Topic 1: Introduction 1 (Version of 1st November 2020) Pierre Flener and Jean-No¨ el Monette Optimisation Group Department of Information Technology Uppsala University Sweden Course 1DL441: Combinatorial Optimisation and Constraint Programming, whose part 1 is Course 1DL451: Modelling for Combinatorial Optimisation 1 Based partly on material by Guido Tack
Optimisation Constraint Problems Combinatorial Optimisation Modelling (in MiniZinc) Solving The MiniZinc Toolchain Course Information Part 1: Modelling for Combinatorial Optimisation Part 2: Combinatorial Optimisation and CP Contact Optimisation is a science of service: to scientists, to engineers, to artists, and to society. COCP/M4CO 1 - 2 -
MiniZinc Challenge 2015: Some Winners Problem & Model Backend & Solver Technology Costas array Mistral CP capacitated VRP iZplus hybrid Constraint GFD schedule Chuffed LCG Problems grid colouring MiniSAT(ID) hybrid Combinatorial Optimisation instruction scheduling Chuffed LCG Modelling large scheduling Google OR-Tools.cp CP (in MiniZinc) Solving application mapping JaCoP CP The MiniZinc multi-knapsack mzn-cplex MIP Toolchain portfolio design fzn-oscar-cbls CBLS Course Information open stacks Chuffed LCG Part 1: Modelling for Combinatorial project planning Chuffed LCG Optimisation Part 2: Combinatorial radiation mzn-gurobi MIP Optimisation and CP Contact satellite management mzn-gurobi MIP time-dependent TSP G12.FD CP zephyrus configuration mzn-cplex MIP COCP/M4CO 1 - 3 -
Outline 1. Constraint Problems Constraint Problems 2. Combinatorial Optimisation Combinatorial Optimisation 3. Modelling (in MiniZinc) Modelling (in MiniZinc) Solving 4. Solving The MiniZinc Toolchain 5. The MiniZinc Toolchain Course Information Part 1: Modelling for Combinatorial 6. Course Information Optimisation Part 2: Combinatorial Part 1: Modelling for Combinatorial Optimisation Optimisation and CP Contact Part 2: Combinatorial Optimisation and CP Contact COCP/M4CO 1 - 4 -
Outline 1. Constraint Problems Constraint Problems 2. Combinatorial Optimisation Combinatorial Optimisation 3. Modelling (in MiniZinc) Modelling (in MiniZinc) Solving 4. Solving The MiniZinc Toolchain 5. The MiniZinc Toolchain Course Information Part 1: Modelling for Combinatorial 6. Course Information Optimisation Part 2: Combinatorial Part 1: Modelling for Combinatorial Optimisation Optimisation and CP Contact Part 2: Combinatorial Optimisation and CP Contact COCP/M4CO 1 - 5 -
Example (Agricultural experiment design) plot1 plot2 plot3 plot4 plot5 plot6 plot7 barley Constraint corn Problems millet Combinatorial oats Optimisation rye Modelling (in MiniZinc) spelt Solving wheat The MiniZinc Toolchain Constraints to be satisfied : Course Information 1 Equal growth load: Every plot grows 3 grains. Part 1: Modelling for Combinatorial Optimisation 2 Equal sample size: Every grain is grown in 3 plots. Part 2: Combinatorial Optimisation and CP 3 Balance: Every grain pair is grown in 1 common plot. Contact Instance : 7 plots, 7 grains, 3 grains/plot, 3 plots/grain, balance 1. COCP/M4CO 1 - 6 -
Example (Agricultural experiment design) plot1 plot2 plot3 plot4 plot5 plot6 plot7 barley – – – – ✓ ✓ ✓ Constraint corn – – – – ✓ ✓ ✓ Problems millet – – – – ✓ ✓ ✓ Combinatorial oats – – – – Optimisation ✓ ✓ ✓ rye – – – – Modelling ✓ ✓ ✓ (in MiniZinc) spelt – – – – ✓ ✓ ✓ Solving wheat – – – – ✓ ✓ ✓ The MiniZinc Toolchain Constraints to be satisfied : Course Information 1 Equal growth load: Every plot grows 3 grains. Part 1: Modelling for Combinatorial Optimisation 2 Equal sample size: Every grain is grown in 3 plots. Part 2: Combinatorial Optimisation and CP 3 Balance: Every grain pair is grown in 1 common plot. Contact Instance : 7 plots, 7 grains, 3 grains/plot, 3 plots/grain, balance 1. COCP/M4CO 1 - 6 -
Example (Doctor rostering) Mon Tue Wed Thu Fri Sat Sun Doctor A Doctor B Constraint Doctor C Problems Doctor D Combinatorial Optimisation Doctor E Modelling Constraints to be satisfied : (in MiniZinc) Solving 1 #on-call doctors / day = 1 The MiniZinc 2 #operating drs / weekday ≤ 2 Toolchain Course 3 #operating drs / week ≥ 7 Information Part 1: Modelling for 4 #appointed drs / week ≥ 4 Combinatorial Optimisation Part 2: Combinatorial 5 day off after operation day Optimisation and CP Contact 6 . . . Objective function to be minimised : Cost: . . . COCP/M4CO 1 - 7 -
Example (Doctor rostering) Mon Tue Wed Thu Fri Sat Sun Doctor A call none oper none oper none none Doctor B appt call none oper none none call Constraint Doctor C oper none call appt appt call none Problems Doctor D appt oper none call oper none none Combinatorial Optimisation Doctor E oper none oper none call none none Modelling Constraints to be satisfied : (in MiniZinc) Solving 1 #on-call doctors / day = 1 The MiniZinc 2 #operating drs / weekday ≤ 2 Toolchain Course 3 #operating drs / week ≥ 7 Information Part 1: Modelling for 4 #appointed drs / week ≥ 4 Combinatorial Optimisation Part 2: Combinatorial 5 day off after operation day Optimisation and CP Contact 6 . . . Objective function to be minimised : Cost: . . . COCP/M4CO 1 - 7 -
Example (Vehicle routing: parcel delivery) Given a depot with parcels for clients and a vehicle fleet, find which vehicle visits which client when. Constraints to be satisfied : Constraint Problems 1 All parcels are delivered on time. Combinatorial Optimisation 2 No vehicle is overloaded. Modelling (in MiniZinc) 3 Driver regulations are respected. Solving 4 . . . The MiniZinc Objective function to be minimised : Toolchain Course Cost: the total fuel consumption and driver salary. Information Part 1: Modelling for Combinatorial Optimisation Part 2: Combinatorial Optimisation and CP Example (Travelling salesperson: optimisation TSP) Contact Given a map and cities, find a shortest route visiting each city once and returning to the starting city. COCP/M4CO 1 - 8 -
Applications in Air Traffic Management Demand vs capacity Airspace sectorisation Constraint Problems Combinatorial Optimisation Modelling (in MiniZinc) Solving The MiniZinc Contingency planning Workload balancing Toolchain Course Flow Time Span Hourly Rate Information From: Arlanda 00:00 – 09:00 3 Part 1: Modelling for To: west, south 09:00 – 18:00 5 Combinatorial 18:00 – 24:00 2 Optimisation From: Arlanda 00:00 – 12:00 4 Part 2: Combinatorial Optimisation and CP To: east, north 12:00 – 24:00 3 Contact . . . . . . . . . COCP/M4CO 1 - 9 -
Example (Air-traffic demand-capacity balancing) Reroute flights, in height and speed, so as to balance the workload of air traffic controllers in a multi-sector airspace: Constraint Problems Combinatorial Optimisation Modelling (in MiniZinc) Solving The MiniZinc Toolchain Course Information Part 1: Modelling for Combinatorial Optimisation Part 2: Combinatorial Optimisation and CP Contact COCP/M4CO 1 - 10 -
Example (Airspace sectorisation) Given an airspace split Find a colouring of the cells into c cells, a targeted into s connected convex sec- number s of sectors, and tors, with minimal imbalance Constraint Problems flight schedules. of the workloads of their air Combinatorial traffic controllers. Optimisation Modelling (in MiniZinc) Solving The MiniZinc Toolchain Course Information Part 1: Modelling for Combinatorial Optimisation Part 2: Combinatorial Optimisation and CP Contact There are s c possible colourings, but very few optimally satisfy the constraints: is intelligent search necessary? COCP/M4CO 1 - 11 -
Applications in Biology and Medicine Phylogenetic supertree Haplotype inference Constraint Problems Combinatorial Optimisation Modelling (in MiniZinc) Solving The MiniZinc Toolchain Medical image analysis Doctor rostering Course Information Part 1: Modelling for Combinatorial Optimisation Part 2: Combinatorial Optimisation and CP Contact COCP/M4CO 1 - 12 -
Example (What supertree is maximally consistent with several given trees that share some species?) Constraint Problems Combinatorial Optimisation Modelling (in MiniZinc) Solving The MiniZinc Toolchain Course Information Part 1: Modelling for Combinatorial Optimisation Part 2: Combinatorial Optimisation and CP Contact COCP/M4CO 1 - 13 -
Example (Haplotype inference by pure parsimony) Given n child genotypes, with homo- & heterozygous sites: · · · Constraint A C / G T C A / T C Problems · · · Combinatorial A / T G T C / G A C Optimisation · · · Modelling (in MiniZinc) find a minimal set of (at most 2 · n ) parent haplotypes: Solving The MiniZinc Toolchain · · · A C T C T C Course Information · · · Part 1: Modelling for Combinatorial A G T C A C Optimisation Part 2: Combinatorial · · · Optimisation and CP T G T G A C Contact · · · so that each given genotype conflates 2 found haplotypes. COCP/M4CO 1 - 14 -
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