Top problems of linear programming using the graphical method, presentation before an algebra lesson on the topic. Linear programming Presentation of the graphical method of linear programming

Virus of linear irregularity systems

Anxiety

Linear irregularities – those of the form ∑ai xi +b≥c

The creation of a system of linear irregularities with two or three unknowns means the creation of a rounded rich-cut area on the plane or, apparently, a rounded rich-faceted body in space.

Beginning in the mid-1940s, a new area of applied mathematics—linear programming—developed with important applications in economics and technology. Complete linear programming is just one of the branches (albeit even more important) of the theory of linear inequity systems.

Let's take a look at the level of the first step with two unknown x and y

Tlumachuyuchi x and y as coordinates of the point

on the square, we can say that

The multiplicity of points that are assigned to the lines (1) is a straight line on the plane.

Geometric sense level of the first stage

Similarly for unevenness ax+by+c≥0. (2)

If b≠0, then this inequality is induced to one of the types y≥kh+p or y≤kh+p.

The first of these inequalities is satisfied with all the points that lie “higher” behind the straight line y=khx+p or on this straight line, and the other is satisfied with all the points that lie “lower” behind the straight line y=khx+p or on this straight line.

If b=0, then the unevenness is induced in one of the types x≥h or x≤h. The first is satisfied with all the points that lie “to the right” of the straight line x = h or on this straight line, the other is satisfied with all the points that lie “to the left” of the straight line x = h or on this straight line.

Geometric sense of linear irregularity system

Let there be a system of inequalities from two unknowns, x and y. a1 x+b1 y+c1 ≥0,

a2 x+b2 y+c2 ≥0,

………….........

am x+bm y+cm ≥0.

The first unevenness of the system means on the coordinate plane xOy the surface of the plane P1, the other - the plane of P2, etc. If a pair of numbers x, y satisfies all the unevenness of the system, then the corresponding point M (x, y) lies on all the planes P1, P2, ..., Pm at the same time. In other words, point M lies on the crossbar (the back part) of the designated surfaces. It is easy to understand that the cross section of the end number of surfaces is a rich area.

Butt

Use strokes along the outline of the image area to go to the middle of the area. The stench immediately indicates which side of the line the surface plane lies on; those same ones are assigned to the additional arrows.

The area is not demarcated

The area before is called the area of decoupling of the system of irregularities. It is immediately significant that the decision area will always be demarcated; As a result, the webbing on several surfaces may collapse and the area may not be bordered.

By clicking on the "Download archives" button, you will download the file you need without any costs.
Before accessing this file, find out about those good abstracts, tests, coursework, dissertations, statistics and other documents that are not required on your computer. It is your duty to take part in the development of marriage and bring wealth to people. Find out your work and submit your knowledge to the database.
We and all students, postgraduate students, young people, who have a strong knowledge base in their new job, will be most grateful to you.

To add archives to a document, in the lower-shaded field, enter a five-digit number and click the “Own archives” button.

Linear programming

Methods of linear programming are used in forecasting processes, when planning and organizing production processes. Linear programming is a field of mathematics that involves methods for investigating and identifying extreme values of a given linear function, based on the arguments for applying linear boundaries.

Slide 3

Such a linear function is called goal-oriented, and a set of numerous interchangeable relations that express the songs of economical task in the appearance of equalities and inequalities is called There is a system of exchange. The word programming was introduced in connection with the fact that unknown changes mean a program or plan of a given subject.

Slide 4

The set of relations that combines the target function with its arguments is called a mathematical model of optimization. The ZLP is recorded in the official form as follows: when deformed

Slide 5

Here - unknown - are given constant quantities. The boundaries may be set by equalities. Most often, problems arise as follows: є resources during exchanges. It is necessary to calculate the obligations of these resources, for which purpose the maximum (minimum) function can be achieved, in order to know the optimal distribution of shared resources. Whose natural limit is >0.

Slide 6

At which the extremum of the objective function is found on the admissible solution multiplier, which is determined by the boundary system, and all or any inequalities in the boundary system can be recorded in a similar manner.

Slide 7

The short recording of the ZLP has the form: when deformed

Slide 8

To develop a mathematical model of the ZLP, it is necessary to: 1) designate the changes; 2) set the goal function; 3) record the system of delineation according to the specification; 4) write down a system of interrelations with the management of the minds of the demonstrators. Since all tasks are interconnected by variables, a model of this type is called canonical. Although at the same time there is uncertainty, the model is non-canonical.

Slide 9

Make an effort to get promoted to the PPL.

assigning an optimal distribution of resources when planning the production of products for the enterprise (assortment assignment); focusing on maximum production output for a given assortment; information about sums (diet, diet, etc.); transport department; knowledge about rational vikoristannya of obvious efforts; Zavdannya about recognition.

Slide 10

1. Providing the optimal distribution of resources.

It is acceptable that this enterprise produces various viruses. Their production requires different types of resources (syrup, labor and machine hours, additional materials). These resources are exchanged and planning is made for the period of mental units. There are also technological coefficients that indicate how many units of the i-th resource are required for the production of the j-th species of virus. Let the profit that is generated by the enterprise upon the sale of one virus of the jth species, ancient. During the period that is planned, all displays are transferred permanently.

Slide 11

It is necessary to put together such a plan for the production of products, when realizing which income of the enterprise will be the greatest. Economic and mathematical model of the problem

Slide 12

The goal function is the total income from the sale of products of all types. This model allows for advanced optimization with an additional selection of the most popular product types. Exchange means that from any of its resources the total cost of production of all types of products is collected from its reserves.

Slide 13

Apply it

Slide 14
It is acceptable that species A, species B and species C will be produced. To produce such a number of species, it will be necessary to spend machine-years of milling equipment. The fragments of the working hour fund of work stations of this type cannot exceed 120, then inequality may arise

Slide 15
Dimensions and similarly, you can fold the boundary system

Slide 16
Now let's create a goal function. Income from the type A in the warehouse 10, from the sale - from the type B -14 and from the sale - from the type C-12 The total income from the sale of all the viruses in the warehouse

Slide 17
In this way, we come to the onset of the ZLP: It is necessary, among all the unknown solutions to the system of inequities, to know at which point the goal function reaches its maximum value.

Slide 18
Butt 2

The dairy's products include milk, kefir and sour cream, packaged in containers. To produce 1 ton of milk, kefir and sour cream, a total of 1010.1010 and 9450 kg of milk is required. At this working hour, when bottling 1 ton of milk and kefir, it becomes 0.18 and 0.19 machine-years. To pack 1 ton of sour cream, special machines take 3.25 years.

Slide 19
In order to produce whole milk products, the plant can produce 136,000 kg of milk. The main equipment can last for 21.4 machine-years, and the sour cream filling machine can last for 16.25 machine-years. The income from 1 ton of milk, kefir and sour cream is consistently 30, 22 and 136 rubles. The plant is guilty of almost 100 tons of milk, packaged at the dance floor. There are no restrictions on the production of other products.

Slide 20
It is necessary to determine which products and in what quantities are produced by the plant on a daily basis, so that the profit from sales is maximized. Fold the mathematical model of the plant.

Slide 21
Decision

Let the plant produce milk, kefir and sour cream. I need kg of milk. Since the plant can produce no more than 136,000 kg of milk today, inequality may arise

Slide 22
We are constantly working on packing milk and kefir and packing sour cream. The fragments today may produce less than 100 tons of milk, then. For an economical price

Slide 23
The total income from the sale of all products is rubles. In this way, we arrive at the onset of the task: when exchanging the fragments, the goal function is linear and the boundary is set by a system of inequalities, then the task is ZLP.

Slide 24
A story about sumishi.

Є two types of products that contain vital substances (fats, proteins, etc.)

Slide 25
Table
Slide 26
Decision

A healthy diet while maintaining the necessary minimum of living speech

Slide 27
Mathematical formulation of the task: adjust the daily diet so that it satisfies the metabolic system and minimizes the goal function. The secret view of the group about the sums seems to be the task of finding the cheapest set of the latest output materials that will ensure the recovery of the sums from the authorities. We remove all the necessary components from our warehouse in large quantities, and the components themselves are stored in parts of the output materials.

Slide 28
We have entered the following designations: - the quantity of the j-th material to be included; - price for material of the jth type; - this is the minimally necessary place for the i-th component of the sumisha. Coefficients show the nutritional value of the i-th component in a unit of the j-th material

Slide 29
Economic-mathematical model of the plant.

The purposeful function is the total performance of the sum, and the functional exchange is interchanged with the components of the sum: the sum is obliged to place the components in the duties, no less than the meanings.

Slide 30
The story about the opening

At a sewing factory, fabric can be cut into pieces using methods for producing the required parts for sewing machines. Let the j-th opening option produce parts of the i-th type, and the amount of output with this opening option is the same. come out. Fold the mathematical model of the plant.

Slide 31
Decision. Let’s assume that the jth option requires a hundred fabrics to be cut. When cutting the fabric, the i-th type of parts will come out in the j-th option; for all the cutting options, the fabric will be removed from the fabric.
Slide 35
Main Director of LP

Def.4. The main or canonical ZLP is called the task, which lies at the designated value of the goal function for the mind, so the system of exchange is presented in the form of a system of ranks:

Slide 36
If it is necessary for clarity or instead of the task to go from one record form to another, proceed like this. If you need to know the minimum of a function, you can go to find the maximum by multiplying the goal function by (-1). Limitation - inequality of the species can be transformed into jealousy added to the left part of the non-negative additional change, and boundary inequality of the species - into restriction - jealousy of the ones from the left part of the additional non-negative no change.

Slide 41
The basic plane is called non-virtuous, since it is capable of displacing m positive components. It is also called virogenim. The plan for which the objective function of the PLP reaches its maximum (minimum) value is called optimal.

View all slides

Description of the presentation with the following slides:

1 slide

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004. Linear programming To which class of linear programming (75% of the tasks assigned by Americans) is the task of the purpose of the function Wm(x), m=1,2,..., M, boundary as equalities hk(x)=0, k=1,2...K, and irregularities gj(x)>0, j=1,2,...J, are linear and have no mathematical solution. Possible topics for the LP department: rational selection of raw materials and materials; I will reveal the required optimization; optimization of production programs of enterprises; optimal placement and concentration of production; to develop a rational plan for transportation, robotic transport; inventory management; There are many others that fall within the scope of optimal planning. Statement of the LP problem (the importance of showing the effectiveness of variable tasks, specifying a linear objective function W(x), which promotes minimization or maximization of functional hk(x), gj( x) ta oblasnih xli
2 slide

Slide description:
The theory of the decision of PetrSU, A.P. Moshchevikin, 2004 Application of the LP problem Application - Optimization of the placement of by-products of the forestry The forestry has 24 hectares of free land under fallow and connected winding you get income from it. You can feel the sajants of a hybrid of the new native yalina, which grows rapidly, which reach the stems in one river, or the saplings, which have raised part of the earth for pasture. The trees are widely available and sold in batches of 1000 pieces. It takes 1.5 hectares to grow one batch of trees and 4 hectares to grow one stick. Forestry can spend up to 200 years per river on its secondary production. Practice shows that it takes 20 years. for cultivation, pruning, felling and packaging of one batch of trees. It also takes 20 years to look after one whip. Forestry can be spent at a cost of 6 thousand. crb. River costs for one batch of trees amount to 150 krb. that 1.2 thousand. rub. for one whip. The contract for the supply of 2 whips has already been laid out. For prices, one new yalina brings a profit of 2.5 rubles, one whip - 5 thousand. rub.

3 slide

Slide description:
Theory of making a decision PetrSU, A.P. Moshchevikin, 2004 Setting the task 1. As an indicator of the effectiveness of taking the profit for the operation (real income from the land in rubles). 2. In a bowl of ceramic slabs, take: x1 - the number of whips to be prepared, per river; x2 - the number of batches of rotating, fast-growing new eggs, 1000 pcs. skin on the river 3. Goal function: 5000 x1 + 2500 x2  max, where 5000 is the net income from one whip, rub.; 2500 – net income from one batch of trees (1000 pieces for 2.5 rubles). 4. Limitation: 4.1. Land allocation, hectares: 4 x1 + 1.5 x2  24 4.2. According to the budget, rub.: 1200 x1 + 150 x2  6000 4.3. For labor resources, year: 20 x1 + 20 x2  200 4.4. Zobov'yazannya for the contract, note: x1  2 4.5. Area of limitation: x1  0, x2  0

4 slide

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004. Graphically, the most important task of the LP is displayed on the graph, which corresponds to the current level, 4 x1 + 1.5 x2 = 24 1200 x1 + 150 x2 = 6000 20 x1 = 2 x2 = 0 shaded area, at the points ї all exchanges are concluded. Each such point is called admissible decisions, and the set of all admissible decisions is called admissible area. Obviously, the unraveling of the LP problem lies in the search for the shortest solution in the admissible region, which, in its own way, is called optimal. The analyzed application has an optimal solution and acceptable solutions that maximize the function W=5000 x1 + 2500 x2. The value of the objective function, which indicates an optimal solution, is called the optimal value of the given LP.

5 slide

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004. Graphic design of LP

6 slide

Slide description:
The theory for making a solution PetrSU, A.P. Moshchevikin, 2004 Graphical solution of the LP problem Iterate over all critical points in the area of acceptable solutions to achieve maximum income in the amount of 34 thousand. rub. (W=5000x1+2500x2), where the forestry can be drawn, turning 3.6 sticks and 6.4 batches of new-growing yalinkas. Integer methods (for example, enumeration) give x1=3 and x2=6, which leads to an income of 30 thousand. rub., x1=4 and x2=5 to achieve a greater than optimal result of 32.5 thousand. rub., point x1 =3 and x2=7 lead to a similar result. The graphical method, through the great size of real practical tasks of the LP, rarely becomes stagnant, and allows one to clearly understand one of the main powers of the LP - if the LP problem has an optimal solution, then we take one of the vertices of the admissible region and optimal solutions. Regardless of the fact that the permissible area of the given LP consists of an infinite number of points, the optimal solution can later be found by directly searching through the terminal number of its vertices. This simplex method of solving the LP problem is based on this fundamental power.

7 slide

Slide description:
One of the new functions of the MS Excel spreadsheet editor (you must check the box when installing MS Office) is “Search for a solution.” This package allows you to easily change the settings of linear and nonlinear programming.

8 slide

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004. The setting of the LP in the standard form The setting of the LP in the standard form with m boundaries and n changes looks like this: W = c1x1 + c2x2 + ... + cnxn  min (max) with boundaries a11x1+a12x2+...+a1nxn=b1; a21x1 + a22x2 + ... + a2nxn = b2; ... am1x1 + am2x2 + ... + amnxn = bm; x10; x20;...; xn0 b10; b20;...; bm0 In the matrix form W = cx  min (max) with displacement Ax = b; x0; b0, where A is the matrix of dimensions m*n, x is the vector series of variable dimensions n*1, b is the vector series of resources of dimension m*1, h is the vector row of estimates for the LP problem 1*n.

Slide 9

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004. Reworking of inaccuracies Exchanges as inaccuracies can be reworked evenly with the help of the addition of so-called excess and over-excess changes. The equalization from the front butt 4x1 + 1.5x2  24 can be converted to equalization using the additional non-negative change 4x1 + 1.5x2 + x3 = 24. The change x3 is non-negative and represents a difference right and left parts of inequality. Similarly, x1  2 can be reversed by the path of provoking the supermundane change x4: x1 - x4 = 2.

10 slide

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004 Transformation undefined. by the sign of the changes Reversal of non-interchangeable values by the sign of the changes Changes that appear as positive and negative values are replaced by the difference of two non-negative ones: x = x+ - x-; x+0; x-  0. App. 3x1+4x2+5x3+4x4  max 2x1+x2+3x3+5x4  5 5x1+3x2+x3+2x4  20 4x1+2x2+3x3+x4 = 15 x10; x20; x3 0; x4 =  sign -3x1-4x2+5x3-4x4  min 2x1+x2-3x3+5x4-x5 = 5 5x1+3x2-x3+2x4+x6 = 20 4x1+2x2-3x3+x4 = 15 x1; x20; x30; x4 =  sign; x4 =x8-x7 -3x1-4x2+5x3-4x8+4x7 min 2x1+x2-3x3+5x8-5x7-x5 = 5 5x1+3x2-x3+2x8-2x7+x6 = 20 4x1+2x2-3x3+8 -x7 = 15 x1, x2, x3, x5, x6, x7, x8 0; x4=x8 -3x1-4x2+5x3-4x4+4x7 min 2x1+x2-3x3+5x4-5x7-x5 = 5 5x1+3x2-x3+2x4-2x7+x6 = 20 4x1+2x2-3x3+x4-x7 = 15 x1, x2, x3, x4, x5, x6, x70; x8 was replaced by x4

11 slide

Slide description:
The theory of decision making PetrSU, A.P. Moshchevikin, 2004 Simplex method of LPs The simplex method is an iterative procedure for completing the tasks of LPs written in a standard form, the system of equations in which, with the help of elementary operations on matrices, is reduced to the canon literal form: x1 + a1 , m+1xm+1 + ... + a1sxs+...+ a1nxn = b1; x2 + a2, m+1xm+1 + ... + a2sxs+...+ a2nxn = b2; ...xm+am,m+1xm+1+...+amsxs+...+amnxn = bm. Variations x1, x2,...,xm, which enter with single coefficients of only one level of the system and with zero ones - to others, are called basic. In the canonical system, the skin level is represented by one basic change. Other n-m variables (xm+1, ..., xn) are called non-basic variables. To bring the system to a canonical form, you can perform two types of elementary operations on rows: Increasing the system's level by positive and negative numbers. An addition to any level of the other level of the system, multiplied by a positive or negative number.

12 slide

Slide description:
Theory of solution PetrSU, A.P.Moshchevikin, 2004 Simplex LP method Recording the problem in the form x1 + a1,m+1xm+1 + ... + a1sxs+...+ a1nxn = b1; x2 + a2, m+1xm+1 + ... + a2sxs+...+ a2nxn = b2; ...xm+am,m+1xm+1+...+amsxs+...+amnxn = bm. The same notation looks like the matrix 1 0 .. 0 a1,m+1 .. a1s .. a1n x1 b1 0 1 .. 0 a2,m+1 .. a2s .. a2n x2 = b2 . . .. . . .. . .. . .. .. 0 0 .. 1 am,m+1 .. ams .. amn xn bm

Slide 13

Slide description:
Theory for making a solution PetrSU, A.P. Moshchevikin, 2004 Algorithm of the simplex method 1. Select an admissible basis for the solution. Basic solutions are called solutions that are removed at zero values of non-basic changes, then. xi=0, i=m+1,...,n. A basic solution is called an admissible basic solution if the values of the inputs of the new basic solutions are non-negative. xj = bj  0, j=1,2,...,m. In this case, the target function will now appear: W=cbxb=c1b1+c2b2+...+cmbm. Let’s remember the cob table using the simplex method:

Slide 14

Slide description:
The theory of decision making PetrSU, A.P. Moshchevikin, 2004 Algorithm of the simplex method 2. We calculate the vector of advanced estimates c using the additional scalar rule cj = cj - cbSj, where cb is the vector of estimates of the basic variables; Sj is the jth class of coefficients aij in the canonical system, which corresponds to the analyzed basis. Additionally, the cob table with - in a row.

15 slide

Slide description:
The theory will be accepted by PetrSU, A.P. Moshchevikin, 2004. Algorithm of the simplex method 3. If all estimates are cj  0 (cj  0), i=1,...,n, then, more precisely, the permissible solution is maximum (minimal). The solution has been found. 4. Otherwise, it is necessary to enter into the basis a non-basic variable xr with the largest values cj instead of one of the basic variables (div. table).

16 slide

Slide description:
The theory will be solved by PetrSU, A.P. Moshchevikin, 2004 Algorithm of the simplex method 5. Using the additional rule of minimum ratio min(bi/air), the change xp is determined, which is derived from the basis. Since the coefficient air is negative, bi/air=. As a result of the transition, where the non-basic variable xr is entered, and the rows where the basic variable xp is displayed, the position of the wire element in the table is determined. 6. Elementary changes are necessary to create a new permissible basic solution and a new table. As a result, the conductive element must be set to 1 and other elements in connection with the conductive element will take on zero values. 7. We calculate new final estimates from the vicors of the rule of scalar transformation and move on to chapter 4.

Slide 17

Slide description:
The theory will be solved PetrSU, A.P. Moshchevikin, 2004 Application of the solution using the simplex method Application - Optimization of the placement of by-products of forestry 3. Objective function: 5000 x1 + 2500 x2  max, 4. nya: . Land allocation, hectares: 4 x1 + 1.5 x2  24 4.2. According to the budget, rub.: 1200 x1 + 150 x2  6000 4.3. For labor resources, year: 20 x1 + 20 x2  200 4.4. Zobov'yazannya for the contract, note: x1  2 4.5. Areas of delineation: x1  0, x2  0 We bring the data to the standard form: 4 x1 + 1.5 x2 +x3= 24 1200 x1 + 150 x2 +x4= 6000 20 x1 + 20 x2 +x5= 200 x1 – x1 .. . x6  0 The first three levels follow the basic exchange rate x3, x4, x5, but the fourth one has a day through those that with the change x6 have a negative single coefficient. To bring the system to its canonical form, we use the method of piecemeal changes. x1 – x6+x7= 2, we sent away the piece change x7.