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     1.2 +++ b/deps/glpk/doc/glpk03.tex	Sun Nov 06 20:59:10 2011 +0100
     1.3 @@ -0,0 +1,1577 @@
     1.4 +%* glpk03.tex *%
     1.5 +
     1.6 +\chapter{Utility API routines}
     1.7 +
     1.8 +\section{Problem data reading/writing routines}
     1.9 +
    1.10 +\subsection{glp\_read\_mps---read problem data in MPS format}
    1.11 +
    1.12 +\subsubsection*{Synopsis}
    1.13 +
    1.14 +\begin{verbatim}
    1.15 +int glp_read_mps(glp_prob *lp, int fmt, const void *parm,
    1.16 +      const char *fname);
    1.17 +\end{verbatim}
    1.18 +
    1.19 +\subsubsection*{Description}
    1.20 +
    1.21 +The routine \verb|glp_read_mps| reads problem data in MPS format from a
    1.22 +text file. (The MPS format is described in Appendix \ref{champs}, page
    1.23 +\pageref{champs}.)
    1.24 +
    1.25 +The parameter \verb|fmt| specifies the MPS format version as follows:
    1.26 +
    1.27 +\begin{tabular}{@{}ll}
    1.28 +\verb|GLP_MPS_DECK| & fixed (ancient) MPS format; \\
    1.29 +\verb|GLP_MPS_FILE| & free (modern) MPS format. \\
    1.30 +\end{tabular}
    1.31 +
    1.32 +The parameter \verb|parm| is reserved for use in the future and must be
    1.33 +specified as \verb|NULL|.
    1.34 +
    1.35 +The character string \verb|fname| specifies a name of the text file to
    1.36 +be read in. (If the file name ends with suffix `\verb|.gz|', the file is
    1.37 +assumed to be compressed, in which case the routine \verb|glp_read_mps|
    1.38 +decompresses it ``on the fly''.)
    1.39 +
    1.40 +Note that before reading data the current content of the problem object
    1.41 +is completely erased with the routine \verb|glp_erase_prob|.
    1.42 +
    1.43 +\subsubsection*{Returns}
    1.44 +
    1.45 +If the operation was successful, the routine \verb|glp_read_mps|
    1.46 +returns zero. Otherwise, it prints an error message and returns
    1.47 +non-zero.
    1.48 +
    1.49 +\subsection{glp\_write\_mps---write problem data in MPS format}
    1.50 +
    1.51 +\subsubsection*{Synopsis}
    1.52 +
    1.53 +\begin{verbatim}
    1.54 +int glp_write_mps(glp_prob *lp, int fmt, const void *parm,
    1.55 +      const char *fname);
    1.56 +\end{verbatim}
    1.57 +
    1.58 +\subsubsection*{Description}
    1.59 +
    1.60 +The routine \verb|glp_write_mps| writes problem data in MPS format to a
    1.61 +text file. (The MPS format is described in Appendix \ref{champs}, page
    1.62 +\pageref{champs}.)
    1.63 +
    1.64 +The parameter \verb|fmt| specifies the MPS format version as follows:
    1.65 +
    1.66 +\begin{tabular}{@{}ll}
    1.67 +\verb|GLP_MPS_DECK| & fixed (ancient) MPS format; \\
    1.68 +\verb|GLP_MPS_FILE| & free (modern) MPS format. \\
    1.69 +\end{tabular}
    1.70 +
    1.71 +The parameter \verb|parm| is reserved for use in the future and must be
    1.72 +specified as \verb|NULL|.
    1.73 +
    1.74 +The character string \verb|fname| specifies a name of the text file to
    1.75 +be written out. (If the file name ends with suffix `\verb|.gz|', the
    1.76 +file is assumed to be compressed, in which case the routine
    1.77 +\verb|glp_write_mps| performs automatic compression on writing it.)
    1.78 +
    1.79 +\subsubsection*{Returns}
    1.80 +
    1.81 +If the operation was successful, the routine \verb|glp_write_mps|
    1.82 +returns zero. Otherwise, it prints an error message and returns
    1.83 +non-zero.
    1.84 +
    1.85 +\subsection{glp\_read\_lp---read problem data in CPLEX LP format}
    1.86 +
    1.87 +\subsubsection*{Synopsis}
    1.88 +
    1.89 +\begin{verbatim}
    1.90 +int glp_read_lp(glp_prob *lp, const void *parm,
    1.91 +      const char *fname);
    1.92 +\end{verbatim}
    1.93 +
    1.94 +\subsubsection*{Description}
    1.95 +
    1.96 +The routine \verb|glp_read_lp| reads problem data in CPLEX LP format
    1.97 +from a text file. (The CPLEX LP format is described in Appendix
    1.98 +\ref{chacplex}, page \pageref{chacplex}.)
    1.99 +
   1.100 +The parameter \verb|parm| is reserved for use in the future and must be
   1.101 +specified as \verb|NULL|.
   1.102 +
   1.103 +The character string \verb|fname| specifies a name of the text file to
   1.104 +be read in. (If the file name ends with suffix `\verb|.gz|', the file is
   1.105 +assumed to be compressed, in which case the routine \verb|glp_read_lp|
   1.106 +decompresses it ``on the fly''.)
   1.107 +
   1.108 +Note that before reading data the current content of the problem object
   1.109 +is completely erased with the routine \verb|glp_erase_prob|.
   1.110 +
   1.111 +\subsubsection*{Returns}
   1.112 +
   1.113 +If the operation was successful, the routine \verb|glp_read_lp| returns
   1.114 +zero. Otherwise, it prints an error message and returns non-zero.
   1.115 +
   1.116 +\subsection{glp\_write\_lp---write problem data in CPLEX LP format}
   1.117 +
   1.118 +\subsubsection*{Synopsis}
   1.119 +
   1.120 +\begin{verbatim}
   1.121 +int glp_write_lp(glp_prob *lp, const void *parm,
   1.122 +      const char *fname);
   1.123 +\end{verbatim}
   1.124 +
   1.125 +\subsubsection*{Description}
   1.126 +
   1.127 +The routine \verb|glp_write_lp| writes problem data in CPLEX LP format
   1.128 +to a text file. (The CPLEX LP format is described in Appendix
   1.129 +\ref{chacplex}, page \pageref{chacplex}.)
   1.130 +
   1.131 +The parameter \verb|parm| is reserved for use in the future and must be
   1.132 +specified as \verb|NULL|.
   1.133 +
   1.134 +The character string \verb|fname| specifies a name of the text file to
   1.135 +be written out. (If the file name ends with suffix `\verb|.gz|', the
   1.136 +file is assumed to be compressed, in which case the routine
   1.137 +\verb|glp_write_lp| performs automatic compression on writing it.)
   1.138 +
   1.139 +\subsubsection*{Returns}
   1.140 +
   1.141 +If the operation was successful, the routine \verb|glp_write_lp|
   1.142 +returns zero. Otherwise, it prints an error message and returns
   1.143 +non-zero.
   1.144 +
   1.145 +\subsection{glp\_read\_prob---read problem data in GLPK format}
   1.146 +
   1.147 +\subsubsection*{Synopsis}
   1.148 +
   1.149 +\begin{verbatim}
   1.150 +int glp_read_prob(glp_prob *P, int flags, const char *fname);
   1.151 +\end{verbatim}
   1.152 +
   1.153 +\subsubsection*{Description}
   1.154 +
   1.155 +The routine \verb|glp_read_prob| reads problem data in the GLPK LP/MIP
   1.156 +format from a text file. (For description of the GLPK LP/MIP format see
   1.157 +below.)
   1.158 +
   1.159 +The parameter \verb|flags| is reserved for use in the future and should
   1.160 +be specified as zero.
   1.161 +
   1.162 +The character string \verb|fname| specifies a name of the text file to
   1.163 +be read in. (If the file name ends with suffix `\verb|.gz|', the file
   1.164 +is assumed to be compressed, in which case the routine
   1.165 +\verb|glp_read_prob| decompresses it ``on the fly''.)
   1.166 +
   1.167 +Note that before reading data the current content of the problem object
   1.168 +is completely erased with the routine \verb|glp_erase_prob|.
   1.169 +
   1.170 +\subsubsection*{Returns}
   1.171 +
   1.172 +If the operation was successful, the routine \verb|glp_read_prob|
   1.173 +returns zero. Otherwise, it prints an error message and returns
   1.174 +non-zero.
   1.175 +
   1.176 +\subsubsection*{GLPK LP/MIP format}
   1.177 +
   1.178 +The GLPK LP/MIP format is a DIMACS-like format.\footnote{The DIMACS
   1.179 +formats were developed by the Center for Discrete Mathematics and
   1.180 +Theoretical Computer Science (DIMACS) to facilitate exchange of problem
   1.181 +data. For details see: {\tt <http://dimacs.rutgers.edu/Challenges/>}. }
   1.182 +The file in this format is a plain ASCII text file containing lines of
   1.183 +several types described below. A line is terminated with the end-of-line
   1.184 +character. Fields in each line are separated by at least one blank
   1.185 +space. Each line begins with a one-character designator to identify the
   1.186 +line type.
   1.187 +
   1.188 +The first line of the data file must be the problem line (except
   1.189 +optional comment lines, which may precede the problem line). The last
   1.190 +line of the data file must be the end line. Other lines may follow in
   1.191 +arbitrary order, however, duplicate lines are not allowed.
   1.192 +
   1.193 +\paragraph{Comment lines.} Comment lines give human-readable
   1.194 +information about the data file and are ignored by GLPK routines.
   1.195 +Comment lines can appear anywhere in the data file. Each comment line
   1.196 +begins with the lower-case character \verb|c|.
   1.197 +
   1.198 +\begin{verbatim}
   1.199 +   c This is an example of comment line
   1.200 +\end{verbatim}
   1.201 +
   1.202 +\paragraph{Problem line.} There must be exactly one problem line in the
   1.203 +data file. This line must appear before any other lines except comment
   1.204 +lines and has the following format:
   1.205 +
   1.206 +\begin{verbatim}
   1.207 +   p CLASS DIR ROWS COLS NONZ
   1.208 +\end{verbatim}
   1.209 +
   1.210 +The lower-case letter \verb|p| specifies that this is the problem line.
   1.211 +
   1.212 +The \verb|CLASS| field defines the problem class and can contain either
   1.213 +the keyword \verb|lp| (that means linear programming problem) or
   1.214 +\verb|mip| (that means mixed integer programming problem).
   1.215 +
   1.216 +The \verb|DIR| field defines the optimization direction (that is, the
   1.217 +objective function sense) and can contain either the keyword \verb|min|
   1.218 +(that means minimization) or \verb|max| (that means maximization).
   1.219 +
   1.220 +The \verb|ROWS|, \verb|COLS|, and \verb|NONZ| fields contain
   1.221 +non-negative integer values specifying, respectively, the number of
   1.222 +rows (constraints), columns (variables), and non-zero constraint
   1.223 +coefficients in the problem instance. Note that \verb|NONZ| value does
   1.224 +not account objective coefficients.
   1.225 +
   1.226 +\paragraph{Row descriptors.} There must be at most one row descriptor
   1.227 +line in the data file for each row (constraint). This line has one of
   1.228 +the following formats:
   1.229 +
   1.230 +\begin{verbatim}
   1.231 +   i ROW f
   1.232 +   i ROW l RHS
   1.233 +   i ROW u RHS
   1.234 +   i ROW d RHS1 RHS2
   1.235 +   i ROW s RHS
   1.236 +\end{verbatim}
   1.237 +
   1.238 +The lower-case letter \verb|i| specifies that this is the row
   1.239 +descriptor line.
   1.240 +
   1.241 +The \verb|ROW| field specifies the row ordinal number, an integer
   1.242 +between 1 and $m$, where $m$ is the number of rows in the problem
   1.243 +instance.
   1.244 +
   1.245 +The next lower-case letter specifies the row type as follows:
   1.246 +
   1.247 +\verb|f| --- free (unbounded) row: $-\infty<\sum a_jx_j<+\infty$;
   1.248 +
   1.249 +\verb|l| --- inequality constraint of `$\geq$' type:
   1.250 +$\sum a_jx_j\geq b$;
   1.251 +
   1.252 +\verb|u| --- inequality constraint of `$\leq$' type:
   1.253 +$\sum a_jx_j\leq b$;
   1.254 +
   1.255 +\verb|d| --- double-sided inequality constraint:
   1.256 +$b_1\leq\sum a_jx_j\leq b_2$;
   1.257 +
   1.258 +\verb|s| --- equality constraint: $\sum a_jx_j=b$.
   1.259 +
   1.260 +The \verb|RHS| field contains a floaing-point value specifying the
   1.261 +row right-hand side. The \verb|RHS1| and \verb|RHS2| fields contain
   1.262 +floating-point values specifying, respectively, the lower and upper
   1.263 +right-hand sides for the double-sided row.
   1.264 +
   1.265 +If for some row its descriptor line does not appear in the data file,
   1.266 +by default that row is assumed to be an equality constraint with zero
   1.267 +right-hand side.
   1.268 +
   1.269 +\paragraph{Column descriptors.} There must be at most one column
   1.270 +descriptor line in the data file for each column (variable). This line
   1.271 +has one of the following formats depending on the problem class
   1.272 +specified in the problem line:
   1.273 +
   1.274 +\bigskip
   1.275 +
   1.276 +\begin{tabular}{@{}l@{\hspace*{40pt}}l}
   1.277 +LP class & MIP class \\
   1.278 +\hline
   1.279 +\verb|j COL f|           & \verb|j COL KIND f|           \\
   1.280 +\verb|j COL l BND|       & \verb|j COL KIND l BND|       \\
   1.281 +\verb|j COL u BND|       & \verb|j COL KIND u BND|       \\
   1.282 +\verb|j COL d BND1 BND2| & \verb|j COL KIND d BND1 BND2| \\
   1.283 +\verb|j COL s BND|       & \verb|j COL KIND s BND|       \\
   1.284 +\end{tabular}
   1.285 +
   1.286 +\bigskip
   1.287 +
   1.288 +The lower-case letter \verb|j| specifies that this is the column
   1.289 +descriptor line.
   1.290 +
   1.291 +The \verb|COL| field specifies the column ordinal number, an integer
   1.292 +between 1 and $n$, where $n$ is the number of columns in the problem
   1.293 +instance.
   1.294 +
   1.295 +The \verb|KIND| field is used only for MIP problems and specifies the
   1.296 +column kind as follows:
   1.297 +
   1.298 +\verb|c| --- continuous column;
   1.299 +
   1.300 +\verb|i| --- integer column;
   1.301 +
   1.302 +\verb|b| --- binary column (in this case all remaining fields must be
   1.303 +omitted).
   1.304 +
   1.305 +The next lower-case letter specifies the column type as follows:
   1.306 +
   1.307 +\verb|f| --- free (unbounded) column: $-\infty<x<+\infty$;
   1.308 +
   1.309 +\verb|l| --- column with lower bound: $x\geq l$;
   1.310 +
   1.311 +\verb|u| --- column with upper bound: $x\leq u$;
   1.312 +
   1.313 +\verb|d| --- double-bounded column: $l\leq x\leq u$;
   1.314 +
   1.315 +\verb|s| --- fixed column: $x=s$.
   1.316 +
   1.317 +The \verb|BND| field contains a floating-point value that specifies the
   1.318 +column bound. The \verb|BND1| and \verb|BND2| fields contain
   1.319 +floating-point values specifying, respectively, the lower and upper
   1.320 +bounds for the double-bounded column.
   1.321 +
   1.322 +If for some column its descriptor line does not appear in the file, by
   1.323 +default that column is assumed to be non-negative (in case of LP class)
   1.324 +or binary (in case of MIP class).
   1.325 +
   1.326 +\paragraph{Coefficient descriptors.} There must be exactly one
   1.327 +coefficient descriptor line in the data file for each non-zero
   1.328 +objective or constraint coefficient. This line has the following format:
   1.329 +
   1.330 +\begin{verbatim}
   1.331 +   a ROW COL VAL
   1.332 +\end{verbatim}
   1.333 +
   1.334 +The lower-case letter \verb|a| specifies that this is the coefficient
   1.335 +descriptor line.
   1.336 +
   1.337 +For objective coefficients the \verb|ROW| field must contain 0. For
   1.338 +constraint coefficients the \verb|ROW| field specifies the row ordinal
   1.339 +number, an integer between 1 and $m$, where $m$ is the number of rows
   1.340 +in the problem instance.
   1.341 +
   1.342 +The \verb|COL| field specifies the column ordinal number, an integer
   1.343 +between 1 and $n$, where $n$ is the number of columns in the problem
   1.344 +instance.
   1.345 +
   1.346 +If both the \verb|ROW| and \verb|COL| fields contain 0, the line
   1.347 +specifies the constant term (``shift'') of the objective function
   1.348 +rather than objective coefficient.
   1.349 +
   1.350 +The \verb|VAL| field contains a floating-point coefficient value (it is
   1.351 +allowed to specify zero value in this field).
   1.352 +
   1.353 +The number of constraint coefficient descriptor lines must be exactly
   1.354 +the same as specified in the field \verb|NONZ| of the problem line.
   1.355 +
   1.356 +\paragraph{Symbolic name descriptors.} There must be at most one
   1.357 +symbolic name descriptor line for the problem instance, objective
   1.358 +function, each row (constraint), and each column (variable). This line
   1.359 +has one of the following formats:
   1.360 +
   1.361 +\begin{verbatim}
   1.362 +   n p NAME
   1.363 +   n z NAME
   1.364 +   n i ROW NAME
   1.365 +   n j COL NAME
   1.366 +\end{verbatim}
   1.367 +
   1.368 +The lower-case letter \verb|n| specifies that this is the symbolic name
   1.369 +descriptor line.
   1.370 +
   1.371 +The next lower-case letter specifies which object should be assigned a
   1.372 +symbolic name:
   1.373 +
   1.374 +\verb|p| --- problem instance;
   1.375 +
   1.376 +\verb|z| --- objective function;
   1.377 +
   1.378 +\verb|i| --- row (constraint);
   1.379 +
   1.380 +\verb|j| --- column (variable).
   1.381 +
   1.382 +The \verb|ROW| field specifies the row ordinal number, an integer
   1.383 +between 1 and $m$, where $m$ is the number of rows in the problem
   1.384 +instance.
   1.385 +
   1.386 +The \verb|COL| field specifies the column ordinal number, an integer
   1.387 +between 1 and $n$, where $n$ is the number of columns in the problem
   1.388 +instance.
   1.389 +
   1.390 +The \verb|NAME| field contains the symbolic name, a sequence from 1 to
   1.391 +255 arbitrary graphic ASCII characters, assigned to corresponding
   1.392 +object.
   1.393 +
   1.394 +\paragraph{End line.} There must be exactly one end line in the data
   1.395 +file. This line must appear last in the file and has the following
   1.396 +format:
   1.397 +
   1.398 +\begin{verbatim}
   1.399 +   e
   1.400 +\end{verbatim}
   1.401 +
   1.402 +The lower-case letter \verb|e| specifies that this is the end line.
   1.403 +Anything that follows the end line is ignored by GLPK routines.
   1.404 +
   1.405 +\subsubsection*{Example of data file in GLPK LP/MIP format}
   1.406 +
   1.407 +The following example of a data file in GLPK LP/MIP format specifies
   1.408 +the same LP problem as in Subsection ``Example of MPS file''.
   1.409 +
   1.410 +\begin{center}
   1.411 +\footnotesize\tt
   1.412 +\begin{tabular}{l@{\hspace*{50pt}}}
   1.413 +p lp min 8 7 48   \\
   1.414 +n p PLAN          \\
   1.415 +n z VALUE         \\
   1.416 +i 1 f             \\
   1.417 +n i 1 VALUE       \\
   1.418 +i 2 s 2000        \\
   1.419 +n i 2 YIELD       \\
   1.420 +i 3 u 60          \\
   1.421 +n i 3 FE          \\
   1.422 +i 4 u 100         \\
   1.423 +n i 4 CU          \\
   1.424 +i 5 u 40          \\
   1.425 +n i 5 MN          \\
   1.426 +i 6 u 30          \\
   1.427 +n i 6 MG          \\
   1.428 +i 7 l 1500        \\
   1.429 +n i 7 AL          \\
   1.430 +i 8 d 250 300     \\
   1.431 +n i 8 SI          \\
   1.432 +j 1 d 0 200       \\
   1.433 +n j 1 BIN1        \\
   1.434 +j 2 d 0 2500      \\
   1.435 +n j 2 BIN2        \\
   1.436 +j 3 d 400 800     \\
   1.437 +n j 3 BIN3        \\
   1.438 +j 4 d 100 700     \\
   1.439 +n j 4 BIN4        \\
   1.440 +j 5 d 0 1500      \\
   1.441 +n j 5 BIN5        \\
   1.442 +n j 6 ALUM        \\
   1.443 +n j 7 SILICON     \\
   1.444 +a 0 1 0.03        \\
   1.445 +a 0 2 0.08        \\
   1.446 +a 0 3 0.17        \\
   1.447 +a 0 4 0.12        \\
   1.448 +a 0 5 0.15        \\
   1.449 +a 0 6 0.21        \\
   1.450 +a 0 7 0.38        \\
   1.451 +a 1 1 0.03        \\
   1.452 +a 1 2 0.08        \\
   1.453 +a 1 3 0.17        \\
   1.454 +a 1 4 0.12        \\
   1.455 +a 1 5 0.15        \\
   1.456 +a 1 6 0.21        \\
   1.457 +\end{tabular}
   1.458 +\begin{tabular}{|@{\hspace*{80pt}}l}
   1.459 +a 1 7 0.38        \\
   1.460 +a 2 1 1           \\
   1.461 +a 2 2 1           \\
   1.462 +a 2 3 1           \\
   1.463 +a 2 4 1           \\
   1.464 +a 2 5 1           \\
   1.465 +a 2 6 1           \\
   1.466 +a 2 7 1           \\
   1.467 +a 3 1 0.15        \\
   1.468 +a 3 2 0.04        \\
   1.469 +a 3 3 0.02        \\
   1.470 +a 3 4 0.04        \\
   1.471 +a 3 5 0.02        \\
   1.472 +a 3 6 0.01        \\
   1.473 +a 3 7 0.03        \\
   1.474 +a 4 1 0.03        \\
   1.475 +a 4 2 0.05        \\
   1.476 +a 4 3 0.08        \\
   1.477 +a 4 4 0.02        \\
   1.478 +a 4 5 0.06        \\
   1.479 +a 4 6 0.01        \\
   1.480 +a 5 1 0.02        \\
   1.481 +a 5 2 0.04        \\
   1.482 +a 5 3 0.01        \\
   1.483 +a 5 4 0.02        \\
   1.484 +a 5 5 0.02        \\
   1.485 +a 6 1 0.02        \\
   1.486 +a 6 2 0.03        \\
   1.487 +a 6 5 0.01        \\
   1.488 +a 7 1 0.7         \\
   1.489 +a 7 2 0.75        \\
   1.490 +a 7 3 0.8         \\
   1.491 +a 7 4 0.75        \\
   1.492 +a 7 5 0.8         \\
   1.493 +a 7 6 0.97        \\
   1.494 +a 8 1 0.02        \\
   1.495 +a 8 2 0.06        \\
   1.496 +a 8 3 0.08        \\
   1.497 +a 8 4 0.12        \\
   1.498 +a 8 5 0.02        \\
   1.499 +a 8 6 0.01        \\
   1.500 +a 8 7 0.97        \\
   1.501 +e o f             \\
   1.502 +\\
   1.503 +\end{tabular}
   1.504 +\end{center}
   1.505 +
   1.506 +\newpage
   1.507 +
   1.508 +\subsection{glp\_write\_prob---write problem data in GLPK format}
   1.509 +
   1.510 +\subsubsection*{Synopsis}
   1.511 +
   1.512 +\begin{verbatim}
   1.513 +int glp_write_prob(glp_prob *P, int flags, const char *fname);
   1.514 +\end{verbatim}
   1.515 +
   1.516 +\subsubsection*{Description}
   1.517 +
   1.518 +The routine \verb|glp_write_prob| writes problem data in the GLPK
   1.519 +LP/MIP format to a text file. (For description of the GLPK LP/MIP
   1.520 +format see Subsection ``Read problem data in GLPK format''.)
   1.521 +
   1.522 +The parameter \verb|flags| is reserved for use in the future and should
   1.523 +be specified as zero.
   1.524 +
   1.525 +The character string \verb|fname| specifies a name of the text file to
   1.526 +be written out. (If the file name ends with suffix `\verb|.gz|', the
   1.527 +file is assumed to be compressed, in which case the routine
   1.528 +\verb|glp_write_prob| performs automatic compression on writing it.)
   1.529 +
   1.530 +\subsubsection*{Returns}
   1.531 +
   1.532 +If the operation was successful, the routine \verb|glp_read_prob|
   1.533 +returns zero. Otherwise, it prints an error message and returns
   1.534 +non-zero.
   1.535 +
   1.536 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   1.537 +
   1.538 +\newpage
   1.539 +
   1.540 +\section{Routines for processing MathProg models}
   1.541 +
   1.542 +\subsection{Introduction}
   1.543 +
   1.544 +GLPK supports the {\it GNU MathProg modeling language}.\footnote{The
   1.545 +GNU MathProg modeling language is a subset of the AMPL language. For
   1.546 +its detailed description see the document ``Modeling Language GNU
   1.547 +MathProg: Language Reference'' included in the GLPK distribution.}
   1.548 +As a rule, models written in MathProg are solved with the GLPK LP/MIP
   1.549 +stand-alone solver \verb|glpsol| (see Appendix D) and do not need any
   1.550 +programming with API routines. However, for various reasons the user
   1.551 +may need to process MathProg models directly in his/her application
   1.552 +program, in which case he/she may use API routines described in this
   1.553 +section. These routines provide an interface to the {\it MathProg
   1.554 +translator}, a component of GLPK, which translates MathProg models into
   1.555 +an internal code and then interprets (executes) this code.
   1.556 +
   1.557 +The processing of a model written in GNU MathProg includes several
   1.558 +steps, which should be performed in the following order:
   1.559 +
   1.560 +\begin{enumerate}
   1.561 +\item{\it Allocating the workspace.}
   1.562 +The translator allocates the workspace, an internal data structure used
   1.563 +on all subsequent steps.
   1.564 +\item{\it Reading model section.} The translator reads model section
   1.565 +and, optionally, data section from a specified text file and translates
   1.566 +them into the internal code. If necessary, on this step data section
   1.567 +may be ignored.
   1.568 +\item{\it Reading data section(s).} The translator reads one or more
   1.569 +data sections from specified text file(s) and translates them into the
   1.570 +internal code.
   1.571 +\item{\it Generating the model.} The translator executes the internal
   1.572 +code to evaluate the content of the model objects such as sets,
   1.573 +parameters, variables, constraints, and objectives. On this step the
   1.574 +execution is suspended at the solve statement.
   1.575 +\item {\it Building the problem object.} The translator obtains all
   1.576 +necessary information from the workspace and builds the standard
   1.577 +problem object (that is, the program object of type \verb|glp_prob|).
   1.578 +\item{\it Solving the problem.} On this step the problem object built
   1.579 +on the previous step is passed to a solver, which solves the problem
   1.580 +instance and stores its solution back to the problem object.
   1.581 +\item{\it Postsolving the model.} The translator copies the solution
   1.582 +from the problem object to the workspace and then executes the internal
   1.583 +code from the solve statement to the end of the model. (If model has
   1.584 +no solve statement, the translator does nothing on this step.)
   1.585 +\item{\it Freeing the workspace.} The translator frees all the memory
   1.586 +allocated to the workspace.
   1.587 +\end{enumerate}
   1.588 +
   1.589 +Note that the MathProg translator performs no error correction, so if
   1.590 +any of steps 2 to 7 fails (due to errors in the model), the application
   1.591 +program should terminate processing and go to step 8.
   1.592 +
   1.593 +\subsubsection*{Example 1}
   1.594 +
   1.595 +In this example the program reads model and data sections from input
   1.596 +file \verb|egypt.mod|\footnote{This is an example model included in
   1.597 +the GLPK distribution.} and writes the model to output file
   1.598 +\verb|egypt.mps| in free MPS format (see Appendix B). No solution is
   1.599 +performed.
   1.600 +
   1.601 +\begin{small}
   1.602 +\begin{verbatim}
   1.603 +/* mplsamp1.c */
   1.604 +
   1.605 +#include <stdio.h>
   1.606 +#include <stdlib.h>
   1.607 +#include <glpk.h>
   1.608 +
   1.609 +int main(void)
   1.610 +{     glp_prob *lp;
   1.611 +      glp_tran *tran;
   1.612 +      int ret;
   1.613 +      lp = glp_create_prob();
   1.614 +      tran = glp_mpl_alloc_wksp();
   1.615 +      ret = glp_mpl_read_model(tran, "egypt.mod", 0);
   1.616 +      if (ret != 0)
   1.617 +      {  fprintf(stderr, "Error on translating model\n");
   1.618 +         goto skip;
   1.619 +      }
   1.620 +      ret = glp_mpl_generate(tran, NULL);
   1.621 +      if (ret != 0)
   1.622 +      {  fprintf(stderr, "Error on generating model\n");
   1.623 +         goto skip;
   1.624 +      }
   1.625 +      glp_mpl_build_prob(tran, lp);
   1.626 +      ret = glp_write_mps(lp, GLP_MPS_FILE, NULL, "egypt.mps");
   1.627 +      if (ret != 0)
   1.628 +         fprintf(stderr, "Error on writing MPS file\n");
   1.629 +skip: glp_mpl_free_wksp(tran);
   1.630 +      glp_delete_prob(lp);
   1.631 +      return 0;
   1.632 +}
   1.633 +
   1.634 +/* eof */
   1.635 +\end{verbatim}
   1.636 +\end{small}
   1.637 +
   1.638 +\subsubsection*{Example 2}
   1.639 +
   1.640 +In this example the program reads model section from file
   1.641 +\verb|sudoku.mod|\footnote{This is an example model which is included
   1.642 +in the GLPK distribution along with alternative data file
   1.643 +{\tt sudoku.dat}.} ignoring data section in this file, reads alternative
   1.644 +data section from file \verb|sudoku.dat|, solves the problem instance
   1.645 +and passes the solution found back to the model.
   1.646 +
   1.647 +\begin{small}
   1.648 +\begin{verbatim}
   1.649 +/* mplsamp2.c */
   1.650 +
   1.651 +#include <stdio.h>
   1.652 +#include <stdlib.h>
   1.653 +#include <glpk.h>
   1.654 +
   1.655 +int main(void)
   1.656 +{     glp_prob *mip;
   1.657 +      glp_tran *tran;
   1.658 +      int ret;
   1.659 +      mip = glp_create_prob();
   1.660 +      tran = glp_mpl_alloc_wksp();
   1.661 +      ret = glp_mpl_read_model(tran, "sudoku.mod", 1);
   1.662 +      if (ret != 0)
   1.663 +      {  fprintf(stderr, "Error on translating model\n");
   1.664 +         goto skip;
   1.665 +      }
   1.666 +      ret = glp_mpl_read_data(tran, "sudoku.dat");
   1.667 +      if (ret != 0)
   1.668 +      {  fprintf(stderr, "Error on translating data\n");
   1.669 +         goto skip;
   1.670 +      }
   1.671 +      ret = glp_mpl_generate(tran, NULL);
   1.672 +      if (ret != 0)
   1.673 +      {  fprintf(stderr, "Error on generating model\n");
   1.674 +         goto skip;
   1.675 +      }
   1.676 +      glp_mpl_build_prob(tran, mip);
   1.677 +      glp_simplex(mip, NULL);
   1.678 +      glp_intopt(mip, NULL);
   1.679 +      ret = glp_mpl_postsolve(tran, mip, GLP_MIP);
   1.680 +      if (ret != 0)
   1.681 +         fprintf(stderr, "Error on postsolving model\n");
   1.682 +skip: glp_mpl_free_wksp(tran);
   1.683 +      glp_delete_prob(mip);
   1.684 +      return 0;
   1.685 +}
   1.686 +
   1.687 +/* eof */
   1.688 +\end{verbatim}
   1.689 +\end{small}
   1.690 +
   1.691 +\subsection{glp\_mpl\_alloc\_wksp---allocate the translator workspace}
   1.692 +
   1.693 +\subsubsection*{Synopsis}
   1.694 +
   1.695 +\begin{verbatim}
   1.696 +glp_tran *glp_mpl_alloc_wksp(void);
   1.697 +\end{verbatim}
   1.698 +
   1.699 +\subsubsection*{Description}
   1.700 +
   1.701 +The routine \verb|glp_mpl_alloc_wksp| allocates the MathProg translator
   1.702 +work\-space. (Note that multiple instances of the workspace may be
   1.703 +allocated, if necessary.)
   1.704 +
   1.705 +\subsubsection*{Returns}
   1.706 +
   1.707 +The routine returns a pointer to the workspace, which should be used in
   1.708 +all subsequent operations.
   1.709 +
   1.710 +\subsection{glp\_mpl\_read\_model---read and translate model section}
   1.711 +
   1.712 +\subsubsection*{Synopsis}
   1.713 +
   1.714 +\begin{verbatim}
   1.715 +int glp_mpl_read_model(glp_tran *tran, const char *fname,
   1.716 +      int skip);
   1.717 +\end{verbatim}
   1.718 +
   1.719 +\subsubsection*{Description}
   1.720 +
   1.721 +The routine \verb|glp_mpl_read_model| reads model section and,
   1.722 +optionally, data section, which may follow the model section, from a
   1.723 +text file, whose name is the character string \verb|fname|, performs
   1.724 +translation of model statements and data blocks, and stores all the
   1.725 +information in the workspace.
   1.726 +
   1.727 +The parameter \verb|skip| is a flag. If the input file contains the
   1.728 +data section and this flag is non-zero, the data section is not read as
   1.729 +if there were no data section and a warning message is printed. This
   1.730 +allows reading data section(s) from other file(s).
   1.731 +
   1.732 +\subsubsection*{Returns}
   1.733 +
   1.734 +If the operation is successful, the routine returns zero. Otherwise
   1.735 +the routine prints an error message and returns non-zero.
   1.736 +
   1.737 +\subsection{glp\_mpl\_read\_data---read and translate data section}
   1.738 +
   1.739 +\subsubsection*{Synopsis}
   1.740 +
   1.741 +\begin{verbatim}
   1.742 +int glp_mpl_read_data(glp_tran *tran, const char *fname);
   1.743 +\end{verbatim}
   1.744 +
   1.745 +\subsubsection*{Description}
   1.746 +
   1.747 +The routine \verb|glp_mpl_read_data| reads data section from a text
   1.748 +file, whose name is the character string \verb|fname|, performs
   1.749 +translation of data blocks, and stores the data read in the translator
   1.750 +workspace. If necessary, this routine may be called more than once.
   1.751 +
   1.752 +\subsubsection*{Returns}
   1.753 +
   1.754 +If the operation is successful, the routine returns zero. Otherwise
   1.755 +the routine prints an error message and returns non-zero.
   1.756 +
   1.757 +\subsection{glp\_mpl\_generate---generate the model}
   1.758 +
   1.759 +\subsubsection*{Synopsis}
   1.760 +
   1.761 +\begin{verbatim}
   1.762 +int glp_mpl_generate(glp_tran *tran, const char *fname);
   1.763 +\end{verbatim}
   1.764 +
   1.765 +\subsubsection*{Description}
   1.766 +
   1.767 +The routine \verb|glp_mpl_generate| generates the model using its
   1.768 +description stored in the translator workspace. This operation means
   1.769 +generating all variables, constraints, and objectives, executing check
   1.770 +and display statements, which precede the solve statement (if it is
   1.771 +presented).
   1.772 +
   1.773 +The character string \verb|fname| specifies the name of an output text
   1.774 +file, to which output produced by display statements should be written.
   1.775 +If \verb|fname| is \verb|NULL|, the output is sent to the terminal.
   1.776 +
   1.777 +\subsubsection*{Returns}
   1.778 +
   1.779 +If the operation is successful, the routine returns zero. Otherwise
   1.780 +the routine prints an error message and returns non-zero.
   1.781 +
   1.782 +\subsection{glp\_mpl\_build\_prob---build problem instance from the
   1.783 +model}
   1.784 +
   1.785 +\subsubsection*{Synopsis}
   1.786 +
   1.787 +\begin{verbatim}
   1.788 +void glp_mpl_build_prob(glp_tran *tran, glp_prob *prob);
   1.789 +\end{verbatim}
   1.790 +
   1.791 +\subsubsection*{Description}
   1.792 +
   1.793 +The routine \verb|glp_mpl_build_prob| obtains all necessary information
   1.794 +from the translator workspace and stores it in the specified problem
   1.795 +object \verb|prob|. Note that before building the current content of
   1.796 +the problem object is erased with the routine \verb|glp_erase_prob|.
   1.797 +
   1.798 +\subsection{glp\_mpl\_postsolve---postsolve the model}
   1.799 +
   1.800 +\subsubsection*{Synopsis}
   1.801 +
   1.802 +\begin{verbatim}
   1.803 +int glp_mpl_postsolve(glp_tran *tran, glp_prob *prob,
   1.804 +      int sol);
   1.805 +\end{verbatim}
   1.806 +
   1.807 +\subsubsection*{Description}
   1.808 +
   1.809 +The routine \verb|glp_mpl_postsolve| copies the solution from the
   1.810 +specified problem object \verb|prob| to the translator workspace and
   1.811 +then executes all the remaining model statements, which follow the
   1.812 +solve statement.
   1.813 +
   1.814 +The parameter \verb|sol| specifies which solution should be copied
   1.815 +from the problem object to the workspace as follows:
   1.816 +
   1.817 +\begin{tabular}{@{}ll}
   1.818 +\verb|GLP_SOL| & basic solution; \\
   1.819 +\verb|GLP_IPT| & interior-point solution; \\
   1.820 +\verb|GLP_MIP| & mixed integer solution. \\
   1.821 +\end{tabular}
   1.822 +
   1.823 +\subsubsection*{Returns}
   1.824 +
   1.825 +If the operation is successful, the routine returns zero. Otherwise
   1.826 +the routine prints an error message and returns non-zero.
   1.827 +
   1.828 +\subsection{glp\_mpl\_free\_wksp---free the translator workspace}
   1.829 +
   1.830 +\subsubsection*{Synopsis}
   1.831 +
   1.832 +\begin{verbatim}
   1.833 +void glp_mpl_free_wksp(glp_tran *tran);
   1.834 +\end{verbatim}
   1.835 +
   1.836 +\subsubsection*{Description}
   1.837 +
   1.838 +The routine \verb|glp_mpl_free_wksp| frees all the memory allocated to
   1.839 +the translator workspace. It also frees all other resources, which are
   1.840 +still used by the translator.
   1.841 +
   1.842 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   1.843 +
   1.844 +\newpage
   1.845 +
   1.846 +\section{Problem solution reading/writing routines}
   1.847 +
   1.848 +\subsection{glp\_print\_sol---write basic solution in printable format}
   1.849 +
   1.850 +\subsubsection*{Synopsis}
   1.851 +
   1.852 +\begin{verbatim}
   1.853 +int glp_print_sol(glp_prob *lp, const char *fname);
   1.854 +\end{verbatim}
   1.855 +
   1.856 +\subsubsection*{Description}
   1.857 +
   1.858 +The routine \verb|glp_print_sol writes| the current basic solution of
   1.859 +an LP problem, which is specified by the pointer \verb|lp|, to a text
   1.860 +file, whose name is the character string \verb|fname|, in printable
   1.861 +format.
   1.862 +
   1.863 +Information reported by the routine \verb|glp_print_sol| is intended
   1.864 +mainly for visual analysis.
   1.865 +
   1.866 +\subsubsection*{Returns}
   1.867 +
   1.868 +If no errors occurred, the routine returns zero. Otherwise the routine
   1.869 +prints an error message and returns non-zero.
   1.870 +
   1.871 +\subsection{glp\_read\_sol---read basic solution from text file}
   1.872 +
   1.873 +\subsubsection*{Synopsis}
   1.874 +
   1.875 +\begin{verbatim}
   1.876 +int glp_read_sol(glp_prob *lp, const char *fname);
   1.877 +\end{verbatim}
   1.878 +
   1.879 +\subsubsection*{Description}
   1.880 +
   1.881 +The routine \verb|glp_read_sol| reads basic solution from a text file
   1.882 +whose name is specified by the parameter \verb|fname| into the problem
   1.883 +object.
   1.884 +
   1.885 +For the file format see description of the routine \verb|glp_write_sol|.
   1.886 +
   1.887 +\subsubsection*{Returns}
   1.888 +
   1.889 +On success the routine returns zero, otherwise non-zero.
   1.890 +
   1.891 +\newpage
   1.892 +
   1.893 +\subsection{glp\_write\_sol---write basic solution to text file}
   1.894 +
   1.895 +\subsubsection*{Synopsis}
   1.896 +
   1.897 +\begin{verbatim}
   1.898 +int glp_write_sol(glp_prob *lp, const char *fname);
   1.899 +\end{verbatim}
   1.900 +
   1.901 +\subsubsection*{Description}
   1.902 +
   1.903 +The routine \verb|glp_write_sol| writes the current basic solution to a
   1.904 +text file whose name is specified by the parameter \verb|fname|. This
   1.905 +file can be read back with the routine \verb|glp_read_sol|.
   1.906 +
   1.907 +\subsubsection*{Returns}
   1.908 +
   1.909 +On success the routine returns zero, otherwise non-zero.
   1.910 +
   1.911 +\subsubsection*{File format}
   1.912 +
   1.913 +The file created by the routine \verb|glp_write_sol| is a plain text
   1.914 +file, which contains the following information:
   1.915 +
   1.916 +\begin{verbatim}
   1.917 +   m n
   1.918 +   p_stat d_stat obj_val
   1.919 +   r_stat[1] r_prim[1] r_dual[1]
   1.920 +   . . .
   1.921 +   r_stat[m] r_prim[m] r_dual[m]
   1.922 +   c_stat[1] c_prim[1] c_dual[1]
   1.923 +   . . .
   1.924 +   c_stat[n] c_prim[n] c_dual[n]
   1.925 +\end{verbatim}
   1.926 +
   1.927 +\noindent
   1.928 +where:
   1.929 +
   1.930 +\noindent
   1.931 +$m$ is the number of rows (auxiliary variables);
   1.932 +
   1.933 +\noindent
   1.934 +$n$ is the number of columns (structural variables);
   1.935 +
   1.936 +\noindent
   1.937 +\verb|p_stat| is the primal status of the basic solution
   1.938 +(\verb|GLP_UNDEF| = 1, \verb|GLP_FEAS| = 2, \verb|GLP_INFEAS| = 3, or
   1.939 +\verb|GLP_NOFEAS| = 4);
   1.940 +
   1.941 +\noindent
   1.942 +\verb|d_stat| is the dual status of the basic solution
   1.943 +(\verb|GLP_UNDEF| = 1, \verb|GLP_FEAS| = 2, \verb|GLP_INFEAS| = 3, or
   1.944 +\verb|GLP_NOFEAS| = 4);
   1.945 +
   1.946 +\noindent
   1.947 +\verb|obj_val| is the objective value;
   1.948 +
   1.949 +\noindent
   1.950 +\verb|r_stat[i]|, $i=1,\dots,m$, is the status of $i$-th row
   1.951 +(\verb|GLP_BS| = 1, \verb|GLP_NL| = 2, \verb|GLP_NU| = 3,
   1.952 +\verb|GLP_NF| = 4, or \verb|GLP_NS| = 5);
   1.953 +
   1.954 +\noindent
   1.955 +\verb|r_prim[i]|, $i=1,\dots,m$, is the primal value of $i$-th row;
   1.956 +
   1.957 +\noindent
   1.958 +\verb|r_dual[i]|, $i=1,\dots,m$, is the dual value of $i$-th row;
   1.959 +
   1.960 +\noindent
   1.961 +\verb|c_stat[j]|, $j=1,\dots,n$, is the status of $j$-th column
   1.962 +(\verb|GLP_BS| = 1, \verb|GLP_NL| = 2, \verb|GLP_NU| = 3,
   1.963 +\verb|GLP_NF| = 4, or \verb|GLP_NS| = 5);
   1.964 +
   1.965 +\noindent
   1.966 +\verb|c_prim[j]|, $j=1,\dots,n$, is the primal value of $j$-th column;
   1.967 +
   1.968 +\noindent
   1.969 +\verb|c_dual[j]|, $j=1,\dots,n$, is the dual value of $j$-th column.
   1.970 +
   1.971 +\subsection{glp\_print\_ipt---write interior-point solution in
   1.972 +printable format}
   1.973 +
   1.974 +\subsubsection*{Synopsis}
   1.975 +
   1.976 +\begin{verbatim}
   1.977 +int glp_print_ipt(glp_prob *lp, const char *fname);
   1.978 +\end{verbatim}
   1.979 +
   1.980 +\subsubsection*{Description}
   1.981 +
   1.982 +The routine \verb|glp_print_ipt| writes the current interior point
   1.983 +solution  of an LP problem, which the parameter \verb|lp| points to, to
   1.984 +a text file, whose name is the character string \verb|fname|, in
   1.985 +printable format.
   1.986 +
   1.987 +Information reported by the routine \verb|glp_print_ipt| is intended
   1.988 +mainly for visual analysis.
   1.989 +
   1.990 +\subsubsection*{Returns}
   1.991 +
   1.992 +If no errors occurred, the routine returns zero. Otherwise the routine
   1.993 +prints an error message and returns non-zero.
   1.994 +
   1.995 +\subsection{glp\_read\_ipt---read interior-point solution from text
   1.996 +file}
   1.997 +
   1.998 +\subsubsection*{Synopsis}
   1.999 +
  1.1000 +\begin{verbatim}
  1.1001 +int glp_read_ipt(glp_prob *lp, const char *fname);
  1.1002 +\end{verbatim}
  1.1003 +
  1.1004 +\subsubsection*{Description}
  1.1005 +
  1.1006 +The routine \verb|glp_read_ipt| reads interior-point solution from a
  1.1007 +text file whose name is specified by the parameter \verb|fname| into the
  1.1008 +problem object.
  1.1009 +
  1.1010 +For the file format see description of the routine \verb|glp_write_ipt|.
  1.1011 +
  1.1012 +\subsubsection*{Returns}
  1.1013 +
  1.1014 +On success the routine returns zero, otherwise non-zero.
  1.1015 +
  1.1016 +\subsection{glp\_write\_ipt---write interior-point solution to text
  1.1017 +file}
  1.1018 +
  1.1019 +\subsubsection*{Synopsis}
  1.1020 +
  1.1021 +\begin{verbatim}
  1.1022 +int glp_write_ipt(glp_prob *lp, const char *fname);
  1.1023 +\end{verbatim}
  1.1024 +
  1.1025 +\subsubsection*{Description}
  1.1026 +
  1.1027 +The routine \verb|glp_write_ipt| writes the current interior-point
  1.1028 +solution to a text file whose name is specified by the parameter
  1.1029 +\verb|fname|. This file can be read back with the routine
  1.1030 +\verb|glp_read_ipt|.
  1.1031 +
  1.1032 +\subsubsection*{Returns}
  1.1033 +
  1.1034 +On success the routine returns zero, otherwise non-zero.
  1.1035 +
  1.1036 +\subsubsection*{File format}
  1.1037 +
  1.1038 +The file created by the routine \verb|glp_write_ipt| is a plain text
  1.1039 +file, which contains the following information:
  1.1040 +
  1.1041 +\begin{verbatim}
  1.1042 +   m n
  1.1043 +   stat obj_val
  1.1044 +   r_prim[1] r_dual[1]
  1.1045 +   . . .
  1.1046 +   r_prim[m] r_dual[m]
  1.1047 +   c_prim[1] c_dual[1]
  1.1048 +   . . .
  1.1049 +   c_prim[n] c_dual[n]
  1.1050 +\end{verbatim}
  1.1051 +
  1.1052 +\noindent
  1.1053 +where:
  1.1054 +
  1.1055 +\noindent
  1.1056 +$m$ is the number of rows (auxiliary variables);
  1.1057 +
  1.1058 +\noindent
  1.1059 +$n$ is the number of columns (structural variables);
  1.1060 +
  1.1061 +\noindent
  1.1062 +\verb|stat| is the solution status (\verb|GLP_UNDEF| = 1 or
  1.1063 +\verb|GLP_OPT| = 5);
  1.1064 +
  1.1065 +\noindent
  1.1066 +\verb|obj_val| is the objective value;
  1.1067 +
  1.1068 +\noindent
  1.1069 +\verb|r_prim[i]|, $i=1,\dots,m$, is the primal value of $i$-th row;
  1.1070 +
  1.1071 +\noindent
  1.1072 +\verb|r_dual[i]|, $i=1,\dots,m$, is the dual value of $i$-th row;
  1.1073 +
  1.1074 +\noindent
  1.1075 +\verb|c_prim[j]|, $j=1,\dots,n$, is the primal value of $j$-th column;
  1.1076 +
  1.1077 +\noindent
  1.1078 +\verb|c_dual[j]|, $j=1,\dots,n$, is the dual value of $j$-th column.
  1.1079 +
  1.1080 +\subsection{glp\_print\_mip---write MIP solution in printable format}
  1.1081 +
  1.1082 +\subsubsection*{Synopsis}
  1.1083 +
  1.1084 +\begin{verbatim}
  1.1085 +int glp_print_mip(glp_prob *lp, const char *fname);
  1.1086 +\end{verbatim}
  1.1087 +
  1.1088 +\subsubsection*{Description}
  1.1089 +
  1.1090 +The routine \verb|glp_print_mip| writes a best known integer solution
  1.1091 +of a MIP problem, which is specified by the pointer \verb|lp|, to a text
  1.1092 +file, whose name is the character string \verb|fname|, in printable
  1.1093 +format.
  1.1094 +
  1.1095 +Information reported by the routine \verb|glp_print_mip| is intended
  1.1096 +mainly for visual analysis.
  1.1097 +
  1.1098 +\subsubsection*{Returns}
  1.1099 +
  1.1100 +If no errors occurred, the routine returns zero. Otherwise the routine
  1.1101 +prints an error message and returns non-zero.
  1.1102 +
  1.1103 +\newpage
  1.1104 +
  1.1105 +\subsection{glp\_read\_mip---read MIP solution from text file}
  1.1106 +
  1.1107 +\subsubsection*{Synopsis}
  1.1108 +
  1.1109 +\begin{verbatim}
  1.1110 +int glp_read_mip(glp_prob *mip, const char *fname);
  1.1111 +\end{verbatim}
  1.1112 +
  1.1113 +\subsubsection*{Description}
  1.1114 +
  1.1115 +The routine \verb|glp_read_mip| reads MIP solution from a text file
  1.1116 +whose name is specified by the parameter \verb|fname| into the problem
  1.1117 +object.
  1.1118 +
  1.1119 +For the file format see description of the routine \verb|glp_write_mip|.
  1.1120 +
  1.1121 +\subsubsection*{Returns}
  1.1122 +
  1.1123 +On success the routine returns zero, otherwise non-zero.
  1.1124 +
  1.1125 +\subsection{glp\_write\_mip---write MIP solution to text file}
  1.1126 +
  1.1127 +\subsubsection*{Synopsis}
  1.1128 +
  1.1129 +\begin{verbatim}
  1.1130 +int glp_write_mip(glp_prob *mip, const char *fname);
  1.1131 +\end{verbatim}
  1.1132 +
  1.1133 +\subsubsection*{Description}
  1.1134 +
  1.1135 +The routine \verb|glp_write_mip| writes the current MIP solution to a
  1.1136 +text file whose name is specified by the parameter \verb|fname|. This
  1.1137 +file can be read back with the routine \verb|glp_read_mip|.
  1.1138 +
  1.1139 +\subsubsection*{Returns}
  1.1140 +
  1.1141 +On success the routine returns zero, otherwise non-zero.
  1.1142 +
  1.1143 +\subsubsection*{File format}
  1.1144 +
  1.1145 +The file created by the routine \verb|glp_write_sol| is a plain text
  1.1146 +file, which contains the following information:
  1.1147 +
  1.1148 +\begin{verbatim}
  1.1149 +   m n
  1.1150 +   stat obj_val
  1.1151 +   r_val[1]
  1.1152 +   . . .
  1.1153 +   r_val[m]
  1.1154 +   c_val[1]
  1.1155 +   . . .
  1.1156 +   c_val[n]
  1.1157 +\end{verbatim}
  1.1158 +
  1.1159 +\noindent
  1.1160 +where:
  1.1161 +
  1.1162 +\noindent
  1.1163 +$m$ is the number of rows (auxiliary variables);
  1.1164 +
  1.1165 +\noindent
  1.1166 +$n$ is the number of columns (structural variables);
  1.1167 +
  1.1168 +\noindent
  1.1169 +\verb|stat| is the solution status (\verb|GLP_UNDEF| = 1,
  1.1170 +\verb|GLP_FEAS| = 2, \verb|GLP_NOFEAS| = 4, or \verb|GLP_OPT| = 5);
  1.1171 +
  1.1172 +\noindent
  1.1173 +\verb|obj_val| is the objective value;
  1.1174 +
  1.1175 +\noindent
  1.1176 +\verb|r_val[i]|, $i=1,\dots,m$, is the value of $i$-th row;
  1.1177 +
  1.1178 +\noindent
  1.1179 +\verb|c_val[j]|, $j=1,\dots,n$, is the value of $j$-th column.
  1.1180 +
  1.1181 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  1.1182 +
  1.1183 +\newpage
  1.1184 +
  1.1185 +\section{Post-optimal analysis routines}
  1.1186 +
  1.1187 +\subsection{glp\_print\_ranges---print sensitivity analysis report}
  1.1188 +
  1.1189 +\subsubsection*{Synopsis}
  1.1190 +
  1.1191 +\begin{verbatim}
  1.1192 +int glp_print_ranges(glp_prob *P, int len, const int list[],
  1.1193 +      int flags, const char *fname);
  1.1194 +\end{verbatim}
  1.1195 +
  1.1196 +\subsubsection*{Description}
  1.1197 +
  1.1198 +The routine \verb|glp_print_ranges| performs sensitivity analysis of
  1.1199 +current optimal basic solution and writes the analysis report in
  1.1200 +human-readable format to a text file, whose name is the character
  1.1201 +string {\it fname}. (Detailed description of the report structure is
  1.1202 +given below.)
  1.1203 +
  1.1204 +The parameter {\it len} specifies the length of the row/column list.
  1.1205 +
  1.1206 +The array {\it list} specifies ordinal number of rows and columns to be
  1.1207 +analyzed. The ordinal numbers should be passed in locations
  1.1208 +{\it list}[1], {\it list}[2], \dots, {\it list}[{\it len}]. Ordinal
  1.1209 +numbers from 1 to $m$ refer to rows, and ordinal numbers from $m+1$ to
  1.1210 +$m+n$ refer to columns, where $m$ and $n$ are, resp., the total number
  1.1211 +of rows and columns in the problem object. Rows and columns appear in
  1.1212 +the analysis report in the same order as they follow in the array list.
  1.1213 +
  1.1214 +It is allowed to specify $len=0$, in which case the array {\it list} is
  1.1215 +not used (so it can be specified as \verb|NULL|), and the routine
  1.1216 +performs analysis for all rows and columns of the problem object.
  1.1217 +
  1.1218 +The parameter {\it flags} is reserved for use in the future and must be
  1.1219 +specified as zero.
  1.1220 +
  1.1221 +On entry to the routine \verb|glp_print_ranges| the current basic
  1.1222 +solution must be optimal and the basis factorization must exist.
  1.1223 +The application program can check that with the routine
  1.1224 +\verb|glp_bf_exists|, and if the factorization does
  1.1225 +not exist, compute it with the routine \verb|glp_factorize|. Note that
  1.1226 +if the LP preprocessor is not used, on normal exit from the simplex
  1.1227 +solver routine \verb|glp_simplex| the basis factorization always exists.
  1.1228 +
  1.1229 +\subsubsection*{Returns}
  1.1230 +
  1.1231 +If the operation was successful, the routine \verb|glp_print_ranges|
  1.1232 +returns zero. Otherwise, it prints an error message and returns
  1.1233 +non-zero.
  1.1234 +
  1.1235 +\subsubsection*{Analysis report example}
  1.1236 +
  1.1237 +An example of the sensitivity analysis report is shown on the next two
  1.1238 +pages. This example corresponds to the example of LP problem described
  1.1239 +in Subsection ``Example of MPS file''.
  1.1240 +
  1.1241 +\subsubsection*{Structure of the analysis report}
  1.1242 +
  1.1243 +For each row and column specified in the array {\it list} the routine
  1.1244 +prints two lines containing generic information and analysis
  1.1245 +information, which depends on the status of corresponding row or column.
  1.1246 +
  1.1247 +Note that analysis of a row is analysis of its auxiliary variable,
  1.1248 +which is equal to the row linear form $\sum a_jx_j$, and analysis of
  1.1249 +a column is analysis of corresponding structural variable. Therefore,
  1.1250 +formally, on performing the sensitivity analysis there is no difference
  1.1251 +between rows and columns.
  1.1252 +
  1.1253 +\bigskip
  1.1254 +
  1.1255 +\noindent
  1.1256 +{\it Generic information}
  1.1257 +
  1.1258 +\medskip
  1.1259 +
  1.1260 +\noindent
  1.1261 +{\tt No.} is the row or column ordinal number in the problem object.
  1.1262 +Rows are numbered from 1 to $m$, and columns are numbered from 1 to $n$,
  1.1263 +where $m$ and $n$ are, resp., the total number of rows and columns in
  1.1264 +the problem object.
  1.1265 +
  1.1266 +\medskip
  1.1267 +
  1.1268 +\noindent
  1.1269 +{\tt Row name} is the symbolic name assigned to the row. If the row has
  1.1270 +no name assigned, this field contains blanks.
  1.1271 +
  1.1272 +\medskip
  1.1273 +
  1.1274 +\noindent
  1.1275 +{\tt Column name} is the symbolic name assigned to the column. If the
  1.1276 +column has no name assigned, this field contains blanks.
  1.1277 +
  1.1278 +\medskip
  1.1279 +
  1.1280 +\noindent
  1.1281 +{\tt St} is the status of the row or column in the optimal solution:
  1.1282 +
  1.1283 +{\tt BS} --- non-active constraint (row), basic column;
  1.1284 +
  1.1285 +{\tt NL} --- inequality constraint having its lower right-hand side
  1.1286 +active (row), non-basic column having its lower bound active;
  1.1287 +
  1.1288 +{\tt NU} --- inequality constraint having its upper right-hand side
  1.1289 +active (row), non-basic column having its upper bound active;
  1.1290 +
  1.1291 +{\tt NS} --- active equality constraint (row), non-basic fixed column.
  1.1292 +
  1.1293 +{\tt NF} --- active free row, non-basic free (unbounded) column. (This
  1.1294 +case means that the optimal solution is dual degenerate.)
  1.1295 +
  1.1296 +\medskip
  1.1297 +
  1.1298 +\noindent
  1.1299 +{\tt Activity} is the (primal) value of the auxiliary variable (row) or
  1.1300 +structural variable (column) in the optimal solution.
  1.1301 +
  1.1302 +\medskip
  1.1303 +
  1.1304 +\noindent
  1.1305 +{\tt Slack} is the (primal) value of the row slack variable.
  1.1306 +
  1.1307 +\medskip
  1.1308 +
  1.1309 +\noindent
  1.1310 +{\tt Obj coef} is the objective coefficient of the column (structural
  1.1311 +variable).
  1.1312 +
  1.1313 +\begin{landscape}
  1.1314 +\begin{scriptsize}
  1.1315 +\begin{verbatim}
  1.1316 +GLPK 4.42 - SENSITIVITY ANALYSIS REPORT                                                                         Page   1
  1.1317 +
  1.1318 +Problem:    PLAN
  1.1319 +Objective:  VALUE = 296.2166065 (MINimum)
  1.1320 +
  1.1321 +   No. Row name     St      Activity         Slack   Lower bound       Activity      Obj coef  Obj value at Limiting
  1.1322 +                                          Marginal   Upper bound          range         range   break point variable
  1.1323 +------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------
  1.1324 +     1 VALUE        BS     296.21661    -296.21661          -Inf      299.25255      -1.00000        .      MN
  1.1325 +                                            .               +Inf      296.21661          +Inf          +Inf
  1.1326 +
  1.1327 +     2 YIELD        NS    2000.00000        .         2000.00000     1995.06864          -Inf     296.28365 BIN3
  1.1328 +                                           -.01360    2000.00000     2014.03479          +Inf     296.02579 CU
  1.1329 +
  1.1330 +     3 FE           NU      60.00000        .               -Inf       55.89016          -Inf     306.77162 BIN4
  1.1331 +                                          -2.56823      60.00000       62.69978       2.56823     289.28294 BIN3
  1.1332 +
  1.1333 +     4 CU           BS      83.96751      16.03249          -Inf       93.88467       -.30613     270.51157 MN
  1.1334 +                                            .          100.00000       79.98213        .21474     314.24798 BIN5
  1.1335 +
  1.1336 +     5 MN           NU      40.00000        .               -Inf       34.42336          -Inf     299.25255 BIN4
  1.1337 +                                           -.54440      40.00000       41.68691        .54440     295.29825 BIN3
  1.1338 +
  1.1339 +     6 MG           BS      19.96029      10.03971          -Inf       24.74427      -1.79618     260.36433 BIN1
  1.1340 +                                            .           30.00000        9.40292        .28757     301.95652 MN
  1.1341 +
  1.1342 +     7 AL           NL    1500.00000        .         1500.00000     1485.78425       -.25199     292.63444 CU
  1.1343 +                                            .25199          +Inf     1504.92126          +Inf     297.45669 BIN3
  1.1344 +
  1.1345 +     8 SI           NL     250.00000      50.00000     250.00000      235.32871       -.48520     289.09812 CU
  1.1346 +                                            .48520     300.00000      255.06073          +Inf     298.67206 BIN3
  1.1347 +\end{verbatim}
  1.1348 +\end{scriptsize}
  1.1349 +\end{landscape}
  1.1350 +
  1.1351 +\begin{landscape}
  1.1352 +\begin{scriptsize}
  1.1353 +\begin{verbatim}
  1.1354 +GLPK 4.42 - SENSITIVITY ANALYSIS REPORT                                                                         Page   2
  1.1355 +
  1.1356 +Problem:    PLAN
  1.1357 +Objective:  VALUE = 296.2166065 (MINimum)
  1.1358 +
  1.1359 +   No. Column name  St      Activity      Obj coef   Lower bound       Activity      Obj coef  Obj value at Limiting
  1.1360 +                                          Marginal   Upper bound          range         range   break point variable
  1.1361 +------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------
  1.1362 +     1 BIN1         NL        .             .03000        .           -28.82475       -.22362     288.90594 BIN4
  1.1363 +                                            .25362     200.00000       33.88040          +Inf     304.80951 BIN4
  1.1364 +
  1.1365 +     2 BIN2         BS     665.34296        .08000        .           802.22222        .01722     254.44822 BIN1
  1.1366 +                                            .         2500.00000      313.43066        .08863     301.95652 MN
  1.1367 +
  1.1368 +     3 BIN3         BS     490.25271        .17000     400.00000      788.61314        .15982     291.22807 MN
  1.1369 +                                            .          800.00000     -347.42857        .17948     300.86548 BIN5
  1.1370 +
  1.1371 +     4 BIN4         BS     424.18773        .12000     100.00000      710.52632        .10899     291.54745 MN
  1.1372 +                                            .          700.00000     -256.15524        .14651     307.46010 BIN1
  1.1373 +
  1.1374 +     5 BIN5         NL        .             .15000        .          -201.78739        .13544     293.27940 BIN3
  1.1375 +                                            .01456    1500.00000       58.79586          +Inf     297.07244 BIN3
  1.1376 +
  1.1377 +     6 ALUM         BS     299.63899        .21000        .           358.26772        .18885     289.87879 AL
  1.1378 +                                            .               +Inf      112.40876        .22622     301.07527 MN
  1.1379 +
  1.1380 +     7 SILICON      BS     120.57762        .38000        .           124.27093        .14828     268.27586 BIN5
  1.1381 +                                            .               +Inf       85.54745        .46667     306.66667 MN
  1.1382 +
  1.1383 +End of report
  1.1384 +\end{verbatim}
  1.1385 +\end{scriptsize}
  1.1386 +\end{landscape}
  1.1387 +
  1.1388 +\noindent
  1.1389 +{\tt Marginal} is the reduced cost (dual activity) of the auxiliary
  1.1390 +variable (row) or structural variable (column).
  1.1391 +
  1.1392 +\medskip
  1.1393 +
  1.1394 +\noindent
  1.1395 +{\tt Lower bound} is the lower right-hand side (row) or lower bound
  1.1396 +(column). If the row or column has no lower bound, this field contains
  1.1397 +{\tt -Inf}.
  1.1398 +
  1.1399 +\medskip
  1.1400 +
  1.1401 +\noindent
  1.1402 +{\tt Upper bound} is the upper right-hand side (row) or upper bound
  1.1403 +(column). If the row or column has no upper bound, this field contains
  1.1404 +{\tt +Inf}.
  1.1405 +
  1.1406 +\bigskip
  1.1407 +
  1.1408 +\noindent
  1.1409 +{\it Sensitivity analysis of active bounds}
  1.1410 +
  1.1411 +\medskip
  1.1412 +
  1.1413 +\noindent
  1.1414 +The sensitivity analysis of active bounds is performed only for rows,
  1.1415 +which are active constraints, and only for non-basic columns, because
  1.1416 +inactive constraints and basic columns have no active bounds.
  1.1417 +
  1.1418 +For every auxiliary (row) or structural (column) non-basic variable the
  1.1419 +routine starts changing its active bound in both direction. The first
  1.1420 +of the two lines in the report corresponds to decreasing, and the
  1.1421 +second line corresponds to increasing of the active bound. Since the
  1.1422 +variable being analyzed is non-basic, its activity, which is equal to
  1.1423 +its active bound, also starts changing. This changing leads to changing
  1.1424 +of basic (auxiliary and structural) variables, which depend on the
  1.1425 +non-basic variable. The current basis remains primal feasible and
  1.1426 +therefore optimal while values of all basic variables are primal
  1.1427 +feasible, i.e. are within their bounds. Therefore, if some basic
  1.1428 +variable called the {\it limiting variable} reaches its (lower or
  1.1429 +upper) bound first, before any other basic variables, it thereby limits
  1.1430 +further changing of the non-basic variable, because otherwise the
  1.1431 +current basis would become primal infeasible. The point, at which this
  1.1432 +happens, is called the {\it break point}. Note that there are two break
  1.1433 +points: the lower break point, which corresponds to decreasing of the
  1.1434 +non-basic variable, and the upper break point, which corresponds to
  1.1435 +increasing of the non-basic variable.
  1.1436 +
  1.1437 +In the analysis report values of the non-basic variable (i.e. of its
  1.1438 +active bound) being analyzed at both lower and upper break points are
  1.1439 +printed in the field `{\tt Activity range}'. Corresponding values of
  1.1440 +the objective function are printed in the field `{\tt Obj value at
  1.1441 +break point}', and symbolic names of corresponding limiting basic
  1.1442 +variables are printed in the field `{\tt Limiting variable}'.
  1.1443 +If the active bound can decrease or/and increase unlimitedly, the field
  1.1444 +`{\tt Activity range}' contains {\tt -Inf} or/and {\tt +Inf}, resp.
  1.1445 +
  1.1446 +For example (see the example report above), row SI is a double-sided
  1.1447 +constraint, which is active on its lower bound (right-hand side), and
  1.1448 +its activity in the optimal solution being equal to the lower bound is
  1.1449 +250. The activity range for this row is $[235.32871,255.06073]$. This
  1.1450 +means that the basis remains optimal while the lower bound is
  1.1451 +increasing up to 255.06073, and further increasing is limited by
  1.1452 +(structural) variable BIN3. If the lower bound reaches this upper break
  1.1453 +point, the objective value becomes equal to 298.67206.
  1.1454 +
  1.1455 +Note that if the basis does not change, the objective function depends
  1.1456 +on the non-basic variable linearly, and the per-unit change of the
  1.1457 +objective function is the reduced cost (marginal value) of the
  1.1458 +non-basic variable.
  1.1459 +
  1.1460 +\bigskip
  1.1461 +
  1.1462 +\noindent
  1.1463 +{\it Sensitivity analysis of objective coefficients at non-basic
  1.1464 +variables}
  1.1465 +
  1.1466 +\medskip
  1.1467 +
  1.1468 +\noindent
  1.1469 +The sensitivity analysis of the objective coefficient at a non-basic
  1.1470 +variable is quite simple, because in this case change in the objective
  1.1471 +coefficient leads to equivalent change in the reduced cost (marginal
  1.1472 +value).
  1.1473 +
  1.1474 +For every auxiliary (row) or structural (column) non-basic variable the
  1.1475 +routine starts changing its objective coefficient in both direction.
  1.1476 +(Note that auxiliary variables are not included in the objective
  1.1477 +function and therefore always have zero objective coefficients.) The
  1.1478 +first of the two lines in the report corresponds to decreasing, and the
  1.1479 +second line corresponds to increasing of the objective coefficient.
  1.1480 +This changing leads to changing of the reduced cost of the non-basic
  1.1481 +variable to be analyzed and does affect reduced costs of all other
  1.1482 +non-basic variables. The current basis remains dual feasible and
  1.1483 +therefore optimal while the reduced cost keeps its sign. Therefore, if
  1.1484 +the reduced cost reaches zero, it limits further changing of the
  1.1485 +objective coefficient (if only the non-basic variable is non-fixed).
  1.1486 +
  1.1487 +In the analysis report minimal and maximal values of the objective
  1.1488 +coefficient, on which the basis remains optimal, are printed in the
  1.1489 +field `\verb|Obj coef range|'. If the objective coefficient can
  1.1490 +decrease or/and increase unlimitedly, this field contains {\tt -Inf}
  1.1491 +or/and {\tt +Inf}, resp.
  1.1492 +
  1.1493 +For example (see the example report above), column BIN5 is non-basic
  1.1494 +having its lower bound active. Its objective coefficient is 0.15, and
  1.1495 +reduced cost in the optimal solution 0.01456. The column lower bound
  1.1496 +remains active while the column reduced cost remains non-negative,
  1.1497 +thus, minimal value of the objective coefficient, on which the current
  1.1498 +basis still remains optimal, is $0.15-0.01456=0.13644$, that is
  1.1499 +indicated in the field `\verb|Obj coef range|'.
  1.1500 +
  1.1501 +\bigskip
  1.1502 +
  1.1503 +\noindent
  1.1504 +{\it Sensitivity analysis of objective coefficients at basic variables}
  1.1505 +
  1.1506 +\medskip
  1.1507 +
  1.1508 +\noindent
  1.1509 +To perform sensitivity analysis for every auxiliary (row) or structural
  1.1510 +(column) variable the routine starts changing its objective coefficient
  1.1511 +in both direction. (Note that auxiliary variables are not included in
  1.1512 +the objective function and therefore always have zero objective
  1.1513 +coefficients.) The first of the two lines in the report corresponds to
  1.1514 +decreasing, and the second line corresponds to increasing of the
  1.1515 +objective coefficient. This changing leads to changing of reduced costs
  1.1516 +of non-basic variables. The current basis remains dual feasible and
  1.1517 +therefore optimal while reduced costs of all non-basic variables
  1.1518 +(except fixed variables) keep their signs. Therefore, if the reduced
  1.1519 +cost of some non-basic non-fixed variable called the {\it limiting
  1.1520 +variable} reaches zero first, before reduced cost of any other
  1.1521 +non-basic non-fixed variable, it thereby limits further changing of the
  1.1522 +objective coefficient, because otherwise the current basis would become
  1.1523 +dual infeasible (non-optimal). The point, at which this happens, is
  1.1524 +called the {\it break point}. Note that there are two break points: the
  1.1525 +lower break point, which corresponds to decreasing of the objective
  1.1526 +coefficient, and the upper break point, which corresponds to increasing
  1.1527 +of the objective coefficient. Let the objective coefficient reach its
  1.1528 +limit value and continue changing a bit further in the same direction
  1.1529 +that makes the current basis dual infeasible (non-optimal). Then the
  1.1530 +reduced cost of the non-basic limiting variable becomes ``a bit'' dual
  1.1531 +infeasible that forces the limiting variable to enter the basis
  1.1532 +replacing there some basic variable, which leaves the basis to keep its
  1.1533 +primal feasibility. It should be understood that if we change the
  1.1534 +current basis in this way exactly at the break point, both the current
  1.1535 +and adjacent bases will be optimal with the same objective value,
  1.1536 +because at the break point the limiting variable has zero reduced cost.
  1.1537 +On the other hand, in the adjacent basis the value of the limiting
  1.1538 +variable changes, because there it becomes basic, that leads to
  1.1539 +changing of the value of the basic variable being analyzed. Note that
  1.1540 +on determining the adjacent basis the bounds of the analyzed basic
  1.1541 +variable are ignored as if it were a free (unbounded) variable, so it
  1.1542 +cannot leave the current basis.
  1.1543 +
  1.1544 +In the analysis report lower and upper limits of the objective
  1.1545 +coefficient at the basic variable being analyzed, when the basis
  1.1546 +remains optimal, are printed in the field `{\tt Obj coef range}'.
  1.1547 +Corresponding values of the objective function at both lower and upper
  1.1548 +break points are printed in the field `{\tt Obj value at break point}',
  1.1549 +symbolic names of corresponding non-basic limiting variables are
  1.1550 +printed in the field `{\tt Limiting variable}', and values of the basic
  1.1551 +variable, which it would take on in the adjacent bases (as was
  1.1552 +explained above) are printed in the field `{\tt Activity range}'.
  1.1553 +If the objective coefficient can increase or/and decrease unlimitedly,
  1.1554 +the field `{\tt Obj coef range}' contains {\tt -Inf} and/or {\tt +Inf},
  1.1555 +resp. It also may happen that no dual feasible adjacent basis exists
  1.1556 +(i.e. on entering the basis the limiting variable can increase or
  1.1557 +decrease unlimitedly), in which case the field `{\tt Activity range}'
  1.1558 +contains {\tt -Inf} and/or {\tt +Inf}.
  1.1559 +
  1.1560 +\newpage
  1.1561 +
  1.1562 +For example (see the example report above), structural variable
  1.1563 +(column) BIN3 is basic, its optimal value is 490.25271, and its
  1.1564 +objective coefficient is 0.17. The objective coefficient range for this
  1.1565 +column is $[0.15982,0.17948]$. This means that the basis remains
  1.1566 +optimal while the objective coefficient is decreasing down to 0.15982,
  1.1567 +and further decreasing is limited by (auxiliary) variable MN. If we
  1.1568 +make the objective coefficient a bit less than 0.15982, the limiting
  1.1569 +variable MN will enter the basis, and in that adjacent basis the
  1.1570 +structural variable BIN3 will take on new optimal value 788.61314. At
  1.1571 +the lower break point, where the objective coefficient is exactly
  1.1572 +0.15982, the objective function takes on the value 291.22807 in both
  1.1573 +the current and adjacent bases.
  1.1574 +
  1.1575 +Note that if the basis does not change, the objective function depends
  1.1576 +on the objective coefficient at the basic variable linearly, and the
  1.1577 +per-unit change of the objective function is the value of the basic
  1.1578 +variable.
  1.1579 +
  1.1580 +%* eof *%