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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

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Lecture 5 project planning 2

Lecture 5: Project Planning 2

© J. Christopher Beck 2005


Outline
Outline

  • Time/Cost Tradeoffs

    • Linear and non-linear

  • Adding Workforce Constraints

  • Slides borrowed from Twente & Iowa

    • See Pinedo CD

© J. Christopher Beck 2005


Time cost trade offs
Time/CostTrade-Offs

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

    • More money  shorter processing time

    • Run machine at higher speed

© J. Christopher Beck 2005


Linear costs
Linear Costs

Money

Marginal

cost

Processing

time

© J. Christopher Beck 2005


Problem
Problem

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

Cost per activity

“Overhead” cost

© J. Christopher Beck 2005


Solution methods
Solution Methods

  • Objective: minimum cost of project

  • Time/Cost Trade-off Heuristic

    • Good schedules

    • Works also for non-linear costs

  • Linear programming formulation

    • Optimal schedules

    • Non-linear version not easily solved

© J. Christopher Beck 2005


Sources sinks cuts

Cut set

Sink node

Source (dummy) node

Minimal cut set

Sources, Sinks, & Cuts

© J. Christopher Beck 2005


Time cost trade off heuristic
Time/Cost Trade-off Heuristic

  • Step 1:

    • Set all processing times at their maximum

    • Determine all critical paths

    • Construct the graph Gcp of critical paths

© J. Christopher Beck 2005


Time cost trade off heuristic1
Time/Cost Trade-off Heuristic

  • 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

© J. Christopher Beck 2005


Time/Cost Trade-Off Heuristic

  • 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

© J. Christopher Beck 2005


Time/Cost Trade-Off Heuristic

  • 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

© J. Christopher Beck 2005


Example 4.4.2

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

© J. Christopher Beck 2005


Step 1 maximum processing times find g cp

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Step 1: Maximum Processing Times, Find Gcp

© J. Christopher Beck 2005


Step 1 maximum processing times find g cp1

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Step 1: Maximum Processing Times, Find Gcp

Cost = overhead + job costs

= co * Cmax + Σcaj

= 6 * 56 + 350

= 686

© J. Christopher Beck 2005


Step 2 3 min cut sets in g cp lowest cost

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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

© J. Christopher Beck 2005


Step 4 1 reduce processing time for each job by 1

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

Cost = overhead + processing

= c0 * Cmax + Σjob costs

= 6 * 55 + 352

= 682

© J. Christopher Beck 2005


Step 2 3 min cut sets in g cp lowest cost1

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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

© J. Christopher Beck 2005


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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

Cost = overhead + processing

= co * Cmax + Σjob costs

= 6 * 52 + 355

= 667

© J. Christopher Beck 2005


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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?

© J. Christopher Beck 2005


After more iterations

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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

© J. Christopher Beck 2005


Linear programming formulation
Linear Programming Formulation

  • The heuristic does not guarantee optimum

    • See example 4.4.3

  • Here total cost is linear so use LP

  • Want to minimize

© J. Christopher Beck 2005


Linear program
Linear Program

Minimize

subject to

earliest start

time of job k

processing

time of job k

© J. Christopher Beck 2005


Can also have non linear costs
Can Also Have Non-linear Costs

  • 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)?

© J. Christopher Beck 2005


What if jobs require resources
What If Jobs Require Resources?

  • Back to fixed durations

    • Without resources → easy

    • With resources → hard

  • Resource Constraint Project Scheduling Problem (RCPSP)

© J. Christopher Beck 2005


Rcpsp example

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RCPSP Example

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Resource requirements

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© J. Christopher Beck 2005


What if r 1 4
What if R1 = 4?

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© J. Christopher Beck 2005


Rcpsp
RCPSP

  • 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

© J. Christopher Beck 2005


Rcpsp example1
RCPSP Example

© J. Christopher Beck 2005


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