%* glpk03.tex *%

 \chapter{Utility API routines}

 \section{Problem data reading/writing routines}

 \subsection{glp\_read\_mps---read problem data in MPS format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_read_mps(glp_prob *lp, int fmt, const void *parm,

       const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_read_mps| reads problem data in MPS format from a

 text file. (The MPS format is described in Appendix \ref{champs}, page

 \pageref{champs}.)

 The parameter \verb|fmt| specifies the MPS format version as follows:

 \begin{tabular}{@{}ll}

 \verb|GLP_MPS_DECK| & fixed (ancient) MPS format; \\

 \verb|GLP_MPS_FILE| & free (modern) MPS format. \\

 \end{tabular}

 The parameter \verb|parm| is reserved for use in the future and must be

 specified as \verb|NULL|.

 The character string \verb|fname| specifies a name of the text file to

 be read in. (If the file name ends with suffix `\verb|.gz|', the file is

 assumed to be compressed, in which case the routine \verb|glp_read_mps|

 decompresses it ``on the fly''.)

 Note that before reading data the current content of the problem object

 is completely erased with the routine \verb|glp_erase_prob|.

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_read_mps|

 returns zero. Otherwise, it prints an error message and returns

 non-zero.

 \subsection{glp\_write\_mps---write problem data in MPS format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_write_mps(glp_prob *lp, int fmt, const void *parm,

       const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_write_mps| writes problem data in MPS format to a

 text file. (The MPS format is described in Appendix \ref{champs}, page

 \pageref{champs}.)

 The parameter \verb|fmt| specifies the MPS format version as follows:

 \begin{tabular}{@{}ll}

 \verb|GLP_MPS_DECK| & fixed (ancient) MPS format; \\

 \verb|GLP_MPS_FILE| & free (modern) MPS format. \\

 \end{tabular}

 The parameter \verb|parm| is reserved for use in the future and must be

 specified as \verb|NULL|.

 The character string \verb|fname| specifies a name of the text file to

 be written out. (If the file name ends with suffix `\verb|.gz|', the

 file is assumed to be compressed, in which case the routine

 \verb|glp_write_mps| performs automatic compression on writing it.)

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_write_mps|

 returns zero. Otherwise, it prints an error message and returns

 non-zero.

 \subsection{glp\_read\_lp---read problem data in CPLEX LP format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_read_lp(glp_prob *lp, const void *parm,

       const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_read_lp| reads problem data in CPLEX LP format

 from a text file. (The CPLEX LP format is described in Appendix

 \ref{chacplex}, page \pageref{chacplex}.)

 The parameter \verb|parm| is reserved for use in the future and must be

 specified as \verb|NULL|.

 The character string \verb|fname| specifies a name of the text file to

 be read in. (If the file name ends with suffix `\verb|.gz|', the file is

 assumed to be compressed, in which case the routine \verb|glp_read_lp|

 decompresses it ``on the fly''.)

 Note that before reading data the current content of the problem object

 is completely erased with the routine \verb|glp_erase_prob|.

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_read_lp| returns

 zero. Otherwise, it prints an error message and returns non-zero.

 \subsection{glp\_write\_lp---write problem data in CPLEX LP format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_write_lp(glp_prob *lp, const void *parm,

       const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_write_lp| writes problem data in CPLEX LP format

 to a text file. (The CPLEX LP format is described in Appendix

 \ref{chacplex}, page \pageref{chacplex}.)

 The parameter \verb|parm| is reserved for use in the future and must be

 specified as \verb|NULL|.

 The character string \verb|fname| specifies a name of the text file to

 be written out. (If the file name ends with suffix `\verb|.gz|', the

 file is assumed to be compressed, in which case the routine

 \verb|glp_write_lp| performs automatic compression on writing it.)

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_write_lp|

 returns zero. Otherwise, it prints an error message and returns

 non-zero.

 \subsection{glp\_read\_prob---read problem data in GLPK format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_read_prob(glp_prob *P, int flags, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_read_prob| reads problem data in the GLPK LP/MIP

 format from a text file. (For description of the GLPK LP/MIP format see

 below.)

 The parameter \verb|flags| is reserved for use in the future and should

 be specified as zero.

 The character string \verb|fname| specifies a name of the text file to

 be read in. (If the file name ends with suffix `\verb|.gz|', the file

 is assumed to be compressed, in which case the routine

 \verb|glp_read_prob| decompresses it ``on the fly''.)

 Note that before reading data the current content of the problem object

 is completely erased with the routine \verb|glp_erase_prob|.

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_read_prob|

 returns zero. Otherwise, it prints an error message and returns

 non-zero.

 \subsubsection*{GLPK LP/MIP format}

 The GLPK LP/MIP format is a DIMACS-like format.\footnote{The DIMACS

 formats were developed by the Center for Discrete Mathematics and

 Theoretical Computer Science (DIMACS) to facilitate exchange of problem

 data. For details see: {\tt <http://dimacs.rutgers.edu/Challenges/>}. }

 The file in this format is a plain ASCII text file containing lines of

 several types described below. A line is terminated with the end-of-line

 character. Fields in each line are separated by at least one blank

 space. Each line begins with a one-character designator to identify the

 line type.

 The first line of the data file must be the problem line (except

 optional comment lines, which may precede the problem line). The last

 line of the data file must be the end line. Other lines may follow in

 arbitrary order, however, duplicate lines are not allowed.

 \paragraph{Comment lines.} Comment lines give human-readable

 information about the data file and are ignored by GLPK routines.

 Comment lines can appear anywhere in the data file. Each comment line

 begins with the lower-case character \verb|c|.

 \begin{verbatim}

    c This is an example of comment line

 \end{verbatim}

 \paragraph{Problem line.} There must be exactly one problem line in the

 data file. This line must appear before any other lines except comment

 lines and has the following format:

 \begin{verbatim}

    p CLASS DIR ROWS COLS NONZ

 \end{verbatim}

 The lower-case letter \verb|p| specifies that this is the problem line.

 The \verb|CLASS| field defines the problem class and can contain either

 the keyword \verb|lp| (that means linear programming problem) or

 \verb|mip| (that means mixed integer programming problem).

 The \verb|DIR| field defines the optimization direction (that is, the

 objective function sense) and can contain either the keyword \verb|min|

 (that means minimization) or \verb|max| (that means maximization).

 The \verb|ROWS|, \verb|COLS|, and \verb|NONZ| fields contain

 non-negative integer values specifying, respectively, the number of

 rows (constraints), columns (variables), and non-zero constraint

 coefficients in the problem instance. Note that \verb|NONZ| value does

 not account objective coefficients.

 \paragraph{Row descriptors.} There must be at most one row descriptor

 line in the data file for each row (constraint). This line has one of

 the following formats:

 \begin{verbatim}

    i ROW f

    i ROW l RHS

    i ROW u RHS

    i ROW d RHS1 RHS2

    i ROW s RHS

 \end{verbatim}

 The lower-case letter \verb|i| specifies that this is the row

 descriptor line.

 The \verb|ROW| field specifies the row ordinal number, an integer

 between 1 and $m$, where $m$ is the number of rows in the problem

 instance.

 The next lower-case letter specifies the row type as follows:

 \verb|f| --- free (unbounded) row: $-\infty<\sum a_jx_j<+\infty$;

 \verb|l| --- inequality constraint of `$\geq$' type:

 $\sum a_jx_j\geq b$;

 \verb|u| --- inequality constraint of `$\leq$' type:

 $\sum a_jx_j\leq b$;

 \verb|d| --- double-sided inequality constraint:

 $b_1\leq\sum a_jx_j\leq b_2$;

 \verb|s| --- equality constraint: $\sum a_jx_j=b$.

 The \verb|RHS| field contains a floaing-point value specifying the

 row right-hand side. The \verb|RHS1| and \verb|RHS2| fields contain

 floating-point values specifying, respectively, the lower and upper

 right-hand sides for the double-sided row.

 If for some row its descriptor line does not appear in the data file,

 by default that row is assumed to be an equality constraint with zero

 right-hand side.

 \paragraph{Column descriptors.} There must be at most one column

 descriptor line in the data file for each column (variable). This line

 has one of the following formats depending on the problem class

 specified in the problem line:

 \bigskip

 \begin{tabular}{@{}l@{\hspace*{40pt}}l}

 LP class & MIP class \\

 \hline

 \verb|j COL f|           & \verb|j COL KIND f|           \\

 \verb|j COL l BND|       & \verb|j COL KIND l BND|       \\

 \verb|j COL u BND|       & \verb|j COL KIND u BND|       \\

 \verb|j COL d BND1 BND2| & \verb|j COL KIND d BND1 BND2| \\

 \verb|j COL s BND|       & \verb|j COL KIND s BND|       \\

 \end{tabular}

 \bigskip

 The lower-case letter \verb|j| specifies that this is the column

 descriptor line.

 The \verb|COL| field specifies the column ordinal number, an integer

 between 1 and $n$, where $n$ is the number of columns in the problem

 instance.

 The \verb|KIND| field is used only for MIP problems and specifies the

 column kind as follows:

 \verb|c| --- continuous column;

 \verb|i| --- integer column;

 \verb|b| --- binary column (in this case all remaining fields must be

 omitted).

 The next lower-case letter specifies the column type as follows:

 \verb|f| --- free (unbounded) column: $-\infty<x<+\infty$;

 \verb|l| --- column with lower bound: $x\geq l$;

 \verb|u| --- column with upper bound: $x\leq u$;

 \verb|d| --- double-bounded column: $l\leq x\leq u$;

 \verb|s| --- fixed column: $x=s$.

 The \verb|BND| field contains a floating-point value that specifies the

 column bound. The \verb|BND1| and \verb|BND2| fields contain

 floating-point values specifying, respectively, the lower and upper

 bounds for the double-bounded column.

 If for some column its descriptor line does not appear in the file, by

 default that column is assumed to be non-negative (in case of LP class)

 or binary (in case of MIP class).

 \paragraph{Coefficient descriptors.} There must be exactly one

 coefficient descriptor line in the data file for each non-zero

 objective or constraint coefficient. This line has the following format:

 \begin{verbatim}

    a ROW COL VAL

 \end{verbatim}

 The lower-case letter \verb|a| specifies that this is the coefficient

 descriptor line.

 For objective coefficients the \verb|ROW| field must contain 0. For

 constraint coefficients the \verb|ROW| field specifies the row ordinal

 number, an integer between 1 and $m$, where $m$ is the number of rows

 in the problem instance.

 The \verb|COL| field specifies the column ordinal number, an integer

 between 1 and $n$, where $n$ is the number of columns in the problem

 instance.

 If both the \verb|ROW| and \verb|COL| fields contain 0, the line

 specifies the constant term (``shift'') of the objective function

 rather than objective coefficient.

 The \verb|VAL| field contains a floating-point coefficient value (it is

 allowed to specify zero value in this field).

 The number of constraint coefficient descriptor lines must be exactly

 the same as specified in the field \verb|NONZ| of the problem line.

 \paragraph{Symbolic name descriptors.} There must be at most one

 symbolic name descriptor line for the problem instance, objective

 function, each row (constraint), and each column (variable). This line

 has one of the following formats:

 \begin{verbatim}

    n p NAME

    n z NAME

    n i ROW NAME

    n j COL NAME

 \end{verbatim}

 The lower-case letter \verb|n| specifies that this is the symbolic name

 descriptor line.

 The next lower-case letter specifies which object should be assigned a

 symbolic name:

 \verb|p| --- problem instance;

 \verb|z| --- objective function;

 \verb|i| --- row (constraint);

 \verb|j| --- column (variable).

 The \verb|ROW| field specifies the row ordinal number, an integer

 between 1 and $m$, where $m$ is the number of rows in the problem

 instance.

 The \verb|COL| field specifies the column ordinal number, an integer

 between 1 and $n$, where $n$ is the number of columns in the problem

 instance.

 The \verb|NAME| field contains the symbolic name, a sequence from 1 to

 255 arbitrary graphic ASCII characters, assigned to corresponding

 object.

 \paragraph{End line.} There must be exactly one end line in the data

 file. This line must appear last in the file and has the following

 format:

 \begin{verbatim}

    e

 \end{verbatim}

 The lower-case letter \verb|e| specifies that this is the end line.

 Anything that follows the end line is ignored by GLPK routines.

 \subsubsection*{Example of data file in GLPK LP/MIP format}

 The following example of a data file in GLPK LP/MIP format specifies

 the same LP problem as in Subsection ``Example of MPS file''.

 \begin{center}

 \footnotesize\tt

 \begin{tabular}{l@{\hspace*{50pt}}}

 p lp min 8 7 48   \\

 n p PLAN          \\

 n z VALUE         \\

 i 1 f             \\

 n i 1 VALUE       \\

 i 2 s 2000        \\

 n i 2 YIELD       \\

 i 3 u 60          \\

 n i 3 FE          \\

 i 4 u 100         \\

 n i 4 CU          \\

 i 5 u 40          \\

 n i 5 MN          \\

 i 6 u 30          \\

 n i 6 MG          \\

 i 7 l 1500        \\

 n i 7 AL          \\

 i 8 d 250 300     \\

 n i 8 SI          \\

 j 1 d 0 200       \\

 n j 1 BIN1        \\

 j 2 d 0 2500      \\

 n j 2 BIN2        \\

 j 3 d 400 800     \\

 n j 3 BIN3        \\

 j 4 d 100 700     \\

 n j 4 BIN4        \\

 j 5 d 0 1500      \\

 n j 5 BIN5        \\

 n j 6 ALUM        \\

 n j 7 SILICON     \\

 a 0 1 0.03        \\

 a 0 2 0.08        \\

 a 0 3 0.17        \\

 a 0 4 0.12        \\

 a 0 5 0.15        \\

 a 0 6 0.21        \\

 a 0 7 0.38        \\

 a 1 1 0.03        \\

 a 1 2 0.08        \\

 a 1 3 0.17        \\

 a 1 4 0.12        \\

 a 1 5 0.15        \\

 a 1 6 0.21        \\

 \end{tabular}

 \begin{tabular}{|@{\hspace*{80pt}}l}

 a 1 7 0.38        \\

 a 2 1 1           \\

 a 2 2 1           \\

 a 2 3 1           \\

 a 2 4 1           \\

 a 2 5 1           \\

 a 2 6 1           \\

 a 2 7 1           \\

 a 3 1 0.15        \\

 a 3 2 0.04        \\

 a 3 3 0.02        \\

 a 3 4 0.04        \\

 a 3 5 0.02        \\

 a 3 6 0.01        \\

 a 3 7 0.03        \\

 a 4 1 0.03        \\

 a 4 2 0.05        \\

 a 4 3 0.08        \\

 a 4 4 0.02        \\

 a 4 5 0.06        \\

 a 4 6 0.01        \\

 a 5 1 0.02        \\

 a 5 2 0.04        \\

 a 5 3 0.01        \\

 a 5 4 0.02        \\

 a 5 5 0.02        \\

 a 6 1 0.02        \\

 a 6 2 0.03        \\

 a 6 5 0.01        \\

 a 7 1 0.7         \\

 a 7 2 0.75        \\

 a 7 3 0.8         \\

 a 7 4 0.75        \\

 a 7 5 0.8         \\

 a 7 6 0.97        \\

 a 8 1 0.02        \\

 a 8 2 0.06        \\

 a 8 3 0.08        \\

 a 8 4 0.12        \\

 a 8 5 0.02        \\

 a 8 6 0.01        \\

 a 8 7 0.97        \\

 e o f             \\

 \\

 \end{tabular}

 \end{center}

 \newpage

 \subsection{glp\_write\_prob---write problem data in GLPK format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_write_prob(glp_prob *P, int flags, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_write_prob| writes problem data in the GLPK

 LP/MIP format to a text file. (For description of the GLPK LP/MIP

 format see Subsection ``Read problem data in GLPK format''.)

 The parameter \verb|flags| is reserved for use in the future and should

 be specified as zero.

 The character string \verb|fname| specifies a name of the text file to

 be written out. (If the file name ends with suffix `\verb|.gz|', the

 file is assumed to be compressed, in which case the routine

 \verb|glp_write_prob| performs automatic compression on writing it.)

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_read_prob|

 returns zero. Otherwise, it prints an error message and returns

 non-zero.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \newpage

 \section{Routines for processing MathProg models}

 \subsection{Introduction}

 GLPK supports the {\it GNU MathProg modeling language}.\footnote{The

 GNU MathProg modeling language is a subset of the AMPL language. For

 its detailed description see the document ``Modeling Language GNU

 MathProg: Language Reference'' included in the GLPK distribution.}

 As a rule, models written in MathProg are solved with the GLPK LP/MIP

 stand-alone solver \verb|glpsol| (see Appendix D) and do not need any

 programming with API routines. However, for various reasons the user

 may need to process MathProg models directly in his/her application

 program, in which case he/she may use API routines described in this

 section. These routines provide an interface to the {\it MathProg

 translator}, a component of GLPK, which translates MathProg models into

 an internal code and then interprets (executes) this code.

 The processing of a model written in GNU MathProg includes several

 steps, which should be performed in the following order:

 \begin{enumerate}

 \item{\it Allocating the workspace.}

 The translator allocates the workspace, an internal data structure used

 on all subsequent steps.

 \item{\it Reading model section.} The translator reads model section

 and, optionally, data section from a specified text file and translates

 them into the internal code. If necessary, on this step data section

 may be ignored.

 \item{\it Reading data section(s).} The translator reads one or more

 data sections from specified text file(s) and translates them into the

 internal code.

 \item{\it Generating the model.} The translator executes the internal

 code to evaluate the content of the model objects such as sets,

 parameters, variables, constraints, and objectives. On this step the

 execution is suspended at the solve statement.

 \item {\it Building the problem object.} The translator obtains all

 necessary information from the workspace and builds the standard

 problem object (that is, the program object of type \verb|glp_prob|).

 \item{\it Solving the problem.} On this step the problem object built

 on the previous step is passed to a solver, which solves the problem

 instance and stores its solution back to the problem object.

 \item{\it Postsolving the model.} The translator copies the solution

 from the problem object to the workspace and then executes the internal

 code from the solve statement to the end of the model. (If model has

 no solve statement, the translator does nothing on this step.)

 \item{\it Freeing the workspace.} The translator frees all the memory

 allocated to the workspace.

 \end{enumerate}

 Note that the MathProg translator performs no error correction, so if

 any of steps 2 to 7 fails (due to errors in the model), the application

 program should terminate processing and go to step 8.

 \subsubsection*{Example 1}

 In this example the program reads model and data sections from input

 file \verb|egypt.mod|\footnote{This is an example model included in

 the GLPK distribution.} and writes the model to output file

 \verb|egypt.mps| in free MPS format (see Appendix B). No solution is

 performed.

 \begin{small}

 \begin{verbatim}

 /* mplsamp1.c */

 #include <stdio.h>

 #include <stdlib.h>

 #include <glpk.h>

 int main(void)

 {     glp_prob *lp;

       glp_tran *tran;

       int ret;

       lp = glp_create_prob();

       tran = glp_mpl_alloc_wksp();

       ret = glp_mpl_read_model(tran, "egypt.mod", 0);

       if (ret != 0)

       {  fprintf(stderr, "Error on translating model\n");

          goto skip;

       }

       ret = glp_mpl_generate(tran, NULL);

       if (ret != 0)

       {  fprintf(stderr, "Error on generating model\n");

          goto skip;

       }

       glp_mpl_build_prob(tran, lp);

       ret = glp_write_mps(lp, GLP_MPS_FILE, NULL, "egypt.mps");

       if (ret != 0)

          fprintf(stderr, "Error on writing MPS file\n");

 skip: glp_mpl_free_wksp(tran);

       glp_delete_prob(lp);

       return 0;

 }

 /* eof */

 \end{verbatim}

 \end{small}

 \subsubsection*{Example 2}

 In this example the program reads model section from file

 \verb|sudoku.mod|\footnote{This is an example model which is included

 in the GLPK distribution along with alternative data file

 {\tt sudoku.dat}.} ignoring data section in this file, reads alternative

 data section from file \verb|sudoku.dat|, solves the problem instance

 and passes the solution found back to the model.

 \begin{small}

 \begin{verbatim}

 /* mplsamp2.c */

 #include <stdio.h>

 #include <stdlib.h>

 #include <glpk.h>

 int main(void)

 {     glp_prob *mip;

       glp_tran *tran;

       int ret;

       mip = glp_create_prob();

       tran = glp_mpl_alloc_wksp();

       ret = glp_mpl_read_model(tran, "sudoku.mod", 1);

       if (ret != 0)

       {  fprintf(stderr, "Error on translating model\n");

          goto skip;

       }

       ret = glp_mpl_read_data(tran, "sudoku.dat");

       if (ret != 0)

       {  fprintf(stderr, "Error on translating data\n");

          goto skip;

       }

       ret = glp_mpl_generate(tran, NULL);

       if (ret != 0)

       {  fprintf(stderr, "Error on generating model\n");

          goto skip;

       }

       glp_mpl_build_prob(tran, mip);

       glp_simplex(mip, NULL);

       glp_intopt(mip, NULL);

       ret = glp_mpl_postsolve(tran, mip, GLP_MIP);

       if (ret != 0)

          fprintf(stderr, "Error on postsolving model\n");

 skip: glp_mpl_free_wksp(tran);

       glp_delete_prob(mip);

       return 0;

 }

 /* eof */

 \end{verbatim}

 \end{small}

 \subsection{glp\_mpl\_alloc\_wksp---allocate the translator workspace}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 glp_tran *glp_mpl_alloc_wksp(void);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_alloc_wksp| allocates the MathProg translator

 work\-space. (Note that multiple instances of the workspace may be

 allocated, if necessary.)

 \subsubsection*{Returns}

 The routine returns a pointer to the workspace, which should be used in

 all subsequent operations.

 \subsection{glp\_mpl\_read\_model---read and translate model section}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_mpl_read_model(glp_tran *tran, const char *fname,

       int skip);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_read_model| reads model section and,

 optionally, data section, which may follow the model section, from a

 text file, whose name is the character string \verb|fname|, performs

 translation of model statements and data blocks, and stores all the

 information in the workspace.

 The parameter \verb|skip| is a flag. If the input file contains the

 data section and this flag is non-zero, the data section is not read as

 if there were no data section and a warning message is printed. This

 allows reading data section(s) from other file(s).

 \subsubsection*{Returns}

 If the operation is successful, the routine returns zero. Otherwise

 the routine prints an error message and returns non-zero.

 \subsection{glp\_mpl\_read\_data---read and translate data section}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_mpl_read_data(glp_tran *tran, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_read_data| reads data section from a text

 file, whose name is the character string \verb|fname|, performs

 translation of data blocks, and stores the data read in the translator

 workspace. If necessary, this routine may be called more than once.

 \subsubsection*{Returns}

 If the operation is successful, the routine returns zero. Otherwise

 the routine prints an error message and returns non-zero.

 \subsection{glp\_mpl\_generate---generate the model}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_mpl_generate(glp_tran *tran, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_generate| generates the model using its

 description stored in the translator workspace. This operation means

 generating all variables, constraints, and objectives, executing check

 and display statements, which precede the solve statement (if it is

 presented).

 The character string \verb|fname| specifies the name of an output text

 file, to which output produced by display statements should be written.

 If \verb|fname| is \verb|NULL|, the output is sent to the terminal.

 \subsubsection*{Returns}

 If the operation is successful, the routine returns zero. Otherwise

 the routine prints an error message and returns non-zero.

 \subsection{glp\_mpl\_build\_prob---build problem instance from the

 model}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 void glp_mpl_build_prob(glp_tran *tran, glp_prob *prob);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_build_prob| obtains all necessary information

 from the translator workspace and stores it in the specified problem

 object \verb|prob|. Note that before building the current content of

 the problem object is erased with the routine \verb|glp_erase_prob|.

 \subsection{glp\_mpl\_postsolve---postsolve the model}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_mpl_postsolve(glp_tran *tran, glp_prob *prob,

       int sol);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_postsolve| copies the solution from the

 specified problem object \verb|prob| to the translator workspace and

 then executes all the remaining model statements, which follow the

 solve statement.

 The parameter \verb|sol| specifies which solution should be copied

 from the problem object to the workspace as follows:

 \begin{tabular}{@{}ll}

 \verb|GLP_SOL| & basic solution; \\

 \verb|GLP_IPT| & interior-point solution; \\

 \verb|GLP_MIP| & mixed integer solution. \\

 \end{tabular}

 \subsubsection*{Returns}

 If the operation is successful, the routine returns zero. Otherwise

 the routine prints an error message and returns non-zero.

 \subsection{glp\_mpl\_free\_wksp---free the translator workspace}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 void glp_mpl_free_wksp(glp_tran *tran);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_mpl_free_wksp| frees all the memory allocated to

 the translator workspace. It also frees all other resources, which are

 still used by the translator.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \newpage

 \section{Problem solution reading/writing routines}

 \subsection{glp\_print\_sol---write basic solution in printable format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_print_sol(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_print_sol writes| the current basic solution of

 an LP problem, which is specified by the pointer \verb|lp|, to a text

 file, whose name is the character string \verb|fname|, in printable

 format.

 Information reported by the routine \verb|glp_print_sol| is intended

 mainly for visual analysis.

 \subsubsection*{Returns}

 If no errors occurred, the routine returns zero. Otherwise the routine

 prints an error message and returns non-zero.

 \subsection{glp\_read\_sol---read basic solution from text file}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_read_sol(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_read_sol| reads basic solution from a text file

 whose name is specified by the parameter \verb|fname| into the problem

 object.

 For the file format see description of the routine \verb|glp_write_sol|.

 \subsubsection*{Returns}

 On success the routine returns zero, otherwise non-zero.

 \newpage

 \subsection{glp\_write\_sol---write basic solution to text file}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_write_sol(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_write_sol| writes the current basic solution to a

 text file whose name is specified by the parameter \verb|fname|. This

 file can be read back with the routine \verb|glp_read_sol|.

 \subsubsection*{Returns}

 On success the routine returns zero, otherwise non-zero.

 \subsubsection*{File format}

 The file created by the routine \verb|glp_write_sol| is a plain text

 file, which contains the following information:

 \begin{verbatim}

    m n

    p_stat d_stat obj_val

    r_stat[1] r_prim[1] r_dual[1]

    . . .

    r_stat[m] r_prim[m] r_dual[m]

    c_stat[1] c_prim[1] c_dual[1]

    . . .

    c_stat[n] c_prim[n] c_dual[n]

 \end{verbatim}

 \noindent

 where:

 \noindent

 $m$ is the number of rows (auxiliary variables);

 \noindent

 $n$ is the number of columns (structural variables);

 \noindent

 \verb|p_stat| is the primal status of the basic solution

 (\verb|GLP_UNDEF| = 1, \verb|GLP_FEAS| = 2, \verb|GLP_INFEAS| = 3, or

 \verb|GLP_NOFEAS| = 4);

 \noindent

 \verb|d_stat| is the dual status of the basic solution

 (\verb|GLP_UNDEF| = 1, \verb|GLP_FEAS| = 2, \verb|GLP_INFEAS| = 3, or

 \verb|GLP_NOFEAS| = 4);

 \noindent

 \verb|obj_val| is the objective value;

 \noindent

 \verb|r_stat[i]|, $i=1,\dots,m$, is the status of $i$-th row

 (\verb|GLP_BS| = 1, \verb|GLP_NL| = 2, \verb|GLP_NU| = 3,

 \verb|GLP_NF| = 4, or \verb|GLP_NS| = 5);

 \noindent

 \verb|r_prim[i]|, $i=1,\dots,m$, is the primal value of $i$-th row;

 \noindent

 \verb|r_dual[i]|, $i=1,\dots,m$, is the dual value of $i$-th row;

 \noindent

 \verb|c_stat[j]|, $j=1,\dots,n$, is the status of $j$-th column

 (\verb|GLP_BS| = 1, \verb|GLP_NL| = 2, \verb|GLP_NU| = 3,

 \verb|GLP_NF| = 4, or \verb|GLP_NS| = 5);

 \noindent

 \verb|c_prim[j]|, $j=1,\dots,n$, is the primal value of $j$-th column;

 \noindent

 \verb|c_dual[j]|, $j=1,\dots,n$, is the dual value of $j$-th column.

 \subsection{glp\_print\_ipt---write interior-point solution in

 printable format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_print_ipt(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_print_ipt| writes the current interior point

 solution  of an LP problem, which the parameter \verb|lp| points to, to

 a text file, whose name is the character string \verb|fname|, in

 printable format.

 Information reported by the routine \verb|glp_print_ipt| is intended

 mainly for visual analysis.

 \subsubsection*{Returns}

 If no errors occurred, the routine returns zero. Otherwise the routine

 prints an error message and returns non-zero.

 \subsection{glp\_read\_ipt---read interior-point solution from text

 file}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_read_ipt(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_read_ipt| reads interior-point solution from a

 text file whose name is specified by the parameter \verb|fname| into the

 problem object.

 For the file format see description of the routine \verb|glp_write_ipt|.

 \subsubsection*{Returns}

 On success the routine returns zero, otherwise non-zero.

 \subsection{glp\_write\_ipt---write interior-point solution to text

 file}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_write_ipt(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_write_ipt| writes the current interior-point

 solution to a text file whose name is specified by the parameter

 \verb|fname|. This file can be read back with the routine

 \verb|glp_read_ipt|.

 \subsubsection*{Returns}

 On success the routine returns zero, otherwise non-zero.

 \subsubsection*{File format}

 The file created by the routine \verb|glp_write_ipt| is a plain text

 file, which contains the following information:

 \begin{verbatim}

    m n

    stat obj_val

    r_prim[1] r_dual[1]

    . . .

    r_prim[m] r_dual[m]

    c_prim[1] c_dual[1]

    . . .

    c_prim[n] c_dual[n]

 \end{verbatim}

 \noindent

 where:

 \noindent

 $m$ is the number of rows (auxiliary variables);

 \noindent

 $n$ is the number of columns (structural variables);

 \noindent

 \verb|stat| is the solution status (\verb|GLP_UNDEF| = 1 or

 \verb|GLP_OPT| = 5);

 \noindent

 \verb|obj_val| is the objective value;

 \noindent

 \verb|r_prim[i]|, $i=1,\dots,m$, is the primal value of $i$-th row;

 \noindent

 \verb|r_dual[i]|, $i=1,\dots,m$, is the dual value of $i$-th row;

 \noindent

 \verb|c_prim[j]|, $j=1,\dots,n$, is the primal value of $j$-th column;

 \noindent

 \verb|c_dual[j]|, $j=1,\dots,n$, is the dual value of $j$-th column.

 \subsection{glp\_print\_mip---write MIP solution in printable format}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_print_mip(glp_prob *lp, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_print_mip| writes a best known integer solution

 of a MIP problem, which is specified by the pointer \verb|lp|, to a text

 file, whose name is the character string \verb|fname|, in printable

 format.

 Information reported by the routine \verb|glp_print_mip| is intended

 mainly for visual analysis.

 \subsubsection*{Returns}

 If no errors occurred, the routine returns zero. Otherwise the routine

 prints an error message and returns non-zero.

 \newpage

 \subsection{glp\_read\_mip---read MIP solution from text file}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_read_mip(glp_prob *mip, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_read_mip| reads MIP solution from a text file

 whose name is specified by the parameter \verb|fname| into the problem

 object.

 For the file format see description of the routine \verb|glp_write_mip|.

 \subsubsection*{Returns}

 On success the routine returns zero, otherwise non-zero.

 \subsection{glp\_write\_mip---write MIP solution to text file}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_write_mip(glp_prob *mip, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_write_mip| writes the current MIP solution to a

 text file whose name is specified by the parameter \verb|fname|. This

 file can be read back with the routine \verb|glp_read_mip|.

 \subsubsection*{Returns}

 On success the routine returns zero, otherwise non-zero.

 \subsubsection*{File format}

 The file created by the routine \verb|glp_write_sol| is a plain text

 file, which contains the following information:

 \begin{verbatim}

    m n

    stat obj_val

    r_val[1]

    . . .

    r_val[m]

    c_val[1]

    . . .

    c_val[n]

 \end{verbatim}

 \noindent

 where:

 \noindent

 $m$ is the number of rows (auxiliary variables);

 \noindent

 $n$ is the number of columns (structural variables);

 \noindent

 \verb|stat| is the solution status (\verb|GLP_UNDEF| = 1,

 \verb|GLP_FEAS| = 2, \verb|GLP_NOFEAS| = 4, or \verb|GLP_OPT| = 5);

 \noindent

 \verb|obj_val| is the objective value;

 \noindent

 \verb|r_val[i]|, $i=1,\dots,m$, is the value of $i$-th row;

 \noindent

 \verb|c_val[j]|, $j=1,\dots,n$, is the value of $j$-th column.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \newpage

 \section{Post-optimal analysis routines}

 \subsection{glp\_print\_ranges---print sensitivity analysis report}

 \subsubsection*{Synopsis}

 \begin{verbatim}

 int glp_print_ranges(glp_prob *P, int len, const int list[],

       int flags, const char *fname);

 \end{verbatim}

 \subsubsection*{Description}

 The routine \verb|glp_print_ranges| performs sensitivity analysis of

 current optimal basic solution and writes the analysis report in

 human-readable format to a text file, whose name is the character

 string {\it fname}. (Detailed description of the report structure is

 given below.)

 The parameter {\it len} specifies the length of the row/column list.

 The array {\it list} specifies ordinal number of rows and columns to be

 analyzed. The ordinal numbers should be passed in locations

 {\it list}[1], {\it list}[2], \dots, {\it list}[{\it len}]. Ordinal

 numbers from 1 to $m$ refer to rows, and ordinal numbers from $m+1$ to

 $m+n$ refer to columns, where $m$ and $n$ are, resp., the total number

 of rows and columns in the problem object. Rows and columns appear in

 the analysis report in the same order as they follow in the array list.

 It is allowed to specify $len=0$, in which case the array {\it list} is

 not used (so it can be specified as \verb|NULL|), and the routine

 performs analysis for all rows and columns of the problem object.

 The parameter {\it flags} is reserved for use in the future and must be

 specified as zero.

 On entry to the routine \verb|glp_print_ranges| the current basic

 solution must be optimal and the basis factorization must exist.

 The application program can check that with the routine

 \verb|glp_bf_exists|, and if the factorization does

 not exist, compute it with the routine \verb|glp_factorize|. Note that

 if the LP preprocessor is not used, on normal exit from the simplex

 solver routine \verb|glp_simplex| the basis factorization always exists.

 \subsubsection*{Returns}

 If the operation was successful, the routine \verb|glp_print_ranges|

 returns zero. Otherwise, it prints an error message and returns

 non-zero.

 \subsubsection*{Analysis report example}

 An example of the sensitivity analysis report is shown on the next two

 pages. This example corresponds to the example of LP problem described

 in Subsection ``Example of MPS file''.

 \subsubsection*{Structure of the analysis report}

 For each row and column specified in the array {\it list} the routine

 prints two lines containing generic information and analysis

 information, which depends on the status of corresponding row or column.

 Note that analysis of a row is analysis of its auxiliary variable,

 which is equal to the row linear form $\sum a_jx_j$, and analysis of

 a column is analysis of corresponding structural variable. Therefore,

 formally, on performing the sensitivity analysis there is no difference

 between rows and columns.

 \bigskip

 \noindent

 {\it Generic information}

 \medskip

 \noindent

 {\tt No.} is the row or column ordinal number in the problem object.

 Rows are numbered from 1 to $m$, and columns are numbered from 1 to $n$,

 where $m$ and $n$ are, resp., the total number of rows and columns in

 the problem object.

 \medskip

 \noindent

 {\tt Row name} is the symbolic name assigned to the row. If the row has

 no name assigned, this field contains blanks.

 \medskip

 \noindent

 {\tt Column name} is the symbolic name assigned to the column. If the

 column has no name assigned, this field contains blanks.

 \medskip

 \noindent

 {\tt St} is the status of the row or column in the optimal solution:

 {\tt BS} --- non-active constraint (row), basic column;

 {\tt NL} --- inequality constraint having its lower right-hand side

 active (row), non-basic column having its lower bound active;

 {\tt NU} --- inequality constraint having its upper right-hand side

 active (row), non-basic column having its upper bound active;

 {\tt NS} --- active equality constraint (row), non-basic fixed column.

 {\tt NF} --- active free row, non-basic free (unbounded) column. (This

 case means that the optimal solution is dual degenerate.)

 \medskip

 \noindent

 {\tt Activity} is the (primal) value of the auxiliary variable (row) or

 structural variable (column) in the optimal solution.

 \medskip

 \noindent

 {\tt Slack} is the (primal) value of the row slack variable.

 \medskip

 \noindent

 {\tt Obj coef} is the objective coefficient of the column (structural

 variable).

 \begin{landscape}

 \begin{scriptsize}

 \begin{verbatim}

 GLPK 4.42 - SENSITIVITY ANALYSIS REPORT                                                                         Page   1

 Problem:    PLAN

 Objective:  VALUE = 296.2166065 (MINimum)

    No. Row name     St      Activity         Slack   Lower bound       Activity      Obj coef  Obj value at Limiting

                                           Marginal   Upper bound          range         range   break point variable

 ------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------

      1 VALUE        BS     296.21661    -296.21661          -Inf      299.25255      -1.00000        .      MN

                                             .               +Inf      296.21661          +Inf          +Inf

      2 YIELD        NS    2000.00000        .         2000.00000     1995.06864          -Inf     296.28365 BIN3

                                            -.01360    2000.00000     2014.03479          +Inf     296.02579 CU

      3 FE           NU      60.00000        .               -Inf       55.89016          -Inf     306.77162 BIN4

                                           -2.56823      60.00000       62.69978       2.56823     289.28294 BIN3

      4 CU           BS      83.96751      16.03249          -Inf       93.88467       -.30613     270.51157 MN

                                             .          100.00000       79.98213        .21474     314.24798 BIN5

      5 MN           NU      40.00000        .               -Inf       34.42336          -Inf     299.25255 BIN4

                                            -.54440      40.00000       41.68691        .54440     295.29825 BIN3

      6 MG           BS      19.96029      10.03971          -Inf       24.74427      -1.79618     260.36433 BIN1

                                             .           30.00000        9.40292        .28757     301.95652 MN

      7 AL           NL    1500.00000        .         1500.00000     1485.78425       -.25199     292.63444 CU

                                             .25199          +Inf     1504.92126          +Inf     297.45669 BIN3

      8 SI           NL     250.00000      50.00000     250.00000      235.32871       -.48520     289.09812 CU

                                             .48520     300.00000      255.06073          +Inf     298.67206 BIN3

 \end{verbatim}

 \end{scriptsize}

 \end{landscape}

 \begin{landscape}

 \begin{scriptsize}

 \begin{verbatim}

 GLPK 4.42 - SENSITIVITY ANALYSIS REPORT                                                                         Page   2

 Problem:    PLAN

 Objective:  VALUE = 296.2166065 (MINimum)

    No. Column name  St      Activity      Obj coef   Lower bound       Activity      Obj coef  Obj value at Limiting

                                           Marginal   Upper bound          range         range   break point variable

 ------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------

      1 BIN1         NL        .             .03000        .           -28.82475       -.22362     288.90594 BIN4

                                             .25362     200.00000       33.88040          +Inf     304.80951 BIN4

      2 BIN2         BS     665.34296        .08000        .           802.22222        .01722     254.44822 BIN1

                                             .         2500.00000      313.43066        .08863     301.95652 MN

      3 BIN3         BS     490.25271        .17000     400.00000      788.61314        .15982     291.22807 MN

                                             .          800.00000     -347.42857        .17948     300.86548 BIN5

      4 BIN4         BS     424.18773        .12000     100.00000      710.52632        .10899     291.54745 MN

                                             .          700.00000     -256.15524        .14651     307.46010 BIN1

      5 BIN5         NL        .             .15000        .          -201.78739        .13544     293.27940 BIN3

                                             .01456    1500.00000       58.79586          +Inf     297.07244 BIN3

      6 ALUM         BS     299.63899        .21000        .           358.26772        .18885     289.87879 AL

                                             .               +Inf      112.40876        .22622     301.07527 MN

      7 SILICON      BS     120.57762        .38000        .           124.27093        .14828     268.27586 BIN5

                                             .               +Inf       85.54745        .46667     306.66667 MN

 End of report

 \end{verbatim}

 \end{scriptsize}

 \end{landscape}

 \noindent

 {\tt Marginal} is the reduced cost (dual activity) of the auxiliary

 variable (row) or structural variable (column).

 \medskip

 \noindent

 {\tt Lower bound} is the lower right-hand side (row) or lower bound

 (column). If the row or column has no lower bound, this field contains

 {\tt -Inf}.

 \medskip

 \noindent

 {\tt Upper bound} is the upper right-hand side (row) or upper bound

 (column). If the row or column has no upper bound, this field contains

 {\tt +Inf}.

 \bigskip

 \noindent

 {\it Sensitivity analysis of active bounds}

 \medskip

 \noindent

 The sensitivity analysis of active bounds is performed only for rows,

 which are active constraints, and only for non-basic columns, because

 inactive constraints and basic columns have no active bounds.

 For every auxiliary (row) or structural (column) non-basic variable the

 routine starts changing its active bound in both direction. The first

 of the two lines in the report corresponds to decreasing, and the

 second line corresponds to increasing of the active bound. Since the

 variable being analyzed is non-basic, its activity, which is equal to

 its active bound, also starts changing. This changing leads to changing

 of basic (auxiliary and structural) variables, which depend on the

 non-basic variable. The current basis remains primal feasible and

 therefore optimal while values of all basic variables are primal

 feasible, i.e. are within their bounds. Therefore, if some basic

 variable called the {\it limiting variable} reaches its (lower or

 upper) bound first, before any other basic variables, it thereby limits

 further changing of the non-basic variable, because otherwise the

 current basis would become primal infeasible. The point, at which this

 happens, is called the {\it break point}. Note that there are two break

 points: the lower break point, which corresponds to decreasing of the

 non-basic variable, and the upper break point, which corresponds to

 increasing of the non-basic variable.

 In the analysis report values of the non-basic variable (i.e. of its

 active bound) being analyzed at both lower and upper break points are

 printed in the field `{\tt Activity range}'. Corresponding values of

 the objective function are printed in the field `{\tt Obj value at

 break point}', and symbolic names of corresponding limiting basic

 variables are printed in the field `{\tt Limiting variable}'.

 If the active bound can decrease or/and increase unlimitedly, the field

 `{\tt Activity range}' contains {\tt -Inf} or/and {\tt +Inf}, resp.

 For example (see the example report above), row SI is a double-sided

 constraint, which is active on its lower bound (right-hand side), and

 its activity in the optimal solution being equal to the lower bound is

 250. The activity range for this row is $[235.32871,255.06073]$. This

 means that the basis remains optimal while the lower bound is

 increasing up to 255.06073, and further increasing is limited by

 (structural) variable BIN3. If the lower bound reaches this upper break

 point, the objective value becomes equal to 298.67206.

 Note that if the basis does not change, the objective function depends

 on the non-basic variable linearly, and the per-unit change of the

 objective function is the reduced cost (marginal value) of the

 non-basic variable.

 \bigskip

 \noindent

 {\it Sensitivity analysis of objective coefficients at non-basic

 variables}

 \medskip

 \noindent

 The sensitivity analysis of the objective coefficient at a non-basic

 variable is quite simple, because in this case change in the objective

 coefficient leads to equivalent change in the reduced cost (marginal

 value).

 For every auxiliary (row) or structural (column) non-basic variable the

 routine starts changing its objective coefficient in both direction.

 (Note that auxiliary variables are not included in the objective

 function and therefore always have zero objective coefficients.) The

 first of the two lines in the report corresponds to decreasing, and the

 second line corresponds to increasing of the objective coefficient.

 This changing leads to changing of the reduced cost of the non-basic

 variable to be analyzed and does affect reduced costs of all other

 non-basic variables. The current basis remains dual feasible and

 therefore optimal while the reduced cost keeps its sign. Therefore, if

 the reduced cost reaches zero, it limits further changing of the

 objective coefficient (if only the non-basic variable is non-fixed).

 In the analysis report minimal and maximal values of the objective

 coefficient, on which the basis remains optimal, are printed in the

 field `\verb|Obj coef range|'. If the objective coefficient can

 decrease or/and increase unlimitedly, this field contains {\tt -Inf}

 or/and {\tt +Inf}, resp.

 For example (see the example report above), column BIN5 is non-basic

 having its lower bound active. Its objective coefficient is 0.15, and

 reduced cost in the optimal solution 0.01456. The column lower bound

 remains active while the column reduced cost remains non-negative,

 thus, minimal value of the objective coefficient, on which the current

 basis still remains optimal, is $0.15-0.01456=0.13644$, that is

 indicated in the field `\verb|Obj coef range|'.

 \bigskip

 \noindent

 {\it Sensitivity analysis of objective coefficients at basic variables}

 \medskip

 \noindent

 To perform sensitivity analysis for every auxiliary (row) or structural

 (column) variable the routine starts changing its objective coefficient

 in both direction. (Note that auxiliary variables are not included in

 the objective function and therefore always have zero objective

 coefficients.) The first of the two lines in the report corresponds to

 decreasing, and the second line corresponds to increasing of the

 objective coefficient. This changing leads to changing of reduced costs

 of non-basic variables. The current basis remains dual feasible and

 therefore optimal while reduced costs of all non-basic variables

 (except fixed variables) keep their signs. Therefore, if the reduced

 cost of some non-basic non-fixed variable called the {\it limiting

 variable} reaches zero first, before reduced cost of any other

 non-basic non-fixed variable, it thereby limits further changing of the

 objective coefficient, because otherwise the current basis would become

 dual infeasible (non-optimal). The point, at which this happens, is

 called the {\it break point}. Note that there are two break points: the

 lower break point, which corresponds to decreasing of the objective

 coefficient, and the upper break point, which corresponds to increasing

 of the objective coefficient. Let the objective coefficient reach its

 limit value and continue changing a bit further in the same direction

 that makes the current basis dual infeasible (non-optimal). Then the

 reduced cost of the non-basic limiting variable becomes ``a bit'' dual

 infeasible that forces the limiting variable to enter the basis

 replacing there some basic variable, which leaves the basis to keep its

 primal feasibility. It should be understood that if we change the

 current basis in this way exactly at the break point, both the current

 and adjacent bases will be optimal with the same objective value,

 because at the break point the limiting variable has zero reduced cost.

 On the other hand, in the adjacent basis the value of the limiting

 variable changes, because there it becomes basic, that leads to

 changing of the value of the basic variable being analyzed. Note that

 on determining the adjacent basis the bounds of the analyzed basic

 variable are ignored as if it were a free (unbounded) variable, so it

 cannot leave the current basis.

 In the analysis report lower and upper limits of the objective

 coefficient at the basic variable being analyzed, when the basis

 remains optimal, are printed in the field `{\tt Obj coef range}'.

 Corresponding values of the objective function at both lower and upper

 break points are printed in the field `{\tt Obj value at break point}',

 symbolic names of corresponding non-basic limiting variables are

 printed in the field `{\tt Limiting variable}', and values of the basic

 variable, which it would take on in the adjacent bases (as was

 explained above) are printed in the field `{\tt Activity range}'.

 If the objective coefficient can increase or/and decrease unlimitedly,

 the field `{\tt Obj coef range}' contains {\tt -Inf} and/or {\tt +Inf},

 resp. It also may happen that no dual feasible adjacent basis exists

 (i.e. on entering the basis the limiting variable can increase or

 decrease unlimitedly), in which case the field `{\tt Activity range}'

 contains {\tt -Inf} and/or {\tt +Inf}.

 \newpage

 For example (see the example report above), structural variable

 (column) BIN3 is basic, its optimal value is 490.25271, and its

 objective coefficient is 0.17. The objective coefficient range for this

 column is $[0.15982,0.17948]$. This means that the basis remains

 optimal while the objective coefficient is decreasing down to 0.15982,

 and further decreasing is limited by (auxiliary) variable MN. If we

 make the objective coefficient a bit less than 0.15982, the limiting

 variable MN will enter the basis, and in that adjacent basis the

 structural variable BIN3 will take on new optimal value 788.61314. At

 the lower break point, where the objective coefficient is exactly

 0.15982, the objective function takes on the value 291.22807 in both

 the current and adjacent bases.

 Note that if the basis does not change, the objective function depends

 on the objective coefficient at the basic variable linearly, and the

 per-unit change of the objective function is the value of the basic

 variable.

 %* eof *%