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## 1.1 Linear programming problem

In MathProg it is assumed that the linear programming (LP) problem has the following statement:

Minimize (or maximize)

z=c1x1 +c2x2 + … +cn xn+c0subject to linear constraints

L1 <=a11x1 +a12x2 + … +a1nxn<=U1

L2 <=a21x1 +a22x2 + … +a2nxn<=U2

. . . . .

Lm<=am1x1 +am2x2 + … +amnxn<=Umand bounds of variables

l1 <=x1 <=u1

l2 <=x2 <=u2

. . . . .

ln<=xn<=un

where: | |

| are variables; |

| is the objective function; |

| are coefficients of the objective function; |

| is the constant term (“shift”) of the objective function; |

| are constraint coefficients; |

| are lower constraint bounds; |

| are upper constraint bounds; |

| are lower bounds of variables; |

| are upper bounds of variables. |

Bounds of variables and constraint bounds can be finite as well as infinite. Besides, lower bounds can be equal to corresponding upper bounds. Thus, the following types of variables and constraints are allowed:

-inf <

x< +infFree (unbounded) variable

x>=lVariable with lower bound

x<=uVariable with upper bound

l<=x<=uDouble-bounded variable

x=l(=u)Fixed variable

-inf < sum

ajxj< +infFree (unbounded) linear form

sum

ajxj>=LInequality constraint “greater than or equal to”

sum

ajxj<=UInequality constraint “less than or equal to”

L<= sumajxj<=UDouble-bounded inequality constraint

sum

ajxj=L(=U)Equality constraint

In addition to pure LP problems MathProg allows mixed integer linear programming (MIP) problems, where some (or all) structural variables are restricted to be integer.

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