Project Management - PERT/CPM What is project management? Consider building a house: Step A: Prepare site. (5 days) Step B: Build foundation. (8 days) Step C: Frame walls and roof. (15 days) Step D: Rough in Plumbing (12 days) Step E: Rough in Electrical (10 days) Step F: HVAC Venting (8 days) Step G: Drywall (11 days) Step H: Finish Electrical (5 days) Step I: Finish Plumbing (4 days)Step M: Paint (5 days) Step J: Finish HVAC (2 days)Step N: Landscape (5 days) Step K: Install Kitchen (8 days) Step L: Install Baths (14 days)

Project Management - PERT/CPM Let each node represent a project event/milestone (node 1 is start of project, node 11 is end of project). Let each arc represent a project task/job. Each arc is identified by a job letter and duration. Note the dummy jobs indicating precedence that jobs H and I must complete before K or L begins. 2 A, B,8C,15 D,12 F,8 E,10 G,11 H,5 I,4 J,2 K,8 L,14 M,5 N,5 0 0

Project Management - PERT/CPM What questions might project managers be interested in? How long will the project take? Can I add manpower or tools to reduce the overall project length? To which tasks should I add manpower? What tasks are on the critical path? Is the project on schedule? When should materials and personnel be in place to begin a task? Other?…

Project Management - Examples University Convocation Center Windsor Engine Plant Other major construction projects Large defense contracts NASA projects (space shuttle) Maintenance planning of oil refineries, power plants, etc… other…

Project Management – Minimum Completion Time 2 A, C,4 D,2B,1 E,5 0 LP Solution:Let t i be the time of event i. Min Z = t 5 – t 1 s.t.t 2 – t 1 >= 3 t 3 – t 2 >= 0 t 3 – t 1 >= 1 t 4 – t 2 >= 4 t 4 – t 3 >= 2 t 5 – t 4 >= 5 t i >= 0 for all i

Project Management – Critical Path 2 A, C,4 D,2B,1 E,5 0 LP Solution:insert Lindo Solution here How do you find the critical path from the Lindo solution?

Project Management – Minimum Completion Time and Critical Path 2 A, C,4 D,2B,1 E,5 0 Solution by Network Analysis: Let earliest time of node j, U j, be the earliest time at which event j can occur. Set U 1 = 0 thenU 2 = U 1 + t 12 = = 3 U 3 = Max{U 1 + t 13, U 2 + t 23 } = Max{1,3} = 3 U 4 = Max{U 3 + t 34, U 2 + t 24 } = Max{5,7} = 7 U 5 = U 4 + t 45 = 12

Project Management – Minimum Completion Time and Critical Path 2 A, C,4 D,2B,1 E,5 0 Solution by Network Analysis: Let latest time of node j, V j, be the latest time at which event j can occur while still completing project by minimum the minimum completion time, U m. Set V 5 = U 5 = 12 thenV 4 = V 5 - t 45 = = 7 V 3 = V 4 - t 34 = 7 – 2 = 5 V 2 = Min{V 4 - t 24,V } = 3 V 1 = Min{V 2 - t 12,V 3 – t 13 } = 0

Project Management – Minimum Completion Time and Critical Path 2 A, C,4 D,2B,1 E,5 0 Solution by Network Analysis: To find the critical path, solve for slack time = V j - U j. All events with slack time equal to 0, and tasks connecting these events are on the critical path. V 5 - U 5 = 12 – 12 = 0 V 4 - U 4 = 7 – 7 = 0 V 3 - U 3 = 5 – 3 = 2 V 2 - U 2 = 3 – 3 = 0 V 1 - U 1 = 0 – 0 = 0 Critical Path: 1->2->4->5

CPM – Critical Path Method Can normal task times be reduced? Is there an increase in direct costs? Additional manpower Additional machines Overtime, etc… Can there be a reduction in indirect costs? Less overhead costs Less daily rental charges Bonus for early completion Avoid penalties for running late Avoid cost of late startup CPM addresses these cost trade-offs.

CPM – Critical Path Method Example: Overhead cost = $5/day

CPM – Critical Path Method Enumerative Approach: Reduce job H by 1 day: Total Cost improves by $5 - $4 = $1. Reduce job A by 2 days: Total cost improves by $10 - $8 = $2. Reduce job A by an additional day, and job B by a day? Total cost improves by $5 - $4 - $2 = -$1. Therefore do not take this action. Reduce job A by an additional day, and job C by a day? Total cost improves by $5 - $4 - $2 = -$1. Therefore do not take this action. Evaluate combinations of reducing path and by one day. D & E = $5 - $3 - $3 = -$1F & E = $5 - $5 - $3 = -$3 D & G = $5 - $3 - $1 = $1F & G = $5 - $5 - $1 = -$1 Therefore, reduce job D & G by 1 day: TC improves by $5 - $3 -$1 = $1. Overall improvement: $1 + $2 + $1 = $4.

CPM – Critical Path Method LP Approach: Let t ij – decision variable for time to complete task connecting events i and j. k ij – normal completion time of task connecting events i and j. l ij – minimum completion time of task connecting events i and j. C ij – incremental cost of reducing task connecting events i and j. Model I: Given project must be complete by some time T, which tasks should be reduced to minimize the total cost? Min s.t. for all jobs (i,j) for all i

CPM – Critical Path Method LP Approach: Model II: Given an additional budget of $B for “crashing” tasks, what minimum project completion time can be obtained while staying within your budget? Min s.t. for all jobs (i,j) for all i