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740 lines
26 KiB
740 lines
26 KiB
%* glpk08.tex *%
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\chapter{MPS Format}
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\label{champs}
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\section{Fixed MPS Format}
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\label{secmps}
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The MPS format\footnote{The MPS format was developed in 1960's by IBM
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as input format for their mathematical programming system MPS/360.
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Today the MPS format is a most widely used format understood by most
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mathematical programming packages. This appendix describes only the
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features of the MPS format, which are implemented in the GLPK package.}
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is intended for coding LP/MIP problem data. This format assumes the
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formulation of LP/MIP problem (1.1)---(1.3) (see Section \ref{seclp},
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page \pageref{seclp}).
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{\it MPS file} is a text file, which contains two types of
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cards\footnote{In 1960's MPS file was a deck of 80-column punched
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cards, so the author decided to keep the word ``card'', which may be
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understood as ``line of text file''.}: indicator cards and data cards.
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Indicator cards determine a kind of succeeding data. Each indicator
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card has one word in uppercase letters beginning in column 1.
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Data cards contain problem data. Each data card is divided into six
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fixed fields:
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\begin{center}
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\begin{tabular}{lcccccc}
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& Field 1 & Field 2 & Field 3 & Field 4 & Field 5 & Field 6 \\
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\hline
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Columns & 2---3 & 5---12 & 15---22 & 25---36 & 40---47 & 50---61 \\
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Contents & Code & Name & Name & Number & Name & Number \\
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\end{tabular}
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\end{center}
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On a particular data card some fields may be optional.
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Names are used to identify rows, columns, and some vectors (see below).
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Aligning the indicator code in the field 1 to the left margin is
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optional.
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All names specified in the fields 2, 3, and 5 should contain from 1 up
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to 8 arbitrary characters (except control characters). If a name is
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placed in the field 3 or 5, its first character should not be the dollar
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sign `\verb|$|'. If a name contains spaces, the spaces are ignored.
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All numerical values in the fields 4 and 6 should be coded in the form
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$sxx$\verb|E|$syy$, where $s$ is the plus `\verb|+|' or the minus
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`\verb|-|' sign, $xx$ is a real number with optional decimal point,
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$yy$ is an integer decimal exponent. Any number should contain up to 12
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characters. If the sign $s$ is omitted, the plus sign is assumed. The
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exponent part is optional. If a number contains spaces, the spaces are
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ignored.
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%\newpage
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If a card has the asterisk `\verb|*|' in the column 1, this card is
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considered as a comment and ignored. Besides, if the first character in
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the field 3 or 5 is the dollar sign `\verb|$|', all characters from the
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dollar sign to the end of card are considered as a comment and ignored.
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MPS file should contain cards in the following order:
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%\vspace*{-8pt}
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%\begin{itemize}
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\Item{---}NAME indicator card;
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\Item{---}ROWS indicator card;
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\Item{---}data cards specifying rows (constraints);
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\Item{---}COLUMNS indicator card;
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\Item{---}data cards specifying columns (structural variables) and
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constraint coefficients;
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\Item{---}RHS indicator card;
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\Item{---}data cards specifying right-hand sides of constraints;
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\Item{---}RANGES indicator card;
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\Item{---}data cards specifying ranges for double-bounded constraints;
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\Item{---}BOUNDS indicator card;
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\Item{---}data cards specifying types and bounds of structural
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variables;
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\Item{---}ENDATA indicator card.
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%\end{itemize}
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%\vspace*{-8pt}
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{\it Section} is a group of cards consisting of an indicator card and
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data cards succeeding this indicator card. For example, the ROWS section
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consists of the ROWS indicator card and data cards specifying rows.
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The sections RHS, RANGES, and BOUNDS are optional and may be omitted.
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\section{Free MPS Format}
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{\it Free MPS format} is an improved version of the standard (fixed)
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MPS format described above.\footnote{This format was developed in the
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beginning of 1990's by IBM as an alternative to the standard fixed MPS
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format for Optimization Subroutine Library (OSL).} Note that all
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changes in free MPS format concern only the coding of data while the
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structure of data is the same for both fixed and free versions of the
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MPS format.
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In free MPS format indicator and data records\footnote{{\it Record} in
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free MPS format has the same meaning as {\it card} in fixed MPS format.}
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may have arbitrary length not limited to 80 characters. Fields of data
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records have no predefined positions, i.e. the fields may begin in any
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position, except position 1, which must be blank, and must be separated
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from each other by one or more blanks. However, the fields must appear
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in the same order as in fixed MPS format.
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%\newpage
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Symbolic names in fields 2, 3, and 5 may be longer than 8
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characters\footnote{GLPK allows symbolic names having up to 255
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characters.} and must not contain embedded blanks.
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Numeric values in fields 4 and 6 are limited to 12 characters and must
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not contain embedded blanks.
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Only six fields on each data record are used. Any other fields are
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ignored.
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If the first character of any field (not necessarily fields 3 and 5)
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is the dollar sign (\$), all characters from the dollar sign to the end
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of record are considered as a comment and ignored.
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\newpage
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\section{NAME indicator card}
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The NAME indicator card should be the first card in the MPS file
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(except optional comment cards, which may precede the NAME card). This
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card should contain the word \verb|NAME| in the columns 1---4 and the
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problem name in the field 3. The problem name is optional and may be
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omitted.
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\section{ROWS section}
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\label{secrows}
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The ROWS section should start with the indicator card, which contains
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the word \verb|ROWS| in the columns 1---4.
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Each data card in the ROWS section specifies one row (constraint) of
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the problem. All these data cards have the following format.
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`\verb|N|' in the field 1 means that the row is free (unbounded):
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$$-\infty < x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n}
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< +\infty;$$
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`\verb|L|' in the field 1 means that the row is of ``less than or equal
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to'' type:
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$$-\infty < x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n}
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\leq b_i;$$
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`\verb|G|' in the field 1 means that the row is of ``greater than or
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equal to'' type:
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$$b_i \leq x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n}
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< +\infty;$$
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`\verb|E|' in the field 1 means that the row is of ``equal to'' type:
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$$x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} \leq
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b_i,$$
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where $b_i$ is a right-hand side. Note that each constraint has a
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corresponding implictly defined auxiliary variable ($x_i$ above), whose
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value is a value of the corresponding linear form, therefore row bounds
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can be considered as bounds of such auxiliary variable.
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The filed 2 specifies a row name (which is considered as the name of
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the corresponding auxiliary variable).
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%\newpage
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The fields 3, 4, 5, and 6 are not used and should be empty.
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Numerical values of all non-zero right-hand sides $b_i$ should be
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specified in the RHS section (see below). All double-bounded (ranged)
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constraints should be specified in the RANGES section (see below).
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\section{COLUMNS section}
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The COLUMNS section should start with the indicator card, which
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contains the word \verb|COLUMNS| in the columns 1---7.
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Each data card in the COLUMNS section specifies one or two constraint
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coefficients $a_{ij}$ and also introduces names of columns, i.e. names
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of structural variables. All these data cards have the following
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format.
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The field 1 is not used and should be empty.
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The field 2 specifies a column name. If this field is empty, the column
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name from the immediately preceeding data card is assumed.
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The field 3 specifies a row name defined in the ROWS section.
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The field 4 specifies a numerical value of the constraint coefficient
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$a_{ij}$, which is placed in the corresponding row and column.
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The fields 5 and 6 are optional. If they are used, they should contain
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a second pair ``row name---constraint coefficient'' for the same column.
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Elements of the constraint matrix (i.e. constraint coefficients) should
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be enumerated in the column wise manner: all elements for the current
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column should be specified before elements for the next column. However,
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the order of rows in the COLUMNS section may differ from the order of
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rows in the ROWS section.
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Constraint coefficients not specified in the COLUMNS section are
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considered as zeros. Therefore zero coefficients may be omitted,
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although it is allowed to explicitly specify them.
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\section{RHS section}
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The RHS section should start with the indicator card, which contains the
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word \verb|RHS| in the columns 1---3.
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Each data card in the RHS section specifies one or two right-hand sides
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$b_i$ (see Section \ref{secrows}, page \pageref{secrows}). All these
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data cards have the following format.
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The field 1 is not used and should be empty.
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The field 2 specifies a name of the right-hand side (RHS)
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vector\footnote{This feature allows the user to specify several RHS
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vectors in the same MPS file. However, before solving the problem a
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particular RHS vector should be chosen.}. If this field is empty, the
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RHS vector name from the immediately preceeding data card is assumed.
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%\newpage
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The field 3 specifies a row name defined in the ROWS section.
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The field 4 specifies a right-hand side $b_i$ for the row, whose name is
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specified in the field 3. Depending on the row type $b_i$ is a lower
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bound (for the row of \verb|G| type), an upper bound (for the row of
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\verb|L| type), or a fixed value (for the row of \verb|E|
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type).\footnote{If the row is of {\tt N} type, $b_i$ is considered as
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a constant term of the corresponding linear form. Should note, however,
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this convention is non-standard.}
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The fields 5 and 6 are optional. If they are used, they should contain
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a second pair ``row name---right-hand side'' for the same RHS vector.
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All right-hand sides for the current RHS vector should be specified
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before right-hand sides for the next RHS vector. However, the order of
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rows in the RHS section may differ from the order of rows in the ROWS
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section.
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Right-hand sides not specified in the RHS section are considered as
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zeros. Therefore zero right-hand sides may be omitted, although it is
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allowed to explicitly specify them.
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\newpage
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\section{RANGES section}
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The RANGES section should start with the indicator card, which contains
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the word \verb|RANGES| in the columns 1---6.
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Each data card in the RANGES section specifies one or two ranges for
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double-side constraints, i.e. for constraints that are of the types
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\verb|L| and \verb|G| at the same time:
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$$l_i \leq x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n}
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\leq u_i,$$
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where $l_i$ is a lower bound, $u_i$ is an upper bound. All these data
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cards have the following format.
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The field 1 is not used and should be empty.
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The field 2 specifies a name of the range vector\footnote{This feature
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allows the user to specify several range vectors in the same MPS file.
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However, before solving the problem a particular range vector should be
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chosen.}. If this field is empty, the range vector name from the
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immediately preceeding data card is assumed.
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The field 3 specifies a row name defined in the ROWS section.
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The field 4 specifies a range value $r_i$ (see the table below) for the
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row, whose name is specified in the field 3.
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The fields 5 and 6 are optional. If they are used, they should contain
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a second pair ``row name---range value'' for the same range vector.
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All range values for the current range vector should be specified before
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range values for the next range vector. However, the order of rows in
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the RANGES section may differ from the order of rows in the ROWS
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section.
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For each double-side constraint specified in the RANGES section its
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lower and upper bounds are determined as follows:
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%\newpage
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\begin{center}
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\begin{tabular}{cccc}
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Row type & Sign of $r_i$ & Lower bound & Upper bound \\
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\hline
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{\tt G} & $+$ or $-$ & $b_i$ & $b_i + |r_i|$ \\
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{\tt L} & $+$ or $-$ & $b_i - |r_i|$ & $b_i$ \\
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{\tt E} & $+$ & $b_i$ & $b_i + |r_i|$ \\
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{\tt E} & $-$ & $b_i - |r_i|$ & $b_i$ \\
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\end{tabular}
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\end{center}
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\noindent
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where $b_i$ is a right-hand side specified in the RHS section (if $b_i$
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is not specified, it is considered as zero), $r_i$ is a range value
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specified in the RANGES section.
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\section{BOUNDS section}
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\label{secbounds}
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The BOUNDS section should start with the indicator card, which contains
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the word \verb|BOUNDS| in the columns 1---6.
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Each data card in the BOUNDS section specifies one (lower or upper)
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bound for one structural variable (column). All these data cards have
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the following format.
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The indicator in the field 1 specifies the bound type:
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\verb|LO| --- lower bound;
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\verb|UP| --- upper bound;
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\verb|FX| --- fixed variable (lower and upper bounds are equal);
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\verb|FR| --- free variable (no bounds);
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\verb|MI| --- no lower bound (lower bound is ``minus infinity'');
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\verb|PL| --- no upper bound (upper bound is ``plus infinity'').
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The field 2 specifies a name of the bound vector\footnote{This feature
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allows the user to specify several bound vectors in the same MPS file.
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However, before solving the problem a particular bound vector should be
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chosen.}. If this field is empty, the bound vector name from the
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immediately preceeding data card is assumed.
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The field 3 specifies a column name defined in the COLUMNS section.
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The field 4 specifies a bound value. If the bound type in the field 1
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differs from \verb|LO|, \verb|UP|, and \verb|FX|, the value in the field
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4 is ignored and may be omitted.
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The fields 5 and 6 are not used and should be empty.
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All bound values for the current bound vector should be specified before
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bound values for the next bound vector. However, the order of columns in
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the BOUNDS section may differ from the order of columns in the COLUMNS
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section. Specification of a lower bound should precede specification of
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an upper bound for the same column (if both the lower and upper bounds
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are explicitly specified).
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By default, all columns (structural variables) are non-negative, i.e.
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have zero lower bound and no upper bound. Lower ($l_j$) and upper
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($u_j$) bounds of some column (structural variable $x_j$) are set in the
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following way, where $s_j$ is a corresponding bound value explicitly
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specified in the BOUNDS section:
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%\newpage
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\verb|LO| sets $l_j$ to $s_j$;
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\verb|UP| sets $u_j$ to $s_j$;
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\verb|FX| sets both $l_j$ and $u_j$ to $s_j$;
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\verb|FR| sets $l_j$ to $-\infty$ and $u_j$ to $+\infty$;
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\verb|MI| sets $l_j$ to $-\infty$;
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\verb|PL| sets $u_j$ to $+\infty$.
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\section{ENDATA indicator card}
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The ENDATA indicator card should be the last card of MPS file (except
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optional comment cards, which may follow the ENDATA card). This card
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should contain the word \verb|ENDATA| in the columns 1---6.
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\section{Specifying objective function}
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It is impossible to explicitly specify the objective function and
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optimization direction in the MPS file. However, the following implicit
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rule is used by default: the first row of \verb|N| type is considered
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as a row of the objective function (i.e. the objective function is the
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corresponding auxiliary variable), which should be {\it minimized}.
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GLPK also allows specifying a constant term of the objective function
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as a right-hand side of the corresponding row in the RHS section.
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\section{Example of MPS file}
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\label{secmpsex}
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To illustrate what the MPS format is, consider the following example of
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LP problem:
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\def\arraystretch{1.2}
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\noindent\hspace{.5in}minimize
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$$
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value = .03\ bin_1 + .08\ bin_2 + .17\ bin_3 + .12\ bin_4 + .15\ bin_5
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+ .21\ al + .38\ si
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$$
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\noindent\hspace{.5in}subject to linear constraints
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$$
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\begin{array}{@{}l@{\:}l@{}}
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yield &= \ \ \ \ \;bin_1 + \ \ \ \ \;bin_2 + \ \ \ \ \;bin_3 +
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\ \ \ \ \;bin_4 + \ \ \ \ \;bin_5 + \ \ \ \ \;al +
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\ \ \ \ \;si \\
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FE &= .15\ bin_1 + .04\ bin_2 + .02\ bin_3 + .04\ bin_4 + .02\ bin_5
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+ .01\ al + .03\ si \\
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CU &= .03\ bin_1 + .05\ bin_2 + .08\ bin_3 + .02\ bin_4 + .06\ bin_5
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+ .01\ al \\
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MN &= .02\ bin_1 + .04\ bin_2 + .01\ bin_3 + .02\ bin_4 + .02\ bin_5
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\\
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MG &= .02\ bin_1 + .03\ bin_2
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + .01\ bin_5 \\
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AL &= .70\ bin_1 + .75\ bin_2 + .80\ bin_3 + .75\ bin_4 + .80\ bin_5
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+ .97\ al \\
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SI &= .02\ bin_1 + .06\ bin_2 + .08\ bin_3 + .12\ bin_4 + .02\ bin_5
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+ .01\ al + .97\ si \\
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\end{array}
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$$
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\noindent\hspace{.5in}and bounds of (auxiliary and structural)
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variables
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$$
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\begin{array}{r@{\ }l@{\ }l@{\ }l@{\ }rcr@{\ }l@{\ }l@{\ }l@{\ }r}
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&&yield&=&2000&&0&\leq&bin_1&\leq&200\\
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-\infty&<&FE&\leq&60&&0&\leq&bin_2&\leq&2500\\
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-\infty&<&CU&\leq&100&&400&\leq&bin_3&\leq&800\\
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-\infty&<&MN&\leq&40&&100&\leq&bin_4&\leq&700\\
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-\infty&<&MG&\leq&30&&0&\leq&bin_5&\leq&1500\\
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1500&\leq&AL&<&+\infty&&0&\leq&al&<&+\infty\\
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250&\leq&SI&\leq&300&&0&\leq&si&<&+\infty\\
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\end{array}
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$$
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\def\arraystretch{1}
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A complete MPS file which specifies data for this example is shown
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below (the first two comment lines show card positions).
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\newpage
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\begin{footnotesize}
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\begin{verbatim}
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*000000001111111111222222222233333333334444444444555555555566
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*234567890123456789012345678901234567890123456789012345678901
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NAME PLAN
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ROWS
|
|
N VALUE
|
|
E YIELD
|
|
L FE
|
|
L CU
|
|
L MN
|
|
L MG
|
|
G AL
|
|
L SI
|
|
COLUMNS
|
|
BIN1 VALUE .03000 YIELD 1.00000
|
|
FE .15000 CU .03000
|
|
MN .02000 MG .02000
|
|
AL .70000 SI .02000
|
|
BIN2 VALUE .08000 YIELD 1.00000
|
|
FE .04000 CU .05000
|
|
MN .04000 MG .03000
|
|
AL .75000 SI .06000
|
|
BIN3 VALUE .17000 YIELD 1.00000
|
|
FE .02000 CU .08000
|
|
MN .01000 AL .80000
|
|
SI .08000
|
|
BIN4 VALUE .12000 YIELD 1.00000
|
|
FE .04000 CU .02000
|
|
MN .02000 AL .75000
|
|
SI .12000
|
|
BIN5 VALUE .15000 YIELD 1.00000
|
|
FE .02000 CU .06000
|
|
MN .02000 MG .01000
|
|
AL .80000 SI .02000
|
|
ALUM VALUE .21000 YIELD 1.00000
|
|
FE .01000 CU .01000
|
|
AL .97000 SI .01000
|
|
SILICON VALUE .38000 YIELD 1.00000
|
|
FE .03000 SI .97000
|
|
RHS
|
|
RHS1 YIELD 2000.00000 FE 60.00000
|
|
CU 100.00000 MN 40.00000
|
|
SI 300.00000
|
|
MG 30.00000 AL 1500.00000
|
|
RANGES
|
|
RNG1 SI 50.00000
|
|
BOUNDS
|
|
UP BND1 BIN1 200.00000
|
|
UP BIN2 2500.00000
|
|
LO BIN3 400.00000
|
|
UP BIN3 800.00000
|
|
LO BIN4 100.00000
|
|
UP BIN4 700.00000
|
|
UP BIN5 1500.00000
|
|
ENDATA
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
%\vspace*{-6pt}
|
|
|
|
\section{MIP features}
|
|
|
|
%\vspace*{-4pt}
|
|
|
|
The MPS format provides two ways for introducing integer variables into
|
|
the problem.
|
|
|
|
The first way is most general and based on using special marker cards
|
|
INTORG and INTEND. These marker cards are placed in the COLUMNS section.
|
|
The INTORG card indicates the start of a group of integer variables
|
|
(columns), and the card INTEND indicates the end of the group. The MPS
|
|
file may contain arbitrary number of the marker cards.
|
|
|
|
The marker cards have the same format as the data cards (see Section
|
|
\ref{secmps}, page \pageref{secmps}).
|
|
|
|
The fields 1, 2, and 6 are not used and should be empty.
|
|
|
|
The field 2 should contain a marker name. This name may be arbitrary.
|
|
|
|
The field 3 should contain the word \verb|'MARKER'| (including
|
|
apostrophes).
|
|
|
|
The field 5 should contain either the word \verb|'INTORG'| (including
|
|
apostrophes) for the marker card, which begins a group of integer
|
|
columns, or the word \verb|'INTEND'| (including apostrophes) for the
|
|
marker card, which ends the group.
|
|
|
|
The second way is less general but more convenient in some cases. It
|
|
allows the user declaring integer columns using three additional types
|
|
of bounds, which are specified in the field 1 of data cards in the
|
|
BOUNDS section (see Section \ref{secbounds}, page \pageref{secbounds}):
|
|
|
|
\verb|LI| --- lower integer. This bound type specifies that the
|
|
corresponding column (structural variable), whose name is specified in
|
|
field 3, is of integer kind. In this case an lower bound of the
|
|
column should be specified in field 4 (like in the case of \verb|LO|
|
|
bound type).
|
|
|
|
\verb|UI| --- upper integer. This bound type specifies that the
|
|
corresponding column (structural variable), whose name is specified in
|
|
field 3, is of integer kind. In this case an upper bound of the
|
|
column should be specified in field 4 (like in the case of \verb|UP|
|
|
bound type).
|
|
|
|
\verb|BV| --- binary variable. This bound type specifies that the
|
|
corresponding column (structural variable), whose name is specified in
|
|
the field 3, is of integer kind, its lower bound is zero, and its upper
|
|
bound is one (thus, such variable being of integer kind can have only
|
|
two values zero and one). In this case a numeric value specified in the
|
|
field 4 is ignored and may be omitted.
|
|
|
|
Consider the following example of MIP problem:
|
|
|
|
\noindent
|
|
\hspace{1in} minimize
|
|
$$Z = 3 x_1 + 7 x_2 - x_3 + x4$$
|
|
\hspace{1in} subject to linear constraints
|
|
$$
|
|
\begin{array}{c}
|
|
\nonumber r_1 = 2 x_1 - \ \ x_2 + \ \ x_3 - \ \;x_4 \\
|
|
\nonumber r_2 = \ \;x_1 - \ \;x_2 - 6 x_3 + 4 x_4 \\
|
|
\nonumber r_3 = 5 x_1 + 3 x_2 \ \ \ \ \ \ \ \ \ + \ \ x_4 \\
|
|
\end{array}
|
|
$$
|
|
\hspace{1in} and bound of variables
|
|
$$
|
|
\begin{array}{cccl}
|
|
\nonumber 1 \leq r_1 < +\infty && 0 \leq x_1 \leq 4 &{\rm(continuous)}\\
|
|
\nonumber 8 \leq r_2 < +\infty && 2 \leq x_2 \leq 5 &{\rm(integer)} \\
|
|
\nonumber 5 \leq r_3 < +\infty && 0 \leq x_3 \leq 1 &{\rm(integer)} \\
|
|
\nonumber && 3 \leq x_4 \leq 8 &{\rm(continuous)}\\
|
|
\end{array}
|
|
$$
|
|
|
|
The corresponding MPS file may look like follows:
|
|
|
|
\newpage
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
NAME SAMP1
|
|
ROWS
|
|
N Z
|
|
G R1
|
|
G R2
|
|
G R3
|
|
COLUMNS
|
|
X1 R1 2.0 R2 1.0
|
|
X1 R3 5.0 Z 3.0
|
|
MARK0001 'MARKER' 'INTORG'
|
|
X2 R1 -1.0 R2 -1.0
|
|
X2 R3 3.0 Z 7.0
|
|
X3 R1 1.0 R2 -6.0
|
|
X3 Z -1.0
|
|
MARK0002 'MARKER' 'INTEND'
|
|
X4 R1 -1.0 R2 4.0
|
|
X4 R3 1.0 Z 1.0
|
|
RHS
|
|
RHS1 R1 1.0
|
|
RHS1 R2 8.0
|
|
RHS1 R3 5.0
|
|
BOUNDS
|
|
UP BND1 X1 4.0
|
|
LO BND1 X2 2.0
|
|
UP BND1 X2 5.0
|
|
UP BND1 X3 1.0
|
|
LO BND1 X4 3.0
|
|
UP BND1 X4 8.0
|
|
ENDATA
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
%\newpage
|
|
\vspace{-3pt}
|
|
|
|
The same example may be coded without INTORG/INTEND markers using the
|
|
bound type UI for the variable $x_2$ and the bound type BV for the
|
|
variable $x_3$:
|
|
|
|
%\medskip
|
|
|
|
\begin{footnotesize}
|
|
\begin{verbatim}
|
|
NAME SAMP2
|
|
ROWS
|
|
N Z
|
|
G R1
|
|
G R2
|
|
G R3
|
|
COLUMNS
|
|
X1 R1 2.0 R2 1.0
|
|
X1 R3 5.0 Z 3.0
|
|
X2 R1 -1.0 R2 -1.0
|
|
X2 R3 3.0 Z 7.0
|
|
X3 R1 1.0 R2 -6.0
|
|
X3 Z -1.0
|
|
X4 R1 -1.0 R2 4.0
|
|
X4 R3 1.0 Z 1.0
|
|
RHS
|
|
RHS1 R1 1.0
|
|
RHS1 R2 8.0
|
|
RHS1 R3 5.0
|
|
BOUNDS
|
|
UP BND1 X1 4.0
|
|
LO BND1 X2 2.0
|
|
UI BND1 X2 5.0
|
|
BV BND1 X3
|
|
LO BND1 X4 3.0
|
|
UP BND1 X4 8.0
|
|
ENDATA
|
|
\end{verbatim}
|
|
\end{footnotesize}
|
|
|
|
%\section{Specifying predefined basis}
|
|
%\label{secbas}
|
|
%
|
|
%The MPS format can also be used to specify some predefined basis for an
|
|
%LP problem, i.e. to specify which rows and columns are basic and which
|
|
%are non-basic.
|
|
%
|
|
%The order of a basis file in the MPS format is:
|
|
%
|
|
%$\bullet$ NAME indicator card;
|
|
%
|
|
%$\bullet$ data cards (can appear in arbitrary order);
|
|
%
|
|
%$\bullet$ ENDATA indicator card.
|
|
%
|
|
%Each data card specifies either a pair "basic column---non-basic row"
|
|
%or a non-basic column. All the data cards have the following format.
|
|
%
|
|
%`\verb|XL|' in the field 1 means that a column, whose name is given in
|
|
%the field 2, is basic, and a row, whose name is given in the field 3,
|
|
%is non-basic and placed on its lower bound.
|
|
%
|
|
%`\verb|XU|' in the field 1 means that a column, whose name is given in
|
|
%the field 2, is basic, and a row, whose name is given in the field 3,
|
|
%is non-basic and placed on its upper bound.
|
|
%
|
|
%`\verb|LL|' in the field 1 means that a column, whose name is given in
|
|
%the field 3, is non-basic and placed on its lower bound.
|
|
%
|
|
%`\verb|UL|' in the field 1 means that a column, whose name is given in
|
|
%the field 3, is non-basic and placed on its upper bound.
|
|
%
|
|
%The field 2 contains a column name.
|
|
%
|
|
%If the indicator given in the field 1 is `\verb|XL|' or `\verb|XU|',
|
|
%the field 3 contains a row name. Otherwise, if the indicator is
|
|
%`\verb|LL|' or `\verb|UL|', the field 3 is not used and should be
|
|
%empty.
|
|
%
|
|
%The field 4, 5, and 6 are not used and should be empty.
|
|
%
|
|
%A basis file in the MPS format acts like a patch: it doesn't specify
|
|
%a basis completely, instead that it is just shows in what a given basis
|
|
%differs from the "standard" basis, where all rows (auxiliary variables)
|
|
%are assumed to be basic and all columns (structural variables) are
|
|
%assumed to be non-basic.
|
|
%
|
|
%As an example here is a basis file that specifies an optimal basis
|
|
%for the example LP problem given in Section \ref{secmpsex},
|
|
%Page \pageref{secmpsex}:
|
|
%
|
|
%\pagebreak
|
|
%
|
|
%\begin{verbatim}
|
|
%*000000001111111111222222222233333333334444444444555555555566
|
|
%*234567890123456789012345678901234567890123456789012345678901
|
|
%NAME PLAN
|
|
% XL BIN2 YIELD
|
|
% XL BIN3 FE
|
|
% XL BIN4 MN
|
|
% XL ALUM AL
|
|
% XL SILICON SI
|
|
% LL BIN1
|
|
% LL BIN5
|
|
%ENDATA
|
|
%\end{verbatim}
|
|
|
|
%* eof *%
|