Variables & Domains

Variables represent the mathematical decision variables in your optimization model. WaSHI provides domain-specific factory methods on the Model instance: m.continuous(), m.integer(), m.binary(), and m.bool().

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Decision Variable Types

1. Continuous Variables (m.continuous)

Can take any real number value within [lb, ub]. Ideal for physical measurements, flow rates, budgets, and weights:

import { Model } from "@ohtaman/washi";

const m = new Model();

// Single continuous variable bounded in [0, 1000]
const fuel = m.continuous("fuel", { lb: 0, ub: 1000 });

2. Integer Variables (m.integer)

Restricted to integers. Ideal for countable assets (machines, trucks, staff count):

// Single integer variable bounded in [0, 15]
const trucks = m.integer("trucks", { lb: 0, ub: 15 });

3. Binary Variables (m.binary)

Restricted to $\{0, 1\}$ as mathematical 0-1 values. Ideal for MIP selection, yes/no decisions, and facility selections:

// Single binary variable
const openWarehouse = m.binary("open_warehouse");
// lb is automatically 0, ub is automatically 1

4. Propositional Boolean Variables (m.bool)

Pure boolean logic propositions for SMT / SAT modeling with Z3:

// Single propositional boolean variable
const isNight = m.bool("isNight");
const alarmTriggered = m.bool("alarmTriggered");

// First-class boolean operators: and, or, not, implies, xor
m.constraint(alarmTriggered.implies(isNight), "alarm_policy");

💡 Modeling Semi-Continuous Decisions

Semi-continuous variables take values in $\{0\} \cup [L, U]$ (either zero or between a positive lower and upper bound, useful for minimum production batches or operational thresholds).

While not a primitive VarType in Model, semi-continuous behavior is cleanly modeled by combining a continuous variable with a binary indicator variable and linking constraints ($L \cdot b \le x \le U \cdot b$):

import { Model, c } from "@ohtaman/washi";

const m = new Model();
const batch = m.continuous("batch", { lb: 0, ub: 100 });
const isActive = m.binary("is_active");

// If isActive === 0, batch is forced to 0; if 1, batch is within [10, 100]
m.constraint(c`${batch} >= 10 * ${isActive}`, "min_batch_run");
m.constraint(c`${batch} <= 100 * ${isActive}`, "max_batch_run");

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Multi-Dimensional & Array Creation

The domain factory methods (continuous, integer, binary, bool) support count, arrays of keys, and row/column keys to create 1D arrays, typed dictionaries, and 2D matrices in a single call.

By Count

Pass a number to create an array of variables:

const x = m.continuous(5, { lb: 0 }); // Var[] (x_0, ..., x_4)
const flags = m.bool(3);              // BoolVar[] (p_0, p_1, p_2)

1D Variable Dictionaries

Pass an array of keys to get a typed dictionary Record<string, Var> or Record<string, BoolVar>:

const items = ["laptop", "phone", "tablet"];

// Record<"laptop" | "phone" | "tablet", Var>
const selected = m.binary(items);

const perms = m.bool(["read", "write", "execute"]);
// perms.read, perms.write, perms.execute

2D Variable Matrices

Pass row keys and column keys to create a 2D matrix Record<string, Record<string, Var>> (or Record<string, Record<string, BoolVar>>):

const factories = ["Tokyo", "Osaka"];
const stores = ["StoreA", "StoreB", "StoreC"];

// ship[factory][store]
const ship = m.continuous(factories, stores, { lb: 0 });

// 2D boolean grid for staff permissions
const grid = m.bool(["alice", "bob"], ["admin", "viewer"]);
// grid.alice.admin

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Variable Options

Option	Type	Default	Description
lb	number	0	Lower bound. Defaults to 0 (non-negative).
ub	number	Infinity (or 1 for binary)	Upper bound.
type	"Continuous" | "Integer" | "Binary"	Inferred from method	Underlying variable domain category.

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Accessing Solution Values

After calling await m.solve(), optimal values are accessible via variable.value or the bulk accessor m.getValues():

await m.solve();

// Single variable
console.log(`trucks: ${trucks.value}`);

// Bulk extraction for 1D or 2D collections
const selectedValues = m.getValues(selected);
// { laptop: 1, phone: 0, tablet: 1 }

const flowValues = m.getValues(ship);
// { Tokyo: { StoreA: 50, ... }, Osaka: { ... } }