Example 5: Knapsack Problem

Problem Description

There are several items, each with corresponding value and weight.

	Item A	Item B	Item C	Item D	Item E
Weight	$2kg$	$2kg$	$6kg$	$5kg$	$4kg$
Value	$6$	$3$	$5$	$4$	$6$

Objective: Select a subset of these items to maximize total value, while satisfying the following condition:

1. The total weight of selected items does not exceed $10kg$.

Mathematical Model

Variables

$x_c$: Indicates whether item $c$ is selected.

Intermediate Values

1. Total Value

$$\text{Value}=\sum _{c\in C}{\text{Value}}_c\cdot x_c$$

2. Total Weight

$$\text{Weight}=\sum _{c\in C}{\text{Weight}}_c\cdot x_c$$

Objective Function

1. Maximize Total Value

$$max\text{Value}$$

Constraints

1. Total Weight Does Not Exceed Maximum Weight

$$\text{s.t.}\text{Weight}\le {\text{Weight}}^{\text{Max}}$$

Expected Results

Select items A, B, and E.

Code Implementation

kotlin

import fuookami.ospf.kotlin.utils.math.*
import fuookami.ospf.kotlin.utils.concept.*
import fuookami.ospf.kotlin.utils.functional.*
import fuookami.ospf.kotlin.utils.multi_array.*
import fuookami.ospf.kotlin.core.frontend.variable.*
import fuookami.ospf.kotlin.core.frontend.expression.monomial.*
import fuookami.ospf.kotlin.core.frontend.expression.polynomial.*
import fuookami.ospf.kotlin.core.frontend.expression.symbol.*
import fuookami.ospf.kotlin.core.frontend.inequality.*
import fuookami.ospf.kotlin.core.frontend.model.mechanism.*
import fuookami.ospf.kotlin.core.backend.plugins.scip.*

data class Cargo(
    val weight: UInt64,
    val value: UInt64
) : AutoIndexed(Cargo::class)

val cargos: List<Cargo> = ... // Item list
val maxWeight = UInt64(10U)

// Create model instance
val metaModel = LinearMetaModel("demo5")

// Define variables
val x = BinVariable1("x", Shape1(cargos.size))
for (c in cargos) {
    x[c].name = "${x.name}_${c.index}"
}
metaModel.add(x)

// Define intermediate values
val cargoValue = LinearExpressionSymbol(sum(cargos) { c -> c.value * x[c] }, "value")
metaModel.add(cargoValue)

val cargoWeight = LinearExpressionSymbol(sum(cargos) { c -> c.weight * x[c] }, "weight")
metaModel.add(cargoWeight)

// Define objective function
metaModel.maximize(cargoValue, "value")

// Define constraints
metaModel.addConstraint(
    cargoWeight leq maxWeight,
    "weight"
)

// Call solver to solve
val solver = ScipLinearSolver()
when (val ret = solver(metaModel)) {
    is Ok -> {
        metaModel.tokens.setSolution(ret.value.solution)
    }

    is Failed -> {}
}

// Parse results
val solution = HashSet<Cargo>()
for (token in metaModel.tokens.tokens) {
    if (token.result!! eq Flt64.one && token.variable.belongsTo(x)) {
        solution.add(cargos[token.variable.vectorView[0]])
    }
}

Complete Implementation Reference:

• Kotlin