Slack

Function Form

$$z=\text{slack}(x,y)=\{\begin{array}{ll}max(0,x-y),& \text{compute positive slack}\\ \\ max(0,y-x),& \text{compute negative slack}\\ \\ |x-y|,& \text{compute both positive and negative slack}\end{array}$$

Additional Variables

$neg\in \mathbb{R}-{\mathbb{R}}^{-}$: Negative slack.

$pos\in \mathbb{R}-{\mathbb{R}}^{-}$: Positive slack.

Derived Symbol

$$z=neg+pos$$

Mathematical Model

Compute Both Positive and Negative Slack

$$\begin{array}{rl}\text{s.t.}& x+neg-pos=y\end{array}$$

Compute Positive Slack

$$\begin{array}{rl}\text{s.t.}& x-pos\le y\end{array}$$

Compute Negative Slack

$$\begin{array}{rl}\text{s.t.}& x+neg\ge y\end{array}$$

Code Example

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 = RealVar("x")
x.range.leq(Flt64.two)
x.range.geq(-Flt64.three)
val slack = SlackFunction(
    x = x,
    y = Flt64.five,
    name = "slack"
)

val model = LinearMetaModel()
model.add(x)
model.add(slack)
model.minimize(slack)

val solver = ScipLinearSolver()
val result = runBlocking { solver(model) }
assert(result.value!!.obj eq Flt64.three)
assert(result.value!!.solution[0] eq Flt64.two)

Complete Implementation Reference:

• Kotlin

Complete Example Reference:

• Kotlin