Chapter 9. Introduction to Red Hat build of OptaPlanner
OptaPlanner is a lightweight, embeddable planning engine that optimizes planning problems. It helps normal Java programmers solve planning problems efficiently, and it combines optimization heuristics and metaheuristics with very efficient score calculations.
For example, OptaPlanner helps solve various use cases:
- Employee/Patient Rosters: It helps create timetables for nurses and keeps track of patient bed management.
- Educational Timetables: It helps schedule lessons, courses, exams, and conference presentations.
- Shop Schedules: It tracks car assembly lines, machine queue planning, and workforce task planning.
- Cutting Stock: It minimizes waste by reducing the consumption of resources such as paper and steel.
Every organization faces planning problems; that is, they provide products and services with a limited set of constrained resources (employees, assets, time, and money).
OptaPlanner is open source software under the Apache Software License 2.0. It is 100% pure Java and runs on most Java virtual machines (JVMs).
9.1. Planning problems
A planning problem has an optimal goal, based on limited resources and under specific constraints. Optimal goals can be any number of things, such as:
- Maximized profits - the optimal goal results in the highest possible profit.
- Minimized ecological footprint - the optimal goal has the least amount of environmental impact.
- Maximized satisfaction for employees or customers - the optimal goal prioritizes the needs of employees or customers.
The ability to achieve these goals relies on the number of resources available. For example, the following resources might be limited:
- Number of people
- Amount of time
- Budget
- Physical assets, for example, machinery, vehicles, computers, buildings
You must also take into account the specific constraints related to these resources, such as the number of hours a person works, their ability to use certain machines, or compatibility between pieces of equipment.
Red Hat build of OptaPlanner helps Java programmers solve constraint satisfaction problems efficiently. It combines optimization heuristics and metaheuristics with efficient score calculation.
9.2. NP-completeness in planning problems
The provided use cases are probably NP-complete or NP-hard, which means the following statements apply:
- It is easy to verify a specific solution to a problem in reasonable time.
- There is no simple way to find the optimal solution of a problem in reasonable time.
The implication is that solving your problem is probably harder than you anticipated, because the two common techniques do not suffice:
- A brute force algorithm (even a more advanced variant) takes too long.
- A quick algorithm, for example in the bin packing problem, putting in the largest items first returns a solution that is far from optimal.
By using advanced optimization algorithms, OptaPlanner finds a good solution in reasonable time for such planning problems.
9.3. Solutions to planning problems
A planning problem has a number of solutions.
Several categories of solutions are:
- Possible solution
- A possible solution is any solution, whether or not it breaks any number of constraints. Planning problems often have an incredibly large number of possible solutions. Many of those solutions are not useful.
- Feasible solution
- A feasible solution is a solution that does not break any (negative) hard constraints. The number of feasible solutions are relative to the number of possible solutions. Sometimes there are no feasible solutions. Every feasible solution is a possible solution.
- Optimal solution
- Optimal solutions are the solutions with the highest scores. Planning problems usually have a few optimal solutions. They always have at least one optimal solution, even in the case that there are no feasible solutions and the optimal solution is not feasible.
- Best solution found
- The best solution is the solution with the highest score found by an implementation in a specified amount of time. The best solution found is likely to be feasible and, given enough time, it’s an optimal solution.
Counterintuitively, the number of possible solutions is huge (if calculated correctly), even with a small data set.
In the examples provided in the planner-engine distribution folder, most instances have a large number of possible solutions. As there is no guaranteed way to find the optimal solution, any implementation is forced to evaluate at least a subset of all those possible solutions.
OptaPlanner supports several optimization algorithms to efficiently wade through that incredibly large number of possible solutions.
Depending on the use case, some optimization algorithms perform better than others, but it is impossible to know in advance. Using OptaPlanner, you can switch the optimization algorithm by changing the solver configuration in a few lines of XML or code.
9.4. Constraints on planning problems
Usually, a planning problem has minimum two levels of constraints:
A (negative) hard constraint must not be broken.
For example, one teacher can not teach two different lessons at the same time.
A (negative) soft constraint should not be broken if it can be avoided.
For example, Teacher A does not like to teach on Friday afternoons.
Some problems also have positive constraints:
A positive soft constraint (or reward) should be fulfilled if possible.
For example, Teacher B likes to teach on Monday mornings.
Some basic problems only have hard constraints. Some problems have three or more levels of constraints, for example, hard, medium, and soft constraints.
These constraints define the score calculation (otherwise known as the fitness function) of a planning problem. Each solution of a planning problem is graded with a score. With OptaPlanner, score constraints are written in an object oriented language such as Java, or in Drools rules.
This type of code is flexible and scalable.
9.5. Examples provided with Red Hat build of OptaPlanner
Several Red Hat build of OptaPlanner examples are shipped with Red Hat Decision Manager. You can review the code for examples and modify it as necessary to suit your needs.
Red Hat does not provide support for the example code included in the Red Hat Decision Manager distribution.
Some of the OptaPlanner examples solve problems that are presented in academic contests. The Contest column in the following table lists the contests. It also identifies an example as being either realistic or unrealistic for the purpose of a contest. A realistic contest is an official, independent contest that meets the following standards:
- Clearly defined real-world use cases
- Real-world constraints
- Multiple real-world datasets
- Reproducible results within a specific time limit on specific hardware
- Serious participation from the academic and/or enterprise Operations Research community.
Realistic contests provide an objective comparison of OptaPlanner with competitive software and academic research.
| Example | Domain | Size | Contest | Directory name |
|---|---|---|---|---|
| 1 entity class (1 variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Pointless (cheatable) |
| |
| 1 entity class (1 variable) |
Entity ⇐
Value ⇐
Search space ⇐ | No (Defined by us) |
| |
| 1 entity class (1 chained variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Unrealistic TSP web |
| |
| 1 entity class (1 variable) |
Entity ⇐
Value ⇐
Search space ⇐ | No (Defined by us) |
| |
| 1 entity class (2 variables) |
Entity ⇐
Value ⇐
Search space ⇐ | No (Defined by us) |
| |
| 1 entity class (2 variables) |
Entity ⇐
Value ⇐
Search space ⇐ | Realistic ITC 2007 track 3 |
| |
| 1 entity class (1 variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Nearly realistic ROADEF 2012 |
| |
| 1 entity class (1 chained variable) 1 shadow entity class (1 automatic shadow variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Unrealistic VRP web |
| |
| Vehicle routing with time windows | All of Vehicle routing (1 shadow variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Unrealistic VRP web |
|
| 1 entity class (2 variables) (1 shadow variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Nearly realistic MISTA 2013 |
| |
| 1 entity class (1 chained variable) (1 shadow variable) 1 shadow entity class (1 automatic shadow variable) |
Entity ⇐
Value ⇐
Search space ⇐ | No Defined by us |
| |
| 2 entity classes (same hierarchy) (2 variables) |
Entity ⇐
Value ⇐
Search space ⇐ | Realistic ITC 2007 track 1 |
| |
| 1 entity class (1 variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Realistic INRC 2010 |
| |
| 1 entity class (1 variable) |
Entity ⇐
Value ⇐
Search space ⇐ | Unrealistic TTP |
| |
| 1 entity class (2 variables) |
Entity ⇐
Value ⇐
Search space ⇐ | Nearly realistic ICON Energy |
| |
| 1 entity class (1 variable) |
Entity ⇐
Value =
Search space ⇐ | No Defined by us |
| |
| 1 entity class (2 variables) |
Entity ⇐
Value ⇐
Search space ⇐ | No Defined by us |
| |
| 1 entity class (1 chained variable) (4 shadow variables) 1 shadow entity class (1 automatic shadow variable) |
Entity ⇐
Value ⇐
Search space ⇐ | No Defined by us |
| |
| 1 entity class (1 variable) 1 shadow entity class (1 automatic shadow variable) |
Entity ⇐
Value ⇐
Search space ⇐ | No Defined by us |
|
9.6. N queens
Place n number of queens on an n sized chessboard so that no two queens can attack each other. The most common n queens puzzle is the eight queens puzzle, with n = 8:
Constraints:
- Use a chessboard of n columns and n rows.
- Place n queens on the chessboard.
- No two queens can attack each other. A queen can attack any other queen on the same horizontal, vertical, or diagonal line.
This documentation heavily uses the four queens puzzle as the primary example.
A proposed solution could be:
Figure 9.1. A wrong solution for the four queens puzzle
The above solution is wrong because queens A1 and B0 can attack each other (so can queens B0 and D0). Removing queen B0 would respect the "no two queens can attack each other" constraint, but would break the "place n queens" constraint.
Below is a correct solution:
Figure 9.2. A correct solution for the Four queens puzzle
All the constraints have been met, so the solution is correct.
Note that most n queens puzzles have multiple correct solutions. We will focus on finding a single correct solution for a specific n, not on finding the number of possible correct solutions for a specific n.
Problem size
4queens has 4 queens with a search space of 256. 8queens has 8 queens with a search space of 10^7. 16queens has 16 queens with a search space of 10^19. 32queens has 32 queens with a search space of 10^48. 64queens has 64 queens with a search space of 10^115. 256queens has 256 queens with a search space of 10^616.
The implementation of the n queens example has not been optimized because it functions as a beginner example. Nevertheless, it can easily handle 64 queens. With a few changes it has been shown to easily handle 5000 queens and more.
9.6.1. Domain model for N queens
This example uses the domain model to solve the four queens problem.
Creating a Domain Model
A good domain model will make it easier to understand and solve your planning problem.
This is the domain model for the n queens example:
public class Column { private int index; // ... getters and setters }public class Row { private int index; // ... getters and setters }public class Queen { private Column column; private Row row; public int getAscendingDiagonalIndex() {...} public int getDescendingDiagonalIndex() {...} // ... getters and setters }Calculating the Search Space.
A
Queeninstance has aColumn(for example: 0 is column A, 1 is column B, …) and aRow(its row, for example: 0 is row 0, 1 is row 1, …).The ascending diagonal line and the descending diagonal line can be calculated based on the column and the row.
The column and row indexes start from the upper left corner of the chessboard.
public class NQueens { private int n; private List<Column> columnList; private List<Row> rowList; private List<Queen> queenList; private SimpleScore score; // ... getters and setters }Finding the Solution
A single
NQueensinstance contains a list of allQueeninstances. It is theSolutionimplementation which will be supplied to, solved by, and retrieved from the Solver.
Notice that in the four queens example, the NQueens getN() method will always return four.
Figure 9.3. A solution for Four Queens
| columnIndex | rowIndex | ascendingDiagonalIndex (columnIndex + rowIndex) | descendingDiagonalIndex (columnIndex - rowIndex) | |
|---|---|---|---|---|
| A1 | 0 | 1 | 1 (**) | -1 |
| B0 | 1 | 0 (*) | 1 (**) | 1 |
| C2 | 2 | 2 | 4 | 0 |
| D0 | 3 | 0 (*) | 3 | 3 |
When two queens share the same column, row or diagonal line, such as (*) and (**), they can attack each other.
9.7. Cloud balancing
For information about this example, see Red Hat build of OptaPlanner quick start guides.
9.8. Traveling salesman (TSP - Traveling Salesman Problem)
Given a list of cities, find the shortest tour for a salesman that visits each city exactly once.
The problem is defined by Wikipedia. It is one of the most intensively studied problems in computational mathematics. Yet, in the real world, it is often only part of a planning problem, along with other constraints, such as employee shift rostering constraints.
Problem size
dj38 has 38 cities with a search space of 10^43. europe40 has 40 cities with a search space of 10^46. st70 has 70 cities with a search space of 10^98. pcb442 has 442 cities with a search space of 10^976. lu980 has 980 cities with a search space of 10^2504.
Problem difficulty
Despite TSP’s simple definition, the problem is surprisingly hard to solve. Because it is an NP-hard problem (like most planning problems), the optimal solution for a specific problem dataset can change a lot when that problem dataset is slightly altered:
9.9. Tennis club scheduling
Every week the tennis club has four teams playing round robin against each other. Assign those four spots to the teams fairly.
Hard constraints:
- Conflict: A team can only play once per day.
- Unavailability: Some teams are unavailable on some dates.
Medium constraints:
- Fair assignment: All teams should play an (almost) equal number of times.
Soft constraints:
- Evenly confrontation: Each team should play against every other team an equal number of times.
Problem size
munich-7teams has 7 teams, 18 days, 12 unavailabilityPenalties and 72 teamAssignments with a search space of 10^60.
Figure 9.4. Domain model
9.10. Meeting scheduling
Assign each meeting to a starting time and a room. Meetings have different durations.
Hard constraints:
- Room conflict: Two meetings must not use the same room at the same time.
- Required attendance: A person cannot have two required meetings at the same time.
- Required room capacity: A meeting must not be in a room that doesn’t fit all of the meeting’s attendees.
- Start and end on same day: A meeting shouldn’t be scheduled over multiple days.
Medium constraints:
- Preferred attendance: A person cannot have two preferred meetings at the same time, nor a preferred and a required meeting at the same time.
Soft constraints:
- Sooner rather than later: Schedule all meetings as soon as possible.
- A break between meetings: Any two meetings should have at least one time grain break between them.
- Overlapping meetings: To minimize the number of meetings in parallel so people don’t have to choose one meeting over the other.
- Assign larger rooms first: If a larger room is available any meeting should be assigned to that room in order to accommodate as many people as possible even if they haven’t signed up to that meeting.
- Room stability: If a person has two consecutive meetings with two or less time grains break between them they better be in the same room.
Problem size
50meetings-160timegrains-5rooms has 50 meetings, 160 timeGrains and 5 rooms with a search space of 10^145. 100meetings-320timegrains-5rooms has 100 meetings, 320 timeGrains and 5 rooms with a search space of 10^320. 200meetings-640timegrains-5rooms has 200 meetings, 640 timeGrains and 5 rooms with a search space of 10^701. 400meetings-1280timegrains-5rooms has 400 meetings, 1280 timeGrains and 5 rooms with a search space of 10^1522. 800meetings-2560timegrains-5rooms has 800 meetings, 2560 timeGrains and 5 rooms with a search space of 10^3285.
9.11. Course timetabling (ITC 2007 Track 3 - Curriculum Course Scheduling)
Schedule each lecture into a timeslot and into a room.
Hard constraints:
- Teacher conflict: A teacher must not have two lectures in the same period.
- Curriculum conflict: A curriculum must not have two lectures in the same period.
- Room occupancy: Two lectures must not be in the same room in the same period.
- Unavailable period (specified per dataset): A specific lecture must not be assigned to a specific period.
Soft constraints:
- Room capacity: A room’s capacity should not be less than the number of students in its lecture.
- Minimum working days: Lectures of the same course should be spread out into a minimum number of days.
- Curriculum compactness: Lectures belonging to the same curriculum should be adjacent to each other (so in consecutive periods).
- Room stability: Lectures of the same course should be assigned to the same room.
The problem is defined by the International Timetabling Competition 2007 track 3.
Problem size
comp01 has 24 teachers, 14 curricula, 30 courses, 160 lectures, 30 periods, 6 rooms and 53 unavailable period constraints with a search space of 10^360. comp02 has 71 teachers, 70 curricula, 82 courses, 283 lectures, 25 periods, 16 rooms and 513 unavailable period constraints with a search space of 10^736. comp03 has 61 teachers, 68 curricula, 72 courses, 251 lectures, 25 periods, 16 rooms and 382 unavailable period constraints with a search space of 10^653. comp04 has 70 teachers, 57 curricula, 79 courses, 286 lectures, 25 periods, 18 rooms and 396 unavailable period constraints with a search space of 10^758. comp05 has 47 teachers, 139 curricula, 54 courses, 152 lectures, 36 periods, 9 rooms and 771 unavailable period constraints with a search space of 10^381. comp06 has 87 teachers, 70 curricula, 108 courses, 361 lectures, 25 periods, 18 rooms and 632 unavailable period constraints with a search space of 10^957. comp07 has 99 teachers, 77 curricula, 131 courses, 434 lectures, 25 periods, 20 rooms and 667 unavailable period constraints with a search space of 10^1171. comp08 has 76 teachers, 61 curricula, 86 courses, 324 lectures, 25 periods, 18 rooms and 478 unavailable period constraints with a search space of 10^859. comp09 has 68 teachers, 75 curricula, 76 courses, 279 lectures, 25 periods, 18 rooms and 405 unavailable period constraints with a search space of 10^740. comp10 has 88 teachers, 67 curricula, 115 courses, 370 lectures, 25 periods, 18 rooms and 694 unavailable period constraints with a search space of 10^981. comp11 has 24 teachers, 13 curricula, 30 courses, 162 lectures, 45 periods, 5 rooms and 94 unavailable period constraints with a search space of 10^381. comp12 has 74 teachers, 150 curricula, 88 courses, 218 lectures, 36 periods, 11 rooms and 1368 unavailable period constraints with a search space of 10^566. comp13 has 77 teachers, 66 curricula, 82 courses, 308 lectures, 25 periods, 19 rooms and 468 unavailable period constraints with a search space of 10^824. comp14 has 68 teachers, 60 curricula, 85 courses, 275 lectures, 25 periods, 17 rooms and 486 unavailable period constraints with a search space of 10^722.
Figure 9.5. Domain model
9.12. Machine reassignment (Google ROADEF 2012)
Assign each process to a machine. All processes already have an original (unoptimized) assignment. Each process requires an amount of each resource (such as CPU or RAM). This is a more complex version of the Cloud Balancing example.
Hard constraints:
- Maximum capacity: The maximum capacity for each resource for each machine must not be exceeded.
- Conflict: Processes of the same service must run on distinct machines.
- Spread: Processes of the same service must be spread out across locations.
- Dependency: The processes of a service depending on another service must run in the neighborhood of a process of the other service.
- Transient usage: Some resources are transient and count towards the maximum capacity of both the original machine as the newly assigned machine.
Soft constraints:
- Load: The safety capacity for each resource for each machine should not be exceeded.
- Balance: Leave room for future assignments by balancing the available resources on each machine.
- Process move cost: A process has a move cost.
- Service move cost: A service has a move cost.
- Machine move cost: Moving a process from machine A to machine B has another A-B specific move cost.
The problem is defined by the Google ROADEF/EURO Challenge 2012.
Figure 9.6. Value proposition
Problem size
model_a1_1 has 2 resources, 1 neighborhoods, 4 locations, 4 machines, 79 services, 100 processes and 1 balancePenalties with a search space of 10^60. model_a1_2 has 4 resources, 2 neighborhoods, 4 locations, 100 machines, 980 services, 1000 processes and 0 balancePenalties with a search space of 10^2000. model_a1_3 has 3 resources, 5 neighborhoods, 25 locations, 100 machines, 216 services, 1000 processes and 0 balancePenalties with a search space of 10^2000. model_a1_4 has 3 resources, 50 neighborhoods, 50 locations, 50 machines, 142 services, 1000 processes and 1 balancePenalties with a search space of 10^1698. model_a1_5 has 4 resources, 2 neighborhoods, 4 locations, 12 machines, 981 services, 1000 processes and 1 balancePenalties with a search space of 10^1079. model_a2_1 has 3 resources, 1 neighborhoods, 1 locations, 100 machines, 1000 services, 1000 processes and 0 balancePenalties with a search space of 10^2000. model_a2_2 has 12 resources, 5 neighborhoods, 25 locations, 100 machines, 170 services, 1000 processes and 0 balancePenalties with a search space of 10^2000. model_a2_3 has 12 resources, 5 neighborhoods, 25 locations, 100 machines, 129 services, 1000 processes and 0 balancePenalties with a search space of 10^2000. model_a2_4 has 12 resources, 5 neighborhoods, 25 locations, 50 machines, 180 services, 1000 processes and 1 balancePenalties with a search space of 10^1698. model_a2_5 has 12 resources, 5 neighborhoods, 25 locations, 50 machines, 153 services, 1000 processes and 0 balancePenalties with a search space of 10^1698. model_b_1 has 12 resources, 5 neighborhoods, 10 locations, 100 machines, 2512 services, 5000 processes and 0 balancePenalties with a search space of 10^10000. model_b_2 has 12 resources, 5 neighborhoods, 10 locations, 100 machines, 2462 services, 5000 processes and 1 balancePenalties with a search space of 10^10000. model_b_3 has 6 resources, 5 neighborhoods, 10 locations, 100 machines, 15025 services, 20000 processes and 0 balancePenalties with a search space of 10^40000. model_b_4 has 6 resources, 5 neighborhoods, 50 locations, 500 machines, 1732 services, 20000 processes and 1 balancePenalties with a search space of 10^53979. model_b_5 has 6 resources, 5 neighborhoods, 10 locations, 100 machines, 35082 services, 40000 processes and 0 balancePenalties with a search space of 10^80000. model_b_6 has 6 resources, 5 neighborhoods, 50 locations, 200 machines, 14680 services, 40000 processes and 1 balancePenalties with a search space of 10^92041. model_b_7 has 6 resources, 5 neighborhoods, 50 locations, 4000 machines, 15050 services, 40000 processes and 1 balancePenalties with a search space of 10^144082. model_b_8 has 3 resources, 5 neighborhoods, 10 locations, 100 machines, 45030 services, 50000 processes and 0 balancePenalties with a search space of 10^100000. model_b_9 has 3 resources, 5 neighborhoods, 100 locations, 1000 machines, 4609 services, 50000 processes and 1 balancePenalties with a search space of 10^150000. model_b_10 has 3 resources, 5 neighborhoods, 100 locations, 5000 machines, 4896 services, 50000 processes and 1 balancePenalties with a search space of 10^184948.
Figure 9.7. Domain model
9.13. Vehicle routing
Using a fleet of vehicles, pick up the objects of each customer and bring them to the depot. Each vehicle can service multiple customers, but it has a limited capacity.
Besides the basic case (CVRP), there is also a variant with time windows (CVRPTW).
Hard constraints:
- Vehicle capacity: a vehicle cannot carry more items then its capacity.
Time windows (only in CVRPTW):
- Travel time: Traveling from one location to another takes time.
- Customer service duration: A vehicle must stay at the customer for the length of the service duration.
- Customer ready time: A vehicle may arrive before the customer’s ready time, but it must wait until the ready time before servicing.
- Customer due time: A vehicle must arrive on time, before the customer’s due time.
Soft constraints:
- Total distance: Minimize the total distance driven (fuel consumption) of all vehicles.
The capacitated vehicle routing problem (CVRP) and its time-windowed variant (CVRPTW) are defined by the VRP web.
Figure 9.8. Value proposition
Problem size
CVRP instances (without time windows):
belgium-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. belgium-road-km-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-road-km-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-road-km-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-road-km-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-road-km-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. belgium-road-time-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-road-time-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-road-time-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-road-time-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-road-time-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. belgium-d2-n50-k10 has 2 depots, 10 vehicles and 48 customers with a search space of 10^74. belgium-d3-n100-k10 has 3 depots, 10 vehicles and 97 customers with a search space of 10^170. belgium-d5-n500-k20 has 5 depots, 20 vehicles and 495 customers with a search space of 10^1168. belgium-d8-n1000-k20 has 8 depots, 20 vehicles and 992 customers with a search space of 10^2607. belgium-d10-n2750-k55 has 10 depots, 55 vehicles and 2740 customers with a search space of 10^8380. A-n32-k5 has 1 depots, 5 vehicles and 31 customers with a search space of 10^40. A-n33-k5 has 1 depots, 5 vehicles and 32 customers with a search space of 10^41. A-n33-k6 has 1 depots, 6 vehicles and 32 customers with a search space of 10^42. A-n34-k5 has 1 depots, 5 vehicles and 33 customers with a search space of 10^43. A-n36-k5 has 1 depots, 5 vehicles and 35 customers with a search space of 10^46. A-n37-k5 has 1 depots, 5 vehicles and 36 customers with a search space of 10^48. A-n37-k6 has 1 depots, 6 vehicles and 36 customers with a search space of 10^49. A-n38-k5 has 1 depots, 5 vehicles and 37 customers with a search space of 10^49. A-n39-k5 has 1 depots, 5 vehicles and 38 customers with a search space of 10^51. A-n39-k6 has 1 depots, 6 vehicles and 38 customers with a search space of 10^52. A-n44-k7 has 1 depots, 7 vehicles and 43 customers with a search space of 10^61. A-n45-k6 has 1 depots, 6 vehicles and 44 customers with a search space of 10^62. A-n45-k7 has 1 depots, 7 vehicles and 44 customers with a search space of 10^63. A-n46-k7 has 1 depots, 7 vehicles and 45 customers with a search space of 10^65. A-n48-k7 has 1 depots, 7 vehicles and 47 customers with a search space of 10^68. A-n53-k7 has 1 depots, 7 vehicles and 52 customers with a search space of 10^77. A-n54-k7 has 1 depots, 7 vehicles and 53 customers with a search space of 10^79. A-n55-k9 has 1 depots, 9 vehicles and 54 customers with a search space of 10^82. A-n60-k9 has 1 depots, 9 vehicles and 59 customers with a search space of 10^91. A-n61-k9 has 1 depots, 9 vehicles and 60 customers with a search space of 10^93. A-n62-k8 has 1 depots, 8 vehicles and 61 customers with a search space of 10^94. A-n63-k9 has 1 depots, 9 vehicles and 62 customers with a search space of 10^97. A-n63-k10 has 1 depots, 10 vehicles and 62 customers with a search space of 10^98. A-n64-k9 has 1 depots, 9 vehicles and 63 customers with a search space of 10^99. A-n65-k9 has 1 depots, 9 vehicles and 64 customers with a search space of 10^101. A-n69-k9 has 1 depots, 9 vehicles and 68 customers with a search space of 10^108. A-n80-k10 has 1 depots, 10 vehicles and 79 customers with a search space of 10^130. F-n45-k4 has 1 depots, 4 vehicles and 44 customers with a search space of 10^60. F-n72-k4 has 1 depots, 4 vehicles and 71 customers with a search space of 10^108. F-n135-k7 has 1 depots, 7 vehicles and 134 customers with a search space of 10^240.
CVRPTW instances (with time windows):
belgium-tw-d2-n50-k10 has 2 depots, 10 vehicles and 48 customers with a search space of 10^74. belgium-tw-d3-n100-k10 has 3 depots, 10 vehicles and 97 customers with a search space of 10^170. belgium-tw-d5-n500-k20 has 5 depots, 20 vehicles and 495 customers with a search space of 10^1168. belgium-tw-d8-n1000-k20 has 8 depots, 20 vehicles and 992 customers with a search space of 10^2607. belgium-tw-d10-n2750-k55 has 10 depots, 55 vehicles and 2740 customers with a search space of 10^8380. belgium-tw-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-tw-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-tw-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-tw-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-tw-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. Solomon_025_C101 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_C201 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_R101 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_R201 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_RC101 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_RC201 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_100_C101 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_C201 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_R101 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_R201 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_RC101 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_RC201 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Homberger_0200_C1_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_C2_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_R1_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_R2_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_RC1_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_RC2_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0400_C1_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_C2_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_R1_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_R2_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_RC1_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_RC2_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0600_C1_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_C2_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_R1_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_R2_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_RC1_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_RC2_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0800_C1_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_C2_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_R1_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_R2_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_RC1_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_RC2_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_1000_C110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_C210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_R110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_R210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_RC110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_RC210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840.
9.13.1. Domain model for Vehicle routing
The vehicle routing with time windows domain model makes heavy use of the shadow variable feature. This allows it to express its constraints more naturally, because properties such as arrivalTime and departureTime, are directly available on the domain model.
Road Distances Instead of Air Distances
In the real world, vehicles cannot follow a straight line from location to location: they have to use roads and highways. From a business point of view, this matters a lot:
For the optimization algorithm, this does not matter much, as long as the distance between two points can be looked up (and are preferably precalculated). The road cost does not even need to be a distance. It can also be travel time, fuel cost, or a weighted function of those. There are several technologies available to precalculate road costs, such as GraphHopper (embeddable, offline Java engine), Open MapQuest (web service) and Google Maps Client API (web service).
There are also several technologies to render it, such as Leaflet and Google Maps for developers.
It is even possible to render the actual road routes with GraphHopper or Google Map Directions, but because of route overlaps on highways, it can become harder to see the standstill order:
Take special care that the road costs between two points use the same optimization criteria as the one used in OptaPlanner. For example, GraphHopper will by default return the fastest route, not the shortest route. Don’t use the km (or miles) distances of the fastest GPS routes to optimize the shortest trip in OptaPlanner: this leads to a suboptimal solution as shown below:
Contrary to popular belief, most users do not want the shortest route: they want the fastest route instead. They prefer highways over normal roads. They prefer normal roads over dirt roads. In the real world, the fastest and shortest route are rarely the same.
9.14. Project job scheduling
Schedule all jobs in time and execution mode to minimize project delays. Each job is part of a project. A job can be executed in different ways: each way is an execution mode that implies a different duration but also different resource usages. This is a form of flexible job shop scheduling.
Hard constraints:
- Job precedence: a job can only start when all its predecessor jobs are finished.
Resource capacity: do not use more resources than available.
- Resources are local (shared between jobs of the same project) or global (shared between all jobs)
- Resources are renewable (capacity available per day) or nonrenewable (capacity available for all days)
Medium constraints:
- Total project delay: minimize the duration (makespan) of each project.
Soft constraints:
- Total makespan: minimize the duration of the whole multi-project schedule.
The problem is defined by the MISTA 2013 challenge.
Problem size
Schedule A-1 has 2 projects, 24 jobs, 64 execution modes, 7 resources and 150 resource requirements. Schedule A-2 has 2 projects, 44 jobs, 124 execution modes, 7 resources and 420 resource requirements. Schedule A-3 has 2 projects, 64 jobs, 184 execution modes, 7 resources and 630 resource requirements. Schedule A-4 has 5 projects, 60 jobs, 160 execution modes, 16 resources and 390 resource requirements. Schedule A-5 has 5 projects, 110 jobs, 310 execution modes, 16 resources and 900 resource requirements. Schedule A-6 has 5 projects, 160 jobs, 460 execution modes, 16 resources and 1440 resource requirements. Schedule A-7 has 10 projects, 120 jobs, 320 execution modes, 22 resources and 900 resource requirements. Schedule A-8 has 10 projects, 220 jobs, 620 execution modes, 22 resources and 1860 resource requirements. Schedule A-9 has 10 projects, 320 jobs, 920 execution modes, 31 resources and 2880 resource requirements. Schedule A-10 has 10 projects, 320 jobs, 920 execution modes, 31 resources and 2970 resource requirements. Schedule B-1 has 10 projects, 120 jobs, 320 execution modes, 31 resources and 900 resource requirements. Schedule B-2 has 10 projects, 220 jobs, 620 execution modes, 22 resources and 1740 resource requirements. Schedule B-3 has 10 projects, 320 jobs, 920 execution modes, 31 resources and 3060 resource requirements. Schedule B-4 has 15 projects, 180 jobs, 480 execution modes, 46 resources and 1530 resource requirements. Schedule B-5 has 15 projects, 330 jobs, 930 execution modes, 46 resources and 2760 resource requirements. Schedule B-6 has 15 projects, 480 jobs, 1380 execution modes, 46 resources and 4500 resource requirements. Schedule B-7 has 20 projects, 240 jobs, 640 execution modes, 61 resources and 1710 resource requirements. Schedule B-8 has 20 projects, 440 jobs, 1240 execution modes, 42 resources and 3180 resource requirements. Schedule B-9 has 20 projects, 640 jobs, 1840 execution modes, 61 resources and 5940 resource requirements. Schedule B-10 has 20 projects, 460 jobs, 1300 execution modes, 42 resources and 4260 resource requirements.
9.15. Task assigning
Assign each task to a spot in an employee’s queue. Each task has a duration which is affected by the employee’s affinity level with the task’s customer.
Hard constraints:
- Skill: Each task requires one or more skills. The employee must possess all these skills.
Soft level 0 constraints:
- Critical tasks: Complete critical tasks first, sooner than major and minor tasks.
Soft level 1 constraints:
Minimize makespan: Reduce the time to complete all tasks.
- Start with the longest working employee first, then the second longest working employee and so forth, to create fairness and load balancing.
Soft level 2 constraints:
- Major tasks: Complete major tasks as soon as possible, sooner than minor tasks.
Soft level 3 constraints:
- Minor tasks: Complete minor tasks as soon as possible.
Figure 9.9. Value proposition
Problem size
24tasks-8employees has 24 tasks, 6 skills, 8 employees, 4 task types and 4 customers with a search space of 10^30. 50tasks-5employees has 50 tasks, 5 skills, 5 employees, 10 task types and 10 customers with a search space of 10^69. 100tasks-5employees has 100 tasks, 5 skills, 5 employees, 20 task types and 15 customers with a search space of 10^164. 500tasks-20employees has 500 tasks, 6 skills, 20 employees, 100 task types and 60 customers with a search space of 10^1168.
Figure 9.10. Domain model
9.16. Exam timetabling (ITC 2007 track 1 - Examination)
Schedule each exam into a period and into a room. Multiple exams can share the same room during the same period.
Hard constraints:
- Exam conflict: Two exams that share students must not occur in the same period.
- Room capacity: A room’s seating capacity must suffice at all times.
- Period duration: A period’s duration must suffice for all of its exams.
Period related hard constraints (specified per dataset):
- Coincidence: Two specified exams must use the same period (but possibly another room).
- Exclusion: Two specified exams must not use the same period.
- After: A specified exam must occur in a period after another specified exam’s period.
Room related hard constraints (specified per dataset):
- Exclusive: One specified exam should not have to share its room with any other exam.
Soft constraints (each of which has a parametrized penalty):
- The same student should not have two exams in a row.
- The same student should not have two exams on the same day.
- Period spread: Two exams that share students should be a number of periods apart.
- Mixed durations: Two exams that share a room should not have different durations.
- Front load: Large exams should be scheduled earlier in the schedule.
- Period penalty (specified per dataset): Some periods have a penalty when used.
- Room penalty (specified per dataset): Some rooms have a penalty when used.
It uses large test data sets of real-life universities.
The problem is defined by the International Timetabling Competition 2007 track 1. Geoffrey De Smet finished 4th in that competition with a very early version of OptaPlanner. Many improvements have been made since then.
Problem Size
exam_comp_set1 has 7883 students, 607 exams, 54 periods, 7 rooms, 12 period constraints and 0 room constraints with a search space of 10^1564. exam_comp_set2 has 12484 students, 870 exams, 40 periods, 49 rooms, 12 period constraints and 2 room constraints with a search space of 10^2864. exam_comp_set3 has 16365 students, 934 exams, 36 periods, 48 rooms, 168 period constraints and 15 room constraints with a search space of 10^3023. exam_comp_set4 has 4421 students, 273 exams, 21 periods, 1 rooms, 40 period constraints and 0 room constraints with a search space of 10^360. exam_comp_set5 has 8719 students, 1018 exams, 42 periods, 3 rooms, 27 period constraints and 0 room constraints with a search space of 10^2138. exam_comp_set6 has 7909 students, 242 exams, 16 periods, 8 rooms, 22 period constraints and 0 room constraints with a search space of 10^509. exam_comp_set7 has 13795 students, 1096 exams, 80 periods, 15 rooms, 28 period constraints and 0 room constraints with a search space of 10^3374. exam_comp_set8 has 7718 students, 598 exams, 80 periods, 8 rooms, 20 period constraints and 1 room constraints with a search space of 10^1678.
9.16.1. Domain model for exam timetabling
The following diagram shows the main examination domain classes:
Figure 9.11. Examination domain class diagram
Notice that we’ve split up the exam concept into an Exam class and a Topic class. The Exam instances change during solving (this is the planning entity class), when their period or room property changes. The Topic, Period and Room instances never change during solving (these are problem facts, just like some other classes).
9.17. Nurse rostering (INRC 2010)
For each shift, assign a nurse to work that shift.
Hard constraints:
- No unassigned shifts (built-in): Every shift need to be assigned to an employee.
- Shift conflict: An employee can have only one shift per day.
Soft constraints:
Contract obligations. The business frequently violates these, so they decided to define these as soft constraints instead of hard constraints.
- Minimum and maximum assignments: Each employee needs to work more than x shifts and less than y shifts (depending on their contract).
- Minimum and maximum consecutive working days: Each employee needs to work between x and y days in a row (depending on their contract).
- Minimum and maximum consecutive free days: Each employee needs to be free between x and y days in a row (depending on their contract).
- Minimum and maximum consecutive working weekends: Each employee needs to work between x and y weekends in a row (depending on their contract).
- Complete weekends: Each employee needs to work every day in a weekend or not at all.
- Identical shift types during weekend: Each weekend shift for the same weekend of the same employee must be the same shift type.
- Unwanted patterns: A combination of unwanted shift types in a row, for example a late shift followed by an early shift followed by a late shift.
Employee wishes:
- Day on request: An employee wants to work on a specific day.
- Day off request: An employee does not want to work on a specific day.
- Shift on request: An employee wants to be assigned to a specific shift.
- Shift off request: An employee does not want to be assigned to a specific shift.
- Alternative skill: An employee assigned to a skill should have a proficiency in every skill required by that shift.
The problem is defined by the International Nurse Rostering Competition 2010.
Figure 9.12. Value proposition
Problem size
There are three dataset types:
- Sprint: must be solved in seconds.
- Medium: must be solved in minutes.
- Long: must be solved in hours.
toy1 has 1 skills, 3 shiftTypes, 2 patterns, 1 contracts, 6 employees, 7 shiftDates, 35 shiftAssignments and 0 requests with a search space of 10^27. toy2 has 1 skills, 3 shiftTypes, 3 patterns, 2 contracts, 20 employees, 28 shiftDates, 180 shiftAssignments and 140 requests with a search space of 10^234. sprint01 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint02 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint03 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint04 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint05 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint06 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint07 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint08 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint09 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint10 has 1 skills, 4 shiftTypes, 3 patterns, 4 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_hint01 has 1 skills, 4 shiftTypes, 8 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_hint02 has 1 skills, 4 shiftTypes, 0 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_hint03 has 1 skills, 4 shiftTypes, 8 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_late01 has 1 skills, 4 shiftTypes, 8 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_late02 has 1 skills, 3 shiftTypes, 4 patterns, 3 contracts, 10 employees, 28 shiftDates, 144 shiftAssignments and 139 requests with a search space of 10^144. sprint_late03 has 1 skills, 4 shiftTypes, 8 patterns, 3 contracts, 10 employees, 28 shiftDates, 160 shiftAssignments and 150 requests with a search space of 10^160. sprint_late04 has 1 skills, 4 shiftTypes, 8 patterns, 3 contracts, 10 employees, 28 shiftDates, 160 shiftAssignments and 150 requests with a search space of 10^160. sprint_late05 has 1 skills, 4 shiftTypes, 8 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_late06 has 1 skills, 4 shiftTypes, 0 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_late07 has 1 skills, 4 shiftTypes, 0 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. sprint_late08 has 1 skills, 4 shiftTypes, 0 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 0 requests with a search space of 10^152. sprint_late09 has 1 skills, 4 shiftTypes, 0 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 0 requests with a search space of 10^152. sprint_late10 has 1 skills, 4 shiftTypes, 0 patterns, 3 contracts, 10 employees, 28 shiftDates, 152 shiftAssignments and 150 requests with a search space of 10^152. medium01 has 1 skills, 4 shiftTypes, 0 patterns, 4 contracts, 31 employees, 28 shiftDates, 608 shiftAssignments and 403 requests with a search space of 10^906. medium02 has 1 skills, 4 shiftTypes, 0 patterns, 4 contracts, 31 employees, 28 shiftDates, 608 shiftAssignments and 403 requests with a search space of 10^906. medium03 has 1 skills, 4 shiftTypes, 0 patterns, 4 contracts, 31 employees, 28 shiftDates, 608 shiftAssignments and 403 requests with a search space of 10^906. medium04 has 1 skills, 4 shiftTypes, 0 patterns, 4 contracts, 31 employees, 28 shiftDates, 608 shiftAssignments and 403 requests with a search space of 10^906. medium05 has 1 skills, 4 shiftTypes, 0 patterns, 4 contracts, 31 employees, 28 shiftDates, 608 shiftAssignments and 403 requests with a search space of 10^906. medium_hint01 has 1 skills, 4 shiftTypes, 7 patterns, 4 contracts, 30 employees, 28 shiftDates, 428 shiftAssignments and 390 requests with a search space of 10^632. medium_hint02 has 1 skills, 4 shiftTypes, 7 patterns, 3 contracts, 30 employees, 28 shiftDates, 428 shiftAssignments and 390 requests with a search space of 10^632. medium_hint03 has 1 skills, 4 shiftTypes, 7 patterns, 4 contracts, 30 employees, 28 shiftDates, 428 shiftAssignments and 390 requests with a search space of 10^632. medium_late01 has 1 skills, 4 shiftTypes, 7 patterns, 4 contracts, 30 employees, 28 shiftDates, 424 shiftAssignments and 390 requests with a search space of 10^626. medium_late02 has 1 skills, 4 shiftTypes, 7 patterns, 3 contracts, 30 employees, 28 shiftDates, 428 shiftAssignments and 390 requests with a search space of 10^632. medium_late03 has 1 skills, 4 shiftTypes, 0 patterns, 4 contracts, 30 employees, 28 shiftDates, 428 shiftAssignments and 390 requests with a search space of 10^632. medium_late04 has 1 skills, 4 shiftTypes, 7 patterns, 3 contracts, 30 employees, 28 shiftDates, 416 shiftAssignments and 390 requests with a search space of 10^614. medium_late05 has 2 skills, 5 shiftTypes, 7 patterns, 4 contracts, 30 employees, 28 shiftDates, 452 shiftAssignments and 390 requests with a search space of 10^667. long01 has 2 skills, 5 shiftTypes, 3 patterns, 3 contracts, 49 employees, 28 shiftDates, 740 shiftAssignments and 735 requests with a search space of 10^1250. long02 has 2 skills, 5 shiftTypes, 3 patterns, 3 contracts, 49 employees, 28 shiftDates, 740 shiftAssignments and 735 requests with a search space of 10^1250. long03 has 2 skills, 5 shiftTypes, 3 patterns, 3 contracts, 49 employees, 28 shiftDates, 740 shiftAssignments and 735 requests with a search space of 10^1250. long04 has 2 skills, 5 shiftTypes, 3 patterns, 3 contracts, 49 employees, 28 shiftDates, 740 shiftAssignments and 735 requests with a search space of 10^1250. long05 has 2 skills, 5 shiftTypes, 3 patterns, 3 contracts, 49 employees, 28 shiftDates, 740 shiftAssignments and 735 requests with a search space of 10^1250. long_hint01 has 2 skills, 5 shiftTypes, 9 patterns, 3 contracts, 50 employees, 28 shiftDates, 740 shiftAssignments and 0 requests with a search space of 10^1257. long_hint02 has 2 skills, 5 shiftTypes, 7 patterns, 3 contracts, 50 employees, 28 shiftDates, 740 shiftAssignments and 0 requests with a search space of 10^1257. long_hint03 has 2 skills, 5 shiftTypes, 7 patterns, 3 contracts, 50 employees, 28 shiftDates, 740 shiftAssignments and 0 requests with a search space of 10^1257. long_late01 has 2 skills, 5 shiftTypes, 9 patterns, 3 contracts, 50 employees, 28 shiftDates, 752 shiftAssignments and 0 requests with a search space of 10^1277. long_late02 has 2 skills, 5 shiftTypes, 9 patterns, 4 contracts, 50 employees, 28 shiftDates, 752 shiftAssignments and 0 requests with a search space of 10^1277. long_late03 has 2 skills, 5 shiftTypes, 9 patterns, 3 contracts, 50 employees, 28 shiftDates, 752 shiftAssignments and 0 requests with a search space of 10^1277. long_late04 has 2 skills, 5 shiftTypes, 9 patterns, 4 contracts, 50 employees, 28 shiftDates, 752 shiftAssignments and 0 requests with a search space of 10^1277. long_late05 has 2 skills, 5 shiftTypes, 9 patterns, 3 contracts, 50 employees, 28 shiftDates, 740 shiftAssignments and 0 requests with a search space of 10^1257.
Figure 9.13. Domain model
9.18. Traveling tournament problem (TTP)
Schedule matches between n number of teams.
Hard constraints:
- Each team plays twice against every other team: once home and once away.
- Each team has exactly one match on each timeslot.
- No team must have more than three consecutive home or three consecutive away matches.
- No repeaters: no two consecutive matches of the same two opposing teams.
Soft constraints:
- Minimize the total distance traveled by all teams.
The problem is defined on Michael Trick’s website (which contains the world records too).
Problem size
1-nl04 has 6 days, 4 teams and 12 matches with a search space of 10^5. 1-nl06 has 10 days, 6 teams and 30 matches with a search space of 10^19. 1-nl08 has 14 days, 8 teams and 56 matches with a search space of 10^43. 1-nl10 has 18 days, 10 teams and 90 matches with a search space of 10^79. 1-nl12 has 22 days, 12 teams and 132 matches with a search space of 10^126. 1-nl14 has 26 days, 14 teams and 182 matches with a search space of 10^186. 1-nl16 has 30 days, 16 teams and 240 matches with a search space of 10^259. 2-bra24 has 46 days, 24 teams and 552 matches with a search space of 10^692. 3-nfl16 has 30 days, 16 teams and 240 matches with a search space of 10^259. 3-nfl18 has 34 days, 18 teams and 306 matches with a search space of 10^346. 3-nfl20 has 38 days, 20 teams and 380 matches with a search space of 10^447. 3-nfl22 has 42 days, 22 teams and 462 matches with a search space of 10^562. 3-nfl24 has 46 days, 24 teams and 552 matches with a search space of 10^692. 3-nfl26 has 50 days, 26 teams and 650 matches with a search space of 10^838. 3-nfl28 has 54 days, 28 teams and 756 matches with a search space of 10^999. 3-nfl30 has 58 days, 30 teams and 870 matches with a search space of 10^1175. 3-nfl32 has 62 days, 32 teams and 992 matches with a search space of 10^1367. 4-super04 has 6 days, 4 teams and 12 matches with a search space of 10^5. 4-super06 has 10 days, 6 teams and 30 matches with a search space of 10^19. 4-super08 has 14 days, 8 teams and 56 matches with a search space of 10^43. 4-super10 has 18 days, 10 teams and 90 matches with a search space of 10^79. 4-super12 has 22 days, 12 teams and 132 matches with a search space of 10^126. 4-super14 has 26 days, 14 teams and 182 matches with a search space of 10^186. 5-galaxy04 has 6 days, 4 teams and 12 matches with a search space of 10^5. 5-galaxy06 has 10 days, 6 teams and 30 matches with a search space of 10^19. 5-galaxy08 has 14 days, 8 teams and 56 matches with a search space of 10^43. 5-galaxy10 has 18 days, 10 teams and 90 matches with a search space of 10^79. 5-galaxy12 has 22 days, 12 teams and 132 matches with a search space of 10^126. 5-galaxy14 has 26 days, 14 teams and 182 matches with a search space of 10^186. 5-galaxy16 has 30 days, 16 teams and 240 matches with a search space of 10^259. 5-galaxy18 has 34 days, 18 teams and 306 matches with a search space of 10^346. 5-galaxy20 has 38 days, 20 teams and 380 matches with a search space of 10^447. 5-galaxy22 has 42 days, 22 teams and 462 matches with a search space of 10^562. 5-galaxy24 has 46 days, 24 teams and 552 matches with a search space of 10^692. 5-galaxy26 has 50 days, 26 teams and 650 matches with a search space of 10^838. 5-galaxy28 has 54 days, 28 teams and 756 matches with a search space of 10^999. 5-galaxy30 has 58 days, 30 teams and 870 matches with a search space of 10^1175. 5-galaxy32 has 62 days, 32 teams and 992 matches with a search space of 10^1367. 5-galaxy34 has 66 days, 34 teams and 1122 matches with a search space of 10^1576. 5-galaxy36 has 70 days, 36 teams and 1260 matches with a search space of 10^1801. 5-galaxy38 has 74 days, 38 teams and 1406 matches with a search space of 10^2042. 5-galaxy40 has 78 days, 40 teams and 1560 matches with a search space of 10^2301.
9.19. Cheap time scheduling
Schedule all tasks in time and on a machine to minimize power cost. Power prices differ in time. This is a form of job shop scheduling.
Hard constraints:
- Start time limits: Each task must start between its earliest start and latest start limit.
- Maximum capacity: The maximum capacity for each resource for each machine must not be exceeded.
- Startup and shutdown: Each machine must be active in the periods during which it has assigned tasks. Between tasks it is allowed to be idle to avoid startup and shutdown costs.
Medium constraints:
Power cost: Minimize the total power cost of the whole schedule.
- Machine power cost: Each active or idle machine consumes power, which infers a power cost (depending on the power price during that time).
- Task power cost: Each task consumes power too, which infers a power cost (depending on the power price during its time).
- Machine startup and shutdown cost: Every time a machine starts up or shuts down, an extra cost is incurred.
Soft constraints (addendum to the original problem definition):
- Start early: Prefer starting a task sooner rather than later.
The problem is defined by the ICON challenge.
Problem size
sample01 has 3 resources, 2 machines, 288 periods and 25 tasks with a search space of 10^53. sample02 has 3 resources, 2 machines, 288 periods and 50 tasks with a search space of 10^114. sample03 has 3 resources, 2 machines, 288 periods and 100 tasks with a search space of 10^226. sample04 has 3 resources, 5 machines, 288 periods and 100 tasks with a search space of 10^266. sample05 has 3 resources, 2 machines, 288 periods and 250 tasks with a search space of 10^584. sample06 has 3 resources, 5 machines, 288 periods and 250 tasks with a search space of 10^673. sample07 has 3 resources, 2 machines, 288 periods and 1000 tasks with a search space of 10^2388. sample08 has 3 resources, 5 machines, 288 periods and 1000 tasks with a search space of 10^2748. sample09 has 4 resources, 20 machines, 288 periods and 2000 tasks with a search space of 10^6668. instance00 has 1 resources, 10 machines, 288 periods and 200 tasks with a search space of 10^595. instance01 has 1 resources, 10 machines, 288 periods and 200 tasks with a search space of 10^599. instance02 has 1 resources, 10 machines, 288 periods and 200 tasks with a search space of 10^599. instance03 has 1 resources, 10 machines, 288 periods and 200 tasks with a search space of 10^591. instance04 has 1 resources, 10 machines, 288 periods and 200 tasks with a search space of 10^590. instance05 has 2 resources, 25 machines, 288 periods and 200 tasks with a search space of 10^667. instance06 has 2 resources, 25 machines, 288 periods and 200 tasks with a search space of 10^660. instance07 has 2 resources, 25 machines, 288 periods and 200 tasks with a search space of 10^662. instance08 has 2 resources, 25 machines, 288 periods and 200 tasks with a search space of 10^651. instance09 has 2 resources, 25 machines, 288 periods and 200 tasks with a search space of 10^659. instance10 has 2 resources, 20 machines, 288 periods and 500 tasks with a search space of 10^1657. instance11 has 2 resources, 20 machines, 288 periods and 500 tasks with a search space of 10^1644. instance12 has 2 resources, 20 machines, 288 periods and 500 tasks with a search space of 10^1637. instance13 has 2 resources, 20 machines, 288 periods and 500 tasks with a search space of 10^1659. instance14 has 2 resources, 20 machines, 288 periods and 500 tasks with a search space of 10^1643. instance15 has 3 resources, 40 machines, 288 periods and 500 tasks with a search space of 10^1782. instance16 has 3 resources, 40 machines, 288 periods and 500 tasks with a search space of 10^1778. instance17 has 3 resources, 40 machines, 288 periods and 500 tasks with a search space of 10^1764. instance18 has 3 resources, 40 machines, 288 periods and 500 tasks with a search space of 10^1769. instance19 has 3 resources, 40 machines, 288 periods and 500 tasks with a search space of 10^1778. instance20 has 3 resources, 50 machines, 288 periods and 1000 tasks with a search space of 10^3689. instance21 has 3 resources, 50 machines, 288 periods and 1000 tasks with a search space of 10^3678. instance22 has 3 resources, 50 machines, 288 periods and 1000 tasks with a search space of 10^3706. instance23 has 3 resources, 50 machines, 288 periods and 1000 tasks with a search space of 10^3676. instance24 has 3 resources, 50 machines, 288 periods and 1000 tasks with a search space of 10^3681. instance25 has 3 resources, 60 machines, 288 periods and 1000 tasks with a search space of 10^3774. instance26 has 3 resources, 60 machines, 288 periods and 1000 tasks with a search space of 10^3737. instance27 has 3 resources, 60 machines, 288 periods and 1000 tasks with a search space of 10^3744. instance28 has 3 resources, 60 machines, 288 periods and 1000 tasks with a search space of 10^3731. instance29 has 3 resources, 60 machines, 288 periods and 1000 tasks with a search space of 10^3746. instance30 has 4 resources, 70 machines, 288 periods and 2000 tasks with a search space of 10^7718. instance31 has 4 resources, 70 machines, 288 periods and 2000 tasks with a search space of 10^7740. instance32 has 4 resources, 70 machines, 288 periods and 2000 tasks with a search space of 10^7686. instance33 has 4 resources, 70 machines, 288 periods and 2000 tasks with a search space of 10^7672. instance34 has 4 resources, 70 machines, 288 periods and 2000 tasks with a search space of 10^7695. instance35 has 4 resources, 80 machines, 288 periods and 2000 tasks with a search space of 10^7807. instance36 has 4 resources, 80 machines, 288 periods and 2000 tasks with a search space of 10^7814. instance37 has 4 resources, 80 machines, 288 periods and 2000 tasks with a search space of 10^7764. instance38 has 4 resources, 80 machines, 288 periods and 2000 tasks with a search space of 10^7736. instance39 has 4 resources, 80 machines, 288 periods and 2000 tasks with a search space of 10^7783. instance40 has 4 resources, 90 machines, 288 periods and 4000 tasks with a search space of 10^15976. instance41 has 4 resources, 90 machines, 288 periods and 4000 tasks with a search space of 10^15935. instance42 has 4 resources, 90 machines, 288 periods and 4000 tasks with a search space of 10^15887. instance43 has 4 resources, 90 machines, 288 periods and 4000 tasks with a search space of 10^15896. instance44 has 4 resources, 90 machines, 288 periods and 4000 tasks with a search space of 10^15885. instance45 has 4 resources, 100 machines, 288 periods and 5000 tasks with a search space of 10^20173. instance46 has 4 resources, 100 machines, 288 periods and 5000 tasks with a search space of 10^20132. instance47 has 4 resources, 100 machines, 288 periods and 5000 tasks with a search space of 10^20126. instance48 has 4 resources, 100 machines, 288 periods and 5000 tasks with a search space of 10^20110. instance49 has 4 resources, 100 machines, 288 periods and 5000 tasks with a search space of 10^20078.
9.20. Investment asset class allocation (Portfolio Optimization)
Decide the relative quantity to invest in each asset class.
Hard constraints:
Risk maximum: the total standard deviation must not be higher than the standard deviation maximum.
- Total standard deviation calculation takes asset class correlations into account by applying Markowitz Portfolio Theory.
- Region maximum: Each region has a quantity maximum.
- Sector maximum: Each sector has a quantity maximum.
Soft constraints:
- Maximize expected return.
Problem size
de_smet_1 has 1 regions, 3 sectors and 11 asset classes with a search space of 10^4. irrinki_1 has 2 regions, 3 sectors and 6 asset classes with a search space of 10^3.
Larger datasets have not been created or tested yet, but should not pose a problem. A good source of data is this Asset Correlation website.
9.21. Conference scheduling
Assign each conference talk to a timeslot and a room. Timeslots can overlap. Read and write to and from an *.xlsx file that can be edited with LibreOffice or Excel.
Hard constraints:
- Talk type of timeslot: The type of a talk must match the timeslot’s talk type.
- Room unavailable timeslots: A talk’s room must be available during the talk’s timeslot.
- Room conflict: Two talks can’t use the same room during overlapping timeslots.
- Speaker unavailable timeslots: Every talk’s speaker must be available during the talk’s timeslot.
- Speaker conflict: Two talks can’t share a speaker during overlapping timeslots.
Generic purpose timeslot and room tags:
- Speaker required timeslot tag: If a speaker has a required timeslot tag, then all of his or her talks must be assigned to a timeslot with that tag.
- Speaker prohibited timeslot tag: If a speaker has a prohibited timeslot tag, then all of his or her talks cannot be assigned to a timeslot with that tag.
- Talk required timeslot tag: If a talk has a required timeslot tag, then it must be assigned to a timeslot with that tag.
- Talk prohibited timeslot tag: If a talk has a prohibited timeslot tag, then it cannot be assigned to a timeslot with that tag.
- Speaker required room tag: If a speaker has a required room tag, then all of his or her talks must be assigned to a room with that tag.
- Speaker prohibited room tag: If a speaker has a prohibited room tag, then all of his or her talks cannot be assigned to a room with that tag.
- Talk required room tag: If a talk has a required room tag, then it must be assigned to a room with that tag.
- Talk prohibited room tag: If a talk has a prohibited room tag, then it cannot be assigned to a room with that tag.
- Talk mutually-exclusive-talks tag: Talks that share such a tag must not be scheduled in overlapping timeslots.
- Talk prerequisite talks: A talk must be scheduled after all its prerequisite talks.
Soft constraints:
- Theme track conflict: Minimize the number of talks that share a theme tag during overlapping timeslots.
- Sector conflict: Minimize the number of talks that share a same sector tag during overlapping timeslots.
- Content audience level flow violation: For every content tag, schedule the introductory talks before the advanced talks.
- Audience level diversity: For every timeslot, maximize the number of talks with a different audience level.
- Language diversity: For every timeslot, maximize the number of talks with a different language.
Generic purpose timeslot and room tags:
- Speaker preferred timeslot tag: If a speaker has a preferred timeslot tag, then all of his or her talks should be assigned to a timeslot with that tag.
- Speaker undesired timeslot tag: If a speaker has an undesired timeslot tag, then none of his or her talks should be assigned to a timeslot with that tag.
- Talk preferred timeslot tag: If a talk has a preferred timeslot tag, then it should be assigned to a timeslot with that tag.
- Talk undesired timeslot tag: If a talk has an undesired timeslot tag, then it should not be assigned to a timeslot with that tag.
- Speaker preferred room tag: If a speaker has a preferred room tag, then all of his or her talks should be assigned to a room with that tag.
- Speaker undesired room tag: If a speaker has an undesired room tag, then none of his or her talks should be assigned to a room with that tag.
- Talk preferred room tag: If a talk has a preferred room tag, then it should be assigned to a room with that tag.
- Talk undesired room tag: If a talk has an undesired room tag, then it should not be assigned to a room with that tag.
- Same day talks: All talks that share a theme tag or content tag should be scheduled in the minimum number of days (ideally in the same day).
Figure 9.14. Value proposition
Problem size
18talks-6timeslots-5rooms has 18 talks, 6 timeslots and 5 rooms with a search space of 10^26. 36talks-12timeslots-5rooms has 36 talks, 12 timeslots and 5 rooms with a search space of 10^64. 72talks-12timeslots-10rooms has 72 talks, 12 timeslots and 10 rooms with a search space of 10^149. 108talks-18timeslots-10rooms has 108 talks, 18 timeslots and 10 rooms with a search space of 10^243. 216talks-18timeslots-20rooms has 216 talks, 18 timeslots and 20 rooms with a search space of 10^552.
9.22. Rock tour
Drive the rock bank bus from show to show, but schedule shows only on available days.
Hard constraints:
- Schedule every required show.
- Schedule as many shows as possible.
Medium constraints:
- Maximize revenue opportunity.
- Minimize driving time.
- Visit sooner than later.
Soft constraints:
- Avoid long driving times.
Problem size
47shows has 47 shows with a search space of 10^59.
9.23. Flight crew scheduling
Assign flights to pilots and flight attendants.
Hard constraints:
- Required skill: each flight assignment has a required skill. For example, flight AB0001 requires 2 pilots and 3 flight attendants.
- Flight conflict: each employee can only attend one flight at the same time
- Transfer between two flights: between two flights, an employee must be able to transfer from the arrival airport to the departure airport. For example, Ann arrives in Brussels at 10:00 and departs in Amsterdam at 15:00.
- Employee unavailability: the employee must be available on the day of the flight. For example, Ann is on PTO on 1-Feb.
Soft constraints:
- First assignment departing from home
- Last assignment arriving at home
- Load balance flight duration total per employee
Problem size
175flights-7days-Europe has 2 skills, 50 airports, 150 employees, 175 flights and 875 flight assignments with a search space of 10^1904. 700flights-28days-Europe has 2 skills, 50 airports, 150 employees, 700 flights and 3500 flight assignments with a search space of 10^7616. 875flights-7days-Europe has 2 skills, 50 airports, 750 employees, 875 flights and 4375 flight assignments with a search space of 10^12578. 175flights-7days-US has 2 skills, 48 airports, 150 employees, 175 flights and 875 flight assignments with a search space of 10^1904.
