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.
Blending or Mixing Problem
