Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 1 Outline Questions? Exam results – very good 19 grades improved 6 grades decreased 18 stayed the same Class GPA increased by 0.05 No official homework, but do exercises of your own Quiz on 11/24 Project scheduling

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 2 Exam Results

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 3 Project Scheduling Intermittent systems Examples: Construction of a plant Aircraft carrier Large airplanes Complex and large, thousands of tasks and interdependencies Objectives:Complete on time Minimize cost Minimize time Meet customers’ requirements

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 4 Project Scheduling Analyze -- Plan -- Schedule -- represented as a network of activities Most used: PERT - Program Evaluation and Review Technique CPM - Critical Path Method

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 5 Project Scheduling PERT – Activities on arc CPM – Activities on nodes

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 6 My first PERT experience

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 7 Project Scheduling Predecessor and successor relationships between activities If there is no such relationship, the activity is independent Durations are independent

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 8 Project Scheduling Activity = Task = Job Has a beginning and an end Has a duration = elapsed time = process time Uses resources

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 9 Project Scheduling Planning and scheduling steps Identify activities Precedence constraints Construct the network Estimate durations Assign starting times Analyze resources Once project starts, check progress against plan Reschedule

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 10 Project Scheduling - Networks Network = directed graph Finite number of nodes (n) i,j, ….. = N subset of ordered pairs (i,j) = arcs = A To draw a network: from each i of N, draw arrow to j, if (i,j) is in A where arrow = (i,j) or name of activity i - starting event j - ending event j end i Start Activity

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 11 Project Scheduling - Networks (cont) Rules: The length of the arrow has no significance At a node, the outgoing activity cannot start until all incoming activities are complete A B C D

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 12 Project Scheduling - Networks (cont) Rules (continued) Only one initial node (no predecessors) and only one terminal node (no successors) An activity is uniquely identified by start and end events -no duplicate node numbers -at most one arrow between nodes For every arrow (i,j) such that i <j No closed loops!

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 13 Project Scheduling - Networks (cont) Multiple paths can be avoided with dummies Dummy

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 14 Project Scheduling - Network example

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 15 Project Scheduling - Example - List List activity only if its predecessor is complete - nondecreasing i or j numbers - topological order

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 16 Small example - Activity on Node (AON)

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 17 Project Scheduling – Activity on Arc (AOA) B - E - F = 7 A - D - E - F = 8 A - C - F = 9 Critical path, similar to bottleneck idea We’ll generate all possible schedules to get the concepts A, 2 B, 3 D,2 C, 5 F, 2 E, 2

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 18 Project Scheduling - another example (cont) All activities start as soon as possible( plotted to scale): Use up total slack of E Use up free slack of B Use up both

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 19 Project Scheduling - another example (cont) Use up total slack of E and D If we move B to the right by one unit, we will have used up all slack and everything starts at the latest start time

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 20 Project Scheduling - More definitions Earliest start time ES ij or ES A delayed = start after earliest start time Latest start time LS ij or LS A Delay without affecting start of successors = free slack = Fs ij Delay that affects start of successors - Total slack - TS ij Free slack <= Total slack Critical activities have the least total slack, usually 0

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 21 Project Scheduling - More definitions(cont) EF = Earliest finish LF = Latest finish Y = duration T i = earliest occurrence of node i Forward pass to determine ES Topological order - a task is listed only if all its predecessors have been listed

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 22 Project Scheduling - Forward Pass

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 23 Project Scheduling - More definitions(cont) Backward Pass Reverse topological order Free slack = scheduling flexibility with respect to its immediate successors

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 24 Project Scheduling - Backward Pass

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 25 Project Scheduling Free slack - scheduling flexibility with respect to its immediate successors FS ij = min [ ES of all immediate successors] - EF ij FS A = min [ES D, ES C ] - EF A = min[2, 2] - 2 = 0 FS B = ES E - EF B = = 1 FS C = ES F - EF C = = 0 FS D = ES E - EF D = = 1 FS E = ES F - EF E = = 1 FS F =

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 26 Project Scheduling Total Slack - scheduling flexibility relative to the project completion time TS ij = LS ij - ES ij = LF ij - Efij TS A = = 0 TS B = = 2 TS C = = 0 TS D = = 1 TS E = = 1 TS F = = 0 Note that the activities on the critical path have 0 total slack

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 27 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.

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 28 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.

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 29 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

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 30 Beta Distribution with max and min n/unit/project/beta.html

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 31 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.

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 32 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

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 33 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.

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 34 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 and xbar is the average I have reproduced a normal table for you on the following page. Here the probability is between z =0 and z.

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 35

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 36 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

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 37 PERT - continued If you assume the three values, you can then estimate the mean and the variance by (a simplification of the beta distribution):

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 38 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

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 39 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

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 40 PERT - Example (continued) If we applied the Monte Carlo technique to this problem 5 times: Criticality Indices:

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 41 Resource Allocation Assume a single resource, that is, people for each task:

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 42 Resource Allocation By delaying tasks D and E to their latest start time, we can level the resource usage somewhat:

Session 25 University of Southern California ISE514 November 17, 2015 Geza P. Bottlik Page 43 Project Scheduling - References Morton and Pentico, pages 425 to 503 Kerzner, “Project Management”, 5th Ed. ITP 1995, pages (out of 1152!) Hax and Candria “Production and Inventory Management”, Prentice Hall 1984 Pages 325 to 359