Линейное программирование. Практикум. Учебное пособие для бакалавриата

2. Introduction to LinearProgramming

Here are images of convex and non-convex figures for comparison.

Fig. 1

Remark. The feasible set of the linear programming problem is a closed convex polyhedral region.

We give some properties of convex sets.

Remark. Intersection of convex sets is a convex set. Remark. A linear inequality

a1x1 +a2x2 + +anxn ≤ b

specifies in Rn a half-space. A half-space is convex. Remark. A linear equation

a1x1 +a2x2 + +anxn = b

specifies in Rn a hyperplane. A hyperplane is convex. Linear programming problems can be parted on the

following four groups.

1. The infeasible linear programming problems. The constraints are inconsistent.

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2. Introduction to LinearProgramming

Example

There are no solutions to the problem because there are no solutions to the system of constraints (the feasible set is empty)

2. The unbounded linear programming problems.

The objective function is unbounded on feasible set.

Example

There is no solution of the problem.

3. The linear programming problems having a unique solution.

Example

f = x2 → max

x1 +x2 ≤1,x1 ≥ 0,x2 ≥ 0.

The solution of the problem is X* = 0 .

1

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2. Introduction to LinearProgramming

4. The linear programming problems having infinite solutions.

Example

Theorem. If the feasible set is bounded and not empty, then there exists an optimal solution of the linear programming problem.

Proof. The Weierstrass theorem states that a continuous function defined on closed bounded set assumes the greatest and the least values. Since a linear function is continuous and the feasible set is closed and bounded then there exists an optimal solution of the linear programming problem. +

Remark. The objective function assumes its optimum (maximum or minimum) value at a corner point (vertex) of the feasible set (provided the optimum exists). Occasionally (see 4-th group of the linear programming problems), the optimum occurs along a face (entire edge) of the feasible set, but in this case the optimum occurs at a corner point as well.

Remark. If a linear programming problem possesses several optimal solutions X1*, ,Xk* , then a convex combination of them

Y1* = s1X1* + +skXk*

where

si ≥ 0 (i =1; k);

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2. Introduction to LinearProgramming

s1 + +sk =1

is also an optimal solution.

Definition. A linear programming problem is said to be in standard form when it is written as

and is said to be in canonicals form if

f = CT X +C0 → max

AX = B,

X ≥ 0.

Remark. Any linear programming problem can be transformed to the canonicals form.

Indeed, a linear programming problem

f = CT X → max

AX ≤ B,

DX = Q,X ≥ 0.

is equivalent to a problem written in canonicals form

f = CT X +C0 → max

AX +Y = B,

DX = Q,

X ≥ 0,

Y ≥ 0.

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2. Introduction to LinearProgramming

The variables y1, ,yn are called slack variables. Example. Transform linear programming linear

programming problem to the canonicals form. f =2x1 −x2 −x3 → max

x1 +2x2 −5x3 = 3,2x1 −2x2 ≤1,x1 +x2 +x3 ≤2,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

Solution. Using the slack variables y1,y2 we get

Remark. If a linear programming problem is written in the canonicals form and the system of basic constraints reduced to the unit basis, then the linear programming problem can be transformed to the standard form.

Example. Transform the linear programming problem to the standard form.

f =2x1 −3x2 +x3 −2x4 → max

x1 +2x2 +x3 =2,2x1 −x2 +x4 =1,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

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2. Introduction to LinearProgramming

9)How is the feasible set of a linear programming problem defined?

10)What is the feasible linear programming problem?

11)How many solutions can a linear programming problem have?

12)What is a convex set?

13)Is the feasible set of a linear programming problem

convex?

14)What is the canonicals form of a linear programming problem?

15)Is it possible to transform a linear programming problem to a canonicals form?

16)What is the standard form of a linear programming problem?

17)In which case can a linear programming problem be transformed to a standard form?

Exercises for independent work

1) Transform the linear programming problems to the canonicals form.

f = x1 −5x2 +2x3 → max

x1 −x2 +4x3 =7,

a)x1 −4x2 ≤2,

x1 +2x2 +3x3 ≥ 6,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

f = x1 +2x2 −x4 → min

3= 3,

b)2x1 −2x2 +6x4 ≤1,x1 −x2 +2x3 −x4 ≤2,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0,x4 ≥ 0.x1 +2x2 −5x

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2.Introduction to LinearProgramming

2)Transform the linear programming problems to the standard form.

f= x1 +3x2 +x3 −x4 → max

a)x1 −x2 +x3 =1,3x1 −2x2 +x4 =5,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

f=2x1 −x2 +x3 → min

b)x1 +2x2 +x3 =1,2x1 −x2 +x4 =5,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

3) Write the linear programming problem in the matrix

form

f = x1 −x2 +2x3 → min

x1 +x2 −2x3 ≤ 3,

a)x1 −2x2 +4x3 ≤1,x1 −5x2 +2x3 ≤7,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0,x4 ≥ 0.

f = x1 −3x2 +x3 +4x4 → min

b)x1 −x2 +3x3 −2x4 =2,x1 −3x2 +4x3 −2x4 =5,

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

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3. GEOMETRIC METHOD

This method is used for linear programming problems written in the standard form (all constraints are inequalities) with 2 variables.

Algorithm for finding the feasible region in the case of 2 variables.

1) Find the boundaries of the feasible region.

They are the straight lines given by equations obtained by replacing the inequality sign with an equal sign in the system of constraints. You can build them by finding two points through which this line passes. These points can be obtained by substituting a fixed value of one of the variables into the equation and finding the corresponding value of the other variable.

2) Find the half-planes containing the feasible region. The straight lines of the boundaries divide the plane

into two half-planes. Substituting a test point (for example, a point O(0;0)), we can determine which half-plane containing the feasible region. If, when substituting the test point into the inequality defining the half-plane of the constraint, we get the correct statement, then the feasible region belongs to the half-plane containing the test point. If, when substituting a test point into an inequality defining a half-plane of the constraint, we get an incorrect statement, then the feasible region belongs to a half-plane that does not contain a test point.

3) Find the intersection of all half-planes containing the feasible region.

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3. Geometric Method

Example. The Figure 2 shows the region for the following linear programming problem.

Fig. 2

Definition. The vector n = (n1,n2 )T is called the gradient vector of the objective function f =n1x1 +n2x2 +n0 .

Definition. The straight line n1x1 +n2x2 = c

where c is a constant, is called the level line of the function f =n1x1 +n2x2 +n0 .

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