Generating RCPSP instances with Known Optimal Solutions

1
Generating RCPSP instances with Known Optimal
Solutions

• José Coelho
• jcoelho_at_univ-ab.pt
• Generator and generated instances in
• http//jcoelho.m6.net/edicao3.asp?pa3569

2
Index

• Introduction
• Generation Method
• Tests
• Conclusions

3
Introduction

• There are several algorithms for solving the
  RCPSP
• The comparison of algorithms requires an instance
  set
• The average performance is extrapolated to all
  RCPSP instances
• A diversified instance set minimizes
  extrapolation errors
• To generate a diversified instance set, one needs
  to
• Identify the relevant indicators
• However these indicators are not consensual
• There are several generators, with different
  kinds of arguments
• The resources are generated randomly, in such a
  way that resource indicators are verified, but
  not getting the optimal solution
• The most widely used instance set is PSPLIB

4
Introduction

• Available generators do not give any optimal
  solution
• RCPSP belongs to NP-hard
• It is impossible to calculate an optimal solution
  for hard instances
• For PSPLIB, only the J30 subset has optimal
  solutions for all instances
• Without optimal solutions
• The comparisons need to be relative to the best
  lower or upper bound
• Or relative to a predefined lower bound
  (CPM-value)
• With optimal solutions
• The calculation of the exact performance of
  algorithms is possible
• The study of instance's complexity will have a
  more solid base
• The proposed generator GenRes
• Uses a network and desired resource indicators
  RF/RC
• Generates resource information
• Returns a RCPSP instance with an optimal solution

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2. Generation Method

• Generation arguments
• A Network
• K of extra precedence relations
• SR of saturated resources
• Optional (default value is read from input
  instance)
• Number R of resources
• Desired resource indicators RF/RC
• Desired sum of all processing times
• Phases
• 1. Extra precedence relations
• 2. Resource usage
• 3. Processing times

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2. Generation MethodPhase 1. Extra precedence
relations

• Set processing times to 1
• Calculate the earliest start schedule (ESS)
• Add K extra precedence relations
• Select at random two activities A, B, to add a
  precedence relation
• Accept precedence relation from A to B if
• Adding the precedence relation change the ESS
• Total processing time does not exceed L
• If precedence relation is accepted, update the
  ESS, otherwise try another two activities

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2. Generation MethodPhase 2. Resource Usage

• For the first SR saturated resources
• Arrange activities in random order
• Set unary use of resource for the first
  activities with different start instants
• Set unary use of resource for the first
  activities that does not use the resource, until
  RF is attained
• For other non saturated resources
• Arrange activities in random order
• Set unary use of resource for the first
  activities, until RF is attained
• Until the resource is saturated, or RC is
  archived
• Select at random an activity with a resource
  usage
• Increase resource usage of activity if resource
  usage in activity start instant is less than
  resource capacity

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2. Generation MethodPhase 3. Processing Times

• Repeat until the sum of all processing times is
  attained
• Select a start instance at random
• Increase processing times of activities that
  start at that instant
• Calculate the ESS and save it as an optimal
  schedule
• Discard the extra precedence relations and return
  the original network with generated resources

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3. Tests

• Questions
• Is GenRes capable of generating instances of all
  types?
• Does the complexity of generated instances go
  from easy to hard?
• What is the influence of generator arguments, SR,
  K and R, on the results?
• The GenRes was tested using PSPLIB instances as
  argument
• The K used is 100, and SR is set to 1
• Optional arguments R/RF/RC are not set, the
  generator will try to match instance values

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3. TestsI. Is GenRes capable of generating
instances of all types?
A

• Figures
• A - RF (blue) and RC (green) original values of
  PSPLIB, versus values of generated instances
• B - RF versus RC of original (blue) and generated
  (green)
• Comments
• For all instances, original RF and RC values are
  accomplished
• The answer is yes, if RF and RC cover all types
  of resource instances

B
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3. TestsII. Does the complexity of generated
instances go from easy to hard?

• Figures
• C - parallel versus serial scheduling of LST in
  PSPLIB
• D - serial LST rule value, relative to the best
  upper bound (blue), best lower bound (green), and
  to the optimal solution (red) for generated
  instances
• Comments
• Using lower or upper bound may lead to very
  different conclusions
• About half of PSPLIB instances are easy
• Generated instances are more equally distributed
  from easy to hard

C
D
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3. TestsIII. What is the influence of generator
argument SR on the results? (1/3)

• Figure
• E - performance of serial LST rule for the
  generated resources with SR equal to 1 (blue), 2
  (green) and 3 (red)
• Comments
• The average complexity of instances increase when
  SR increase
• There are always some easy instances

E
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3. TestsIII. What is the influence of generator
argument K on the results? (2/3)

• Figures
• F - LST rule for K from 100 to 4
• G - LST rule for K equal to 100, 2 and 1
• Comments
• High value for K does not make much difference
• Value 1 or 2 for K increase the number of easy
  instances
• The number of hard instances does not decrease
  very much even with K1

F
G
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3. TestsIII. What is the influence of generator
argument R on the results? (3/3)
H

• Figures
• H - LST rule for R from 2 to 16
• I - scatter plot of R2 vs R4 and R8 vs R16
• Comments
• Increasing R makes the number of instances of
  average difficulty decrease and the number of
  hard instances increase
• An instance that is hard with R8 is hard with
  R16

I
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4. Conclusions

• The GenRes generator that returns an optimal
  solution was presented
• Lower and upper bounds can be very different in
  hard instances
• The generator can produce instances diverse in
  RF/RC
• The distribution of instance complexity is well
  distributed
• Increasing SR and R increases the instance
  hardness/difficulty
• K appears to have no effect on the complexity,
  except for very low values

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4. Future Work

• Research on instance complexity
• What makes an instance hard?
• Is it possible to explain the instance hardness
  with a set of indicators?
• Has morphology something to do with it?
• Generation of an instance set
• Diverse not only in RF/RC but also in R and SR
• Diverse in all indicators related with complexity
• Comparing algorithms
• Average performance of an algorithm
• Average performance of worst 5 instances of an
  algorithm (some type of worst case analysis)

17
Generating RCPSP instances with Known Optimal
Solutions

• José Coelho
• jcoelho_at_univ-ab.pt
• Generator and generated instances in
• http//jcoelho.m6.net/edicao3.asp?pa3569