Logical AND

Function Form

$$y=\text{And}(x_1,x_2,\dots ,x_i)=\{\begin{array}{ll}1,& \underset{i}{\bigwedge }{x}_i\\ \\ 0,& \mathrm{\neg }\underset{i}{\bigwedge }{x}_i\end{array}$$

Binary Case

Additional Variables

$y^{\prime }\in \{0,1\}$: logical AND value.

Basic Formula

$$y=y^{\prime }$$

Mathematical Model

$$\begin{array}{rlrlr}\text{s.t.}& y& \le & x_i,& \mathrm{\forall }i\in P\\ \\ & y& \ge & \sum _{i\in P}{x}_i-|P|+1\end{array}$$

Non-Binary Case

Basic Formula

$$y=\text{Bin}(min(x_1,x_2,\dots ,x_i))$$

where $\text{Bin}(x)$ can refer to Binarization, and $min(x)$ can refer to Minimum Value.

Code Example

Binary Case

kotlin

import kotlinx.coroutines.*
import fuookami.ospf.kotlin.utils.math.*
import fuookami.ospf.kotlin.core.frontend.variable.*
import fuookami.ospf.kotlin.core.frontend.expression.polynomial.*
import fuookami.ospf.kotlin.core.frontend.expression.symbol.linear_function.*
import fuookami.ospf.kotlin.core.frontend.inequality.*
import fuookami.ospf.kotlin.core.frontend.model.mechanism.*
import fuookami.ospf.kotlin.core.backend.plugins.scip.*

val x = BinVar("x")
val y = BinVar("y")
val and = AndFunction(
    listOf(x, y),
    "and"
)
val solver = ScipLinearSolver()

val model1 = LinearMetaModel()
model1.add(x)
model1.add(y)
model1.add(and)
model1.addConstraint(!and)
model1.maximize(x + y)
val result1 = runBlocking { solver(model1) }
assert(result1.value!!.obj eq Flt64.one)

val model2 = LinearMetaModel()
model2.add(x)
model2.add(y)
model2.add(and)
model2.addConstraint(and)
model2.minimize(x + y)
val result2 = runBlocking { solver(model2) }
assert(result2.value!!.obj eq Flt64.two)

Non-Binary Case

kotlin

import kotlinx.coroutines.*
import fuookami.ospf.kotlin.utils.math.*
import fuookami.ospf.kotlin.core.frontend.variable.*
import fuookami.ospf.kotlin.core.frontend.expression.polynomial.*
import fuookami.ospf.kotlin.core.frontend.expression.symbol.linear_function.*
import fuookami.ospf.kotlin.core.frontend.inequality.*
import fuookami.ospf.kotlin.core.frontend.model.mechanism.*
import fuookami.ospf.kotlin.core.backend.plugins.scip.*

val x = UIntVar("x")
x.range.leq(UInt64.one)
val y = UIntVar("y")
y.range.leq(UInt64.two)
val and = AndFunction(
    listOf(x, y),
    "and"
)
val solver = ScipLinearSolver()

val model1 = LinearMetaModel()
model1.add(x)
model1.add(y)
model1.add(and)
model1.addConstraint(and)
model1.maximize(x + y)
val result1 = runBlocking { solver(model1) }
assert(result1.value!!.obj eq Flt64.three)

val model2 = LinearMetaModel()
model2.add(x)
model2.add(y)
model2.add(and)
model2.addConstraint(and)
model2.minimize(x + y)
val result2 = runBlocking { solver(model2) }
assert(result2.value!!.obj eq Flt64.two)

Complete Implementation Reference:

• Kotlin

Complete Example Reference:

• Kotlin