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424 lines
13 KiB
424 lines
13 KiB
%* glpk09.tex *%
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\chapter{CPLEX LP Format}
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\label{chacplex}
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\section{Prelude}
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The CPLEX LP format\footnote{The CPLEX LP format was developed in
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the end of 1980's by CPLEX Optimization, Inc. as an input format for
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the CPLEX linear programming system. Although the CPLEX LP format is
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not as widely used as the MPS format, being row-oriented it is more
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convenient for coding mathematical programming models by human. This
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appendix describes only the features of the CPLEX LP format which are
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implemented in the GLPK package.} is intended for coding LP/MIP problem
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data. It is a row-oriented format that assumes the formulation of
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LP/MIP problem (1.1)---(1.3) (see Section \ref{seclp}, page
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\pageref{seclp}).
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CPLEX LP file is a plain text file written in CPLEX LP format. Each
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text line of this file may contain up to 255 characters\footnote{GLPK
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allows text lines of arbitrary length.}. Blank lines are ignored.
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If a line contains the backslash character ($\backslash$), this
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character and everything that follows it until the end of line are
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considered as a comment and also ignored.
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An LP file is coded by the user using the following elements: keywords,
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symbolic names, numeric constants, delimiters, and blanks.
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{\it Keywords} which may be used in the LP file are the following:
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\begin{verbatim}
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minimize minimum min
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maximize maximum max
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subject to such that s.t. st. st
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bounds bound
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general generals gen
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integer integers int
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binary binaries bin
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infinity inf
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free
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end
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\end{verbatim}
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\noindent
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All the keywords are case insensitive. Keywords given above on the same
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line are equivalent. Any keyword (except \verb|infinity|, \verb|inf|,
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and \verb|free|) being used in the LP file must start at the beginning
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of a text line.
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\newpage
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{\it Symbolic names} are used to identify the objective function,
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constraints (rows), and variables (columns). All symbolic names are case
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sensitive and may contain up to 16 alphanumeric characters\footnote{GLPK
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allows symbolic names having up to 255 characters.} (\verb|a|, \dots,
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\verb|z|, \verb|A|, \dots, \verb|Z|, \verb|0|, \dots, \verb|9|) as well
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as the following characters:
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\begin{verbatim}
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! " # $ % & ( ) / , . ; ? @ _ ` ' { } | ~
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\end{verbatim}
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\noindent
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with exception that no symbolic name can begin with a digit or
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a period.
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{\it Numeric constants} are used to denote constraint and objective
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coefficients, right-hand sides of constraints, and bounds of variables.
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They are coded in the standard form $xx$\verb|E|$syy$, where $xx$ is
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a real number with optional decimal point, $s$ is a sign (\verb|+| or
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\verb|-|), $yy$ is an integer decimal exponent. Numeric constants may
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contain arbitrary number of characters. The exponent part is optional.
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The letter `\verb|E|' can be coded as `\verb|e|'. If the sign $s$ is
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omitted, plus is assumed.
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{\it Delimiters} that may be used in the LP file are the following:
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\begin{verbatim}
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:
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+
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-
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< <= =<
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> >= =>
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=
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\end{verbatim}
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\noindent
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Delimiters given above on the same line are equivalent. The meaning of
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the delimiters will be explained below.
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{\it Blanks} are non-significant characters. They may be used freely to
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improve readability of the LP file. Besides, blanks should be used to
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separate elements from each other if there is no other way to do that
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(for example, to separate a keyword from a following symbolic name).
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The order of an LP file is:
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\vspace*{-8pt}
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\begin{itemize}
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\item objective function definition;
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\item constraints section;
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\item bounds section;
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\item general, integer, and binary sections (can appear in arbitrary
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order);
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\item end keyword.
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\end{itemize}
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\vspace*{-8pt}
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These components are discussed in following sections.
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\section{Objective function definition}
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The objective function definition must appear first in the LP file.
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It defines the objective function and specifies the optimization
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direction.
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The objective function definition has the following form:
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$$
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\left\{
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\begin{array}{@{}c@{}}
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{\tt minimize} \\ {\tt maximize}
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\end{array}
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\right\}\ f\ {\tt :}\ s\ c\ x\ s\ c\ x\ \dots\ s\ c\ x
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$$
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where $f$ is a symbolic name of the objective function, $s$ is a sign
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\verb|+| or \verb|-|, $c$ is a numeric constant that denotes an
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objective coefficient, $x$ is a symbolic name of a variable.
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If necessary, the objective function definition can be continued on as
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many text lines as desired.
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The name of the objective function is optional and may be omitted
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(together with the semicolon that follows it). In this case the default
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name `\verb|obj|' is assigned to the objective function.
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If the very first sign $s$ is omitted, the sign plus is assumed. Other
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signs cannot be omitted.
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If some objective coefficient $c$ is omitted, 1 is assumed.
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Symbolic names $x$ used to denote variables are recognized by context
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and therefore needn't to be declared somewhere else.
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Here is an example of the objective function definition:
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\begin{verbatim}
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Minimize Z : - x1 + 2 x2 - 3.5 x3 + 4.997e3x(4) + x5 + x6 +
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x7 - .01x8
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\end{verbatim}
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\section{Constraints section}
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The constraints section must follow the objective function definition.
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It defines a system of equality and/or inequality constraints.
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The constraint section has the following form:
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\begin{center}
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\begin{tabular}{l}
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\verb|subject to| \\
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{\it constraint}$_1$ \\
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{\it constraint}$_2$ \\
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\hspace{20pt}\dots \\
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{\it constraint}$_m$ \\
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\end{tabular}
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\end{center}
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\noindent where {\it constraint}$_i, i=1,\dots,m,$ is a particular
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constraint definition.
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Each constraint definition can be continued on as many text lines as
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desired. However, each constraint definition must begin on a new line
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except the very first constraint definition which can begin on the same
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line as the keyword `\verb|subject to|'.
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Constraint definitions have the following form:
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$$
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r\ {\tt:}\ s\ c\ x\ s\ c\ x\ \dots\ s\ c\ x
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\ \left\{
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\begin{array}{@{}c@{}}
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\mbox{\tt<=} \\ \mbox{\tt>=} \\ \mbox{\tt=}
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\end{array}
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\right\}\ b
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$$
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where $r$ is a symbolic name of a constraint, $s$ is a sign \verb|+| or
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\verb|-|, $c$ is a numeric constant that denotes a constraint
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coefficient, $x$ is a symbolic name of a variable, $b$ is a right-hand
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side.
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The name $r$ of a constraint (which is the name of the corresponding
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auxiliary variable) is optional and may be omitted (together with the
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semicolon that follows it). In this case the default names like
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`\verb|r.nnn|' are assigned to unnamed constraints.
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The linear form $s\ c\ x\ s\ c\ x\ \dots\ s\ c\ x$ in the left-hand
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side of a constraint definition has exactly the same meaning as in the
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case of the objective function definition (see above).
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After the linear form one of the following delimiters that indicates
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the constraint sense must be specified:
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\verb|<=| \ means `less than or equal to'
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\verb|>=| \ means `greater than or equal to'
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\verb|= | \ means `equal to'
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The right hand side $b$ is a numeric constant with an optional sign.
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Here is an example of the constraints section:
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\begin{verbatim}
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Subject To
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one: y1 + 3 a1 - a2 - b >= 1.5
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y2 + 2 a3 + 2
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a4 - b >= -1.5
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two : y4 + 3 a1 + 4 a5 - b <= +1
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.20y5 + 5 a2 - b = 0
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1.7 y6 - a6 + 5 a777 - b >= 1
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\end{verbatim}
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Should note that it is impossible to express ranged constraints in the
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CPLEX LP format. Each a ranged constraint can be coded as two
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constraints with identical linear forms in the left-hand side, one of
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which specifies a lower bound and other does an upper one of the
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original ranged constraint. Another way is to introduce a slack
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double-bounded variable; for example, the
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constraint
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$$10\leq x+2y+3z\leq 50$$
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can be written as follows:
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$$x+2y+3z+t=50,$$
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where $0\leq t\leq 40$ is a slack variable.
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\section{Bounds section}
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The bounds section is intended to define bounds of variables. This
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section is optional; if it is specified, it must follow the constraints
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section. If the bound section is omitted, all variables are assumed to
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be non-negative (i.e. that they have zero lower bound and no upper
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bound).
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The bounds section has the following form:
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\begin{center}
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\begin{tabular}{l}
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\verb|bounds| \\
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{\it definition}$_1$ \\
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{\it definition}$_2$ \\
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\hspace{20pt}\dots \\
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{\it definition}$_p$ \\
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\end{tabular}
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\end{center}
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\noindent
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where {\it definition}$_k, k=1,\dots,p,$ is a particular bound
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definition.
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Each bound definition must begin on a new line\footnote{The GLPK
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implementation allows several bound definitions to be placed on the
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same line.} except the very first bound definition which can begin on
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the same line as the keyword `\verb|bounds|'.
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\newpage
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Syntactically constraint definitions can have one of the following six
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forms:
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\begin{center}
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\begin{tabular}{ll}
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$x$ \verb|>=| $l$ & specifies a lower bound \\
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$l$ \verb|<=| $x$ & specifies a lower bound \\
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$x$ \verb|<=| $u$ & specifies an upper bound \\
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$l$ \verb|<=| $x$ \verb|<=| $u$ &specifies both lower and upper bounds\\
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$x$ \verb|=| $t$ &specifies a fixed value \\
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$x$ \verb|free| &specifies free variable
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\end{tabular}
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\end{center}
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\noindent
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where $x$ is a symbolic name of a variable, $l$ is a numeric constant
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with an optional sign that defines a lower bound of the variable or
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\verb|-inf| that means that the variable has no lower bound, $u$ is a
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numeric constant with an optional sign that defines an upper bound of
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the variable or \verb|+inf| that means that the variable has no upper
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bound, $t$ is a numeric constant with an optional sign that defines a
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fixed value of the variable.
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By default all variables are non-negative, i.e. have zero lower bound
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and no upper bound. Therefore definitions of these default bounds can
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be omitted in the bounds section.
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Here is an example of the bounds section:
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\begin{verbatim}
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Bounds
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-inf <= a1 <= 100
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-100 <= a2
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b <= 100
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x2 = +123.456
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x3 free
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\end{verbatim}
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\section{General, integer, and binary sections}
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The general, integer, and binary sections are intended to define
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some variables as integer or binary. All these sections are optional
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and needed only in case of MIP problems. If they are specified, they
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must follow the bounds section or, if the latter is omitted, the
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constraints section.
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All the general, integer, and binary sections have the same form as
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follows:
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\begin{center}
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\begin{tabular}{l}
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$
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\left\{
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\begin{array}{@{}l@{}}
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\verb|general| \\
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\verb|integer| \\
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\verb|binary | \\
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\end{array}
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\right\}
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$ \\
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\hspace{10pt}$x_1$ \\
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\hspace{10pt}$x_2$ \\
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\hspace{10pt}\dots \\
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\hspace{10pt}$x_q$ \\
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\end{tabular}
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\end{center}
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\noindent
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where $x_k$ is a symbolic name of variable, $k=1,\dots,q$.
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Each symbolic name must begin on a new line\footnote{The GLPK
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implementation allows several symbolic names to be placed on the same
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line.} except the very first symbolic name which can begin on the same
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line as the keyword `\verb|general|', `\verb|integer|', or
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`\verb|binary|'.
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\newpage
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If a variable appears in the general or the integer section, it is
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assumed to be general integer variable. If a variable appears in the
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binary section, it is assumed to be binary variable, i.e. an integer
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variable whose lower bound is zero and upper bound is one. (Note that
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if bounds of a variable are specified in the bounds section and then
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the variable appears in the binary section, its previously specified
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bounds are ignored.)
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Here is an example of the integer section:
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\begin{verbatim}
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Integer
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z12
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z22
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z35
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\end{verbatim}
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\section{End keyword}
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The keyword `\verb|end|' is intended to end the LP file. It must begin
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on a separate line and no other elements (except comments and blank
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lines) must follow it. Although this keyword is optional, it is strongly
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recommended to include it in the LP file.
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\section{Example of CPLEX LP file}
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Here is a complete example of CPLEX LP file that corresponds to the
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example given in Section \ref{secmpsex}, page \pageref{secmpsex}.
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\medskip
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\begin{footnotesize}
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\begin{verbatim}
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\* plan.lp *\
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Minimize
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value: .03 bin1 + .08 bin2 + .17 bin3 + .12 bin4 + .15 bin5 +
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.21 alum + .38 silicon
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Subject To
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yield: bin1 + bin2 + bin3 + bin4 + bin5 +
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alum + silicon = 2000
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fe: .15 bin1 + .04 bin2 + .02 bin3 + .04 bin4 + .02 bin5 +
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.01 alum + .03 silicon <= 60
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cu: .03 bin1 + .05 bin2 + .08 bin3 + .02 bin4 + .06 bin5 +
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.01 alum <= 100
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mn: .02 bin1 + .04 bin2 + .01 bin3 + .02 bin4 + .02 bin5 <= 40
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mg: .02 bin1 + .03 bin2 + .01 bin5 <= 30
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al: .70 bin1 + .75 bin2 + .80 bin3 + .75 bin4 + .80 bin5 +
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.97 alum >= 1500
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si1: .02 bin1 + .06 bin2 + .08 bin3 + .12 bin4 + .02 bin5 +
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.01 alum + .97 silicon >= 250
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si2: .02 bin1 + .06 bin2 + .08 bin3 + .12 bin4 + .02 bin5 +
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.01 alum + .97 silicon <= 300
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Bounds
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bin1 <= 200
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bin2 <= 2500
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400 <= bin3 <= 800
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100 <= bin4 <= 700
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bin5 <= 1500
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End
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\* eof *\
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\end{verbatim}
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\end{footnotesize}
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%* eof *%
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