Inferential Methods and the Computer-Based Approach to Project Management

Gordana DUKIĆ Josip Juraj Strossmayer University of Osijek, Faculty of Philosophy Department of Information Sciences Lorenza Jägera 9, 31000 Osijek, Croatia

Mate SESAR Ph.D. Student Josip Juraj Strossmayer University of Osijek, Faculty of Agriculture Department of Agroeconomics Trg Sv. Trojstva 3, 31000 Osijek, Croatia

Ivana SESAR Student Josip Juraj Strossmayer University of Osijek, Faculty of Agriculture Department of Agroeconomics Trg Sv. Trojstva 3, 31000 Osijek, Croatia

ABSTRACT INTRODUCTION

Project management is a discipline of outstanding Successful operation of any organization is directly importance, especially in cases when a certain task related to project realization. Projects as such can serve requires substantial human, technical and financial different purposes, e.g. they can result in shortening the resources to be accomplished. When estimating project time required for a certain activity, cost reduction, quality duration and its costs, as well as identifying its critical improvement, increase in sales, or improved working path activities, project managers can gain crucial support conditions. There are numerous determinants of the from computer simulation. The decision-making model concept 'project', two of which will be pointed out here. presented in this paper is based on the assumption that G.H. Blackiston [9] refers to projects when speaking of activity durations can be defined as random variables that activities that result in new or changed products, services, follow a triangular or beta PERT distribution. In order to environments, processes and organizations. According to estimate project duration as accurately as possible, the H.A. Levine [12], a project is a group of tasks, performed proposed computer-based model envisages a higher in a definable time period, in order to meet a specific set number of simulation sets. By using the analysis of of objectives. This author believes that a project is (ANOVA) and post hoc comparisons we are characterized by being a one-time program, by its life able to determine simulation sets whose averages of cycle, work areas that can be categorized in certain tasks, project duration have no statistically significant budget, using multiple resources, and possibly the need to differences. After their frequency distribution is formed, establish a special organization or to cross the boundaries we can calculate the confidence interval for the mean of the traditional one. Although the issues of project project duration, which is one of the key pieces of management is directly connected to the development of information in project management. human civilization, modern project management, based on scientific approach and systematic application of Keywords: project management, computer-based technological achievements, came into being only in mid- approach, decision-making model, computer simulation, 20th century. Over the past few decades, different random variable, ANOVA, post hoc comparisons, organizations, including non-profit, public and confidence interval for the mean project duration government institutions, have recognized the importance of project management. In today's turbulent conditions, efficient realization of any complex task is inconceivable without this discipline.

Project management can be defined as the application of stochastic one. The procedures used within these methods relevant logic and tools to planning, directing, and to determine the critical path, the expected project controlling a temporary endeavor means of planning and duration, and the slack are available in numerous control (M.M. Helms (eds.) [5]). In line with the Project publications (R. Bronson, G. Naadimuthu [2], F.S. Management Body of Knowledge (PMBOK), J.P. Lewis Hillier, G.J. Lieberman [7], H.A. Taha [17]). Within the [13] defines project management as the application of proposed decision-making model, activity durations are knowledge, skills, tools and techniques to project defined as random variables that follow a triangular or activities to achieve project requirements. It follows from beta PERT distribution (F.E. Williams [19], C. the above that project management encompasses a wide Hendrickson [6], J. Fente, K. Knutson, C. Schexnayder spectrum of different concepts, tools and techniques [4], P.A. Jensen [8]). In order to estimate project duration designed to facilitate the efficient achievement of planned as accurately as possible, and to identify the activities on objectives. the critical path, the model envisages repeating a number of simulation sets. This procedure is inconceivable According to E. Verzuh [18], project management can be without a computer and appropriate program applications. divided into five main functions: selection, definition, planning, control and close out. During selection, a Project managers can significantly improve their project is assessed regarding its purpose and benefits, its decision-making by using inferential statistics methods objectives are being related to the organization's strategic (D.-E. Lee, D. Arditi [10], D.-E. Lee, J.J. Shi [11], G. goals, and resource constraints are determined. Having Dukić, D. Dukić, M. Sesar [3]). In the model analyzed selected the project, the organization identifies all the here, the simulation sets whose averages of project stakeholders interested in its performance, appoints duration have no statistically significant differences are people in charge, and draws up the required documents. determined by means of the analysis of variance They list the objectives, constraints, management (ANOVA) and post hoc comparisons. These are then the structure, performance benchmarks, and all the other facts basis for calculating the confidence interval for the mean or assumptions that are important for completing the project duration, which is one of the most valuable pieces project. In this way, the project has been defined. This is of information for project managers. followed by the planning stage, including a detailed description of the activities, their schedule, an estimate of project costs and duration, as well as other required PROJECT PLANNING USING CPM/PERT resources, especially personnel. Within the control function, the project progress is monitored, establishing The aim of the CPM and PERT methods is to provide an possible departures from the set plan, which requires analytical framework for scheduling the activities. feedback between particular phases. If there are Therefore, there are four key values to be determined departures from the plan, the project team takes within both of these methods: the earliest start time (ES), corrective action. An important aspect of this function is the earliest finish time (EF), the latest start time (LS) and communication of the control bodies with the project the latest finish time (LF). They are calculated on the team and all the other stakeholders. Close out is the last basis of estimated duration of each activity (Dij). In this phase in project implementation. As a function, it way we can obtain the elements required to create a includes disbanding the team that worked on the project network diagram. An example of such a diagram, where and dismantling the project infrastructure. There should the nodes are marked with A, is given in Figure 1. be a systematic follow-up of the outcomes achieved in the realization phase. If these outcomes do not meet the organization's requirements, project management needs to A(1) be put in place again. As its efficiency is closely related 1 ES1 LF1 D 0 1 to the available management information, an area that D D 4 1 A(0)3 A(4) merits special attention within project management is the D D 34 modeling and construction of an adequate database. ES0 LF0 A(3) ES4 LF4 02 D 23 A(2) ES3 LF3

ES LF RESEARCH METHODOLOGY 2 2

The decision-making model presented in this paper is Figure 1. An example of a network diagram based on project management techniques. The two best known, CPM (Critical Path Method) and PERT (Project The basic difference between the CPM and PERT Evaluation and Review Technique), are the starting point methods lies in the way how activity duration is for the model we have developed. The former belongs to determined. Whereas CPM is based on a single time the group of deterministic methods, whereas the latter is a estimate, within PERT, three time estimates are given for every activity - optimistic (a), most likely (m) and ⎛ T − μ{}E( j ) ⎞ pessimistic (b). In such a case, the expected duration of P()D{}E( j ) ≤ T = P⎜ z ≤ ⎟ ⎜ σ 2 {}⎟ each activity (i, j) is calculated by means of the formula: ⎝ E( j ) ⎠

1 Dij = ()a + 4m + b With the increasing number of activities the parameter 6 approximation is improving, and with it also the estimate If the earliest start time of the first activity is marked with of probable completion of an event within the set ES0, then it must be ES0 = 0. The earliest start time of any timeframe. activity (ESj) is determined in the following way:

{} ES j = max ESi + Dij , for all i entering into j. SIMULATION OF ACTIVITY DURATION i BASED ON A TRIANGULAR AND BETA PERT DISTRIBUTION The latest finish time of an activity (LFi) can be calculated as follows: In comparison to the CPM method, the usage of PERT can only slightly decrease the risk of inaccurate estimate LFi = min{}LF j − Dij , for all j leaving from i. j of project duration and its costs, as well as of the incorrect identification of activities on the critical path. The activity is critical if it satisfies the following The reason for this is that, when using PERT in project conditions: planning, only one time estimate is taken into account, namely the one obtained by calculating the expected

1. ESi = LFi duration of an activity. What is disregarded in this way are all the possible subcritical paths, which could be 2. ES j = LFj generated due to wrong estimates of this value. Hence, − = − = 3. ES j ESi LFj LFi Dij the initial model based on the PERT method needs to include the concept of computer simulation. The time remaining after completing an activity is called the slack. It can be determined in the following way: PERT method is based on an assumption that project activity duration can be defined as a random variable that

TFij = LFj − ESi − Dij follows one of the theoretical distributions. The distribution choice will depend on various factors that Activities on the critical path have no slack, i.e. for them project managers have to take into account when making a decision. Because of their characteristics, the triangular TFij =0. and beta PERT distributions seem to be an appropriate

choice for this purpose. In contrast to CPM, within PERT method it is possible to determine the of completing any activity, and The is defined by three parameters, the project as a whole, within the planned schedule. If it which can be perceived as three time estimates in the area is assumed that activity durations are independent random of project planning - optimistic (a), most likely (m) and variables, then the mean and variance of the selected path pessimistic (b). The triangular probability density to the node A( j) are as follows: function is:

μ{}A( j ) = ES j ⎧ 2 x − a , a ≤ x ≤ m σ 2 {}= σ 2 ⎪ A( j ) k ⎪b − a m − a ∑ f ( x ) = k ⎨ − ⎪ 2 b x ≤ ≤ , m x b ⎩⎪b − a b − m In the above expression, k refers to the defined activities along the longest path leading to the node A( j). The mean and variance of activity duration (i, j), defined as a random variable that follows a triangular distribution, In accordance with the central limit theorem, we can are as follows: assume that the project duration until the observed node is normally distributed. In this case, the probability of + + completing an event E( j) within time T can be calculated a m b μij = with the following formula: 3 2 + 2 + 2 − − − 2 a b m ab am mb σ ij = 18 f(x) Figure 2 shows three different shapes of the probability (α = 2,β = 4) (α = 4,β = 2) density function of the triangular distribution, depending 2.0 on the parameters a, m and b.

1.5 ) f(x) 2 = β , (=2,=,amb 4 =6) 2 0.5 = (=2,=3,=6)amb (=2,=,amb 5 =6) (α 1.0 0.4 0.5 0.3

0.2 0.2 0.4 0.6 0.8 1.0 x

0.1 Figure 3. Three different shapes of the beta probability density function, depending on the parameters α and β 2 34 56 x Assuming that three time estimates are known, the mean Figure 2. Three different shapes of the triangular and variance of activity duration can be determined in the probability density function, depending on the following way: parameters a, m and b 1 μij = ()a + 4m + b A distribution with the parameter m = 3 reflects the 6 optimistic attitude of the project manager regarding the − 2 duration of an activity. On the other hand, a distribution 2 ⎛ b a ⎞ σ ij = ⎜ ⎟ with the parameter m = 5, which is negatively ⎝ 6 ⎠ asymmetrical, reflects the pessimistic attitude. The middle curve indicates that the project manager is To carry out the procedure of simulating the activity undecided in this respect. durations on the basis of a random variable generated from the beta PERT distribution, it is necessary to Beta PERT distribution is based on the , estimate the values of parameters α and β: which has the following probability density function:

μ − μ − − μ − α −1 − β −1 ⎛ ij a ⎞⎛ ( ij a )(b ij ) ⎞ = ( x a ) ( b x ) α = ⎜ ⎟⎜ −1⎟ f ( x ) α +β − ⎜ − ⎟⎜ σ 2 ⎟ Β(α ,β )(b − a ) 1 ⎝ b a ⎠⎝ ij ⎠

≤ ≤ α β > ⎛ b − μij ⎞ a x b , , 0 β = ⎜ ⎟α ⎜ μ − ⎟ ⎝ ij a ⎠ In the above expression, Β (α,β) denotes the , whereas a denotes the lower, and b the upper The generation of random values of activity durations can bound of the distribution. The mean and variance of a start once the theoretical distribution has been chosen and beta distribution are: its parameters determined. It should be noted that, compared to the triangular distribution, the beta PERT αb + βa distribution to a lesser degree results in generating μ = α + β extreme values. Thus, if a project manager is leaning towards a more optimistic or a more pessimistic estimate 2 ⎛ b − a ⎞ αβ of activity duration, (s)he should opt for the triangular σ 2 = ⎜ ⎟ ⎜ ⎟ distribution. If the opposite is the case, the beta PERT ⎝ α + β ⎠ α + β +1 distribution should be used.

Figure 3 shows three different shapes of the beta Because of their complexity and iterative character, probability density function, depending on the parameters performing simulations requires the use of computers and α and β. appropriate software. The development of information

technologies was the basic prerequisite for broad implementation of simulations in different fields of human activities. USING INFERENTIAL STATISTICS manager as additional information in the decision making METHODS IN PROJECT PLANNING process.

In the decision making model we have developed here, it If the results of testing carried out by means of the is assumed that the simulation process takes place through analysis of variance show that there is at least one mean a particular number of sets. Their number, as well as the project duration that is statistically significantly different number of simulations within particular sets, should be from the others, we need to proceed with the post hoc determined by the project manager. It is advisable to carry comparisons. This refers to an analysis that uses a out at least 125 simulations in each set, since for n >125 particular statistical test in order to compare all the the critical value of the t-distribution is changed by less possible pairs of the means, so as to identify the groups than 1%. with the biggest influence on accepting the hypothesis that there are differences. There are several such tests that Random numbers are generated from the chosen a project manager can use (LSD test, Scheffé test, theoretical distribution with a computer. The process of Duncan test, Tukey HSD test, Student-Newman-Keuls simulation can be broken down into several basic phases, test, Dunnett’s one-tailed t-test, Bonferroni t-test, Waller- which depends primarily on the features of program Duncan t-test, Sidak t-test, Gabriel’s pairwise application used for this purpose. As a rule, the first phase comparisons test, etc.). Given the limited scope of this includes choosing a random number from the interval paper, only the LSD test will be briefly presented below. (0,1). In this way we determine the probability of completing the analyzed activity. After that, we need to LSD test is equivalent to the t-test for independent find the value of the triangular or beta PERT distribution samples. When all the samples in this test are of the equal which corresponds to this probability, and represents the size, the least significant difference is calculated expected duration of the activity. In the proposed decision according to the following formula: making model, the above procedure should be carried out for all project activities. The values obtained in this way 2MSE LSD = tα / 2,k( n−1) are the basis for creating a network diagram, which is used n to identify the critical path, and to estimate the slack and project duration. Pertmaster, Project Planner, Where RationalPlan, PERT Chart EXPERT, Planner Suite, tα / 2,k( n−1) = Critical value of the t-distribution Microsoft Project, RiskyProject and MinuteManPlus Management Software are only a few of project k = Number of simulation sets which are management software applications that can be used for compared, i.e. the number of means these purposes. n = Common number of simulation MSE = Error mean square To determine the simulation sets whose mean project durations have no statistically significant differences, it is The decision to accept or reject the hypothesis that mean necessary to conduct the analysis of variance (ANOVA) project durations are equal is made by comparing their and post hoc comparisons. ANOVA is used to test the differences with LSD value. If this difference is smaller hypothesis that three or more population means are equal. that the LSD value, we should accept the assumption that In addition to the F-value and p-level, on the basis of the means of the corresponding pair do not differ which we make the decision whether to accept or reject statistically significantly. Otherwise, this hypothesis is the set hypothesis, the ANOVA table contains also some rejected. statistical-analytical values, which can serve the project

SOURCE df SUM OF SQUARES MEAN SQUARE F-value p-level

n SSR = ˆ − 2 = REGRESSION k SSR ∑( yi y ) MSR i=1 k MSR F = n SSE MSE RESIDUAL n − ( k +1) SSE = ( y − ˆy )2 MSE = ∑ i i n − ( k + 1) i=1 n − = − 2 TOTAL n 1 SST ∑( yi y ) i=1

Table 1: ANOVA table The confidence interval for the mean project duration is Stiffler, V., Hljuz Dubric, V., Bekic, Z. (eds.): formed on the basis of average values for which no Proceedings of the ITI 2008, SRCE University statistically significant differences have been established: Computing Centre, University of Zagreb, Zagreb, 2008., pp. 203-208. [4] Fente, J., Knutson, K., Schexnayder, C.: Defining a ⎛ sT sT ⎞ P⎜ x − tα < μ < x + tα ⎟ = ()1− α ⎜ T / 2 T / 2 ⎟ Beta Distribution Function for Construction ⎝ kT kT ⎠ Simulation, http://www.informs-cs.org/wsc99papers /146.PDF Where [5] Helms, M.M. (ed.): Encyclopedia of Management,

xT = Mean of the average project durations for which Fifth Edition, Thomson Gale, Detroit, 2006. no statistically significant differences have been [6] Hendrickson, C.: Advanced Scheduling established Techniques, http://www.ce.cmu.edu/pmbook/11_

tα / 2 = Critical value of the t-distribution Advanced_Scheduling_Techniques.html [7] Hillier, F.S., Lieberman, G.J.: Introduction to sT = Estimate of the standard deviation Operations Research, Seventh Edition, McGraw- k = Number of simulation sets, i.e. the number of Hill, Boston, 2001. means [8] Jensen, P.A.: Project Management - Beta

Distribution, http://www.me.utexas.edu/~jansen/ Numerous different purpose program applications that are ORMM/omie/computation/unit/project/beta.html available to project managers make it possible to conduct [9] Kerzner, H.: Strategic Planning for Project other forms of analyses within the project planning stage. Management Using a Project Management

Maturity Model, John Wiley & Sons, Inc., New

York, 2001. CONCLUSIONS [10] Lee, D.-E., Arditi, D.: Automated Statistical

Analysis in Stochastic Project Scheduling The paper has presented a decision-making model for Simulation, http://www.iit.edu/~leedong1/SPSSII. project management, which is based on a computer pdf approach and inferential statistics methods. Its [11] Lee, D.-E., Shi, J.J.: Statistical Analyses for implementation creates the conditions for improved Simulating Schedule Networks, http://www. decision making by project managers, which leads to a informs-cs.org/wsc04papers/167.pdf more efficient project planning. We should emphasize [12] Levine, H.A.: Practical Project Management - that the proposed model cannot eliminate all the risks Tips, Tactics, and Tools, John Wiley & Sons, Inc., connected to project management; however, they can be New York, 2002. reduced to a degree, by improving information base. [13] Lewis, J.P.: Fundamentals of Project Management, Third Edition, AMACOM, New This model has not dealt specifically with the area of cost York, 2007. management in a project. Given the importance of costs, [14] Montgomery, D.C., Runger, G.C.: Applied further research should focus on this area. Furthermore, Statistics and Probability for Engineers, Third there is a need to develop program applications that Edition, John Wiley & Sons, Inc., New York, 2003. would combine all the elements of the presented model. [15] Mora, M., Forgionne, G.A., Gupta, J.N.D. (eds.):

Decision Making Support Systems -

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