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 PRFGERT (Parallel and ReversibleFuzzy 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 reversibletoitself 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 noloop 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.
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 [
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 [
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 [
Project management, in particular, project planning, is a critical factor in the success or failure of new product development (NPD) [
When fuzzy theory was introduced and developed by Zadeh [
Some new analytical methods for determining the completion time of GERTtype networks have been proposed by Shibanov [
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 [
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,
to present methods for equivalent making of parallel and series branches in fuzzy networks considering the meanings and concepts of deterministic and probabilistic nodes,
to provide a new mathematical model for removing delay or reversibletoitself branches in fuzzy networks,
to provide a new mathematical model for removing simple and subindependent complex loops in fuzzy networks based on transformation to reversibletoitself branches,
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,
to provide a new mathematical model for estimating of the probability of project success.
In Section
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 [
The main assumption of the research is the existence of three parameters
Concerning GERT networks’ features with fuzzy times as well as presented assumptions and parameters, the firsttimeexecuted 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.
The assumptions and parameters used in this paper are as follows.
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.
Input side of a node was applied of And, Exclusiveor, and Inclusiveor types; output side of a node was applied of deterministic and probabilistic types.
Uncertainty in estimating duration of activities is of positive trapezoidal fuzzy number.
Uncertainty in occurrence of activities is considered of probable type.
Uncertainty in success of activities is considered of probable type.
In the project activities network, loops are considered.
Maximum number of parallel branches between two nodes is three.
The number of loops iteration is uncertain.
Establishing the law of independence between existing loops in the network.
The parameters are as follows:
Regarding
In this section, new parallel and reversible branches in the fuzzy GERT (PRFGERT) are presented to solve the problem defined in Section
A graphical presentation of the proposed approach.
Given Figure
A display of delay branches.
(i) Computing the average of added time in other separating branches from node
According to Table
Added time in other branches separating from node (
Qyt. of branch ( 
Occurrence probability  Added time 

0 

0 
1 


2 


⋮  ⋮  ⋮ 



⋮  ⋮  ⋮ 
According to the
By exiting one
We assume that
By substituting of (
Therefore, after elimination of node
(ii) Computing the change of success probability in other separating branches from node
To study the change of success probability in other branches from node
Success probability of each branch from node (
Qyt. of branch ( 
Occurrence probability  Success probability of branch ( 

0 


1 


2 


⋮  ⋮  ⋮ 



⋮  ⋮  ⋮ 
Six states of parallel branches between two different nodes.
State  Definition  Figure 

1  Parallel branches between two nodes with “Probabilistic” output and “ExclusiveOr” input 



2  Parallel branches between two nodes with “Deterministic” output and “InclusiveOr” input 



3  Parallel branches between two nodes with “Deterministic” output and “And” input 



4  Parallel branches between two nodes with “Probabilistic” output and “And” input 



5  Parallel branches between two nodes with “Deterministic” output and “ExclusiveOr” input 



6  Parallel branches between two nodes with “Probabilistic” output and “InclusiveOr” input 

According to Table
Now, by expanding of (
So we can conclude that success probability of each branch from node
(iii) Computing the change of occurrence probability in other separating branches from node
In order to calculate sum of probabilities of branches separating from node
According to assumption in present paper, there are maximum three,
Similarly, other branches (
Concerning occurrence probability and success probability of equivalent branch, since branches are parallel and independent, occurrence probability is equal to probabilities aggregation (
As we see, to achieve equivalent activity time, first, the time of current branches are getting defuzzy (based on (
Defuzzy operation is a reversing process which returns a fuzzy distance to its crisp number:
Then, these activities will be sorted from the lowest to the highest based on defuzzy values. Now, we assign 1 for the lowest defuzzy value and mark other values:
Therefore, we have
Concerning occurrence probability and success probability, when activities are parallel and independent, occurrence probability is equal to the aggregation of standardized probabilities and success probability is a similar trend to activities duration:
On the other hand, based on the approach, the success probability of equivalent branch resulted from multiple success probability of all activities (
A set of branches is called series when the path between their source nodes includes no diversion path (
A display of series branches.
Existence of reversible branches can be eventuated to form a loop and to execute one or several activities leading to delay and increase in time of progress and performance of activities (Figure
View of a loop formation.
Parameters equivalent to a loop are equal to parameters of equivalent branch to the paths between the two nodes of source (
In order to reduce amount of computations, first, the parallel and series branches between two nodes of (
In the next stage, the set of branches and nodes between two nodes of (
If the equivalent branch connecting two nodes of (
The reviewed complicated loops are of subindependent type in which all subsets of the main loop are in condition of independence relating to each other and the main loop. To start solution and application of a complex loop impact, the following shall be followed:
Classification of the loops is as follows:
To assume the main loop as a loop at zero level.
To classify the internal loops into two groups: (a) Level 1 simple loops and (b) Level 1 complex loops.
To classify the internal loops of Level 1 complex loops into two groups: (a) Level 2 simple loops and (b) Level 2 complex loops.
To continue the process of classification until levels of all loops are specified.
Priority of the loops is as follows:
The main assumption on priority of loops shall be as follows:
The priority shall belong to loops in which the longest distance of their source node to last level source node involves the minimum rate.
Selection and solution of reviewed loop based on rendered priority in step (b) are as follows:
First, the lowest classified level in the loop shall be prioritized and based on such priority, the loops of this level shall be solved in the selected complex loop, and their impacts shall be applied based on step (6). Then, process of prioritization shall be executed for the loop at one upper level in this complex loop. The related impacts shall be computed and applied on the basis of priority of the loops at this level of complex loop again. This process shall continue until reaching the main loop of the selected complex loop.
Review of main loop is as follows:
After reviewing of all loops existing in Level 1 and application of their impacts, the impact of main loop of complex loop (Level zero) shall be computed and then we will go to step (5). Thus, the delay impacts resulting from existence of the loops (simple and complex) in the network can be applied, and their equivalent can be made and they shall be removed from the network.
Based on the parameters of the branch equalized for the paths between nodes (
For the application of the impacts of the loop (
Also, to fix the probability of the occurrences in the output branches from node (
Introduced problem solving method is based on equivalent making of parallel and series branches in each step of simplifying the fuzzy network and moving from the source node to the sink (end) node by surveying different paths between two nodes and to achieve a step in which only one equal branch is built between the two nodes of source and sink. Operation flowchart is shown in Figure
The proposed algorithm to resolve noloop fuzzy networks.
This part of the project is conducted for the first time in a research institute without any ambiguity in defining the actions and estimating the time and probability of success was considered as an applied example. In addition to supporting model validity, one can also determine the deficiencies and even inabilities of current methods in treating with parallel paths and combining them with loops as well as drawing and resolving the network. According to Figure
Input parameters of activity network.
Code of activity  Duration (week)  Success probability of activity/loop  Occurrence probability of activity/loop 

01  (4, 5, 6, 7) 


12  (7, 9, 10, 11) 


23  (2, 3, 4, 5) 


34  (8, 9, 10, 11) 


313  (12, 14, 15, 18) 


45 (a)  (2, 3, 4, 5) 


45 (b)  (2, 4, 5, 7) 


45 (c)  (6, 7, 8, 10) 


53  (0, 0, 0, 0) 


52  (0, 0, 0, 0) 


513  (1, 2, 2, 3) 


1315  (1, 1, 2, 2) 


06  (4, 5, 6, 8) 


67  (2, 3, 3, 4) 


78  (4, 5, 6, 7) 


89  (3, 4, 4, 5) 


98  (0, 0, 0, 0) 


910 (d)  (3, 4, 5, 6) 


910 (e)  (2, 4, 5, 7) 


1014  (1, 1, 1, 1) 


611  (2, 3, 3, 4) 


1112  (6, 7, 8, 9) 


1211  (0, 0, 0, 0) 


1212  (1, 2, 3, 3) 


1214  (1, 2, 2, 3) 


1415  (2, 3, 4, 4) 


1516  (4, 5, 6, 7) 


161  (0, 0, 0, 0) 


1617  (1, 2, 2, 3) 


Activity network of the R&D project.
Summary of solution procedure based on the presented algorithm is as follows: classification and prioritization of the loops existing in the network, based on steps (a) and (b), are as follows:
In the first step and based on Section
Input parameters of activity network without any loops.
Code of activity  Initial fuzzy duration of activity performance  Final fuzzy duration of activity performance  Success probability of activity  Occurrence probability of activity 

01  (4, 5, 6, 7)  (4, 5, 6, 7) 


12  (7, 9, 10, 11)  (16.8, 21.31, 24.38, 27.85) 


23  (2, 3, 4, 5)  (7.57, 10.01, 12.23, 14.85) 


34  (8, 9, 10, 11)  (11.53, 13.33, 15.03, 16.89) 


313  (12, 14, 15, 18)  (15.53, 18.33, 20.03, 23.89) 


45  (3.17, 4.71, 5.94, 7.66)  (3.17, 4.71, 5.94, 7.66) 


513  (1, 2, 2, 3)  (1, 2, 2, 3) 


1315  (1, 1, 2, 2)  (1, 1, 2, 2) 


06  (4, 5, 6, 8)  (4, 5, 6, 8) 


67  (2, 3, 3, 4)  (2, 3, 3, 4) 


78  (4, 5, 6, 7)  (4, 5, 6, 7) 


89  (3, 4, 4, 5)  (5, 6.67, 6.67, 8.33) 


910 (d)  (3, 4, 5, 6)  (3, 4, 5, 6) 


910 (e)  (2, 4, 5, 7)  (2, 4, 5, 7) 


1014  (1, 1, 1, 1)  (1, 1, 1, 1) 


611  (2, 3, 3, 4)  (2, 3, 3, 4) 


1112  (6, 7, 8, 9)  (8.91, 10.6, 12.29, 13.73) 


1214  (1, 2, 2, 3)  (1.54, 3.08, 3.62, 4.62) 


1415  (2, 3, 4, 4)  (2, 3, 4, 4) 


1516  (4, 5, 6, 7)  (4, 5, 6, 7) 


1617  (1, 2, 2, 3)  (1, 2, 2, 3) 


Activity network of the R&D project without any loops.
After three times, the network without loop (Figure
Deterministic methods are used to schedule the project so that limitation in drawing the network is proportionate to actual realities (such as regressive and parallel paths). Therefore, there will be timescales with huge differences. To resolve output problem, fuzzy CPM is used (10.9). To compare outputs from proposed algorithm and other methods, average of duration is computed and percentage of deviation can be computed by real time of executing the project.
According to Table
Comparative results and percentage of deviation.
Fuzzy duration (week)  Average of duration (month)  Percentage of deviation  

Fuzzy CPM  (34, 44, 50, 59)  10.9 

Proposed model  (54.1, 62.8, 80, 94.4)  17.3 

Real completion time  (102, 102, 102, 102)  23.8  — 
It is noteworthy that the probability of project success was computable in none of the current methods while it was computable in the proposed model and its obtained value (for case study) was 59%.
In this paper, for sensitivity analysis, the impact of loops occurrence probability on the project completion time (PCT) and the success probability are examined. According to the logic of GERT with loops, it is expected that, with increasing the probability of loop occurrence, the PCT increases and the success probability decreases and vice versa. To do this, the loop number 53
Results of sensitivity analysis.
Variation percentage of occurrence probability  Variation percentage of the PCT  Variation percentage of success probability 

−70%  −14.4%  6.3% 
−50%  −11.3%  4.9% 
−30%  −7.4%  3.2% 
−10%  −2.8%  1.2% 
10%  3.2%  −1.4% 
30%  11.2%  −5.1% 
50%  22.9%  −10.2% 
70%  43.0%  −18.6% 
Impact of variation probability of loop occurrence on success probability.
Impact of variation probability of loop occurrence on the PCT.
To prove appropriate performance of the algorithm presented in this paper, the results obtained by 3 articles including scheduling of fuzzy GERT networks were selected and the proposed algorithm’s output was compared with the outputs of the articles. These methods include Itakura’s method [
Proposed algorithm’s results, Gavareshki method’s results, and CPM solution.
Proposed algorithm’s result  Gavareshki method’s result  CPM solution 

(3.6, 4.5, 4.5, 6.3)  (4.5, 6.5, 6.5, 10.6)  3.5 
Results of Hashemin’s solution.
Activity  Fuzzy duration of activity  Occurrence probability of activity 

05  (6.06, 9.90, 16.13, 19.74)  0.121 
07  (5.67, 10.04, 15.52, 19.31)  0.879 
Fuzzy time of Itakura and the proposed algorithm.
In the article presented by Hashemin [
However, Hashemin’s method [
Since use of probabilistic distribution techniques in the projects including large activity network and a dynamic environment full of uncertainty has been rather complicated, difficult, and limited, during the recent decades, attention to simulation and fuzzy approaches has been considerably intensified and development of techniques in these fields is expanded.
Therefore,
In future, proposed algorithm (method) would need research and development in the below areas:
On complex loops, yet is opportunity for more research and development on subdependent complex loops.
One can work on changing provided algorithm to flowcharts and create software.
It is possible to use fuzzy numbers in estimating occurrence probability of activities and loops instead of probabilistic functions.
One can use analysis of sensitivity and update scheduling of activities network with loops.
To estimate cost of each activity and project completion based on the proposed algorithm.
The authors declare that there is no conflict of interests regarding the publication of this paper.