Operations Research Lecture 7.

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Presentation transcript:

Operations Research Lecture 7

Operations Research CPM / PERT

OR PROCESS OBSERVATION PROBLEM DEFINITION MODEL CONSTRUCTION SOLUTION H N I Q U S MODEL CONSTRUCTION SOLUTION IMPLEMENTATION

Critical Path Method (CPM) Networks Critical Path Method (CPM) Program Evaluation and Review Technique (PERT) Network Flow

Example Obtain Stamp Write a Letter Put in Env. Address & Stamp Post 3 8 16 12 14 Obtain Envelope 7

Dummy Activities There is a need for dummy activities when the project contains groups of two or more jobs which have common predecessors. The time taken for the dummy activities is zero.

J A C D B F H D3 D1 D2 E G

H 5 7 L D I J A B E G K M 4 1 2 3 6 8 9 10 N F 11 C

Looping 8 9 10 9 10

Dangling 10 8 9 9 11

Merge Node 8 11 9 9 10 10

Burst Node 10 5 9 11 12

Example EVENT 16 20 20 + 15 16 + 15, OR 20 + 10 OR 20 + 15 + 3 30, OR 35 + 16, OR 38 + 12

Cont! EARLIEST TIME: By Longest Chain LATEST TIME: Backward Pass 51, 39, 35, 20, 24, 0

TIMES (ACTIVITY) EARLIEST START TIME Earliest possible time at which an activity can start and is given by the earliest time of the Tail Event.

TIMES (ACTIVITY) EARLIEST FINISH TIME Earliest possible time at which an activity can finish and is given by adding the duration time to the earliest start time.

TIMES (ACTIVITY) LATEST FINISH TIME Latest Event Time of the Head Event.

TIMES (ACTIVITY) LATEST START TIME Latest possible time by which an activity start and is given by subtracting the duration time from the Latest Finish Time.

Example

Cont! ACTIVITY DURATION START FINISH E L E L 1-2 16 0 8 16 24 1-3 20 0 0 20 20 1-11 30 0 21 30 51 2-8 15 16 24 31 39 3-7 15 20 20 35 35 3-8 10 20 29 30 39 7-8 3 35 36 38 39 7-11 16 35 35 51 51 8-11 12 38 39 50 51

Notations Earliest Time of Tail Event i= iE Latest Time of Tail Event i = iL Earliest Time of Head Event j= jE Latest Time of Head Event j = jL

FLOATS INDEPENDENT FLOAT Time by which an activity can expand without affecting other (PREV or SUBSEQ) I = jE - iL - D If Negative, take I = 0

Cont ! FREE FLOAT Time by which an activity can expand without affecting subsequent activity. F = jE - iE - D

Cont ! TOTAL FLOAT Time by which an activity can expand without affecting the overall duration of the project. T = jL - iE - D

Example Activity 2-8 T = jL – iE – D = 39 – 16 – 15 = 9 F = jE – iE – D = 38 – 16 – 15 = 7 I = jE – iL – D = 38 – 24 – 15 = -1

Example 3 11 4 7 5 10 1 2 5 8 10 6 4

Cont ! E=12 L=12 E=23 E=0 E=5 L=5 L=0 L=23 E=13 L=13 3 11 4 7 5 10 1 2 8 6 E=13 4 L=13

For Activity 1 – 3 T = jL - iE – D = 12- 0- 4 = 8 F = jE - iE – D = 12- 0- 4 = 8 I = jE - iL – D = 12- 0- 4 = 8

ACTIVITY DURATION FLOATS T F I 1 - 2 5 0 0 0 1 - 3 4 8 8 8 1 - 4 6 7 7 7 2 - 3 7 0 0 0 2 - 4 8 0 0 0 2 - 5 10 8 8 8 3 - 5 11 0 0 0 4 - 5 10 0 0 0 1 – 2 – 3 – 5 ! 1 – 2 – 4 – 5 ! CRITICAL PATH

Cont ! E=12 L=12 E=23 E=0 E=5 L=5 L=0 L=23 E=13 L=13 3 11 4 7 5 10 1 2 8 6 E=13 4 L=13

Example 2 2 5 3 1 6 6 4 8 4 7 1 9 8 4 5 3 6

Example 2 5 4 7 8 1 3 6 E=11 E=1 2 L=12 L=10 1 3 E=15 E=21 E=0 E=6 6 4 9 L=15 8 L=21 L=0 E=15 4 E=8 5 3 6 L=10 L=15

Cont ! ACT DUR START FINISH FLOAT E L E L T F I 1-2 1 0 9 1 10 9 0 0 1-2 1 0 9 1 10 9 0 0 1-3 8 0 2 8 10 2 0 0 1-4 6 0 0 6 6 0 0 0 2-5 2 1 10 3 12 9 8 1 3-6 5 8 10 13 15 2 2 0 4-5 5 6 7 11 12 1 0 0 4-6 9 6 6 15 15 0 0 0 4-7 4 6 11 10 15 5 5 5 5-7 3 11 12 14 15 1 1 1 6-8 4 15 17 19 21 2 2 0 7-8 6 15 15 21 21 0 0 0

Cont ! 2 5 4 7 8 1 3 6 E=11 E=1 2 L=12 L=10 1 3 E=15 E=21 E=0 E=6 6 4 9 L=15 8 L=21 L=0 E=15 4 E=8 5 3 6 L=10 L=15

Cont ! 2 5 4 7 8 1 3 6 E=11 E=1 2 L=12 L=10 1 3 E=15 E=21 E=0 E=6 6 4 9 L=15 8 L=21 L=0 E=15 4 E=8 5 3 6 L=10 L=15

Example 6 4 5 25 40 2 3 4 7 20 5 10 50 55 3 9 2 8 15 30 45 12 35

Cont ! E=11 E=17 5 25 40 L=16 L=22 E=26 E=11 E=18 E=0 E=4 2 3 4 7 20 5 10 50 55 L=23 L=4 L=16 L=0 L=26 E=6 3 9 E=4 2 E=18 8 15 30 45 L=6 L=18 L=4 E=6 12 35 L=6

Cont ! ACT DUR START FINISH FLOAT E L E L T F I 5-10 4 0 0 4 4 0 0 0 5-10 4 0 0 4 4 0 0 0 5-15 3 0 1 3 4 1 1 1 5-25 5 0 11 5 16 11 6 6 10-15 0 4 4 4 4 0 0 0 10-20 7 4 9 11 16 5 0 0 15-30 2 4 4 6 6 0 0 0 20-25 0 11 16 11 16 5 0 0 20-50 2 11 21 13 23 10 5 0

Cont ! ACT DUR START FINISH FLOAT E L E L T F I 25-40 6 11 16 17 22 5 0 0 30-35 0 6 8 6 6 0 0 0 30-45 9 6 9 15 18 3 2 3 35-45 12 6 6 18 18 0 0 0 40-55 4 17 22 21 26 5 5 0 45-50 0 18 23 18 23 5 0 0 45-55 8 18 18 26 26 0 0 0 50-55 3 18 23 21 26 5 5 0

Cont ! 6 4 5 25 40 2 3 4 7 20 5 10 50 55 3 9 2 8 15 30 45 12 35

Operations Research Lecture 7