Blending Problem

The Blending Problem optimizes the mixing of ingredients to satisfy restrictions while minimizing cost. The restrictions are that certain properties of using the ingredients must be within their minimum and maximum values allowed. For example, ingredients in food may have nutrional properties, and this problem could force Calcium content of the mixture to be within a lower and upper bound (in terms of proportions). This problem deals with proportions, so it must another constraint to ensure that the sum of the decision variables (proportion of each ingredient to use) equals 1. A matrix-like parameter shows the numeric properties of each of the ingredients (IngredientProperties). This type of problem arises often in the food, feed, and oil refinement industries. The diet problem is a well-studied example of an application of this problem class.

Definitions

Sets

  • Ingredients - Set of ingredients that are available for use

    • i in Ingredients or \(i \in I\)

  • Properties - Set of properties that exist in the ingredients

    • p in Properties or \(p \in P\)

Parameters

  • Cost - measure of cost of using one unit of Ingredient i

    • Cost[i] for i in Ingredients or \(C_i \enspace \forall i \in I\)

  • IngredientProperties - measure of how much Property p is in Ingredient i

    • IngredientProperties[i, p] for i in Ingredients for p in Properties or \(N_{i,p} \enspace \forall i \in I, p \in P\)

  • MinProperty - minimum amount of Property p needed in the blend

    • MinProperty[p] for p in Properties or \(L_p \enspace \forall p \in P\)

  • MaxProperty - maximum amount of Property p allowed in the blend

    • MaxProperty[p] for p in Properties or \(U_p \enspace \forall p \in P\)

Decision Variables

  • Blend - proportion of Ingredient i to include in the blend.

    • Blend[i] for i in Ingredients or \(X_i \enspace \forall i \in I\)

Objective

Minimize total cost of the ingredients in the blend.

\[\text{Min} \sum_{i \in I} C_iX_i\]

Constraints

  • The Blend must have its Properties within the upper and lower bounds, MinProperty[p] and MaxProperty[p].

\[L_p \leq \sum_{i \in I}N_{i,p}X_i \leq U_p \quad \forall p \in P\]

  • The Blend decision variables are proportions of the ingredients to include, and thus, the decision variables must add up to 1. Additionally, these decision variables must all be greater than or equal to zero.

\[ \begin{align}\begin{aligned}\sum_{i \in I} X_i = 1\\X_i \geq 0 \enspace \forall i \in I\end{aligned}\end{align} \]

API Reference

See the corresponding section in the API Library Reference to learn more about how to use the API for this problem class.