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## Chapter 3. Use cases and examples

## 3.1. Examples overview

OptaPlanner has several examples. In this manual we explain OptaPlanner mainly using the n queens example and cloud balancing example. So it's advisable to read at least those sections.

The source code of all these examples is available in the distribution zip under `examples/sources` and also in git under `optaplanner/optaplanner-examples`.

Table 3.1. Examples overview

SpeExample | Domain | Size | Competition? | Special features used
--- | --- | --- | --- | ---
N queens | 1 entity class 1 variable | Entity <= `256` Value <= 256 Search space <= `10^616` | Pointless (cheatable) | None
Cloud balancing | 1 entity class 1 variable | Entity <= `2400` Value <= 800 Search space <= `10^6967` | No Definied by us | Real-time planning
Traveling salesman | 1 entity class 1 chained variable | Entity <= `980` Value <= 980 Search space <= `10^2927` | Unrealistic TSP web | Real-time planning
Dinner party | 1 entity class 1 variable | Entity <= `144` Value <= 72 Search space <= `10^310` | Unrealistic | Decision Table spreadsheet for score constraints
Tennis club scheduling | 1 entity class 1 variable | Entity <= `72` Value <= 7 Search space <= `10^60` | No Definied by us | Fairness score constraints Immovable entities
Course timetabling | 1 entity class 2 variables | Entity <= `434` Value <= 25 and <= `20` Search space <= `10^1171` | Realistic ITC 2007 track 3 | Immovable entities
Machine reassignment | 1 entity class 1 variable | Entity <= `50000` Value <= 5000 Search space <= `10^184948` | Nearly realistic ROADEF 2012 | Real-time planning
Vehicle routing | 1 entity class 1 chained variable 1 shadow entity class 1 automatic shadow variable | Entity <= `134` Value <= 141 Search space <= `10^285` | Unrealistic VRP web | VariableListener Real-time planning
Vehicle routing with time windows | Extra on Vehicle routing: 1 shadow variable | Entity <= `1000` Value <= 1250 Search space <= `10^3000` | Unrealistic VRP web | VariableListener Real-time planning
Project job scheduling | 1 entity class 2 variables 1 shadow variable | Entity <= `640` Value <= ? and <= `?` Search space <= `?` | Nearly realistic MISTA 2013 | Bendable score VariableListener
Hospital bed planning | 1 entity class 1 nullable variable | Entity <= `2750` Value <= 471 Search space <= `10^6851` | Unrealistic Kaho PAS | Overconstrained planning
Exam timetabling | 2 entity classes (same hierarchy) 2 variables | Entity <= `1096` Value <= 80 and <= `49` Search space <= `10^3374` | Realistic ITC 2007 track 1 | VariableListener
Employee rostering | 1 entity class 1 variable | Entity <= `752` Value <= 50 Search space <= `10^1277` | Realistic INRC 2010 | Continuous planning Real-time planning
Traveling tournament | 1 entity class 1 variable | Entity <= `1560` Value <= 78 Search space <= `10^2951` | Unrealistic TTP | None
Cheap time scheduling | 1 entity class 2 variables | Entity <= `500` Value <= 100 and <= `288` Search space <= `10^20078` | Nearly realistic ICON | None

A realistic competition is an official, independent competition:

- that clearly defines a real-word use case

- with real-world constraints

- with multiple, real-world datasets

- that expects reproducible results within a specific time limit on specific hardware

- that has had serious participation from the academic and/or enterprise Operations Research community

These realistic competitions provide an objective comparison of OptaPlanner with competitive software and academic research.

## 3.2. Basic examples

### 3.2.1. N queens

#### 3.2.1.1. Problem statement

Place n queens on a n sized chessboard so no 2 queens can attach each other. The most common n queens puzzle is the 8 queens puzzle, with n = 8:

Constraints:

- Use a chessboard of n columns and n rows.

- Place n queens on the chessboard.

- No 2 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 4 queens puzzle as the primary example.

A proposed solution could be:

Figure 3.1. A wrong solution for the 4 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 2 queens can attack each other" constraint, but would break the "place n queens" constraint.

Below is a correct solution:

Figure 3.2. A correct solution for the 4 queens puzzle

All the constraints have been met, so the solution is correct. Note that most n queens puzzles have multiple correct solutions. We'll focus on finding a single correct solution for a given n, not on finding the number of possible correct solutions for a given n.

#### 3.2.1.2. 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.

#### 3.2.1.3. Domain model

Use a good domain model: it will be 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

    }

A `Queen` instance has a `Column` (for example: 0 is column A, 1 is column B, ...) and a `Row` (its row, for example: 0 is row 0, 1 is row 1, ...). Based on the column and the row, the ascending diagonal line as well as the descending diagonal line can be calculated. The column and row indexes start from the upper left corner of the chessboard.

    public class NQueens implements Solution<SimpleScore> {

        private int n;

        private List<Column> columnList;

        private List<Row> rowList;

        private List<Queen> queenList;

        private SimpleScore score;

        // ... getters and setters

    }

A single `NQueens` instance contains a list of all `Queen` instances. It is the `Solution` implementation which will be supplied to, solved by and retrieved from the Solver. Notice that in the 4 queens example, NQueens's `getN()` method will always return 4.

Table 3.2. A solution for 4 queens shown in the domain model

A solution | Queen | 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 2 queens share the same column, row or diagonal line, such as (*) and (**), they can attack each other.

### 3.2.2. Cloud balancing

This example is explained in a tutorial.

### 3.2.3. Traveling salesman (TSP - Traveling salesman problem)

#### 3.2.3.1. Problem statement

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's often only part of a planning problem, along with other constraints, such as employee shift rostering constraints.

#### 3.2.3.2. Problem size

    dj38     has  38 cities with a search space of   10^58.
    europe40 has  40 cities with a search space of   10^62.
    st70     has  70 cities with a search space of  10^126.
    pcb442   has 442 cities with a search space of 10^1166.
    lu980    has 980 cities with a search space of 10^2927.

#### 3.2.3.3. Problem difficulty

Despite TSP's simple definition, the problem is surprisingly hard to solve. Because it's 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:

### 3.2.4. Dinner party

#### 3.2.4.1. Problem statement

Miss Manners is throwing another dinner party.

- This time she invited 144 guests and prepared 12 round tables with 12 seats each.

- Every guest should sit next to someone (left and right) of the opposite gender.

- And that neighbour should have at least one hobby in common with the guest.

- At every table, there should be 2 politicians, 2 doctors, 2 socialites, 2 coaches, 2 teachers and 2 programmers.

- And the 2 politicians, 2 doctors, 2 coaches and 2 programmers shouldn't be the same kind at a table.

Drools Expert also has the normal Miss Manners example (which is much smaller) and employs an exhaustive heuristic to solve it. OptaPlanner's implementation is far more scalable because it uses heuristics to find the best solution and Drools Expert to calculate the score of each solution.

#### 3.2.4.2. Problem size

    wedding01 has 18 jobs, 144 guests, 288 hobby practicians, 12 tables and 144 seats with a search space of 10^310.

### 3.2.5. Tennis club scheduling

#### 3.2.5.1. Problem statement

Every week the tennis club has 4 teams playing round robin against each other. Assign those 4 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.

#### 3.2.5.2. Problem size

    munich-7teams has 7 teams, 18 days, 12 unavailabilityPenalties and 72 teamAssignments with a search space of 10^60.

## 3.3. Real examples

### 3.3.1. Course timetabling (ITC 2007 track 3 - Curriculum course scheduling)

#### 3.3.1.1. Problem statement

Schedule each lecture into a timeslot and into a room.

Hard constraints:

- Teacher conflict: A teacher must not have 2 lectures in the same period.

- Curriculum conflict: A curriculum must not have 2 lectures in the same period.

- Room occupancy: 2 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 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 the same room.

The problem is defined by the International Timetabling Competition 2007 track 3.

#### 3.3.1.2. 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.

#### 3.3.1.3. Domain model

### 3.3.2. Machine reassignment (Google ROADEF 2012)

#### 3.3.2.1. Problem statement

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, RAM, ...). This is 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 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.

#### 3.3.2.2. 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.

#### 3.3.2.3. Domain model

### 3.3.3. Vehicle routing

#### 3.3.3.1. Problem statement

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 in 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 it's timewindowed variant (CVRPTW) are defined by the VRP web.

#### 3.3.3.2. Problem size

CVRP instances (without time windows):

    A-n32-k5  has 1 depots,  5 vehicles and  31 customers with a search space of  10^46.
    A-n33-k5  has 1 depots,  5 vehicles and  32 customers with a search space of  10^48.
    A-n33-k6  has 1 depots,  6 vehicles and  32 customers with a search space of  10^48.
    A-n34-k5  has 1 depots,  5 vehicles and  33 customers with a search space of  10^50.
    A-n36-k5  has 1 depots,  5 vehicles and  35 customers with a search space of  10^54.
    A-n37-k5  has 1 depots,  5 vehicles and  36 customers with a search space of  10^56.
    A-n37-k6  has 1 depots,  6 vehicles and  36 customers with a search space of  10^56.
    A-n38-k5  has 1 depots,  5 vehicles and  37 customers with a search space of  10^58.
    A-n39-k5  has 1 depots,  5 vehicles and  38 customers with a search space of  10^60.
    A-n39-k6  has 1 depots,  6 vehicles and  38 customers with a search space of  10^60.
    A-n44-k7  has 1 depots,  7 vehicles and  43 customers with a search space of  10^70.
    A-n45-k6  has 1 depots,  6 vehicles and  44 customers with a search space of  10^72.
    A-n45-k7  has 1 depots,  7 vehicles and  44 customers with a search space of  10^72.
    A-n46-k7  has 1 depots,  7 vehicles and  45 customers with a search space of  10^74.
    A-n48-k7  has 1 depots,  7 vehicles and  47 customers with a search space of  10^78.
    A-n53-k7  has 1 depots,  7 vehicles and  52 customers with a search space of  10^89.
    A-n54-k7  has 1 depots,  7 vehicles and  53 customers with a search space of  10^91.
    A-n55-k9  has 1 depots,  9 vehicles and  54 customers with a search space of  10^93.
    A-n60-k9  has 1 depots,  9 vehicles and  59 customers with a search space of 10^104.
    A-n61-k9  has 1 depots,  9 vehicles and  60 customers with a search space of 10^106.
    A-n62-k8  has 1 depots,  8 vehicles and  61 customers with a search space of 10^108.
    A-n63-k10 has 1 depots, 10 vehicles and  62 customers with a search space of 10^111.
    A-n63-k9  has 1 depots,  9 vehicles and  62 customers with a search space of 10^111.
    A-n64-k9  has 1 depots,  9 vehicles and  63 customers with a search space of 10^113.
    A-n65-k9  has 1 depots,  9 vehicles and  64 customers with a search space of 10^115.
    A-n69-k9  has 1 depots,  9 vehicles and  68 customers with a search space of 10^124.
    A-n80-k10 has 1 depots, 10 vehicles and  79 customers with a search space of 10^149.
    F-n135-k7 has 1 depots,  7 vehicles and 134 customers with a search space of 10^285.
    F-n45-k4  has 1 depots,  4 vehicles and  44 customers with a search space of  10^72.
    F-n72-k4  has 1 depots,  4 vehicles and  71 customers with a search space of 10^131.

CVRPTW instances (with time windows):

    Solomon_025_C101       has 1 depots,  25 vehicles and   25 customers with a search space of   10^34.
    Solomon_025_C201       has 1 depots,  25 vehicles and   25 customers with a search space of   10^34.
    Solomon_025_R101       has 1 depots,  25 vehicles and   25 customers with a search space of   10^34.
    Solomon_025_R201       has 1 depots,  25 vehicles and   25 customers with a search space of   10^34.
    Solomon_025_RC101      has 1 depots,  25 vehicles and   25 customers with a search space of   10^34.
    Solomon_025_RC201      has 1 depots,  25 vehicles and   25 customers with a search space of   10^34.
    Solomon_100_C101       has 1 depots,  25 vehicles and  100 customers with a search space of  10^200.
    Solomon_100_C201       has 1 depots,  25 vehicles and  100 customers with a search space of  10^200.
    Solomon_100_R101       has 1 depots,  25 vehicles and  100 customers with a search space of  10^200.
    Solomon_100_R201       has 1 depots,  25 vehicles and  100 customers with a search space of  10^200.
    Solomon_100_RC101      has 1 depots,  25 vehicles and  100 customers with a search space of  10^200.
    Solomon_100_RC201      has 1 depots,  25 vehicles and  100 customers with a search space of  10^200.
    Homberger_0200_C1_2_1  has 1 depots,  50 vehicles and  200 customers with a search space of  10^460.
    Homberger_0200_C2_2_1  has 1 depots,  50 vehicles and  200 customers with a search space of  10^460.
    Homberger_0200_R1_2_1  has 1 depots,  50 vehicles and  200 customers with a search space of  10^460.
    Homberger_0200_R2_2_1  has 1 depots,  50 vehicles and  200 customers with a search space of  10^460.
    Homberger_0200_RC1_2_1 has 1 depots,  50 vehicles and  200 customers with a search space of  10^460.
    Homberger_0200_RC2_2_1 has 1 depots,  50 vehicles and  200 customers with a search space of  10^460.
    Homberger_0400_C1_4_1  has 1 depots, 100 vehicles and  400 customers with a search space of 10^1040.
    Homberger_0400_C2_4_1  has 1 depots, 100 vehicles and  400 customers with a search space of 10^1040.
    Homberger_0400_R1_4_1  has 1 depots, 100 vehicles and  400 customers with a search space of 10^1040.
    Homberger_0400_R2_4_1  has 1 depots, 100 vehicles and  400 customers with a search space of 10^1040.
    Homberger_0400_RC1_4_1 has 1 depots, 100 vehicles and  400 customers with a search space of 10^1040.
    Homberger_0400_RC2_4_1 has 1 depots, 100 vehicles and  400 customers with a search space of 10^1040.
    Homberger_0600_C1_6_1  has 1 depots, 150 vehicles and  600 customers with a search space of 10^1666.
    Homberger_0600_C2_6_1  has 1 depots, 150 vehicles and  600 customers with a search space of 10^1666.
    Homberger_0600_R1_6_1  has 1 depots, 150 vehicles and  600 customers with a search space of 10^1666.
    Homberger_0600_R2_6_1  has 1 depots, 150 vehicles and  600 customers with a search space of 10^1666.
    Homberger_0600_RC1_6_1 has 1 depots, 150 vehicles and  600 customers with a search space of 10^1666.
    Homberger_0600_RC2_6_1 has 1 depots, 150 vehicles and  600 customers with a search space of 10^1666.
    Homberger_0800_C1_8_1  has 1 depots, 200 vehicles and  800 customers with a search space of 10^2322.
    Homberger_0800_C2_8_1  has 1 depots, 200 vehicles and  800 customers with a search space of 10^2322.
    Homberger_0800_R1_8_1  has 1 depots, 200 vehicles and  800 customers with a search space of 10^2322.
    Homberger_0800_R2_8_1  has 1 depots, 200 vehicles and  800 customers with a search space of 10^2322.
    Homberger_0800_RC1_8_1 has 1 depots, 200 vehicles and  800 customers with a search space of 10^2322.
    Homberger_0800_RC2_8_1 has 1 depots, 200 vehicles and  800 customers with a search space of 10^2322.
    Homberger_1000_C110_1  has 1 depots, 250 vehicles and 1000 customers with a search space of 10^3000.
    Homberger_1000_C210_1  has 1 depots, 250 vehicles and 1000 customers with a search space of 10^3000.
    Homberger_1000_R110_1  has 1 depots, 250 vehicles and 1000 customers with a search space of 10^3000.
    Homberger_1000_R210_1  has 1 depots, 250 vehicles and 1000 customers with a search space of 10^3000.
    Homberger_1000_RC110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^3000.
    Homberger_1000_RC210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^3000.

#### 3.3.3.3. Domain model

The vehicle routing with timewindows domain model makes heavily use of shadow variables. This allows it to express its constraints more naturally, because properties such as `arrivalTime` and `departureTime`, are directly available on the domain model.

#### 3.3.3.4. Road distances instead of air distances

In the real world, vehicles can't 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 doesn't matter much, as long as the distance between 2 points can be looked up (and are preferably precalculated). The road cost doesn't 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) and Open MapQuest (webservice). There are also several technologies to render it, such as Leaflet.

### 3.3.4. Project job scheduling

#### 3.3.4.1. Problem statement

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 then available.

  - Resources are local (shared between jobs of the same project) or global (shared between all jobs)

  - Resource 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.

#### 3.3.4.2. 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.

### 3.3.5. Hospital bed planning (PAS - Patient admission scheduling)

#### 3.3.5.1. Problem statement

Assign each patient (that will come to the hospital) into a bed for each night that the patient will stay in the hospital. Each bed belongs to a room and each room belongs to a department. The arrival and departure dates of the patients is fixed: only a bed needs to be assigned for each night.

This problem features overconstrained datasets.

Hard constraints:

- 2 patients must not be assigned to the same bed in the same night. Weight: `-1000hard * conflictNightCount`.

- A room can have a gender limitation: only females, only males, the same gender in the same night or no gender limitation at all. Weight: `-50hard * nightCount`.

- A department can have a minimum or maximum age. Weight: `-100hard * nightCount`.

- A patient can require a room with specific equipment(s). Weight: `-50hard * nightCount`.

Medium constraints:

- Assign every patient to a bed, unless the dataset is overconstrained. Weight: `-1medium * nightCount`.

Soft constraints:

- A patient can prefer a maximum room size, for example if he/she want a single room. Weight: `-8soft * nightCount`.

- A patient is best assigned to a department that specializes in his/her problem. Weight: `-10soft * nightCount`.

- A patient is best assigned to a room that specializes in his/her problem. Weight: `-20soft * nightCount`.

  - That room specialism should be priority 1. Weight: `-10soft * (priority - 1) * nightCount`.

- A patient can prefer a room with specific equipment(s). Weight: `-20soft * nightCount`.

The problem is a variant on Kaho's Patient Scheduling and the datasets come from real world hospitals.

#### 3.3.5.2. Problem size

    testdata01 has 4 specialisms, 2 equipments, 4 departments,  98 rooms, 286 beds, 14 nights,  652 patients and  652 admissions with a search space of 10^1601.
    testdata02 has 6 specialisms, 2 equipments, 6 departments, 151 rooms, 465 beds, 14 nights,  755 patients and  755 admissions with a search space of 10^2013.
    testdata03 has 5 specialisms, 2 equipments, 5 departments, 131 rooms, 395 beds, 14 nights,  708 patients and  708 admissions with a search space of 10^1838.
    testdata04 has 6 specialisms, 2 equipments, 6 departments, 155 rooms, 471 beds, 14 nights,  746 patients and  746 admissions with a search space of 10^1994.
    testdata05 has 4 specialisms, 2 equipments, 4 departments, 102 rooms, 325 beds, 14 nights,  587 patients and  587 admissions with a search space of 10^1474.
    testdata06 has 4 specialisms, 2 equipments, 4 departments, 104 rooms, 313 beds, 14 nights,  685 patients and  685 admissions with a search space of 10^1709.
    testdata07 has 6 specialisms, 4 equipments, 6 departments, 162 rooms, 472 beds, 14 nights,  519 patients and  519 admissions with a search space of 10^1387.
    testdata08 has 6 specialisms, 4 equipments, 6 departments, 148 rooms, 441 beds, 21 nights,  895 patients and  895 admissions with a search space of 10^2366.
    testdata09 has 4 specialisms, 4 equipments, 4 departments, 105 rooms, 310 beds, 28 nights, 1400 patients and 1400 admissions with a search space of 10^3487.
    testdata10 has 4 specialisms, 4 equipments, 4 departments, 104 rooms, 308 beds, 56 nights, 1575 patients and 1575 admissions with a search space of 10^3919.
    testdata11 has 4 specialisms, 4 equipments, 4 departments, 107 rooms, 318 beds, 91 nights, 2514 patients and 2514 admissions with a search space of 10^6291.
    testdata12 has 4 specialisms, 4 equipments, 4 departments, 105 rooms, 310 beds, 84 nights, 2750 patients and 2750 admissions with a search space of 10^6851.
    testdata13 has 5 specialisms, 4 equipments, 5 departments, 125 rooms, 368 beds, 28 nights,  907 patients and 1109 admissions with a search space of 10^2845.

## 3.4. Difficult examples

### 3.4.1. Exam timetabling (ITC 2007 track 1 - Examination)

#### 3.4.1.1. Problem statement

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: 2 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: 2 specified exams must use the same period (but possibly another room).

  - Exclusion: 2 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: 1 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 2 exams in a row.

- The same student should not have 2 exams on the same day.

- Period spread: 2 exams that share students should be a number of periods apart.

- Mixed durations: 2 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.

#### 3.4.1.2. 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.

#### 3.4.1.3. Domain model

Below you can see the main examination domain classes:

Figure 3.3. 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).

### 3.4.2. Employee rostering (INRC 2010 - Nurse rostering)

#### 3.4.2.1. Problem statement

For each shift, assign a nurse to work that shift.

The problem is defined by the International Nurse Rostering Competition 2010.

#### 3.4.2.2. Problem size

There are 3 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.

### 3.4.3. Traveling tournament problem (TTP)

#### 3.4.3.1. Problem statement

Schedule matches between n teams.

Hard constraints:

- Each team plays twice against every other team: once home and once away.

- Each team has exactly 1 match on each timeslot.

- No team must have more than 3 consecutive home or 3 consecutive away matches.

- No repeaters: no 2 consecutive matches of the same 2 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).

#### 3.4.3.2. Problem size

    1-nl04     has  6 days,  4 teams and   12 matches with a search space of    10^9.
    1-nl06     has 10 days,  6 teams and   30 matches with a search space of   10^30.
    1-nl08     has 14 days,  8 teams and   56 matches with a search space of   10^64.
    1-nl10     has 18 days, 10 teams and   90 matches with a search space of  10^112.
    1-nl12     has 22 days, 12 teams and  132 matches with a search space of  10^177.
    1-nl14     has 26 days, 14 teams and  182 matches with a search space of  10^257.
    1-nl16     has 30 days, 16 teams and  240 matches with a search space of  10^354.
    2-bra24    has 46 days, 24 teams and  552 matches with a search space of  10^917.
    3-nfl16    has 30 days, 16 teams and  240 matches with a search space of  10^354.
    3-nfl18    has 34 days, 18 teams and  306 matches with a search space of  10^468.
    3-nfl20    has 38 days, 20 teams and  380 matches with a search space of  10^600.
    3-nfl22    has 42 days, 22 teams and  462 matches with a search space of  10^749.
    3-nfl24    has 46 days, 24 teams and  552 matches with a search space of  10^917.
    3-nfl26    has 50 days, 26 teams and  650 matches with a search space of 10^1104.
    3-nfl28    has 54 days, 28 teams and  756 matches with a search space of 10^1309.
    3-nfl30    has 58 days, 30 teams and  870 matches with a search space of 10^1534.
    3-nfl32    has 62 days, 32 teams and  992 matches with a search space of 10^1778.
    4-super04  has  6 days,  4 teams and   12 matches with a search space of    10^9.
    4-super06  has 10 days,  6 teams and   30 matches with a search space of   10^30.
    4-super08  has 14 days,  8 teams and   56 matches with a search space of   10^64.
    4-super10  has 18 days, 10 teams and   90 matches with a search space of  10^112.
    4-super12  has 22 days, 12 teams and  132 matches with a search space of  10^177.
    4-super14  has 26 days, 14 teams and  182 matches with a search space of  10^257.
    5-galaxy04 has  6 days,  4 teams and   12 matches with a search space of    10^9.
    5-galaxy06 has 10 days,  6 teams and   30 matches with a search space of   10^30.
    5-galaxy08 has 14 days,  8 teams and   56 matches with a search space of   10^64.
    5-galaxy10 has 18 days, 10 teams and   90 matches with a search space of  10^112.
    5-galaxy12 has 22 days, 12 teams and  132 matches with a search space of  10^177.
    5-galaxy14 has 26 days, 14 teams and  182 matches with a search space of  10^257.
    5-galaxy16 has 30 days, 16 teams and  240 matches with a search space of  10^354.
    5-galaxy18 has 34 days, 18 teams and  306 matches with a search space of  10^468.
    5-galaxy20 has 38 days, 20 teams and  380 matches with a search space of  10^600.
    5-galaxy22 has 42 days, 22 teams and  462 matches with a search space of  10^749.
    5-galaxy24 has 46 days, 24 teams and  552 matches with a search space of  10^917.
    5-galaxy26 has 50 days, 26 teams and  650 matches with a search space of 10^1104.
    5-galaxy28 has 54 days, 28 teams and  756 matches with a search space of 10^1309.
    5-galaxy30 has 58 days, 30 teams and  870 matches with a search space of 10^1534.
    5-galaxy32 has 62 days, 32 teams and  992 matches with a search space of 10^1778.
    5-galaxy34 has 66 days, 34 teams and 1122 matches with a search space of 10^2041.
    5-galaxy36 has 70 days, 36 teams and 1260 matches with a search space of 10^2324.
    5-galaxy38 has 74 days, 38 teams and 1406 matches with a search space of 10^2628.
    5-galaxy40 has 78 days, 40 teams and 1560 matches with a search space of 10^2951.

### 3.4.4. Cheap time scheduling

#### 3.4.4.1. Problem statement

Schedule all tasks in time and on a machine to minimize power cost. Power prices differs 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 inflicted.

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.

#### 3.4.4.2. 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.

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