Outline

1. Outline • Questions? Comments? • Discuss homework • Discuss Quiz • Give Assignment • Conclude scheduling • Session – Chapter 9 • Crashing • Resource allocation problem • Resource Loading • Resource leveling • Constrained resource scheduling • Multi-project scheduling and resource allocation • Goldratt’s critical chain

2. Project Scheduling - More definitions(cont) • Backward Pass • Reverse topological order • Free slack = scheduling flexibility with respect to its immediate successors

3. Project Scheduling - Backward Pass

4. Project Scheduling • Free slack - scheduling flexibility with respect to its immediate successors • FSij = min [ ES of all immediate successors] - EFij • FSA= min [ESD, ESC] - EFA = min[2, 2] - 2 = 0 • FSB= ESE - EFB = 4 - 3 = 1 • FSC= ESF - EFC = 7 - 7 = 0 • FSD= ESE - EFD = 4 - 4 = 0 • FSE= ESF - EFE = 7 - 6 = 1 • FSF = 9 - 9 - 0

5. Project Scheduling • Total Slack - scheduling flexibility relative to the project completion time • TSij = LSij - ESij = LFij - Efij • TSA = 0 - 0 = 0 • TSB = 2 - 0 = 2 • TSC = 2 - 2 = 0 • TSD = 3 - 2 = 1 • TSE = 5 - 4 = 1 • TSF = 7 - 7 = 0 • Note that the activities on the critical path have 0 total slack

6. Question – Chapter 8 • Professor Bottlik, have you ever utilized the “as-late-as-possible” approach for starting tasks described in the book? • I tried to avoid it – it is best to let slack use itself up, rather than helping it along • CY

7. Statistics Review • Distributions • All measurable things vary, even if we assume that they are constant. This is why we call them random variables. • A random variable can be described by its mean and its standard deviation and the shape of its distribution • Most natural phenomena are normally distributed. The normal distribution extends to plus and minus infinity, so it is not useful for variables that have definite minima and maxima • The beta distribution does have these cutoffs.

8. Statistics Review - Continued • We specify the beta distribution by its minimum, maximum, and two parameters, usually denoted by alpha and beta. In the equation below, we use nu1 and nu2: • Excel uses alpha and beta and allows intervals other than 0,1.

9. Statistics Review - Continued • The mean and standard deviation of the beta distribution can be expressed in terms of its parameters: • So it is possible to find (by trial and error), the parameters from a mean and a standard deviation

10. Beta Distribution with max and min • http://www.me.utexas.edu/~jensen/ORMM/omie/computation/unit/project/beta.html

11. Beta Distribution with max and min • For project analysis we may be given the mode and require values of the shape parameters, alpha and beta, to specify the Beta distribution. Formulas for two cases are below. In each case we must choose one parameter and solve for the other.

12. Statistics Review - Continued • One other very important statistical fact that we need is the central limit theorem: • 1. The distribution of the mean of a normal population (with standard deviation s) will be distributed normally with standard deviation s/sqrt(n), where n is the sample size • 2. If n is large enough this will be true even if the population is not normally distributed • This allows us to assume that the completion time of a project is normally distributed

13. Statistics Review - Continued • One more statistical fact: • When adding up distributions: • 1. The mean of the sum is the sum of the means • 2. The variance of the sum is the sum of the variances • This allows us to get a mean and a standard deviation of the critical path of a project • Note: The standard deviation is the square root of the variance.

14. Statistics Review - Continued • The standard normal distribution(z) is tabulated in all statistics books, but you must be careful to ascertain the exact meaning of the tables. You map back and forth from your variable (x) to the standard table with the equation: • z = (x - xbar)/s, where s is the standard deviation • I have reproduced a normal table for you on the following page. Here the probability is between z =0 and z.

16. PERT • Probabilistic methods • Instead of one duration, assume a worst, most likely, and a best possible value. • You can, of course, use other ways of approximating the distribution of the duration time. • The beta distribution is the most popular for this because it can be shaped to one’s liking and has a definite minimum and maximum • The normal distribution is not a good choice because in simulations it could yield very short or very long processing times as it is not limited at either end

17. PERT - continued • If you assume the three values, you can then estimate the mean and the variance by (a simplification of the beta distribution):

18. PERT - Example • Continuing with our previous example: • Our critical path ACF still has average length of 9, but with a standard deviation of 0.8 • If we look at path ADEF, the average is 8, with a standard deviation of 1.92

19. PERT - Example (continued) • Calculating the probability of a completion time of 10 or less for each path: • ACF: z =(10-9)/0.8 = 1.2 P(<=10) = 0.89 • ADEF: z =(10-8)/1.92 = 1.04 P(<=10) = 0.85 • That is, the “shorter” path is more likely to cause a delay

20. PERT - Example (continued) • If we applied the Monte Carlo technique to this problem 5 times: • Criticality Indices:

21. Resource Allocation • Assume a single resource, that is, people for each task:

22. Resource Allocation • By delaying tasks D and E to their latest start time, we can level the resource usage somewhat:

23. Resource Allocation • Critical Path - Crashing • Resource • Allocation • Loading • Leveling • Constrained Resource Scheduling • Multi-project Scheduling • Goldratt

24. Critical path - crashing • In CPM, one can specify two time/cost pairs – normal and crash • For example 5 days at $1000/day or 3 days at $2000/day • Resource availability must be considered • Expediting tends to create problems • Cost/Time slope =(crash cost – normal cost)/(crash time – normal time) • For the above example = $1000/(-2) = -$500/day • Frequent discontinuities in time and cost

25. Crashing • Start with the critical path • Crash selected activities, one at a time at minimum additional cost

26. Input • When one might chooses to crash a project? Is crashing a project common in the industry? Los Angeles earthquake example was an emergency situation. Do these kinds of situations arise in the industry too often? • As a due date for any activity approaches, people realize that they will have difficulty meeting it and attempt to accelerate their efforts. (As I sitting here this morning getting ready for tonight’s class) Thus, almost every project is crashed, even if people are unfamiliar with the term or the concepts.

27. T. • I have never worked on a project that was crashed, but it would seem that by rushing aspects of the project, quality would suffer. How should that risk be mitigated? In a crashed schedule there is no room for error. How do you drive down the error rate, and speed up schedule when majority of the resources involved are people, not hardware? • I think that you have correctly identified the effects of crashing – it is intended to meet a due date at the expense of cost and specifications

28. More on crashing • I believe that crashing, resource loading, and resource leveling are all very prevalent in the consulting industry, at least based on my own personal experience.  I have worked on many projects where consultants or subcontractors are brought onto the project in the midst of the project lifecycle.   • As a result, this creates some inefficiencies since these resources must deal with a steep learning curve and must adapt to a new, emerging/evolving work environment.  I think almost all projects, however, are crashed.   There seem to always be unexpected issues, new requirements, etc. which provoke this action to take place.   • Another way I have witnessed leveling/loading is that consultants are told suddenly that deliverables have changed, for instance, and we are now told to bill more hours to the client in order to complete these new tasks/requirements.  

29. Fast tracking • Overlapping of activities – allow a successor to start before a predecessor is complete • Results • Possibly an earlier completion • Increased change orders • Loss of productivity • Increased cost • Loss of time • Use when early steps are fairly routine

30. The Resource Allocation problem • Resource must be described specifically by • Individual labor • Specific machines • Specific facilities • Specific materials • Since time cannot be inventoried or renewed, the timing of the use of resources must be specified. It is not sufficient to say that I need a total of 50 hours this months and have a 160 available. • Almost all trade offs involve additional costs • Two types: Time limited or resource limited

31. Resource loading and leveling • Amount of a specific resource required by a project during a specific time (e.g. I am required for this class from 6 P.M. to 9:10 P.M. on Thursdays) • The example I used last week is repeated in the next two slides – it assumes that the resources are interchangeable and can be measured by a common unit – Grocery store check out clerks would be one example • Large fluctuations in utilization are undesirable • Leveling aims to reduce the variations by shifting tasks within their slack allowances • Leveling resource usage also levels cash flow • Do not schedule beyond 85 to 90% capacity due to uncertainties – the book implies that this is not necessary for manufacturing but experience says otherwise!

32. Resource Allocation • Assume a single resource, that is, people for each task:

33. Resource Allocation • By delaying tasks D and E to their latest start time, we can level the resource usage somewhat:

34. Input On resource leveling, the authors casually mentioned that engineers are hired to work 40 hours per week but they usually work 50-60 hours per week without extra pay. The authors then used 55 hours per week to calculate the engineering labor-hours capacity and declared the resultant number sufficient for the example. I hope none of my managers read this section. It is bad enough engineers have to work overtime periodically to recover from falling behind. It is unbelievable that a project manager would use 55 hours per week as the standard to calculate resource loading. If the project falls behind are the engineers expected to work 70-80 hours per week now to catch up? In a responsible organization one might schedule overtime, but only if it is paid. Scheduling more than 35 hours of billable work a week per person is foolish. The author does suggest that compensatory time can also be used

35. T. • It looks like a tradeoff of resource leveling is the elimination of some portion of slack. How do you know the right balance? You want to level resources, but you also still need to “always be prepared” for the unexpected. • I would use the resource loading as a guide – up to 80% max when using up slack. • If the resource is leveled and the critical path must be adjusted, I would use 90%

36. Heuristics and Optimization • Heuristic – accepted rules of thumb approaches that are believed to yield good results • Optimization –mathematical techniques that yield optimal solutions • Large, non-linear, complex problems can generally not be solved optimally in a reasonable amount of time • In effect, most heuristics are simulations based on priority rules: • When faced with a choice of which task to assign to a scarce resource, the choice is made according to a priori rules or on randomized choices of rules or randomly

37. Common priority rules • As soon as possible • As late as possible • Shortest first • Most resources first • Least slack first • Most critical followers • Most successors • Random • Higher priority project by definition or value to the client or organization • (The authors report least slack as a good rule – In job shop scheduling it performs poorly)

38. Heuristics • Simulation will end when: • All activities are scheduled • Runs out of resources before all activities are scheduled • Simulations are frequently run in reverse – back from the due date. If the latest start is before the current time, the project cannot be done on to time

39. Goldratt • Known for Theory of Constraints – One activity controls the throughput of a system • Problems: • Project’s scope is changed without consultation or warning, without change in schedule or budget • Due dates are set without regard to resources • Impossible to achieve objectives within budget • Work load and due dates set externally without regard to the nature of the project and resources required • Due dates are used as an incentive

40. Goldratt (cont) • What creates optimistic schedules? • Optimism without thinking • Setting capacity near demand • Postponing to start to latest start • Switching between jobs • Ignoring variation due to complexity • Assuming people will work harder under impossible targets • Padding estimates and responding by deleting requested resources

41. Goldratt (cont) • Do early finishes cancel out late ones? • Not usually, because people will postpone the start of successors of predecessors that finish early

42. Input – “Student Syndrome” • Goldratt describes the “student syndrome”, and states that it is common for activities with high slack to be delayed until the slack is gone. Based on your experiences, could this theory be used to justify scheduling project completion for an extremely optimistic time duration, in effort to prevent expending all of the project’s slack? • I think you mean pessimistic- that way the duration will take the place of the slack. I don’t think it will work because most people see through the ruse and delay the start anyway. Goldratt does suggest doing this • “Never do today what you can postpone to tomorrow”

43. Goldratt – critical chain • Create buffer time at end of the project • Create buffer in front of the critical resource (keep it working) • Order tasks with resource dependencies sequentially

44. My last project (Dec 2007)