Combinatorial optimization @ CO@Work 2020, Pawel Lichocki, - PowerPoint PPT Presentation

Combinatorial optimization @ CO@Work 2020, Pawel Lichocki, 25.09.2020 htups://developers.google.com/optimization

Introduction Deep dive Benefits Challenges

Introduction Deep dive Benefits Challenges

Combinatorial optimization 3 2 8 4 6 5 3 1 6 3 5 4 1 6 4 8 7 9 9 5 8 7 9 6 2 8

Solvers 3 2 8 4 6 5 3 1 6 3 5 4 1 6 4 8 7 9 9 5 8 7 9 6 2 8 VRP CP MIP LP SAT Graph

OR-tools 3 2 8 4 6 5 3 1 6 3 5 4 1 6 4 8 7 9 9 5 8 7 9 6 2 8 VRP CP MIP LP SAT Graph

Image stabilization

Datacenter optimization

Street view

Loon

Introduction Deep dive Benefits Challenges

MIP model min/max c 0 + c T x lb ct ≤ Ax ≤ ub ct lb var ≤ x ≤ ub var x j ∈ Z, j ∈ J

Place items

Indices Variables Constants Item i = 1..I double Required(i, r) Bin b = 1..B double Available(b, r) Resource r = 1..R Constraints Objective

Indices Variables Constants Item i = 1..I place(i, b) in {0, 1} double Required(i, r) Bin b = 1..B double Available(b, r) Resource r = 1..R Constraints Objective

Indices Variables Constants Item i = 1..I place(i, b) in {0, 1} double Required(i, r) Bin b = 1..B double Available(b, r) Resource r = 1..R Constraints for item i = 1..I: ∑ b = 1..B place(i, b) = 1 for resource r = 1..R: for bin b = 1..B: ∑ i = 1..I Required(i, r) * place(i, b) ≤ Available(b, r) Objective

Redundancy

Indices Variables Constants Item i = 1..I place(i, b) in [0..Copies(i)] int Copies(i) Bin b = 1..B double Required(i, r) Resource r = 1..R double Available(b, r) Constraints for item i = 1..I: ∑ b = 1..B place(i, b) = Copies(i) for resource r = 1..R: for bin b = 1..B: ∑ i = 1..I Required(i, r) * place(i, b) ≤ Available(b, r) Objective

Fault tolerance

Indices Variables Constants Item i = 1..I place(i, b) in {0, 1} int Copies(i) Bin b = 1..B double Required(i, r) Resource r = 1..R double Available(b, r) Constraints for item i = 1..I: ∑ b = 1..B place(i, b) = Copies(i) for resource r = 1..R: for bin b = 1..B: ∑ i = 1..I Required(i, r) * place(i, b) ≤ Available(b, r) Objective

Balance

Indices Variables Constants Item i = 1..I place(i, b) in {0, 1} int Copies(i) Bin b = 1..B surplus(b) in [0, +inf) double Required(i, r) Resource r = 1..R double Available(b, r) Constraints for item i = 1..I: ∑ b = 1..B place(i, b) = Copies(i) for resource r = 1..R: for bin b = 1..B: ∑ i = 1..I Required(i, r) * place(i, b) ≤ Available(b, r) for bin b = 1..B: ∑ i = 1..I place(i, b) - ∑ i = 1..I Copies(i) / B ≤ surplus(b) Objective min ∑ b = 1..B surplus(b)

Minimize churn

Indices Variables Constants Item i = 1..I place(i, b) in {0, 1} int Copies(i) Bin b = 1..B surplus(b) in [0, +inf) double Required(i, r) Resource r = 1..R double Available(b, r) int MaxChange Constraints bool Placed(i, b) for item i = 1..I: ∑ b = 1..B place(i, b) = Copies(i) for resource r = 1..R: for bin b = 1..B: ∑ i = 1..I Required(i, r) * place(i, b) ≤ Available(b, r) for bin b = 1..B: ∑ i = 1..I place(i, b) - ∑ i = 1..I Copies(i) / B ≤ surplus(b) ∑ b = 1..B ∑ i = 1..I Placed(i, b) * (1 - place(i, b)) ≤ MaxChange Objective min ∑ b = 1..B surplus(b)

Multi-dimensional multi-packing with redundancy, fault tolerance, balancing, and reducing churn

Introduction Deep dive Benefits Challenges

Imperative vs Declarative Input Algorithm Action Input Model Solver Action

Minimalistic min/max c 0 + c T x lb ct ≤ Ax ≤ ub ct lb var ≤ x ≤ ub var x j ∈ Z j ∈ J

Modular Constraints for item i = 1..I: ∑ b = 1..B place(i, b) = Copies(i) for resource r = 1..R: for bin b = 1..B: ∑ i = 1..I Required(i, r) * place(i, b) ≤ Available(b, r) for bin b = 1..B: ∑ i = 1..I place(i, b) - ∑ i = 1..I Copies(i) / B ≤ surplus(b) surplus(b) ≤ max_surplus ∑ b = 1..B ∑ i = 1..I Placed(i, b) * (1 - place(i, b)) ≤ MaxChange

Encapsulated Input Algorithm Action

Encapsulated Input Input Input Algorithm Action Action Action

Encapsulated Input Input Model Solver Input Action Action Action

Introduction Deep dive Benefits Challenges

Meet Define Prototype

Meet Solve Collect data Define Prototype Implement Verify

Meet Solve Collect data Define Prototype Implement Verify

Meet Solve Redefine Collect data Define Prototype Implement Verify

Meet Solve Land Redefine Deploy Collect data Define Prototype Implement Monitor Verify

Meet Solve Land Maintain Redefine Deploy Collect data Define Add features Prototype Implement Monitor Verify

Meet Solve Land Maintain Redefine Deploy Collect data Define Add features Prototype Implement Monitor Verify

Meet Solve Land Maintain 1 month +1 year 3 months forever :-) Redefine Deploy Collect data Define Add features Prototype Implement Monitor Verify

Time spent Investigate Implement GET DATA DISCUSSION CODE EXPLAIN RESULTS

Thank you! CO@Work 2020, Pawel Lichocki, 25.09.2020 htups://developers.google.com/optimization