\in [a,b],$$ (26)

$$\gamma =1+4\frac{b-a}{c-a},\theta =1+4\frac{c-b}{c-a},E(T)=\frac{a+4b+c}{6},$$ (27)

$$V(T)=\frac{(E(T)-a)(c-E(T))}{7}.$$ (28)

The fuzzy–PERT set is a beta-based set that uses the same parameters of the triangular distribution, but it lacks of the inherent bias problem of its mean and high sensitivity of its standard deviation since the beta function is smoother around the mode and produces a systematically lower standard deviation.

By extension of Eq. (26), it is possible to define an Interval Type–2 fuzzy–PERT set through its lower and upper membership functions. This fuzzy–PERT representation helps either to represent nonlinear perceptions of a group of experts who disagree in their particular perceptions or to deal with both ambiguity and imprecision in the definition of a $t_{ij}$. This way, a fuzzy–PERT set measures imprecision and an Interval Type–2 fuzzy–PERT set measures uncertainty around $t_{ij}$ (seen as disagreement and/or ambiguity).

Definition 9

The Interval Type–2 fuzzy–PERT set $\tilde{T}$ of the duration of an activity $t⩾0$ is characterized by a lower $\underset{\_}{T}$ and a upper $\bar{T}$ fuzzy set, as follows:

$$\bar{T}(t)=\frac{{(t-a_1)}^{{\gamma }_1-1}{(c_1-t)}^{{\theta }_1-1}}{{(b-a_1)}^{{\gamma }_1-1}{(c_1-b)}^{{\theta }_1-1}};t\in [a_1,c_1],$$ (29)

$$\underset{\_}{T}(t)=\frac{{(t-a_2)}^{{\gamma }_2-1}{(c_2-t)}^{{\theta }_2-1}}{{(b-a_2)}^{{\gamma }_2-1}{(c_2-b)}^{{\theta }_2-1}};t\in [a_2,c_2],{\gamma }_1=1+4\frac{b-a_1}{c_1-a_1};{\gamma }_2=1+4\frac{b-a_2}{c_2-a_2},{\theta }_1=1+4\frac{c_1-b}{c_1-a_1};{\theta }_2=1+4\frac{c_2-b}{c_2-a_2}.0⩽a_1⩽a_2⩽b⩽c_2⩽c_1.$$ (30)

Unlike $E(T)$ and $V(T)$ (see Eqs. (27) and (28)) the computation of $C(\tilde{T})$ and $V(\tilde{T})$ have no closed forms (see Definition 5, Definition 6) but they can be computed with iterative methods (see Wu & Mendel [39]). Fig. 4 displays an Interval Type–2 fuzzy–PERT set $\tilde{T}$ (see Definition 9).

Figure 4.

Interval Type-2 fuzzy–PERT set $\tilde{T}$.

3.2. The fuzzy project network model

A fuzzy project model as proposed by Chen [8] namely $\tilde{P}$ is a triplet of n vertexes V, a set of feasible arcs/activities $(i,j)\in A$ with fuzzy activities duration $T_{ij}\in {\mathcal{F}}_1({\mathbb{R}}^{+})$ i.e. $\tilde{P}=\langle V,A,T_{ij}\rangle$. Its extension to a Type-2 fuzzy environment is given in cases where experts of the project do not agree on their perceptions/opinions about activity times. A Type-2 fuzzy project network ${\tilde{P}}_2$ is a triplet of n vertexes V, a set of feasible arcs/activities $(i,j)\in A$ and its Type-2 fuzzy activities duration ${\tilde{T}}_{ij}\in {\mathcal{F}}_2({\mathbb{R}}^{+})$ i.e. ${\tilde{P}}_2=\langle V,A,{\tilde{T}}_{ij}\rangle$ whose mathematical programming model is

$$\begin{gathered} \tilde{Z}=\mathrm{Max}\sum _i\sum _j{\tilde{T}}_{ij}{x}_{ij} \\ s.t\sum _j{x}_{1j}=1, \end{gathered}$$ (31)

$$\sum _j{x}_{ij}-\sum _k{x}_{ki}=0,i\notin \{1,n\}$$ (32)

$$\sum _k{x}_{kn}=1,x_{ij}\in \{0,1\},(i,j)\in A$$ (33)

where $x_{ij}$ is a decision variable of activating flow from node i to node j.

The activity times are fuzzy sets represented by IT2FNs $\tilde{T}:A\to {\mathcal{F}}_2(R^{+}),{\mathcal{F}}_2(R^{+})$ is the set of non-negative fuzzy numbers and $\tilde{Z}:{\mathcal{F}}_2({\mathbb{R}}_{nn}^{+})\times {\mathcal{F}}_2({\mathbb{R}}_{nn}^{+})\to {\mathcal{F}}_2({\mathbb{R}}^{+})$ is the Type-2 fuzzy set of all possible durations of the project.

Definition 10

Let us denote constraints (31), (32) and (33) as $g(ij)$ and the α–cut of a compact activity time ${}^{\alpha }\tilde{T}_{ij}$ be decomposed into ${\check{T}}^u(\alpha ),{\check{T}}^l(\alpha ),{\hat{T}}^l(\alpha ),{\hat{T}}^u(\alpha )\in \mathbb{R}$ (see (20) and (21)) which reach the following optimal values $z^{⁎}\in \mathbb{R}$, namely

$${\check{z}}^u(\alpha )\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _j{\check{T}}_{ij}^u(\alpha )x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (34)

$${\check{z}}^l(\alpha )\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _j{\check{T}}_{ij}^l(\alpha )x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (35)

$${\hat{z}}^l(\alpha )\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _j{\hat{T}}_{ij}^l(\alpha )x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (36)

$${\hat{z}}^u(\alpha )\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _j{\hat{T}}_{ij}^u(\alpha )x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (37)

which are binary ordered i.e.

$${\check{z}}^u(\alpha )⩽{\check{z}}^l(\alpha )⩽{\hat{z}}^l(\alpha )⩽{\hat{z}}^u(\alpha ).$$ (38)

Now, $\tilde{Z}$ is the algebraic sum of all fuzzy activity times $\tilde{T}$ which can be decomposed into a set of $L\in [0,1]$ α–cuts, so every value $\alpha \in L$ leads to set of possible values ${}^{\alpha }\tilde{Z}$ defined as follows:

$${}^{\alpha }\tilde{Z}=[{\check{z}}^u(\alpha ),{\check{z}}^l(\alpha ),{\hat{z}}^l(\alpha ),{\hat{z}}^u(\alpha )]$$

so as the algebraic sum of all $\tilde{T}$ is also binary ordered for any $\alpha \in L$ i.e.

$$\sum _i\sum _j{\check{T}}_{ij}^u(\alpha )⩽\sum _i\sum _j{\check{T}}_{ij}^l(\alpha )⩽\sum _i\sum _j{\hat{T}}_{ij}^l(\alpha )⩽\sum _i\sum _j{\hat{T}}_{ij}^u(\alpha ).$$

In this approach, an amount of α–cuts is selected to then compute ${}^{\alpha }T_{ij}^u(\alpha )$ i.e. ${\check{T}}_{ij}^u(\alpha ),{\check{T}}_{ij}^l(\alpha )$, ${\hat{T}}_{ij}^l(\alpha ),{\hat{T}}_{ij}^u(\alpha )$ and solve ${\check{z}}^u(\alpha ),{\check{z}}^l(\alpha ),{\hat{z}}^l(\alpha ),{\hat{z}}^u(\alpha )$ as shown in Eqs. (34)-(37) to finally compose the set $\tilde{Z}$ as the union of all those α–level sets (see Eqs. (11) and (12)).

3.3. Interval set of expected solutions of a uncertain project $\tilde{P}$

Another way to compute solutions for $\tilde{P}$ is by using a centroid-based model (see Figueroa-García [14]). Its main idea is to defuzzify each ${\tilde{T}}_{ij}$ using its centroid $C({\tilde{T}}_{ij})$ which is the set of all possible centroids $C(\tilde{T})\in \mathcal{P}(\mathbb{R})$ to then obtain the expected set of optimal values of $\tilde{P}$. Wu & Mendel [29], [40] and Melgarejo [6] has proposed a more efficient version and its computation using α-cuts has been defined by Figueroa-García [12], [13]. Now, the mathematical programming models that obtain the set of central values of the project $Z_c$ based on the centroid/expected/average values of $\tilde{T}$ is as follows:

$$Z_c\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _{j}C({\tilde{T}}_{ij})x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (39)

which is fully characterized by its upper/lower boundaries (see Figueroa-García [14]) as follows

$${\check{z}}_c\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _j{C}_l({\tilde{T}}_{ij})x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (40)

$${\hat{z}}_c\triangleq \underset{x_{ij}}{\mathrm{Max}}\{\sum _i\sum _j{C}_r({\tilde{T}}_{ij})x_{ij}:g(ij),x_{ij}\in \{0,1\},(i,j)\in A,\alpha \in [0,1]\}$$ (41)

Thus, the central set of optimal values $Z_c\in \mathcal{P}(\mathbb{R})$ is composed by two boundaries ${\check{z}}_c,{\hat{z}}_c$ and a central value ${\overline{z}}_c$ that comes from Eqs. (40) and (41):

$$Z_c=[{\check{z}}_c,{\hat{z}}_c]\equiv [\sum _i\sum _j{C}_l({\tilde{T}}_{ij}){\check{x}}_c^{⁎},\sum _i\sum _j{C}_r({\tilde{T}}_{ij}){\hat{x}}_c^{⁎}]$$

where ${\check{x}}_c^{⁎}$ and ${\hat{x}}_c^{⁎}$ are the optimal routes/solutions for (40) and (41) and its central value ${\overline{z}}_c$ is

$$z_c=({\check{z}}_c+{\hat{z}}_c)/2.$$

In this centroid-based approach, the centroids of all activity times $C({\tilde{T}}_{ij})$ are computed to then obtain a set of central optimal durations of the project namely $Z_c$ which provides an apriori expectation of the duration of the project in advance. Unlike the α-cuts approach less MLPs are needed, but the problem of obtaining $C({\tilde{T}}_{ij})$ still needs additional computations.

4. Illustrative examples

Now, two classical project network examples are solved. The first one is a CPM problem proposed by Taha [34] whose activities duration are deterministic, so we consider them as fuzzy activity durations with different nonlinear shapes (triangular, Gaussian and exponential). The second example is a classical PERT proposed by Lyndwood & Montgomery [19] where each activity duration is considered as PERT distributed, so we first define the fuzzy–PERT distribution to then solve a uncertain fuzzy–PERT problem.

4.1. Taha example

This example is a fuzzy version of a classical CPM method introduced by Taha [34]. Rather than using a single shape for all activities which is the most common approach addressed in the literature, this example uses three different shapes: a classical linear triangular fuzzy sets which uses three parameters (optimistic a, modal b and pessimistic c) and are widely used in project scheduling, nonlinear Gaussian membership functions which are commonly used to represent human-like perceptions and nonlinear exponential membership functions which are often related to time variables. The core namely c of all deterministic activity times of the original example $t_{ij}$ has been preserved. The network of the fuzzy project $\tilde{P}$ is displayed in Fig. 5.

Figure 5.

Type-2 fuzzy network of Example 1 (Taha).

Triangular IT2FNs (see Fig. 6.) are defined as follows.

$$\bar{T}(t)=\max (\min (\frac{t-a_1}{c-a_1},\frac{b_1-t}{b_1-c}));t\in [a_1,b_1]\underset{\_}{T}(t)=\max (\min (\frac{t-a_2}{c-a_2},\frac{b_2-t}{b_2-c}));t\in [a_2,b_2]a_2<a_1;b_2<b_1.$$

Gaussian IT2FNs (see Fig. 7) are defined as follows

$$\begin{gathered} \bar{T}(t)=\{\begin{array}{cc}exp(-\frac{1}{2}{((t-c)/a_1)}^2);\hfill & t⩽c\hfill \\ exp(-\frac{1}{2}{((t-c)/b_1)}^2);\hfill & t>c\hfill \end{array} \\ \underset{\_}{T}(t)=\{\begin{array}{cc}exp(-\frac{1}{2}{((t-c)/a_2)}^2);\hfill & t⩽c\hfill \\ exp(-\frac{1}{2}{((t-c)/b_2)}^2);\hfill & t>c\hfill \end{array} \\ {a}_2<a_1;b_2<b_1. \end{gathered}$$

and exponential IT2FNs (see Fig. 8) are defined as follows

$$\begin{gathered} \bar{T}(t)=\{\begin{array}{cc}exp(-(c-t)/a_1);\hfill & t⩽c\hfill \\ exp(-(t-c)/b_1);\hfill & t>c\hfill \end{array} \\ \underset{\_}{T}(t)=\{\begin{array}{cc}exp(-(c-t)/a_2);\hfill & t⩽c\hfill \\ exp(-(t-c)/b_2);\hfill & t>c\hfill \end{array} \\ {a}_2<a_1;b_2<b_1. \end{gathered}$$

Figure 6.

Triangular IT2FS $\tilde{A}$.

Figure 7.

Symmetric Gaussian IT2FS $\tilde{A}$.

Figure 8.

Exponential IT2FS $\tilde{A}$.

The values of $S(\tilde{A})$ of a β-truncated fuzzy set used in Table 1 for a Gaussian IT2FS are:

$$\begin{gathered} {\check{T}}^u=c+\sqrt{\ln (\beta )}\cdot 2a_1 \\ {\check{T}}^l=c+\sqrt{\ln (\beta )}\cdot 2a_2 \\ {\hat{T}}^l=c-\sqrt{\ln (\beta )}\cdot 2b_1 \\ {\hat{T}}^u=c-\sqrt{\ln (\beta )}\cdot 2b_2 \end{gathered}$$

and for an exponential IT2FS are:

$$\begin{gathered} {\check{T}}^u=c+\ln (\beta )\cdot a_1 \\ {\check{T}}^l=c-\ln (\beta )\cdot a_2 \\ {\hat{T}}^l=c+ln(\beta )\cdot b_1 \\ {\hat{T}}^u=c-\ln (\beta )\cdot b_2 \end{gathered}$$

Table 1.

Parameters of example 1 (Taha).

Set	Shape	a1	a2	c	b2	b1	${\check{T}}^u$	${\check{T}}^l$	${\hat{T}}^l$	${\hat{T}}^u$	$C_l(\tilde{T})$	$C_r(\tilde{T})$
${\tilde{T}}_{12}$	Gaussian	1.5	1	5	1.5	2	1.328	2.552	8.672	9.895	5	5.720
${\tilde{T}}_{13}$	Triangular	1	3	6	8	9	1	3	8	9	5	6
${\tilde{T}}_{23}$	Gaussian	1	0.75	3	0.75	1.5	0.552	1.164	4.836	6.672	2.820	3.541
${\tilde{T}}_{24}$	Exponential	2	1.5	8	1	1.5	2.009	3.506	10.996	12.494	7.277	8
${\tilde{T}}_{35}$	Triangular	0.5	1	2	3	5	0.5	1	3	5	1.833	2.667
${\tilde{T}}_{36}$	Gaussian	3	2	11	1	2	3.657	6.105	13.448	15.895	9.557	11
${\tilde{T}}_{46}$	Triangular	0.1	0.25	1	1.5	2	0.1	0.25	1.5	2	0.867	1.083
${\tilde{T}}_{56}$	Exponential	2	1	12	2	3	6.009	9.004	17.991	20.987	12	13.458

The parameters of every activity duration ${\tilde{T}}_{ij}$ for $\beta =0.05$ are rescaled according to Definition 2, Definition 7 and the bounds ${\check{T}}^u,{\check{T}}^l,{\hat{T}}^l,{\hat{T}}^u$ of the support for the β–truncated fuzzy sets are shown in Table 1 (note that only Gaussian and exponential shapes need to be β–truncated since they are unbounded fuzzy sets).

The obtained set of optimal durations of the project for eleven α–cuts i.e. $\alpha =\{0,0.1,\cdots ,1\}$ (see Definition 10) is shown in Fig. 9.

Figure 9.

Fuzzy set of optimal values of the project (Taha).

All activity times are shown in Table B.5 (see Appendix B) and the obtained optimal durations of the project per α are summarized in Table 2 where the optimal route $(1,2)\to (2,4)\to (5,6)$ holds for all α.

Table 2.

Results of Example 1 (Taha).

α	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
${\check{z}}^u(\alpha )$	9.35	14.33	16.76	18.41	19.68	20.72	21.62	22.43	23.18	23.93	25
${\check{z}}^l(\alpha )$	15.06	18.21	19.74	20.79	21.59	22.25	22.83	23.34	23.82	24.3	25
${\hat{z}}^l(\alpha )$	37.66	33.74	31.82	30.5	29.48	28.64	27.9	27.24	26.61	25.97	25
${\hat{z}}^u(\alpha )$	43.38	37.62	34.8	32.88	31.4	30.17	29.11	28.15	27.25	26.34	25

The expected set of optimal values $Z_c$ (see Section 3.3) comes from the centroid of every $\tilde{T}$ i.e. $C(\tilde{T})=[C_l(\tilde{T}),C_r(\tilde{T})]$ which are summarized in Table 1 where $C_l(\tilde{T}),C_r(\tilde{T})$ have been computed using the type reduction method proposed by Figueroa–García et al. [17]. The obtained set $Z_c$ is as follows:

$$Z_c=[{\check{z}}_c^{⁎},{\hat{z}}_c^{⁎}]=[24.277,27.178]$$

whose central value is $z_c=25.727$ and the optimal routes are ${\check{x}}_c^{⁎}={\hat{x}}_c^{⁎}=(1,2)\to (2,4)\to (5,6)$.

4.1.1. Analysis of results

It is clear that Type–2 fuzzy activity durations lead to a Type–2 fuzzy set of possible durations as shown in Fig. A.12. The set of most possible durations is represented by the expected set of possible durations $Z_c$ and the central expected value of the project $z_c$ is the mean value of $Z_c$ which can be seen as a deterministic value to refer to the overall mean value of the project while the mode value of the project is in this case, the core of durations of the project i.e. $K(\tilde{Z})=25$ (we recall that this value was intended to correspond to the original example). It is important to note that the α-cuts approach needs to solve $4(n-1)\cdot \alpha +1$ optimal values for single-core IT2FSs and $4n\cdot \alpha$ MLPs for interval-core IT2FSs (see Eqs. (34)-(37)).

Figure A.12.

Earliest/latest starting times of the non–critical activity (2,3) for Example 1.

On the other hand, the most optimistic/pessimistic durations of the project are given by the support of $\tilde{Z}$ i.e. $S(\tilde{Z})=[{\check{z}}^u,{\hat{z}}^u]=[9.35,43.38]$ which means that the smallest possible duration of the project is 9.35 and the largest possible duration of the project is 43.38 given the expectations provided by the experts. It is important to recall the importance of obtaining β–truncated fuzzy activity durations (see Definition 7) since their original shapes lead to infinity bounded optimistic/pessimistic project durations.

It is observed that the earliest starting times of every non–critical activity are well defined functions of α but, their latest start and their slacks are not a function of α. This is a consequence of the nature of the max operator involved in the computation of the late start and the slack of every non–critical activity. The earliest finishing time (in black) and the latest starting time (in blue) for the non–critical activity $(2,3)$ are displayed in Fig. A.12 (see Appendix A).

4.2. Johnson & Montgomery example

This example is a fuzzy version of a classical PERT method introduced by Lyndwood & Montgomery [19]. The network of the fuzzy project $\tilde{P}$ is displayed in Fig. 10. The parameters of the Interval Type-2 fuzzy–PERT activity durations (see Definition 9 and Fig. 4) are shown in Table 3. Note that the original values b have been preserved as the core of every $\tilde{T}$.

Figure 10.

Type-2 fuzzy network of Example 2 (Johnson & Montgomery).

Table 3.

Parameters of example 2 (Lyndwood & Montgomery).

Set	a1	a2	b	c1	c2	$C_l(\tilde{T})$	$C_u(\tilde{T})$	$V_l(\tilde{A})$	$V_r(\tilde{A})$
${\tilde{T}}_{12}$	1	2.5	4	5	8	3.667	4.417	0.214	2.079
${\tilde{T}}_{24}$	2	4.5	7	9	13	6.5	7.583	0.704	4.945
${\tilde{T}}_{23}$	3	6.5	7	13	15	7.333	8.25	0.909	6.014
${\tilde{T}}_{25}$	1	3.5	6	8	11	5.5	6.417	0.704	3.965
${\tilde{T}}_{46}$	8	12.5	14	21	26	14.167	15.75	1.967	12.846
${\tilde{T}}_{35}$	1	2.5	10	11	15	8.667	9.583	1.774	7.681
${\tilde{T}}_{56}$	4	6.5	9	11	14	8.5	9.417	0.704	3.974
${\tilde{T}}_{67}$	3	5.5	7	14	18	7.5	8.583	1.999	7.988

The PERT method is based on computing the centroid $C(\tilde{T})=[C_l(\tilde{T}),C_r(\tilde{T})]$ and variance $V(\tilde{T})=[V_l(\tilde{T}),V_r(\tilde{T})]$ of every activity which are summarized in Table 3 to then compute set of optimal values $Z_c$ (see Section 3.3). The obtained set $Z_c$ is as follows:

$$Z_c=[{\check{z}}_c^{⁎},{\hat{z}}_c^{⁎}]=[35.17,40.83]$$

whose central value is $z_c=34.84$, their optimal routes are ${\check{x}}_c^{⁎}={\hat{x}}_c^{⁎}=(1,2)\to (2,3)\to (3,5)\to (5,6)\to (6,7)$ and the obtained minimum and maximum variances namely ${\check{v}}_c^{⁎},{\hat{v}}_c^{⁎}$ are computed as the sum of individual variances of each optimal route i.e.

$$\begin{gathered} {\check{v}}_c^{⁎}=\min \{[0.214,2.079]+[0.909,6.014]+[1.774,7.681]+[0.704,3.974]+[1.999,7.988]\}=5.6 \\ {\hat{v}}_c^{⁎}=\max \{[0.214,2.079]+[0.909,6.014]+[1.774,7.681]+[0.704,3.974]+[1.999,7.988]\}=27.736 \end{gathered}$$

The obtained optimal durations of the project for every α are summarized in Table 4 and displayed in Figure 11 where two optimal routes $a=(1,2)\to (2,3)\to (4,6)\to (6,7)$ and $b=(1,2)\to (2,3)\to (3,5)\to (5,6)\to (6,7)$ were obtained.

Table 4.

Results of Example 2 (Lyndwood & Montgomery).

α	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
${\check{z}}^u(\alpha )$	15(a)	16.44(a)	18.03	19.71	21.31	22.9	24.55	26.32	28.35	30.91	37
${\check{z}}^l(\alpha )$	27(a)	27.5	28.65	29.54	30.33	31.08	31.82	32.59	33.45	34.52	37
${\hat{z}}^l(\alpha )$	54	48.28	46.74	45.57	44.56	43.63	42.72	41.8	40.8	39.61	37
${\hat{z}}^u(\alpha )$	70	62.24	59.32	56.96	54.81	52.76	50.71	48.58	46.22	43.36	37

Figure 11.

Fuzzy set of optimal values of the project (Johnson & Montgomery).

4.2.1. Analysis of results

As expected, Fig. 11 shows that the project has a fuzzy–PERT shape since all activity times are fuzzy–PERT shaped. The set $Z_c$ is the set of expected durations of the project and central expected value of the project $z_c$ is the overall mean value of the project, the core $K(\tilde{Z})=37$ is the modal value of durations of the project. Again, $K(\tilde{Z})$ was intended to correspond to the original example and at least $4(n-1)\cdot \alpha +1$ MLPs are required to compose $\tilde{Z}$.

The most optimistic/pessimistic durations of the project are given by $S(\tilde{Z})=[{\check{z}}^u,{\hat{z}}^u]=[15,70]$ so the smallest possible duration of the project is 15 and the largest possible duration of the project is 70 given the fuzzy–PERT perception of the experts. No β–truncated sets were needed in this example.

Two critical paths were obtained: $a=(1,2)\to (2,3)\to (4,6)\to (6,7)$ which was obtained for values ${\check{z}}^u(\alpha ),\alpha ⩽0.1$ and ${\check{z}}^l(\alpha ),\alpha =0$ to then switch to path $b=(1,2)\to (2,3)\to (3,5)\to (5,6)\to (6,7)$ for the remaining α (see Table 4). This means that the project tends to flow through path (a) for optimistic durations and switch to path (b) for central/modal and pessimistic durations so the analyst can have an idea of the possible path changes ahead of time.

5. Concluding remarks

Fuzzy sets theory is useful in projects where no statistical information to estimate activity durations is available, and interval Type–2 fuzzy sets helps to deal with uncertainty coming from multiple expert estimates which are incorporated into a mathematical programming model (see Section 3.2). Two solution methods are proposed: an α–cuts representation method which provides a detailed representation of the duration of the project (see Definition 10) and a centroid–based method which provides an expectation of the duration of the project (see Eqs. (39), (40) and (41)).

The use of Interval Type–2 fuzzy sets to involve uncertainties in project network allow analysts to obtain a set of possible project durations $\tilde{Z}$, its optimal values and routes in advance. The set $\tilde{Z}$ also provides optimistic/modal/pessimistic estimates of the duration of the project in order to support decision making.

The presented examples are intended to illustrate two approaches: a CPM example originally introduced by Taha [34] which is focused to provide the set of optimal durations $\tilde{Z}$ and a fuzzy-PERT example originally proposed by Lyndwood & Montgomery [19] which also obtain the expected/variance of the project in an interval Type–2 fuzzy environment. It is important to note that $\tilde{Z}$ needs at least $4(n-1)\cdot \alpha +1$ optimal MLPs and the computation of $C(\tilde{T})$ and $V(\tilde{T})$ are still open problems without closed forms (see Mendel & Liu [29], Wu & Mendel [40] and Mendel [41]) so efficient computations are a forthcoming issue to be addressed.

Future work

The use of general Type-2 fuzzy sets is a natural step in the analysis of fuzzy uncertainty and its application to fuzzy optimization/scheduling. Other shapes that can be used in the analysis include the Gau–Angle IT2FS proposed by Wagner [37] and other non symmetric membership functions that can help to represent nonlinear activity durations.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

CRediT authorship contribution statement

Juan Carlos Figueroa–García: conceived and designed the experiments, wrote the paper.

Germán Hernández—Pérez: analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Jennifer Soraya Ramos–Cuesta: performed the experiments; Contributed reagents, materials, analysis tools or data.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Earliest/latest starting times

The obtained values of the earliest/latests starting times ${\tilde{e}}_{ij}$ and ${\tilde{l}}_{ij}$ for the non–critical activity (2,3) of the Example 1 (Taha) are displayed in Fig. A.12.

The behavior of the earliest starting time (in black) of non–critical activities is clearly binary ordered which means that it is a function of α i.e.

$${\check{e}}_{ij}^u(\alpha )⩽{\check{e}}_{ij}^l(\alpha )⩽{\hat{e}}_{ij}^l(\alpha )⩽{\hat{e}}_{ij}^u(\alpha );\alpha \in [0,1]$$

while its late starting time (in blue) is clearly non–binary ordered i.e. it is not a function of α–cuts.

Appendix B. Values used in Example 4.1

Table B.5.

Values of α-cuts for Example 4.1.

Activity		${\tilde{T}}_{12}$	${\tilde{T}}_{13}$	${\tilde{T}}_{23}$	${\tilde{T}}_{24}$	${\tilde{T}}_{35}$	${\tilde{T}}_{36}$	${\tilde{T}}_{46}$	${\tilde{T}}_{56}$
α = 0	${\check{T}}^u(\alpha )$	1,33	1,00	0,55	2,01	0,50	3,66	0,10	6,01
	${\check{T}}^l(\alpha )$	2,55	3,00	1,16	3,51	1,00	6,10	0,25	9,00
	${\hat{T}}^l(\alpha )$	8,67	6,00	4,84	11,00	3,00	13,45	1,00	17,99
	${\hat{T}}^u(\alpha )$	9,90	9,00	6,67	12,49	5,00	15,90	1,50	20,99
α = 0.1	${\check{T}}^u(\alpha )$	2,05	1,50	1,03	4,14	0,65	5,10	0,19	8,14
	${\check{T}}^l(\alpha )$	3,03	3,30	1,53	5,10	1,10	7,07	0,33	10,07
	${\hat{T}}^l(\alpha )$	7,95	6,00	4,47	9,93	2,90	12,97	1,00	15,86
	${\hat{T}}^u(\alpha )$	8,93	8,70	5,95	10,90	4,70	14,93	1,45	17,79
α = 0.2	${\check{T}}^u(\alpha )$	2,47	2,00	1,31	5,15	0,80	5,93	0,28	9,15
	${\check{T}}^l(\alpha )$	3,31	3,60	1,73	5,86	1,20	7,62	0,40	10,57
	${\hat{T}}^l(\alpha )$	7,53	6,00	4,27	9,43	2,80	12,69	1,00	14,85
	${\hat{T}}^u(\alpha )$	8,38	8,40	5,53	10,14	4,40	14,38	1,40	16,28
α = 0.3	${\check{T}}^u(\alpha )$	2,78	2,50	1,52	5,81	0,95	6,56	0,37	9,81
	${\check{T}}^l(\alpha )$	3,52	3,90	1,89	6,36	1,30	8,04	0,48	10,91
	${\hat{T}}^l(\alpha )$	7,22	6,00	4,11	9,09	2,70	12,48	1,00	14,19
	${\hat{T}}^u(\alpha )$	7,96	8,10	5,22	9,64	4,10	13,96	1,35	15,28
α = 0.4	${\check{T}}^u(\alpha )$	3,05	3,00	1,70	6,31	1,10	7,10	0,46	10,31
	${\check{T}}^l(\alpha )$	3,70	4,20	2,03	6,73	1,40	8,40	0,55	11,16
	${\hat{T}}^l(\alpha )$	6,95	6,00	3,97	8,84	2,60	12,30	1,00	13,69
	${\hat{T}}^u(\alpha )$	7,60	7,80	4,95	9,27	3,80	13,60	1,30	14,53
α = 0.5	${\check{T}}^u(\alpha )$	3,30	3,50	1,86	6,71	1,25	7,59	0,55	10,71
	${\check{T}}^l(\alpha )$	3,86	4,50	2,15	7,03	1,50	8,73	0,63	11,36
	${\hat{T}}^l(\alpha )$	6,70	6,00	3,85	8,64	2,50	12,14	1,00	13,29
	${\hat{T}}^u(\alpha )$	7,27	7,50	4,70	8,97	3,50	13,27	1,25	13,93
α = 0.6	${\check{T}}^u(\alpha )$	3,53	4,00	2,02	7,04	1,40	8,07	0,64	11,04
	${\check{T}}^l(\alpha )$	4,02	4,80	2,27	7,28	1,60	9,04	0,70	11,52
	${\hat{T}}^l(\alpha )$	6,47	6,00	3,73	8,48	2,40	11,98	1,00	12,96
	${\hat{T}}^u(\alpha )$	6,96	7,20	4,47	8,72	3,20	12,96	1,20	13,43
α = 0.7	${\check{T}}^u(\alpha )$	3,77	4,50	2,18	7,33	1,55	8,54	0,73	11,33
	${\check{T}}^l(\alpha )$	4,18	5,10	2,39	7,50	1,70	9,36	0,78	11,66
	${\hat{T}}^l(\alpha )$	6,23	6,00	3,61	8,34	2,30	11,82	1,00	12,67
	${\hat{T}}^u(\alpha )$	6,64	6,90	4,23	8,50	2,90	12,64	1,15	13,01
α = 0.8	${\check{T}}^u(\alpha )$	4,03	5,00	2,35	7,58	1,70	9,05	0,82	11,58
	${\check{T}}^l(\alpha )$	4,35	5,40	2,51	7,68	1,80	9,70	0,85	11,79
	${\hat{T}}^l(\alpha )$	5,97	6,00	3,49	8,21	2,20	11,65	1,00	12,42
	${\hat{T}}^u(\alpha )$	6,30	6,60	3,97	8,32	2,60	12,30	1,10	12,63
α = 0.9	${\check{T}}^u(\alpha )$	4,33	5,50	2,55	7,80	1,85	9,66	0,91	11,80
	${\check{T}}^l(\alpha )$	4,55	5,70	2,66	7,85	1,90	10,11	0,93	11,90
	${\hat{T}}^l(\alpha )$	5,67	6,00	3,34	8,10	2,10	11,45	1,00	12,20
	${\hat{T}}^u(\alpha )$	5,89	6,30	3,67	8,15	2,30	11,89	1,05	12,30
α = 1	${\check{T}}^u(\alpha )$	5	6	3	8	2	11	1	12
	${\check{T}}^l(\alpha )$	5	6	3	8	2	11	1	12
	${\hat{T}}^l(\alpha )$	5	6	3	8	2	11	1	12
	${\hat{T}}^u(\alpha )$	5	6	3	8	2	11	1	12

Data availability

Data included in article/supplementary material/referenced in article.