Boolean Variables in Economic Models Solved by Linear Programming

• Author: Lixandroiu, D.
• Position: Faculty of Economic Sciences and Business Administration, Transilvania University of Brasov
• Pages: 305-312

Bulletin of the Transilvania University of Braşov

Series V: Economic Sciences • Vol. 7 (56) No. 2 - 2014

BOOLEAN VARIABLES IN ECONOMIC

MODELS SOLVED BY LINEAR

PROGRAMMING

Dorin LIX;NDROIU1

Abstract: The article analyses the use of logical variables in economic

models solved by linear programming. Focus is given to the presentation of

the way logical constraints are obtained and of th e definition rules based on

predicate logic. Emphasis is also put on the possibility to use logical

variables in constructing a linear objective function on intervals. Such

functions are encountered when costs or unitary receipts are different on

disjunct intervals of production volumes achieved or sold. Other uses of

Boolean variables are connected to constraint systems with conditions and

the case of a variable which takes values from a finite set of integers.

Key Words: linear programming, Boolean variable, economic model.

1 Faculty of Economic Sciences and Business Administration, Transilvania University of Braşov.

1. Introduction

A wide class of optimization models in

economics are solved by means of linear

programming. The linear programming

problem is comprised within the general

mathematical programming models and is

characterized by the fact that both the

objective function and the constraints are

expressed mathematically by linear

functions [2].

The general form of the linear

programming problem (LPP) in matrix

notation is:

Max [Min]

()

XCXf t⋅=

BXA ≤⋅ (1)

0≥

X

where:

()

nmA , – is the matrix of coefficients of

the constraint system

(

)

1,mB – is the column vector of free

terms

(

)

1,nX – is the column vector of the n

variables

(

)

nC t,1 – is the transposed column

vector (whose components

determine the unknown

coefficients of the objective

function).

In the general form of the LPP, it is

considered that variables are real numbers.

There are many economic applications of

great importance which lead to models

which also impose other conditions on

variables.

If the unknowns are Boolean variables,

i.e. the final solutions for the linear

programming problem is 0/1, a Boolean

linear programming problem is obtained.

There is also the possibility that only some

of the variables are Boolean.