Lecture7

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Unformatted text preview: . All tasks with zero total float are critical. 2 . All critical tasks have zero free float. 3 . All critical tasks have zero total float. 4 Statements 2-4 are all true. Statement 1 is not true by the following example: C has zero free float but 1 total float. C (8) S=[5,6] F=[13,14] START A (5) S=[0,0] F=[5,5] F (11) S=[13,14] F=[24,25] D (8) S=[5,5] F=[13,13] G (1) S=[13,24] F=[14,25] END H (12) S=[13,13] F=[25,25] J. G. Carlsson, U of MN ISyE Lecture 7: PERT & CPM October 3/17, 2013 27 / 38 Crashing Crashing an activity refers to taking special costly measures to reduce the duration of an activity below its normal value E.g. “if we pay $50,000, we can reduce the duration of activity D from 8 to 7” Naturally, it is only worth considering crashing activities on the critical path Crashing may change the critical path by reducing the duration of critical activities J. G. Carlsson, U of MN ISyE Lecture 7: PERT & CPM October 3/17, 2013 28 / 38 Crashing as an LP Suppose that each activity i can be crashed by paying pi to reduce the duration by 1 day (or whatever time unit we like) If we want to crash by xi days, then we pay pi xi We have a given deadline that we have to meet: how can we crash in the cheapest way possible? B (3) pB = 3 C (8) pC = 8 START D (11) pD = 8 E (1) pE = 4 A (5) pA = 1 0 END F (12) pF = 6 J. G. Carlsson, U of MN ISyE Lecture 7: PERT & CPM October 3/17, 2013 29 / 38 Crashing as an LP Say we need to complete by 15 days: minimize 10xA + 3xB + 8xC + 8xD + 4xE + 6xF yA yB s.t. ≥ 0 + 5 − xA ≥ yA + 3 − xB yC yD ≥ yA + 8 − xC ≥ yB + 11 − xD yD yE yF ≥ yC + 11 − xD ≥ yC + 1 − xE ≥ yC + 12 − xF yA , yB , yC , yD , yE , yF xA , xB , xC , xD , xE , xF ≤ 15 ≥0 yA , yB , yC , yD , yE , yF ≥0 Here the yi ’s represent completion times and the xi ’s represent how much we crash J. G. Carlsson, U of MN ISyE Lecture 7: PERT & CPM October 3/17, 2013 30 / 38 Optimal solution At right, a “cost-duration tradeoff curve” J. G. Carlsson, U of MN ISyE Lecture 7: PERT & CPM October 3/17, 2013 31 / 38 PERT Networks It is natural to try to take some uncertainty into account when designing project networks A PERT network estimates times based on three input values: The optimistic time a occurs if execution goes extremely well The most likely time m is just what it sounds like The pessimistic time b occurs if execution goes extremely poorly ¯ The average duration time D and the variance ν are defined by ¯ D ν J. G. Carlsson, U of MN ISyE a + 4m + b 6 ( )2 b−a = 6 = Lecture 7: PERT & CPM October 3/17, 2013 32 / 38 PDM The precedence diagramming method (PDM) is an extension of PERT/CPM in which mutually dependent activities can be performed partially instead of sequentially In PERT/CPM, we assume that an activity must be completely finished before we can work on its successors PDM generalizes this by allowing the following additional relationships: SSAB (start-to-start) lead: activity B cannot start until activity A has been in progress for at least SS time...
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This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

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