1 / 28

# Lecture 5 - PowerPoint PPT Presentation

Lecture 5: Project Planning 2. Outline. Time/Cost Tradeoffs Linear and non-linear Adding Workforce Constraints Slides borrowed from Twente & Iowa See Pinedo CD. Time/Cost Trade-Offs. What if you could spend money to reduce the job duration More money  shorter processing time

Related searches for Lecture 5

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Lecture 5' - oshin

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Lecture 5: Project Planning 2

• Linear and non-linear

• Slides borrowed from Twente & Iowa

• See Pinedo CD

• What if you could spend money to reduce the job duration

• More money  shorter processing time

• Run machine at higher speed

Money

Marginal

cost

Processing

time

• Spend money to reduce processing times so as to minimize:

Cost per activity

• Objective: minimum cost of project

• Good schedules

• Works also for non-linear costs

• Linear programming formulation

• Optimal schedules

• Non-linear version not easily solved

Sink node

Source (dummy) node

Minimal cut set

Sources, Sinks, & Cuts

• Step 1:

• Set all processing times at their maximum

• Determine all critical paths

• Construct the graph Gcp of critical paths

• Step 2:

• Determine all minimum cut sets in Gcp

• Consider those sets where all processing times are larger than their minimum

• If no such set STOP; otherwise continue to Step 3

• Step 3:

• For each minimum cut set:

• Compute the cost of reducing all processing times by one time unit.

• Take the minimum cut set with the lowest cost

• If this is less than the overhead per time unit go on to Step 4; otherwise STOP

• Step 4:

• Reduce all processing times in the minimum cut set by one time unit

• Determine the new set of critical paths

• Revise graph Gcp and go back to Step 2

Overhead: co = 6 (cost of project per time unit)

2

3

6

9

5

8

4

7

11

10

12

14

13

Step 1: Maximum Processing Times, Find Gcp

4

5

9

7

3

2

1

6

13

11

10

12

14

Step 1: Maximum Processing Times, Find Gcp

Cost = overhead + job costs

= co * Cmax + Σcaj

= 6 * 56 + 350

= 686

3

6

9

11

12

14

Step 2 & 3: Min. Cut Sets in Gcp & Lowest Cost

c1=7

c12=2

c6=3

c9=4

c14=8

c11=2

c3=4

Cut sets: {1},{3},{6},{9},

{11},{12},{14}.

Minimum cut

set with lowest cost

4

5

9

7

3

2

1

6

13

11

10

12

14

Step 4 & 1: Reduce Processing Time for Each Job by 1

= c0 * Cmax + Σjob costs

= 6 * 55 + 352

= 682

3

6

9

14

11

12

13

Step 2 & 3: Min. Cut Sets in Gcp & Lowest Cost

c1=7

c12=2

c6=3

c9=4

c14=8

c11=2

c13=4

c3=4

Cut sets: {1},{3},{6},{9},

{11},{12,13},{14}.

Minimum cut

set with lowest cost

3

6

9

11

12

14

13

Next 3 Iterations

c1=7

c12=2

c6=3

c9=4

c14=8

c11=2

c13= 4

c3=4

Next 3 iterations

reduce processing

time from 7 to 4

= co * Cmax + Σjob costs

= 6 * 52 + 355

= 667

3

6

9

11

12

14

13

Step 1,2, & 3

c1=7

c12=2

c6=3

c9=4

c14=8

c11=2

c13= 4

c3=4

Reduce processing time

next on job 6

Q: why not 12?

1

3

6

9

7

2

12

13

11

14

10

After More Iterations …

c2=2 c4=3 c7=4

c10=5

c1=7

c12=2

c6=3

c9=4

c14=8

c11=2

c13= 4

c3=4

• The heuristic does not guarantee optimum

• See example 4.4.3

• Here total cost is linear so use LP

• Want to minimize

Minimize

subject to

earliest start

time of job k

processing

time of job k

• Arbitrary function cj(pj) → cost of setting job j to processing time pj

• LP doesn’t work!

• See Section 4.5

• A question I like:

• Given processing times and cj(pj), which algorithm would you use (heuristic or LP)?

• Back to fixed durations

• Without resources → easy

• With resources → hard

• Resource Constraint Project Scheduling Problem (RCPSP)

5

2

1

6

4

RCPSP Example

6

5

4

3

2

1

Resource requirements

1

2

3

4

5

6

7

8

What if R1 = 4?

6

5

4

2

3

1

5

6

2

3

4

1

1

2

3

4

5

6

7

8

9

10

• n: jobs j=1,…,n

• N: resources i=1,…,N

• Rk: availability of resource k

• pj: duration of job j

• Rkj: requirement of job j for resource k

• Pj: (immediate) predecessors of job j

• Minimize Cmax