Optimal Dispatching of Welding Robots
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1 Optimal Dispatching of Welding Robots Cornelius Schwarz and Jörg Rambau Lehrstuhl für Wirtschaftsmathematik Universität Bayreuth Germany Aussois January 2009
2 Application: Laser Welding in Car Body Shops
3 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?
4 The Laser Sharing Problem (LSP)
5 The Laser Sharing Problem (LSP) Given Robots, jobs, laser sources
6 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
7 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)
8 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
9 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
10 Goal
11 Goal Solve LSP of industrial scale 30 jobs, 6 robots, 6 laser sources
12 Goal Solve LSP of industrial scale 30 jobs, 6 robots, 6 laser sources So far: [Tuchscherer et. al. 2006]: 34 jobs for fixed robot paths with MILP and cplex [Schneider 2006]: 14 jobs, 3 robots, 3 laser sources (3 4 days CPU-time)
13 Goal Solve LSP of industrial scale 30 jobs, 6 robots, 6 laser sources So far: [Tuchscherer et. al. 2006]: 34 jobs for fixed robot paths with MILP and cplex [Schneider 2006]: 14 jobs, 3 robots, 3 laser sources (3 4 days CPU-time) New: [R. & Schwarz 2008] 34 jobs, 3 robots, 1 3 laser sources solved to optimality in 16 min 32 jobs, 3 robots, 1 3 laser sources solved to optimality in 340 h (optimal solution found in 17 min with provable gap < 1 %)
14 LP Bounds LP Bounds for 2 to 34 jobs linear ordering time expanded network 300 gap in % number of jobs
15 LP Bounds LP Bounds for 2 to 34 jobs linear ordering time expanded network 300 gap in % number of jobs Makespan weak LP Bounds
16 1-Server Problem welding lines possible transversal lines
17 1-Server Problem welding lines possible transversal lines 1 robot only rural postman problem
18 1-Server Problem welding lines possible transversal lines 1 robot only rural postman problem NP-hard, but fast solvable for the scale of the LSP ( 30 jobs)
19 1-Server Problem welding lines possible transversal lines 1 robot only rural postman problem NP-hard, but fast solvable for the scale of the LSP ( 30 jobs) (e.g., by concorde [Applegate, Bixby, Chvátal, Cook])
20 Combinatorial Bounds Idea LSP with Fixed Assignments Assume: robot-laser assignment (r) {laser sources} job-robot assignment (r) {jobs} LSP(, )
21 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 or,(p 1,q 1,t 1 ),...,(p nr,q nr,t nr ) completion by solving 1-server problem through remaining jobs lower bound for LSP(, ).
22 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 or,(p 1,q 1,t 1 ),...,(p nr,q nr,t nr ) completion by solving 1-server problem through remaining jobs lower bound for LSP(, ). Branch-and-Bound (B&B) over partial scheduled tours
23 Combinatorial bounds Subproblem b 1 partial scheduled tours a R welding lines given transversal lines S R S a b time?
24 Combinatorial bounds Subproblem subproblem b 1 a R start position end position welding lines given transversal lines possible transversal lines S R S a b time?
25 Combinatorial Bounds Subproblem Solution solution of subproblem b 1 a R start position end position welding lines given transversal lines S transversal lines of 1-server solution R S a b time
26 Combinatorial Bounds Subproblem Solution solution of subproblem b 1 a R start position end position welding lines given transversal lines S transversal lines of 1-server solution R S a b time not feasible!
27 Combinatorial Bounds Subproblem Solution feasible heuristic solution b 1 a R welding lines transversal lines S R S a b time Bonus: Fixed tour algorithm (e.g., Tuchscherer) feasible solution
28 Computational Results for Fixed Assignments LP Bounds for 2 to 34 jobs for fixed assignment linear ordering time expanded network combinatorial relaxation gap in % number of jobs
29 Computational Results for Fixed Assignments LP Bounds for 2 to 34 jobs for fixed assignment linear ordering time expanded network combinatorial relaxation gap in % number of jobs Observations Combinatorial bounds yield much better gaps. Evaluation very fast (< 0.1 s) using concorde.
30 Solving the LSP
31 Solving the LSP Finding an optimal assignment robot-laser assignment by simple enumeration (few candidates) job-robot assignment by B&B
32 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:
33 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: 1. partial assignments nodes
34 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: 1. partial assignments nodes 2. 1-server problems lower bounds
35 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: 1. partial assignments nodes 2. 1-server problems lower bounds TSP again!
36 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: 1. partial assignments nodes 2. 1-server problems lower bounds 3. schedule fixed tours upper bounds (only leaves)
37 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: 1. partial assignments nodes 2. 1-server problems lower bounds 3. schedule fixed tours upper bounds (only leaves) 4. leaf lower bound upper bound candidate
38 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: 1. partial assignments nodes 2. 1-server problems lower bounds 3. schedule fixed tours upper bounds (only leaves) 4. leaf lower bound upper bound candidate 5. node lower bound > upper bound pruning
39 Collisions
40 Collision model Collisions
41 Collision model Collisions line-line collisions: pair of robot movements that must not overlap in time
42 Collision model Collisions 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!)
43 Collision model Collisions 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:
44 Collision model Collisions 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)
45 Collision model Collisions 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)
46 Collision model Collisions 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 (1-server solutions for the rest)
47 Collision model Collisions 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 (1-server solutions for the rest) Upper bound by (collision-free Tuchscherer scheduling of (partial tours plus 1-server solutions for the rest))
48 Conclusions Benefit of combinatorial bounds Sometimes better bounds than LP based ones Key observation: Large scale (original problem) small scale (subproblem)
49 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, )
50 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!
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