# A Generalized PERT/CPM Implementation in a Spreadsheet

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2 Using the approaches found in most textbooks (e.g. Hillier et. al, 2000, Ragsdale 2001), the approximate expected duration and approximate variance of activity duration for each of the activities are calculated and are shown in last two columns of Table 1. Two additional worksheets are used after solving the problem for displaying a Gantt chart for the project and the range names used in the spreadsheet. Methods The project network is shown in Figure 1. The method used for calculating the critical path revolves around the simple concept of representing a network with a matrix. Any graph or network consisting of n nodes can be represented by an n x n matrix where each row and each column of the matrix represents a node in the graph and an entry of 1 in a cell of the matrix represents a directed arc between the corresponding nodes. An example of the matrix representation is provided in Table 2 for the project network given in Figure 1. The network representation here uses the activityon-node convention. Figure 1: Project Network We will adopt the convention of representing the predecessors in the columns and successors in rows throughout this paper. The implementation of the PERT/CPM method in the spreadsheet is achieved by using a total of four worksheets in an Excel Workbook. The first one is used for the input data; the next two are used respectively to calculate the Earliest Start (EST) and the Earliest Finish Time (EFT), and the Latest Start (LST) and Latest Finish Time (LFT) of the activities; and the fourth sheet is used for reporting the critical path as well as for calculating the average project completion time and the project variance. Table 2: Matrix representation of the project network and the predecessor relationships The first step of the process is to organize the input data in the spreadsheet. This entails laying out all the given information in Table 1 as well as representation of the network with the predecessor relationships as shown in Table 2, in the spreadsheet. The spreadsheet layouts of the inputs for the example problem are shown in Figures 2 and 3 below. After entering the input data, the procedure starts calculating the EST and the EFT pairs for the activities in the network. This is accomplished by copying the network matrix into another sheet (named EST-EFT) and then adding two more columns to keep track of the final EST and EFT values of the activities. The layout of this sheet is shown in Figure 4. The earliest start time of an activity is the maximum of all the EFTs of its immediate predecessor activities. The spreadsheet is constructed following that principle. For any activity pair i and j, i in the column and j in the row, the cell(i,j) in the matrix contains the EFT of activity i if i is an immediate predecessor of j, zero otherwise. We will look at activity D in the EST-EFT sheet as an example. The cell E7, representing the pair A and D, has a value of 3.00, the EFT of A, because A is an immediate predecessor of D. The cell F7, on the other hand, contains zero because B is not an immediate predecessor of D. Conditional formatting is used to make the zero entries appear as hatched cells to improve 17

3 Figure 2: Layout of the Input data in the spreadsheet Figure 3: The Predecessor Relationship Representation in Spreadsheet the appearance of the EST-EFT results. The formula used for calculating the entries in the cells of the matrix is shown below using the cell E7, representing the activity pair A and D, as an example. Cell E7: =IF(Inputs!I6=1,VLOOKUP(E\$7,EST_EFTtable,3,False),0) Here EST_EFTtable refers to range A4:C203 in the EST-EFT sheet. The range up to C203 is used to accommodate up to 200 activities in the spreadsheet without renaming the range. The EST for an activity is the maximum of all the EFTs of its immediate predecessors. Therefore, the EST for activity A in cell B4 is given by the formula Cell B4: =MAX(E4:GV4), which is copied to the cells B4 through B11 for finding the EST of the remaining activities ( ite.informs.org/vol2no1/seal/pertcpm.xls). Once the ESTs are obtained, the EFT is found by adding the expected activity time to the EST. For example, the formula used for finding the EFT for activity A is Cell C43: =B4+VLOOKUP(A4,Inputs,7,False) 18

4 Figure 4: The EST-EFT Calculations Figure 5: The LST-LFT Calculations which is then copied to the cells C4 through C11. The VLOOKUP function is used to extract the expected completion time of the activity from the input data in the range A3:H203 (named as Inputs). The next step in the process is to find the latest start time (LST) and latest finish time (LFT) for each activity. Noting that the LFT of an activity with successors must be the minimum of the LSTs of all the successor activities, we can use the same logic that we employed to calculate the EST-EFT pairs for an activity, only in reverse. This is accomplished in the LST-LFT sheet shown in Figure 5. Once again a matrix with the activities in the rows as well as in the columns is created. The activities in the rows represent the successors. This is consistent with the predecessor representation followed in the paper. Two extra rows are inserted in this sheet to store the values of the LST-LFT pairs for each activity. In 19

6 Usage and Extendibility The method presented in this paper of using a spreadsheet to solve PERT/CPM problems by the critical path algorithm is a useful addition to professors teaching MS/OR courses. Spreadsheet programs are widely available now and managers are familiar and comfortable with the software. As a result, MS/OR courses in many business schools are being taught using spreadsheets as the software tool. This paper makes a definite contribution to that effort. Depending on the goals, level, and the pace of the course, the model can either be used as a black box or as a detailed illustration of the mechanics of the PERT/CPM method. Unlike the existing spreadsheet based PERT/CPM approach, which require the rebuilding of the model for each new network through a new linear programming formulation, our model is easily extendable Figure 6: The Final CPM Table Table 3: The formulas used in the "CPM Table" sheet 21

9 Appendix - Example of Extendibility Let us assume that the example problem presented in this paper needs to be revised with the addition of one more activity "G" and the resulting project network after the addition of this new activity is as shown in Figure A1. It is further assumed that the optimistic, most probable, and the pessimistic time estimates for the completion of "G" are given and they are 5, 10, and 15 weeks, respectively. Figure A1: Project Network with Additional Activity G The following steps need to be carried out to extend the example problem with this new activity. In the Inputs sheet, the content of row 8, the row with the current last activity (activity F) is copied to the next row down (to row 9) and the column with the current last activity F (column N) is copied to its right (to column O). This extends the input matrix for accommodating one more activity. The activity name in the new row (row 9) is changed from F to G and the time estimates and predecessor matrix are updated with proper information for this new activity. The new "Inputs" sheet is shown in Figure A2 and Figure A3. In the "EST-EFT" sheet, row 9, the current last row with an activity in it, is copied to row 10 and then column J, the current last column with activity F, is copied to column H. The order of copying is important to avoid circular reference. Since the activity names in this sheet are connected to the activity names on the input sheet (the formula in cell A4 of this sheet is "Inputs!A3" which is copied down the rows for the other activities and the formula for cell E2 is "Inputs!I2" which refers to the activity names at the top of the predecessor matrix in the "Inputs" sheet. It is then copied across the other columns), no other editing is necessary. The resulting "EST-EFT" sheet with the updated EST-EFT pairs for all the activities is shown in Figure A4. In the "LST-LFT" sheet, the current last column with an activity (Column H) is copied to the next column (column I) and then row 11 is copied down to the next row. Once again the order of copying must be Figure A2: Input Sheet with New Row for activity G 24

10 Figure A3: Input Sheet with New Column for activity G Figure A4: Updated EST-EFT after addition of Activity G maintained to avoid circular references. Again no extra editing is needed to update the names of the activities since the cells used for the designating the names of the activities in the "LST-LFT" sheet are connected to the corresponding cells in the "Inputs" sheet. The resulting "LST-LFT" sheet with the updated LST-LFT pairs for all the activities is shown in Figure A5. Finally the CPM Table is updated by copying of the current last row (row 10) in the "PERT-CPM Table" sheet to the next row down (row 11). The updated result is shown in Figure A6. For the expansion of the network with the addition of more activities, the range-names do not have to be redefined because they are initially set up to accommodate a network of 200 activities. The 25

11 Figure A5: Updated LST-LFT sheet after addition of activity G Figure A6: Updated CPM Table after addition of activity G maximum number of activities in a project network that can be solved in a spreadsheet using this approach and the layout presented in this paper is the maximum number of columns possible in the predecessor matrix. Since the predecessor matrix can start only at column I, (the previous columns are needed for other information on the input data) the upper limit on the number of activities possible in modern spreadsheets is 248 (column IV - column I). 26

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