Линейное программирование. Ч.1. Учебное пособие для бакалавриата

1919

Федеральное государственное образовательное бюджетное учреждение высшего образования «ФИНАНСОВЫЙ УНИВЕРСИТЕТ

ПРИ ПРАВИТЕЛЬСТВЕ РОССИЙСКОЙ ФЕДЕРАЦИИ» (Финансовый университет)

Департамент математики

И.Г. Шандра

ЛИНЕЙНОЕ ПРОГРАММИРОВАНИЕ

Часть 1

Учебное пособие для бакалавриата экономических и математических вузов и факультетов

МОСКВА

2023

1919

FEDERAL STATE EDUCATIONAL BUDGETARY INSTITUTION

OF HIGHER EDUCATION

“FINANCIAL UNIVERSITY

UNDER THE GOVERNMENT OF THE RUSSIAN FEDERATION”

(Financial University)

Department of Mathematics

I.G. Shandra

LINEAR PROGRAMMING

Part 1

Manual for undergraduate students

of economic and mathematical universities and faculties

Moscow

2023

CONTENT

1.Preface . . . . . . . . . . . . . . . . . . . . 6

2.Introduction to linear programing. . . . . . . . . . 7

— 5 —

1. PREFACE

Linear programming as a science appeared due to the need to solve applied economic problems.

The founder of linear programming was the Soviet mathematician Leonid Kantorovich. In 1939, his first work was published on the application of linear programming to solve applied economic problems. In 1975, Leonid Kantorovich was awarded the Nobel Prize for the development of linear programming methods.

The American mathematician George Bernard Danzig also made a great contribution to the development of linear programming. In 1949, he developed an effective method for solving linear programming problems — the simplex method.

Thе manual consists of four parts; introduction to linear programming, geometric method, simplex method, artificial variables method. The manual can be used to read selected chapters of the courses “Mathematics”, “Linear Algebra”, “Linear Programming”, Optimization Methods”, “Methods of optimal solutions” for bachelor’s and master’s degrees in economics and mathematics. The manual may also be of interest to graduate students, postgraduates, researchers, teachers, university professors.

— 6 —

2. INTRODUCTION TO LINEAR

PROGRAMMING

Linear programming isabranchofappliedmathematics that deals with solving problems of maximization or minimization of a linear function subject to linear constraints (it means that constraints are linear equations or linear inequalities).

Here is an example of a linear programming problem in mathematical form.

Example.

f =2x1 −x2 +2x3 +3 →max

Subject to

x1 +6x2 −2x3 =4,5x1 +3x2 ≥5,

x1 ≥0,x2 ≥0,x3 ≥0.

Now let us consider important economic problems that lead to linear programming problems.

unit of food Fj . A diet is healthy if it has daily at least q1, ,qn quantities of the nutrients N1, ,Nn respectively. To compose such a diet, we need to buy quantities x1, ,xm of the foods having prices p1, , pm . The problem is to supply the required nutrients at minimum cost.

— 7 —

2. Introduction to LinearProgramming

f = p1x1 + + pmxm →mina11x1 + +a1mxm ≥q1,

,

an1x1 + +anmxm ≥qn,

x1 ≥0, ,xm ≥0.

Or in the compact matrix form

f = PT X →min

AX ≥ Q,

X ≥0.

Example. (The planning production problem).

Suppose a firm produces m types of goods: G1, ,Gm

of resource Ri used in the production of one unit of good Gj and q1, ,qn be the supplies of resources. The problem is to find the outputs of productions (the optimal plan) x1, ,xm maximizing the total income.

f = p1x1 + + pmxm →maxa11x1 + +a1mxm ≤q1,

,

an1x1 + +anmxm ≤qn,

x1 ≥0, ,xm ≥0.

Or in the compact matrix form

f = PT X →max

AX ≤ Q,

X ≥0.

Example. (The transportation problem). Let there be m producers of some good: A1, , Am (points of departure) and n consumers B1, ,Bn (points of destination). It is

— 8 —

2. Introduction to LinearProgramming

given for each producer the supply ai ≥0 (i =1, ,m) and for each customer the demand bj ≥0 (j =1, ,n) . It is known the unit cost cij ≥0 of transportation of the good from the i-th producer to the j-th customer. It is required to find the volume of transportation xij from the i-th producer to the j-th customer minimizing the total cost of transportation.

f = CT X +C0 →max (min)

AX ≤ B,

DX = Q,

X ≥0.

— 9 —

2. Introduction to LinearProgramming

The linear function f to be maximized or minimized is called the objective function.

Remark. If we have the problem of finding the maximum of the objective function f , then the problem for the objective function g = −f under the same constraints will be the minimization problem and vice versa.

A solution X* =(x1*,...,xn* )T of the linear programming problem is called the optimal solution (vector).

The constraints (1), (2) are called the main constraints and the constraints (3) are called the non-negativity constraints.

A vector X =(x1,...,xn )T satisfying the constraints of the linear programming problem is called a feasible solution (vector).

The set of all feasible solutions is called the feasible set

(region).

A linear programming problem is said to be feasible if the feasible set is not empty otherwise it is called infeasible.

Definition. A set M is said to be convex if for any x,y M and any t [0;1] it follows that tx +(1−t)y M .

From the geometrical point of view of geometry convexity of set means that if two points belong to the set then the segment connecting these points also belongs to the set.

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

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.

— 10 —