Simplifying PERT Network Simulation with ARENA William J. Cosgrove California State Polytechnic University, Pomona, CA The Arena simulation software package is one of the most advanced and sophisticated simulation tools employed in business and industrial engineering. It includes “drop and drag” features that make it suitable for a visual network model construction that parallels the nodes and arcs of a typical network representation. This study employs these features to simplify model construction for activity time networks based on PERT. This reduction in the burden of model construction should enhance the use of Arena in network simulation, permitting users to tap additional modeling features not typically found in more specialized network simulation packages.

I. INTRODUCTION

Activity time networks have been a staple in project management literature since the development of PERT (Program Evaluation and Review Technique) during the 1950’s when the U. S. Navy was hard pressed to develop the submarine launched Polaris missile system. The basic design of activity networks included arcs representing the passage of time that were bounded by nodes representing instantaneous events, such as boundary points that marked the beginning and end of activities (Malcolm et al., 1959). As PERT applications evolved, more attention was given to PERT bias which was described as a sometimes significant understatement of project network completion times (e.g., MacCrimmon and Ryavec, 1964). While several analytical approaches were proposed to adjust for PERT bias, it was evident that Monte Carlo methods and simulation offered the best solution in that they generated completion time distributions that were free from PERT bias (e.g., Van Slyke, 1966). A number of studies on network simulation approaches began to appear following Van Slyke’s study. GERT (Graphical Evaluation and Review Technique) simulation was an

enhancement to PERT which permitted probabilistic branching on AOA (activity on arrow) networks (Pritsker and Happ, 1966). PLANET (Project Length Analysis and Evaluation Technique) was a modification of GERT which accommodated criticality indices (Kennedy and Thrall, 1976). VERT (Venture Evaluation and Review Technique) shares many of the features of GERT and was independently developed for the military (Moeller and Digman, 1981). CAPERTSIM (Computer Assisted PERT SIMULATION) was developed primarily as a project management teaching tool and accommodated cost-time tradeoffs (Ameen, 1987). STARC was developed to allow a duration risk factor to be measured as a percentage over a time range for extended activities (Badiru, 1991). Other studies receiving attention in the literature (e.g., Herbert, 1979; Johnson and Schou, 1990) extended PERT simulation to stochastic forms of CPM (Critical Path Method). More recent studies that focus on particular simulation packages include an animation of a PERT network in Arena as a pedagogical tool (Cosgrove, 2006), a set of project management tools which employ Crystal Ball spreadsheets (e.g., Meredith and Mantel, 2002), and a Java based CPM simulation

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package referred to as SPSS (Stochastic Project Scheduling Simulation) developed by Lee (2005). The purpose of this study is to propose an Arena based approach to PERT simulation as a subset of Arena (Kelton et al., 2007) and SIMAN (Pedgen et al., 1995). Unlike the other PERT simulation approaches described above, this study utilizes the “drop and drag” and “cut and paste” features of Arena modules to develop a set of modules and module groups that are especially suitable for activity time network simulation. Model construction is simplified in that the Arena modules and module groups closely parallel the arcs and nodes on a network. This contrasts sharply with approaches based on GERT, VERT, PLANET, CAPERTSIM, and STARC which depend on numerous text statements, or blocks corresponding to text statements (e.g., GERT and VERT), which are inserted into an executable file which sometimes requires the specification of external files for input and output. Arena also permits model construction with spreadsheets which are automatically generated from Arena modules. However, unlike Crystal Ball, Arena spreadsheets are an option in model construction and tend to be employed for large networks by advanced users. Modules and module groups offer a better representation of model logic because they permit model construction within a model window that provides a diagram that corresponds directly with the network, as opposed to a tabular representation of the model required with the exclusive use of spreadsheets. Both Arena and SPSS employ a model window for displaying the network. A key advantage of SPSS over Arena is that SPSS is more suitable to run over the Internet. However, key advantages of Arena over SPSS follow from the defaults on Arena modules which ease the burden of model construction. Arena also includes features for computer animation and dynamic counters which use Arena variables to plot completion time distributions as the simulation progresses, even permitting the user

to control the speed of the simulation. For advanced Arena users, construction of a core network model can easily be extended to add more advanced and sophisticated features from Arena and/or SIMAN. While extended features of Java can be added to SPSS as well, such features would be more difficult to employ given that Java is a general purpose language, while Arena’s features are based on the SIMAN simulation language which is tailored specifically for simulation modeling. Rockwell International is the vendor for Arena, and its current policy on the use of the academic edition of Arena is quite flexible. Installation of the academic version is permitted at no charge on computers in university labs and on personal computers owned by faculty and students. Rockwell International markets both basic and professional editions of Arena with current prices ranging from approximately $800 to $15,000. The academic edition is similar to the professional version, and differs primarily on the size of the simulation problem. Methods employed in this study are suitable for all current editions of Arena. II. ARENA SETUP

The academic version of Arena is available on a single CD. Installation requires a Windows operating system (Windows 98 or above) with the user having Administrator privileges for later versions of Windows. The installation is simple and requires only a few minutes with the Wizard. Once installed, the user can open the program from the Windows START button or set up the Arena icon on the desktop. With default settings in place, opening Arena leads to the screen in Figure 1 which includes a menu bar and toolbars which are typical of Windows applications. The project bar provides the modules that are dragged and dropped as needed to construct the model in the model window flowchart view area, which in this study will be referred to as the “model window.”

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FIGURE 1: ARENA WINDOW Menu Bar

Toolbars

Project Bar

Model Window Flowchart View

FIGUR

Model Window Spreadsheet View

FIGURE 2: SELECTED ARENA TOOLBARS

Arena is typical of high end software in that it is overloaded with features, most of which are distracting and unnecessary for most users interested in PERT simulation. Fortunately, and perhaps by chance, a number of default settings in Arena can be exploited to work in favor of the

PERT user by limiting the number of required features to the bare necessities for network modeling. Figures 2 and 3 provide a listing of the selected toolbars and modules which are employed in this study. These items are discussed later in this study with the

FIGURE 3: SELECTED ARENA MODULES Create Module

Assign Module

Separate Module

Process Module

Batch Module Histogram Dispose Module

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construction of a PERT network simulation example. To setup Arena for modeling a PERT network, an Arena file (with the doe extension) is created which includes all of the modules in Figure 3 plus a modular group. The file is created by opening the Arena program where the screen in Figure 1 appears. It is suggested that the toggle split screen button (Figure 2) be pressed to eliminate the split screen, leaving more space in the model window by removing the spreadsheet view. Note that several modules displayed in the Project Bar (with the title “Basic Process” above the modules) match the modules in Figure 3. A series of drag and drops from the Project Bar into the model window are performed such that all seven modules in Figure 3 appear in the model window. If necessary, adjustments for centering the modules and sizing the model window can be made using the centering, zoom in, and zoom out buttons on the toolbars (Figure 2); and/or using the scroll bars. The next step is to create a set of modular groups consisting of Separate modules that are layered with one on top of the other in the same space. Referring to Figure 4, perform a copy with the toolbar on the Separate module and paste six Separate modules as shown on the left of the figure (ignore the default names given to the modules). Each module has three nodes, with one on the left and two on the right. Connect some of the modules with the connect toolbar (Figure 2) as shown in Figure 4.

With a careful manipulation with the mouse, it is suggested to allow a small gap between the body of the module and the nodes. Now layer the two sets of modules that are connected by highlighting them with the mouse and moving one on top of the other. The new configurations for the six modules are shown on the right of Figure 4. Note that an entity entering one of these modules/module groups will separate into two, three, or four separate duplicate entities. For networks with N branches emanating from a PERT node, simply follow the above process using N-1 Separate modules, or even combine the above modular groups in series to achieve the required number of branches. The final step is to save the file which defaults with the doe extension. An appropriate file name would be Modules.doe. This file consists of all the modules in Figure 3 plus two modular groups, and represents a prefabrication of all the nodes necessary to construct a PERT model in Arena. III. PERT NETWORK EXAMPLE

Consider the AOA PERT network in Figure 5 with the activity times specified by one constant (Activity 4) and seven triangular distributions, the latter with ranges from a (optimistic time estimates) to b (pessimistic time estimates) and corresponding modes m (most likely time estimates). While there are several approaches to model this network in the model window, it is suggested to simply open the Modules.doe file and group the modules and module groups on the left of the model window using the centering and zoom buttons (Figure 2). Additional space can be added by clicking the close button for the Project Bar (which can be recovered by clicking Project Bar in the View menu). Referring to Figure 6, note that the dark numbers and circles were added to the Arena network using buttons on the drawing toolbars in Figure 2. These added items illustrate how the network elements in Figure 5 correspond to the Arena modules and

FIGURE 4: CREATING A MODULAR GROUP Separate 1

Separate 2

Separate 1

Separate 3

Separate 23

Separate 4

Separate 5

Separ ate 6

Separate Separate Separate 546

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FIGURE 5: PERT EXAMPLE 2 1

Activity 1 2 3 4 5 6 7 8

3 4

5

6

8 7

modular groups in Figure 6. To construct the model in Figure 6, it is suggested that the modules be placed in the model window using the modules from the Modules.doe file with the copy and paste buttons (ignore the Arena feature that renames the modules), and then be connected with the connect toolbar (Figure 2) so as to parallel the structure of the PERT network in Figure 5. The next step is to delete all the original modules and modular groups that were copied in constructing the Arena network, and save the remaining modules/module groups in the model window as a new file named PERT Network.doe. In summary, the network was built in the same file containing the modules and modular groups, but saved in a different file after the Arena network was built. This preserves the Modules.doe file for future use in constructing other Arena networks.

a 0 1 0 0 0 0 1 1

m 2 6 4 1 5 2 6 4

b 4 9 7 0 8 5 9 6

Referring to the shapes of the modules, Activity 1 is bounded by nodes consisting of a single Separate module and a Separate module group that shows three emanating branches. Batch modules are employed following Activities 4 and 6 5 and 7, and 3 and 8. The rectangular shapes are Process modules corresponding to activities with the exception of the Process module following Activity 2. With a left double click of the mouse on any Process module, a dialog box appears with a pull down menu which permits the selection of at least four distributions and an option titled Expression, which adds an additional six distributions and an option to generate either continuous or discrete user created empirical distributions. For our purpose, the dialog box has a field for Delay Type where the triangular distribution is selected

FIGURE 6: ARENA MODEL OF THE PERT EXAMPLE

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(Figure 7 illustrates the dialog box for Activity 1, where the name Process 2 is ignored since it is an Arena generated name). Fields are then created for minimum (a), most likely (m), and maximum (b), where the parameters are entered from the data in Figure 5. (Note that a field for Units specifies hours. This is a default that should be ignored; otherwise this field would have to be changed to other time units for all Process modules in the model window.) These dialog boxes illustrate the ease in changing the Delay field to reflect changes in activity times. The settings for the dialog box from Activity 4 should be constant for the Delay Type field and 1 for the Value field. Since the Process module following Activity 2 is being treated as a node, the Delay Type field should be constant and the Value field should be 0. Dialog boxes for the Dispose and Separate modules (and Separate modular groups) are not opened because their default settings do not impact the Arena network. For the Batch module’s dialog box (Figure 7), the batch size field should be set to 2 since this number equals the number of branches connected to the input

side of the module. The dialog box for the Create module should be modified by setting constant in the Type field and at least 100 in the Value field. The latter is determined by the expected upper limit of the completion time distribution. It assures that each entity created by the Create Module, after splitting and recombining, exits the simulation before the next entity is created. The Max Arrivals field sets the total number of entities for the simulation run. A number of 10,000 or more is suggested to assure the accuracy of the histogram for the completion time distribution. The last task prior to running the simulation is to construct the completion time histogram. By left clicking the histogram toolbar (Figure 2), a crosshair is created that is controlled by the mouse. A left click and hold in the model window creates a box which sets the size of the histogram. The result is a histogram similar to the illustration on the upper right of Figure 6. After sizing the histogram box, double click on the histogram to obtain its dialog box (Figure 7). In the Expression field, enter TVALUE(Source Entity.TotalTime), which is an

FIGURE 7: DIALOG BOX EXAMPLES

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Arena generated variable attached to entities to track the total time duration that an entity is active in the simulation (i.e., the time it spends after its creation in the Creation Module to its demise in the Dispose Module). For the fields minimum, maximum, and # Cells, enter 5, 25, and 20. The first two fields represent estimates of the range of the completion time histogram, and the last field gives the number of time intervals on the histogram. A set of toolbars is used to run the simulation (Figure 2). The go toolbar starts the simulation in the default mode which shows some animation, including the dynamic development of the histogram for the completion time distribution. The fast forward button turns off the animation and accelerates the simulation run time on the PC. A pause button is available to pause the simulation, perhaps useful for long simulation runs that can be delayed so as not to slow other running PC applications. When the simulation is completed, the end button serves to reset the model for another simulation run. Since the Create and Dispose modules have counters on their perimeters, a check can be made on the number of entities entering and leaving the simulation for a single run. In this example, both counters are expected to register

10,000 at the end of the simulation as entered in the Create module dialog box. IV. ENHANCEMENTS

Enhancements can be added to the model by expanding the number of module groups in the Modules.doe file. The author’s Modules.doe file includes modular groups to develop completion time distributions with the layout in Figure 8. The time interval probabilities follow from the variable toolbar in Figure 2. It requires only one connection from the Arena network to the module group. However, some knowledge of using Arena variables is required, and the time interval numbers have to be typed in using the drawing toolbar (Figure 2). Critical path probabilities, criticality indices, and GERT network modeling tend to be a greater challenge, particularly because they have to be tailored to the structure of the Arena network. One approach to critical paths is to assign a single attribute on each entity which sums numbers assigned to each activity with the Assign module (Figure 3). If the sums are unique for each path, the critical path can be determined from the attribute when the entity leaves the simulation.

FIGURE 8: DISTRIBUTION SPECIFYING TIME PROBABILITIES

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For criticality indices, an array of attributes assigned to each entity can record a number for criticality indices, an array of attributes assigned to each entity can record a number for each element in the array, allowing for the tracking of the longest time entity for each entity created by the Create module. GERT networks can be developed by employing the Decide Module (Figure 3) which accommodates multiple path probabilistic branching. However, the batch size field in the Batch module is now a variable. This can be addressed by using numerous Assign modules and an embedded control network within the Arena network.

author is committed to providing enhanced features modules to interested parties as they develop. VI. REFERENCES

Ameen, D.A., “A Computer Assisted PERT Simulation,” Journal of Systems Management, April, 1987, 6-9. Badiru, A. B., “A Simulation Approach to Network Analysis,” Simulation, 57(4), 1991, 245-255. Cosgrove, W. J., “An Animated Simulation of a PERT Network with Arena,” California Journal of Operations Management, 4(1), 2006, 47-51. Herbert, J. E., “Critical Path Analysis for Stochastic Project Network Models,” Proceedings of the American Institute for Decision Sciences, 1980, 136-138. Johnson, G. A. and C. D. Schou, “Expediting Projects in PERT with Stochastic Time Estimates,” Project Management Journal, 21(2), 1990, 29-32. Kelton, W. D., Sadowski, R. P., and D. T. Sturrock, Simulation with Arena, 4th Edition, McGraw-Hill, Boston, 2007. Kennedy, K. W. and R. M. Trall, “PLANET: A Simulation Approach to PERT,” Computers and Operations Research, 3, 1976, 313-325. Lee, D. E., “Probability of Project Completion Using Stochastic Project Scheduling Simulation,” Journal of Construction Engineering and Management, 131(3), 2005, 310-318. MacCrimmon, K. R. and C. A. Ryavec, “An Analytical Study of PERT Assumptions,” Operations Research, 12(1), 1964, 16-37. Malcolm, D. G., Roseboom, J. H., Clark, C. E., and W. Fazar, “Applications of a Technique for R and D Program Evaluation (PERT),” Operations Research, 7(5), 1959, 646-669.

V. CONCLUSIONS

This study has shown that use of a selected subset of Arena features offers an alternative to basic PERT simulation with a “quick and dirty” approach to the construction of PERT models, and for the generation of meaningful completion time distributions free of the limitations of classical PERT. The author’s experience suggests that most applications in instructional settings from models discussed in management texts can be developed in Arena over several minutes, providing a more enhanced learning environment for students when such models can be developed from scratch as classroom demonstrations. While employing the enhanced features requires some working knowledge of Arena, such knowledge need not go beyond the Basic Process panel in the Project Bar, and can be limited in scope to a narrow subset of topics in Chapters 3 and 4 in Kelton et al. (2007). Given that Arena’s underlying language is SIMAN, opportunities to integrate PERT and GERT networks into more advanced simulation settings is enhanced for the serious researcher and practitioner. This paper concludes by expressing the author’s goal for an on-going development of shared module groups with easy access through Internet downloads. To that end, the

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Pritsker, A. A. B. and W. W. Happ, “GERT: Graphical Evaluation and Review Technique, Part I—Fundamentals,” Journal of Industrial Engineering, 17(5), 1966, 267-274. Van Slyke, R. M., “”Monte Carlo Methods and the PERT Problem,” Operations Research, 11(5), 1966, 839-860.

Meredith, J. R. and S. J. Mantel, Project Management: A Managerial Approach, Wiley and Sons, New York, 2002. Moeller, G. L. and L. A. Digman, “Operations Planning with VERT,” Operations Research, 29(4), 1981, 676-697. Pedgen, C. D., Shannon, R. E., and R. P. Swdowski, Introduction to Simulation Using SIMAN, 2nd Edition, McGraw-Hill, New York, 1995.

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