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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

slide10
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

slide11
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

slide12
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

slide18
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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

slide19
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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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