DECISION MODELING WITH MICROSOFT EXCEL Chapter 14 P E R T and C P M Part 1 Copyright 2001 Prentice Hall

I NTRODUCTION The task of managing major projects is an ancient and honorable art. In about 2600 B.C., the Egyptians build the Great Pyramid for King Khufu. The Greek historian Herodotus claimed that 400,000 men worked for 20 years to build this structure. Modern projects ranging from building a suburban shopping center to putting a man on the moon are amazingly large, complex, and costly. Completing such projects on time and within budget is not an easy task.

Indeed, the complicated problems of scheduling such projects are often structured by the interdependence of activities. Typically, certain activities may not be initiated before others have been completed. Some key questions to be answered in this chapter are: 1. What is the expected project completion date? 2. What is the potential “variability” in this date? 3. What are the scheduled start and completion dates for each specific activity?

4. What activities are critical in the sense that they must be completed exactly as scheduled in order to meet the target for overall project completion? 5. How long can noncritical activities be delayed before a delay in the overall completion date is incurred? 6. How might resources be concentrated most effectively on activities in order to speed up project completion? 7. What controls can be exercised on the flows of expenditures for the various activities throughout the duration of the project in order that the overall budget can be adhered to?

To answer these questions, we will use the methods PERT (Program Evaluation Review Technique) and CPM (Critical Path Method). Both of these approaches to scheduling represents a project as a network. When a project involves uncertain elements, the representation of the project requires a stochastic network. PERT was developed in the late 1950s by the Navy Special Projects Office in cooperation with the management consulting firm of Booz, Allen, and Hamilton. The technique was used in the engineering and development program of the Polaris missile.

Many firms and government agencies today require all contractors to use PERT. CPM was developed in 1957 by J. E. Kelly of Remington Rand and M. R. Walker of Du Pont. CPM differs from PERT in the details of how time and cost are treated. The implementation of PERT and CPM had an immediate impact on scheduling projects because it allowed the practice of “management by exception.”

TYPICAL PROJECT: A THE GLOBAL OIL CREDIT CARD OPERATION Moving the Global Oil credit card operation to Des Moines, Iowa, from the home office in Dallas is an important project. Global’s board of directors has set a firm deadline of 22 weeks for the move and has put the Operations Analysis Group in charge of the move. The move is difficult to coordinate because it involves many different divisions within the company. Real estate must select one of three available office sites.

Personnel has to determine which employees from Dallas will move, how many new employees to hire, and who will train them. The systems group and the treasurer’s office must organize and implement the operating procedures and the financial arrangements for the new operation. The architects will have to design the interior space and oversee needed structural improvements. Office partitions, computer facilities, furnishings, and so on, must all be provided for the existing building.

A second complicating factor is the inter-dependence of activities (i A second complicating factor is the inter-dependence of activities (i.e., some parts of the project cannot be started until other parts are completed). Consider two obvious examples: Global cannot construct the interior of an office before it has been designed. Global cannot hire new employees until it has determined its personnel requirements. THE ACTIVITY LIST PERT and CPM are specifically designed for projects of this sort. The first step is to define the activities in the project and to establish precedence relationships.

The first activity list prepared for the move is shown below. This is the most important part of any PERT or CPM project and must be a group effort so that no important activities are overlooked. Note that the columns labeled Time and Resources are indications of things to come.

Each activity (represented by a letter) is placed on a separate line. An activity’s immediate predecessors are recorded on the same line. An immediate predecessor is an activity that must be completed prior to the start of the activity in question (e.g., Global cannot start activity C until activity B is completed).

THE GANTT CHART Before discussing PERT and CPM, let’s look at another graphical approach, the Gantt chart (developed by Henry L. Gantt in 1918). Activities Time (weeks)

Each bar represents the anticipated as well as actual duration. The horizontal axis is time while each activity is listed on the vertical axis. The beginning of the bar represents the earliest possible starting time for the activity. Each bar represents the anticipated as well as actual duration. Activities Time (weeks) As each activity is completed, the appropriate bar is shaded.

As of week 13, activities D, E, and H are behind schedule. At any point in time, it is clear which activities are on schedule and which are not. As of week 13, activities D, E, and H are behind schedule. Activity G has actually been completed and hence is ahead of schedule. Activities Time (weeks)

The previous example shows how the Gantt chart is mainly used as a record-keeping device for following the progression in time of the subtasks of a project. With the Gantt chart, we can see which tasks are on, behind or ahead of schedule. It is important to note that in the context of the Gantt chart, “on schedule” means “it has been completed no later than the earliest possible completion time.” However, this is too simple a concept for whether an activity is on schedule. The appropriate point of view should be whether the overall project is being delayed in terms of a target completion date.

The Gantt chart also fails to reveal which activities are immediate predecessors of other activities. This information is of vital importance in determining project completion time. We will now see that the network representation contains the immediate predecessor information that we need. THE NETWORK DIAGRAM In a PERT network diagram, each activity is represented by an arrow that is called a branch or arc. the beginning and end of each activity (an event) is indicated by a circle that is called a node.

When an activity is completed, the event occurs. Constructing the Network Diagram The following network diagram shows activities A through C. Note that the numbers assigned to the nodes are arbitrary. They are simply used to identify events and do no imply anything about precedence relationships. 1 2 3 4 A B C In the network diagram, each activity must start at the node in which its immediate predecessors ended. For example, activity C starts at node because its immediate predecessor, activity B, ended there. 3

However, adding activity F creates new problems. Now we need to add activity D to the network. However, note that activities A and C are both immediate predecessors to activity D. Therefore, nodes and must be combined so that activity D can start from it. 2 4 1 3 2 4 A B C 5 D E Note that activity E, which has only D as an immediate predecessor, can be added with no difficulty. However, adding activity F creates new problems.

Since F has C as an immediate predecessor, it would emanate from node Since F has C as an immediate predecessor, it would emanate from node . However, this would imply that A is also an immediate predecessor to F, which is incorrect. 3 The Use of Dummy Activities This dilemma is solved by introducing a dummy activity, which is represented by a dashed line in the network diagram. The dummy activity is fictitious in the sense that it requires no time or resources. It merely provides a pedagogical device that enables us to draw a network representation that correctly maintains the appropriate precedence relationships.

Here is the resulting network diagram with the dummy activity. 1 2 3 5 A B C 7 D E 4 F 6

The procedure is generalized as follows: Suppose that we wish to add an activity A to the network starting at node N, but not all of the activities that enter node N are immediate predecessors of activity A. Create a new node M with a dummy activity running from node M to node N. Take those activities that are currently entering node N and that are immediate predecessors of activity A and reroute them to enter node M. Now, make activity A start at node M.

This network diagram shows the complete activity list and all of the precedence relationships. Select site Plan Personnel requirements personnel Financial arrangements Design Construct Move Hire Train 1 2 3 5 A B C 7 D E 4 F 6 8 H G J I

Some computer programs may have difficulty with activities H and G since they both start at node and end at node . 6 7 These activities may be read by the computer program as one instead of two. 1 2 3 5 A B C 7 D E 4 F 6 8 H G J I

A dummy activity can be used to cure this condition. Select Site [3] Plan [5] Personnel Requirements [3] Personnel [2] Financial Arrangements [5] Design [4] Construct [8] Move [2] Hire [4] Train [3] 1 2 3 5 A B C 7 D E 4 F 6 9 H G J I 8 Expected activity completion times.

An Activity-on-Nodes Example The previous network was an example of an Activity-on-Arc (or AOA) approach. In the Activity-on-Nodes (AON) approach, the activities are associated with the nodes of the network while the arcs of the network display the precedence relationships. The Global Oil network would be represented as follows: A D B E J C F I H G

T HE CRITICAL PATH - MEETING THE BOARD’S DEADLINE The activity list and an appropriate network diagram are useful devices for representing the precedence relationships among the activities in a project. Recall that the board has set a firm goal of 22 weeks for the overall project to be completed. The time estimates must first be incorporated before we can tell if this goal can be reached. The PERT-CPM procedure requires management to produce an estimate of the expected time it will take to complete each activity on the activity list.

THE CRITICAL PATH CALCULATION These time estimates are given below: THE CRITICAL PATH CALCULATION If you added up all of the expected activity times in the above table, the total working time required to complete all the individual activities would be 3 + 5 + 3 + 4 + 8 + 2 + 4 + 2 + 5 + 3 = 39 weeks

However, this does not take into account those activities which can be performed simultaneously. For example, activities A (3 weeks) and B (5 weeks) can be initiated at the same time. Therefore, the total time to complete both activities would be 5 weeks. To obtain a prediction of the minimum calendar time required for overall project duration, we must find the critical path in the network. A path is a sequence of connected activities that leads from the starting node to the completion node (e.g., path B-C-D-E-J). 1 9 To complete the project, the activities on all paths must be completed (all paths must be traversed).

The task is to analyze the total amount of calendar time required for all paths to be traversed and to find the longest path from start to finish. The longest path through the network is called the critical path and will determine the overall project duration (because no other path will be longer). The activities on the critical path are called critical activities of the project since if they are delayed, then the entire project will be delayed. It is this subset of activities that must be kept on schedule.

“What is the earliest time at which activity D can start?” Earliest Start and Earliest Finish Times Now let’s look at the steps employed in finding a critical path. Fundamental in this process is the earliest start time for each activity. To illustrate this idea, consider activity D, “design facility.” Now assume that the project starts at time zero and ask yourself: “What is the earliest time at which activity D can start?” Clearly, it cannot start until activity A is complete (3 weeks). However, it also cannot start before the dummy activity is complete (0 weeks). Since the dummy cannot start until B and C are complete (a total of 8 weeks), we see that D cannot start until 8 weeks have passed.

A is complete after 3 weeks. C is complete after 5 + 3 = 8 weeks In this calculation, it is crucial to note that activities A and B both start at time 0. A is complete after 3 weeks. C is complete after 5 + 3 = 8 weeks B requires another 2 weeks B is complete after 5 weeks and C can start Select Site [3] Plan [5] Personnel Requirements [3] Personnel [2] Financial Arrangements [5] Design [4] Construct [8] Move [2] Hire [4] Train [3] 1 2 3 5 A B C 7 D E 4 F 6 9 H G J I 8

earliest start for activity D = 8 weeks Thus, after 8 weeks, both A and C are complete and D can start. In other words, earliest start for activity D = 8 weeks Another important concept is earliest finish time for each activity. If we let ES = earliest start time for a given activity EF = earliest finish time for a given activity t = expected activity time for a given activity Then, for a given activity, the relation between earliest start time and earliest finish time is EF = ES + t

Earliest Start Time Rule Consider the following rule: Earliest Start Time Rule The ES time for an activity leaving a particular node is the largest of the EF times for all activities entering the node.

Looking at nodes , , , and of the previous network, the result is shown in brackets below [ES, EF]. 1 2 3 4 The ES rule says that the ES for activity D is equal to the larger EF for all activities entering node 2. 1 2 3 B [0,5] C [5,8] D [8,12] 4 F [8,10] I [5,10] A [0,3] Continuing to each node in a forward pass through the entire network, the values [ES, EF] are then computed for each activity.

The network with the ES and EF values is given below: 1 2 3 5 A [0,3] B [0,5] C [5,8] 7 D [8,12] E [12,20] 4 F [8,10] 6 9 H [12,12] J [20,23] I [5,10] 8 G [10,14] Note that the earliest finish time for J is 23 weeks. Therefore, the earliest completion time for the entire project is 23 weeks. We have now answered the 1st question.

Latest Start and Latest Finish Times We now proceed with a backward pass calculation in order to identify possible start and completion dates, the activities on the critical path, and how long noncritical activities may be delayed without affecting the overall completion date (answering the 3rd, 4th, and 5th questions). Now that we have the target completion date of 23 weeks, we can work backward from this date, determining the latest date each activity can finish without delaying the entire project. The backward pass begins at the completion node, node . Then, trace back through the network computing the latest start time and latest finish time for each activity. 9

LS = latest start time for a given activity LF = latest finish time for a given activity The relation between these quantities is LS = LF - t The general rule is Latest Finish Time Rule The LF time for an activity entering a particular node is the smallest of the LS times for all activities leaving that node.

The complete network with the LS and LF values (given below the [ES,EF] values) is shown below: 1 2 3 5 A [0,3] B [0,5] C [5,8] 7 D [8,12] E [12,20] 4 F [8,10] 6 9 H [12,12] J [20,23] I [5,10] 8 G [10,14] [5,8] [0,5] [8,12] [12,20] [14,16] [18,23] [20,23] [18,20] [16,20]

Slack for G = LS for G – ES for G Slack and the Critical Path The next step of the algorithm is to identify the amount of slack, or free time, associated with each activity. Slack is the amount of time an activity can be delayed without affecting the completion date for the overall project. Slack is the same concept covered in linear programming and is the extra time that could be spent on that path without affecting the length of the critical path. For each activity, the slack is computed as: Slack = LS – ES = LF – EF For example, the slack for activity G is: Slack for G = LS for G – ES for G = 16 – 10 = 6 weeks

The critical path activities are those with 0 slack. Every activity on the critical path should have a slack of 0. This means that the activity cannot be delayed without affecting the entire project. Therefore, any activity with a slack of 0 is a critical activity and is on the critical path. The critical path activities are those with 0 slack.

Spreadsheet Approach for the Network The spreadsheet solution of this problem is most easily done with an activity-on-the-node (AON) approach. =D2+C2 =G2-C2 =MIN(F5) =F2-D2 =IF(H2=0,”YES”,”NO”) =MIN(F4,F10) =MAX(E3) =MIN(F5,F7) =MAX(E2,E4) =MIN(F6) =MAX(E5) =MIN(F11) =MAX(E4) =MIN(F8,F9) =MAX(E7) =MIN(F11) =MAX(E7) =MIN(F11) =MAX(E3) =E13 =MAX(E6,E8,E9) =E13 =MAX(E2:E11)

We have now answered the following questions: 1. What is the expected project completion date? Answer: 23 weeks 3. What are the scheduled start and completion dates for each specific activity? Answer: An activity may be scheduled to start at any date between “earliest start” and “latest start.” The scheduled completion date will be “start date + expected activity time.”

4. What activities are critical in the sense that they must be completed exactly as scheduled in order to meet the target for overall project completion? Answer: The activities on the critical path are: B, C, D, E, J. 5. How long can noncritical activities be delayed before a delay in the overall completion date is incurred? Answer: Any activity may be started as late as the “latest start” date without delaying the overall project completion. The remaining questions will be answered later.

WAYS OF REDUCING PROJECT DURATION It is clear from the critical path analysis that we have a problem. The board of directors wants to start operating in Des Moines in 22 weeks, and with the current plan 23 weeks are required. WAYS OF REDUCING PROJECT DURATION There are two basic approaches to reducing the time required to complete a project: 1. A strategic analysis: Here the analyst asks: “Does this project have to be done the way it is currently diagrammed?” In particular, “Do all of the activities on the critical path have to be done in the specified order?” Can we make arrangements to accomplish some of these activities in a different way not on the critical path?

2. A tactical approach: In this approach, the analyst assumes that the current diagram is appropriate and works at reducing the time of certain activities on the critical path by devoting more resources to them. The current expected times assume a certain allocation of resources. For example, the 8 weeks for construction (activity E) assumes a regular 8-hour workday. The contractor can complete the job more rapidly by working overtime, but at increased costs. The tactical approach takes into consideration CPM models, which will be discussed later. For now, let’s focus on the so-called strategic questions.

A Strategic Analysis This analysis is analogous to “What if A Strategic Analysis This analysis is analogous to “What if?” analysis done with spreadsheets. After reviewing the network, it is discovered that the current network assumes that activity J, the training of new employees, must be carried out in the new building (after E is complete), and after records and key personnel have been moved (after H is complete). Perhaps these requirements can be changed. First, J can be accomplished independently of H. Moreover, an alternative training facility can be secured by arranging to use surplus classroom space in Des Moines at a minimal cost. The new employees can be trained and ready to start the moment that construction ends.

On the other hand, a new activity will have to be added to the activity list: secure a training facility (activity K). All of these changes to the network may have created a new critical path with a still unsatisfactory minimum time (i.e., one greater than 22 weeks).

Spreadsheet Output for the Redefined Network This redefined activity list is shown in the form of the activity-on-the-arc (AOA) diagram. 1 2 3 5 A B C 7 D E 4 F 9 H G J I 8 6 K Select Site [3] Plan [5] Personnel Requirements [3] Personnel [2] Financial Arrangements [5] Design [4] Construct [8] Move [2] Hire [4] Train [3] Secure Training [3]

Here is the same network using the Activity-on-the-Node (AON) approach. B E J C F I H G K

The modified spreadsheet is shown below: =D2+C2 =G2-C2 =MIN(F5) =F2-D2 =IF(H2=0,”YES”,”NO”) =MIN(F4,F10) =MAX(E3) =MIN(F5,F7) =MAX(E2,E4) =MIN(F6) =MAX(E5) =E14 =MAX(E4) =MIN(F8,F9,F12) =MAX(E4) =MIN(F11) =MAX(E7) =E14 =MAX(E3) =E14 =MAX(E8,E12) =E14 =MAX(E7) =MIN(F11) =MAX(E2:E12) Note that the redefined project completion time is 20 weeks and the new critical path is B-C-D-E.

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