CHAPTER Getting from A to B – project planning using networks 14 Chapter objectives This chapter will help you to: ■ construct network diagrams ■ apply critical path analysis (CPA) ■ identify slack time available for project activities ■ use the Program Evaluation and Review Technique (PERT) ■ conduct cost analysis of projects using network diagrams ■ become acquainted with business uses of CPA/PERT Many business operations involve planning and coordinating a project – a series of interlinked tasks or activities all of which have to be performed in the correct sequence and within the least amount of time in order to achieve a successful conclusion to the venture. This is not only typical of large-scale projects such as you would find in industries like construction and shipbuilding, but also occurs on a more modest basis in administra- tive processes such as organizing events like conferences and concerts. To help plan and execute projects successfully managers can turn to network analysis, a system of representing and linking the activities involved in a project. Once they have designed a network they can use 436 Quantitative methods for business Chapter 14 critical path analysis (CPA) to establish the minimum duration of the project by identifying those tasks whose completion on time is essential, the critical activities. Beyond this they can bring into consideration the probability distributions that reflect the chances of the activities being completed by specific times using the program evaluation and review technique (PERT). In this chapter we will look at these techniques. 14.1 Network analysis We can apply network analysis to any project that consists of series of distinct activities provided that we know which activities must be com- pleted before other activities can commence. This is called the prece- dence of the activities because it involves identifying the activities that must precede each activity. These are crucial because the point of a network diagram is to show how the activities in a project are linked. The diagrams used for network analysis are built up using a single arrow for each activity. The arrow begins at a circle representing the point in time at which the activity begins and finishes at another circle that represents the completion of the activity. These circles are known as event nodes. We would represent a single activity as shown in Figure 14.1. In Figure 14.1 the direction of the arrow tells us that the event node on the right marks the completion of the activity. The activity is labelled using a letter and the events nodes are numbered. Network diagrams consist of many arrows and many event nodes, so logical layout and labelling is important. Networks should be set out so that the beginning of the project is rep- resented by the event node on the extreme left of the diagram and the completion by the event node on the extreme right. In compiling them you should try to ensure that the event nodes are labelled sequentially from left to right from event number 1, the start of the project. All the arrows should point from left to right either directly or at angle and certainly not from right to left. This is usually straightfor- ward because the purpose of an arrow in a network is to represent the position of the activity in relation to other activities and not its duration. A network diagram is not intended to be to scale. Figure 14.1 A single activity 1 2 A Chapter 14 Getting from A to B – project planning using networks 437 Example 14.1 Avia Petitza is an independent airline flying short-haul routes in South America. The new general manager is keen to improve efficiency and believes that one aspect of their operations that can be improved is the time it takes to service planes between flights. She has identified the activities involved and their preceding activities and compiled the following table: Produce a network diagram to portray this project. Activity A has no preceding activities so we can start the network with this as shown in Figure 14.2(a). Four subsequent activities, B, C, D and I depend on activity A being completed so we can extend the network as shown in Figure 14.2(b). Activity G must follow activity C and activity J follows activity I as depicted in Figure 14.2(c). Activity E follows activity B and activity F follows activity E as shown in Figure 14.2(d). Before activity K can take place activity H must also have finished. This presents a problem because we cannot have two or more activities starting at the same event node and finishing at the same event node. If we did we would not be able to use the diagram for the sort of scheduling analysis you will meet in the next section of this chapter. To avoid activities F and H having the same event node marking their start and the same event node marking their completion we can introduce a dummy activity, an activ- ity that takes no time nor uses any resources. Its role is merely to help us distinguish two activities that might otherwise be confused in later analysis. To distinguish between a dummy activity and real activities it is portrayed as a dotted line. You can see a dummy activity in Figure 14.2(e) used to ensure that the event node that marks the conclusion of activity H is not the same as that marking the end of activity F. Activity Description Precedence A Drive service vehicles to plane None B Attach stairway A C Unload baggage A D Refuel A E Passengers disembark B F Clean cabin E G Load baggage C H Load food and beverages E I Stock up water tanks A J Service toilets I K Passengers embark F, H L Detach stairway K M Drive service vehicles from plane D, G, J, L 438 Quantitative methods for business Chapter 14 (d) E F B D CG I J A (c) D B CG I J A (b) B D I C A (a) A The dummy activity in Figure 14.2(e) helps us to identify two sep- arate activities. A dummy activity used in this way is an identity dummy. Dummy activities are also used to resolve logical difficulties that arise in some networks. A dummy activity used to do this is a logical dummy. Chapter 14 Getting from A to B – project planning using networks 439 (f) 7 8 6 5 4 3 2 1 9 10 11 HE F K AD B CG IJ M L (e) H E F B D CG I J A The final diagram, Figure 14.2(f), incorporates activities K, L and M. Activities K and L share the same finishing event as activities D and J. This event represents the begin- ning of activity M, whose closing event marks the conclusion of the project. We can also number the events now the diagram is complete. Figure 14.2 Stages in the compilation of the network diagram for aircraft servicing in Example 14.1 Both types of dummy activity serve only to clarify the network and avoid ambiguity. The important distinction between real and dummy activities is that the latter do not take time or use resources. Compiling network diagrams often brings an understanding of the way in which activities that make up a project fit together. It enables you to clarify uncertainties about the planning of the project. This is some- thing usually achieved by constructing drafts of the network before producing the final version. Whilst an understanding of the sequence is important, there is much more to successful project planning. Ascertaining the minimum time in which the project can be completed and scheduling the activities over 440 Quantitative methods for business Chapter 14 Example 14.2 Two building workers, Bru and Chai, like a mug of black tea in their morning break at work. Bru takes sugar, Chai does not. Making their tea involves four activities: It is tempting to represent these activities in the form of the network in Figure 14.3(a), but this implies that activity C, putting hot water in Chai’s mug, depends on activity B, putting sugar in Bru’s mug, which is not the case. To get around this we can include a dummy activity as shown in Figure 14.3 (b). (a) (b) AC D B AC DB Figure 14.3 Incorrect (a) and correct (b) networks for Example 14.2 Activity Description Precedence A Put tea bags in the mugs None B Put sugar in Bru’s mug None C Pour hot water in Chai’s mug A D Pour hot water in Bru’s mug A, B time are typically the central issues. The technique that enables us to do this is critical path analysis. 14.2 Critical path analysis If we know the duration of each activity we can use a network diagram to find out the least amount of time it will take to finish the project. A network diagram shows the way that activities are linked; in effect it portrays the project as a series of paths of activities. Since every activity must be finished for the project to be completed, the minimum dur- ation of the project is the length of time required to carry out the most time-consuming path of activities. This path is known as the critical path since any delay in the completion of activities along it will prolong the entire project. Finishing those activities on the critical path on time is therefore critical for the project; they are known as critical activities. Critical path analysis involves increasing the amount of information in the network by enhancing the role of the event nodes. In drawing a network diagram the event nodes are in effect the punctuation marks in the diagram; they bring order to the sequence of arrows. In critical path analysis they become distinct points in time. For each event node we assign an earliest event time (EET) and a latest event time (LET). These are written in the circle that represents the event node beneath the event number, as illustrated in Figure 14.4. The circle on the left in Figure 14.4 is the event node that marks the point in time when activity X begins. The number in the upper part of the circle is the event number, 20. The number below it to the left is the earliest event time and the number below it on the right is the latest event time. These numbers tell us that the earliest time that activity X can start is time period 9 and the latest time it can start is time period 11. From the equivalent figures in the event node to the right of activity X you can tell that the earliest time the activity can be completed is time period 13 and the latest time it can be completed is time period 15. To work out the earliest and latest times for the events in a network we use the activity durations. Starting with the event node at the beginning Chapter 14 Getting from A to B – project planning using networks 441 Figure 14.4 Earliest and latest event times for a single activity X 21 13 15 20 11 9 of the network and working through the network from left to right we write in the earliest time that each event node in the network can occur. This is referred to as making a forward pass through the network. When doing this you need to remember that where there are two or 442 Quantitative methods for business Chapter 14 Example 14.3 The durations of the activities undertaken during the ground servicing operation in Example 14.1 are given in the following table: Using these figures we can enter the earliest event times in the network. These are included in Figure 14.5. 7 8 6 7 48 5 5 13 33 33 27 57 59 4 3 2 2 1 0 9 10 11 H E F K A D B C G IJ L M Figure 14.5 Network for ground servicing with earliest event times Activity Description Duration (minutes) A Drive service vehicles to plane 2 B Attach stairway 3 C Unload baggage 25 D Refuel 15 E Passengers disembark 8 F Clean cabin 20 G Load baggage 30 H Load food and beverages 10 I Stock up water tanks 5 J Service toilets 5 K Passengers embark 15 L Detach stairway 3 M Drive service vehicles from plane 2 more activities leading to the same event node it is the duration of the longest of the activities that determines the earliest time for the event since all the activities must be completed before the event is reached. We write the earliest event times in the lower left-hand side of the event nodes. If you look carefully at Figure 14.5 you will see that the earliest event time for event 1 is 0 reflecting the fact that 0 minutes of project time have elapsed at the beginning of the project. The earliest event time for event 2 is 2, reflecting the 2 minutes needed for the completion of the only activity between event node 1 and event node 2, activity B, driving the service vehicles to the plane. The earliest event time for event 8 is perhaps less obvious. The figure entered, 33, has been worked out based on the longest route to it. The activities leading to event node 8 are activity F, with its associated dummy activity between events 7 and 8, and activity H. The earliest time that activity F can start is the earliest event time on the event marking its beginning; the 13 in event 4. If we add the 20 minutes that activity F, cleaning the cabin, takes to the earliest event time for event 4 we get 33 minutes as the earliest time activity F can be completed. Event 4 also marks the beginning of activity H, loading the food and beverages. If we add the time this activity takes, 10 minutes, to the earliest event time of activity 4 we get 23 minutes. This would be the earliest event time for event 8 if we did not need to complete activity F, but since both activity F and activity H have to be completed before event 8 can occur, the earli- est time we can get there must allow for the longer activity, activity F, to be concluded, hence the earliest event time for event 8 is 33 and not 23. The event node indicating the completion of the project on the extreme right of the network has an earliest event time of 59 minutes. This is the minimum duration of the entire project; given the activity durations it cannot be completed in a lesser amount of time. We now need to turn our attention to the latest event times, as comparing these with the earliest event times will enable us to identify the critical path for the project. Once we have established the minimum project duration, in the case of Example 14.3, 59 minutes, we assume that the project manager will want to complete it in that time. We now undertake the same sort of task to find the latest event times as we used to find the earliest event times, but this time we start on the right-hand side and work back through the network ascertaining the latest time each event can occur if the project is to be finished in the minimum time. This is referred to as making a backward pass through the network. The latest event times for Example 14.3 are included in Figure 14.6. Each event now has a number entered in the lower right-hand side of its node. Chapter 14 Getting from A to B – project planning using networks 443 Looking at Figure 14.6 you can see that the latest event time for the last event, event 11, is 59 minutes. If the project is to be completed in 59 minutes then the latest time we must reach this point is 59 minutes. The latest event time of event 10 is 57 minutes, 59 minutes less the 2 minutes we must allow for activity M to be completed. Several activities conclude at event 10, including activities G and L. Activity G begins at event 5. The latest time event time of event 5 is 27 min- utes, sufficient to allow the 30 minutes necessary for the completion of activity G in time for event 10 to be reached in 57 minutes, which in turn allows for the 2 minutes to complete activity M and thus conclude the project in 59 minutes. Activity L begins at event 9. Since activity L, detach- ing the stairway, takes 3 minutes, the latest event time for event 9 is 54 minutes, sufficient to allow the 3 minutes for the completion of activity L so that event 10 can be reached in 57 minutes and thus leave 2 minutes for activity M to finish in time for completing the project in 59 minutes. If you study Figure 14.6 carefully you will see that some events, such as event 5, have the same earliest and latest event times while other events, such as event 6, have different ones. In the case of event 6 the earliest event time is 7 minutes whereas the latest event time is 52 min- utes. This implies that activity I can be completed by 7 minutes but it doesn’t have to be finished until 52 minutes have elapsed. In other words, there is time to spare for the completion of activities I and J; they have what is called slack or float. 444 Quantitative methods for business Chapter 14 Figure 14.6 Network for ground servicing with latest event times 7 8 6 7 48 5 511 1319 33 33 39 39 27 27 57 52 57 54 4 3 2 22 1 00 9 10 H E F K A D B CG IJ M L 59 59 11 [...]... the mean and the standard deviation of the pit stop duration (b) The performance target for the team is 22 seconds What is the probability that they will make it? (c) Within how many seconds will 90% of pit stops be completed? 462 Quantitative methods for business Chapter 14 14 .14 The brand manager at Slattkey Sweets in question 14. 4 has more details of the tasks involved in the re-launch of the Zubirot... 1 14. 17* Bibb and Tukka from question 14. 1 have identified the potential for crashing some of the tasks involved in opening their new outlet The details are: Activity B C E F Normal duration (days) Crash duration (days) Crash cost (£) 8 3 10 7 4 1 8 5 800 300 600 400 (a) Use your network diagram for the project to work out the total float for each activity 464 Quantitative methods for business Chapter. .. Figure 14. 6 This is float associated with the path rather than an individual activity; once used up by, say, taking 9 minutes rather than 3 minutes for activity B, attaching the stairway, it would not be available for the subsequent activities E, F, K and L, which as a consequence would become as critical as the activities on the critical path 446 Quantitative methods for business Chapter 14 Example 14. 4... implemented first Example 14. 8 The general manager in Example 14. 1 has found that three of the activities that make up the ground servicing operation can be crashed; activities C, unloading the baggage, F, cleaning the cabin, and G, loading the baggage The crash durations and crash costs of these activities are set out in Table 14. 3 452 Quantitative methods for business Chapter 14 Table 14. 3 Crash durations... use made of quantitative methods in US businesses, Kathawala (1988) found that 54% of companies in his survey reported that they made moderate, frequent or extensive use of critical path analysis and PERT More recently such planning tools have been used in projects like the expansion of the Kings Cross underground station in London (Lane, 2003) 454 Quantitative methods for business Chapter 14 Morris... packaging Precedence — — Duration (weeks) 8 4 (Continued) 456 Quantitative methods for business Activity C D E F G H I 14. 5 Chapter 14 Description Precedence Duration (weeks) Build pilot production line Trial production run Consumer panel tests Main pilot production Design promotional material Test market redesigned product Produce report for the Board A, B 13 C D E E F, G H 1 1 6 4 10 2 Draw a network... 1.282␴CP ϭ 58.5 ϩ 1.282 * 1.958 ϭ 61.010 minutes At this point you may find it useful to try Review Questions 14. 12 to 14. 16 at the end of the chapter Chapter 14 Getting from A to B – project planning using networks 451 14. 4 Cost analysis: crashing the project duration PERT is designed to allow for random fluctuations in the duration of activities These are by definition in large part difficult or impossible... activities involved in building the summerhouse and identify the predecessor activities for each of them as well as their durations (b) Draw a network diagram for the building of the summerhouse and from it identify the critical path and the minimum duration of the project 458 Quantitative methods for business 14. 8 The Dom Stila fashion house intend to stage a fashion show to promote their autumn... 5 8 2 12 1 4 2 4 6 6 J 1 Draw a network to portray this enterprise and use it to find the minimum duration of the project and those activities that are on the critical path 460 Quantitative methods for business Chapter 14 14.11 Members of the Keeshka Dining Club meet once a month to enjoy a meal with a set two-course menu at the club premises while listening to a string quartet Planning the meal entails... you may find it useful to try Review Questions 14. 1 to 14. 11 at the end of the chapter 14. 3 The Program Evaluation and Review Technique (PERT) So far we have assumed that the activities making up a project each have fixed durations In reality this is unlikely to be the case In the Chapter 14 Getting from A to B – project planning using networks 447 0 Figure 14. 7 Examples of the beta distribution 5 Duration . the 446 Quantitative methods for business Chapter 14 Example 14. 4 Find the total floats for the ground servicing activities in Example 14. 1. Table 14. 1 Total floats for activities in Example 14. 1 Activity. there is time to spare for the completion of activities I and J; they have what is called slack or float. 444 Quantitative methods for business Chapter 14 Figure 14. 6 Network for ground servicing. referred to as making a forward pass through the network. When doing this you need to remember that where there are two or 442 Quantitative methods for business Chapter 14 Example 14. 3 The durations