Engineering Finance

The Model

Past lessons have focused on cash flows. We have considered methods for estimating costs and revenues that occur throughout a project. We have used these estimates to decide whether or not to proceed with a project, and which of several alternative solutions to choose. Our decisions were based on economic criteria. Time has played an important role because of the time value of money. The economic evaluation considers time through the measures of NPW, NAW and IRR. In most of the previous lessons the timing of events is given. Thus we prescribe when investments occur, how long the project will last, and when costs and revenues are realized.

In this and the following lessons, the timing of events is the major question. We consider primarily the construction or installation phase of a project. It might be the construction of a building, the design of a product or the programming of new software, or any of a variety of tasks that have a fixed goal and a collection of activities required to meet the goal. The problem addressed in this section is to schedule the activities of the project over time.

As an example, a builder has the task of constructing a building to meet the specifications of a design provided by an architect. The builder identifies a set of activities that must be performed to accomplish this goal such as clear the land, lay access roads, pour the foundation, erect the walls, and so on. Each of the activities requires time, money and resources. Often there are precedence relations that restrict the start of some activity until others are completed. For example, the erection of the walls cannot begin before the foundation is ready.

Before the project begins, the builder needs to schedule the activities involved. The primary purposes of a schedule are to:

  • predict when the project will be completed
  • predict when a subcontract should be let to perform an activity
  • determine when limited resources should be obtained
  • predict the cash flow over time for project expenditures and revenue

After the project begins, it is necessary to track the performance of the various activities to assure that the project is on schedule and on budget. Some activities may require more time than originally estimated, while others may require less. Resources thought to be available may be delayed. Any change from projected values might require a new schedule to be prepared or corrective actions to be taken.

This lesson describes the model we use for project scheduling. The model includes a list of activities that comprise the project, a list of the immediate predecessors for each activity, and estimates of the time, money and resources required for each activity. Of course an actual problem may not exactly fit this model, but it is the one we use for description and analysis. Most introductory and educational discussions make the same simplifications as the ones described here, but commercial project scheduling software has features that describe more complex situations.

Because of the limited time available for the class, not all the topics alluded to below are required for the course. See the syllabus for a list of the required topics.

Browser Page
  • Given a word description of a project, identify the activities and the immediate predecessors of each activity. Also identify resources used by the project and cash flows associated with each activity.
  • Construct an activity-on-node network describing the project.
  • For a given project, use the Project Management add-in to find the critical path and describe the associated schedule with activity start and end times.
  • Use the Project Management add-in to find schedules that satisfy restrictions on resources. If it is impossible to satisfy restrictions, find a solution that minimizes some measure of infeasibility.
  • For a given project and schedule, use the Project Management add-in to find the cash flow associated with the schedule.

Chapter 9 supports several lessons in this part of the course. Section 9.1 introduces the project scheduling problem, provides a historical perspective and relates the subject of the chapter to the work breakdown schedule (WBS). Read Section 9.2 about estimating activity durations. We have used some of these techniques earlier for estimating costs. For this and the next few lessons we assume durations are fixed, but we later consider durations modeled as random variables. Section 9.3 is about the learning effect. You should able to apply these methods (used earlier for costs) to time estimation for WBS activities. Precedence relations, discussed in Section 9.4 are of primary importance for project scheduling.

9.1-9.4 Project Scheduling

The discussion below mentions Chapters 10, 11 and 12 from the text. These readings are not required, but the references provide a more complete discussion of topics relevant to Project Management than the lessons on this site.


For homework problems we will use the Project Management add-in to simplify computations and address problems of meaningful size. Students should become familiar with all the aspects of the add-in as discussed in the documents reached by clicking the link below. Many of the graphics in this section are screen shots of Excel spreadsheets constructed by the add-in.

Project Management Add-in

The model for project scheduling must include all the physical tasks necessary to complete the project. Earlier we described the WBS that breaks down a project into work packages (WP). Necessary characteristics of the WBS is that WPs cannot overlap in function and that they must collectively describe everything that must be done to complete the project. When discussing project scheduling we usually use the term activities to describe the work contents of a project; however, sometimes we use the word tasks. In our discussions we use the terms WP, activities or tasks interchangeably. The important thing is that the activities do not overlap in function and they collectively describe all the work that must be done.

To illustrate we use an example from earlier in this course called the Oil Collection Problem.

. Oil Collection Problem
Oil Collection Problem

The first step in the analysis is to create a list of activities that comprise the project. We list them below and assign an alphabetic designation to each. The list will always include two artificial activities: a Start activity that precedes all other activities, and an End activity that follows all other activities.

The set of activities must describe all of the steps necessary to complete a project. Activities cannot overlap in function.

Precedence Relations

The oil collection narrative describes various ordering restrictions in terms of which activity must precede another or which activity should follow another. An obvious example is that the "Pour Slab" activity that must precede the "Cure Slab" activity. These are called precedence relations.

The predecessors of an activity must be completed before the activity can begin.

The predecessors for the example are given below. Some of the activities have more than one predecessor.

It is only necessary to identify immediate predecessors. Although we see that D (Cure Slab) and E (Deliver Pump) precede H (Mount Pump), it is also true that the predecessors of D and E, activities A (Pour Slab) and B (Order Pump), also precede H. We do not list A and B as predecessors of H because these relations are implied by the other entries. Although the analysis does not fail if other than immediate predecessors are listed, it is inefficient to do so.

Indirectly, the table also shows successors of the activities. Whenever an activity appears in the list of predecessors, the rows in which the name appears identify the successors of the activity. For example, we see C (Cut Path) listed as a predecessor of F (Run Flow Line) and G (Run Electric Line). This implies that the immediate successors of C are activities F and G.

The successors of an activity cannot begin until the activity is completed.

Although not shown, this table implies relations between the start and end activities as well as the remaining activities. An activity that has no listed predecessors (A, B and C) is an immediate successor of the start activity. Similarly, an activity that has no listed successors (M and N) is an immediate predecessor of the end activity.

Project Network

The list of activities and precedence relations describes the structure of the problem. An equivalent description of the project is the project network. Click the icon below to see the project network for the example problem. The circles, called nodes, represent the activities. The lines connecting the nodes represent the precedence relations and are called arcs. The graphical aid of the project network is helpful when building the project description and for checking the validity of precedence relationships.

Project Network

To better understand the network we often use a simplified version that shows the names rather than the descriptions. In the next figure, the entries below the names are the activity times, discussed in the next paragraph, and the red nodes and arcs show the critical path, also described later. This is the format of the network constructed by the add-in.

Simplified Project Network

This form of the network is called the activity-on-node network. Another form is called the activity-on-arc network. We choose the former because it is easier to construct and is used by the Project Management add-in. The process of constructing this network is called networking, and it is an important part of project management.


Time is the primary consideration of project scheduling. The activity time or duration is the time required from the moment the activity begins to the moment when it ends. Since the analysis takes place before the project begins, the time required for an activity must be an estimate. Of course when the activity is actually performed the time may turn out to be different than the estimate. The estimate times for our example are in the column labeled Time in the chart below. The time measurement in this case is days.

A schedule is a prescription of start times for each activity. A feasible schedule is one where each activity begins after all its predecessors are complete. A feasible schedule is shown in the last two columns of the above table. The scheduled finish time for an activity is the scheduled start time plus the activity duration. For the example, we see that A, B and C all start at time 0. Activity D starts at time 2, immediately after its predecessor A is complete. Activity E starts at time 1 after its precedence B is complete. Activity H, with predecessors D and E, starts at time 5 when both D and E are complete. Although D finishes at time 3, activity H must wait to begin until both D and E are finished at time 5.

An alternative representation of the schedule is shown in the Gantt chart below. Each activity is depicted as a colored bar drawn from the start to the finish time. The particular schedule shown results in the project completion time of 12. This is the earliest possible time.

The goal of project scheduling is to find the schedule that results in the smallest possible completion time. The activities shown in bright red on the Gantt chart are the critical activities. We say these activities are on the critical path. They must be started exactly when specified or the project will be delayed beyond the minimum of 12 days. The activities colored in maroon may be delayed without increasing the project completion time. We address the problem of finding the critical path in the Lesson 21. Horizontal lines in the chart separate the critical and noncritical activities.

Most of the analyses in this section assume activity times are fixed constants. An alternative is to describe an activity time as a random variable with a specified probability distribution function. Lesson 22 considers the problem of estimating the critical activities when the duration times are uncertain.


Limited resources complicate the scheduling problem. In the example, a crew of fixed size is assigned to each activity as shown in the Crew column below. A limited number of workers could make a schedule infeasible.

Using the earliest time schedule given in the Gantt chart, the pattern of worker usage over time is shown in the figure below. For illustration we set the number of workers available to 8. We see that at times 1, 2, 4 and 5, the number of workers used is greater than the number available indicating a shortage in workers. The schedule is infeasible. Assigning a cost of 1 for each unit of shortage, we find that this solution has an infeasibility cost of 7.

Although the schedule is infeasible with respect to the labor resource, it may still be possible to implement it. The shortage in workers may be remedied by assigning overtime to workers or by hiring temporary workers. The shortage cost could be set to reflect overtime charges or the extra costs of temporary workers. Our goal will be to minimize shortages, but it may be impossible to eliminate them entirely. The add-in allows the project to be delayed beyond the minimum time at a prescribed cost per unit delayed. This added flexibility allows schedules that reduce shortages. Although the chart above includes 14 days of data, the example was limited to a completion time of 12 days so the last two days are irrelevant.

When resources are limited, finding a schedule that is feasible or optimal is theoretically a very difficult problem. The Project Management add-in has a heuristic search procedure that attempts to find better solutions. For the example, we obtained a better schedule which is shown below. The activities Cut Path and Run Electric Line have been delayed as indicated by the black bands.

The schedule still has the same finish time, but with a reduced infeasibility cost of 3.

When resources are part of the model, the question must be answered: how should the project be scheduled so that available resources are not exceeded? Because of time limitations, we do not answer this question in this class but the add-in provides tools for estimating resource usage as a function of the schedule. Automatic search methods look for schedules that minimize resource constraint violations.

The add-in also offers an alternative to the objective of minimizing shortages subject to fixed resource limits. The alternative objective is to level the amount of resources used and is achieved by minimizes the sum of the squares of the difference between the resources available and the amount used. This is similar to minimizing the variance of resource utilization over the project duration. The link below opens the instructions for the add-in with regard to resources.

Project Management-Resources

The textbook provides a chapter on the management of resources in project scheduling.

10 Resource Management
Cash Flow

For some projects, the cash flow is important. The table below shows the oil collection problem expanded to include cash flow data. The earliest time schedule is also shown. For a given activity,

  • The entry in the column labeled Initial C.F. is the amount expended at the start time of the activity.
  • The entry in the column labeled Final C.F. is the amount expended at the end time of the activity.
  • The entry in the column labeled Uniform C.F. is the amount expended in each period from the start time to the end time. For the start and end times, the uniform amount is added to the initial and final amounts, respectively.

The term buckets refers to the time intervals for the cash flow analysis. For the example, each bucket is one day in length. For other problems buckets may span several time periods or even factional time periods.

To illustrate the cash flow data consider the Pour Slab activity. The initial cash flow, 1000, is the cost of obtaining materials and setting up the forms to hold the slab concrete. The uniform cost, 80, is the cost of hiring the four person crew per day. This is expended during each day of pouring, 2 days for the example. The final cost, 200, is the cost of removing the forms and cleaning up the area. Because the pour slab operation requires two days, the cost of the first day is the initial cost plus the uniform cost for one day, 1080. The cost for the second day is the final cost plus the uniform cost for one day, 280. Similar justification could be provided for each of the cost entries.

The table below shows the cash flow expenditures in each period for the example problem using the earliest time schedule. This might be important to the person in charge of the project's finances. The first three rows of the table describe the cumulative initial, uniform, and final payments. The fourth row is the total cumulative payments in each time bucket. The last row shows the cash amounts in the individual buckets. The example has activities A, B and C scheduled to begin at time 1. Thus we see that the total initial cost is 1050, (1000 + 50 + 0). The uniform cost in the first period is 220, (80 + 20 + 120). The final cost at time 1 is 0. Although activity B ends in time 1 since it only lasts for one period, its final cost is 0. The table reflects a time range of 14 days, but the schedule shown has a 12-day duration. The maximum time in the table is 15 days to capture the cumulative values for the 14-day time horizon.

The cash flow depends on the schedule. In the example, which uses the earliest time schedule, most of the costs are at the beginning of the project. Costs (and revenues) are delayed as activities are delayed. Because of the time value of money, there may be benefits in delaying activities that involve costs and moving forward activities that involve revenue. The add-in allows an objective to be defined that reflects the net present value (NPW) of the cash flow. By minimizing that criterion, the scheduling activity will reflect the financial measures that could be important. The particular example only involves costs and a small time period, but there are projects that last many years and involve large economic impacts where the financial measures are important.

In the graph below, two cumulative cash flows are shown. The cash flow for the current schedule is shown in green (it is also the early schedule which is normally shown in orange). The cash flow shown in red is for a schedule that allowed the project to be completed in 14 days. To meet this deadline, it was necessary to schedule the activities as late as possible. This allowed the activity expenditures to be delayed. In general, if you plot a schedule with delays, it will be different than the early schedule.

The Project Management add-in provides tools for estimating and evaluating cash flows.

Project Management-Cash Flow

The textbook provides a chapter on the management of budgets for project scheduling.

11 Project Budget
Project Control

Critical path analysis is a planning technique for scheduling future activities. After the project actually begins, activities start and become "in progress." Eventually activities finish. At any given time, some activities are finished, others are in progress and others have yet to begin. Finished activities may have required different durations than originally estimated, and estimates for in-progress activities may also require adjustment. The process of tracking the progress of an ongoing project and making adjustments to estimates and schedules is called Project Control. This is an extremely important in practice because projects rarely proceed as planned. We cannot provide an extensive discussion in this brief introduction, but the interested student can read further on the topic in the references below.

The Project Management add-in has a control procedure called Update. It can be used to monitor the project schedule and update time and resource estimates.

Project Management-Update

The textbook provides a chapter on the control of projects.

12 Project Control
. Browser Page
Project Schedule Model Summary

Navigation Front Page Lessons Resources

Return to Top of Page

Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

Front Page Lessons Resources