Optimal Dispatching of Welding Robots Cornelius Schwarz and Jrg - - PowerPoint PPT Presentation

Optimal Dispatching of Welding Robots

Cornelius Schwarz and Jörg Rambau

Lehrstuhl für Wirtschaftsmathematik Universität Bayreuth Germany

Aussois January 

Application: Laser Welding in Car Body Shops

Application: Laser Welding in Car Body Shops

Question

How many laser sources are needed to supply the robots with energy – only one at a time – such that all weldings jobs can be processed in a given time?

The Laser Sharing Problem (LSP)

The Laser Sharing Problem (LSP)

Given

Robots, jobs, laser sources

The Laser Sharing Problem (LSP)

Given

Robots, jobs, laser sources

Task

Find an assignment of jobs to robots and robots to laser sources and a scheduled tour (i. e., an order of job start and end points with start and end times) for every robot so that

• all jobs are served
• robots assigned to identical laser sources

do not weld simultaneously

• the makespan is minimized

The Laser Sharing Problem (LSP)

Given

Robots, jobs, laser sources

Task

Find an assignment of jobs to robots and robots to laser sources and a scheduled tour (i. e., an order of job start and end points with start and end times) for every robot so that

• all jobs are served
• robots assigned to identical laser sources

do not weld simultaneously

• the makespan is minimized

Observation: LSP is NP-hard (reduction of TSP)

The Laser Sharing Problem (LSP)

Given

Robots, jobs, laser sources

Task

Find an assignment of jobs to robots and robots to laser sources and a scheduled tour (i. e., an order of job start and end points with start and end times) for every robot so that

• all jobs are served
• robots assigned to identical laser sources

do not weld simultaneously

• the makespan is minimized

Observation: LSP is NP-hard (reduction of TSP) Remark: Collision avoidance skipped for the moment

The Laser Sharing Problem (LSP)

Given

Robots, jobs, laser sources

Task

Find an assignment of jobs to robots and robots to laser sources and a scheduled tour (i. e., an order of job start and end points with start and end times) for every robot so that

• all jobs are served
• robots assigned to identical laser sources

do not weld simultaneously

• the makespan is minimized

Observation: LSP is NP-hard (reduction of TSP) Remark: Collision avoidance skipped for the moment Problem: Driving times ⇒ artificial data for computational results

Goal

Goal

Solve LSP of industrial scale ≈  jobs, ≤  robots, ≤  laser sources

Goal

Solve LSP of industrial scale ≈  jobs, ≤  robots, ≤  laser sources

So far:

• [Tuchscherer et. al. ]:  jobs for fixed robot paths with

MILP and cplex

• [Schneider ]: ≈  jobs,  robots, ≤  laser sources

(– days CPU-time)

Goal

Solve LSP of industrial scale ≈  jobs, ≤  robots, ≤  laser sources

So far:

• [Tuchscherer et. al. ]:  jobs for fixed robot paths with

MILP and cplex

• [Schneider ]: ≈  jobs,  robots, ≤  laser sources

(– days CPU-time)

New: [R. & Schwarz ]

•  jobs,  robots, – laser sources

solved to optimality in  min

•  jobs,  robots, – laser sources

solved to optimality in  h (optimal solution found in  min with provable gap <  )

LP Bounds

50 100 150 200 250 300 350 400 5 10 15 20 25 30 gap in % number of jobs LP Bounds for 2 to 34 jobs linear ordering time expanded network

LP Bounds

50 100 150 200 250 300 350 400 5 10 15 20 25 30 gap in % number of jobs LP Bounds for 2 to 34 jobs linear ordering time expanded network

Makespan ⇒ weak LP Bounds

-Server Problem

welding lines possible transversal lines

-Server Problem

welding lines possible transversal lines

 robot only → rural postman problem

-Server Problem

welding lines possible transversal lines

 robot only → rural postman problem NP-hard, but fast solvable for the scale of the LSP (≈  jobs)

-Server Problem

welding lines possible transversal lines

 robot only → rural postman problem NP-hard, but fast solvable for the scale of the LSP (≈  jobs) (e.g., by concorde [Applegate, Bixby, Chvátal, Cook])

Combinatorial Bounds – Idea

LSP with Fixed Assignments

Assume:

• robot-laser assignment (r) ∈ {laser sources}
• job-robot assignment (r) ⊆ {jobs}

→ LSP( , )

Combinatorial Bounds – Idea

LSP with Fixed Assignments

Assume:

• robot-laser assignment (r) ∈ {laser sources}
• job-robot assignment (r) ⊆ {jobs}

→ LSP( , ) Given a partial scheduled tour

• r,(p,q,t),...,(pnr,qnr,tnr)

→ completion by solving -server problem through remaining jobs → lower bound for LSP( , ).

Combinatorial Bounds – Idea

LSP with Fixed Assignments

Assume:

• robot-laser assignment (r) ∈ {laser sources}
• job-robot assignment (r) ⊆ {jobs}

→ LSP( , ) Given a partial scheduled tour

• r,(p,q,t),...,(pnr,qnr,tnr)

→ completion by solving -server problem through remaining jobs → lower bound for LSP( , ). → Branch-and-Bound (B&B) over partial scheduled tours

Combinatorial bounds – Subproblem

welding lines given transversal lines

R S R S

time

?

1 2 3

1 2 3 4

4 a

a

b

b

partial scheduled tours

Combinatorial bounds – Subproblem

welding lines given transversal lines start position end position possible transversal lines

R S R S

time

?

1 2 3

1 2 3 4

4 a

a

b

b

subproblem

Combinatorial Bounds – Subproblem Solution

welding lines given transversal lines start position end position transversal lines of 1-server solution

R S R S

time

1 2 3

1 2 3 4

4 a

a

b

b

solution of subproblem

Combinatorial Bounds – Subproblem Solution

welding lines given transversal lines start position end position transversal lines of 1-server solution

R S R S

time

1 2 3

1 2 3 4

4 a

a

b

b

solution of subproblem

not feasible!

Combinatorial Bounds – Subproblem Solution

welding lines transversal lines

R S R S

time

1 2 3

1 2 3 4

4 a

a

b

b

feasible heuristic solution

Bonus: Fixed tour algorithm (e.g., Tuchscherer) → feasible solution

Computational Results for Fixed Assignments

10 20 30 40 50 5 10 15 20 25 30 gap in % number of jobs LP Bounds for 2 to 34 jobs for fixed assignment linear ordering time expanded network combinatorial relaxation

Computational Results for Fixed Assignments

10 20 30 40 50 5 10 15 20 25 30 gap in % number of jobs LP Bounds for 2 to 34 jobs for fixed assignment linear ordering time expanded network combinatorial relaxation

Observations

• Combinatorial bounds yield much better gaps.
• Evaluation very fast (< . s) using concorde.

Solving the LSP

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

. partial assignments → nodes

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

. partial assignments → nodes . -server problems → lower bounds

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

. partial assignments → nodes . -server problems → lower bounds TSP again!

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

. partial assignments → nodes . -server problems → lower bounds . schedule fixed tours → upper bounds (only leaves)

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

. partial assignments → nodes . -server problems → lower bounds . schedule fixed tours → upper bounds (only leaves) . leaf lower bound ≤ upper bound → candidate

Solving the LSP

Finding an optimal assignment

• robot-laser assignment by simple enumeration

(few candidates)

• job-robot assignment by B&B

B&B for job-robot assignment:

. partial assignments → nodes . -server problems → lower bounds . schedule fixed tours → upper bounds (only leaves) . leaf lower bound ≤ upper bound → candidate . node lower bound > upper bound → pruning

Collisions

Collisions

Collision model

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

• line-point collisions: robot movement and a robot position

that must not overlap in time (important, since waiting robots are obstacles!)

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

• line-point collisions: robot movement and a robot position

that must not overlap in time (important, since waiting robots are obstacles!)

Very important:

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

• line-point collisions: robot movement and a robot position

that must not overlap in time (important, since waiting robots are obstacles!)

Very important:

• Tuchscherer scheduling + collision-avoidance

→ collision-free Tuchscherer scheduling (MILP)

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

• line-point collisions: robot movement and a robot position

that must not overlap in time (important, since waiting robots are obstacles!)

Very important:

• Tuchscherer scheduling + collision-avoidance

→ collision-free Tuchscherer scheduling (MILP)

• New B&B node: partial tours for each robot (not scheduled)

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

• line-point collisions: robot movement and a robot position

that must not overlap in time (important, since waiting robots are obstacles!)

Very important:

• Tuchscherer scheduling + collision-avoidance

→ collision-free Tuchscherer scheduling (MILP)

• New B&B node: partial tours for each robot (not scheduled)

→ Lower bound by (Tuchscherer scheduling of partial tours) plus (-server solutions for the rest)

Collisions

Collision model

• line-line collisions: pair of robot movements

that must not overlap in time

• line-point collisions: robot movement and a robot position

that must not overlap in time (important, since waiting robots are obstacles!)

Very important:

• Tuchscherer scheduling + collision-avoidance

→ collision-free Tuchscherer scheduling (MILP)

• New B&B node: partial tours for each robot (not scheduled)

→ Lower bound by (Tuchscherer scheduling of partial tours) plus (-server solutions for the rest) → Upper bound by (collision-free Tuchscherer scheduling

• f (partial tours plus -server solutions for the rest))

Conclusions

Benefit of combinatorial bounds

• Sometimes better bounds than LP based ones
• Key observation:

Large scale (original problem) ↔ small scale (subproblem)

Conclusions

Benefit of combinatorial bounds

• Sometimes better bounds than LP based ones
• Key observation:

Large scale (original problem) ↔ small scale (subproblem)

Future directions:

• Verify results with realistic robot driving times

(e. g., calculated by SimPro from Kuka)

• Other applications (winter gritting, …)

Conclusions

Benefit of combinatorial bounds

• Sometimes better bounds than LP based ones
• Key observation:

Large scale (original problem) ↔ small scale (subproblem)

Future directions:

• Verify results with realistic robot driving times

(e. g., calculated by SimPro from Kuka)

• Other applications (winter gritting, …)

Thank you!