Index. optimal package assignments, 143 optimal package assignments, symmetrybreaking,

Transcription

1 A Abstract approach, 79 Amphibian coexistence aquarium, 7 executes, 16 model, modeling languages, MPSolver, 12 OR-Tools library, 12 preys, 7 run a model, solution, 17 three-step approach, 7 10 Attendant theory, 4 B Bin packing constraints, decision variables, 139 description, 137 array of weights, 142 bound-trucks, 142 constraints, 143 decision variables, 143, 145 degeneracy, 144 optimal package assignments, 143 optimal package assignments, symmetrybreaking, 147 optimal truck loads, symmetry-breaking, 147 over-constraining, 146 packages and trucks, packages list, weights, 141 simple heuristic to bound, trucks, 148 symmetry-breaking constraints, 144, symmetry_break parameter, tab, 143 truck numbers, 143 truck selection variables, 144 objective, 141 packages, 138 trucks, weight capacity, 138 variations capital budgeting, 149 knapsack, 149 objective function, 149 symmetry-breaking, 148 Serge Kruk 2018 S. Kruk, Practical Python AI Projects, 271

2 Blending problem vs. mixing problems, 32 raw/crude gasolines (see Raw/ crude gasolines) C Continuity equation, 33 Convex function, 63 Curve fitting abstract approach, 79 data and fitted curves, doubling variables approach, 81 Euclidean distance, inequalities, 80 Least-Squares approach, 77 normal equations, 78 optimal solution, 83 polynomial, putative function, 79 soft constraints, 84 Cutting stock constraints, 195 consumer rolls, decision variables, bounds computation, 198 customer demand, 197 decision variables, 198 lower and upper bounds, 196 objective function, 197 optimal solution, 199 pattern search, 196 post-process, 199 symmetry-breaking constraint, 197 objective, 194 paper waste, 192 preallocation better patterns, 200, 202 column generation, distinct patterns, 200 DualValues, 202 linear program, 202 new pattern, optimal solution, 202 optimizers, 200 termination criteria, 202 processors and solvers, 193 product rolls, 192 D, E Decision variables, 8 Diet problem create solver instance, 24 data and solution, 22 heuristic method, 21 minimal cost diet, 23 OR-Tools objects, 25 product mix problems, variations, Dijkstra s algorithm, 118 Discrete linear programs, see Integer programs (IP) 272

3 F Facility location constraints big-m approach, 165 cost, 165 supply and demand, 164 variables, 164 decision variables, 163 distribution cost, 162, objective, 163 plant building costs, Solar-1138, 162 variations, 167 Floyd-Warshall algorithm, 123 Full-time employees (FTE), G Google s Operations Research Tools (OR-Tools), 4 5, 12 H Heuristic method, 21 Hyperplane cell data and separation, 58 classification, directions, 61 equation, maximizing the margins, 86, 88 I Indicator variables, 126 Integer programs (IP) bin packing (see Bin packing) decision variables, 126 description, 125 elements, 125 indicator variables, 126 integral variables, 126 pseudo-integral models, 125 set cover problem, 126 set packing, 134 TSP (see Travelling salesman problem (TSP)) J, K Job shop scheduling constraints, 185 decision variable, 185 description, 184, graphical representation, 189 machine and duration, 185 machines, 184 objective, 186 variables, 186 variations, 189 L The Lady or the Tiger, 257 Least-Squares approach,

4 Linear continuous models elements, 19 objective function, 20 product mix problems, project management (see Project management) M Maximum flow (maxflow) assign workers to jobs, conservation of flow, 95 decision variables, 92, 95 96, 98 network-related problems, 90 objective functions chained sources and cycles, dual objective, 94 optimal value, 93 reversal, 94 sources to sinks, 92 substance flows, 90 variations, 98 visual representation, network flow, 91 water, oil, and electricity, 91 Minimax problem, 44 Minimum cost (mincost) flow bipartite graph, constraints, 102 decision variables, 101 electrical distribution cost, 100, objectives, 102 power distribution model, power plants, 99 Solar-1138 Inc., 99 supply and demand, 102 variations, Mixed integer programs (MIP) classical problems, 161 facility location, 161 job shop scheduling, 184 multi-commodity flow, 168 staffing level, 176 Multi-commodity flow all-pairs shortest paths, constraints, 170 cost matrices, 168 decision variables, 169 description, 168, instances, 176 objective, 170 Multiple linear and integer solvers (MPSolver), 12 N Network models constraints, 90 Erdös number, 89 integrality, 90 maximum flow (maxflow), 90 minimum cost (mincost) flow, 99 movie buffs,

5 rounding, 90 shortest paths (see Shortest paths) structural description, 89 transshipment, 106 Non-convex piecewise function binary variables, cost function, 205 decision variables, 206 incorrect solution, 207 maximax and minimin, select k adjacent variables out of n, select k constraints out of n binary variables, 215, 220 bounds_on_box, if and only if condition, 221 indicator variable, 215, 217 integer variables, 216 logical expressions, 216 parameters, 216 reify_force, 218 reify_raise, upper bound and lower bound, 217 select k out of n variables, O Objective function, 20 Optimization problems best route, 2 goal, 2 model, 2 OR-Tools MPSolver all_different predicate, 269 bounds extraction, 269 decision variables array declaration, 265 continuous, D array, 265 integer, generic constraint declaration, 265 higher-level constraints, library ddeclaration, 263 objective function, 266 optimal value and solution, 267 simple algebraic constraints, 265 solver instance, 263, 264 solver invocation, 266 sum operator, 266 wrapper functions, optimal value and solutions, 267 P, Q Part-time employee (PTE), 181, 183 Piecewise function constraint, 67 convex cost function, 66 defined, 65, minimizing non-linear functions, non-convex, objective function,

6 Principle of Optimality, 123 Product mix problems, Project management absolute value problems, 44 alternate solution, 42 decision variables, 40, graphical representation, 43 house construction, 38 minimax problem, 44 multiple objectives, 43 optimal solution, properties, 38 tasks, 39 Pseudo-chess problems attacking rooks, 246 columns and rows extraction utility, 247 diagonal extraction helper functions, maxpiece general model, 249 maxrook, objective function, 246 positions, rooks, 246 queens and bishops, runtime, increasing board size, Puzzles description, 191, 245 The Lady or the Tiger, 257 pseudo-chess problems, 246 Send More Money!, Sudoku, 251 Python expression in math, 6 models, 5 text, 3 R Raw/crude gasolines blending problem, 31 constraints, 34 continuity equation, 33 model, 34 non-constraining equations, 33 octane rating, 31, 34 oils flavor, 37 refined gasolines, 31 solution, 36 stab, 32 two-dimensional, 36 S Send More Money puzzle, Set cover problem binary/indicator variable, 128 Boolean variables, 128 constraints, 129 contracts, suppliers, 128 electric cars, 127 binary variable, 131 COIN-OR project, 131 cost function,

7 decision variable declaration, 130 IntVar and NumVar, 132 optimal solution, 133 solver instantiation, 130 suppliers and parts, 132 two-dimensional array, 131 General Engine, objective, 129 supplier and parts, 127, 129 variations crew scheduling problem, 133 fire stations, 134 telecommunications, 134 virus detection, 134 Set packing airline crew scheduling, constraints, 136 decision variables, 135, objective, 136 optimal solution, 137 pilots, 135 universal set and subsets, 134 variations, 137 Shortest paths decision variables, 115 definition, 113 Dijkstra s algorithm, 118 distance matrix, , 117, 115, 117 Google Maps, 113 objective function, 115, 117 optimal solution, 117 variations all-pairs function, cycle-free directed graph, 119 Floyd-Warshall algorithm, 123 longest paths, 118 minimization to maximization, 119 Principle of Optimality, 123 product, 118 project management, critical paths, tree model, Soap and oils acid content (Aj), 46 constraints, cost of oils, 47 decision variables, 48 executable code, 51 52, 54 fatty acid content targets, 47 initial inventory, 48 objective function, 51 variations, 54, Sports timetabling constraints, 235 data, 234 decision variables, 234 additional cuts, constraints, 241 decision variables, 238, 240 instances,

8 Sports timetabling (cont.) numbers of teams and games per week, 241 One game per week per team, 240 optimal solution, ordered pairs generation, 244 parameters, 237 professional league, 240 relaxation, 240 subtour elimination constraints, 244 objective functions, 236 parameters, 234 relaxation tightening, 234 schedule of games, 234 variations, 245 Staffing level airlines, 178 constraints FTE must work, 180 minimum full-time staff, 180 PTE and FTE, 181 time intervals, 179 decision variables, 179 decomposition, 178 description, 176, integer program, 178 objective, 179 requirement matrix, 177 shifts, 177 shifts types, 177 solar system, 178 variations, 184 Staff scheduling constraints, 228 course sections, 224 decision variables, 227, instructors, 224 list of preferences instructors, 226 pairs, 227 sets, 227 list of sections offered, 225 objective, preferences pairs, variations, 233 Sudoku binary variables, 251 constraints, 251 decision variables, 252 description, 251 grid position, 251 helper functions, 252 indicator variables, 254 objective function, 252 solution, 254 variable declaration, 252 T, U, V, W, X, Y, Z Text, 3 4 Transshipment characteristics, 106 conservation of flow,

9 constraints, data, 107 distribution cost, , mininimum cost flow, 107 nodes, 109 objective, 109 optimal solution, 112 supply and demand, 107 two-dimensional variable, 108 variations, Travelling salesman problem (TSP) circuit design, 150 constraints, decision variables, 151 description, 150 distance matrix, iterations and subtours, 155 main loop, 155 subtour elimination constraints, , 157 subtour extraction, 154 successive (partial) solutions, 156 objective, 152 power supply, 150 variations,