Combining Models and Algorithms

1
Combining Models and Algorithms

• - from Alchemy to Cookery

2
Divide and Conquer

• Decompose Problem
• Devise Solver for each Subproblem
• Combine Solvers

3
Architecture
PhD
Solver Platform
4
Overview

• Two Example Problems
• Steps from Conceptual Model to Design Model
• Schemas for Subproblem Interaction
• Transformations
• Design Model Requirements

5
Construction Scheduling
6
Construction Scheduling with Maintenance
Constraints

• No resource is used more than a fixed time limit
  without maintenance
• No more than a fixed number of resources are in
  maintenance at the same time

7
Construction Scheduling Complexity Results

• Resource problem Polynomial
• (maximum flow of minimum cost)
• Resource problem with maintenance NP-hard
• (multiprocessor scheduling problem)

8
Construction Scheduling

• Decomposition
• assign resources to tasks
• schedule resource maintenance
• Hybridisation
• pass maintenance intervals to (B)
• report success or failure to (A)
• (A) after failure, extend duration of each task
  by one maintenance interval

9
Logistics with Depots
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Logistics with Depots

• Collection and delivery time windows
• Vehicle capacity data
• Travel time data
• Vehicle load/unload time data
• Driver shift data
• Minimise cost

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Logistics with Depots

• Decomposition
• (A) consignment routing
• (B) load consolidation
• vehicle routing
• inter-depot consolidation
• vehicle assignment
• driver allocation
• Hybridisation
• route for consignment - to (B)
• consolidate with other consignments, cost - to
  (A)
• (A) choose cheapest alternative - to (B)
• (B) optimise consolidation for all consignments

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Overview

• Two Example Problems
• Steps from Conceptual Model to Design Model
• Schemas for Subproblem Interaction
• Transformations
• Design Model Requirements

13
Steps to Design Model

• Decompose
• Transform
• Add Inference
• Specify Search

• Specify Subproblem Interaction

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A Toy Example
problem1 - c1(X,Y), c2(Y,Z), c3(Z,X), c4(X,U,
V).
c2
Y
Z
c1
c3
X
U
V
c4
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Toy Example Decomposition
sub_p
problem - sub_p(X), sub_q(X). sub_p(X)
- c1(X,Y), c2(Y,Z), c3(Z,X). sub_q(X)
- c4(X,U,V).
c2
Y
Z
c1
c3
X
U
V
c4
sub_q
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Toy Example Transformation

sub_p
problem - sub_p(X), sub_q(X). sub_p(X)
- c1(X,Y), c2(Y,Z), c3(Z,X). sub_q(X)
- c5(T,X), c6(T,U), c7(T,V).
c2
Y
Z
c1
c3
X
c5
T
c6
c7
U
V
sub_q
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Toy Example Inference
problem - sub_p(X), sub_q(X). sub_p(X) -
X,Y,Z1..10, infer( c1(X,Y) ,fc) , infer(
c2(Y,Z) ,fc) , infer( c3(Z,X) ,fc) . sub_q(X)
- X,U,V1..10, infer( c5(T,X) ,fc) ,
infer( c6(T,U) ,ac) , infer( c7(T,V) ,ac) .
sub_p
c2fc
Y
Z
c1fc
c3fc
X
c5fc
T
c6ac
c7ac
U
V
sub_q
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Toy Example Search
problem - sub_p(X),sub_q(X). sub_p(X) -
X,Y,Z1..10, infer( c1(X,Y) ,fc) , infer(
c2(Y,Z) ,fc) , infer( c3(Z,X) ,fc), label(X),
check(Y) . sub_q(X) - X,U,V1..10, infer(
c5(T,X) ,fc) , infer( c6(T,U) ,ac) , infer(
c7(T,V) ,ac) .
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Steps to Design ModelConstruction Scheduling

• Decompose
• Assign resources to tasks
• Schedule resource maintenance
• Transform
• Introduce assignment booleans
• Add Inference
• Optimise linear relaxation
• FD propagation
• Specify Search
• MIP
• Partially order maintenance tasks

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Steps to Design ModelLogistics with Depots

• Decompose
• Consignment routing
• Load consolidation
• Transform
• Introduce minimisation of routes per collection
  point
• Add Inference
• Interval consistency
• Specify Search
• BB interleaved with consignment routing final
  BB

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Overview

• Two Example Problems
• Steps from Conceptual Model to Design Model
• Schemas for Subproblem Interaction
• Transformations
• Design Model Requirements

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Subproblem Interaction Direct
Subproblems solved concurrently
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Subproblem Interaction Via Search Agent
sub_p
Search agent alternates with subproblem solvers
global search agent
sub_q
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Subproblem Interaction Via Intermediates
Constraint store is blackboard forsubproblem
solvers
sub_p
global search agent
sub_q
constraint store
sub_r
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Subproblem Composition
problem
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Subproblem Interaction What is Communicated?

• Perfect Relaxation
• Via Search Agent
• Via Intermediates
• Direct

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Perfect Relaxation - A Toy Example
W
c3
c2
problem1 - c1(X,Y), c2(Y,W), c3(W,Z), c4(Z,X)
, c5(X,U,V).
Z
Y
c4
c1
X
U
V
c5
Naïve worst-case complexity o(d6)
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Perfect Relaxation No Subproblem Interaction!
problem1 - sub_1(X), sub_4(X). sub_1(X)
- sub_2(X,W), sub_3(W,X). sub_2(X,W)
- c1(X,Y),c2(Y,W). sub_3(W,X) - c3(W,Z),
c4(Z,X). sub_4(X) - c5(X,U,V).
Naïve worst-case complexity o(d3)
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Subproblem Interaction What is Communicated?

• Perfect Relaxation
• set of solutions
• Via Search Agent
• Via Intermediates
• Direct

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Reporting Success or Failure
sub_p
search agent
assignment
sub_q
NB Search method influences communication
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Reporting Cost
sub_p
search agent
assignment
sub_q
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Returning Value Choice Heuristics
sub_p
Example Limited Discrepancy Search
search agent
assignment
sub_q
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Returning Variable Choice Heuristics
sub_p
search agent
assignment
Tree Search
sub_q
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Variable Choice Heuristics Updating the
Meta-Data
sub_p
assignment
result
search agent
Constraint Names Scopes
result
sub_q
assignment
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Subproblem Interaction What is Communicated?

• Perfect Relaxation
• set of solutions
• Via Search Agent
• assignment/result/constraint scopes
• Via Intermediates
• Direct

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Info Communicated with Intermediates
sub_p
search agent
sub_q
constraint store
sub_r
NB Search method influences communication
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Info Communicated with Intermediate Repair
Search
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Info Communicated with Intermediate
Incremental Search
sub_p
fd
search agent
finite domains
assignment
fd
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Info Communicated with FD Intermediate
sub_p MAC
fd
search agent
finite domains
assignment
fd
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Deleteff
sub_p
search agent
finite domains
sub_q
sub_r
constraint graph
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Search by Adding Constraints
sub_p
search agent
finite domains
sub_q
constraint
sub_r
constraint graph
newsub
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Propagation by Adding Constraints
sub_p
search agent
finite domains
sub_q
sub_r
constraint graph
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Info Communicated with Linear Intermediate
44
Info Communicated with Linear Intermediate
linear constraint solver
linear constraint
sub_p
assignment
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Combining Solvers - Probing
sub_p
alternative constraints
variablelabels
conflicts
searchagent
assignment
linearconstraintsolver
sub_q
linear constraint
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Combining Solvers FD and Linear
sub_p
finite domains
search agent
sub_q
sub_r
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Subproblem Interaction What is Communicated?

• Perfect Relaxation
• set of solutions
• Via Search Agent
• assignment/result/constraint scopes
• Via Intermediates
• assignment/result/ constraint scopes/
• constraints/conflict
• Direct

48
Communicating Infeasible Assignments
Relaxed Problem
Full Problem
assignment
success/failure
Solve relaxed problem
Extend relaxed solution to a solution to the
full problem
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Communicating Infeasible Assignments -
Construction Scheduling
assign resources to tasks
schedule resource maintenance
assignment
success/failure
On failure, add one maintenance periodto each
task, and solve again
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Communicating Infeasible Assignments -
Construction Scheduling
create fdconstraints
assignment
finite domains
sub_p
sub_q
sub_r
linear constraint solver
search agent
createlinearconstraints
failure
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Communicating Infeasible Assignments
Relaxed Problem
Full Problem
assignment
constraint
Search for a solution
Find violated constraints
Constraint always holds for the full problem
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Communicating Infeasible Assignments Via an
Intermediate
search agent
assignment
sub_p
constraint store
constraint
Constraint posted at root of search tree
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Communicating Infeasible Assignments Via a
Linear Solver
sub_q
linear constraint solver
linear constraints
sub_r
search agent
assignment
variable labels
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Communicating Tentative Assignments
Partial Problem
Full Problem
assignment
constraint
Generate a solution
Use assignment as a search heuristic
Constraint only holds below current node in
search tree
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Communicating Tentative Assignments - Probing
assignmentconflicts
variablelabels
sub_q
constraints
sub_p
finite domains
searchagent
linearconstraintsolver
constraints
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Communicating Assignments and Costs
sub_p
sub_q
assignment
cost
Search for a feasible assignment
Evaluate cost
Local or Constructive Search
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Logistics with Depots
consignmentrouting
loadconsolidation
assignment
cost
Propose assignment
Evaluate cost
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Logistics with Depots
tentative label
search consignmentrouting
variablelabels
search loadconsolidation
cost
finitedomains
chosen label
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Local Search Under Constraints
Sub Problem
Full Problem
assignment
cost
Find a feasible neighbour
Calculate its cost bound
Skeleton and Flesh
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Local Search Under Constraints Workforce
Scheduling
createMIPconstraints
sub_p
assignment
search engineersrouting
linearconstraints
search allocate tasks to engineers
cost
variablelabels
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Lagrangian Relaxation
Main Problem With Soft Constraints
Sub Soft Constraints
assignment
cost function
Evaluate violations of soft constraints
Search for a solution
Aim Find cost function such that optimal
solution of relaxed problem is optimal solution
of full problem
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Lagrangian Relaxation Interdependence of
Subproblems

• Aim only possible if Main problem has necessary
  properties
• Termination of iteration requires further
  properties of Main
• Sub only makes sense for solving Main
• Sub cannot be tackled independently of Main

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Lagrangian Relaxation
assignment
Remainder Of Problem
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Lagrangian Relaxation
lagrangian relaxation
hard constraints
linear constraints
soft constraints
assignment
Remainder Of Problem
For cookery treat as one solver
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Column Generation
Slave Problem
Master Problem
assignment
cost
Use assignment to improve previous solutions
Search for solutions under cost constraint
Done when no admissible solutions can be found
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Column Generation Interdependence of Subproblems

• Decomposition only makes sense for column
  generation
• Master problem solution is a linear combination
  of slave problem solutions
• Slave problem can be handled independently of
  master

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Column Generation Construction Scheduling
find optimal combination for all resources
find sequence of activities for a resource
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Column Generation
linear constraints
master constraints
sub_p
dual costs
new columns
column generator
cost
slave constraints
slave problem
assignment
Still Alchemy!
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Subproblem Interaction What is Communicated?

• Perfect Relaxation
• set of solutions
• Via Search Agent
• assignment/result/constraint scopes
• Via Intermediates
• assignment/result/ constraint scopes/
• constraints/conflict
• Direct
• assignment/result/ constraint scopes/
• constraints/conflict
• transformed assignments/constraints

70
Overview

• Two Example Problems
• Steps from Conceptual Model to Design Model
• Schemas for subproblem interaction
• Transformations
• Design Model Requirements

71
Steps to Design Model

• Decompose
• Transform
• Infer
• Search

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Transformations

• N-ary to Binary
• CSP -gt MIP
• CSP Dual
• Linear Dual

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Decomposition and Transformation
sub_q
sub_p
Transformation
t_sub_p
???
74
N-ary to Binary
sub_n
Variables remain Constraints changed New
variables added
N-ary to binary
Alchemy New constraints and new variables
exported
sub_2
Cookery Only old variables exported
75
CSP to MIP

• Numeric variables remain
• Constraints reified
• Boolean variables added
• Symbolic values mapped to numeric

csp_p
CSP to MIP
Alchemy New constraints, new variables, and new
values exported
mip_p
Cookery Only old variables exported. New numeric
values automatically mapped to symbolic
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CSP Dual
sub
Variables changed Constraints changed
CSP Dual
Alchemy New constraints and new variables
exported
dual_sub
Cookery Exported information automatically
undualised!
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Linear Dual
sub
This dual is usually hidden Within the linear
solver.
Alchemy Export Shadow Prices And Reduced Costs
Linear Dual
dual_sub
Cookery Hide dual info. inside specialised agents
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Overview

• Two Example Problems
• Steps from Conceptual Model to Design Model
• Schemas for Subproblem Interaction
• Transformations
• Design Model Requirements

79
Design Model Requirements

• Represent problem decomposition
• Map problem model to subproblem models
• Represent subproblem solvers
• Map conceptual model to algorithm
• Capture solver combination
• Map algorithms to hybrid algorithm

80
Research into Design Model

• Start bottom up
• Routing
• Partitioning
• Scheduling
• Collect and analyse algorithms
• Constraint propagation
• Linear constraint solving
• Complete search
• Stochastic search
• Devise (modelling) languages
• For expressing algorithms
• For combining algorithms

81
CSP and Constraint Programming -Language Aims

• Teaching MBAs
• Familiar
• Simple
• Closed
• Developing Advanced Algorithms
• Powerful
• Integrated
• Reconfigurable

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IC-Parc Research Aims the ECLiPSe Language

• Separate modelling from behaviour
• Support an extensive toolset
• Representations
• Algorithms
• Support experimentation
• Representations
• Algorithms
• Hybrids

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IC-Parc and Parc Technologies - Applications

• Transportation
• Internet Support
• Logistics