© J. Christopher Beck 2005 1

Lecture 5: Project Planning 2

© J. Christopher Beck 2005 2

Outline Time/Cost Tradeoffs

Linear and non-linear Adding Workforce Constraints

Slides borrowed from Twente & Iowa See Pinedo CD

© J. Christopher Beck 2005 3

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 4

Linear Costs

Money

Processingtimemax

jp

ajc

bjc

minjp

maxminjjj ppp

minmaxjj

bj

aj

j pp

ccc

Marginal cost

© J. Christopher Beck 2005 5

Problem

Spend money to reduce processing times so as to minimize:

n

jjjj

bj ppccCc

1

maxmax0

“Overhead” cost Cost per activity

© J. Christopher Beck 2005 6

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 7

Source (dummy) node

Sink node

Minimal cut set

Cut set

Sources, Sinks, & Cuts

© J. Christopher Beck 2005 8

Step 1:

Set all processing times at their maximum

Determine all critical paths

Construct the graph Gcp of critical paths

maxjj pp

Time/Cost Trade-off Heuristic

© J. Christopher Beck 2005 9

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

cpjj Gjpp ,min

Time/Cost Trade-off Heuristic

© J. Christopher Beck 2005 10

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 11

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 12

Example 4.4.2

j 1 2 3 4 5 6 7 8 9 10 11 12 13 14

pmaxj 5 6 9 12 7 12 10 6 10 9 7 8 7 5

pminj 3 5 7 9 5 9 8 3 7 6 4 5 5 2

cja 20 25 20 15 30 40 35 25 30 20 25 35 20 10

cj 7 2 4 3 4 3 4 4 4 5 2 2 4 8

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

© J. Christopher Beck 2005 13

Step 1: Maximum Processing Times, Find Gcp

1

2

3

6 9

5 8

4 7

11

10

12 14

13

© J. Christopher Beck 2005 14

Step 1: Maximum Processing Times, Find Gcp

1

2

3

6 9

5 8

4 7

11

10

12 14

13

56max C

Cost = overhead + job costs = co * Cmax + Σca

j

= 6 * 56 + 350 = 686

© J. Christopher Beck 2005 15

1

3

6 9

11

12 14

c1=7

c3=4

c6=3 c9=4

c11=2

c12=2 c14=8

Minimum cutset with lowest cost

Cut sets: {1},{3},{6},{9}, {11},{12},{14}.

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

© J. Christopher Beck 2005 16

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

1

2

3

6 9

5 8

4 7

11

10

12 14

13

55max C

Cost = overhead + processing = c0 * Cmax + Σjob costs = 6 * 55 + 352 = 682

© J. Christopher Beck 2005 17

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

1

3

6 9

11

12 14

c1=7

c3=4

c6=3 c9=4

c11=2

c12=2 c14=8

13

Cut sets: {1},{3},{6},{9}, {11},{12,13},{14}.

c13=4

Minimum cutset with lowest cost

© J. Christopher Beck 2005 18

1

3

6 9

11

12 14

c1=7

c3=4

c6=3 c9=4

c11=2

c12=2 c14=8

13

c13= 4

Next 3 iterationsreduce processingtime from 7 to 4

Cost = overhead + processing = co * Cmax + Σjob costs = 6 * 52 + 355 = 667

Next 3 Iterations

© J. Christopher Beck 2005 19

1

3

6 9

11

12 14

c1=7

c3=4

c6=3 c9=4

c11=2

c12=2 c14=8

13

Reduce processing timenext on job 6

c13= 4

Step 1,2, & 3

Q: why not 12?

© J. Christopher Beck 2005 20

After More Iterations …

1

3

6 9

11

12 14

c1=7

c3=4

c6=3 c9=4

c11=2

c12=2 c14=8

13

c13= 4

2 4 7

10

c2=2 c4=3 c7=4

c10=5

© J. Christopher Beck 2005 21

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

n

jjjj

bj ppccCc

1

maxmax0

n

jjj pcCc

1max0 .

© J. Christopher Beck 2005 22

Minimize

subject to

jpxC

jx

jpp

jpp

Akjxpx

jj

j

jj

jj

jjk

,0

,0

,

,

,0

max

min

max

n

jjj pcCc

1max0 .

earliest start time of job k

processing time of job k

Linear Program

© J. Christopher Beck 2005 23

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 24

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 25

321 4 5 6 7 8

3

2

1

4

5

6

1

2

3

4

5

6

Job p(j) Predecessors S' C'' R(1,j)1 2 - 0 3 32 3 - 0 3 13 1 - 0 6 24 4 1,2 3 7 25 2 2,3 3 8 36 1 4 7 8 3

RCPSP Example

Resource requirements

© J. Christopher Beck 2005 26

321 4 5 6 7 8

3

2

1

4

5

6

9 10

1

2

3 45 6

What if R1 = 4?

© J. Christopher Beck 2005 27

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 28

RCPSP Example

Job p(j) P(j) S' C'' R(1,j) R(2,j)1 2 - 0 3 3 22 3 - 0 3 1 13 1 - 0 6 2 14 4 1,2 3 7 2 15 2 2,3 3 8 3 26 1 4 7 8 3 1

4R1 2R2