19th International CODATA Conference
Category: Poster

Application of Dual Model in Animal Feed Formulation Optimizing System

This study introduced a dual model in an original linear programming to obtain the shadow prices of resources that take part in optimization. In feed formulation, the shadow prices of nutrients resource show degrees of influence on a diet last cost when increasing or decreasing expected diet nutrient values of a diet. The higher the shadow price of a nutrient resource, the more obvious its influence on last cost. When the shadow price of a resource equals to “zero”, it means that reaching this nutrient value does not have influence on a special diet cost within a particular value. At the same time, this paper in China also discusses the development direction of feed formulation optimizing techniques in China.

In general, original linear programming model could be described as below:

Restricted equation set:

a₁₁X₁+ a₁₂X₂+ …+a_1nXn ≥ b₁

a₂₁X1+ a₂₂X₂+ …+a_2nX_n ≥ b₂

… … … … … …

a_m1X₁+ a_m2X₂+ …+a_mnX_n ≥ b_m

X₁ ≥ 0, X₂ ≥ 0, …, X_n ≥0

Where decisive variables X_i ( i=1,2, …, n) indicate the amount of feedstuff included in calculation ;C_i ( j=1,2, …, n) the price of raw material; B_i ( i=1,2, …, m) nutrient requirements expected to be achieved; A_ij ( i=1,2, …, m; j=1,2, …, n) feed composition ie composition coefficient (namely nutrient coefficient). From model (I) are derived the following dual linear programming issue:

( II):Target function: maxg= b₁Y₁+ b₂Y₂+ …+ B_mY_m, restricted equation set can be transformed into :

A₁₁Y₁+ a₂₁Y₂+ …+Am1Y_m ≤ c₁

A₁₂Y₁+ a₂₂Y₂+ …+Am2Y_m ≤ c₂

… … … … … … …

A_1nY₁+ a_2nY₂+ …+ A_mnY_m ≤ c_m

Y₁ ≥ 0, Y₂ ≥ 0, …, Y_n ≥0

We call (II) the linear programming issue as dual one of (I). In the dual linear programming problem, decisive variable Y_i（i=1，2，…, m）is “ shadow price ” or marginal cost of resources b_i ( i=1,2, …, m) which are expected to determine their quantity used in the diet . What is so called will explain in the following example. Composition or nutrient coefficient of problem ( II)  is inverse matrix of the problem ( I).

It can be proved from operational research in theory that : ① problem ( I) and problem( II)  are dual issue from each other ; ② if both of original problem and dual one are feasible, both of two would have optimizing solution and both as the same.

The application of dual theories above to remote feed formula system in China feed information network center have increased creation and application models of dual model, and put “shadow price” of nutrient composition including in the model optimizing output in the result .In the following, an example is to explain for setting up and application of the dual model and especially practical meaning for the calculating results.