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

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

Feb 16, 2024 549 likes •998 views

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

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Combinatorial optimization @

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

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Introduction Deep dive Benefits Challenges

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Introduction Deep dive Benefits Challenges

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Combinatorial optimization

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

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Solvers

MIP LP SAT CP Graph VRP

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

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OR-tools

MIP LP SAT CP Graph VRP

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

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Image stabilization

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Datacenter optimization

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Street view

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Loon

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Introduction Deep dive Benefits Challenges

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min/max c0 + cTx lbct ≤ Ax ≤ ubct lbvar ≤ x ≤ ubvar xj ∈ Z, j∈J

MIP model

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Place items

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

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

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

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Redundancy

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Constants int Copies(i) double Required(i, r) double Available(b, r) Indices Item i = 1..I Bin b = 1..B Resource r = 1..R Variables place(i, b) in [0..Copies(i)] Objective 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)

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Fault tolerance

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

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Balance

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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..ICopies(i) / B ≤ surplus(b) Variables place(i, b) in {0, 1} surplus(b) in [0, +inf) Constants int Copies(i) double Required(i, r) double Available(b, r) Indices Item i = 1..I Bin b = 1..B Resource r = 1..R Objective min ∑b = 1..Bsurplus(b)

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Minimize churn

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

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

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Introduction Deep dive Benefits Challenges

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Algorithm Input

Imperative vs Declarative

Action Input Action Model Solver

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Minimalistic

min/max c0 + cTx lbct ≤ Ax ≤ ubct lbvar ≤ x ≤ ubvar xj∈ Z j∈J

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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..ICopies(i) / B ≤ surplus(b) surplus(b) ≤ max_surplus ∑b = 1..B∑i = 1..IPlaced(i, b) * (1 - place(i, b)) ≤ MaxChange

Modular

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Algorithm Input Action

Encapsulated

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Algorithm Input Action

Encapsulated

Input Input Action Action

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Input Action

Encapsulated

Input Input Action Action Model Solver

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Introduction Deep dive Benefits Challenges

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Define Prototype

Meet

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Define Prototype Collect data Implement Verify

Solve Meet

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Define Prototype Collect data Implement Verify

Solve Meet

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Define Prototype Collect data Implement Verify

Solve Meet

Redefine

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Define Prototype Collect data Implement Verify Deploy Monitor Redefine

Solve Land Meet

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Define Prototype Collect data Implement Verify Deploy Monitor Add features Redefine

Solve Land Maintain Meet

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Define Prototype Collect data Implement Verify Deploy Monitor Add features Redefine

Solve Land Maintain Meet

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Define Prototype Collect data Implement Verify Deploy Monitor Add features Redefine

Solve Land Maintain Meet

1 month +1 year 3 months forever :-)

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Time spent

DISCUSSION GET DATA CODE EXPLAIN RESULTS

Investigate Implement

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Thank you!

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