Terminology Project: Combination of activities that have to be carried out in a certain order Activity: Anything that uses up time and resources CPM: „Critical Path Method“ PERT: „Project Evaluation and Review Technique“

PERT/CPM 4 Phases: Planning of activities Sequencing of activities Time, resource and cost management Controlling

Planning/Sequencing of Activities Representation of activities using arcs Representation of events using nodes An „event“ is a point in time where some activities end and others begin. Each activity is uniquely determined by its start event and its end event.

Planning/Sequencing of Activities Rules for setting up PERT/CPM networks: Rule 1: Each activity is represented by one and only one arc. Rule 2: For each pair of nodes, there is at most one activity connecting these nodes (if necessary, use dummy activities to express precedence relations)

Planning/Sequencing of Activities Rules for setting up PERT/CPM networks: Rule 3: To ensure correct modelling of precedence relations, always consider the following when adding an activity to a PERT/CPM network: a) Which activities must end exactly before the new activity can begin? b) Which activities immediately follow the new activity? c) Which activities can be carried out simultaneously?

Example Set up a PERT/CPM network consisting of the activities A-L, such that the following conditions are satisfied: 1. A, B and C (the first activities of the project) may start simultaneously. 2. A and B precede D. 3. B precedes E, F and H. 4. F and C precede G. 5. E and H precede I and J. 6. C, D, F and J precede K. 7. K precedes L. 8. I, G and L are the terminal activities of the project.

Example Indicate how each of the following additional relations can be incorporated in the network 1. A and B precede G. 2. D precedes G. 3. C precedes D.

Critical Path Calculations Forward pass: compute earliest times for events Backward pass: compute latest times for events Compute slacks and find critical path Activities on the critical path are referred to as „critical activities“, all others as „uncritical activities“

Critical Path Calculations E i...earliest time for event i, i.e., the earliest starting time for all activities rooting in i (usually: E 0 =0) t ij... Duration of activity (i,j) Forward pass: for all activities ending in j:

Critical Path Calculations L i...latest completion time of all activities ending in node i (usually L n =E n, with n the last event in the project) Backward pass: for all activities starting in i:

Critical Path Calculations The slack for an event is the difference between its latest and its earliest times. The slack for an activity (i,j) is L j -E i -t ij. A critical path for a project is a path through the network such that all the activities on this path have zero slack.

Example Consider the following project network. Assume that the time required for each activity is a predictable constant and that it is given by the number along the corresponding arc. Find the earliest time, latest time and slack for each event as well as the slack for each activity. Also identify the critical path!

Properties of Critical Paths A project network always has a critical path, sometimes there is more than one. All activities having zero slack must lie on a critical path, whereas no activities having slack greater than zero can lie on a critical path. All events having zero slack must lie on a critical path, whereas no events having slack greater than zero can lie on a critical path. A path through the network such that the events on this path have zero slack need not be a critical path (because one or more activities on the path may have slack greater than zero!).

PERT-Adding Uncertainty to Project Networks In reality, activity durations are not deterministic For each activity, three estimates are provided in PERT: a) most likely estimate b) optimistic estimate c) pessimistic estimate Additional assumptions: A 1: The spread between a and b is 6 standard deviations. A 2: The probability distribution of each activity time is (approximately) a beta distribution.

PERT-Adding Uncertainty to Project Networks These assumptions lead to the following formulae for expected value and variance of activity times:

PERT-Adding Uncertainty to Project Networks To calculate expected value and variance of the project time, additional assumptions have to be made: A 3: The activity times are statistically independent random variables. A 4: The critical path (calculated using expected times) always requires a longer total elapsed time than any other path. Using these assumptions, expected project time equals the sum of the expected activity times along the critical path. The variance of the project time is the sum of the variances of the activity times on the critical path.

PERT-Adding Uncertainty to Project Networks To make probability estimates for events occurring before a specified time, one additional assumption has to be made: A 5: The probability distribution of project time is (at least approximately) a normal distribution. The larger the project network, the more reasonable this assumption will be!

Example Consider the following project network. The PERT three-estimate approach has been used, and it has led to the following estimates of the expected value and variance of the time required for the respective activities: The scheduled project completion time is 22 months after the start of the project. a) Using expected values, determine the critical path for the project. b) Find the approximate probability that the project will be completed by the scheduled time. c) In addition to the critical path, there are five other paths through the network. For each of these other paths, find the approximate probability that the sum of the activity times along the path is not more than 22 months.

CPM-Adding Time-Cost Tradeoffs Two points are provided for each activity in CPM: a) The normal point (activity time and cost under normal circumstances) b) The crash point (activity time and cost when activity is carried out as fast as technically possible) Additional assumption: Linear time-cost relation

CPM-Adding Time-Cost Tradeoffs Notation: D ij...normal time for activity (i,j) C Dij...normal (direct) cost for activity (i,j) d ij...crash time for activity (i,j) C dij...crash (direct) cost for activity (i,j) LP formulation: x ij...time for activity (i,j) Slope of cost function for activity (i,j):

CPM-Adding Time-Cost Tradeoffs LP formulation: y k...(unknown) earliest time for event k (which is a deterministic function of x ij )

Example Consider the following project network. The CPM method of time-cost trade-offs is to be used to minimize the cost of completing the project within 15 weeks. The relevant data (in weeks and thousands of dollars) are as follows: Formulate a linear programming model for this problem!