Presented by: Meysam rahimi Project crashing Presented by: Meysam rahimi

Problem statement: A town council wishes to construct a small stadium in order to improve the services provided to the people living in the district. After the invitation to tender, a local construction company is awarded the contract and wishes to complete the task within the shortest possible time. All the major tasks are listed in the following table. The durations are expressed in weeks. Some tasks can only start after the completion of certain other tasks.

Problem statement: The town council would like the project to terminate earlier than the time announced by the builder To obtain this, the council is prepared to pay a bonus of BC 30K for every week the work finishes early. The builder needs to employ additional workers and rent more equipment to cut down on the total time. In the preceding table he has summarized the maximum number of weeks he can save per task (column "Max. reduct.") and the associated additional cost per week.

Problem Data:

Precedence graph of construction tasks:

Two important questions: Which is the earliest possible date of completing the construction? When will the project be completed if the builder wishes to maximize his profit?

Model formulation for question 1:

Critical path method:

Model formulation for question 2:

Crashing algorithm: To shorten a project, crash only activities that are critical. Crash from least expensive to most expensive. Each activity can be crashed until -it reaches it’s maximum time reduction -it causes another path to also become critical -it is more expensive to crash than not to crash Continue until no more activities should be crashed.

Example: Bonus=$1400

Example cont. ABD 18 ACD 19 ACE 20

Example cont. E is least expensive to crash. we save $1400 per day the project is shortened and would spend $700 per day to crash E, so it is profitable to crash E. E has maximum time reduction of 3, but if it is crashed by 1, then ACD also becomes a critical path. So we should crash E by 1 period.

Example cont. ABD 18 18 ACD 19 19 * ACE 20 * 19 *

Example cont. Now we have 3 choices to crash: Crash A with the cost of $1000 Crash C with the cost of $2500 Crash both D&E with the cost of $3700 Crashing A is less expensive and laso is profitable because of being less than $1400.

Example cont. We can reduce A to its minimum duration, because no new task will become critical by the crashing of A.

Example cont. To continue, we could crash C or both D and E. But in each case, the cost would be greater than the $1400 savings per day. So, we stop at this point.

Uncertainty in activity durations: In the real word, we may encounter with uncertainty in activity durations. There are three main approaches to deal with uncertainty in activity duration: Fuzzy logic Robust optimization Stochastic optimization

Project scheduling, stochastic approach:

Distribution of completion time:

Expected completion time approximation method: Traditional method: PERT(Program evaluation and review technique) Converts the stochastic model to a deterministic model Then uses CPM to solve it. Is a very optimistic method Upper and lower bounds Monte Carlo simulation

Monte Carlo simulation: Generate random activity durations Find the longest path by CPM the average of completion time in samples is an unbiased estimator for the expected value of completion time. The standard deviation of sample is also an unbiased estimator for

Distribution of completion time example: 6 13 8 7 5 14

Simulation results: PERT estimation:21 Simulation estimation: 21.68

Activity crashing in stochastic networks:

Activity crashing in stochastic networks: Because we don’t have the distribution of completion time, we may get into trouble with the problem. All the methods which has been presented to solve this problem are heuristic methods most critical first Least expensive first Combination of two above methods BBI

Optimization with embedded simulation: Can be used with improvement algorithms Steepest descent Newton method SQP Uses simulation for evaluation of objective function and finding the search direction for the next iteration.

Optimization with embedded simulation: Start with a feasible point Simulation Evaluation of current solution Optimization Finding next solution

Any questions?