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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)
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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. J,2 H,5 7 M,5 D,12 K,8 0 A,5 B,8 C,15 G,11 E,10 11 1 2 3 4 5 6 9 10 0 L,14 N,5 F,8 I,4 8
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 A,3 C,4 E,5 1 2 4 5 0 B,1 D,2 3 LP Solution: Let ti be the time of event i. Min Z = t5 – t1 s.t. t2 – t1 >= 3 t3 – t2 >= 0 t3 – t1 >= 1 t4 – t2 >= 4 t4 – t3 >= 2 t5 – t4 >= 5 ti >= 0 for all i
Project Management – Critical Path A,3 C,4 E,5 1 2 4 5 0 B,1 D,2 3 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 A,3 C,4 E,5 1 2 4 5 0 B,1 D,2 3 Solution by Network Analysis: Let earliest time of node j, Uj, be the earliest time at which event j can occur. Set U1 = 0 thenU2 = U1 + t12 = 0 + 3 = 3 U3 = Max{U1 + t13 , U2 + t23} = Max{1,3} = 3 U4 = Max{U3 + t34 , U2 + t24} = Max{5,7} = 7 U5 = U4 + t45 = 12
Project Management – Minimum Completion Time and Critical Path A,3 C,4 E,5 1 2 4 5 0 B,1 D,2 3 Solution by Network Analysis: Let latest time of node j, Vj, be the latest time at which event j can occur while still completing project by minimum the minimum completion time, Um . Set V5 = U5 = 12 then V4 = V5 - t45 = 12 - 5 = 7 V3 = V4 - t34 = 7 – 2 = 5 V2 = Min{V4 - t24 ,V3- 0} = 3 V1 = Min{V2 - t12 ,V3– t13} = 0
Project Management – Minimum Completion Time and Critical Path A,3 C,4 E,5 1 2 4 5 0 B,1 D,2 3 Solution by Network Analysis: To find the critical path, solve for slack time = Vj - Uj. All events with slack time equal to 0, and tasks connecting these events are on the critical path. V5 - U5 = 12 – 12 = 0 V4 - U4 = 7 – 7 = 0 V3 - U3 = 5 – 3 = 2 V2 - U2 = 3 – 3 = 0 V1 - U1 = 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 3-4-6 and 3-5-6 by one day. D & E = $5 - $3 - $3 = -$1 F & E = $5 - $5 - $3 = -$3 D & G = $5 - $3 - $1 = $1 F & 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 tij – decision variable for time to complete task connecting events i and j. kij – normal completion time of task connecting events i and j. lij – minimum completion time of task connecting events i and j. Cij – 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 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 jobs (i,j) for all i
