Perspective of mathematical optimization and its applications Tokyo University of Marine Science and Technology Mikio Kubo

How to Solve Real Combinatorial Optimization Problems Quickly Mixed Integer Programming (MIP) Solver Constraint Programming (CP) Solver Scheduling Solver (or develop (meta)heuristics)... using Python Language

Why Python? We can do anything by importing some modules Optimization import gurobipy (MIP) import SCOP (CP) Draw graphs import networkX Also fly! import antigravity ? /

Mixed Integer Programming (MIP) Variables x : Real or Integer or Binary Constraints Linear or (convex) Quadratic Expressions minimize c’x+x’Qx (objective function) subject to Ax=b (constraints)

What’s Gurobi? MIP solver Developed by: Zonghao Gu, Edward Rothberg ， Robert Bixby Current version Free academic license

Gurobi Objects Model Variable Constraint LinExprQuadExpr Column Callbacks GRBError addVar addConstr SOS addSOS

Introduction to Gurobi (1) Create a model object model = Model("Wine Blending")

Add variable objects x1 = model.addVar(name="x1") x2 = model.addVar(name="x2") x3 = model.addVar(name="x3") Model update (needed before adding constraints; lazy update!) model.update() Introduction to Gurobi (2)

Introduction to Gurobi (3) Set the objective model.setObjective(15*x1 + 18*x2 + 30*x3, GRB.MAXIMIZE) Add constraints model.addConstr(2*x1 + x2 + x3 <= 60) model.addConstr(x1 + 2*x2 + x3 <= 60) model.addConstr(x3 <= 30) Optimize model.optimize()

Modeling with Lists Add variable objects into list x=[ ] for i in range(1,4): var=model.addVar(name=“x[ %s]”%i ) x.append(var) Add constraint “x1 + x2 + x3 <= 2” model.addConstr( sum(x) <= 2 ) or model.addConstr( quicksum(x) <= 2 ) X[1]X[2]X[3] ・・・

Modeling with Dictionaries Dictionary that maps keys (“Dry”, “Medium”, “Sweet”) to variable objects x={ } x[“Dry”]= model.addVar(name=“Dry”) x[“Medium”]= model.addVar(name=“Medium”) x[“Sweet”]= model.addVar(name=“Sweet”) Key “Hello”, “Dry Wine” Value “Nihao” Variable Object Mapping

Modeling with Dictionaries Add constraint “2 x1 + x2 + x3 <= 30” model.addConstr( 2* x[“Dry”]+ x[“Medium”] +x[“Sweet”] <=30 ) multidict Blends, Profit = multidict({"Dry":15, "Medium":18, "Sweet":30}) => Blends=["Dry", "Medium“, "Sweet“] Profit[“Dry”]=15, Profit[“Medium”]=18,...

Wine Blending with Dictionaries (1) multidict Blends, Profit = multidict({"Dry":15, "Medium":18, "Sweet":30}) => Blends=["Dry", "Medium“, "Sweet“] List of Keys Profit[“Dry”]=15, Profit[“Medium”]=18,... Grapes, Inventory = multidict({"Alfrocheiro":60, "Baga":60, "Castelao":30}) Use = { ("Alfrocheiro","Dry"):2, ("Alfrocheiro","Medium"):1, ("Alfrocheiro","Sweet"):1, ("Baga","Dry"):1,.... }

Wine Blending with Dictionaries (2) x = {} for j in Blends: x[j] = model.addVar(vtype="C", name="x[%s]"%j) model.update() model.setObjective(quicksum(Profit[j]*x[j] for j in Blends), GRB.MAXIMIZE) for i in Grapes: model.addConstr(quicksum(Use[i,j]*x[j] for j in Blends) <= Inventory[i], name="use[%s]"%i) model.optimize()

k-median problem A facility location problem with min-sum objective function Number of customers n=200 Number of facilities selected from customer sites k=20

Formulation week formulation

Python Code （ 1 ） from gurobipy import * model = Model("k-median") x, y = {}, {} # empty dictionaries Key “Hanako”, (1,2) Value “127cm” Variable Object Mapping Dictionary Data Structure

I=range(n) J=range(n) for j in J: y[j] = model.addVar(vtype="B", name="y[%s]"%j) for i in I: x[i,j] =model.addVar( vtype="B",name="x[%s,%s]"%(i,j)) model.update() Python Code （２） model.setObjective(quicksum(c[i,j]*x[i,j] for i in I for j in J)) Add variable objects “B” means binary variable or GRB.BINARY Set the objective

Python Code （３） for i in I: model.addConstr(quicksum(x[i,j] for j in J) = = 1, "Assign[%s]"%i) for j in J: model.addConstr(x[i,j] <= y[j], "Strong[%s,%s]"%(i,j)) model.addConstr(quicksum(y[j] for j in J) = = k, "k_median")

Python Code （４） … model.optimize() print “Opt.value=”,model.ObjVal edge=[] for (i,j) in x: if x[i,j].X= =1: edge.append((i,j)) return edge

Python Code （５） import networkx as NX #networkX module import matplotlib.pyplot as P #prepare drawing P.ion() G = NX.Graph() #graph object G.add_nodes_from(range(n)) #add nodes for (i,j) in edge: #add edges G.add_edge(i,j) NX.draw(G)

Optimize a model with 401 Rows, Columns and NonZeros … Explored 1445 nodes (63581 simplex iterations) in seconds Thread count was 2 (of 2 available processors) Optimal solution found (tolerance 1.00e-04) Best objective e+01, best bound e+01, gap % Opt.value= Weak formulation (result) n=200,k=20

Upper and lower bounds (Weak Formulation)

Optimize a model with Rows, Columns and NonZeros … Explored 0 nodes (1697 simplex iterations) in 3.33 seconds (No branching ！） Thread count was 2 (of 2 available processors) Optimal solution found (tolerance 1.00e-04) Best objective e+01, best bound e+01, gap 0.0% Opt.value= Strong formulation (result)

k-center problem A facility location problem with min-max object n=100 customers ， k=10 facilities k-center (n=30,k=3) k-median (n=30,k=3)

Formulation

Upper and lower bounds (n=100,k=10)

k-Covering Problem =1 if customer is not covered # of uncovered customers parameter that is =1 if distance is less than or equal to θ

k-Covering+Binary Search Upper and Lower Bounds UB ， LB while UB – LB >ε: θ= (UB+LB)/2 if opt. val. of k-covering is 0 then UB = θ else LB = θ

Computational Experiments

Constraint Programming (CP) Variables x has to be selected from a finite domain (set of values) Constraints Linear, (non-convex) Quadratic, All Different or Any Types of Combinatorial Expressions

Assignment Problem (MIP/CP) Assign three jobs A,B,C to three workers 1,2,3 MIP formulation: x ij Binary CP formulation : x[i] with domain {A,B,C} AllDiff ( x[1], x[2], x[3] )

Constraint Programming Solver SCOP (Solver for COnstraint or Programming) developed by Prof. Ibaraki and Prof. Nonobe ITC (International Timetabling Competition) 2007 – Finalist for all 3 tracks (3 rd, 2 nd, 3 rd among 5 finalists) International Nurse Rostering Competition (INRC) 2010 – Finalists for all 3 tracks (2 nd,3 rd, 4 th among 5 finalists)

SCOP Objects Model Variable Linear addVariable(s) Quadratic Alldiff addConstraint

Scheduling Activities (=Variables) x has to be selected from a set of modes (=domain of CP) Resource Renewable or Non-renewable Resources (=Constraints) Temporal Constraints

Scheduling Solver OptSeq II developed by Prof. Ibaraki and Prof. Nonobe Algorithms CPU sec. # of Feasible Solutions Error from Best Known Tabu search / % Priority-rule method / % Our tabu search / % 1. [De Reyck and Herroelen, 1999] 333MHz PC 2. [Heilmann, 2001] 333MHz PC 3. 1GHz PC Multi-mode RCPSP PSPLIB [Kolisch and Sprecher, 1997]

OptSeq Objects Model Attribute Mode addActivity Resource Temporal addResource addTemporal addMode

Applications Train scheduling in a steel plant: Flow optimization using Gurobi + Detailed scheduling using OptSeq Supply Chain Modeling Language (SCML)

What’s SCML? SCML Supply Chain Optimization Models Combinatorial Optimization Models Solvers (metaheuristics, MIP/CP solvers) SCML.py

Supply Chain Optimization Models resource constrained scheduling lot-sizing logistics network design safety stock allocation economic order quantity inventory policy optimization vehicle routing

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