Problem-Solving Examples
(Non-Preemptive Case)
5-1
Outline
• Job-shop scheduling problem (JSSP)
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Problem definition
Basic search procedure
Heuristics
Results
• Resource-constrained project scheduling
problem (RCPSP)
5-2
JSSP: Problem definition
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Set of machines {M1 ... Mm}
Set of jobs {J1 ... Jn}
List of operations Oi1 ... Oim(i) for each job Ji
Processing time pij for each operation Oij
Machine Mij for each operation Oij
5-3
JSSP: Problem variables
• Integer variables start(O) and end(O) for
each operation O
• Optimization criterion
makespan = maxi(end(Oim(i)))
5-4
JSSP: Problem constraints
• Temporal constraints
0  start(Oij)
end(Oij)  start(Oij+1)
• Duration constraints
start(Oij) + pij = end(Oij)
• Exclusion constraints
Mij = Mkl implies
[end(Oij)  start(Okl)] OR [end(Okl)  start(Oij)]
5-5
JSSP: Decision var. complexity
• NP-complete in the strong sense when m = 3 and
m(i)  3 for all i
• Solvable in polynomial time if m = 2 and m(i)  2
for all i
• NP-complete in the ordinary sense when m = 2
and m(i)  3 for all i, or when m = 3 and m(i)  2
for all i
• Flow-shop scheduling: NP-complete in the strong
sense for m = 3
5-6
JSSP: Basic search procedure
• Select a machine M and a set W of operations
which require M
• For each operation O in W, create a branch where
O is set to execute before (or after) all the other
operations of W
• Select a branch, propagate the decision, and iterate
(until a solution is found or no branch remains)
5-7
JSSP: Makespan minimization
• Set makespanmin to an obvious lower bound
• Set makespanmax to an obvious upper bound
• Iterate until makespanmin = makespanmax
– Select a value in [makespanmin makespanmax)
– Constrain the makespan to be smaller than (or equal to)
the chosen value
– Run the search procedure: when a solution is found, set
makespanmax to the makespan of the solution; if there is
no solution, set makespanmin to (value + 1)
5-8
JSSP: Makespan minimization
• Example
– start with makespanmin obtained by propagation
– until a solution has been found, increment the tested
value by min(2i-1, (makespanmax - makespanmin) / 2)
(after the ith iteration)
– after a solution has been found, set the tested value to
(makespanmax - makespanmin) / 2
5-9
JSSP: Heuristics
• Selection of the machine
Find the task interval [startmin(A), endmax(B)] with
the highest demand/supply ratio
TI = {C | startmin(A)  startmin(C) and endmax(C)  endmax(B)}
demand = SCTI duration(C)
supply = endmax(B) - startmin(A)
5-10
JSSP: Heuristics
• Branch exploration ordering
– First or last?
• Always select an operation to execute first
• Always select an operation to execute last
• Use edge-finding rules to determine the number Nf
of operations that can be first and the number Nl of
operations that can be last (and select the option
with the smallest number of candidates)
5-11
JSSP: Heuristics
• Branch exploration ordering
– Select the branch on which the operation with
the smallest endmax is scheduled first (EDD)
– Select the branch on which the operation with
the largest startmin is scheduled last
– Propagate each alternative in turn and select the
one which appears the most constraining
5-12
JSSP: Benchmark results
• [Applegate & Cook 91]
– Ten instances with 10 jobs and 10 machines
(100 operations)
– Results (table) taken from [Le Pape 95]
• [Vaessens et al 94]
– Thirteen instances (one open) with up to 225
operations
5-13
JSSP: Benchmark results
Instance
MT10
ABZ5
ABZ6
LA19
LA20
ORB1
ORB2
ORB3
ORB4
ORB5
BT
13684
19303
6227
18102
40597
22725
31490
36729
13751
12648
CPU
184.8
226.6
80.3
219.2
407.3
323.7
416.4
488.4
169.9
168.5
BT(PR)
4735
4519
312
6561
20626
6261
14123
22138
1916
2658
CPU(PR)
52.8
49.7
3.8
74.7
186.9
85.3
189.3
277.6
19.0
29.7
5-14
Benchmarks (Applegate & Cook)
215
210
25
MT10
ABZ5
ABZ6
LA19
[Applegate & Cook 91]
[Baptiste et al 95] [Le Pape 95]
[Caseau & Laburthe 95]
[Colombani 96]
LA20
ORB1
ORB2
Number of nodes
Number of fails
Number of choices
ORB3
ORB4
ORB5
CPU time (Sparc1)
CPU time (RS6000)
CPU time (Sparc10)
CPU time (Sparc 5)
5-15
Benchmarks (Applegate & Cook)
215
210
25
MT10
ABZ5
ABZ6
LA19
PROOFS OF OPTIMALITY
[Applegate & Cook 91]
[Baptiste et al 95] [Le Pape 95]
[Caseau & Laburthe 95]
LA20
ORB1
ORB2
Number of nodes
Number of fails
Number of fails
ORB3
ORB4
ORB5
CPU time (Sparc1)
CPU time (RS6000)
CPU time (Sparc10)
5-16
Benchmarks (Vaessens et al)
15
10
5
MT10
LA02
LA19
LA21
LA24
[Adams et al 88]
[Nuijten & Le Pape 97]
[Caseau & Laburthe 95]
LA25
LA27
LA29
LA36
LA37
LA38
LA39
LA40
[Mattfeld 96]
[Nowicki & Smutnicki 93]
[Van Laarhoven et al 92]
5-17
Benchmarks (Vaessens et al)
215
210
25
MT10
LA02
LA19
LA21
LA24
LA25
[Adams et al 88] (Vax780)
[Nuijten & Le Pape 97] (RS6000)
[Caseau & Laburthe 95] (Sparc10)
LA27
LA29
LA36
LA37
LA38
LA39
LA40
[Mattfeld 96] (Sparc10)
[Nowicki & Smutnicki 93] (PC)
[Van Laarhoven et al 92] (Vax785)
5-18
RCPSP: Problem definition (1)
• Given:
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m resources R1 ... Rm with given capacities c1 ... cm
n activities A1 ... An with given durations d1 ... dn
precedence constraints between activities
resource requirements qij (for activity i and resource j)
assign start and end times to activities so as
to satisfy the constraints and minimize the
duration of the overall project
5-19
RCPSP: Problem definition (2)
• Given:
– m resources R1 ... Rm with given maximal capacities c1max ... cmmax
– n activities A1 ... An with given minimal durations d1min ... dnmin and
maximal durations d1max ... dnmax
– precedence constraints between activities
– resource requirements qijd (for activity i, resource j and duration d)
assign start and end times to activities so as to
satisfy the constraints and achieve the "best"
tradeoff between the duration of the overall project
and the peak capacity of the resources
5-20
RCPSP: Problem variables
• Integer variables start(Ai) and end(Ai) for
each activity Ai
• Integer variables duration(Ai), cj and qij in
the variable-duration variable-demand case
• Optimization criterion
makespan = maxA(end(A))
5-21
RCPSP: Problem constraints
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Resource capacity constraints (cj)
Duration constraints
Precedence constraints
Resource requirement constraints (qij)
Relations between durations and resource
requirements
5-22
RCPSP: Search procedure (1)
1 Initialize the set of selectable activities W to the complete set.
2 If all activities have fixed start and end times, exit. Otherwise remove
from W those activities which have fixed start and end times.
3 If W is not empty, select an activity from W, create a choice point for
the selected activity (to allow backtracking) and schedule the selected
activity from its earliest start time to its earliest end time.
4 If W is empty, backtrack to the most recent choice point. (If there is no
choice point left, report that there is no problem solution and exit.)
5 Upon backtracking, mark the backtracked activity as not selectable as
long as its earliest start and end times have not changed. Then goto
step 2.
5-23
RCPSP: Search procedure (2)
1 Initialize the set of selectable activities W to the complete set.
2 If all activities have fixed start and end times, exit. Otherwise remove
from W those activities which have fixed start and end times.
3 If W is not empty, select an activity from W, create a choice point for
the duration of the selected activity, select a duration, create a choice
point for the start and end times of the activity and schedule the
selected activity from its earliest start time to its earliest end time.
4 If W is empty, backtrack to the most recent choice point. (If there is no
choice point left, report that there is no problem solution and exit.)
5 Upon backtracking on a start/end time choice point, mark the
backtracked activity as not selectable as long as its earliest start and
end times have not changed. Then goto step 2.
6 Upon backtracking on a duration choice point, select another duration.
5-24
RCPSP: Optimization
• Classical minimization scheme
• Pareto-optimization (find all Pareto-optimal
tradeoffs)
1 Generate a solution which minimizes the first criterion c1. (If there
is no solution, exit.) Let v1 be the optimal value for c1.
2 Constrain c1 to be smaller than or equal to v1. Generate a solution
which minimizes the second criterion c2. Let v2 be the optimal
value for c2. The solution found is Pareto-optimal.
3 Remove the constraint stating that c1 is smaller than or equal to v1.
Replace it by a constraint stating that c2 is strictly smaller than v2.
Goto step 1.
5-25
RCPSP: Heuristics
• Selection of an activity
– Select the activity with the smallest endmin
• Selection of a duration
– Select the shortest possible duration
5-26
RCPSP: Results
• Ship loading problem [Aggoun & Beldiceanu 92]
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1 resource
34 activities with precedence constraints
Fixed-duration fixed-demand fixed-capacity: 0.2s
Fixed-duration fixed-demand Pareto-optimization: 0.4s
Variable-duration variable-demand fixed-capacity: 0.2s to 0.9s
Variable-duration variable-demand Pareto-optimization: 13s
• Industrial problem
– 2 resources
– 40 to 50 activities with precedence constraints
– Variable-duration variable-demand fixed-capacity: 1s
5-27
RCPSP: Results
• However, the efficiency of RCPSP search
procedures is shown to depend a lot on the
structure of the precedence network. Much
harder instances can be obtained by simple
modifications of the ship loading problem.
• Additional constraints of real-life problems
can make the problem much more simple or
much more difficult than the RCPSP.
5-28