Mathematical models of linear programming problems. Linear mathematical models Transition from the limitations of a linear mathematical model

The basis for solving economic problems are mathematical models.

Mathematical model problem is a set of mathematical relationships that describe the essence of the problem.

• selection of problem variables
• drawing up a system of restrictions
• choice of objective function

Task variables are called the quantities X1, X2, Xn, which completely characterize the economic process. They are usually written as a vector: X=(X 1, X 2,...,X n).

System of restrictions problems are a set of equations and inequalities that describe the limited resources in the problem under consideration.

Objective function tasks are called a function of task variables that characterizes the quality of the task and the extremum of which needs to be found.

In general, a linear programming problem can be written as follows:

This entry means the following: find the extremum of the objective function (1) and the corresponding variables X=(X 1 , X 2 ,...,X n) provided that these variables satisfy the system of restrictions (2) and the non-negativity conditions (3) .

Valid solution(plan) of a linear programming problem is any n-dimensional vector X=(X 1 , X 2 ,...,X n) that satisfies the system of constraints and non-negativity conditions.

The set of feasible solutions (plans) of the problem forms region of feasible solutions(ODR).

The optimal solution(plan) of a linear programming problem is such an admissible solution (plan) of the problem in which the objective function reaches an extremum.

Example of compiling a mathematical model

The problem of using resources (raw materials)

Condition: To produce n types of products, m types of resources are used. Create a mathematical model.

Known:

• b i (i = 1,2,3,...,m) — reserves of each i-th type of resource;
• a ij (i = 1,2,3,...,m; j=1,2,3,...,n) - the costs of each i-th type of resource for the production of a unit of volume of the j-th type of product;
• c j (j = 1,2,3,...,n) - profit from the sale of a unit of volume of the j-th type of product.

It is required to draw up a production plan that ensures maximum profit under given restrictions on resources (raw materials).

Solution:

Let us introduce a vector of variables X=(X 1, X 2,...,X n), where x j (j = 1,2,...,n) is the production volume of the j-th type of product.

The costs of the i-th type of resource for the production of a given volume x j of products are equal to a ij x j , therefore the restriction on the use of resources for the production of all types of products has the form:
Profit from the sale of the jth type of product is equal to c j x j, so the objective function is equal to:

Answer- The mathematical model looks like:

Canonical form of a linear programming problem

In the general case, a linear programming problem is written in such a way that the constraints are both equations and inequalities, and the variables can be either non-negative or arbitrarily varying.

In the case where all constraints are equations and all variables satisfy the non-negativity condition, the linear programming problem is called canonical.

It can be represented in coordinate, vector and matrix notation.

The canonical linear programming problem in coordinate notation has the form:

The canonical linear programming problem in matrix notation has the form:

• A is the matrix of coefficients of the system of equations
• X—matrix-column of task variables
• Ао — matrix-column of the right parts of the system of restrictions

Linear programming problems, called symmetric ones, are often used, which in matrix notation have the form:

Reducing a general linear programming problem to canonical form

In most methods for solving linear programming problems, it is assumed that the system of constraints consists of equations and natural conditions for the non-negativity of variables. However, when compiling models of economic problems, constraints are mainly formed in the form of a system of inequalities, so it is necessary to be able to move from a system of inequalities to a system of equations.

Let's take the linear inequality a 1 x 1 +a 2 x 2 +...+a n x n ≤b and add to its left side a certain value x n+1 such that the inequality turns into the equality a 1 x 1 +a 2 x 2 + ...+a n x n +x n+1 =b. Moreover, this value x n+1 is non-negative.

Let's look at everything using an example.

Bring the linear programming problem to canonical form:

Solution:
Let's move on to the problem of finding the maximum of the objective function.
To do this, we change the signs of the coefficients of the objective function.
To transform the second and third inequalities of the system of constraints into equations, we introduce non-negative additional variables x 4 x 5 (in the mathematical model this operation is marked with the letter D).
The variable x 4 is introduced into the left side of the second inequality with the “+” sign, since the inequality has the form “≤”.
The variable x 5 is introduced into the left side of the third inequality with a “-” sign, since the inequality has the form “≥”.
The variables x 4 x 5 are entered into the objective function with a coefficient. equal to zero.
We write the problem in canonical form.

T10. Statement of the linear programming problem

Mathematical model An economic problem is a set of mathematical relationships that describe the economic process.

To compile a mathematical model you need:

1. select task variables;

2. create a system of restrictions;

3. set the objective function.

Task variables are called the quantities x 1, x 2,..., x n, which completely characterize the economic process. They are usually written as a vector X = (x 1, x 2,…, x n).

Problem constraint system is a set of equations and inequalities that are satisfied by the variables of the problem and which follow from the limited resources and other economic conditions, for example, the positivity of the variables. In general, they look like:

The objective function is called function F(X) = f(x 1, x 2,..., x n) of task variables, which characterizes the quality of the task and the extremum of which needs to be found.

General mathematical programming problem is formulated as follows: find the task variables x 1, x 2,..., x n, which provide the extremum of the objective function

F(X) = f(x 1, x 2,…, x n) ® max (min) (2)

and satisfy the system of restrictions (1).

If the objective function (2) and the system of constraints (1) are linear, then the mathematical programming problem is called linear programming problem (LPP).

Vector X (set of problem variables) is called acceptable solution, or the ZLP plan if it satisfies the system of restrictions (1). An admissible plan X that provides an extremum of the objective function is called optimal solution ZLP.

2. Examples of drawing up mathematical models of economic problems

The study of specific production situations leads to ZLP, which are interpreted as problems about the optimal use of limited resources.

1.Optimal production plan problem

To produce two types of products T 1 and T 2, three types of resources S 1, S 2, S 3 are used. Resource reserves, the number of resource units spent on producing a unit of product, as well as profit from the sale of a unit of product are shown in the table:

It is required to find a production plan that will maximize the profit from its sale.

Solution.

Let us denote x 1, x 2 – the number of units of production, respectively T 1 and T 2, planned for production. To produce them, you will need (x 1 + x 2) units of resource S 1, (x 1 + 4x 2) units of resource S 2, (x 1) units of resource S 3. Consumption of resources S 1 , S 2 , S 3 should not exceed their reserves, respectively 8, 20 and 5 units.

Then the economic and mathematical model of the problem can be formulated as follows:

Find the production plan X = (x 1, x 2) that satisfies the system of constraints:

and condition,

at which the function takes the maximum value.

The problem can be easily generalized to the case of producing n types of products using m types of resources.

2.Optimal diet problem

There are two types of food, K 1 and K 2, containing nutrients S 1, S 2 and S 3. The content of the number of nutrient units in 1 kg of each type of feed, the required minimum of nutrients, as well as the cost of 1 kg of feed are shown in the table:

It is necessary to create a daily diet that has a minimum cost, in which the content of each type of nutrient would not be less than the established limit.

Solution.

Let us denote x 1, x 2 – the amount of feed K 1 and K 2 included in the daily diet. Then this diet will include (3x 1 + x 2) units of nutrient S 1, (x 1 + 2x 2) units of nutrient S 2, (x 1 + 6x 2) units of nutrient S 3. Since the content of nutrients S1, S2 and S3 in the diet should be 9, 8 and 12 units, respectively, the economic and mathematical model of the problem can be formulated as follows:

Create a daily ration X = (x 1, x 2) that satisfies the system of restrictions:

and condition,

at which the function takes the minimum value.

PAP recording forms

In the ZLP it is required to find the extremum of the linear objective function:

with restrictions:

and the condition of non-negativity

where a ij, b i, c j ( , ) are given constant values.

This is how the ZLP is written in general form. If the system of restrictions contains only inequalities, then the LLP is presented in standard form. Canonical (main) The ZLP form of notation is a notation when the system of constraints contains only equalities. So the above ZLPs are written in standard form.

The general, standard and canonical forms of the ZLP are equivalent in the sense that each of them can be rewritten in a different form with the help of simple transformations. This means that if there is a way to solve one of the specified problems, then the optimal plan for any of the problems can be determined.

In order to move from one form of recording the PLP to another, one must be able to move from inequality constraints to equality constraints and vice versa.

The inequality constraint (£) can be converted to an equality constraint by adding an additional non-negative variable to its left side, and the inequality constraint (³) can be converted to an equality constraint by subtracting an additional non-negative variable from its left side. The number of introduced additional non-negative variables is equal to the number of transformed inequality constraints.

Input additional variables make some economic sense. Thus, if the limitations of the original PPP reflect the consumption and availability of resources, then the value of the additional PPP variable in canonical form is equal to the volume of the unused corresponding resource.

Example 1. Write in the canonical form of the ZLP:

with restrictions:

Solution.

The objective function remains unchanged:

The system of inequalities is transformed into a system of equalities:

When solving the ZLP by the graphical method, a transition from the canonical form to the standard form is required.

To bring the PPP to a standard form, use Jordan–Gauss method SLAU solutions. Unlike the Gauss method, in which the extended matrix of the system is reduced to a stepwise form, in the Jordan–Gauss method, a unit matrix is ​​formed as part of the extended matrix. Therefore, reversing is not required here.

To convert the original canonical ZLP to the equivalent standard ZLP:

a) in the extended matrix of the constraint system, a non-zero element a qp is selected. This element is called permissive, and q is the i row and pth column are called resolve row and resolve column.

b) the permitting row is rewritten without change, and all elements of the permitting column, except the permitting one, are replaced with zeros. The remaining elements of the extended matrix are determined using the “rectangle rule”:

Let's consider four elements of the extended matrix: element a ij to be transformed, resolving element a qp and elements a i p and a qj. To find the element a ij, from the element a ij, subtract the product of the elements a i p and a qj, located at opposite vertices of the rectangle, divided by the resolving element a qp:

c) at the same time, allowed unknowns are excluded from the objective function. To do this, the coefficients of the objective function are written in the last row of the extended matrix. The calculations take into account that the enabling element in the last line cannot be selected.

Example 2. Reduce to standard form:

Solution.

Using the Jordan–Gauss method, we reduce the system of equations-constraints of the ZLP to an equivalent system of inequalities. Let us select the third element of the first row as the resolving element:

The number -9 obtained in the last column of the last row must be written into the target function with the opposite sign. As a result of transformations, the ZLP takes the form:

Because variables x 2 and x 3 are non-negative, then discarding them, we can write the ZLP in a symmetric form:

In the canonical form of the ZLP, the objective function can be either minimized or maximized. To move from finding the maximum to finding the minimum or vice versa, it is enough to change the signs of the coefficients of the objective function: F 1 = - F. The resulting problem and the original ZLP have the same optimal solution, and the values ​​of the objective functions on this solution differ only in sign.

Properties of ZLP

1. The set of all feasible solutions to the system of constraints of a linear programming problem is convex.

The set of points is called convex, if it contains the entire segment connecting any two points of this set.

According to this definition, the polygon in Fig. 1a is a convex set, but the polygon in Fig. 1b is not, because the segment MN between its two points M and N does not completely belong to this polygon.

Not only polygons can be convex sets. Examples of convex sets are circle, sector, segment, cube, pyramid, etc.

2. If the ZLP has an optimal solution, then the linear function takes a maximum (minimum) value at one of the corner points of the solution polyhedron. If a linear function takes a maximum (minimum) value at more than one corner point, then it takes it at any point that is a convex linear combination of these points.

Point X is called convex linear combination points X 1, X 2,…, X n, if the following conditions are met:

X = α 1 X 1 +α 2 X 2 +…+ α n X n,

α j ≥ 0, Σα j = 1.

Obviously, in the special case with n = 2, a convex linear combination of two points is the segment connecting them.

3. Each admissible basic solution of the system of restrictions of the canonical ZLP corresponds to a corner point of the solution polyhedron, and vice versa, to each corner point of the solution polyhedron there corresponds an admissible basis solution.

From the last two properties it follows: if a ZLP has an optimal solution, then it coincides with at least one of its admissible basic solutions.

Thus, the extremum of the linear ZLP function must be sought among the finite number of its admissible basic solutions.

Basic Modeling Concepts

In the process of human life, ideas about certain properties of real objects and their interactions are developed. These ideas are formed by humans in the form of descriptions of objects for which a description language is used. This can be a verbal description (verbal models), drawing, drawing, graph, layout, etc. All of the above is summarized by one concept model, and the process of building models is modeling.

Modeling is a universal way to study processes and phenomena of the real world. Modeling is of particular importance when studying objects that are inaccessible to direct observation and research. These, in particular, include socio-economic phenomena and processes.

The study of any object, any form of movement is the discovery of not only its qualitative, but also quantitative laws studied by mathematics. The above fully applies to economics.

Economy is a system of social production that carries out the actual production, distribution, exchange and consumption of material goods necessary for society.

Respectively, economic-mathematical model is an economic abstraction expressed in formal mathematical terms, the logical structure of which is determined both by the objective properties of the subject of description and by the subjective target factor of the study for which this description is being undertaken.

Economic and mathematical problems in agriculture are solved using mathematical methods. Among them, the most developed are linear programming (LP) methods. Such methods are used to solve economic and mathematical problems in which quantitative dependencies are expressed linearly, i.e. all conditions are expressed in the form of a system of linear equations and inequalities, and the optimality criterion is expressed in the form of a linear function tending to a minimum or maximum (towards an extremum).

A linear programming problem consists of an objective function, a system of constraints, and the condition for non-negativity of variables.

Let the function be given n variables You need to find the largest or smallest value of this function, provided that the argument

An optimization problem posed in this way is called a mathematical programming problem. Many X is called the set of feasible solutions, and the function is called the objective function or goal function. An admissible solution in which the function takes the largest (or smallest) value is called the optimal solution to the problem.

If the objective function is linear and the set X is given using a system of linear equations and inequalities, the problem is called a linear programming problem (LPP). Thus, the general formulation of the linear programming problem is:

find the extremum of the function

under restrictions

under non-negativity conditions

Let us introduce the following notation:

Reserves i-th type of resource;

cost i-th type of resource for production j-th type of product;

profit from unit sales j-th type of product.

In compact notation, the linear programming problem has the form:

A compact notation shows that the model of a general linear programming problem includes five main elements:

Variables whose value is sought in the process of solving a problem;

Technical and economic coefficients with variables in restrictions;

The volume of the right side of the inequalities, which are called the constants of the problem;

Coefficients of variables in the objective function, which are called estimates of the variables;

Variable index;

Restriction index.

Objective function(goal function) is a mathematical expression for which you need to find an extreme, that is, maximum or minimum, value.

Variablesx j denote types and methods of activity, the dimensions of which are unknown and must be determined in the course of solving the problem. Typically, in agricultural problems, variables mean the required sizes of sectors of the economy, types of feed in the diet, brands of tractors and agricultural machines, etc. In accordance with specific conditions, the same crop or type of livestock can be expressed by several variables. For example, commercial and feed grain; corn for grain, silage, green fodder; perennial grasses for hay, haylage, green fodder, grass meal and seeds, etc.

Variable quantities can change arbitrarily under the conditions of the problem under consideration. Variable , whose coefficients form a unit column is called basic The basic variables form unit basis systems. Variables not included in the unit basis are called free.

The total number of variables included in the task is determined by the nature of the task, specific production conditions, the ability to collect information, etc.

Variables can be expressed in a variety of units of measurement: ha, c, kg, pcs., heads, etc. By nature, variables are divided into basic, additional and auxiliary. The main variables include the types of activity being sought: sectors of the economy, types of feed, brands of cars. Additional variables are those that are formed in the process of turning inequalities into equations. They can mean an underutilized part of the resources, a surplus over the right side of the inequality (if it is an inequality of the “no more” type). Auxiliary variables are included in the problem in order to determine the estimated values ​​of acquired production resources, the estimated values ​​of indicators of economic efficiency of production.

Additional and auxiliary variables always have unity coefficients (+1 or –1).

Technical and economic coefficients (a ij) with variables in the system of restrictions, they express the norms of expenditure of production resources or the norm of output per unit of measurement of the variable.

In both cases, it is necessary that the technical and economic coefficients exactly correspond to the planning period for which the problem is being solved. For example, if the problem is solved for the economic and mathematical analysis of production for the past period, then the coefficients will be calculated based on the reporting data. If it is decided for the future, then the coefficients must be calculated for this future.

Resource consumption rates are most often determined from reference books; they must be adjusted to the relevant specific conditions. Yield ratios are calculated based on planned crop yields and animal productivity.

In cases where it is necessary to provide for predetermined relationships between variables, technical-economic coefficients represent proportionality coefficients. For example, the share of agricultural crops in crop rotation or the share of any feed in the general group of feeds, etc.

Right side of constraints (b i) are called constants, i.e. constant values. These include the volume of production resources - land, labor, equipment, fertilizers, capital investments, etc. Production resources must be determined taking into account their actual state and must take into account the planning period. In addition, those production resources, the use of which is uneven throughout the year, are calculated not only for the year as a whole, but also for individual busy periods or months (labor resources).

Production resources are determined in various units: land - in hectares, labor resources - in man-days or man-hours, equipment - in the number of machine shifts, shift or daily output, etc.

Thus, determining the availability of production resources is not a simple matter. It is necessary to carefully analyze the production activities of the economy, the use of labor, land, technical and other resources, and only then include their volumes in the restrictions.

The right side of the restrictions reflects not only the amount of resources, but also the volume of products produced at the upper or lower level. The lower level is shown in cases where the volume of production is known in advance, less than which the farm should not produce, and the upper level does not allow production above a certain volume. These restrictions are not always required. However, almost no problem involving the determination of a combination of industries can be done without corresponding restrictions on products, otherwise the result will be a one-sided solution. This is due to the fact that the efficiency of industries varies.

In all other restrictions, zeros are placed on the right side, since they formulate conditions for the production and use of products or reflect restrictions on the proportional relationship.

Limitation is a mathematical expression that relates variables in the form of equalities and inequalities. All restrictions form system of restrictions tasks. A system of constraints in mathematical form characterizes the conditions of the problem. The completeness of reflection of these conditions depends on the composition of the restrictions. Therefore, when determining the number of restrictions, two circumstances must be taken into account:

v reflect in the problem only those conditions that really limit the possibilities of production;

v too many constraints increase the size of the problem and make it difficult to solve

Constraints come in three types: equalities (=), inequalities of type less than or equal to (≤), inequalities of type greater than or equal to (≥). For example,

Where i = 1, 2, … , m. The coefficients of the variables are denoted a ij, where index i– restriction number, index j– variable number, free terms (right side of restrictions) are designated b i, index i– restriction number.

Constraints of the first type are called restrictions from above, since the left side of the inequality cannot be higher than a certain value (constant). Restrictions of the third type are called restrictions from below, since the left side of the inequality cannot be below a certain value (constant).

In terms of meaning, all restrictions can be divided into basic, additional and auxiliary.

Main restrictions – these are those that overlap with all or most of the task variables. As a rule, with their help, the main conditions of the problem are reflected - land, labor, feed, nutrients, equipment, etc.

Additional restrictions superimposed on part of the variable quantities or on one variable. These restrictions are introduced in cases where it is necessary to limit the size of individual variables from above or below, for example, taking into account crop rotation requirements or taking into account the physiological limits of saturation of the diet with individual feeds or their groups. Thus, additional restrictions reflect various additional conditions that arise during the modeling process. But every additional restriction narrows the scope of freedom of choice. Therefore, they should be introduced into the problem carefully, within reasonable limits and when necessary.

Auxiliary restrictions As a rule, they do not have independent meaning and are introduced into the problem to formalize individual conditions. These include restrictions that establish a proportional relationship between individual variables or their groups.

Evaluation of variables in the objective function (with j) are coefficients expressing the amount of total income or costs per unit of measurement of a variable. The evaluation of a variable, as a rule, expresses the accepted optimality criterion. It can be presented both in kind and in monetary form, i.e. costs per unit of production (product cost).

The condition for non-negativity of variables is written in the form

x j≥ 0, j = 1, 2, …, n.

In real life production, based on the conditions of the task, according to this record of the structural economic-mathematical model (EMM), a list of variables and restrictions is compiled, initial information is prepared, a detailed EMM of the problem is constructed, which is then written down in the form of a matrix (table) and entered into the computer and the results are calculated and analyzed using the appropriate program.i = 1, …, m, (1.5)

j = 1, …, n. (1.6)

Vector x = (x 1 , x 2 , …, x n), components x j which satisfy constraints (1.2) and (1.3) [or (1.5) and (1.6) in the minimum problem] is called acceptable solution or valid plan LP problems. The set of all admissible plans is called set of admissible plans.

Canonical The form of the linear programming problem is characterized by the fact that it contains the objective function, all restrictions equality, all variables are non-negative.

Any linear programming problem can be reduced to a linear programming problem in canonical form. To do this, in the general case, you need to be able to reduce the maximization problem to the minimization problem; move from inequality constraints to equality constraints and replace variables that do not obey the non-negativity condition.

The rule for bringing a linear programming problem to canonical form is as follows:

1) if in the original problem it is necessary to determine the maximum of a linear function, then you should change the sign and look for the minimum of this function;

2) if the right side of the restrictions is negative, then this restriction should be multiplied by – 1;

3) if there are inequalities among the restrictions, then by introducing additional variables of non-negative variables they are transformed into equalities. For example, additional variables Sj in restrictions of type less than or equal to (£) are entered with a plus sign:

Additional Variables Sj in restrictions of the type greater than or equal to (≥) are entered with a minus sign:

To eliminate the negativity of additional variables − Sj introduce artificial variables with a plus sign + M j with very large values.

Lecture 2

IN canonical form

acceptable decision of the PPP(valid plan).

optimal solution for the ZLP.

Necessity

Example.

Let's write the problem in canonical form

Special situations of the graphical solution of the ZLP

Except when the task has the only optimal solution for and , maybe special situations:

1. the task has an infinite number of optimal solutions – the extremum of the function is achieved on the segment ( alternative optimum)– Figure 2;

2. task not solvable due to the unlimited nature of the ODR, or – Figure 3;

3. ODR - single point Ah, then ;

4. task not solvable , if ODR is an empty area.

A

Figure 2 Figure 3

If the level line is parallel to the side of the region of feasible solutions, then the extremum is reached at all points on the side. The problem has countless optimal solutions - alternative optimum . The optimal solution is found by the formula

where is the parameter. For any value from 0 to 1, you can get all points of the segment, for each of which the function takes the same value. Hence the name - alternative optimum.

Example. Solve graphically a linear programming problem ( alternative optimum):

Questions for self-control

1. Write down the linear programming problem in general form.

2. Write the linear programming problem in canonical and standard forms.

3. What transformations can be used to move from the general or standard form of a linear programming problem to the canonical one?

4. Define admissible and optimal solutions to a linear programming problem.

5. Which of the solutions is the “best” for the problem of minimizing the function if ?

6. Which of the solutions is the “best” for the problem of maximizing the function if ?

7. Write down the standard form of a mathematical model for a linear programming problem with two variables.

8. How to construct a half-plane defined by a linear inequality in two variables ?

9. What is called the solution to a system of linear inequalities in two variables? Construct on the plane a region of admissible solutions of such a system of linear inequalities that:

1) has a unique solution;

2) has an infinite number of solutions;

3) has no solution.

10. Write for a linear function vector gradient, name the type of level lines. How are the gradient and level lines located relative to each other?

11. Formulate an algorithm for a graphical method for solving a standard ZLP with two variables.

12. How to find the coordinates of the solution and the values ​​of , ?

13. Construct the region of feasible solutions, gradient and level lines, for linear programming problems in which:

1) is achieved at a single point, a - on the ODR segment;

2) is achieved at a single point of the ODR, and .

14. Give a geometric illustration of the ZLP if it:

1) has unique optimal solutions for and ;

2) has many optimal solutions for .

Lecture 2

graphical method for finding the optimal solution

1. Forms of linear mathematical models and their transformation

2. Graphical method for solving a linear programming problem

3. Special situations of the graphical solution of the ZLP

4. Graphical solution of economic linear programming problems

Forms of linear mathematical models and their transformation

The mathematical model of a linear programming problem (LPP) can be written in one of three forms.

IN general form of the mathematical model you need to find the maximum or minimum of the objective function; the system of constraints contains inequalities and equations; not all variables can be non-negative.

IN canonical form the mathematical model needs to find the maximum of the objective function; the system of constraints consists only of equations; all variables are non-negative.

In the standard form of a mathematical model, you need to find the maximum or minimum of a function; all constraints are inequalities; all variables are non-negative.

A solution to a system of constraints that satisfies the conditions of non-negativity of variables is called acceptable decision of the PPP(valid plan).

The set of feasible solutions is called area of ​​admissible solutions of the PLP.

An admissible solution at which the objective function reaches an extreme value is called optimal solution for the ZLP.

The three forms of writing the ZLP are equivalent in the sense that each of them can be reduced to another form using mathematical transformations.

Necessity transition from one form of mathematical model to another is associated with methods for solving problems: for example, the simplex method, widely used in linear programming, is applied to a problem written in canonical form, and the graphical method is applied to the standard form of a mathematical model.

Transition to the canonical form of recording ZLP.

Example.

Let's write the problem in canonical form, introducing into the left side of the first inequality of the system of restrictions an additional (balance) variable with a “+” sign, and into the left side of the second inequality an additional variable with a “minus” sign.

The economic meaning of various additional variables may not be the same: it depends on the economic meaning of the restrictions in which these variables are included.

Thus, in the problem of using raw materials, they show the remaining raw materials, and in the problem of choosing optimal technologies, they show the unused time of the enterprise using a certain technology; in a cutting problem – production of blanks of a given length in excess of the plan, etc.