Solving PERT Problems Involving Type-2 Fuzzy Uncertainty: An Approach - Technologies and Applications of Artificial Intelligence - page 136

Solving PERT Problems Involving Type-2
Fuzzy Uncertainty: An Approach

Juan Carlos Figueroa-Garcıa, Ivan Darıo Jimenez-Medina,
and Rausses Danilo Rojas-Olaya

Universidad Distrital Francisco Jose de Caldas, Bogota - Colombia
[e-mail]
[e-mail]
[e-mail]

Abstract. This paper proposes a method for solving the PERT (Program Eval-
uation and Review Technique) problem that involves uncertainty coming from
the perception of multiple experts about the activities. The experts opinions over
the duration of an activity is represented by interval Type-2 Fuzzy Sets (IT2FSs).
Four linear programming models based on the ʱ -cuts done over the duration of
the activities of the project are proposed and solved, keeping fuzzy information
coming from the experts into the solution.

Keywords: PERT, Interval Type-2 Fuzzy Sets, Linear Programming.

1 Introduction

The CPM (Critical Path Method) allows us to plan a project when the activity times are
considered as deterministic, it helps to make a fairly accurate estimate of the duration
of the project (see [1], and Kelley [2]). Duration of activities are not well known in all
projects, so the PERT technique has been independently but contemporaneously devel-
oped with the CPM method (see [3]), as a tool to handle probabilistic uncertainty over
the activities times through the beta distribution, thereby driving a degree of uncertainty.
There are some criticisms about the statistical assumptions of the PERT method, and
some authors have questioned the applicability of the beta distribution to estimate the
duration of activities (see MacCrimmon & Ryavec [4], and Shipley, De korvin & Omer
[5]). As an alternative to estimate the duration of activities, triangular and trapezoidal
distributions has been used for, but inference for activity times requires a measurement
process, or a posterior frequency distribution as pointed out by Chen [6], so in practice
some projects does lack of historical data and it is not possible to infer distributions, this
is the case where subjectivity of the experts comes as the only available information.
Fuzzy set theory has been proposed by Lotfi A. Zadeh in 1965 [7], thus the concept
of fuzzy PERT has been proposed by Fargier, Galvagnon & Dubois [8], Hsiau & Li
[9], Wang & Hao [10] in which activity times are represented by fuzzy sets , these fuzzy
sets come from the opinion of only one expert, or the agreement of several experts.
McCahon [11] developed a fuzzy project network analysis by fuzzifying the activity
times in project-network analysis.