Journal of Applied Mathematics

Volume 2015, Article ID 809216, 15 pages

http://dx.doi.org/10.1155/2015/809216

## Developing a Mathematical Model for Scheduling and Determining Success Probability of Research Projects Considering Complex-Fuzzy Networks

Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran

Received 23 January 2015; Revised 25 June 2015; Accepted 25 June 2015

Academic Editor: Ching-Jong Liao

Copyright © 2015 Gholamreza Norouzi et al. 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

In project management context, time management is one of the most important factors affecting project success. This paper proposes a new method to solve research project scheduling problems (RPSP) containing Fuzzy Graphical Evaluation and Review Technique (FGERT) networks. Through the deliverables of this method, a proper estimation of project completion time (PCT) and success probability can be achieved. So algorithms were developed to cover all features of the problem based on three main parameters “duration, occurrence probability, and success probability.” These developed algorithms were known as PR-FGERT (Parallel and Reversible-Fuzzy GERT networks). The main provided framework includes simplifying the network of project and taking regular steps to determine PCT and success probability. Simplifications include (1) equivalent making of parallel and series branches in fuzzy network considering the concepts of probabilistic nodes, (2) equivalent making of delay or reversible-to-itself branches and impact of changing the parameters of time and probability based on removing related branches, (3) equivalent making of simple and complex loops, and (4) an algorithm that was provided to resolve no-loop fuzzy network, after equivalent making. Finally, the performance of models was compared with existing methods. The results showed proper and real performance of models in comparison with existing methods.

#### 1. Introduction

Research and R&D projects are often conducted initially to design and manufacture a product with certain capabilities. A major part of projects is conducted in research organizations and institutes especially military ones. These projects have certain features including [1, 2] (1) uncertainty in defining the activities, (2) unclear and uncertain results of activities, (3) uncertainty in time and cost of activities, (4) noniterative activities, (5) loops and iterations to previous activities, and (6) parallel paths. Due to such features, a certain technique is needed to deal with such projects under determined time and cost effectively. On the other hand, backwardness and longtime of undertaking project would lead into huge costs not forecasted in initial budget plan and organizations may face numerous problems.

In practice, the projects have been implemented through an uncertain environment that ambiguity is one of the major features of such environments. Uncertainty may be considered as a property of the system which is an indicative defect in human knowledge towards a system and its state of progression [3]. Introduction of uncertainty topic for the first time entitled probability and it was attributed to Aristotle. Meanwhile, there are various types of uncertainty in the real world including fuzzy and random and being uncertain that includes the former two items [4]. Along with randomization, fuzziness is introduced as a fundamental type of subjective uncertainty [5].

In the early 20th century, Henry Gant and Frederick Taylor introduced Gantt chart which showed start and end of projects. However, overall timescale of the projects based on precedence relationships and analyses was not possible by this technique. Therefore, network designing in project control and design to remove raised problems and providing a more comprehensive technique was developed by a group of operation research (OR) scientists in 1950 and different methods, that is, CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique), were introduced and developed [6].

Project management, in particular, project planning, is a critical factor in the success or failure of new product development (NPD) [7] and research and development (R&D) projects [8]. Limited facilities of CPM and PERT methods for modeling of projects having complex and uncertain networks [6] caused the great scientists such as Pritsker and Whitehouse [9] and Pritsker and Happ [10] to introduce the probability GERT (Graphical Evaluation and Review Technique) which is a method consisted of flow graph theory, moment generation functions, and PERT to solve problems with probability activities. Pritsker [11] developed GERT method and used it for different nodes and introduced use of simulation in that method. Of course it is pointed out that Pritsker and Whitehouse [9] used Maison’s role which had been introduced in graph theory to solve probability networks. The pointed out probability methods are all based on probable distributions such as Beta () or normal distribution which are used for estimation of project activities period. Thus, to use probable distributions, random samples, repeatability, and statistical deduction will be required.

When fuzzy theory was introduced and developed by Zadeh [12], scientists’ and researchers’ attention in the field of network and science engineering was gradually caught by the issue that they will be able to solve the problem of uncertainty in problems and also expressed problems by a fuzzy approach. Thus, for the first time, Chanas and Kamburowski [13] introduced a new technique called FPERT (Fuzzy Program Evaluation and Review Technique), in which by using fuzzy theory, the estimation of activities time and project completion time had been shown by triangular fuzzy numbers. Gazdik [14] presented another technique based on fuzzy sets and graph theory called FNET (fuzzy network). This is for estimation of activities time by use of algebraic operators and by deductive method in the projects without data. Itakura and Nishikawa [15] applied fuzzy concepts in GERT networks for the first time, and by replacing fuzzy parameters for probability parameters, presented FGERT (Fuzzy Graphical Evaluation and Review Technique). method of solution is similar to probability GERT, but it is different because fuzzy theory was used to review GERT networks. McCahon [16] presented FPNA (Fuzzy Project Network Analysis) technique to identify critical path and obtain total float value and probability time for project completion. Ten years after presentation of his first measures on fuzzy GERT, Cheng [17, 18] presented another method of solution. Of course this method was presented to solve problems of series systems reliability, in which merely Exclusive-or nodes were dealt with. Basis of his method is the same probable GERT, but it is only different in using fuzzy parameters instead of probable functions. Nasution [19] dealt with development of FNET technique. He solved fuzzy networks with the multicritical paths approach by using interactive differential fuzzy method in the network reversible computations. Chang et al. [20] considered projects of large size planning networks. When such projects are under uncertain conditions, then the techniques presented to solve timed networks become complicated and sometimes unsolvable. Thus, they presented a technique by merging comparison and composite methods and also by using fuzzy Delphi, which was efficient for solving problems with the foregoing features. Shipley et al. [21] presented a technique called Belief in Fuzzy Probabilities of Estimated Time (BFPET) which is based on the fuzzy logic, belief functions, and fuzzy probability distributions and development principle. Wang [22–24] dealt with development of fuzzy sets approach for planning product development projects of limited and imprecise data. The approach he adopted was in line with the works of Buckley [25] and Tatish, that is, use of trapezoidal fuzzy numbers and their computational procedure. Chanas and Zielinski [26], Chen and Huang [27], Chen and Hsueh [28], Fortin et al. [29], and Ke and Liu [30] dealt with the issue of critical path in the network of projects scheduling problem that contains fuzzy duration times of activities. All these studies represent activity duration times described by means of fuzzy sets however without any component of time dependency. Huang et al. [31] have studied the project scheduling problem in CPM networks with activity duration times that are both fuzzy and time dependent.

Some new analytical methods for determining the completion time of GERT-type networks have been proposed by Shibanov [32], Hashemin, and Fatemi Ghomi [33]. The main problem of these methods is high complexity of relations and computations in a network without loops. Kurihara and Nishiuchi [34] solved probability GERT networks by Monte Carlo simulation, and by this method, they could analyze specific kind of nodes present in GERT networks without loops (with loops [35]). Gavareshki [1] and Lachmayer et al. [2, 36, 37] presented a new method to solve fuzzy GERT networks whose basis goes back to the method for solving CPM networks, that is, moving forward and implementation of fuzzy literature and inference of the nodes. Hashemin [38] provides an analytical procedure for this kind of networks by simplifying GERT networks. Shi and Blomquist [39] dealt with using matrix structures for solving time scheduling networks. By combination of fuzzy logic and DSM (Dependency Structure Matrix), he could introduce a new approach for solving problems on time scheduling of projects with uncertainty in periods of activities and uncertainty in overlapping the activities. Martínez León et al. [40, 41] present an analytical framework for effective management of projects with uncertain iterations in probable GERT networks. The framework is based on the Design Structure Matrix. As the introduced approach was new, it is capable of being developed in the fuzzy networks and uncertainty in the definition of activities.

Usual techniques are not able to estimate project completion time (PCT) of the projects that are executed for the first time or projects having computational problems [8, 42, 43]. Therefore, only networking techniques with particular definitions on parallel nods and branches may be regressed and can create loops in fuzzy environment which provide proper timescale for these projects by using the capabilities of fuzzy sets and networks adapted to the attributes and features of these projects. Concerning the bottlenecks of these techniques, a new method is developed to resolve the problems of these networks and to understand them better. Thus, the objectives of this research are(1)to apply three parameters, duration, occurrence probability, and success probability for each of the project activities, as well as use of trapezoidal fuzzy numbers to present duration, and probabilities to occurrence and success of activities, (2)to present methods for equivalent making of parallel and series branches in fuzzy networks considering the meanings and concepts of deterministic and probabilistic nodes, (3)to provide a new mathematical model for removing delay or reversible-to-itself branches in fuzzy networks, (4)to provide a new mathematical model for removing simple and subindependent complex loops in fuzzy networks based on transformation to reversible-to-itself branches, (5)to provide an algorithm for estimating of PCT and scheduling of the simplified (no loops) fuzzy networks, based on equivalent making of parallel and series branches, (6)to provide a new mathematical model for estimating of the probability of project success.

In Section 2, features of studied problem (PR-FGERT) are described. In Section 3, assumptions and parameters will be dealt with which we come across in the process of research stages. In Section 4, the analytical approach and proposed algorithm are described. In Section 5, a numerical case study is given, so that the algorithm efficiency is demonstrated. In Section 6, the proposed algorithm and solution model validation have been presented. Finally, the conclusion and future research are presented in Section 7.

#### 2. Problem Definition and Features of Studied Problem (PR-FGERT)

As mentioned in literature review, the closest and the most compatible type of network to display R&D projects is GERT network. Due to limitation of accessing time information on each activity, we used times of trapezoidal fuzzy. In most of the engineering applications, trapezoidal fuzzy numbers are used as they are simple to represent, are easy to understand, and have a linear membership function so that arithmetic computations can be performed easily [44]. These networks have capabilities such as different types of nodes, parallel branches between two nodes (various ways to do an activity and/or transition from one node to another one, Table 3), the existence of delay or reversible-to-itself branches as well as simple and mixed loops, precedence relationships to several nods, and the existence of several end nodes to describe and to convert a project to a model properly.

The main assumption of the research is the existence of three parameters for each activity where is duration of doing the activities as a trapezoidal fuzzy number, is occurrence probability, and is the probability of ending an activity successfully (this value is independent of other activities for different branches and activities).

Concerning GERT networks’ features with fuzzy times as well as presented assumptions and parameters, the first-time-executed project can be modeled easily. To resolve these networks so that they have high performance in terms of understanding deliverables and functionality, an algorithm is provided to resolve the probability and fuzzy of these networks by using an innovative technique. It also provides an estimation of PCT along with the probability of its successful ending.

#### 3. Model Assumptions and Parameters

The assumptions and parameters used in this paper are as follows.

##### 3.1. The Modeling Procedure Has Been Done according to the Following Assumptions

(i)Activity of the network has a single source and a single sink node. If several initial nodes and some end nodes exist, they should be connected to a dummy node.(ii)Input side of a node was applied of And, Exclusive-or, and Inclusive-or types; output side of a node was applied of deterministic and probabilistic types.(iii)Uncertainty in estimating duration of activities is of positive trapezoidal fuzzy number.(iv)Uncertainty in occurrence of activities is considered of probable type.(v)Uncertainty in success of activities is considered of probable type.(vi)In the project activities network, loops are considered.(vii)Maximum number of parallel branches between two nodes is three.(viii)The number of loops iteration is uncertain.(ix)Establishing the law of independence between existing loops in the network.

##### 3.2. Parameters

The parameters are as follows: : output activity from node and input to node . : output set of activities from node . : fuzzy duration of doing activity . : defuzzy duration of doing activity . : occurrence probability of activity . : success probability of activity . : output loop from node () and input to node (). : loop of level output from node () and input to node (). : occurrence probability of loop . : success probability of loop . : set of activities in loop . : time value of loop unit . : fuzzy duration of activity after impact of first time delay loop. : fuzzy duration of activity after impact of second time delay loop. : set of activities in a path consisting of series branches. : fuzzy duration of equivalent branch to series branches. : occurrence probability of equivalent branch to series branches. : success probability of equivalent branch to series branches. : fuzzy duration of equivalent branch to parallel branches. : occurrence probability of equivalent branch to parallel branches. : success probability of equivalent branch to parallel branches.

##### 3.3. Fuzzy Operations Used in the Proposed Solution Algorithm

Regarding and as two positive trapezoidal fuzzy numbers, the fuzzy operators are calculated as follows [44]:

#### 4. Analytical Approach and Methodology

##### 4.1. Conceptual Model

In this section, new parallel and reversible branches in the fuzzy GERT (PR-FGERT) are presented to solve the problem defined in Section 2. In the presented analytical approach, the proposed algorithm consists of several steps so that a graphical presentation is shown in Figure 1 described in the following subsections. However, five main steps of the algorithm are as follows.