using simulated annealing and evolution strategy scheduling capital products with complex product...
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Using Simulated Annealing and Evolution Strategy scheduling
capital products with complex product structure
By: Dongping SONG
Supervisors: Dr. Chris Hicks and
Prof. Chris F. Earl
Department of MMM Engineering
University of Newcastle, Oct., 2000.
Contents
• Introduction
• Problem formulation
• A discrete event-driven model
• Simulated Annealing
• Evolution Strategy
• Case studies
• Conclusions
Introduction
Production scheduling -- the allocation of resources over time to perform a collection of tasks (Baker, 1974).
Two important points in scheduling:
Sequencing -- in which order to perform tasks on resources
Timing -- when to start and complete tasks
Introduction• The importance of sequencing has been well recognised, because
the optimal schedule can only be characterised by the sequences for performance measures such as mean flow-time, percentage of jobs tardy, mean tardiness, etc.
• However, timing scheduling is necessary
for performance measures such as earliness and tardiness costs, total discrepancy from the due dates, etc.
Introduction - effect of timingExample 1. One machine with three independent jobs.
job 1 job 2
job 2
job 3
job 1 job 3
job 2job 1 job 3
E 1
T 2E 1
T 2 T 3
d 1 d 2 d 3
Total cost =
Total cost =
Total cost =
E 1
T 2E 1
T 2 T 3
+
+
time
Schedule A:
Schedule B:
Schedule C:
Due dates
Introduction - effect of timingExample 2. Two machines four jobs s.t. job 1 becomes job 4 and job 2 becomes job 3 after completion.
Schedule A:
Schedule B:
job 1 job 3
job 4job 2
E 4
job 1 job 3
job 4job 2
Holding cost =
Holding cost =
E 4
E 1
M 1:
M 2:
M 1:
M 2:
time
E 1
Complex product structure
Introduction - a capital product
248:1
8 opers
. . .
244:1
9 opers
. . .7 opers
. . .
240:1
11 opers
. . .
236:1
16 opers. . .12 opers
235:1
10 opers
. . .
241:7
243
238230 242
232:12
226:15
234
235:10
233:12
239
240:11
236:16
229
247
248:8244:9
246
228
231
237 245
226:!
15 opers
. . .
232:1
12 opers
. . .
233:1
. . .
241:1
Introduction• Constraints in our scheduling problem:
– Operation precedence constraints– Resource capacity constraints– Due date constraints– Assembly co-ordination requirements
• Scheduling problem: to find optimal operation sequences and timings to meet above constraints and minimise total cost.
Problem formulation
• Notation: si -- planned start time for operation i; N -- total operation number.
• Solution space of schedules := RN{sequences on resources}.
• Solution space can be simplified to RN, because operations on the same resource have different start times (i.e. timings imply sequences).
Problem formulation
• The schedule problem can be formulated as a numerical optimisation problem.
• Find the optimal {si, i=1,..,N} to minimise the total cost
J(s) = (Work-in-progress holding costs+ product earliness costs
+ product tardiness costs)
Problem formulation
• Questions:
(1) How to execute a schedule that is characterised by {si} ?
(2) How to evaluate the cost function for a given schedule ?
Discrete event-driven model• Two types of events :
– the start of an operation
– the completion of an operation.
• Two constraints to trigger the start events :– Physical constraints : an event cannot occur before all
preceding events are completed.
– Planning constraints : an operation cannot be started before its planned start time si.
Discrete event-driven model
The evolution of the system for a given schedule {si} can be described by:
• If a resource is idle, an operation will be processed as soon as the physical and planning constraints are satisfied.
• If there is a queue of operations ready for processing, the operation with the earliest si will be processed first.
Simulated Annealing
• Neighbourhood of a solution -- by adding a random number to each si.
• Outer loop -- cooling the temperature T until T=0.
• Inner loop -- perform Metropolis simulation with fixed T to find equilibrium state.
Simulated Annealing
• Adjust the solution : – shift the whole schedule (optional)– impose precedence constraints (optional)– make non-negative
• Evaluate cost function :– run the DED model
Simulated AnnealingInitialisation
Metropolis simulation with fixedtemperature T
Adjust the solution
Evaluate cost function
Improvement
Accept newsolution
Accept new solutionwith a probability
Check for equilibrium
Stop criteria at outer loop
Return optimal solution
Coolingtemperature T
Generate new solution
NoYes
Yes
No
Yes
No
Evolution Strategy• Similarity of Genetic Algorithms and ES:
– model organic evolution.– iterative scheme including “selection”,
“crossover” and “mutation”.
• Difference of GA and ES:– GA uses binary or string representations,
suitable for combinatorial optimisation problem.
– ES uses continuous variable, suitable for numerical optimisation problem.
Evolution Strategy
s 1(n ,1) s 2
(n ,1) s 3(n ,1) ... ... s N-2
(n ,1) s N-1(n ,1) s N
(n ,1)
s 1(n ,2) s 2
(n ,2) s 3(n ,2) ... ... s N-2
(n ,2) s N-1(n ,2) s N
(n ,2)
Parent 1
Parent 2
offspring s 1(n ,2) s 2
(n ,1) s 3(n ,1) ... ... s N-2
(n ,2) s N-1(n ,1) s N
(n ,2)
• Crossover -- randomly copy elements from parents column by column.
Evolution Strategy• Mutation -- add a random number from a Normal distribution to each element.
offspring s 1 s 2 s 3 ... ... s N-2 s N-1 s N
offspring s 1 +z 1 s 2 +z 2 s 3 +z 3 ... ... s N-2 +z N -2 s N-1 +z N-1 s N + z N
z i ~ N (0, )
Evolution Strategy
• Adjust the solution :– shift the whole schedule (optional)– impose precedence constraints (optional)– make non-negative
• Evaluate cost function -- run the DED model.
• Selection -- choose a set of best offspring as parents for the next generation
Evolution StrategyInitialisation
Mutation for the offspring
Adjust the offspring
Select a set of best offspringto replace the parent generation
Stop criteria
Return optimal solution
Reduce standarddeviation for mutation
if necessary
Crossover to generate offspring
Evaluate cost function
Select candidate(s) from parentgeneration
Finish offspring generation
No
No
Yes
Yes
Case studies
Case Products Maching/Assembly operation
Resources
1 1 100/13 132 3 210/29 17
Characteristics of scheduling problems
Case study 1MRP+FIFO
MRP+EDD
MRP+SPT
SA ES
Total cost 187.78 185.59 188.56 89.22 87.79
Cost is reduced by 50% for SA and ES.
MRP -- material requirement planning
FIFO -- first in first out
EDD -- earliest due date first
SPT -- shortest processing time first
Case study 1 -- ES methodCost
CPU(s)
Maximum cost at each generation
Maximum cost in all parents
Minimum cost at each generation
Case study 2
MRP+FIFO
MRP+EDD
MRP+SPT
SA ES
Total cost 925.34 927.88 931.16 472.57 416.43
Cost is reduced by 50% for SA and ES.
Case study 2 -- ES methodCost
CPU(s)
Maximum cost at each generation
Maximum cost in all parents
Minimum cost at each generation
Conclusions
• SA and ES can reduce total cost by 50% compared with MRP+dispatching rules.
• ES is generally better than SA in both cost and CPU time.
• ES is more robust to its initial parameter selection than SA.
Conclusions• Suggestions for SA initial parameters:
– T0 and step-size is taken from [d/N, 20*d/N];
– Temperature cooling rate > 0.5 and step-size reduction factors > 0.70;
– No-improvement number at inner loop > N/2;
• Suggestions for ES initial parameters:– Offspring population is from [N/2, 2*N];
– Parent population is 1/10 to 1/5 of offspring;
– Initial standard deviation is from [d/N, 5*d/N].
Further work• Compare our methods with GA (Pongcharoen, et
al.) for the same cost function.
• Develop hybrid optimisation methods by combining SA, ES with Perturbation Analysis or heuristics.
• Extend to stochastic situations such as dynamic customer demand arrivals and processing uncertainties.
1.00.90.80.70.60.5
110
100
90
80
Temperature cooling factor
To
tal c
ost
SA -- effect of parameters
Initial temperature and temperature cooling factor
T0=1
T0=40
T0=20
T0=10