The Linear Programming Problem

  • Manfred Padberg
Part of the Algorithms and Combinatorics book series (AC, volume 12)


The general linear programming problem has e.g. the form
$$\begin{array}{*{20}{c}} {\min imize\quad \sum\nolimits_{{j = 1}}^{n} {{{c}_{j}}{{x}_{j}}} } \\ {subject\,to\quad \sum\nolimits_{{j = 1}}^{n} {a_{j}^{i}{{x}_{j}} \leqslant {{b}_{i}}\quad for\,i = 1, \ldots ,p} } \\ {\sum\nolimits_{{j = 1}}^{n} {a_{j}^{i}{{x}_{j}} = {{b}_{i}}\quad for\,i = p + 1, \ldots ,m} } \\ {{{x}_{j}} \geqslant 0\quad \quad \quad \quad \;for\,j = 1,...,q} \\ \end{array}$$
where qn; that is, some variables (or “decision variables” or “activities”) x j for j = q+1,...,n may be unrestricted in sign (“free variables”). Inequalities of the type
$$\sum\limits_{j = 1}^n {a_j^i{x_j}} {b_i}$$


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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Manfred Padberg
    • 1
  1. 1.Leonard N. Stern School of Business, Statistics and Operations Research DepartmentNew York UniversityNew York CityUSA

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