Operations Research Models and Methods / Models / Linear Programming

Blending or Mixing Problem
 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

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Updated 2/2/00
Operations Research Models and Methods
by Paul A. Jensen and Jon Bard, University of Texas, Copyright by the Authors