Example 3: Ingredient Problem

Problem Description

There are a number of products and raw materials. Each product has a specified production quantity, each raw material has a cost, and each raw material can be used to produce multiple different products.

	Product A	Product B	Product C
Demand	$15000$	$15000$	$10000$

	Material A	Material B	Material C	Material D
Cost	$115$	$97$	$82$	$76$

	Material A	Material B	Material C	Material D
Product A	$30$	$15$	$-$	$15$
Product B	$10$	$-$	$25$	$15$
Product C	$-$	$20$	$15$	$15$

Determine the amount of each raw material used to minimize the total cost.

Mathematical Model

Variables

$x_m$ : usage of raw material $m$。

Intermediate Expressions

1. Total Cost

$$Cost=\sum _{m\in M}Cos{t}_m\cdot x_m$$

2. Product Yield

$$Yiel{d}_p^{Product}=\sum _{m\in M}Yiel{d}_{mp}\cdot x_m,\mathrm{\forall }p\in P$$

Objective Function

1. Minimize Total Cost

$$minCost$$

Constraints

1. Demand Limit

$$s.t.Yiel{d}_p^{Product}=Yiel{d}_p^{Product,Min},\mathrm{\forall }p\in P$$

Expected Result

Raw material A uses $284$ units, raw material B uses $8$ units, raw material C uses $232$ units, raw material D uses $424$ units.

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 Product(
    val minYield: Flt64
) : AutoIndexed(Product::class)

data class Material(
    val cost: Flt64,
    val yieldQuantity: Map<Product, Flt64>
) : AutoIndexed(Material::class)

val products: List<Product> = ...  // product data
val materials: List<Material> = ... // material data

// create a model instance
val metaModel = LinearMetaModel("demo3")

// define variables
val x = = UIntVariable1("x", Shape1(materials.size))
for (c in materials) {
    x[c].name = "${x.name}_${c.index}"
}
metaModel.add(x)

// define intermediate expressions
val cost = LinearExpressionSymbol(
    sum(materials) { it.cost * x[it] }, 
    "cost"
)
metaModel.add(cost)

val yield = LinearIntermediateSymbols1("yield", Shape1(products.size)) { p, _ ->
    val product = products[p]
    LinearExpressionSymbol(
        sum(materials.filter { it.yieldQuantity.contains(product) }) { m ->
            m.yieldQuantity[product]!! * x[m]
        },
        "yieldProduct_${p}"
    )
}
metaModel.add(yield)

// define objective function
metaModel.minimize(cost)

// define constraints
for (p in products) {
    metaModel.addConstraint(yield[p.index] eq p.minYield)
}

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

    is Failed -> {}
}

// parse results
val solution = HashMap<Material, UInt64>()
for (token in metaModel.tokens.tokens) {
    if (token.result!! eq Flt64.one && token.variable.belongsTo(x)) {
        val material = materials[token.variable.vectorView[0]]
        solution[material] = token.result!!.round().toUInt64()
    }
}

For the complete implementation, please refer to:

• Kotlin