Models
Blending or Mixing Problem
 Blending or Mixing Problem
Statement

Another classic problem that can be modeled as a linear program concerns blending or mixing ingredients to obtain a product with certain characteristics or properties. We illustrate this class with the problem of determining the optimum amounts of three ingredients to include in an animal feed mix. The final product must satisfy several nutrient restrictions. The possible ingredients, their nutritive contents (in kilograms of nutrient per kilograms of ingredient) and the unit cost are shown in the following table.

The mixture must meet the following restrictions:

• Calcium — at least 0.8% but not more than 1.2%.
• Protein — at least 22%.
• Fiber — at most 5%.
The problem is to find the composition of the feed mix that satisfies these constraints while minimizing cost.

 Nutritive content and price of ingredients Ingredient Calcium (kg/kg) Protein (kg/kg) Fiber (kg/kg) Unit cost (cents/kg) Limestone 0.38 0.0 0.0 10.0 Corn 0.001 0.09 0.02 30.5 Soybean meal 0.002 0.50 0.08 90.0

Model

Variable Definitions

L, C, S : proportions of limestone, corn, and soybean meal, respectively, in the mixture.

Constraints

The number of hours available on each machine type is 40 times the number of machines. All the constraints are dimensioned in hours. For machine 1, for example, we have 40 hrs/machine ¥ 4 machines = 160 hrs. In writing out the constraints, it is customary to provide a column in the model for each variable.

 Minimum calcium: 0.38L + 0.001C + 0.002S > 0.008 Maximum calcium: 0.38L + 0.001C + 0.002S < 0.012 Minimum protein: + 0.09C + 0.50S > 0.22 Maximum fiber: + 0.02C + 0.08S < 0.05 Conservation: L + C + S = 1

Nonnegativity

L, C, S > 0

Objective Function

Because each decision variable is defined as a fraction of a kilogram, the objective is to minimize the cost of providing one kilogram of feed mix.

Minimize Z = 10L + 30.5C + 90S

Excel Model and Solution

The model was created by the Math Programming add-in and solved with the Jensen LP/IP Solver. The solution shows the optimum proportion of each component of the feed. The objective value is the cost per kilogram of the minimum cost mix that meets the nutruitive requirements.

Sensitivity Analysis

The sensitivity analysis created by the Jensen Solver shows that the upper bound of the calcium constraint and the lower bound of the protein constraint are limiting the solution. The shadow prices are the derviatives of the unit cost of the feed with respect to the tight bounds. For example, if one increases the upper bound of the calcium constraint from 0.012 to 0.013 the cost will be reduced by about \$0.20.

cost increase = shadow price * bound increase = (-19.62)(0.001) = -0.1952.

In like manner, if the lower bound on protein were increased from 0.22 to 0.23 the cost would be increased by about \$1.45.

cost increase = shadow price * bound increase = (145.17)(0.01) = 1.4517.

The fiber constraint is strictly between its bounds, so the shadow price is zero.

Operations Research Models and Methods
Internet
by Paul A. Jensen