Step Functions

IntStepFunction is a piecewise-constant function defined at modeling time. Step functions provide constant data that the solver uses to model calendars, availability windows, time-dependent costs, energy pricing, efficiency rates, and other time-varying quantities. The function itself doesn't change during solving—it's static input data.

Creating Step Functions

Create a step function using model.step_function():

Python

import optalcp as cp

model = cp.Model()

func = model.step_function([
    # Implicitly: value is 0 before time 0
    (0, 10),    # From time 0: value is 10
    (100, 20),  # From time 100: value is 20
    (200, 15),  # From time 200 onwards: value is 15
])

C#

using OptalCP;

var model = new Model();

var func = model.StepFunction(new (int, int)[]
{
    // Implicitly: value is 0 before time 0
    (0, 10),    // From time 0: value is 10
    (100, 20),  // From time 100: value is 20
    (200, 15),  // From time 200 onwards: value is 15
});

TypeScript

import * as CP from '@scheduleopt/optalcp';

const model = new CP.Model();

const func = model.stepFunction([
    // Implicitly: value is 0 before time 0
    [0, 10],    // From time 0: value is 10
    [100, 20],  // From time 100: value is 20
    [200, 15],  // From time 200 onwards: value is 15
]);

Parameters

• values: Iterable of (time, value) pairs defining the function. Must be sorted by time in ascending order.

Semantics

A step function $f$ defined by points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$:

$$f(x) = \begin{cases} 0 & \text{if } x < x_0 \\ y_i & \text{if } x_i \le x < x_{i+1} \\ y_n & \text{if } x \ge x_n \end{cases}$$

The function value at any point is the value from the most recent step at or before that point. Before the first step, the value is 0.

Evaluating Step Functions

Point Evaluation

Evaluate the function at a specific point in time:

Python

# Cost varies by time of day
cost_func = model.step_function([
    (0, 10),    # 0h-8h: low cost ($10)
    (8, 20),    # 8h-17h: high cost ($20)
    (17, 10),   # 17h onwards: low cost ($10)
])

task = model.interval_var(length=5, name="task")

# Evaluate cost at task start
start_cost = model.eval(cost_func, task.start())

model.minimize(start_cost)

C#

// Cost varies by time of day
var costFunc = model.StepFunction(new (int, int)[]
{
    (0, 10),    // 0h-8h: low cost ($10)
    (8, 20),    // 8h-17h: high cost ($20)
    (17, 10),   // 17h onwards: low cost ($10)
});

var task = model.IntervalVar(length: 5, name: "task");

// Evaluate cost at task start
var startCost = model.Eval(costFunc, task.Start());

model.Minimize(startCost);

TypeScript

// Cost varies by time of day
const costFunc = model.stepFunction([
    [0, 10],    // 0h-8h: low cost ($10)
    [8, 20],    // 8h-17h: high cost ($20)
    [17, 10],   // 17h onwards: low cost ($10)
]);

const task = model.intervalVar({ length: 5, name: "task" });

// Evaluate cost at task start
const startCost = model.eval(costFunc, task.start());

model.minimize(startCost);

Integral Over Interval

Compute the sum of function values at each integer point in the interval:

$$\mathtt{integral}(f, [s, e)) = \sum_{t=s}^{e-1} f(t)$$

The interval is half-open: includes start, excludes end. If the interval has zero length ($s = e$), the result is 0. If the interval is absent, the result is absent.

Requirement: The step function must be non-negative.

Python

# Energy rate per time unit
energy_rate = model.step_function([
    (0, 5),     # 0h-8h: off-peak ($5/hour)
    (8, 15),    # 8h-18h: peak ($15/hour)
    (18, 5),    # 18h onwards: off-peak ($5/hour)
])

task = model.interval_var(length=10, name="task")

# Total energy cost = sum of rate over task duration
total_cost = model.integral(energy_rate, task)

model.minimize(total_cost)

C#

// Energy rate per time unit
var energyRate = model.StepFunction(new (int, int)[]
{
    (0, 5),     // 0h-8h: off-peak ($5/hour)
    (8, 15),    // 8h-18h: peak ($15/hour)
    (18, 5),    // 18h onwards: off-peak ($5/hour)
});

var task = model.IntervalVar(length: 10, name: "task");

// Total energy cost = sum of rate over task duration
var totalCost = model.Integral(energyRate, task);

model.Minimize(totalCost);

TypeScript

// Energy rate per time unit
const energyRate = model.stepFunction([
    [0, 5],     // 0h-8h: off-peak ($5/hour)
    [8, 15],    // 8h-18h: peak ($15/hour)
    [18, 5],    // 18h onwards: off-peak ($5/hour)
]);

const task = model.intervalVar({ length: 10, name: "task" });

// Total energy cost = sum of rate over task duration
const totalCost = model.integral(energyRate, task);

model.minimize(totalCost);

Forbid Constraints

Step functions can forbid task placement in time windows where the function value is zero.

Forbid Start / End

Require the function to be non-zero at task start or end:

Python

# Availability: 1 = available, 0 = unavailable
availability = model.step_function([
    (0, 1),     # 0h-8h: available
    (8, 0),     # 8h-12h: break
    (12, 1),    # 12h onwards: available
])

task = model.interval_var(length=5, name="task")

# Task cannot start when availability is 0
task.forbid_start(availability)
# Equivalent to:
model.enforce(availability.eval(task.start()) != 0)

# Task cannot end when availability is 0
task.forbid_end(availability)
# Equivalent to:
model.enforce(availability.eval(task.end()) != 0)

C#

// Availability: 1 = available, 0 = unavailable
var availability = model.StepFunction(new (int, int)[]
{
    (0, 1),     // 0h-8h: available
    (8, 0),     // 8h-12h: break
    (12, 1),    // 12h onwards: available
});

var task = model.IntervalVar(length: 5, name: "task");

// Task cannot start when availability is 0
task.ForbidStart(availability);
// Equivalent to:
model.Enforce(availability.Eval(task.Start()) != 0);

// Task cannot end when availability is 0
task.ForbidEnd(availability);
// Equivalent to:
model.Enforce(availability.Eval(task.End()) != 0);

TypeScript

// Availability: 1 = available, 0 = unavailable
const availability = model.stepFunction([
    [0, 1],     // 0h-8h: available
    [8, 0],     // 8h-12h: break
    [12, 1],    // 12h onwards: available
]);

const task = model.intervalVar({ length: 5, name: "task" });

// Task cannot start when availability is 0
task.forbidStart(availability);
// Equivalent to:
model.enforce(availability.eval(task.start()).ne(0))

// Task cannot end when availability is 0
task.forbidEnd(availability);
// Equivalent to:
model.enforce(availability.eval(task.end()).ne(0))

Forbid Extent

Require the function to be non-zero throughout the entire task:

Python

# Machine availability calendar
availability = model.step_function([
    (0, 1),     # 0h-8h: available
    (8, 0),     # 8h-12h: maintenance
    (12, 1),    # 12h-20h: available
    (20, 0),    # 20h onwards: closed
])

task = model.interval_var(length=5, name="task")

# Task cannot overlap any period where availability is 0
task.forbid_extent(availability)

C#

// Machine availability calendar
var availability = model.StepFunction(new (int, int)[]
{
    (0, 1),     // 0h-8h: available
    (8, 0),     // 8h-12h: maintenance
    (12, 1),    // 12h-20h: available
    (20, 0),    // 20h onwards: closed
});

var task = model.IntervalVar(length: 5, name: "task");

// Task cannot overlap any period where availability is 0
task.ForbidExtent(availability);

TypeScript

// Machine availability calendar
const availability = model.stepFunction([
    [0, 1],     // 0h-8h: available
    [8, 0],     // 8h-12h: maintenance
    [12, 1],    // 12h-20h: available
    [20, 0],    // 20h onwards: closed
]);

const task = model.intervalVar({ length: 5, name: "task" });

// Task cannot overlap any period where availability is 0
task.forbidExtent(availability);

Use Cases

Time-Dependent Costs

Model costs that vary throughout the day:

Python

electricity_cost = model.step_function([
    (0, 5),     # 0h-6h: night ($5/hour)
    (6, 8),     # 6h-9h: morning ($8/hour)
    (9, 15),    # 9h-17h: peak ($15/hour)
    (17, 8),    # 17h-22h: evening ($8/hour)
    (22, 5),    # 22h onwards: night ($5/hour)
])

tasks = [model.interval_var(length=10) for _ in range(5)]

# Minimize total electricity cost
total_cost = model.sum([
    model.integral(electricity_cost, task) for task in tasks
])
model.minimize(total_cost)

C#

var electricityCost = model.StepFunction(new (int, int)[]
{
    (0, 5),     // 0h-6h: night ($5/hour)
    (6, 8),     // 6h-9h: morning ($8/hour)
    (9, 15),    // 9h-17h: peak ($15/hour)
    (17, 8),    // 17h-22h: evening ($8/hour)
    (22, 5),    // 22h onwards: night ($5/hour)
});

var tasks = Enumerable.Range(0, 5)
    .Select(_ => model.IntervalVar(length: 10))
    .ToArray();

// Minimize total electricity cost
var totalCost = model.Sum(
    tasks.Select(task => model.Integral(electricityCost, task))
);
model.Minimize(totalCost);

TypeScript

const electricityCost = model.stepFunction([
    [0, 5],     // 0h-6h: night ($5/hour)
    [6, 8],     // 6h-9h: morning ($8/hour)
    [9, 15],    // 9h-17h: peak ($15/hour)
    [17, 8],    // 17h-22h: evening ($8/hour)
    [22, 5],    // 22h onwards: night ($5/hour)
]);

const tasks = Array.from({ length: 5 }, () => model.intervalVar({ length: 10 }));

// Minimize total electricity cost
const totalCost = model.sum(
    tasks.map(task => model.integral(electricityCost, task))
);
model.minimize(totalCost);

Availability Windows

Model resource availability over time:

Python

# Machine available only during working hours
working_hours = model.step_function([
    (0, 0),     # 0h-8h: closed
    (8, 1),     # 8h-17h: open
    (17, 0),    # 17h onwards: closed
])

tasks = [model.interval_var(length=5) for _ in range(10)]

# All tasks must execute during working hours
for task in tasks:
    task.forbid_extent(working_hours)

model.minimize(model.max([t.end() for t in tasks]))

C#

// Machine available only during working hours
var workingHours = model.StepFunction(new (int, int)[]
{
    (0, 0),     // 0h-8h: closed
    (8, 1),     // 8h-17h: open
    (17, 0),    // 17h onwards: closed
});

var tasks = Enumerable.Range(0, 10)
    .Select(_ => model.IntervalVar(length: 5))
    .ToArray();

// All tasks must execute during working hours
foreach (var task in tasks)
    task.ForbidExtent(workingHours);

model.Minimize(model.Max(tasks.Select(t => t.End())));

TypeScript

// Machine available only during working hours
const workingHours = model.stepFunction([
    [0, 0],     // 0h-8h: closed
    [8, 1],     // 8h-17h: open
    [17, 0],    // 17h onwards: closed
]);

const tasks = Array.from({ length: 10 }, () => model.intervalVar({ length: 5 }));

// All tasks must execute during working hours
tasks.forEach(task => task.forbidExtent(workingHours));

model.minimize(model.max(tasks.map(t => t.end())));

Minimum Collected Value

Require a task to accumulate a minimum total from a step function:

Python

# Sunlight intensity throughout the day
sunlight = model.step_function([
    (0, 0),     # 0h-6h: dark
    (6, 3),     # 6h-9h: morning
    (9, 10),    # 9h-15h: peak sunlight
    (15, 5),    # 15h-18h: afternoon
    (18, 0),    # 18h onwards: dark
])

# Solar charging task with variable length
charging = model.interval_var(length=(10, 50), name="charging")

# Must collect at least 100 units of sunlight
model.enforce(model.integral(sunlight, charging) >= 100)

C#

// Sunlight intensity throughout the day
var sunlight = model.StepFunction(new (int, int)[]
{
    (0, 0),     // 0h-6h: dark
    (6, 3),     // 6h-9h: morning
    (9, 10),    // 9h-15h: peak sunlight
    (15, 5),    // 15h-18h: afternoon
    (18, 0),    // 18h onwards: dark
});

// Solar charging task with variable length
var charging = model.IntervalVar(length: (10, 50), name: "charging");

// Must collect at least 100 units of sunlight
model.Enforce(model.Integral(sunlight, charging) >= 100);

TypeScript

// Sunlight intensity throughout the day
const sunlight = model.stepFunction([
    [0, 0],     // 0h-6h: dark
    [6, 3],     // 6h-9h: morning
    [9, 10],    // 9h-15h: peak sunlight
    [15, 5],    // 15h-18h: afternoon
    [18, 0],    // 18h onwards: dark
]);

// Solar charging task with variable length
const charging = model.intervalVar({ length: [10, 50], name: "charging" });

// Must collect at least 100 units of sunlight
model.enforce(model.integral(sunlight, charging).ge(100));

Maintenance Windows

Forbid task execution during scheduled maintenance:

Python

# Machine status: 1 = available, 0 = maintenance
machine_status = model.step_function([
    (0, 1),     # 0-100: available
    (100, 0),   # 100-120: maintenance
    (120, 1),   # 120-200: available
    (200, 0),   # 200-220: maintenance
    (220, 1),   # 220 onwards: available
])

tasks = [model.interval_var(length=15) for _ in range(10)]

# Tasks cannot run during maintenance
for task in tasks:
    task.forbid_extent(machine_status)

model.no_overlap(tasks)

C#

// Machine status: 1 = available, 0 = maintenance
var machineStatus = model.StepFunction(new (int, int)[]
{
    (0, 1),     // 0-100: available
    (100, 0),   // 100-120: maintenance
    (120, 1),   // 120-200: available
    (200, 0),   // 200-220: maintenance
    (220, 1),   // 220 onwards: available
});

var tasks = Enumerable.Range(0, 10)
    .Select(_ => model.IntervalVar(length: 15))
    .ToArray();

// Tasks cannot run during maintenance
foreach (var task in tasks)
    task.ForbidExtent(machineStatus);

model.NoOverlap(tasks);

TypeScript

// Machine status: 1 = available, 0 = maintenance
const machineStatus = model.stepFunction([
    [0, 1],     // 0-100: available
    [100, 0],   // 100-120: maintenance
    [120, 1],   // 120-200: available
    [200, 0],   // 200-220: maintenance
    [220, 1],   // 220 onwards: available
]);

const tasks = Array.from({ length: 10 }, () => model.intervalVar({ length: 15 }));

// Tasks cannot run during maintenance
tasks.forEach(task => task.forbidExtent(machineStatus));

model.noOverlap(tasks);

Complete Example

Python

import optalcp as cp

model = cp.Model()

# Energy pricing (24-hour cycle)
energy_price = model.step_function([
    (0, 5),     # 0h-6h: low ($5/hour)
    (6, 10),    # 6h-9h: medium ($10/hour)
    (9, 20),    # 9h-17h: peak ($20/hour)
    (17, 10),   # 17h-22h: medium ($10/hour)
    (22, 5),    # 22h onwards: low ($5/hour)
])

# Machine availability
machine_hours = model.step_function([
    (0, 0),     # 0h-6h: closed
    (6, 1),     # 6h-22h: open
    (22, 0),    # 22h onwards: closed
])

tasks = [
    model.interval_var(length=8, name=f"task_{i}")
    for i in range(5)
]

# Tasks must run during machine hours
for task in tasks:
    task.forbid_extent(machine_hours)

# Sequential execution
for i in range(len(tasks) - 1):
    tasks[i].end_before_start(tasks[i + 1])

# Minimize total energy cost
total_cost = model.sum([
    model.integral(energy_price, task) for task in tasks
])
model.minimize(total_cost)

result = model.solve()
if result.solution:
    print(f"Total energy cost: ${result.objective}")
    for task in tasks:
        start = result.solution.get_start(task)
        end = result.solution.get_end(task)
        print(f"{task.name}: {start}-{end}")

C#

using OptalCP;

var model = new Model();

// Energy pricing (24-hour cycle)
var energyPrice = model.StepFunction(new (int, int)[]
{
    (0, 5),     // 0h-6h: low ($5/hour)
    (6, 10),    // 6h-9h: medium ($10/hour)
    (9, 20),    // 9h-17h: peak ($20/hour)
    (17, 10),   // 17h-22h: medium ($10/hour)
    (22, 5),    // 22h onwards: low ($5/hour)
});

// Machine availability
var machineHours = model.StepFunction(new (int, int)[]
{
    (0, 0),     // 0h-6h: closed
    (6, 1),     // 6h-22h: open
    (22, 0),    // 22h onwards: closed
});

var tasks = Enumerable.Range(0, 5)
    .Select(i => model.IntervalVar(length: 8, name: $"task_{i}"))
    .ToArray();

// Tasks must run during machine hours
foreach (var task in tasks)
    task.ForbidExtent(machineHours);

// Sequential execution
for (int i = 0; i < tasks.Length - 1; i++)
    tasks[i].EndBeforeStart(tasks[i + 1]);

// Minimize total energy cost
var totalCost = model.Sum(
    tasks.Select(task => model.Integral(energyPrice, task))
);
model.Minimize(totalCost);

var result = model.Solve();
if (result.Solution != null)
{
    Console.WriteLine($"Total energy cost: ${result.Objective}");
    foreach (var task in tasks)
    {
        int? start = result.Solution.GetStart(task);
        int? end = result.Solution.GetEnd(task);
        Console.WriteLine($"{task.Name}: {start}-{end}");
    }
}

TypeScript

import * as CP from '@scheduleopt/optalcp';

const model = new CP.Model();

// Energy pricing (24-hour cycle)
const energyPrice = model.stepFunction([
    [0, 5],     // 0h-6h: low ($5/hour)
    [6, 10],    // 6h-9h: medium ($10/hour)
    [9, 20],    // 9h-17h: peak ($20/hour)
    [17, 10],   // 17h-22h: medium ($10/hour)
    [22, 5],    // 22h onwards: low ($5/hour)
]);

// Machine availability
const machineHours = model.stepFunction([
    [0, 0],     // 0h-6h: closed
    [6, 1],     // 6h-22h: open
    [22, 0],    // 22h onwards: closed
]);

const tasks = Array.from({ length: 5 }, (_, i) =>
    model.intervalVar({ length: 8, name: `task_${i}` })
);

// Tasks must run during machine hours
tasks.forEach(task => task.forbidExtent(machineHours));

// Sequential execution
for (let i = 0; i < tasks.length - 1; i++) {
    tasks[i].endBeforeStart(tasks[i + 1]);
}

// Minimize total energy cost
const totalCost = model.sum(
    tasks.map(task => model.integral(energyPrice, task))
);
model.minimize(totalCost);

const result = await model.solve();
if (result.solution) {
    console.log(`Total energy cost: $${result.objective}`);
    tasks.forEach(task => {
        const start = result.solution.getStart(task);
        const end = result.solution.getEnd(task);
        console.log(`${task.name}: ${start}-${end}`);
    });
}

Edge Cases

Value Before First Step

Before the first step, the function value is 0:

Python

func = model.step_function([
    # Implicitly: value is 0 before time 10
    (10, 5),    # From time 10: value is 5
    (20, 10),   # From time 20 onwards: value is 10
])

C#

var func = model.StepFunction(new (int, int)[]
{
    // Implicitly: value is 0 before time 10
    (10, 5),    // From time 10: value is 5
    (20, 10),   // From time 20 onwards: value is 10
});

TypeScript

const func = model.stepFunction([
    // Implicitly: value is 0 before time 10
    [10, 5],    // From time 10: value is 5
    [20, 10],   // From time 20 onwards: value is 10
]);

Tip: For forbid start, forbid end, and forbid extent, the constraint forbids times where the function is zero. If you don't define a step at time 0, tasks cannot start/end/run before the first defined step.

Empty Steps

An empty step function is valid—the function is 0 everywhere:

Python

func = model.step_function([])  # func(x) = 0 for all x

C#

var func = model.StepFunction(Array.Empty<(int, int)>());  // func(x) = 0 for all x

TypeScript

const func = model.stepFunction([]);  // func(x) = 0 for all x

See Also

• Intervals - IntervalVar forbid functions
• Expressions - Using evaluated step functions in expressions
• Objectives - Optimizing costs with step functions
• Resources/Cumulative - Time-varying resource capacity
• Tutorial - Step-by-step learning path