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103 CHAPTER 8: PROJECT MANAGEMENT Overview One of the most popular uses of networks is for project analysis. Such projects as the construction of a building, the development of a drug, or the installation of a computer system can be represented as networks. These networks illu...

103 CHAPTER 8: PROJECT MANAGEMENT Overview One of the most popular uses of networks is for project analysis. Such projects as the construction of a building, the development of a drug, or the installation of a computer system can be represented as networks. These networks illustrate the way in which the parts of the project are organized, and they can be used to determine the time duration of the projects. The network techniques that are used for project analysis are CPM and PERT. CPM stands for critical path method, and PERT is an acronym for project evaluation and review technique. These two techniques are very similar. There were originally two primary differences between CPM and PERT. With CPM, a single, or deterministic, estimate for activity time was used, whereas with PERT probabilistic time estimates were employed. The other difference was related to the mechanics of drawing the project network. In PERT, activities were represented as arcs, or arrowed lines, between two nodes, or circles, whereas in CPM, activities were represented as the nodes or circles. However, these were minor differences, and over time CPM and PERT have been effectively merged into a single technique, conventionally referred to as simply CPM/PERT. CPM and PERT were developed at approximately the same time (although independently) during the late 1950s. The fact that they have already been so frequently and widely applied attests to their value as management science techniques. Learning Outcomes At the end of this chapter students are expected to: 1. Define and explain labor relations and how it could be promoted and managed 2. Identify the roles of the state, employee, employer in achieving industrial peace 3. Identify the different processes and procedures in managing projects 4. Able to apply PERT and CPM processes in project management COURSE MATERIALS The Elements of Project Management Management is generally perceived to be concerned with the planning, organization, and control of an ongoing process or activity such as the production of a product or delivery of a service. Project management is different in that it reflects a commitment of resources and people to a typically important activity for a relatively short time frame, after which the management effort is dissolved. Projects do not have the continuity of supervision that is typical in the management of a production process. As such, the features and characteristics of project management tend to be somewhat unique. Figure 8.1 provides an overview of project management, which encompasses three major processes—planning, scheduling, and control. It also includes a number of the more prominent elements of these processes. In the remainder of this section we will discuss some of the features of the project planning process. 104 Basic Elements of Project Planning:  Objectives - A detailed statement of what is to be accomplished by the project, how it will achieve the company’s goals and meet the strategic plan, and an estimate of when it needs to be completed, the cost, and the return.  Project Scope - A discussion of how to approach the project, the technological and resource feasibility, the major tasks involved, and a preliminary schedule; it includes a justification of the project and what constitutes project success.  Contract Requirements - A general structure of managerial, reporting, and performance responsibilities, including a detailed list of staff, suppliers, subcontractors, managerial requirements and agreements, reporting requirements, and a projected organizational structure.  Schedules - A list of all major events, tasks, and subschedules, from which a master schedule is developed.  Resources - The overall project budget for all resource requirements and procedures for budgetary control.  Personnel - Identification and recruitment of personnel required for the project team, including special skills and training.  Control - Procedures for monitoring and evaluating progress and performance, including schedules and cost.  Risk and Problem Management - Anticipation and assessment of uncertainties, problems, and potential difficulties that might increase the risk of project delays and/or failure and threaten project success. Figure 8.1 The Project Management Process 105 Project Return In a business, one of the most popular measures of benefit is return on investment (ROI). ROI is a performance measure that is often used to evaluate the expected outcome of a project or to compare a number of different projects. To calculate ROI, the benefit (return) of a project is divided by the cost of the project; the result is expressed as a percentage or a ratio: If a project does not have a positive ROI, or if another project has a higher ROI, then the project might not be undertaken. ROI is a very popular metric for project planning because of its versatility and simplicity. However, projects sometimes have benefits that cannot be measured in a tangible way with something like an ROI; these benefits are referred to as ―soft‖ returns. Projects undertaken by government agencies and not-for-profit organizations typically do not have an ROI-type benefit; they are undertaken to benefit the ―public good.‖ In general, it may be more appropriate to measure a project’s benefit not just in terms of financial return but also in terms of the positive impact it may have on a company’s employees and customers (i.e., quality improvement). The Project Team A project team typically consists of a group of individuals selected from other areas in the organization, or from consultants outside the organization, because of their special skills, expertise, and experience related to the project activities. Members of the engineering staff, particularly industrial engineering, are often assigned to project work because of their technical skills. The project team may also include various managers and staff personnel from specificareas related to the project. Even workers can be involved on the project team if their jobs are afunction of the project activity. Assignment to a project team is usually temporary and thus can have both positive and neg- ative repercussions. The temporary loss of workers and staff from their permanent jobs can bedisruptive for both the employees and the work area. An employee must sometimes ―serve twomasters,‖in a sense,reporting to both the project manager and a regular supervisor. The most important member of a project team is the project manager. The job of managing a project is subject to a great deal of uncertainty and the distinct possibility of failure. Because each project is unique and usually has not been attempted previously, the outcome is not as certain as the outcome of an ongoing process would be. The project team members are often from diverse areas of the organization and possess different skills, which must be coordinated into a single, focused effort to successfully complete the project. Overall, there is usually more perceived and real pressure associated with project management than in a normal management position. Scope Statement A scope statement is a document that provides a common understanding of a project. It includes a justification for the project that describes what factors have created a need within the 106 company for the project. It also includes an indication of what the expected results of the project will be and what will constitute project success. Further, the scope statement might include a list of the types of planning reports and documents that are part of the project management process. A similar planning document is the statement of work (SOW). In a large project, the SOW is often prepared for individual team members, groups, departments, subcontractors, and suppliers. This statement describes the work in sufficient detail so that the team member responsible for it knows what is required and whether he or she has sufficient resources to accomplish the work successfully and on time. For suppliers and subcontractors, it is often the basis for determining whether they can perform the work and for bidding on it. Some companies require that an SOW be part of an official contract with a supplier or subcontractor. Work Breakdown Structure The work breakdown structure (WBS) is an organizational chart used for project planning. It organizes the work to be done on a project by breaking down the project into its major components, referred to as modules. These components are then subdivided into more detailed subcomponents, which are further broken down into activities, and, finally, into individual tasks. A WBS helps identify activities and determine individual tasks, project workloads, and the resources required. It also helps to identify the relationships between modules and activities and avoid unnecessary duplication of activities. A WBS provides the basis for developing and managing the project schedule, resources, and modifications. The upper levels of the WBS tend to contain the summary activities, major components or functional areas involved in the project that indicate what is to be done. The lower levels tend to describe the detailed work activities of the project within the major components or modules. They typically indicate how things are done. 107 Figure 8.2. Work Breakdown Structure for Computer Order-Processing System Project Figure 8.2 shows a WBS for a project for installing a new computerized order processing system for a manufacturing company that links customers, the manufacturer, and the manufacturer’s suppliers. The WBS is organized according to the three major project categories for the development of the system: hardware, software/system, and personnel. Within each of these categories, the major tasks and activities under those tasks are detailed. For example, under hardware, a major task is installation, and activities required in installation include area preparation, technical/engineering layouts and configurations, wiring, and electrical connections. Responsibility Assignment Matrix After the WBS is developed, to organize the project work into smaller, manageable elements, the project manager assigns the work elements to organizational units—departments, groups, individuals, or subcontractors—by using an organizational breakdown structure (OBS). An OBS is a table or chart that shows which organizational units are responsible for work items. After the OBS is developed, the project manager can then develop a responsibility assignment matrix (RAM). A RAM shows who in the organization is responsible for doing the work in the project. Figure 8.3 shows a RAM for the ―Hardware/Installation‖ category from the WBS for the computerized order processing project shown in Figure 8.2. Notice that there are three levels of work assignments in the matrix, reflecting who is responsible for the work, who actually performs the work, and who performs support activities. As with the WBS, there are many different forms both the OBS and RAM can take, depending on the needs and preferences of the company, project team, and project manager. 108 Project Scheduling A project schedule evolves from the planning documents discussed previously. It is typically the most critical element in the project management process, especially during the implementation phase (i.e., the actual project work), and it is the source of most conflict and problems. One reason is that frequently the single most important criterion for the success of a project is that it be finished on time. If a stadium is supposed to be finished in time for the first game of the season and it’s not, there will be a lot of angry ticket holders; if a new product is not completed by the scheduled launch date, millions of dollars can be lost; and if a new military weapon is not completed on time, it could affect national security. Also, time is a measure of progress that is very visible. It is an absolute with little flexibility; you can spend less money or use fewer people, but you cannot slow down or stop the passage of time. Basic Steps in Developing a Schedule 1. Define the activities that must be performed to complete the project 2. Sequence the activities in the order in which they must be completed 3. Estimate the time required to complete each activity 4. Develop the schedule based on the sequencing and time estimates of the activities. Because scheduling involves a quantifiable measure, time, there are several quantitative techniques available that can be used to develop a project schedule, including the Gantt chart and CPM/PERT networks. There are also various computer software packages that can be used to schedule projects, including the popular Microsoft Project. The Gantt Chart Using a Gantt chart is a traditional management technique for scheduling and planning small projects that have relatively few activities and precedence relationships. This scheduling technique (also called a bar chart) was developed by Henry Gantt, a pioneer in the field of industrial engineering at the artillery ammunition shops of the Frankford Arsenal in 1914. The Gantt chart has been a popular project scheduling tool since its inception and is still widely used today. It is the direct precursor of the CPM/PERT technique, which we will discuss later. The Gantt chart is a graph with a bar representing time for each activity in the project being analyzed. Figure 8.4 illustrates a Gantt chart of a simplified project description for building a 109 house. The project contains only seven general activities, such as designing the house, laying the foundation, ordering materials, and so forth. The first activity is ―design house and obtain financing,‖ and it requires 3 months to complete, shown by the bar from left to right across the chart. After the first activity is finished, the next two activities, ―lay foundation‖ and ―order and receive materials,‖ can start simultaneously. This set of activities demonstrates how a precedence relationship works; the design of the house and the financing must precede the next two activities. The activity ―lay foundation‖ requires 2 months to complete, so it will be finished, at the earliest, at the end of month 5. ―Order and receive materials‖ requires 1 month to complete, and it could be finished after month 4. However, observe that it is possible to delay the start of this activity 1 month, until month 4. This delay would still enable the activity to be completed by the end of month 5, when the next activity, ―build house,‖ is scheduled to start. This extra time for the activity ―order and receive materials‖ is called slack. Slack is the amount of time by which an activity can be delayed without delaying any of the activities that follow it or the project as a whole. The remainder of the Gantt chart is constructed in a similar manner, and the project is scheduled to be completed at the end of month 9. Figure 8.4. A Gantt Chart A Gantt chart provides a visual display of a project schedule, indicating when activities are scheduled to start and to finish and where extra time is available and activities can be delayed. A project manager can use a Gantt chart to monitor the progress of activities and see which ones are ahead of schedule and which ones are behind schedule. A Gantt chart also indicates the precedence relationships between activities; however, these relationships are not always easily discernible. This problem is one of the disadvantages of the Gantt chart method, and it limits the chart’s use to smaller projects with relatively few activities. The CPM/PERT network technique, which we will talk about later, does not suffer this disadvantage. 110 Project Control Project control is the process of making sure a project progresses toward successful completion. It requires that the project be monitored and progress measured so that any deviations from the project plan, and particularly the project schedule, are minimized. If the project is found to be deviating from the plan (i.e., it is not on schedule, cost overruns are occurring, activity results are not as expected), corrective action must be taken. Time management is the process of making sure a project schedule does not slip and that a project is on time. This requires monitoring of individual activity schedules and frequent updates. If the schedule is being delayed to an extent that jeopardizes the project success, it may be necessary for the project manager to shift resources to accelerate critical activities. Some activities may have slack time, so resources can be shifted from them to activities that are not on schedule. This is referred to as time–cost trade-off. However, this can also push the project cost above the budget. In some cases it may be that the work needs to be corrected or made more efficient. In other cases, it may occur that original activity time estimates upon implementation prove to be unrealistic and the schedule must be changed, and the repercussions of such changes on project success must be evaluated. Cost management is often closely tied to time management because of the time–cost trade-off occurrences mentioned previously. If the schedule is delayed, costs tend to go up in order to get the project back on schedule. Also, as a project progresses, some cost estimates may prove to be unrealistic or erroneous. Therefore, it may be necessary to revise cost estimates and develop budget updates. If cost overruns are excessive, corrective actions must be taken. Performance management is the process of monitoring a project and developing timed (i.e., daily, weekly, monthly) status reports to make sure that goals are being met and the plan is being followed. It compares planned target dates for events, milestones, and work completion with dates actually achieved to determine whether the project is on schedule or behind schedule. Key measures of performance include deviation from the schedule, resource usage, and cost overruns. The project manager and individuals and organizational units with performance responsibility develop these status reports. Earned value analysis (EVA) is a specific system for performance management. Activities ―earn value‖ as they are completed. EVA is a recognized standard procedure for numerically measuring a project’s progress, forecasting its completion date and final cost, and providing measures of schedule and budget variation as activities are completed. For example, an EVA metric such as schedule variance compares the work performed during a time period with the work that was scheduled to be performed; a negative variance means the project is behind schedule. Cost variance is the budgeted cost of work performed minus the actual cost of the work; a negative variance means the project is over budget. EVA works best when it is used in conjunction with a WBS that compartmentalizes project work into small packages that are easier to measure. The drawbacks of EVA are that it is sometimes difficult to measure work progress, and the time required for data measurement can be considerable. CPM/PERT Critical path method (CPM) and project evaluation and review technique (PERT) were originally developed as separate techniques. Both are derivatives of the Gantt chart and, as a result, are very similar. There were originally two primary differences between CPM and PERT. With CPM, a single estimate for activity time was used that did not allow for variation; activity times 111 were treated as if they were known with certainty. With PERT, multiple time estimates were used for each activity that reflected variation; activity times were treated as probabilistic. The other difference was in the mechanics of drawing a network. In PERT, activities were represented as arcs, or lines with arrows, between circles called nodes, whereas in CPM activities were represented by the nodes, and the arrows between them showed precedence relationships (i.e., which activity came before another). However, over time CPM and PERT have effectively merged into a single technique conventionally known as CPM/PERT. The advantage of CPM/PERT over the Gantt chart is in the use of a network (instead of a graph) to show the precedence relationships between activities. A Gantt chart does not clearly show precedence relationships, especially for larger networks. Put simply, a CPM/PERT network provides a better picture; it is visually easier to use, which makes using CPM/PERT a popular technique for project planners and managers. AOA Networks A CPM/PERT network is drawn with branches and nodes, as shown in Figure 8.5. As previously mentioned, when CPM and PERT were first developed, they employed different conventions for drawing a network. With CPM, the nodes in Figure 8.5 represent the project activities. The branches with arrows in between the nodes indicate the precedence relationships between activities. For example, in Figure 8.5, activity 1, represented by node 1, precedes activity 2, and 2 must be finished before 3 can start. This approach to constructing a network is called activity-on-node (AON). Alternatively, with PERT, the branches in between the nodes represent activities, and the nodes reflect events or points in time such as the end of one activity and the start of another. This approach is called activity-on-arrow (AOA), and the activities are identified by the node numbers at the start and end of an activity (for example, activity activity 1 → 2, which precedes activity 2 → 3 in Figure 8.5). Figure 8.5. Nodes and Branches To demonstrate how an AOA network is drawn, we will revisit the example of building a house that we used as a Gantt chart example in Figure 8.4. The comparable CPM/PERT network for this project is shown in Figure 8.6. The precedence relationships are reflected in this network by the arrangement of the arrowed (or directed) branches. The first activity in the project is to design the house and obtain financing. This activity must be completed before any subsequent activities can begin. Thus, activity 2 → 3, laying the foundation, and activity 2 → 4 can occur concurrently; neither depends on the other, and both depend only on the completion of activity 1 → 2. Figure 8.6. Expanded network for building a house, showing concurrent activities 112 When the activities of laying the foundation (2 → 3) and ordering and receiving materials (2 → 4) are completed, activities 4 → 5 and 4 → 6 can begin simultaneously. However, notice activity 3 → 4 which is referred to as a dummy activity. A dummy activity is used in an AOA network to show a precedence relationship, but it does not represent any actual passage of time. These activities have the precedence relationship shown in Figure 8.7(a). However, in an AOA network, two or more activities are not allowed to share the same start and ending nodes because Figure 8.7. A dummy activity that would give them the same name designation (i.e., 2 → 3). Activity 3 → 4 is inserted to give the two activities separate end nodes and thus separate identities, as shown in Figure 8.7(b). Returning to the network shown in Figure 8.6, we see that two activities start at node 4. Activity 4 → 6 is the actual building of the house, and activity 4 → 5 is the search for and selection of the paint for the exterior and interior of the house. Activity 4 → 6 and activity 4 → 5 can begin simultaneously and take place concurrently. Following the selection of the paint (activity 4 → 5) and the realization of node 5, the carpet can be selected (activity 5 → 6) because the carpet color is dependent on the paint color. This activity can also occur concurrently with the building of the house (activity 4 → 6). When the building is completed and the paint and carpet are selected, the house can be finished (activity 6 → 7). AON Networks Figure 8.8 shows the comparable AON network to the AOA network in Figure 8.6 for our housebuilding project. Notice that the activities and activity times are on the nodes and not on the activities, as with the AOA network. The branches or arrows simply show the precedence relationships between the activities. Also, notice that there is no dummy activity; dummy activities are not required in an AON network because no two activities have the same start 113 and ending nodes, so they will never be confused. This is one of the advantages of the AON convention, although both AOA and AON have minor advantages and disadvantages. In general, the two methods both accomplish the same thing, and the one that is used is usually a matter of individual preference. However, for our purposes, the AON network has one distinct advantage: It is the convention used in Figure 8.8. AON network for house-building project the popular Microsoft Project software package. Because we will demonstrate how to use that software later in this chapter, we will use AON. The Critical Path A network path is a sequence of connected activities that runs from the start to the end of the network. The network shown in Figure 8.8 has several paths. In fact, close observation of this network shows four paths, identified in Table 8.1 as A, B, C, and D. Table 8.1. Paths through the house-building networks The project cannot be completed (i.e., the house cannot be built) until the longest path in the network is completed. This is the minimum time in which the project can be completed. The longest path is referred to as the critical path. To better understand the relationship between the minimum project time and the critical path, we will determine the length of each of the four paths shown in Figure 8.8. 114 By summing the activity times (shown in Figure 8.8) along each of the four paths, we can compute the length of each path, as follows: Because path A is the longest, it is the critical path; thus, the minimum completion time for the project is 9 months. Now let us analyze the critical path more closely. From Figure 8.9 we can see that activity 3 cannot start until 3 months have passed. It is also easy to see that activity 4 will not start until 5 months have passed (i.e., the sum of activity 1 and 2 times). The start of activity 4 is dependent on the two activities leading into node 4. Activity 2 is completed after 5 months, but activity 3 is completed at the end of 4 months. Thus, we have two possible start times for activity 4: 5 months and 4 months. However, because the activity on node 4 cannot start until all preceding activities have been finished, the soonest node 4 can be started is 5 months. Now, let us consider the activity following node 4. Using the same logic as before, we can see that activity 7 cannot start until after 8 months (5 months at node 4 plus the 3 months required by activity 4), or after 7 months (on path 1-3-5-6-7). Because all activities preceding node 7 must be completed before activity 7 can start, the soonest this can occur is 8 months. Adding 1 month for activity 7 to the time at node 7 gives a project duration of 9 months. Recall that this is the time of the longest path in the network, or the critical path. Activity Scheduling In our analysis of the critical path, we determined the earliest time that each activity could be finished. For example, we found that the earliest time activity 4 could start was at 5 months. This time is referred to as the earliest start time, and it is expressed symbolically as ES. In order to show the earliest start time on the network, as well as some other activity times we will develop in the scheduling process, we will alter our node structure a little. Figure 8.10 shows 115 the structure for node 1, the first activity in our example network for designing a house and obtaining financing. Figure 8.10. Activity on-node configuration To determine the earliest start time for every activity, we make a forward pass through the network. That is, we start at the first node and move forward through the network. The earliest time for an activity is the maximum time for all preceding activities that have been completed— the time when the activity start node is realized. The earliest finish time, EF, for an activity is the earliest start time plus the activity time estimate. For example, if the earliest start time for activity 1 is at time 0, then the earliest finish time is 3 months. In general, the earliest start and finish times for an activity are computed according to the following mathematical formulas: ES = Maximum (EF immediate predecessors) EF = ES + t The earliest start and earliest finish times for all the activities in our project network are shown in 116 Figure 8.11. The earliest start time for the first activity in the network (for which there are no predecessor activities) is always zero, or ES = 0 This enables us to compute the earliest finish time for activity 1 as EF = ES + t = 0 + 3 = 3 months The earliest start time for activity 2 is EF = ES + t = 3 + 2 = 5 months For activity 3 the earliest start time is 3 months, and the earliest finish time is 4 months. Now consider activity 4, which has two predecessor activities. The earliest start time is computed as ES = Max (EF immediate predecessors) = Max (5, 4) = 5 months and the earliest finish time is EF = ES + t = 5 + 3 = 8 months All the remaining earliest start and finish times are computed similarly. Notice in Figure 8.11 that the earliest finish time for activity 7, the last activity in the network, is 9 months, which is the total project duration, or critical path time. Companions to the earliest start and finish are the latest start (LS) and finish (LF) times. The latest start time is the latest time an activity can start without delaying the completion of the project beyond the project’s critical path time. For our example, the project completion time (and earliest finish time) at node 7 is 9 months. Thus, the objective of determining latest times is to see how long each activity can be delayed without the project exceeding 9 months. In general, the latest start and finish times for an activity are computed according to the following formulas: LS = LF – t LF = Minimum (LS immediately following predecessors) Whereas a forward pass through the network is made to determine the earliest times, the latest times are computed using a backward pass. We start at the end of the network at node 7 and work backward, computing the latest time for each activity. Because we want to determine how long each activity in the network can be delayed without extending the project time, the latest 117 finish time at node 7 cannot exceed the earliest finish time. Therefore, the latest finish time at node 7 is 9 months. This and all other latest times are shown in Figure 8.12. Starting at the end of the network, the critical path time, which is also equal to the earliest finish time of activity 7, is 9 months. This automatically becomes the latest finish time for activity 7, or LF = 9 months Using this value, the latest start time for activity 7 can be computed: LS = LF – t = 9 – 1 = 8 months The latest finish time for activity 6 is the minimum of the latest start times for the activities following node 6. Because activity 7 follows node 6, the latest start time is computed as follows: LF = Min (LS following activities) = 8 months The latest start time for activity 6 is LS = LF – t = 8 – 1 = 7 months For activity 4, the latest finish time is 8 months, and the latest start time is 5 months; for activity 5, the latest finish time is 7 months, and the latest start time is 6 months. Now consider activity 3, which has two activities following it. The latest finish time is computed as follows: LF = Min (LS following activities) = Min (5, 6) = 5 months The latest start time is LS = LF – t = 5 – 1 = 4 months All the remaining latest start and latest finish times are computed similarly. Figure 8.12 includes the earliest and latest start times and earliest and latest finish times for all activities. Activity Slack The project network in Figure 8.12, with all activity start and finish times, highlights the critical path (1 → 2 → 4 → 7) we determined earlier by inspection. Notice that for the activities on the critical path, the earliest start times and latest start times are equal. This means that these activities on the critical path must start exactly on time and cannot be delayed at all. If the start of any activity on the critical path is delayed, then the overall project time will be increased. As a result, we now have an alternative way to determine the critical path besides simply inspecting the network. The activities on the critical path can be determined by seeing for which activities ES = LS or EF = LF In Figure 8.12, the activities 1, 2, 4, and 7 all have earliest start times and latest start times that are equal (and EF = LF); thus, they are on the critical path. For those activities not on the critical path, the earliest and latest start times (or earliest and latest finish times) are not equal, and slack time exists. Slack is the amount of time an activity can be delayed without affecting the overall project duration. In effect, it is extra time available for completing an activity. 118 Slack, S, is computed using either of the following formulas: S = LS – ES or S = LF – EF For example, the slack for activity 3 is computed as follows: S = LS – ES = 4 – 3 = 1 month If the start of activity 3 were delayed for 1 month, the activity could still be completed by month 5 without delaying the project completion time. The slack for each activity in our example project network is shown in Table 8.2 and in Figure 8.13. Notice in Figure 8.13 that activity 3 can be delayed 1 month and activity 5, which follows it, can be delayed 1 more month, but then activity 6 cannot be delayed at all, even though it has 1 month of slack. If activity 3 starts late, at month 4 instead of month 3, then it will be completed at month 5, which will not allow activity 5 to start until month 5. If the start of activity 5 is delayed 1 month, then it will be completed at month 7, and activity 6 cannot be delayed at all without exceeding the critical path time. The slack on these three activities is called shared slack. This 119 means that the sequence of activities 3 → 5 → 6 can be delayed 2 months jointly without delaying the project, but not 3 months. Slack is obviously beneficial to a project manager because it enables resources to be temporarily pulled away from activities with slack and used for other activities that might be delayed for various reasons or for which the time estimate has proven to be inaccurate. The times for the network activities are simply estimates for which there is usually not a lot of historical basis (because projects tend to be unique undertakings). As such, activity time estimates are subject to quite a bit of uncertainty. However, the uncertainty inherent in activity time estimates can be reflected to a certain extent by using probabilistic time estimates instead of the single, deterministic estimates we have used so far. Probabilistic Activity Times In the project network for building a house presented in the previous section, all the activity time estimates were single values. By using only a single activity time estimate, we in effect assume that activity times are known with certainty (i.e., they are deterministic). For example, in Figure 8.8, the time estimate for activity 2 (laying the foundation) is shown to be 2 months. Because only this one value is given, we must assume that the activity time does not vary (or varies very little) from 2 months. In reality, however, it is rare that activity time estimates can be made with certainty. Project activities are likely to be unique, and thus there is little historical evidence that can be used as a basis to predict actual times. However, we earlier indicated that one of the primary differences between CPM and PERT was that PERT used probabilistic activity times. Probabilistic Time Estimates To demonstrate the use of probabilistic activity times, we will employ a new example. The Southern Textile Company has decided to install a new computerized order processing system that will link the company with customers and suppliers online. In the past, orders for the cloth the company produces were processed manually, which contributed to delays in delivering orders and resulted in lost sales. The company wants to know how long it will take to install the new system. The network for the installation of the new order processing system is shown in Figure 8.14. We will briefly describe the activities. The network begins with three concurrent activities: The new computer equipment is installed (activity 1); the computerized order processing system is developed (activity 2); and people are recruited to operate the system (activity 3). Once people are hired, they are trained for the job (activity 6), and other personnel in the company, such as marketing, accounting, and production personnel, are introduced to the new system (activity 7). Once the system is developed (activity 2), it is tested manually to make sure that it is logical (activity 5). Following activity 1, the new equipment is tested, and any necessary modifications are made (activity 4), and the newly trained personnel begin training on the computerized system (activity 8). Also, node 9 begins the testing of the system on the computer to check for errors (activity 9). The final activities include a trial run and changeover to the system (activity 11) and final debugging of the computer system (activity 10). At this stage in a project network, we previously assigned a single time estimate to each network activity. In a PERT project network, however, we determine three time estimates for 120 each activity, which will enable us to estimate the mean and variance for a beta distribution of the activity times. The three time estimates for each activity are the most likely time, the optimistic time, and the pessimistic time. The most likely time is the time that would most frequently occur if the activity were repeated many times. The optimistic time is the shortest possible time within which the activity could be completed if everything went right. The pessimistic time is the longest possible time the activity would require to be completed, assuming that everything went wrong. In general, the person most familiar with an activity makes these estimates to the best of his or her knowledge and ability. In other words, the estimate is subjective. These three time estimates can subsequently be used to estimate the mean and variance of a beta distribution. If a = optimistic time possible m = most likely time estimate b = pessimistic time estimate the mean and variance are computed as follows: These formulas provide a reasonable estimate of the mean and variance of the beta distribution, 121 a distribution that is continuous and can take on various shapes—that is, exhibit skewness. The three time estimates for each activity are shown in Figure 8.14and in Table 8.3. The mean and the variance for all the activities in the network shown in Figure 8.14 are also given inTable 8.3. As an example of the computation of the individual activity mean times and variances,con-sider activity 1. The three time estimates (a = 6, m = 8, b = 10) are substituted in our beta distribution formulas as follows: The other values for the mean and variance in Table 8.3are computed similarly.Once the expected activity times have been computed for each activity,we can determine thecritical path the same way we did previously, except that we use the expected activity times, t. Recall that in the project network, we identified the critical path as the one containing those activities with zero slack. This requires the determination of earliest and latest event times, as shown in Figure 8.15. Observing Figure 8.15, we can see that the critical path encompasses activities 2 → 5 → 8 → 11. because these activities have no available slack. We can also see that the expected project completion time (tp) is 25 weeks. However, it is possible to compute the variance for project completion time. To determine the project variance, we sum the variances for those ac-tivities on the critical path. Using the variances computed in Table 8.3and the critical path activities 122 shown in Figure 8.15, we can compute the variance for project duration (vp) as follows: The CPM/PERT method assumes that the activity times are statistically independent, which allows us to sum the individual expected activity times and variances to get an expected project time and variance. It is further assumed that the network mean and variance are normally distributed. This assumption is based on the central limit theorem of probability, which for CPM/PERT analysis and our purposes states that if the number of activities is large enough and the activities are statistically independent, then the sum of the means of the activities along the critical path will approach the mean of a normal distribution. Given these assumptions, we can interpret the expected project time as the mean (μ) and variance (σ2) of a normal distribution: 123 In turn, we can use these statistical parameters to make probabilistic statements about the project. Probability Analysis of the Project Network Using the normal distribution, probabilities can be determined by computing the number of standard deviations (Z) a value is from the mean, as illustrated in Figure 8.16. Figure 8.15. Normal distribution of network duration. The Z value is computed using the following formula: This value is then used to find the corresponding probability in Table A.1 of Appendix A. For example, suppose the textile company manager told customers that the new order processing system would be completely installed in 30 weeks. What is the probability that it will, in fact, be ready by that time? This probability is illustrated as the shaded area in Figure 8.17. Figure 8.17. Probability that the network will be completed in 30 weeks or less 124 To compute the Z value for a time of 30 weeks, we must first compute the standard deviation (σ) from the variance (σ2): Next, we substitute this value for the standard deviation, along with the value for the mean and our proposed project completion time (30 weeks), into the following formula: AZ value of 1.90 corresponds to a probability of.4713 in Table A.1 in Appendix A. This means that there is a.9713 (.5000 +.4713) probability of completing the project in 30 weeks or less. Suppose one customer, frustrated with delayed orders, has told the textile company that if the new ordering system is not working within 22 weeks, she will trade elsewhere. The probability of the project’s being completed within 22 weeks is computed as follows: A Z value of 1.14 (the negative indicates the area is to the left of the mean) corresponds to a probability of.3729 in Table A.1 of Appendix A. Thus, there is only a.1271 (.5000 +.3729) probability that the customer will be retained, as illustrated in Figure 8.18. Figure 8.18. Probability that the network will be completed in 22 weeks or less CPM/PERT Analysis with QM for Windows and Excel QM The capability to perform CPM/PERT network analysis is a standard feature of most management science software packages for the personal computer. To illustrate the application of QM for Windows and Excel QM, we will use our example of installing an order processing system at Southern Textile Company. The QM for Windows solution output is shown in Exhibit 8.1, and the Excel QM solution is shown in Exhibit 8.2. 125 Exhibit 8.1 Exhibit 8.2 Microsoft Project Microsoft Project is a popular and widely used software package for project management andCPM/PERT analysis. It is also relatively easy to use. We will demonstrate how to use Microsoft Project,using the project network for building a house shown in Figure 8.13. When you open Microsoft Project,a screen comes up for a new project, as shown in Exhibit 8.3. Notice that the ―Gantt Chart Tools‖ tab on the toolbar ribbon at the top of thescreen is 126 highlighted. The initial step is to set the project up in this window. First, type in theactivity name, ―Design and finance‖ under the ―Task Name‖ column, and then type in the activity duration in the ―Duration‖ column, which is 3 months for this activity. This only requires that you type ―3 mo‖ and Project will recognize it as 3 months. Next, type in a ―Start‖ date, such as ―May 11, 2011.‖ The start date can also be selected from the drop-down calendar. (Note that a start date must be designated for all starting activities that don’t have predecessors, which in the case of this example is only the first activity.) This first activity has no predecessor, so leave the ―Predecessor‖ cell for this activity blank and drop down to the next line to enter the next activity, ―Lay foundation.‖ Repeat the process followed for the first activity by typing in the duration, ―2 mo‖ but do not type in the start and finish dates; the program will do this automatically when you start identifying activity predecessors. Toggle over to the ―Predecessor‖ cell for this activity and type in ―1‖ (indicating that activity 1 is the predecessor activity for this activity 2). Next, drop down to the next line and type in the information for activity 3, ―Order materials,‖ the duration of ―1 mo‖ and that its ―Predecessor‖ is activity 1. Exhibit 8.4 shows our progress so far. Before proceeding to finishing inputting the project data, we’ll make a few comments. First, note the emerging Gantt chart forming on the calendar part of your window (on the right-hand part of the screen) for the activities we’ve entered. This chart might not be on your screen when you start because the timescale on the calendar does not conform to your start date. If the Gantt chart is not there, you can make it appear by moving the tab at the bottom of the calendar part of the screen ahead (i.e., to the right) until your start date appears, and this view should contain the beginning of your Gantt chart. Also, notice at the top of the window that we have switched to the ―View‖ tab. On this toolbar ribbon, if you click on ―Timescale,‖ you can insert the time units of the project, which in this case is months, or you can accomplish the same thing by right- clicking on the time line just above the Gantt chart and then clicking on ―Timescale‖ from the 127 drop-down menu. You can also condense the size of your Gantt chart to fit on the calendar part of your screen by clicking on the button on the ―Entire Project‖ icon on the toolbar. Microsoft Project uses a ―standard‖ calendar for scheduling project activities; for example, it automatically removes weekend days from the list of working days. Holidays and vacation days can also be designated by first clicking on the ―Project‖ tab and then clicking on the ―Change Working Time‖ icon. This will bring up a window with a calendar and menu for changing work times. The activity predecessor relationships showing, for example, activity 1 precedes activity 2, can also be established by placing the cursor on activity 1, then holding down the ―Ctrl‖ key while clicking on activity 2 and then clicking on the ―link‖ icon (i.e., the little chain link) from the ―Task‖ tab under ―Gantt Chart Tools.‖ Next,we will finish typing in the rest of the information for our project, as shown inExhibit 8.5. Notice on this screen that we have switched back to the ―Format‖tab under― Gantt Chart Tools,‖which allows us to select the ―Critical Tasks‖box to highlight the critical path on the Gantt Chart in red. Note that the ―pins‖in the cells next to each activity inthe ―Task Mode‖ column on our data window indicate that we have manually scheduled ac-tivities (or tasks); the other ―Task Mode‖option is to allow Microsoft Project to automati-cally schedule the activities. For our purposes,manual scheduling is sufficient. In order to show the network diagram of our project, click on ―View‖on the toolbar andthen on ―Network Diagram,‖which results in the screen shown in Exhibit 8.6. (We also used the―Zoom‖ icon to make the network larger.) Notice that the critical path is highlighted in red. Clicking on the ―Gantt Chart‖ icon on the far left of the toolbar ribbon will return you to the Gantt Chart 128 screen,where we typed in our project data. The latest version of Microsoft Project does not have PERT (three time estimate ) capabili- ties,thus to use Microsoft Project when you have three time estimates,you must first manually(or using Excel) compute the mean activity times and use these as single time estimates in Microsoft Project. Exhibit 8.7 shows the Microsoft Project ―Gantt Chart‖ window for our ―Order Processing System‖ project example,where the mean activity time estimates are used. Exhibit 8.8 shows the network diagram. Notice that we gave this project a start date of May 11,2011, which had to be designated as the start date for all three of our starting activities (that donot have predecessors):1, 2, and 3. Microsoft Project also has many additional tools and features,including project updatingand activity completion,resource management,work stoppages,development of work break-down structures (WBS),and the ability to change work times,including shortened days,holi-days,and 129 vacation time. To access information about these features,access the various ―Help‖screens from the different Microsoft Project windows or press the F1 key. Project Crashing and Time–Cost Trade-Off To this point, we have demonstrated the use of CPM and PERT network analysis for determining project time schedules. This in itself is valuable to a manager planning a project. However, in addition to scheduling projects, a project manager is frequently confronted with the problem of having to reduce the scheduled completion time of a project to meet a deadline. In other words, the manager must finish the project sooner than indicated by the CPM or PERT network analysis. Project duration can be reduced by assigning more labor to project activities, often in the form of overtime, and by assigning more resources (material, equipment, etc). However, additional labor and resources cost money and hence increase the overall project cost. Thus, the decision to reduce the project duration must be based on an analysis of the trade-off between time and cost. Project crashing is a method for shortening project duration by reducing the time of one or more of the critical project activities to a time that is less than the normal activity time. This reduction in the normal activity times is referred to as crashing. Crashing is achieved by devoting more resources, measured in terms of dollars, to the activities to be crashed. Figure 8.19. The project network for building a house In Figure 8.19, we will assume that the times (in weeks) shown on the network activities are the normal activity times. For example, normally 12 weeks are required to complete activity 1. Furthermore, we will assume that the cost required to complete this activity in the time indicated is $3,000. This cost is referred to as the normal activity cost. Next, we will assume that the building contractor has estimated that activity 1 can be completed in 7 weeks, but it will cost $5,000 to complete the activity instead of $3,000. This new estimated activity time is known as the crash time, and the revised cost is referred to as the crash cost. Activity 1 can be crashed a total of 5 weeks (normal time – crash time = 12 – 7 = 5 weeks). at a total crash cost of $2,000 (crash cost – normal cost = $5,000 - $3,000 = $2,000). Dividing the 130 total crash cost by the total allowable crash time yields the crash cost per week: If we assume that the relationship between crash cost and crash time is linear, then activity 1can be crashed by any amount of time (not exceeding the maximum allowable crash time) at arate of $400 per week. For example, if the contractor decided to crash activity 1 by only 2 weeks (for an activity time of 10 weeks), the crash cost would be $800 ($400 per week × 2 weeks). The linear relationships between crash cost and crash time and between normal cost and normal timeare illustrated in Figure 8.20. Figure 8.20. Time–cost relationship for crashing activity 1 The normal times and costs, the crash times and costs, the total allowable crash times, and thecrash cost per week for each activity in the network in Figure 8.19 are summarized in Table 8.4. Recall that the critical path for the house-building network encompassed activities 1 → 2 → 4 → 7, and the project duration was 9 months, or 36 weeks. Suppose that the home builder needed the house in 30 weeks and wanted to know how much extra cost would be incurred to complete the house in this time. To analyze this situation, the contractor would crash the project network to 30 weeks, using the information in Table 8.4. Table 8.4. Normal activity and crash data for the network in Figure 8.19 The objective of project crashing is to reduce the project duration while minimizing the cost of crashing. Because the project completion time can be shortened only by crashing activities on the critical path, it may turn out that not all activities have to be crashed. However, as activities 131 are crashed, the critical path may change, requiring crashing of previously noncritical activities to further reduce the project completion time. We start the crashing process by looking at the critical path and seeing which activity has the minimum crash cost per week. Observing Table 8.4 and Figure 8.21, we see that on the critical path activity 1 has the minimum crash cost of $400. Thus, activity 1 will be reduced as much as possible. Table 8.4 shows that the maximum allowable reduction for activity 1 is 5 weeks, but we can reduce activity 1 only to the point where another path becomes critical. When two paths simultaneously become critical, activities on both must be reduced by the same amount. (If we reduce the activity time beyond the point where another path becomes critical, we may incur an unnecessary cost.) This last stipulation means that we must keep up with all the network paths as we reduce individual activities, a condition that makes manual crashing very cumbersome. Later we will demonstrate an alternative method for project crashing, using linear programming; however, for the moment we will pursue this example in order to demonstrate the logic of project crashing. Figure 8.21. Network with normal activity times and weekly activity crashing costs It turns out that activity 1 can be crashed by the total amount of 5 weeks without another path’s becoming critical because activity 1 is included in all four paths in the network. Crashing this activity results in a revised project duration of 31 weeks, at a crash cost of $2,000. The revised network is shown in Figure 8.22. This process must now be repeated. The critical path in Figure 8.22 remains the same, and the new minimum activity crash cost on the critical path is $500 for activity 2. Activity 2 can be crashed a total of 3 weeks, but because the contractor desires to crash the network to only 30 weeks, we need to crash activity 2 by only 1 week. Crashing activity 2 by 1 week does not result in any other path’s becoming critical, so we can safely make this reduction. Crashing activity 2 to 7 weeks (i.e., a 1-week reduction) costs $500 and reduces the project duration to 30 weeks. The extra cost of crashing the project to 30 weeks is $2,500. Thus, the contractor could inform the customer that an additional cost of only $2,500 would be incurred to finish the house in 30 weeks. 132 Figure 8.22. Revised network with activity 1 crashed As indicated earlier, the manual procedure for crashing a network is very cumbersome and generally unacceptable for project crashing. It is basically a trial-and-error approach that is useful for demonstrating the logic of crashing; however, it quickly becomes unmanageable for larger networks. Project Crashing with QM for Windows QM for Windows also has the capability to crash a network completely. In other words, it crashes the network by the maximum amount possible. In our house-building example in the previous section, we crashed the network to only 30 weeks, and we did not consider by how much the network could have actually been crashed. Alternatively, QM for Windows crashes a network by the maximum amount possible. The QM for Windows solution for our housebuilding example is shown in Exhibit 8.9. Notice that the network has been crashed to 24 weeks, at a total crash cost of $31,500. Exhibit 8 9 133 The General Relationship of Time and Cost In our discussion of project crashing, we demonstrated how the project critical path time could be reduced by increasing expenditures for labor and direct resources. The implicit objective of crashing was to reduce the scheduled completion time for its own sake—that is, to reap the results of the project sooner. However, there may be other important reasons for reducing project time. There also may be direct financial penalties for not completing a project on time. For example, many construction contracts and government contracts have penalty clauses for exceeding the project completion date. In general, project crash costs and indirect costs have an inverse relationship; crash costs are highest when the project is shortened, whereas indirect costs increase as the project duration increases. This time–cost relationship is illustrated in Figure 8.23. The best, or optimal, project time is at the minimum point on the total cost curve. Figure 8.23. The time–cost trade-off Formulating the CPM/PERT Network as a Linear Programming Model First we will look at the linear programming formulation of the general CPM/PERT network model and then at the formulation of the project crashing network. We will formulate the linear programming model of a CPM/PERT network, using the AOA convention. As the first step in formulating the linear programming model, we will define the decision variables. In our discussion of CPM/PERT networks using the AOA convention, we designated an activity by its start and ending node numbers. Thus, an activity starting at node 1 and ending at node 2 was referred to as activity 1 → 2. We will use a similar designation to define the decision variables of our linear programming model. We will use a different scheduling convention. Instead of determining the earliest activity start time for each activity, we will use the earliest event time at each node. This is the earliest time that a node (i or j) can be realized. In other words, it is the earliest time that the event the node represents, either the completion of all the activities leading into it or the start of all activities leaving it, can occur. Thus, for an activity i → j the earliest event time of node i or xi, and the earliest event time of node j will be xj. 134 The objective of the project network is to determine the earliest time the project can be completed (i.e., the critical path time). We have already determined from our discussion of CPM/PERT network analysis that the earliest event time of the last node in the network equals the critical path time. If we let xi equal the earliest event times of the nodes in the network, then the objective function can be expressed as Because the value of Z is the sum of all the earliest event times, it has no real meaning; however, it will ensure the earliest event time at each node. Next, we must develop the model constraints. We will define the time for activity i -> j as tij. From our previous discussion of CPM/PERT network analysis, we know that the difference between the earliest event time at node j and the earliest event time at node i must be at least as great as the activity time tij. A set of constraints that expresses this condition is defined as The general linear programming model of formulation of a CPM/PERT network can be summarized as The solution of this linear programming model will indicate the earliest event time of each node in the network and the project duration. As an example of the linear programming model formulation and solution of a project network, we will use the house-building network from Figure 8.6 with times converted to weeks. This network, with activity times in weeks and earliest event times, is shown in Figure 8.24. Figure 8.24. CPM/PERT network for the house-building project, with earliest event times 135 The linear programming model for the network in Figure 8.24 is Notice in this model that there is a constraint for each activity in the network. Solution of the CPM/PERT Linear Programming Model with Excel The linear programming model of the CPM/PERT network in the preceding section enables us to use Excel to schedule the project. Exhibit 8.10 shows an Excel spreadsheet set up to determine the earliest event times for each node—that is, the xi and xj values in our linear programming model for our house-building example. The earliest start times, our decision variables, are in cells B6:B12. Cells F6:F13 contain the model constraints. For example, cell F6 contains the constraint. For example, cell F6 contains the constraint formula =B7 – B6, and cell F7 contains the formula =B8 – B7. These constraint formulas will be set ≥ the activity times in column G when we access Solver. (Also, because activity 3 → 4 is a dummy, a constraint for F9=0 must be added to Solver.) Exhibit 8.10 The Solver Parameters window with the model data is shown in Exhibit 8.11. Notice that the objective is to minimize the sum of the activity times in cell B13. Thus, cell B12 actually contains the project duration, the earliest event time for node 7. 136 Exhibit 8.11 The solution is shown in Exhibit 8.12. Notice that the earliest time at each node is given in cells B6:B12, and the total project duration is 36 weeks. However,this output does not indicatethe critical path. The critical path can be determined by accessing the sensitivity report for this problem. Recall that when you click on ―Solve‖ from Solver, a screen comes up, indicating that Solver has reached a solution. This screen also provides the opportunity to generate several different kinds of reports, including an answer report and a sensitivity report. When we click on ―Sensitivity‖ under the report options, the information shown in Exhibit 8.13 is provided. Exhibit 8.12 137 Exhibit 8.13 The information in which we are interested is the shadow price for each of the activity constraints. The shadow price for each activity will be either 1 or 0. A positive shadow price of 1 for an activity means that you can reduce the overall project duration by an amount with a corresponding decrease by the same amount in the activity duration. Alternatively, a shadow price of 0 means that the project duration will not change, even if you change the activity duration by some amount. This means that those activities with a shadow price of 1 are on the critical path. Cells F6, F7, F9, F11, and F13 have shadow prices of 1, and referring back to Exhibit 8.12, we see that these cells correspond to activities 1 → 2, 2 → 3, 3 → 4, 4 → 6, and 6 → 7 which are the activities on the critical path. Project Crashing with Linear Programming The linear programming model required to perform project crashing analysis differs from thelinear programming model formulated for the general CPM/PERT network in the previous section. The linear programming model for project crashing is somewhat longer and more complex. The objective for our general linear programming model was to minimize project duration; the objective of project crashing is to minimize the cost of crashing, given the limits on how much individual activities can be crashed. As a result, the general linear programming model formulation must be expanded to include crash times and cost. We will continue to define the earliest event times for activity i → j is crashed as yij. Thus, the decision variables are 138 defined as The objective of project crashing is to reduce the project duration at the minimum possible crash cost. For our house-building network, the objective function is written as minimize Z = $400y12 + $500y23 + 3,000y24 + 200y45 + 7,000y46 + 200y56 + 7,000y67 The objective function coefficients are the activity crash costs per week from Table 8.4; the variables yij indicate the number of weeks each activity will be reduced. For example, if activity 1 → 2 is crashed by 2 weeks, then y12 = 2 and a cost of $800 is incurred. The model constraints must specify the limits on the amount of time each activity can be crashed. Using the allowable crash times for each activity from Table 8.4 enables us to develop the following set of constraints: For example, the first constraint, y12 ≤ 5 specifies that the amount of time by which activity 1 → 2 is reduced cannot exceed 5 weeks. The next group of constraints must mathematically represent the relationship between earliest event times for each activity in the network, as the constraint xj – xi ≥ tij did in our original linear programming model. However, we must now reflect the fact that activity times can be crashed by an amount yij. Recall the formulation of the activity 1 → 2 constraint for the general linear programming model formulation in the previous section: x2 – x1 ≥ 12 This constraint can also be written as x1 + 12 ≤ x2 This latter constraint indicates that the earliest event time at node 1 (x1), plus the normal activity time (12 weeks) cannot exceed the earliest event time at node 2 (x2). To reflect the fact that this activity can be crashed, it is necessary only to subtract the amount by which it can be crashed from the left-hand side of the preceding constraint: 139 This revised constraint now indicates that the earliest event time at node 2 (x2) is determined not only by the earliest event time at node 1 plus the activity time but also by the amount the activity is crashed. Each activity in the network must have a similar constraint, as follows: Finally, we must indicate the project duration we are seeking (i.e., the crashed project time). Because the housing contractor wants to crash the project from the 36-week normal critical path time to 30 weeks, our final model constraint specifies that the earliest event time at node 7 should not exceed 30 weeks: x7 ≤ 30 The complete linear programming model formulation is summarized as follows: Project Crashing with Excel 140 Exhibit 8.14 shows a modified version of the Excel spreadsheet we developed earlier, in Exhibit 8.10, to determine the earliest event times for our CPM/PERT network for the house-building project. We have added columns H, I, and J for the activity crash costs, the activity crash times, and the actual activity crash times. Cells J6:J13 correspond to the yij variables in the linear programming model. The constraint formulas for each activity are included in cells F6:F13. For example, cell F6 contains the formula =J6+B7-B6, and cell F7 contains the formula and cell F7 contains the formula =J7+B8-B7. These constraints and the others in column F must be set Ú the activity times in column G. The objective function formula in cell B16 is shown on the formula bar at the top of the spreadsheet. The crashing goal of 30 weeks is included in cell B15. The problem in Exhibit 8.14 is solved by using Solver,as shown in Exhibit 8.15. Notice that there are two sets of decision variables, in cells B6:B12 and in cells J6:J13. The project crashing solution is shown in Exhibit 8.16. Exhibit 8.14 Exhibit 8.15 141 Exhibit 8.16 Activities/Assessments 1. What is a project? 2. Why do you need to manage a project? 3. As future managers/accountants, what do you think will be your role in project management? 142