lemon-project-template-glpk: deps/glpk/doc/gmpl.tex annotate

lemon-project-template-glpk

annotate deps/glpk/doc/gmpl.tex @ 11:4fc6ad2fb8a6

Test GLPK in src/main.cc

author	Alpar Juttner <alpar@cs.elte.hu>
date	Sun, 06 Nov 2011 21:43:29 +0100
parents
children

rev	line source
alpar@9	1 %* gmpl.tex *%
alpar@9	2
alpar@9	3 %***********************************************************************
alpar@9	4 % This code is part of GLPK (GNU Linear Programming Kit).
alpar@9	5 %
alpar@9	6 % Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
alpar@9	7 % 2009, 2010 Andrew Makhorin, Department for Applied Informatics,
alpar@9	8 % Moscow Aviation Institute, Moscow, Russia. All rights reserved.
alpar@9	9 % E-mail: <mao@gnu.org>.
alpar@9	10 %
alpar@9	11 % GLPK is free software: you can redistribute it and/or modify it
alpar@9	12 % under the terms of the GNU General Public License as published by
alpar@9	13 % the Free Software Foundation, either version 3 of the License, or
alpar@9	14 % (at your option) any later version.
alpar@9	15 %
alpar@9	16 % GLPK is distributed in the hope that it will be useful, but WITHOUT
alpar@9	17 % ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
alpar@9	18 % or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
alpar@9	19 % License for more details.
alpar@9	20 %
alpar@9	21 % You should have received a copy of the GNU General Public License
alpar@9	22 % along with GLPK. If not, see <http://www.gnu.org/licenses/>.
alpar@9	23 %***********************************************************************
alpar@9	24
alpar@9	25 \documentclass[10pt]{article}
alpar@9	26 \usepackage[dvipdfm,linktocpage,colorlinks,linkcolor=blue]{hyperref}
alpar@9	27
alpar@9	28 \begin{document}
alpar@9	29
alpar@9	30 \thispagestyle{empty}
alpar@9	31
alpar@9	32 \begin{center}
alpar@9	33
alpar@9	34 \vspace*{1in}
alpar@9	35
alpar@9	36 \begin{huge}
alpar@9	37 \sf\bfseries Modeling Language GNU MathProg
alpar@9	38 \end{huge}
alpar@9	39
alpar@9	40 \vspace{0.5in}
alpar@9	41
alpar@9	42 \begin{LARGE}
alpar@9	43 \sf Language Reference
alpar@9	44 \end{LARGE}
alpar@9	45
alpar@9	46 \vspace{0.5in}
alpar@9	47
alpar@9	48 \begin{LARGE}
alpar@9	49 \sf for GLPK Version 4.45
alpar@9	50 \end{LARGE}
alpar@9	51
alpar@9	52 \vspace{0.5in}
alpar@9	53 \begin{Large}
alpar@9	54 \sf (DRAFT, December 2010)
alpar@9	55 \end{Large}
alpar@9	56
alpar@9	57 \end{center}
alpar@9	58
alpar@9	59 \newpage
alpar@9	60
alpar@9	61 \vspace*{1in}
alpar@9	62
alpar@9	63 \vfill
alpar@9	64
alpar@9	65 \noindent
alpar@9	66 The GLPK package is part of the GNU Project released under the aegis of
alpar@9	67 GNU.
alpar@9	68
alpar@9	69 \medskip\noindent
alpar@9	70 Copyright \copyright{} 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007,
alpar@9	71 2008, 2009, 2010 Andrew Makhorin, Department for Applied Informatics,
alpar@9	72 Moscow Aviation Institute, Moscow, Russia. All rights reserved.
alpar@9	73
alpar@9	74 \medskip\noindent
alpar@9	75 Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
alpar@9	76 MA 02110-1301, USA.
alpar@9	77
alpar@9	78 \medskip\noindent
alpar@9	79 Permission is granted to make and distribute verbatim copies of this
alpar@9	80 manual provided the copyright notice and this permission notice are
alpar@9	81 preserved on all copies.
alpar@9	82
alpar@9	83 \medskip\noindent
alpar@9	84 Permission is granted to copy and distribute modified versions of this
alpar@9	85 manual under the conditions for verbatim copying, provided also that
alpar@9	86 the entire resulting derived work is distributed under the terms of
alpar@9	87 a permission notice identical to this one.
alpar@9	88
alpar@9	89 \medskip\noindent
alpar@9	90 Permission is granted to copy and distribute translations of this
alpar@9	91 manual into another language, under the above conditions for modified
alpar@9	92 versions.
alpar@9	93
alpar@9	94 \newpage
alpar@9	95
alpar@9	96 \tableofcontents
alpar@9	97
alpar@9	98 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	99
alpar@9	100 \newpage
alpar@9	101
alpar@9	102 \section{Introduction}
alpar@9	103
alpar@9	104 {\it GNU MathProg} is a modeling language intended for describing
alpar@9	105 linear mathematical programming models.\footnote{The GNU MathProg
alpar@9	106 language is a subset of the AMPL language. Its GLPK implementation is
alpar@9	107 mainly based on the paper: {\it Robert Fourer}, {\it David M. Gay}, and
alpar@9	108 {\it Brian W. Kernighan}, ``A Modeling Language for Mathematical
alpar@9	109 Programming.'' {\it Management Science} 36 (1990)\linebreak pp. 519-54.}
alpar@9	110
alpar@9	111 Model descriptions written in the GNU MathProg language consist of
alpar@9	112 a set of statements and data blocks constructed by the user from the
alpar@9	113 language elements described in this document.
alpar@9	114
alpar@9	115 In a process called {\it translation}, a program called the {\it model
alpar@9	116 translator} analyzes the model description and translates it into
alpar@9	117 internal data structures, which may be then used either for generating
alpar@9	118 mathematical programming problem instance or directly by a program
alpar@9	119 called the {\it solver} to obtain numeric solution of the problem.
alpar@9	120
alpar@9	121 \subsection{Linear programming problem}
alpar@9	122 \label{problem}
alpar@9	123
alpar@9	124 In MathProg the linear programming (LP) problem is stated as follows:
alpar@9	125
alpar@9	126 \medskip
alpar@9	127
alpar@9	128 \noindent\hspace{.7in}minimize (or maximize)
alpar@9	129 $$z=c_1x_1+c_2x_2+\dots+c_nx_n+c_0\eqno(1.1)$$
alpar@9	130 \noindent\hspace{.7in}subject to linear constraints
alpar@9	131 $$
alpar@9	132 \begin{array}{l@{\ }c@{\ }r@{\ }c@{\ }r@{\ }c@{\ }r@{\ }c@{\ }l}
alpar@9	133 L_1&\leq&a_{11}x_1&+&a_{12}x_2&+\dots+&a_{1n}x_n&\leq&U_1\\
alpar@9	134 L_2&\leq&a_{21}x_1&+&a_{22}x_2&+\dots+&a_{2n}x_n&\leq&U_2\\
alpar@9	135 \multicolumn{9}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}\\
alpar@9	136 L_m&\leq&a_{m1}x_1&+&a_{m2}x_2&+\dots+&a_{mn}x_n&\leq&U_m\\
alpar@9	137 \end{array}\eqno(1.2)
alpar@9	138 $$
alpar@9	139 \noindent\hspace{.7in}and bounds of variables
alpar@9	140 $$
alpar@9	141 \begin{array}{l@{\ }c@{\ }c@{\ }c@{\ }l}
alpar@9	142 l_1&\leq&x_1&\leq&u_1\\
alpar@9	143 l_2&\leq&x_2&\leq&u_2\\
alpar@9	144 \multicolumn{5}{c}{.\ \ .\ \ .\ \ .\ \ .}\\
alpar@9	145 l_n&\leq&x_n&\leq&u_n\\
alpar@9	146 \end{array}\eqno(1.3)
alpar@9	147 $$
alpar@9	148 where $x_1$, $x_2$, \dots, $x_n$ are variables; $z$ is the objective
alpar@9	149 function; $c_1$, $c_2$, \dots, $c_n$ are objective coefficients; $c_0$
alpar@9	150 is the constant term (``shift'') of the objective function; $a_{11}$,
alpar@9	151 $a_{12}$, \dots, $a_{mn}$ are constraint coefficients; $L_1$, $L_2$,
alpar@9	152 \dots, $L_m$ are lower constraint bounds; $U_1$, $U_2$, \dots, $U_m$
alpar@9	153 are upper constraint bounds; $l_1$, $l_2$, \dots, $l_n$ are lower
alpar@9	154 bounds of variables; $u_1$, $u_2$, \dots, $u_n$ are upper bounds of
alpar@9	155 variables.
alpar@9	156
alpar@9	157 Bounds of variables and constraint bounds can be finite as well as
alpar@9	158 infinite. Besides, lower bounds can be equal to corresponding upper
alpar@9	159 bounds. Thus, the following types of variables and constraints are
alpar@9	160 allowed:
alpar@9	161
alpar@9	162 \newpage
alpar@9	163
alpar@9	164 \begin{tabular}{@{}r@{\ }c@{\ }c@{\ }c@{\ }l@{\hspace*{38pt}}l}
alpar@9	165 $-\infty$&$<$&$x$&$<$&$+\infty$&Free (unbounded) variable\\
alpar@9	166 $l$&$\leq$&$x$&$<$&$+\infty$&Variable with lower bound\\
alpar@9	167 $-\infty$&$<$&$x$&$\leq$&$u$&Variable with upper bound\\
alpar@9	168 $l$&$\leq$&$x$&$\leq$&$u$&Double-bounded variable\\
alpar@9	169 $l$&$=$&$x$&=&$u$&Fixed variable\\
alpar@9	170 \end{tabular}
alpar@9	171
alpar@9	172 \bigskip
alpar@9	173
alpar@9	174 \begin{tabular}{@{}r@{\ }c@{\ }c@{\ }c@{\ }ll}
alpar@9	175 $-\infty$&$<$&$\sum a_jx_j$&$<$&$+\infty$&Free (unbounded) linear
alpar@9	176 form\\
alpar@9	177 $L$&$\leq$&$\sum a_jx_j$&$<$&$+\infty$&Inequality constraint ``greater
alpar@9	178 than or equal to''\\
alpar@9	179 $-\infty$&$<$&$\sum a_jx_j$&$\leq$&$U$&Inequality constraint ``less
alpar@9	180 than or equal to''\\
alpar@9	181 $L$&$\leq$&$\sum a_jx_j$&$\leq$&$U$&Double-bounded inequality
alpar@9	182 constraint\\
alpar@9	183 $L$&$=$&$\sum a_jx_j$&=&$U$&Equality constraint\\
alpar@9	184 \end{tabular}
alpar@9	185
alpar@9	186 \bigskip
alpar@9	187
alpar@9	188 In addition to pure LP problems MathProg also allows mixed integer
alpar@9	189 linear programming (MIP) problems, where some or all variables are
alpar@9	190 restricted to be integer or binary.
alpar@9	191
alpar@9	192 \subsection{Model objects}
alpar@9	193
alpar@9	194 In MathProg the model is described in terms of sets, parameters,
alpar@9	195 variables, constraints, and objectives, which are called {\it model
alpar@9	196 objects}.
alpar@9	197
alpar@9	198 The user introduces particular model objects using the language
alpar@9	199 statements. Each model object is provided with a symbolic name that
alpar@9	200 uniquely identifies the object and is intended for referencing purposes.
alpar@9	201
alpar@9	202 Model objects, including sets, can be multidimensional arrays built
alpar@9	203 over indexing sets. Formally, $n$-dimensional array $A$ is the mapping:
alpar@9	204 $$A:\Delta\rightarrow\Xi,\eqno(1.4)$$
alpar@9	205 where $\Delta\subseteq S_1\times\dots\times S_n$ is a subset of the
alpar@9	206 Cartesian product of indexing sets,\linebreak $\Xi$ is a set of array members.
alpar@9	207 In MathProg the set $\Delta$ is called the {\it subscript domain}. Its
alpar@9	208 members are $n$-tuples $(i_1,\dots,i_n)$, where $i_1\in S_1$, \dots,
alpar@9	209 $i_n\in S_n$.
alpar@9	210
alpar@9	211 If $n=0$, the Cartesian product above has exactly one member (namely,
alpar@9	212 \linebreak 0-tuple), so it is convenient to think scalar objects as
alpar@9	213 0-dimensional arrays having one member.
alpar@9	214
alpar@9	215 The type of array members is determined by the type of corresponding
alpar@9	216 model object as follows:
alpar@9	217
alpar@9	218 \medskip
alpar@9	219
alpar@9	220 \noindent\hfil
alpar@9	221 \begin{tabular}{@{}ll@{}}
alpar@9	222 Model object&Array member\\
alpar@9	223 \hline
alpar@9	224 Set&Elemental plain set\\
alpar@9	225 Parameter&Number or symbol\\
alpar@9	226 Variable&Elemental variable\\
alpar@9	227 Constraint&Elemental constraint\\
alpar@9	228 Objective&Elemental objective\\
alpar@9	229 \end{tabular}
alpar@9	230
alpar@9	231 \medskip
alpar@9	232
alpar@9	233 In order to refer to a particular object member the object should be
alpar@9	234 provided with {\it subscripts}. For example, if $a$ is a 2-dimensional
alpar@9	235 parameter defined over $I\times J$, a reference to its particular
alpar@9	236 member can be written as $a[i,j]$, where $i\in I$ and $j\in J$. It is
alpar@9	237 understood that scalar objects being 0-dimensional need no subscripts.
alpar@9	238
alpar@9	239 \subsection{Structure of model description}
alpar@9	240
alpar@9	241 It is sometimes desirable to write a model which, at various points,
alpar@9	242 may require different data for each problem instance to be solved using
alpar@9	243 that model. For this reason in MathProg the model description consists
alpar@9	244 of two parts: the {\it model section} and the {\it data section}.
alpar@9	245
alpar@9	246 The model section is a main part of the model description that contains
alpar@9	247 declarations of model objects and is common for all problems based on
alpar@9	248 the corresponding model.
alpar@9	249
alpar@9	250 The data section is an optional part of the model description that
alpar@9	251 contains data specific for a particular problem instance.
alpar@9	252
alpar@9	253 Depending on what is more convenient the model and data sections can be
alpar@9	254 placed either in one file or in two separate files. The latter feature
alpar@9	255 allows having arbitrary number of different data sections to be used
alpar@9	256 with the same model section.
alpar@9	257
alpar@9	258 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	259
alpar@9	260 \newpage
alpar@9	261
alpar@9	262 \section{Coding model description}
alpar@9	263 \label{coding}
alpar@9	264
alpar@9	265 The model description is coded in plain text format using ASCII
alpar@9	266 character set. Characters valid in the model description are the
alpar@9	267 following:
alpar@9	268
alpar@9	269 \begin{itemize}
alpar@9	270 \item alphabetic characters:\\
alpar@9	271 \verb\|A B C D E F G H I J K L M N O P Q R S T U V W X Y Z\|\\
alpar@9	272 \verb\|a b c d e f g h i j k l m n o p q r s t u v w x y z _\|
alpar@9	273 \item numeric characters:\\
alpar@9	274 \verb\|0 1 2 3 4 5 6 7 8 9\|
alpar@9	275 \item special characters:\\
alpar@9	276 \verb?! " # & ' ( ) * + , - . / : ; < = > [ ] ^ { \| }?
alpar@9	277 \item white-space characters:\\
alpar@9	278 \verb\|SP HT CR NL VT FF\|
alpar@9	279 \end{itemize}
alpar@9	280
alpar@9	281 Within string literals and comments any ASCII characters (except
alpar@9	282 control characters) are valid.
alpar@9	283
alpar@9	284 White-space characters are non-significant. They can be used freely
alpar@9	285 between lexical units to improve readability of the model description.
alpar@9	286 They are also used to separate lexical units from each other if there
alpar@9	287 is no other way to do that.
alpar@9	288
alpar@9	289 Syntactically model description is a sequence of lexical units in the
alpar@9	290 following categories:
alpar@9	291
alpar@9	292 \begin{itemize}
alpar@9	293 \item symbolic names;
alpar@9	294 \item numeric literals;
alpar@9	295 \item string literals;
alpar@9	296 \item keywords;
alpar@9	297 \item delimiters;
alpar@9	298 \item comments.
alpar@9	299 \end{itemize}
alpar@9	300
alpar@9	301 The lexical units of the language are discussed below.
alpar@9	302
alpar@9	303 \subsection{Symbolic names}
alpar@9	304
alpar@9	305 A {\it symbolic name} consists of alphabetic and numeric characters,
alpar@9	306 the first of which must be alphabetic. All symbolic names are distinct
alpar@9	307 (case sensitive).
alpar@9	308
alpar@9	309 \medskip
alpar@9	310
alpar@9	311 \noindent{\bf Examples}
alpar@9	312
alpar@9	313 \medskip
alpar@9	314
alpar@9	315 \noindent\verb\|alpha123\|
alpar@9	316
alpar@9	317 \noindent\verb\|This_is_a_name\|
alpar@9	318
alpar@9	319 \noindent\verb\|_P123_abc_321\|
alpar@9	320
alpar@9	321 \newpage
alpar@9	322
alpar@9	323 Symbolic names are used to identify model objects (sets, parameters,
alpar@9	324 variables, constraints, objectives) and dummy indices.
alpar@9	325
alpar@9	326 All symbolic names (except names of dummy indices) must be unique, i.e.
alpar@9	327 the model description must have no objects with identical names.
alpar@9	328 Symbolic names of dummy indices must be unique within the scope, where
alpar@9	329 they are valid.
alpar@9	330
alpar@9	331 \subsection{Numeric literals}
alpar@9	332
alpar@9	333 A {\it numeric literal} has the form {\it xx}{\tt E}{\it syy}, where
alpar@9	334 {\it xx} is a number with optional decimal point, {\it s} is the sign
alpar@9	335 {\tt+} or {\tt-}, {\it yy} is a decimal exponent. The letter {\tt E} is
alpar@9	336 case insensitive and can be coded as {\tt e}.
alpar@9	337
alpar@9	338 \medskip
alpar@9	339
alpar@9	340 \noindent{\bf Examples}
alpar@9	341
alpar@9	342 \medskip
alpar@9	343
alpar@9	344 \noindent\verb\|123\|
alpar@9	345
alpar@9	346 \noindent\verb\|3.14159\|
alpar@9	347
alpar@9	348 \noindent\verb\|56.E+5\|
alpar@9	349
alpar@9	350 \noindent\verb\|.78\|
alpar@9	351
alpar@9	352 \noindent\verb\|123.456e-7\|
alpar@9	353
alpar@9	354 \medskip
alpar@9	355
alpar@9	356 Numeric literals are used to represent numeric quantities. They have
alpar@9	357 obvious fixed meaning.
alpar@9	358
alpar@9	359 \subsection{String literals}
alpar@9	360
alpar@9	361 A {\it string literal} is a sequence of arbitrary characters enclosed
alpar@9	362 either in single quotes or in double quotes. Both these forms are
alpar@9	363 equivalent.
alpar@9	364
alpar@9	365 If the single quote is part of a string literal enclosed in single
alpar@9	366 quotes, it must be coded twice. Analogously, if the double quote is
alpar@9	367 part of a string literal enclosed in double quotes, it must be coded
alpar@9	368 twice.
alpar@9	369
alpar@9	370 \medskip
alpar@9	371
alpar@9	372 \noindent{\bf Examples}
alpar@9	373
alpar@9	374 \medskip
alpar@9	375
alpar@9	376 \noindent\verb\|'This is a string'\|
alpar@9	377
alpar@9	378 \noindent\verb\|"This is another string"\|
alpar@9	379
alpar@9	380 \noindent\verb\|'1 + 2 = 3'\|
alpar@9	381
alpar@9	382 \noindent\verb\|'That''s all'\|
alpar@9	383
alpar@9	384 \noindent\verb\|"She said: ""No"""\|
alpar@9	385
alpar@9	386 \medskip
alpar@9	387
alpar@9	388 String literals are used to represent symbolic quantities.
alpar@9	389
alpar@9	390 \subsection{Keywords}
alpar@9	391
alpar@9	392 A {\it keyword} is a sequence of alphabetic characters and possibly
alpar@9	393 some special characters.
alpar@9	394
alpar@9	395 All keywords fall into two categories: {\it reserved keywords}, which
alpar@9	396 cannot be used as symbolic names, and {\it non-reserved keywords},
alpar@9	397 which being recognized by context can be used as symbolic names.
alpar@9	398
alpar@9	399 \newpage
alpar@9	400
alpar@9	401 The reserved keywords are the following:
alpar@9	402
alpar@9	403 \medskip
alpar@9	404
alpar@9	405 \noindent\hfil
alpar@9	406 \begin{tabular}{@{}p{.7in}p{.7in}p{.7in}p{.7in}@{}}
alpar@9	407 {\tt and}&{\tt else}&{\tt mod}&{\tt union}\\
alpar@9	408 {\tt by}&{\tt if}&{\tt not}&{\tt within}\\
alpar@9	409 {\tt cross}&{\tt in}&{\tt or}\\
alpar@9	410 {\tt diff}&{\tt inter}&{\tt symdiff}\\
alpar@9	411 {\tt div}&{\tt less}&{\tt then}\\
alpar@9	412 \end{tabular}
alpar@9	413
alpar@9	414 \medskip
alpar@9	415
alpar@9	416 Non-reserved keywords are described in following sections.
alpar@9	417
alpar@9	418 All the keywords have fixed meaning, which will be explained on
alpar@9	419 discussion of corresponding syntactic constructions, where the keywords
alpar@9	420 are used.
alpar@9	421
alpar@9	422 \subsection{Delimiters}
alpar@9	423
alpar@9	424 A {\it delimiter} is either a single special character or a sequence of
alpar@9	425 two special characters as follows:
alpar@9	426
alpar@9	427 \medskip
alpar@9	428
alpar@9	429 \noindent\hfil
alpar@9	430 \begin{tabular}{@{}p{.3in}p{.3in}p{.3in}p{.3in}p{.3in}p{.3in}@{}}
alpar@9	431 {\tt+}&{\tt\textasciicircum}&{\tt==}&{\tt!}&{\tt:}&{\tt)}\\
alpar@9	432 {\tt-}&{\tt\&}&{\tt>=}&{\tt\&\&}&{\tt;}&{\tt[}\\
alpar@9	433 {\tt*}&{\tt<}&{\tt>}&{\tt\|\|}&{\tt:=}&{\tt\|}\\
alpar@9	434 {\tt/}&{\tt<=}&{\tt<>}&{\tt.}&{\tt..}&{\tt\{}\\
alpar@9	435 {\tt**}&{\tt=}&{\tt!=}&{\tt,}&{\tt(}&{\tt\}}\\
alpar@9	436 \end{tabular}
alpar@9	437
alpar@9	438 \medskip
alpar@9	439
alpar@9	440 If the delimiter consists of two characters, there must be no spaces
alpar@9	441 between the characters.
alpar@9	442
alpar@9	443 All the delimiters have fixed meaning, which will be explained on
alpar@9	444 discussion corresponding syntactic constructions, where the delimiters
alpar@9	445 are used.
alpar@9	446
alpar@9	447 \subsection{Comments}
alpar@9	448
alpar@9	449 For documenting purposes the model description can be provided with
alpar@9	450 {\it comments}, which may have two different forms. The first form is
alpar@9	451 a {\it single-line comment}, which begins with the character {\tt\#}
alpar@9	452 and extends until end of line. The second form is a {\it comment
alpar@9	453 sequence}, which is a sequence of any characters enclosed within
alpar@9	454 {\tt/} and {\tt/}.
alpar@9	455
alpar@9	456 \medskip
alpar@9	457
alpar@9	458 \noindent{\bf Examples}
alpar@9	459
alpar@9	460 \medskip
alpar@9	461
alpar@9	462 \noindent\verb\|param n := 10; # This is a comment\|
alpar@9	463
alpar@9	464 \noindent\verb\|/* This is another comment */\|
alpar@9	465
alpar@9	466 \medskip
alpar@9	467
alpar@9	468 Comments are ignored by the model translator and can appear anywhere in
alpar@9	469 the model description, where white-space characters are allowed.
alpar@9	470
alpar@9	471 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	472
alpar@9	473 \newpage
alpar@9	474
alpar@9	475 \section{Expressions}
alpar@9	476
alpar@9	477 An {\it expression} is a rule for computing a value. In model
alpar@9	478 description expressions are used as constituents of certain statements.
alpar@9	479
alpar@9	480 In general case expressions consist of operands and operators.
alpar@9	481
alpar@9	482 Depending on the type of the resultant value all expressions fall into
alpar@9	483 the following categories:
alpar@9	484
alpar@9	485 \begin{itemize}
alpar@9	486 \item numeric expressions;
alpar@9	487 \item symbolic expressions;
alpar@9	488 \item indexing expressions;
alpar@9	489 \item set expressions;
alpar@9	490 \item logical expressions;
alpar@9	491 \item linear expressions.
alpar@9	492 \end{itemize}
alpar@9	493
alpar@9	494 \subsection{Numeric expressions}
alpar@9	495
alpar@9	496 A {\it numeric expression} is a rule for computing a single numeric
alpar@9	497 value represented as a floating-point number.
alpar@9	498
alpar@9	499 The primary numeric expression may be a numeric literal, dummy index,
alpar@9	500 unsubscripted parameter, subscripted parameter, built-in function
alpar@9	501 reference, iterated numeric expression, conditional numeric expression,
alpar@9	502 or another numeric expression enclosed in parentheses.
alpar@9	503
alpar@9	504 \medskip
alpar@9	505
alpar@9	506 \noindent{\bf Examples}
alpar@9	507
alpar@9	508 \medskip
alpar@9	509
alpar@9	510 \noindent
alpar@9	511 \begin{tabular}{@{}ll@{}}
alpar@9	512 \verb\|1.23\|&(numeric literal)\\
alpar@9	513 \verb\|j\|&(dummy index)\\
alpar@9	514 \verb\|time\|&(unsubscripted parameter)\\
alpar@9	515 \verb\|a['May 2003',j+1]\|&(subscripted parameter)\\
alpar@9	516 \verb\|abs(b[i,j])\|&(function reference)\\
alpar@9	517 \verb\|sum{i in S diff T} alpha[i] * b[i,j]\|&(iterated expression)\\
alpar@9	518 \verb\|if i in I then 2 * p else q[i+1]\|&(conditional expression)\\
alpar@9	519 \verb\|(b[i,j] + .5 * c)\|&(parenthesized expression)\\
alpar@9	520 \end{tabular}
alpar@9	521
alpar@9	522 \medskip
alpar@9	523
alpar@9	524 More general numeric expressions containing two or more primary numeric
alpar@9	525 expressions may be constructed by using certain arithmetic operators.
alpar@9	526
alpar@9	527 \medskip
alpar@9	528
alpar@9	529 \noindent{\bf Examples}
alpar@9	530
alpar@9	531 \medskip
alpar@9	532
alpar@9	533 \noindent\verb\|j+1\|
alpar@9	534
alpar@9	535 \noindent\verb\|2 * a[i-1,j+1] - b[i,j]\|
alpar@9	536
alpar@9	537 \noindent\verb\|sum{j in J} a[i,j] * x[j] + sum{k in K} b[i,k] * x[k]\|
alpar@9	538
alpar@9	539 \noindent\verb\|(if i in I then 2 * p else q[i+1]) / (a[i,j] + 1.5)\|
alpar@9	540
alpar@9	541 \subsubsection{Numeric literals}
alpar@9	542
alpar@9	543 If the primary numeric expression is a numeric literal, the resultant
alpar@9	544 value is obvious.
alpar@9	545
alpar@9	546 \subsubsection{Dummy indices}
alpar@9	547
alpar@9	548 If the primary numeric expression is a dummy index, the resultant value
alpar@9	549 is current value assigned to that dummy index.
alpar@9	550
alpar@9	551 \subsubsection{Unsubscripted parameters}
alpar@9	552
alpar@9	553 If the primary numeric expression is an unsubscripted parameter (which
alpar@9	554 must be 0-dimensional), the resultant value is the value of that
alpar@9	555 parameter.
alpar@9	556
alpar@9	557 \subsubsection{Subscripted parameters}
alpar@9	558
alpar@9	559 The primary numeric expression, which refers to a subscripted parameter,
alpar@9	560 has the following syntactic form:
alpar@9	561
alpar@9	562 \medskip
alpar@9	563
alpar@9	564 \noindent\hfil
alpar@9	565 {\it name}{\tt[}$i_1${\tt,} $i_2${\tt,} \dots{\tt,} $i_n${\tt]}
alpar@9	566
alpar@9	567 \medskip
alpar@9	568
alpar@9	569 \noindent where {\it name} is the symbolic name of the parameter,
alpar@9	570 $i_1$, $i_2$, \dots, $i_n$ are subscripts.
alpar@9	571
alpar@9	572 Each subscript must be a numeric or symbolic expression. The number of
alpar@9	573 subscripts in the subscript list must be the same as the dimension of
alpar@9	574 the parameter with which the subscript list is associated.
alpar@9	575
alpar@9	576 Actual values of subscript expressions are used to identify
alpar@9	577 a particular member of the parameter that determines the resultant
alpar@9	578 value of the primary expression.
alpar@9	579
alpar@9	580 \subsubsection{Function references}
alpar@9	581
alpar@9	582 In MathProg there exist the following built-in functions which may be
alpar@9	583 used in numeric expressions:
alpar@9	584
alpar@9	585 \medskip
alpar@9	586
alpar@9	587 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	588 {\tt abs(}$x${\tt)}&$\|x\|$, absolute value of $x$\\
alpar@9	589 {\tt atan(}$x${\tt)}&$\arctan x$, principal value of the arc tangent of
alpar@9	590 $x$ (in radians)\\
alpar@9	591 {\tt atan(}$y${\tt,} $x${\tt)}&$\arctan y/x$, principal value of the
alpar@9	592 arc tangent of $y/x$ (in radians). In this case the signs of both
alpar@9	593 arguments $y$ and $x$ are used to determine the quadrant of the
alpar@9	594 resultant value\\
alpar@9	595 {\tt card(}$X${\tt)}&$\|X\|$, cardinality (the number of elements) of
alpar@9	596 set $X$\\
alpar@9	597 {\tt ceil(}$x${\tt)}&$\lceil x\rceil$, smallest integer not less than
alpar@9	598 $x$ (``ceiling of $x$'')\\
alpar@9	599 {\tt cos(}$x${\tt)}&$\cos x$, cosine of $x$ (in radians)\\
alpar@9	600 {\tt exp(}$x${\tt)}&$e^x$, base-$e$ exponential of $x$\\
alpar@9	601 {\tt floor(}$x${\tt)}&$\lfloor x\rfloor$, largest integer not greater
alpar@9	602 than $x$ (``floor of $x$'')\\
alpar@9	603 \end{tabular}
alpar@9	604
alpar@9	605 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	606 {\tt gmtime()}&the number of seconds elapsed since 00:00:00~Jan~1, 1970,
alpar@9	607 Coordinated Universal Time (for details see Subsection \ref{gmtime},
alpar@9	608 page \pageref{gmtime})\\
alpar@9	609 {\tt length(}$s${\tt)}&$\|s\|$, length of character string $s$\\
alpar@9	610 {\tt log(}$x${\tt)}&$\log x$, natural logarithm of $x$\\
alpar@9	611 {\tt log10(}$x${\tt)}&$\log_{10}x$, common (decimal) logarithm of $x$\\
alpar@9	612 {\tt max(}$x_1${\tt,} $x_2${\tt,} \dots{\tt,} $x_n${\tt)}&the largest
alpar@9	613 of values $x_1$, $x_2$, \dots, $x_n$\\
alpar@9	614 {\tt min(}$x_1${\tt,} $x_2${\tt,} \dots{\tt,} $x_n${\tt)}&the smallest
alpar@9	615 of values $x_1$, $x_2$, \dots, $x_n$\\
alpar@9	616 {\tt round(}$x${\tt)}&rounding $x$ to nearest integer\\
alpar@9	617 {\tt round(}$x${\tt,} $n${\tt)}&rounding $x$ to $n$ fractional decimal
alpar@9	618 digits\\
alpar@9	619 {\tt sin(}$x${\tt)}&$\sin x$, sine of $x$ (in radians)\\
alpar@9	620 {\tt sqrt(}$x${\tt)}&$\sqrt{x}$, non-negative square root of $x$\\
alpar@9	621 {\tt str2time(}$s${\tt,} $f${\tt)}&converting character string $s$ to
alpar@9	622 calendar time (for details see Subsection \ref{str2time}, page
alpar@9	623 \pageref{str2time})\\
alpar@9	624 {\tt trunc(}$x${\tt)}&truncating $x$ to nearest integer\\
alpar@9	625 {\tt trunc(}$x${\tt,} $n${\tt)}&truncating $x$ to $n$ fractional
alpar@9	626 decimal digits\\
alpar@9	627 {\tt Irand224()}&generating pseudo-random integer uniformly distributed
alpar@9	628 in $[0,2^{24})$\\
alpar@9	629 {\tt Uniform01()}&generating pseudo-random number uniformly distributed
alpar@9	630 in $[0,1)$\\
alpar@9	631 {\tt Uniform(}$a${\tt,} $b${\tt)}&generating pseudo-random number
alpar@9	632 uniformly distributed in $[a,b)$\\
alpar@9	633 {\tt Normal01()}&generating Gaussian pseudo-random variate with
alpar@9	634 $\mu=0$ and $\sigma=1$\\
alpar@9	635 {\tt Normal(}$\mu${\tt,} $\sigma${\tt)}&generating Gaussian
alpar@9	636 pseudo-random variate with given $\mu$ and $\sigma$\\
alpar@9	637 \end{tabular}
alpar@9	638
alpar@9	639 \medskip
alpar@9	640
alpar@9	641 Arguments of all built-in functions, except {\tt card}, {\tt length},
alpar@9	642 and {\tt str2time}, must be numeric expressions. The argument of
alpar@9	643 {\tt card} must be a set expression. The argument of {\tt length} and
alpar@9	644 both arguments of {\tt str2time} must be symbolic expressions.
alpar@9	645
alpar@9	646 The resultant value of the numeric expression, which is a function
alpar@9	647 reference, is the result of applying the function to its argument(s).
alpar@9	648
alpar@9	649 Note that each pseudo-random generator function has a latent argument
alpar@9	650 (i.e. some internal state), which is changed whenever the function has
alpar@9	651 been applied. Thus, if the function is applied repeatedly even to
alpar@9	652 identical arguments, due to the side effect different resultant values
alpar@9	653 are always produced.
alpar@9	654
alpar@9	655 \subsubsection{Iterated expressions}
alpar@9	656 \label{itexpr}
alpar@9	657
alpar@9	658 An {\it iterated numeric expression} is a primary numeric expression,
alpar@9	659 which has the following syntactic form:
alpar@9	660
alpar@9	661 \medskip
alpar@9	662
alpar@9	663 \noindent\hfil
alpar@9	664 {\it iterated-operator indexing-expression integrand}
alpar@9	665
alpar@9	666 \medskip
alpar@9	667
alpar@9	668 \noindent where {\it iterated-operator} is the symbolic name of the
alpar@9	669 iterated operator to be performed (see below), {\it indexing-expression}
alpar@9	670 is an indexing expression which introduces dummy indices and controls
alpar@9	671 iterating, {\it integrand} is a numeric expression that participates in
alpar@9	672 the operation.
alpar@9	673
alpar@9	674 In MathProg there exist four iterated operators, which may be used in
alpar@9	675 numeric expressions:
alpar@9	676
alpar@9	677 \medskip
alpar@9	678
alpar@9	679 \noindent\hfil
alpar@9	680 \begin{tabular}{@{}lll@{}}
alpar@9	681 {\tt sum}&summation&$\displaystyle\sum_{(i_1,\dots,i_n)\in\Delta}
alpar@9	682 f(i_1,\dots,i_n)$\\
alpar@9	683 {\tt prod}&production&$\displaystyle\prod_{(i_1,\dots,i_n)\in\Delta}
alpar@9	684 f(i_1,\dots,i_n)$\\
alpar@9	685 {\tt min}&minimum&$\displaystyle\min_{(i_1,\dots,i_n)\in\Delta}
alpar@9	686 f(i_1,\dots,i_n)$\\
alpar@9	687 {\tt max}&maximum&$\displaystyle\max_{(i_1,\dots,i_n)\in\Delta}
alpar@9	688 f(i_1,\dots,i_n)$\\
alpar@9	689 \end{tabular}
alpar@9	690
alpar@9	691 \medskip
alpar@9	692
alpar@9	693 \noindent where $i_1$, \dots, $i_n$ are dummy indices introduced in
alpar@9	694 the indexing expression, $\Delta$ is the domain, a set of $n$-tuples
alpar@9	695 specified by the indexing expression which defines particular values
alpar@9	696 assigned to the dummy indices on performing the iterated operation,
alpar@9	697 $f(i_1,\dots,i_n)$ is the integrand, a numeric expression whose
alpar@9	698 resultant value depends on the dummy indices.
alpar@9	699
alpar@9	700 The resultant value of an iterated numeric expression is the result of
alpar@9	701 applying of the iterated operator to its integrand over all $n$-tuples
alpar@9	702 contained in the domain.
alpar@9	703
alpar@9	704 \subsubsection{Conditional expressions}
alpar@9	705 \label{ifthen}
alpar@9	706
alpar@9	707 A {\it conditional numeric expression} is a primary numeric expression,
alpar@9	708 which has one of the following two syntactic forms:
alpar@9	709
alpar@9	710 \medskip
alpar@9	711
alpar@9	712 \noindent\hfil
alpar@9	713 {\tt if} $b$ {\tt then} $x$ {\tt else} $y$
alpar@9	714
alpar@9	715 \medskip
alpar@9	716
alpar@9	717 \noindent\hspace{126.5pt}
alpar@9	718 {\tt if} $b$ {\tt then} $x$
alpar@9	719
alpar@9	720 \medskip
alpar@9	721
alpar@9	722 \noindent where $b$ is an logical expression, $x$ and $y$ are numeric
alpar@9	723 expressions.
alpar@9	724
alpar@9	725 The resultant value of the conditional expression depends on the value
alpar@9	726 of the logical expression that follows the keyword {\tt if}. If it
alpar@9	727 takes on the value {\it true}, the value of the conditional expression
alpar@9	728 is the value of the expression that follows the keyword {\tt then}.
alpar@9	729 Otherwise, if the logical expression takes on the value {\it false},
alpar@9	730 the value of the conditional expression is the value of the expression
alpar@9	731 that follows the keyword {\it else}. If the second, reduced form of the
alpar@9	732 conditional expression is used and the logical expression takes on the
alpar@9	733 value {\it false}, the resultant value of the conditional expression is
alpar@9	734 zero.
alpar@9	735
alpar@9	736 \subsubsection{Parenthesized expressions}
alpar@9	737
alpar@9	738 Any numeric expression may be enclosed in parentheses that
alpar@9	739 syntactically makes it a primary numeric expression.
alpar@9	740
alpar@9	741 Parentheses may be used in numeric expressions, as in algebra, to
alpar@9	742 specify the desired order in which operations are to be performed.
alpar@9	743 Where parentheses are used, the expression within the parentheses is
alpar@9	744 evaluated before the resultant value is used.
alpar@9	745
alpar@9	746 The resultant value of the parenthesized expression is the same as the
alpar@9	747 value of the expression enclosed within parentheses.
alpar@9	748
alpar@9	749 \subsubsection{Arithmetic operators}
alpar@9	750
alpar@9	751 In MathProg there exist the following arithmetic operators, which may
alpar@9	752 be used in numeric expressions:
alpar@9	753
alpar@9	754 \medskip
alpar@9	755
alpar@9	756 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	757 {\tt +} $x$&unary plus\\
alpar@9	758 {\tt -} $x$&unary minus\\
alpar@9	759 $x$ {\tt +} $y$&addition\\
alpar@9	760 $x$ {\tt -} $y$&subtraction\\
alpar@9	761 $x$ {\tt less} $y$&positive difference (if $x<y$ then 0 else $x-y$)\\
alpar@9	762 $x$ {\tt *} $y$&multiplication\\
alpar@9	763 $x$ {\tt /} $y$&division\\
alpar@9	764 $x$ {\tt div} $y$&quotient of exact division\\
alpar@9	765 $x$ {\tt mod} $y$&remainder of exact division\\
alpar@9	766 $x$ {\tt **} $y$, $x$ {\tt\textasciicircum} $y$&exponentiation (raising
alpar@9	767 to power)\\
alpar@9	768 \end{tabular}
alpar@9	769
alpar@9	770 \medskip
alpar@9	771
alpar@9	772 \noindent where $x$ and $y$ are numeric expressions.
alpar@9	773
alpar@9	774 If the expression includes more than one arithmetic operator, all
alpar@9	775 operators are performed from left to right according to the hierarchy
alpar@9	776 of operations (see below) with the only exception that the
alpar@9	777 exponentiaion operators are performed from right to left.
alpar@9	778
alpar@9	779 The resultant value of the expression, which contains arithmetic
alpar@9	780 operators, is the result of applying the operators to their operands.
alpar@9	781
alpar@9	782 \subsubsection{Hierarchy of operations}
alpar@9	783 \label{hierarchy}
alpar@9	784
alpar@9	785 The following list shows the hierarchy of operations in numeric
alpar@9	786 expressions:
alpar@9	787
alpar@9	788 \medskip
alpar@9	789
alpar@9	790 \noindent\hfil
alpar@9	791 \begin{tabular}{@{}ll@{}}
alpar@9	792 Operation&Hierarchy\\
alpar@9	793 \hline
alpar@9	794 Evaluation of functions ({\tt abs}, {\tt ceil}, etc.)&1st\\
alpar@9	795 Exponentiation ({\tt**}, {\tt\textasciicircum})&2nd\\
alpar@9	796 Unary plus and minus ({\tt+}, {\tt-})&3rd\\
alpar@9	797 Multiplication and division ({\tt*}, {\tt/}, {\tt div}, {\tt mod})&4th\\
alpar@9	798 Iterated operations ({\tt sum}, {\tt prod}, {\tt min}, {\tt max})&5th\\
alpar@9	799 Addition and subtraction ({\tt+}, {\tt-}, {\tt less})&6th\\
alpar@9	800 Conditional evaluation ({\tt if} \dots {\tt then} \dots {\tt else})&
alpar@9	801 7th\\
alpar@9	802 \end{tabular}
alpar@9	803
alpar@9	804 \medskip
alpar@9	805
alpar@9	806 This hierarchy is used to determine which of two consecutive operations
alpar@9	807 is performed first. If the first operator is higher than or equal to
alpar@9	808 the second, the first operation is performed. If it is not, the second
alpar@9	809 operator is compared to the third, etc. When the end of the expression
alpar@9	810 is reached, all of the remaining operations are performed in the
alpar@9	811 reverse order.
alpar@9	812
alpar@9	813 \newpage
alpar@9	814
alpar@9	815 \subsection{Symbolic expressions}
alpar@9	816
alpar@9	817 A {\it symbolic expression} is a rule for computing a single symbolic
alpar@9	818 value represented as a character string.
alpar@9	819
alpar@9	820 The primary symbolic expression may be a string literal, dummy index,
alpar@9	821 unsubscripted parameter, subscripted parameter, built-in function
alpar@9	822 reference, conditional symbolic expression, or another symbolic
alpar@9	823 expression enclosed in parentheses.
alpar@9	824
alpar@9	825 It is also allowed to use a numeric expression as the primary symbolic
alpar@9	826 expression, in which case the resultant value of the numeric expression
alpar@9	827 is automatically converted to the symbolic type.
alpar@9	828
alpar@9	829 \medskip
alpar@9	830
alpar@9	831 \noindent{\bf Examples}
alpar@9	832
alpar@9	833 \medskip
alpar@9	834
alpar@9	835 \noindent
alpar@9	836 \begin{tabular}{@{}ll@{}}
alpar@9	837 \verb\|'May 2003'\|&(string literal)\\
alpar@9	838 \verb\|j\|&(dummy index)\\
alpar@9	839 \verb\|p\|&(unsubscripted parameter)\\
alpar@9	840 \verb\|s['abc',j+1]\|&(subscripted parameter)\\
alpar@9	841 \verb\|substr(name[i],k+1,3)\|&(function reference)\\
alpar@9	842 \verb\|if i in I then s[i,j] else t[i+1]\|&(conditional expression)\\
alpar@9	843 \verb\|((10 * b[i,j]) & '.bis')\|&(parenthesized expression)\\
alpar@9	844 \end{tabular}
alpar@9	845
alpar@9	846 \medskip
alpar@9	847
alpar@9	848 More general symbolic expressions containing two or more primary
alpar@9	849 symbolic expressions may be constructed by using the concatenation
alpar@9	850 operator.
alpar@9	851
alpar@9	852 \medskip
alpar@9	853
alpar@9	854 \noindent{\bf Examples}
alpar@9	855
alpar@9	856 \medskip
alpar@9	857
alpar@9	858 \noindent\verb\|'abc[' & i & ',' & j & ']'\|
alpar@9	859
alpar@9	860 \noindent\verb\|"from " & city[i] & " to " & city[j]\|
alpar@9	861
alpar@9	862 \medskip
alpar@9	863
alpar@9	864 The principles of evaluation of symbolic expressions are completely
alpar@9	865 analogous to the ones given for numeric expressions (see above).
alpar@9	866
alpar@9	867 \subsubsection{Function references}
alpar@9	868
alpar@9	869 In MathProg there exist the following built-in functions which may be
alpar@9	870 used in symbolic expressions:
alpar@9	871
alpar@9	872 \medskip
alpar@9	873
alpar@9	874 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	875 {\tt substr(}$s${\tt,} $x${\tt)}&substring of $s$ starting from
alpar@9	876 position $x$\\
alpar@9	877 {\tt substr(}$s${\tt,} $x${\tt,} $y${\tt)}&substring of $s$ starting
alpar@9	878 from position $x$ and having length $y$\\
alpar@9	879 {\tt time2str(}$t${\tt,} $f${\tt)}&converting calendar time to
alpar@9	880 character string (for details see Subsection \ref{time2str}, page
alpar@9	881 \pageref{time2str})\\
alpar@9	882 \end{tabular}
alpar@9	883
alpar@9	884 \medskip
alpar@9	885
alpar@9	886 The first argument of {\tt substr} must be a symbolic expression while
alpar@9	887 its second and optional third arguments must be numeric expressions.
alpar@9	888
alpar@9	889 The first argument of {\tt time2str} must be a numeric expression, and
alpar@9	890 its second argument must be a symbolic expression.
alpar@9	891
alpar@9	892 The resultant value of the symbolic expression, which is a function
alpar@9	893 reference, is the result of applying the function to its arguments.
alpar@9	894
alpar@9	895 \subsubsection{Symbolic operators}
alpar@9	896
alpar@9	897 Currently in MathProg there exists the only symbolic operator:
alpar@9	898
alpar@9	899 \medskip
alpar@9	900
alpar@9	901 \noindent\hfil
alpar@9	902 {\tt s \& t}
alpar@9	903
alpar@9	904 \medskip
alpar@9	905
alpar@9	906 \noindent where $s$ and $t$ are symbolic expressions. This operator
alpar@9	907 means concatenation of its two symbolic operands, which are character
alpar@9	908 strings.
alpar@9	909
alpar@9	910 \subsubsection{Hierarchy of operations}
alpar@9	911
alpar@9	912 The following list shows the hierarchy of operations in symbolic
alpar@9	913 expressions:
alpar@9	914
alpar@9	915 \medskip
alpar@9	916
alpar@9	917 \noindent\hfil
alpar@9	918 \begin{tabular}{@{}ll@{}}
alpar@9	919 Operation&Hierarchy\\
alpar@9	920 \hline
alpar@9	921 Evaluation of numeric operations&1st-7th\\
alpar@9	922 Concatenation ({\tt\&})&8th\\
alpar@9	923 Conditional evaluation ({\tt if} \dots {\tt then} \dots {\tt else})&
alpar@9	924 7th\\
alpar@9	925 \end{tabular}
alpar@9	926
alpar@9	927 \medskip
alpar@9	928
alpar@9	929 This hierarchy has the same meaning as was explained above for numeric
alpar@9	930 expressions (see Subsection \ref{hierarchy}, page \pageref{hierarchy}).
alpar@9	931
alpar@9	932 \subsection{Indexing expressions and dummy indices}
alpar@9	933 \label{indexing}
alpar@9	934
alpar@9	935 An {\it indexing expression} is an auxiliary construction, which
alpar@9	936 specifies a plain set of $n$-tuples and introduces dummy indices. It
alpar@9	937 has two syntactic forms:
alpar@9	938
alpar@9	939 \medskip
alpar@9	940
alpar@9	941 \noindent\hspace{73.5pt}
alpar@9	942 {\tt\{} {\it entry}$_1${\tt,} {\it entry}$_2${\tt,} \dots{\tt,}
alpar@9	943 {\it entry}$_m$ {\tt\}}
alpar@9	944
alpar@9	945 \medskip
alpar@9	946
alpar@9	947 \noindent\hfil
alpar@9	948 {\tt\{} {\it entry}$_1${\tt,} {\it entry}$_2${\tt,} \dots{\tt,}
alpar@9	949 {\it entry}$_m$ {\tt:} {\it predicate} {\tt\}}
alpar@9	950
alpar@9	951 \medskip
alpar@9	952
alpar@9	953 \noindent where {\it entry}{$_1$}, {\it entry}{$_2$}, \dots,
alpar@9	954 {\it entry}{$_m$} are indexing entries, {\it predicate} is a logical
alpar@9	955 expression that specifies an optional predicate (logical condition).
alpar@9	956
alpar@9	957 Each {\it indexing entry} in the indexing expression has one of the
alpar@9	958 following three forms:
alpar@9	959
alpar@9	960 \medskip
alpar@9	961
alpar@9	962 \noindent\hspace{123pt}
alpar@9	963 $i$ {\tt in} $S$
alpar@9	964
alpar@9	965 \medskip
alpar@9	966
alpar@9	967 \noindent\hfil
alpar@9	968 {\tt(}$i_1${\tt,} $i_2${\tt,} \dots{\tt,}$i_n${\tt)} {\tt in} $S$
alpar@9	969
alpar@9	970 \medskip
alpar@9	971
alpar@9	972 \noindent\hspace{123pt}
alpar@9	973 $S$
alpar@9	974
alpar@9	975 \medskip
alpar@9	976
alpar@9	977 \noindent where $i_1$, $i_2$, \dots, $i_n$ are indices, $S$ is a set
alpar@9	978 expression (discussed in the next section) that specifies the basic set.
alpar@9	979
alpar@9	980 The number of indices in the indexing entry must be the same as the
alpar@9	981 dimension of the basic set $S$, i.e. if $S$ consists of 1-tuples, the
alpar@9	982 first form must be used, and if $S$ consists of $n$-tuples, where
alpar@9	983 $n>1$, the second form must be used.
alpar@9	984
alpar@9	985 If the first form of the indexing entry is used, the index $i$ can be
alpar@9	986 a dummy index only (see below). If the second form is used, the indices
alpar@9	987 $i_1$, $i_2$, \dots, $i_n$ can be either dummy indices or some numeric
alpar@9	988 or symbolic expressions, where at least one index must be a dummy index.
alpar@9	989 The third, reduced form of the indexing entry has the same effect as if
alpar@9	990 there were $i$ (if $S$ is 1-dimensional) or $i_1$, $i_2$, \dots, $i_n$
alpar@9	991 (if $S$ is $n$-dimensional) all specified as dummy indices.
alpar@9	992
alpar@9	993 A {\it dummy index} is an auxiliary model object, which acts like an
alpar@9	994 individual variable. Values assigned to dummy indices are components of
alpar@9	995 $n$-tuples from basic sets, i.e. some numeric and symbolic quantities.
alpar@9	996
alpar@9	997 For referencing purposes dummy indices can be provided with symbolic
alpar@9	998 names. However, unlike other model objects (sets, parameters, etc.)
alpar@9	999 dummy indices need not be explicitly declared. Each {\it undeclared}
alpar@9	1000 symbolic name being used in the indexing position of an indexing entry
alpar@9	1001 is recognized as the symbolic name of corresponding dummy index.
alpar@9	1002
alpar@9	1003 Symbolic names of dummy indices are valid only within the scope of the
alpar@9	1004 indexing expression, where the dummy indices were introduced. Beyond
alpar@9	1005 the scope the dummy indices are completely inaccessible, so the same
alpar@9	1006 symbolic names may be used for other purposes, in particular, to
alpar@9	1007 represent dummy indices in other indexing expressions.
alpar@9	1008
alpar@9	1009 The scope of indexing expression, where implicit declarations of dummy
alpar@9	1010 indices are valid, depends on the context, in which the indexing
alpar@9	1011 expression is used:
alpar@9	1012
alpar@9	1013 \begin{enumerate}
alpar@9	1014 \item If the indexing expression is used in iterated operator, its
alpar@9	1015 scope extends until the end of the integrand.
alpar@9	1016 \item If the indexing expression is used as a primary set expression,
alpar@9	1017 its scope extends until the end of that indexing expression.
alpar@9	1018 \item If the indexing expression is used to define the subscript domain
alpar@9	1019 in declarations of some model objects, its scope extends until the end
alpar@9	1020 of the corresponding statement.
alpar@9	1021 \end{enumerate}
alpar@9	1022
alpar@9	1023 The indexing mechanism implemented by means of indexing expressions is
alpar@9	1024 best explained by some examples discussed below.
alpar@9	1025
alpar@9	1026 Let there be given three sets:
alpar@9	1027
alpar@9	1028 \medskip
alpar@9	1029
alpar@9	1030 \noindent\hspace{33.5pt}
alpar@9	1031 $A=\{4,7,9\}$,
alpar@9	1032
alpar@9	1033 \medskip
alpar@9	1034
alpar@9	1035 \noindent\hfil
alpar@9	1036 $B=\{(1,Jan),(1,Feb),(2,Mar),(2,Apr),(3,May),(3,Jun)\}$,
alpar@9	1037
alpar@9	1038 \medskip
alpar@9	1039
alpar@9	1040 \noindent\hspace{33.5pt}
alpar@9	1041 $C=\{a,b,c\}$,
alpar@9	1042
alpar@9	1043 \medskip
alpar@9	1044
alpar@9	1045 \noindent where $A$ and $C$ consist of 1-tuples (singlets), $B$
alpar@9	1046 consists of 2-tuples (doublets). Consider the following indexing
alpar@9	1047 expression:
alpar@9	1048
alpar@9	1049 \medskip
alpar@9	1050
alpar@9	1051 \noindent\hfil
alpar@9	1052 {\tt\{i in A, (j,k) in B, l in C\}}
alpar@9	1053
alpar@9	1054 \medskip
alpar@9	1055
alpar@9	1056 \noindent where {\tt i}, {\tt j}, {\tt k}, and {\tt l} are dummy
alpar@9	1057 indices.
alpar@9	1058
alpar@9	1059 Although MathProg is not a procedural language, for any indexing
alpar@9	1060 expression an equivalent algorithmic description can be given. In
alpar@9	1061 particular, the algorithmic description of the indexing expression
alpar@9	1062 above could look like follows:
alpar@9	1063
alpar@9	1064 \medskip
alpar@9	1065
alpar@9	1066 \noindent\hfil
alpar@9	1067 \begin{tabular}{@{}l@{}}
alpar@9	1068 {\bf for all} $i\in A$ {\bf do}\\
alpar@9	1069 \hspace{12pt}{\bf for all} $(j,k)\in B$ {\bf do}\\
alpar@9	1070 \hspace{24pt}{\bf for all} $l\in C$ {\bf do}\\
alpar@9	1071 \hspace{36pt}{\it action};\\
alpar@9	1072 \end{tabular}
alpar@9	1073
alpar@9	1074 \newpage
alpar@9	1075
alpar@9	1076 \noindent where the dummy indices $i$, $j$, $k$, $l$ are consecutively
alpar@9	1077 assigned corresponding components of $n$-tuples from the basic sets $A$,
alpar@9	1078 $B$, $C$, and {\it action} is some action that depends on the context,
alpar@9	1079 where the indexing expression is used. For example, if the action were
alpar@9	1080 printing current values of dummy indices, the printout would look like
alpar@9	1081 follows:
alpar@9	1082
alpar@9	1083 \medskip
alpar@9	1084
alpar@9	1085 \noindent\hfil
alpar@9	1086 \begin{tabular}{@{}llll@{}}
alpar@9	1087 $i=4$&$j=1$&$k=Jan$&$l=a$\\
alpar@9	1088 $i=4$&$j=1$&$k=Jan$&$l=b$\\
alpar@9	1089 $i=4$&$j=1$&$k=Jan$&$l=c$\\
alpar@9	1090 $i=4$&$j=1$&$k=Feb$&$l=a$\\
alpar@9	1091 $i=4$&$j=1$&$k=Feb$&$l=b$\\
alpar@9	1092 \multicolumn{4}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}\\
alpar@9	1093 $i=9$&$j=3$&$k=Jun$&$l=b$\\
alpar@9	1094 $i=9$&$j=3$&$k=Jun$&$l=c$\\
alpar@9	1095 \end{tabular}
alpar@9	1096
alpar@9	1097 \medskip
alpar@9	1098
alpar@9	1099 Let the example indexing expression be used in the following iterated
alpar@9	1100 operation:
alpar@9	1101
alpar@9	1102 \medskip
alpar@9	1103
alpar@9	1104 \noindent\hfil
alpar@9	1105 {\tt sum\{i in A, (j,k) in B, l in C\} p[i,j,k,l]}
alpar@9	1106
alpar@9	1107 \medskip
alpar@9	1108
alpar@9	1109 \noindent where {\tt p} is a 4-dimensional numeric parameter or some
alpar@9	1110 numeric expression whose resultant value depends on {\tt i}, {\tt j},
alpar@9	1111 {\tt k}, and {\tt l}. In this case the action is summation, so the
alpar@9	1112 resultant value of the primary numeric expression is:
alpar@9	1113 $$\sum_{i\in A,(j,k)\in B,l\in C}(p_{ijkl}).$$
alpar@9	1114
alpar@9	1115 Now let the example indexing expression be used as a primary set
alpar@9	1116 expression. In this case the action is gathering all 4-tuples
alpar@9	1117 (quadruplets) of the form $(i,j,k,l)$ in one set, so the resultant
alpar@9	1118 value of such operation is simply the Cartesian product of the basic
alpar@9	1119 sets:
alpar@9	1120 $$A\times B\times C=\{(i,j,k,l):i\in A,(j,k)\in B,l\in C\}.$$
alpar@9	1121 Note that in this case the same indexing expression might be written in
alpar@9	1122 the reduced form:
alpar@9	1123
alpar@9	1124 \medskip
alpar@9	1125
alpar@9	1126 \noindent\hfil
alpar@9	1127 {\tt\{A, B, C\}}
alpar@9	1128
alpar@9	1129 \medskip
alpar@9	1130
alpar@9	1131 \noindent because the dummy indices $i$, $j$, $k$, and $l$ are not
alpar@9	1132 referenced and therefore their symbolic names need not be specified.
alpar@9	1133
alpar@9	1134 Finally, let the example indexing expression be used as the subscript
alpar@9	1135 domain in the declaration of a 4-dimensional model object, say,
alpar@9	1136 a numeric parameter:
alpar@9	1137
alpar@9	1138 \medskip
alpar@9	1139
alpar@9	1140 \noindent\hfil
alpar@9	1141 {\tt param p\{i in A, (j,k) in B, l in C\}} \dots {\tt;}
alpar@9	1142
alpar@9	1143 \medskip
alpar@9	1144
alpar@9	1145 \noindent In this case the action is generating the parameter members,
alpar@9	1146 where each member has the form $p[i,j,k,l]$.
alpar@9	1147
alpar@9	1148 As was said above, some indices in the second form of indexing entries
alpar@9	1149 may be numeric or symbolic expressions, not only dummy indices. In this
alpar@9	1150 case resultant values of such expressions play role of some logical
alpar@9	1151 conditions to select only that $n$-tuples from the Cartesian product of
alpar@9	1152 basic sets that satisfy these conditions.
alpar@9	1153
alpar@9	1154 Consider, for example, the following indexing expression:
alpar@9	1155
alpar@9	1156 \medskip
alpar@9	1157
alpar@9	1158 \noindent\hfil
alpar@9	1159 {\tt\{i in A, (i-1,k) in B, l in C\}}
alpar@9	1160
alpar@9	1161 \medskip
alpar@9	1162
alpar@9	1163 \noindent where {\tt i}, {\tt k}, {\tt l} are dummy indices, and
alpar@9	1164 {\tt i-1} is a numeric expression. The algorithmic decsription of this
alpar@9	1165 indexing expression is the following:
alpar@9	1166
alpar@9	1167 \medskip
alpar@9	1168
alpar@9	1169 \noindent\hfil
alpar@9	1170 \begin{tabular}{@{}l@{}}
alpar@9	1171 {\bf for all} $i\in A$ {\bf do}\\
alpar@9	1172 \hspace{12pt}{\bf for all} $(j,k)\in B$ {\bf and} $j=i-1$ {\bf do}\\
alpar@9	1173 \hspace{24pt}{\bf for all} $l\in C$ {\bf do}\\
alpar@9	1174 \hspace{36pt}{\it action};\\
alpar@9	1175 \end{tabular}
alpar@9	1176
alpar@9	1177 \medskip
alpar@9	1178
alpar@9	1179 \noindent Thus, if this indexing expression were used as a primary set
alpar@9	1180 expression, the resultant set would be the following:
alpar@9	1181 $$\{(4,May,a),(4,May,b),(4,May,c),(4,Jun,a),(4,Jun,b),(4,Jun,c)\}.$$
alpar@9	1182 Should note that in this case the resultant set consists of 3-tuples,
alpar@9	1183 not of 4-tuples, because in the indexing expression there is no dummy
alpar@9	1184 index that corresponds to the first component of 2-tuples from the set
alpar@9	1185 $B$.
alpar@9	1186
alpar@9	1187 The general rule is: the number of components of $n$-tuples defined by
alpar@9	1188 an indexing expression is the same as the number of dummy indices in
alpar@9	1189 that expression, where the correspondence between dummy indices and
alpar@9	1190 components on $n$-tuples in the resultant set is positional, i.e. the
alpar@9	1191 first dummy index corresponds to the first component, the second dummy
alpar@9	1192 index corresponds to the second component, etc.
alpar@9	1193
alpar@9	1194 In some cases it is needed to select a subset from the Cartesian
alpar@9	1195 product of some sets. This may be attained by using an optional logical
alpar@9	1196 predicate, which is specified in the indexing expression.
alpar@9	1197
alpar@9	1198 Consider, for example, the following indexing expression:
alpar@9	1199
alpar@9	1200 \medskip
alpar@9	1201
alpar@9	1202 \noindent\hfil
alpar@9	1203 {\tt\{i in A, (j,k) in B, l in C: i <= 5 and k <> 'Mar'\}}
alpar@9	1204
alpar@9	1205 \medskip
alpar@9	1206
alpar@9	1207 \noindent where the logical expression following the colon is a
alpar@9	1208 predicate. The algorithmic description of this indexing expression is
alpar@9	1209 the following:
alpar@9	1210
alpar@9	1211 \medskip
alpar@9	1212
alpar@9	1213 \noindent\hfil
alpar@9	1214 \begin{tabular}{@{}l@{}}
alpar@9	1215 {\bf for all} $i\in A$ {\bf do}\\
alpar@9	1216 \hspace{12pt}{\bf for all} $(j,k)\in B$ {\bf do}\\
alpar@9	1217 \hspace{24pt}{\bf for all} $l\in C$ {\bf do}\\
alpar@9	1218 \hspace{36pt}{\bf if} $i\leq 5$ {\bf and} $l\neq`Mar'$ {\bf then}\\
alpar@9	1219 \hspace{48pt}{\it action};\\
alpar@9	1220 \end{tabular}
alpar@9	1221
alpar@9	1222 \medskip
alpar@9	1223
alpar@9	1224 \noindent Thus, if this indexing expression were used as a primary set
alpar@9	1225 expression, the resultant set would be the following:
alpar@9	1226 $$\{(4,1,Jan,a),(4,1,Feb,a),(4,2,Apr,a),\dots,(4,3,Jun,c)\}.$$
alpar@9	1227
alpar@9	1228 If no predicate is specified in the indexing expression, one, which
alpar@9	1229 takes on the value {\it true}, is assumed.
alpar@9	1230
alpar@9	1231 \subsection{Set expressions}
alpar@9	1232
alpar@9	1233 A {\it set expression} is a rule for computing an elemental set, i.e.
alpar@9	1234 a collection of $n$-tuples, where components of $n$-tuples are numeric
alpar@9	1235 and symbolic quantities.
alpar@9	1236
alpar@9	1237 The primary set expression may be a literal set, unsubscripted set,
alpar@9	1238 subscripted set, ``arithmetic'' set, indexing expression, iterated set
alpar@9	1239 expression, conditional set expression, or another set expression
alpar@9	1240 enclosed in parentheses.
alpar@9	1241
alpar@9	1242 \medskip
alpar@9	1243
alpar@9	1244 \noindent{\bf Examples}
alpar@9	1245
alpar@9	1246 \medskip
alpar@9	1247
alpar@9	1248 \noindent
alpar@9	1249 \begin{tabular}{@{}ll@{}}
alpar@9	1250 \verb\|{(123,'aa'), (i,'bb'), (j-1,'cc')}\|&(literal set)\\
alpar@9	1251 \verb\|I\|&(unsubscripted set)\\
alpar@9	1252 \verb\|S[i-1,j+1]\|&(subscripted set)\\
alpar@9	1253 \verb\|1..t-1 by 2\|&(``arithmetic'' set)\\
alpar@9	1254 \verb\|{t in 1..T, (t+1,j) in S: (t,j) in F}\|&(indexing expression)\\
alpar@9	1255 \verb\|setof{i in I, j in J}(i+1,j-1)\|&(iterated expression)\\
alpar@9	1256 \verb\|if i < j then S[i] else F diff S[j]\|&(conditional expression)\\
alpar@9	1257 \verb\|(1..10 union 21..30)\|&(parenthesized expression)\\
alpar@9	1258 \end{tabular}
alpar@9	1259
alpar@9	1260 \medskip
alpar@9	1261
alpar@9	1262 More general set expressions containing two or more primary set
alpar@9	1263 expressions may be constructed by using certain set operators.
alpar@9	1264
alpar@9	1265 \medskip
alpar@9	1266
alpar@9	1267 \noindent{\bf Examples}
alpar@9	1268
alpar@9	1269 \medskip
alpar@9	1270
alpar@9	1271 \noindent\verb\|(A union B) inter (I cross J)\|
alpar@9	1272
alpar@9	1273 \noindent
alpar@9	1274 \verb\|1..10 cross (if i < j then {'a', 'b', 'c'} else {'d', 'e', 'f'})\|
alpar@9	1275
alpar@9	1276 \subsubsection{Literal sets}
alpar@9	1277
alpar@9	1278 A {\it literal set} is a primary set expression, which has the
alpar@9	1279 following two syntactic forms:
alpar@9	1280
alpar@9	1281 \medskip
alpar@9	1282
alpar@9	1283 \noindent\hspace{39pt}
alpar@9	1284 {\tt\{}$e_1${\tt,} $e_2${\tt,} \dots{\tt,} $e_m${\tt\}}
alpar@9	1285
alpar@9	1286 \medskip
alpar@9	1287
alpar@9	1288 \noindent\hfil
alpar@9	1289 {\tt\{(}$e_{11}${\tt,} \dots{\tt,} $e_{1n}${\tt),}
alpar@9	1290 {\tt(}$e_{21}${\tt,} \dots{\tt,} $e_{2n}${\tt),} \dots{\tt,}
alpar@9	1291 {\tt(}$e_{m1}${\tt,} \dots{\tt,} $e_{mn}${\tt)\}}
alpar@9	1292
alpar@9	1293 \medskip
alpar@9	1294
alpar@9	1295 \noindent where $e_1$, \dots, $e_m$, $e_{11}$, \dots, $e_{mn}$ are
alpar@9	1296 numeric or symbolic expressions.
alpar@9	1297
alpar@9	1298 If the first form is used, the resultant set consists of 1-tuples
alpar@9	1299 (singlets) enumerated within the curly braces. It is allowed to specify
alpar@9	1300 an empty set as {\tt\{\ \}}, which has no 1-tuples. If the second form
alpar@9	1301 is used, the resultant set consists of $n$-tuples enumerated within the
alpar@9	1302 curly braces, where a particular $n$-tuple consists of corresponding
alpar@9	1303 components enumerated within the parentheses. All $n$-tuples must have
alpar@9	1304 the same number of components.
alpar@9	1305
alpar@9	1306 \subsubsection{Unsubscripted sets}
alpar@9	1307
alpar@9	1308 If the primary set expression is an unsubscripted set (which must be
alpar@9	1309 0-dimen\-sional), the resultant set is an elemental set associated with
alpar@9	1310 the corresponding set object.
alpar@9	1311
alpar@9	1312 \newpage
alpar@9	1313
alpar@9	1314 \subsubsection{Subscripted sets}
alpar@9	1315
alpar@9	1316 The primary set expression, which refers to a subscripted set, has the
alpar@9	1317 following syntactic form:
alpar@9	1318
alpar@9	1319 \medskip
alpar@9	1320
alpar@9	1321 \noindent\hfil
alpar@9	1322 {\it name}{\tt[}$i_1${\tt,} $i_2${\tt,} \dots{\tt,} $i_n${\tt]}
alpar@9	1323
alpar@9	1324 \medskip
alpar@9	1325
alpar@9	1326 \noindent where {\it name} is the symbolic name of the set object,
alpar@9	1327 $i_1$, $i_2$, \dots, $i_n$ are subscripts.
alpar@9	1328
alpar@9	1329 Each subscript must be a numeric or symbolic expression. The number of
alpar@9	1330 subscripts in the subscript list must be the same as the dimension of
alpar@9	1331 the set object with which the subscript list is associated.
alpar@9	1332
alpar@9	1333 Actual values of subscript expressions are used to identify a
alpar@9	1334 particular member of the set object that determines the resultant set.
alpar@9	1335
alpar@9	1336 \subsubsection{``Arithmetic'' sets}
alpar@9	1337
alpar@9	1338 The primary set expression, which is an ``arithmetic'' set, has the
alpar@9	1339 following two syntactic forms:
alpar@9	1340
alpar@9	1341 \medskip
alpar@9	1342
alpar@9	1343 \noindent\hfil
alpar@9	1344 $t_0$ {\tt..} $t_1$ {\tt by} $\delta t$
alpar@9	1345
alpar@9	1346 \medskip
alpar@9	1347
alpar@9	1348 \noindent\hspace{138.5pt}
alpar@9	1349 $t_0$ {\tt..} $t_1$
alpar@9	1350
alpar@9	1351 \medskip
alpar@9	1352
alpar@9	1353 \noindent where $t_0$, $t_1$, and $\delta t$ are numeric expressions
alpar@9	1354 (the value of $\delta t$ must not be zero). The second form is
alpar@9	1355 equivalent to the first form, where $\delta t=1$.
alpar@9	1356
alpar@9	1357 If $\delta t>0$, the resultant set is determined as follows:
alpar@9	1358 $$\{t:\exists k\in{\cal Z}(t=t_0+k\delta t,\ t_0\leq t\leq t_1)\}.$$
alpar@9	1359 Otherwise, if $\delta t<0$, the resultant set is determined as follows:
alpar@9	1360 $$\{t:\exists k\in{\cal Z}(t=t_0+k\delta t,\ t_1\leq t\leq t_0)\}.$$
alpar@9	1361
alpar@9	1362 \subsubsection{Indexing expressions}
alpar@9	1363
alpar@9	1364 If the primary set expression is an indexing expression, the resultant
alpar@9	1365 set is determined as described above in Subsection \ref{indexing}, page
alpar@9	1366 \pageref{indexing}.
alpar@9	1367
alpar@9	1368 \subsubsection{Iterated expressions}
alpar@9	1369
alpar@9	1370 An {\it iterated set expression} is a primary set expression, which has
alpar@9	1371 the following syntactic form:
alpar@9	1372
alpar@9	1373 \medskip
alpar@9	1374
alpar@9	1375 \noindent\hfil
alpar@9	1376 {\tt setof} {\it indexing-expression} {\it integrand}
alpar@9	1377
alpar@9	1378 \medskip
alpar@9	1379
alpar@9	1380 \noindent where {\it indexing-expression} is an indexing expression,
alpar@9	1381 which introduces dummy indices and controls iterating, {\it integrand}
alpar@9	1382 is either a single numeric or symbolic expression or a list of numeric
alpar@9	1383 and symbolic expressions separated by commae and enclosed in
alpar@9	1384 parentheses.
alpar@9	1385
alpar@9	1386 If the integrand is a single numeric or symbolic expression, the
alpar@9	1387 resultant set consists of 1-tuples and is determined as follows:
alpar@9	1388 $$\{x:(i_1,\dots,i_n)\in\Delta\},$$
alpar@9	1389 \noindent where $x$ is a value of the integrand, $i_1$, \dots, $i_n$
alpar@9	1390 are dummy indices introduced in the indexing expression, $\Delta$ is
alpar@9	1391 the domain, a set of $n$-tuples specified by the indexing expression,
alpar@9	1392 which defines particular values assigned to the dummy indices on
alpar@9	1393 performing the iterated operation.
alpar@9	1394
alpar@9	1395 If the integrand is a list containing $m$ numeric and symbolic
alpar@9	1396 expressions, the resultant set consists of $m$-tuples and is determined
alpar@9	1397 as follows:
alpar@9	1398 $$\{(x_1,\dots,x_m):(i_1,\dots,i_n)\in\Delta\},$$
alpar@9	1399 where $x_1$, \dots, $x_m$ are values of the expressions in the
alpar@9	1400 integrand list, $i_1$, \dots, $i_n$ and $\Delta$ have the same meaning
alpar@9	1401 as above.
alpar@9	1402
alpar@9	1403 \subsubsection{Conditional expressions}
alpar@9	1404
alpar@9	1405 A {\it conditional set expression} is a primary set expression that has
alpar@9	1406 the following syntactic form:
alpar@9	1407
alpar@9	1408 \medskip
alpar@9	1409
alpar@9	1410 \noindent\hfil
alpar@9	1411 {\tt if} $b$ {\tt then} $X$ {\tt else} $Y$
alpar@9	1412
alpar@9	1413 \medskip
alpar@9	1414
alpar@9	1415 \noindent where $b$ is an logical expression, $X$ and $Y$ are set
alpar@9	1416 expressions, which must define sets of the same dimension.
alpar@9	1417
alpar@9	1418 The resultant value of the conditional expression depends on the value
alpar@9	1419 of the logical expression that follows the keyword {\tt if}. If it
alpar@9	1420 takes on the value {\it true}, the resultant set is the value of the
alpar@9	1421 expression that follows the keyword {\tt then}. Otherwise, if the
alpar@9	1422 logical expression takes on the value {\it false}, the resultant set is
alpar@9	1423 the value of the expression that follows the keyword {\tt else}.
alpar@9	1424
alpar@9	1425 \subsubsection{Parenthesized expressions}
alpar@9	1426
alpar@9	1427 Any set expression may be enclosed in parentheses that syntactically
alpar@9	1428 makes it a primary set expression.
alpar@9	1429
alpar@9	1430 Parentheses may be used in set expressions, as in algebra, to specify
alpar@9	1431 the desired order in which operations are to be performed. Where
alpar@9	1432 parentheses are used, the expression within the parentheses is
alpar@9	1433 evaluated before the resultant value is used.
alpar@9	1434
alpar@9	1435 The resultant value of the parenthesized expression is the same as the
alpar@9	1436 value of the expression enclosed within parentheses.
alpar@9	1437
alpar@9	1438 \subsubsection{Set operators}
alpar@9	1439
alpar@9	1440 In MathProg there exist the following set operators, which may be used
alpar@9	1441 in set expressions:
alpar@9	1442
alpar@9	1443 \medskip
alpar@9	1444
alpar@9	1445 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	1446 $X$ {\tt union} $Y$&union $X\cup Y$\\
alpar@9	1447 $X$ {\tt diff} $Y$&difference $X\backslash Y$\\
alpar@9	1448 $X$ {\tt symdiff} $Y$&symmetric difference $X\oplus Y$\\
alpar@9	1449 $X$ {\tt inter} $Y$&intersection $X\cap Y$\\
alpar@9	1450 $X$ {\tt cross} $Y$&cross (Cartesian) product $X\times Y$\\
alpar@9	1451 \end{tabular}
alpar@9	1452
alpar@9	1453 \medskip
alpar@9	1454
alpar@9	1455 \noindent where $X$ and Y are set expressions, which must define sets
alpar@9	1456 of the identical dimension (except the Cartesian product).
alpar@9	1457
alpar@9	1458 If the expression includes more than one set operator, all operators
alpar@9	1459 are performed from left to right according to the hierarchy of
alpar@9	1460 operations (see below).
alpar@9	1461
alpar@9	1462 The resultant value of the expression, which contains set operators, is
alpar@9	1463 the result of applying the operators to their operands.
alpar@9	1464
alpar@9	1465 The dimension of the resultant set, i.e. the dimension of $n$-tuples,
alpar@9	1466 of which the resultant set consists of, is the same as the dimension of
alpar@9	1467 the operands, except the Cartesian product, where the dimension of the
alpar@9	1468 resultant set is the sum of the dimensions of its operands.
alpar@9	1469
alpar@9	1470 \subsubsection{Hierarchy of operations}
alpar@9	1471
alpar@9	1472 The following list shows the hierarchy of operations in set
alpar@9	1473 expressions:
alpar@9	1474
alpar@9	1475 \medskip
alpar@9	1476
alpar@9	1477 \noindent\hfil
alpar@9	1478 \begin{tabular}{@{}ll@{}}
alpar@9	1479 Operation&Hierarchy\\
alpar@9	1480 \hline
alpar@9	1481 Evaluation of numeric operations&1st-7th\\
alpar@9	1482 Evaluation of symbolic operations&8th-9th\\
alpar@9	1483 Evaluation of iterated or ``arithmetic'' set ({\tt setof}, {\tt..})&
alpar@9	1484 10th\\
alpar@9	1485 Cartesian product ({\tt cross})&11th\\
alpar@9	1486 Intersection ({\tt inter})&12th\\
alpar@9	1487 Union and difference ({\tt union}, {\tt diff}, {\tt symdiff})&13th\\
alpar@9	1488 Conditional evaluation ({\tt if} \dots {\tt then} \dots {\tt else})&
alpar@9	1489 14th\\
alpar@9	1490 \end{tabular}
alpar@9	1491
alpar@9	1492 \medskip
alpar@9	1493
alpar@9	1494 This hierarchy has the same meaning as was explained above for numeric
alpar@9	1495 expressions (see Subsection \ref{hierarchy}, page \pageref{hierarchy}).
alpar@9	1496
alpar@9	1497 \subsection{Logical expressions}
alpar@9	1498
alpar@9	1499 A {\it logical expression} is a rule for computing a single logical
alpar@9	1500 value, which can be either {\it true} or {\it false}.
alpar@9	1501
alpar@9	1502 The primary logical expression may be a numeric expression, relational
alpar@9	1503 expression, iterated logical expression, or another logical expression
alpar@9	1504 enclosed in parentheses.
alpar@9	1505
alpar@9	1506 \medskip
alpar@9	1507
alpar@9	1508 \noindent{\bf Examples}
alpar@9	1509
alpar@9	1510 \medskip
alpar@9	1511
alpar@9	1512 \noindent
alpar@9	1513 \begin{tabular}{@{}ll@{}}
alpar@9	1514 \verb\|i+1\|&(numeric expression)\\
alpar@9	1515 \verb\|a[i,j] < 1.5\|&(relational expression)\\
alpar@9	1516 \verb\|s[i+1,j-1] <> 'Mar'\|&(relational expression)\\
alpar@9	1517 \verb\|(i+1,'Jan') not in I cross J\|&(relational expression)\\
alpar@9	1518 \verb\|S union T within A[i] inter B[j]\|&(relational expression)\\
alpar@9	1519 \verb\|forall{i in I, j in J} a[i,j] < .5 * b\|&(iterated expression)\\
alpar@9	1520 \verb\|(a[i,j] < 1.5 or b[i] >= a[i,j])\|&(parenthesized expression)\\
alpar@9	1521 \end{tabular}
alpar@9	1522
alpar@9	1523 \medskip
alpar@9	1524
alpar@9	1525 More general logical expressions containing two or more primary logical
alpar@9	1526 expressions may be constructed by using certain logical operators.
alpar@9	1527
alpar@9	1528 \newpage
alpar@9	1529
alpar@9	1530 \noindent{\bf Examples}
alpar@9	1531
alpar@9	1532 \medskip
alpar@9	1533
alpar@9	1534 \noindent\verb\|not (a[i,j] < 1.5 or b[i] >= a[i,j]) and (i,j) in S\|
alpar@9	1535
alpar@9	1536 \noindent\verb\|(i,j) in S or (i,j) not in T diff U\|
alpar@9	1537
alpar@9	1538 \subsubsection{Numeric expressions}
alpar@9	1539
alpar@9	1540 The resultant value of the primary logical expression, which is a
alpar@9	1541 numeric expression, is {\it true}, if the resultant value of the
alpar@9	1542 numeric expression is non-zero. Otherwise the resultant value of the
alpar@9	1543 logical expression is {\it false}.
alpar@9	1544
alpar@9	1545 \subsubsection{Relational operators}
alpar@9	1546
alpar@9	1547 In MathProg there exist the following relational operators, which may
alpar@9	1548 be used in logical expressions:
alpar@9	1549
alpar@9	1550 \medskip
alpar@9	1551
alpar@9	1552 \begin{tabular}{@{}ll@{}}
alpar@9	1553 $x$ {\tt<} $y$&test on $x<y$\\
alpar@9	1554 $x$ {\tt<=} $y$&test on $x\leq y$\\
alpar@9	1555 $x$ {\tt=} $y$, $x$ {\tt==} $y$&test on $x=y$\\
alpar@9	1556 $x$ {\tt>=} $y$&test on $x\geq y$\\
alpar@9	1557 $x$ {\tt>} $y$&test on $x>y$\\
alpar@9	1558 $x$ {\tt<>} $y$, $x$ {\tt!=} $y$&test on $x\neq y$\\
alpar@9	1559 $x$ {\tt in} $Y$&test on $x\in Y$\\
alpar@9	1560 {\tt(}$x_1${\tt,}\dots{\tt,}$x_n${\tt)} {\tt in} $Y$&test on
alpar@9	1561 $(x_1,\dots,x_n)\in Y$\\
alpar@9	1562 $x$ {\tt not} {\tt in} $Y$, $x$ {\tt!in} $Y$&test on $x\not\in Y$\\
alpar@9	1563 {\tt(}$x_1${\tt,}\dots{\tt,}$x_n${\tt)} {\tt not} {\tt in} $Y$,
alpar@9	1564 {\tt(}$x_1${\tt,}\dots{\tt,}$x_n${\tt)} {\tt !in} $Y$&test on
alpar@9	1565 $(x_1,\dots,x_n)\not\in Y$\\
alpar@9	1566 $X$ {\tt within} $Y$&test on $X\subseteq Y$\\
alpar@9	1567 $X$ {\tt not} {\tt within} $Y$, $X$ {\tt !within} $Y$&test on
alpar@9	1568 $X\not\subseteq Y$\\
alpar@9	1569 \end{tabular}
alpar@9	1570
alpar@9	1571 \medskip
alpar@9	1572
alpar@9	1573 \noindent where $x$, $x_1$, \dots, $x_n$, $y$ are numeric or symbolic
alpar@9	1574 expressions, $X$ and $Y$ are set expression.
alpar@9	1575
alpar@9	1576 {\it Notes:}
alpar@9	1577
alpar@9	1578 1. In the operations {\tt in}, {\tt not in}, and {\tt !in} the
alpar@9	1579 number of components in the first operands must be the same as the
alpar@9	1580 dimension of the second operand.
alpar@9	1581
alpar@9	1582 2. In the operations {\tt within}, {\tt not within}, and {\tt !within}
alpar@9	1583 both operands must have identical dimension.
alpar@9	1584
alpar@9	1585 All the relational operators listed above have their conventional
alpar@9	1586 mathematical meaning. The resultant value is {\it true}, if
alpar@9	1587 corresponding relation is satisfied for its operands, otherwise
alpar@9	1588 {\it false}. (Note that symbolic values are ordered lexicographically,
alpar@9	1589 and any numeric value precedes any symbolic value.)
alpar@9	1590
alpar@9	1591 \subsubsection{Iterated expressions}
alpar@9	1592
alpar@9	1593 An {\it iterated logical expression} is a primary logical expression,
alpar@9	1594 which has the following syntactic form:
alpar@9	1595
alpar@9	1596 \medskip
alpar@9	1597
alpar@9	1598 \noindent\hfil
alpar@9	1599 {\it iterated-operator} {\it indexing-expression} {\it integrand}
alpar@9	1600
alpar@9	1601 \medskip
alpar@9	1602
alpar@9	1603 \noindent where {\it iterated-operator} is the symbolic name of the
alpar@9	1604 iterated operator to be performed (see below), {\it indexing-expression}
alpar@9	1605 is an indexing expression which introduces dummy indices and controls
alpar@9	1606 iterating, {\it integrand} is a numeric expression that participates in
alpar@9	1607 the operation.
alpar@9	1608
alpar@9	1609 In MathProg there exist two iterated operators, which may be used in
alpar@9	1610 logical expressions:
alpar@9	1611
alpar@9	1612 \medskip
alpar@9	1613
alpar@9	1614 \noindent\hfil
alpar@9	1615 \begin{tabular}{@{}lll@{}}
alpar@9	1616 {\tt forall}&$\forall$-quantification&$\displaystyle
alpar@9	1617 \forall(i_1,\dots,i_n)\in\Delta[f(i_1,\dots,i_n)],$\\
alpar@9	1618 {\tt exists}&$\exists$-quantification&$\displaystyle
alpar@9	1619 \exists(i_1,\dots,i_n)\in\Delta[f(i_1,\dots,i_n)],$\\
alpar@9	1620 \end{tabular}
alpar@9	1621
alpar@9	1622 \medskip
alpar@9	1623
alpar@9	1624 \noindent where $i_1$, \dots, $i_n$ are dummy indices introduced in
alpar@9	1625 the indexing expression, $\Delta$ is the domain, a set of $n$-tuples
alpar@9	1626 specified by the indexing expression which defines particular values
alpar@9	1627 assigned to the dummy indices on performing the iterated operation,
alpar@9	1628 $f(i_1,\dots,i_n)$ is the integrand, a logical expression whose
alpar@9	1629 resultant value depends on the dummy indices.
alpar@9	1630
alpar@9	1631 For $\forall$-quantification the resultant value of the iterated
alpar@9	1632 logical expression is {\it true}, if the value of the integrand is
alpar@9	1633 {\it true} for all $n$-tuples contained in the domain, otherwise
alpar@9	1634 {\it false}.
alpar@9	1635
alpar@9	1636 For $\exists$-quantification the resultant value of the iterated
alpar@9	1637 logical expression is {\it false}, if the value of the integrand is
alpar@9	1638 {\it false} for all $n$-tuples contained in the domain, otherwise
alpar@9	1639 {\it true}.
alpar@9	1640
alpar@9	1641 \subsubsection{Parenthesized expressions}
alpar@9	1642
alpar@9	1643 Any logical expression may be enclosed in parentheses that
alpar@9	1644 syntactically makes it a primary logical expression.
alpar@9	1645
alpar@9	1646 Parentheses may be used in logical expressions, as in algebra, to
alpar@9	1647 specify the desired order in which operations are to be performed.
alpar@9	1648 Where parentheses are used, the expression within the parentheses is
alpar@9	1649 evaluated before the resultant value is used.
alpar@9	1650
alpar@9	1651 The resultant value of the parenthesized expression is the same as the
alpar@9	1652 value of the expression enclosed within parentheses.
alpar@9	1653
alpar@9	1654 \subsubsection{Logical operators}
alpar@9	1655
alpar@9	1656 In MathProg there exist the following logical operators, which may be
alpar@9	1657 used in logical expressions:
alpar@9	1658
alpar@9	1659 \medskip
alpar@9	1660
alpar@9	1661 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	1662 {\tt not} $x$, {\tt!}$x$&negation $\neg\ x$\\
alpar@9	1663 $x$ {\tt and} $y$, $x$ {\tt\&\&} $y$&conjunction (logical ``and'')
alpar@9	1664 $x\;\&\;y$\\
alpar@9	1665 $x$ {\tt or} $y$, $x$ {\tt\|\|} $y$&disjunction (logical ``or'')
alpar@9	1666 $x\vee y$\\
alpar@9	1667 \end{tabular}
alpar@9	1668
alpar@9	1669 \medskip
alpar@9	1670
alpar@9	1671 \noindent where $x$ and $y$ are logical expressions.
alpar@9	1672
alpar@9	1673 If the expression includes more than one logical operator, all
alpar@9	1674 operators are performed from left to right according to the hierarchy
alpar@9	1675 of the operations (see below). The resultant value of the expression,
alpar@9	1676 which contains logical operators, is the result of applying the
alpar@9	1677 operators to their operands.
alpar@9	1678
alpar@9	1679 \subsubsection{Hierarchy of operations}
alpar@9	1680
alpar@9	1681 The following list shows the hierarchy of operations in logical
alpar@9	1682 expressions:
alpar@9	1683
alpar@9	1684 \medskip
alpar@9	1685
alpar@9	1686 \noindent\hfil
alpar@9	1687 \begin{tabular}{@{}ll@{}}
alpar@9	1688 Operation&Hierarchy\\
alpar@9	1689 \hline
alpar@9	1690 Evaluation of numeric operations&1st-7th\\
alpar@9	1691 Evaluation of symbolic operations&8th-9th\\
alpar@9	1692 Evaluation of set operations&10th-14th\\
alpar@9	1693 Relational operations ({\tt<}, {\tt<=}, etc.)&15th\\
alpar@9	1694 Negation ({\tt not}, {\tt!})&16th\\
alpar@9	1695 Conjunction ({\tt and}, {\tt\&\&})&17th\\
alpar@9	1696 $\forall$- and $\exists$-quantification ({\tt forall}, {\tt exists})&
alpar@9	1697 18th\\
alpar@9	1698 Disjunction ({\tt or}, {\tt\|\|})&19th\\
alpar@9	1699 \end{tabular}
alpar@9	1700
alpar@9	1701 \medskip
alpar@9	1702
alpar@9	1703 This hierarchy has the same meaning as was explained above for numeric
alpar@9	1704 expressions (see Subsection \ref{hierarchy}, page \pageref{hierarchy}).
alpar@9	1705
alpar@9	1706 \subsection{Linear expressions}
alpar@9	1707
alpar@9	1708 An {\it linear expression} is a rule for computing so called
alpar@9	1709 a {\it linear form} or simply a {\it formula}, which is a linear (or
alpar@9	1710 affine) function of elemental variables.
alpar@9	1711
alpar@9	1712 The primary linear expression may be an unsubscripted variable,
alpar@9	1713 subscripted variable, iterated linear expression, conditional linear
alpar@9	1714 expression, or another linear expression enclosed in parentheses.
alpar@9	1715
alpar@9	1716 It is also allowed to use a numeric expression as the primary linear
alpar@9	1717 expression, in which case the resultant value of the numeric expression
alpar@9	1718 is automatically converted to a formula that includes the constant term
alpar@9	1719 only.
alpar@9	1720
alpar@9	1721 \medskip
alpar@9	1722
alpar@9	1723 \noindent{\bf Examples}
alpar@9	1724
alpar@9	1725 \medskip
alpar@9	1726
alpar@9	1727 \noindent
alpar@9	1728 \begin{tabular}{@{}ll@{}}
alpar@9	1729 \verb\|z\|&(unsubscripted variable)\\
alpar@9	1730 \verb\|x[i,j]\|&(subscripted variable)\\
alpar@9	1731 \verb\|sum{j in J} (a[i] * x[i,j] + 3 * y)\|&(iterated expression)\\
alpar@9	1732 \verb\|if i in I then x[i,j] else 1.5 * z + 3\|&(conditional expression)\\
alpar@9	1733 \verb\|(a[i,j] * x[i,j] + y[i-1] + .1)\|&(parenthesized expression)\\
alpar@9	1734 \end{tabular}
alpar@9	1735
alpar@9	1736 \medskip
alpar@9	1737
alpar@9	1738 More general linear expressions containing two or more primary linear
alpar@9	1739 expressions may be constructed by using certain arithmetic operators.
alpar@9	1740
alpar@9	1741 \medskip
alpar@9	1742
alpar@9	1743 \noindent{\bf Examples}
alpar@9	1744
alpar@9	1745 \medskip
alpar@9	1746
alpar@9	1747 \noindent\verb\|2 * x[i-1,j+1] + 3.5 * y[k] + .5 * z\|
alpar@9	1748
alpar@9	1749 \noindent\verb\|(- x[i,j] + 3.5 * y[k]) / sum{t in T} abs(d[i,j,t])\|
alpar@9	1750
alpar@9	1751 \subsubsection{Unsubscripted variables}
alpar@9	1752
alpar@9	1753 If the primary linear expression is an unsubscripted variable (which
alpar@9	1754 must be 0-dimensional), the resultant formula is that unsubscripted
alpar@9	1755 variable.
alpar@9	1756
alpar@9	1757 \subsubsection{Subscripted variables}
alpar@9	1758
alpar@9	1759 The primary linear expression, which refers to a subscripted variable,
alpar@9	1760 has the following syntactic form:
alpar@9	1761
alpar@9	1762 \medskip
alpar@9	1763
alpar@9	1764 \noindent\hfil
alpar@9	1765 {\it name}{\tt[}$i_1${\tt,} $i_2${\tt,} \dots{\tt,} $i_n${\tt]}
alpar@9	1766
alpar@9	1767 \medskip
alpar@9	1768
alpar@9	1769 \noindent where {\it name} is the symbolic name of the model variable,
alpar@9	1770 $i_1$, $i_2$, \dots, $i_n$ are subscripts.
alpar@9	1771
alpar@9	1772 Each subscript must be a numeric or symbolic expression. The number of
alpar@9	1773 subscripts in the subscript list must be the same as the dimension of
alpar@9	1774 the model variable with which the subscript list is associated.
alpar@9	1775
alpar@9	1776 Actual values of the subscript expressions are used to identify a
alpar@9	1777 particular member of the model variable that determines the resultant
alpar@9	1778 formula, which is an elemental variable associated with corresponding
alpar@9	1779 member.
alpar@9	1780
alpar@9	1781 \subsubsection{Iterated expressions}
alpar@9	1782
alpar@9	1783 An {\it iterated linear expression} is a primary linear expression,
alpar@9	1784 which has the following syntactic form:
alpar@9	1785
alpar@9	1786 \medskip
alpar@9	1787
alpar@9	1788 \noindent\hfil
alpar@9	1789 {\tt sum} {\it indexing-expression} {\it integrand}
alpar@9	1790
alpar@9	1791 \medskip
alpar@9	1792
alpar@9	1793 \noindent where {\it indexing-expression} is an indexing expression,
alpar@9	1794 which introduces dummy indices and controls iterating, {\it integrand}
alpar@9	1795 is a linear expression that participates in the operation.
alpar@9	1796
alpar@9	1797 The iterated linear expression is evaluated exactly in the same way as
alpar@9	1798 the iterated numeric expression (see Subection \ref{itexpr}, page
alpar@9	1799 \pageref{itexpr}) with exception that the integrand participated in the
alpar@9	1800 summation is a formula, not a numeric value.
alpar@9	1801
alpar@9	1802 \subsubsection{Conditional expressions}
alpar@9	1803
alpar@9	1804 A {\it conditional linear expression} is a primary linear expression,
alpar@9	1805 which has one of the following two syntactic forms:
alpar@9	1806
alpar@9	1807 \medskip
alpar@9	1808
alpar@9	1809 \noindent\hfil
alpar@9	1810 {\tt if} $b$ {\tt then} $f$ {\tt else} $g$
alpar@9	1811
alpar@9	1812 \medskip
alpar@9	1813
alpar@9	1814 \noindent\hspace{127pt}
alpar@9	1815 {\tt if} $b$ {\tt then} $f$
alpar@9	1816
alpar@9	1817 \medskip
alpar@9	1818
alpar@9	1819 \noindent where $b$ is an logical expression, $f$ and $g$ are linear
alpar@9	1820 expressions.
alpar@9	1821
alpar@9	1822 The conditional linear expression is evaluated exactly in the same way
alpar@9	1823 as the conditional numeric expression (see Subsection \ref{ifthen},
alpar@9	1824 page \pageref{ifthen}) with exception that operands participated in the
alpar@9	1825 operation are formulae, not numeric values.
alpar@9	1826
alpar@9	1827 \subsubsection{Parenthesized expressions}
alpar@9	1828
alpar@9	1829 Any linear expression may be enclosed in parentheses that syntactically
alpar@9	1830 makes it a primary linear expression.
alpar@9	1831
alpar@9	1832 Parentheses may be used in linear expressions, as in algebra, to
alpar@9	1833 specify the desired order in which operations are to be performed.
alpar@9	1834 Where parentheses are used, the expression within the parentheses is
alpar@9	1835 evaluated before the resultant formula is used.
alpar@9	1836
alpar@9	1837 The resultant value of the parenthesized expression is the same as the
alpar@9	1838 value of the expression enclosed within parentheses.
alpar@9	1839
alpar@9	1840 \subsubsection{Arithmetic operators}
alpar@9	1841
alpar@9	1842 In MathProg there exists the following arithmetic operators, which may
alpar@9	1843 be used in linear expressions:
alpar@9	1844
alpar@9	1845 \medskip
alpar@9	1846
alpar@9	1847 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	1848 {\tt+} $f$&unary plus\\
alpar@9	1849 {\tt-} $f$&unary minus\\
alpar@9	1850 $f$ {\tt+} $g$&addition\\
alpar@9	1851 $f$ {\tt-} $g$&subtraction\\
alpar@9	1852 $x$ {\tt} $f$, $f$ {\tt} $x$&multiplication\\
alpar@9	1853 $f$ {\tt/} $x$&division
alpar@9	1854 \end{tabular}
alpar@9	1855
alpar@9	1856 \medskip
alpar@9	1857
alpar@9	1858 \noindent where $f$ and $g$ are linear expressions, $x$ is a numeric
alpar@9	1859 expression (more precisely, a linear expression containing only the
alpar@9	1860 constant term).
alpar@9	1861
alpar@9	1862 If the expression includes more than one arithmetic operator, all
alpar@9	1863 operators are performed from left to right according to the hierarchy
alpar@9	1864 of operations (see below). The resultant value of the expression, which
alpar@9	1865 contains arithmetic operators, is the result of applying the operators
alpar@9	1866 to their operands.
alpar@9	1867
alpar@9	1868 \subsubsection{Hierarchy of operations}
alpar@9	1869
alpar@9	1870 The hierarchy of arithmetic operations used in linear expressions is
alpar@9	1871 the same as for numeric expressions (see Subsection \ref{hierarchy},
alpar@9	1872 page \pageref{hierarchy}).
alpar@9	1873
alpar@9	1874 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	1875
alpar@9	1876 \newpage
alpar@9	1877
alpar@9	1878 \section{Statements}
alpar@9	1879
alpar@9	1880 {\it Statements} are basic units of the model description. In MathProg
alpar@9	1881 all statements are divided into two categories: declaration statements
alpar@9	1882 and functional statements.
alpar@9	1883
alpar@9	1884 {\it Declaration statements} (set statement, parameter statement,
alpar@9	1885 variable statement, constraint statement, and objective statement) are
alpar@9	1886 used to declare model objects of certain kinds and define certain
alpar@9	1887 properties of such objects.
alpar@9	1888
alpar@9	1889 {\it Functional statements} (solve statement, check statement, display
alpar@9	1890 statement, printf statement, loop statement) are intended for
alpar@9	1891 performing some specific actions.
alpar@9	1892
alpar@9	1893 Note that declaration statements may follow in arbitrary order, which
alpar@9	1894 does not affect the result of translation. However, any model object
alpar@9	1895 must be declared before it is referenced in other statements.
alpar@9	1896
alpar@9	1897 \subsection{Set statement}
alpar@9	1898
alpar@9	1899 \medskip
alpar@9	1900
alpar@9	1901 \framebox[345pt][l]{
alpar@9	1902 \parbox[c][24pt]{345pt}{
alpar@9	1903 \hspace{6pt} {\tt set} {\it name} {\it alias} {\it domain} {\tt,}
alpar@9	1904 {\it attrib} {\tt,} \dots {\tt,} {\it attrib} {\tt;}
alpar@9	1905 }}
alpar@9	1906
alpar@9	1907 \setlength{\leftmargini}{60pt}
alpar@9	1908
alpar@9	1909 \begin{description}
alpar@9	1910 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	1911 set;
alpar@9	1912 \item[\hspace*{54pt}] {\it alias} is an optional string literal, which
alpar@9	1913 specifies an alias of the set;
alpar@9	1914 \item[\hspace*{54pt}] {\it domain} is an optional indexing expression,
alpar@9	1915 which specifies a subscript domain of the set;
alpar@9	1916 \item[\hspace*{54pt}] {\it attrib}, \dots, {\it attrib} are optional
alpar@9	1917 attributes of the set. (Commae preceding attributes may be omitted.)
alpar@9	1918 \end{description}
alpar@9	1919
alpar@9	1920 \noindent Optional attributes:
alpar@9	1921
alpar@9	1922 \begin{description}
alpar@9	1923 \item[{\tt dimen} $n$\hspace*{19pt}] specifies the dimension of
alpar@9	1924 $n$-tuples, which the set consists of;
alpar@9	1925 \item[{\tt within} {\it expression}]\hspace*{0pt}\\
alpar@9	1926 specifies a superset which restricts the set or all its members
alpar@9	1927 (elemental sets) to be within that superset;
alpar@9	1928 \item[{\tt:=} {\it expression}]\hspace*{0pt}\\
alpar@9	1929 specifies an elemental set assigned to the set or its members;
alpar@9	1930 \item[{\tt default} {\it expression}]\hspace*{0pt}\\
alpar@9	1931 specifies an elemental set assigned to the set or its members whenever
alpar@9	1932 no appropriate data are available in the data section.
alpar@9	1933 \end{description}
alpar@9	1934
alpar@9	1935 \newpage
alpar@9	1936
alpar@9	1937 \noindent{\bf Examples}
alpar@9	1938
alpar@9	1939 \begin{verbatim}
alpar@9	1940 set V;
alpar@9	1941 set E within V cross V;
alpar@9	1942 set step{s in 1..maxiter} dimen 2 := if s = 1 then E else
alpar@9	1943 step[s-1] union setof{k in V, (i,k) in step[s-1], (k,j)
alpar@9	1944 in step[s-1]}(i,j);
alpar@9	1945 set A{i in I, j in J}, within B[i+1] cross C[j-1], within
alpar@9	1946 D diff E, default {('abc',123), (321,'cba')};
alpar@9	1947 \end{verbatim}
alpar@9	1948
alpar@9	1949 The set statement declares a set. If the subscript domain is not
alpar@9	1950 specified, the set is a simple set, otherwise it is an array of
alpar@9	1951 elemental sets.
alpar@9	1952
alpar@9	1953 The {\tt dimen} attribute specifies the dimension of $n$-tuples, which
alpar@9	1954 the set (if it is a simple set) or its members (if the set is an array
alpar@9	1955 of elemental sets) consist of, where $n$ must be unsigned integer from
alpar@9	1956 1 to 20. At most one {\tt dimen} attribute can be specified. If the
alpar@9	1957 {\tt dimen} attribute is not specified, the dimension of\linebreak
alpar@9	1958 $n$-tuples is implicitly determined by other attributes (for example,
alpar@9	1959 if there is a set expression that follows {\tt:=} or the keyword
alpar@9	1960 {\tt default}, the dimension of $n$-tuples of corresponding elemental
alpar@9	1961 set is used). If no dimension information is available, {\tt dimen 1}
alpar@9	1962 is assumed.
alpar@9	1963
alpar@9	1964 The {\tt within} attribute specifies a set expression whose resultant
alpar@9	1965 value is a superset used to restrict the set (if it is a simple set) or
alpar@9	1966 its members (if the set is an array of elemental sets) to be within
alpar@9	1967 that superset. Arbitrary number of {\tt within} attributes may be
alpar@9	1968 specified in the same set statement.
alpar@9	1969
alpar@9	1970 The assign ({\tt:=}) attribute specifies a set expression used to
alpar@9	1971 evaluate elemental set(s) assigned to the set (if it is a simple set)
alpar@9	1972 or its members (if the set is an array of elemental sets). If the
alpar@9	1973 assign attribute is specified, the set is {\it computable} and
alpar@9	1974 therefore needs no data to be provided in the data section. If the
alpar@9	1975 assign attribute is not specified, the set must be provided with data
alpar@9	1976 in the data section. At most one assign or default attribute can be
alpar@9	1977 specified for the same set.
alpar@9	1978
alpar@9	1979 The {\tt default} attribute specifies a set expression used to evaluate
alpar@9	1980 elemental set(s) assigned to the set (if it is a simple set) or its
alpar@9	1981 members (if the set is an array of elemental sets) whenever
alpar@9	1982 no appropriate data are available in the data section. If neither
alpar@9	1983 assign nor default attribute is specified, missing data will cause an
alpar@9	1984 error.
alpar@9	1985
alpar@9	1986 \subsection{Parameter statement}
alpar@9	1987
alpar@9	1988 \medskip
alpar@9	1989
alpar@9	1990 \framebox[345pt][l]{
alpar@9	1991 \parbox[c][24pt]{345pt}{
alpar@9	1992 \hspace{6pt} {\tt param} {\it name} {\it alias} {\it domain} {\tt,}
alpar@9	1993 {\it attrib} {\tt,} \dots {\tt,} {\it attrib} {\tt;}
alpar@9	1994 }}
alpar@9	1995
alpar@9	1996 \setlength{\leftmargini}{60pt}
alpar@9	1997
alpar@9	1998 \begin{description}
alpar@9	1999 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	2000 parameter;
alpar@9	2001 \item[\hspace*{54pt}] {\it alias} is an optional string literal, which
alpar@9	2002 specifies an alias of the parameter;
alpar@9	2003 \item[\hspace*{54pt}] {\it domain} is an optional indexing expression,
alpar@9	2004 which specifies a subscript domain of the parameter;
alpar@9	2005 \item[\hspace*{54pt}] {\it attrib}, \dots, {\it attrib} are optional
alpar@9	2006 attributes of the parameter. (Commae preceding attributes may be
alpar@9	2007 omitted.)
alpar@9	2008 \end{description}
alpar@9	2009
alpar@9	2010 \noindent Optional attributes:
alpar@9	2011
alpar@9	2012 \begin{description}
alpar@9	2013 \item[{\tt integer}\hspace*{18.5pt}] specifies that the parameter is
alpar@9	2014 integer;
alpar@9	2015 \item[{\tt binary}\hspace*{24pt}] specifies that the parameter is
alpar@9	2016 binary;
alpar@9	2017 \item[{\tt symbolic}\hspace*{13.5pt}] specifies that the parameter is
alpar@9	2018 symbolic;
alpar@9	2019 \item[{\it relation expression}]\hspace*{0pt}\\
alpar@9	2020 (where {\it relation} is one of: {\tt<}, {\tt<=}, {\tt=}, {\tt==},
alpar@9	2021 {\tt>=}, {\tt>}, {\tt<>}, {\tt!=})\\
alpar@9	2022 specifies a condition that restricts the parameter or its members to
alpar@9	2023 satisfy that condition;
alpar@9	2024 \item[{\tt in} {\it expression}]\hspace*{0pt}\\
alpar@9	2025 specifies a superset that restricts the parameter or its members to be
alpar@9	2026 in that superset;
alpar@9	2027 \item[{\tt:=} {\it expression}]\hspace*{0pt}\\
alpar@9	2028 specifies a value assigned to the parameter or its members;
alpar@9	2029 \item[{\tt default} {\it expression}]\hspace*{0pt}\\
alpar@9	2030 specifies a value assigned to the parameter or its members whenever
alpar@9	2031 no appropriate data are available in the data section.
alpar@9	2032 \end{description}
alpar@9	2033
alpar@9	2034 \noindent{\bf Examples}
alpar@9	2035
alpar@9	2036 \begin{verbatim}
alpar@9	2037 param units{raw, prd} >= 0;
alpar@9	2038 param profit{prd, 1..T+1};
alpar@9	2039 param N := 20, integer, >= 0, <= 100;
alpar@9	2040 param comb 'n choose k' {n in 0..N, k in 0..n} :=
alpar@9	2041 if k = 0 or k = n then 1 else comb[n-1,k-1] + comb[n-1,k];
alpar@9	2042 param p{i in I, j in J}, integer, >= 0, <= i+j,
alpar@9	2043 in A[i] symdiff B[j], in C[i,j], default 0.5 * (i + j);
alpar@9	2044 param month symbolic default 'May' in {'Mar', 'Apr', 'May'};
alpar@9	2045 \end{verbatim}
alpar@9	2046
alpar@9	2047 The parameter statement declares a parameter. If a subscript domain is
alpar@9	2048 not specified, the parameter is a simple (scalar) parameter, otherwise
alpar@9	2049 it is a $n$-dimensional array.
alpar@9	2050
alpar@9	2051 The type attributes {\tt integer}, {\tt binary}, and {\tt symbolic}
alpar@9	2052 qualify the type of values that can be assigned to the parameter as
alpar@9	2053 shown below:
alpar@9	2054
alpar@9	2055 \medskip
alpar@9	2056
alpar@9	2057 \noindent\hfil
alpar@9	2058 \begin{tabular}{@{}ll@{}}
alpar@9	2059 Type attribute&Assigned values\\
alpar@9	2060 \hline
alpar@9	2061 (not specified)&Any numeric values\\
alpar@9	2062 {\tt integer}&Only integer numeric values\\
alpar@9	2063 {\tt binary}&Either 0 or 1\\
alpar@9	2064 {\tt symbolic}&Any numeric and symbolic values\\
alpar@9	2065 \end{tabular}
alpar@9	2066
alpar@9	2067 \newpage
alpar@9	2068
alpar@9	2069 The {\tt symbolic} attribute cannot be specified along with other type
alpar@9	2070 attributes. Being specified it must precede all other attributes.
alpar@9	2071
alpar@9	2072 The condition attribute specifies an optional condition that restricts
alpar@9	2073 values assigned to the parameter to satisfy that condition. This
alpar@9	2074 attribute has the following syntactic forms:
alpar@9	2075
alpar@9	2076 \medskip
alpar@9	2077
alpar@9	2078 \begin{tabular}{@{}ll@{}}
alpar@9	2079 {\tt<} $v$&check for $x<v$\\
alpar@9	2080 {\tt<=} $v$&check for $x\leq v$\\
alpar@9	2081 {\tt=} $v$, {\tt==} $v$&check for $x=v$\\
alpar@9	2082 {\tt>=} $v$&check for $x\geq v$\\
alpar@9	2083 {\tt>} $v$&check for $x\geq v$\\
alpar@9	2084 {\tt<>} $v$, {\tt!=} $v$&check for $x\neq v$\\
alpar@9	2085 \end{tabular}
alpar@9	2086
alpar@9	2087 \medskip
alpar@9	2088
alpar@9	2089 \noindent where $x$ is a value assigned to the parameter, $v$ is the
alpar@9	2090 resultant value of a numeric or symbolic expression specified in the
alpar@9	2091 condition attribute. Arbitrary number of condition attributes can be
alpar@9	2092 specified for the same parameter. If a value being assigned to the
alpar@9	2093 parameter during model evaluation violates at least one of specified
alpar@9	2094 conditions, an error is raised. (Note that symbolic values are ordered
alpar@9	2095 lexicographically, and any numeric value precedes any symbolic value.)
alpar@9	2096
alpar@9	2097 The {\tt in} attribute is similar to the condition attribute and
alpar@9	2098 specifies a set expression whose resultant value is a superset used to
alpar@9	2099 restrict numeric or symbolic values assigned to the parameter to be in
alpar@9	2100 that superset. Arbitrary number of the {\tt in} attributes can be
alpar@9	2101 specified for the same parameter. If a value being assigned to the
alpar@9	2102 parameter during model evaluation is not in at least one of specified
alpar@9	2103 supersets, an error is raised.
alpar@9	2104
alpar@9	2105 The assign ({\tt:=}) attribute specifies a numeric or symbolic
alpar@9	2106 expression used to compute a value assigned to the parameter (if it is
alpar@9	2107 a simple parameter) or its member (if the parameter is an array). If
alpar@9	2108 the assign attribute is specified, the parameter is {\it computable}
alpar@9	2109 and therefore needs no data to be provided in the data section. If the
alpar@9	2110 assign attribute is not specified, the parameter must be provided with
alpar@9	2111 data in the data section. At most one assign or {\tt default} attribute
alpar@9	2112 can be specified for the same parameter.
alpar@9	2113
alpar@9	2114 The {\tt default} attribute specifies a numeric or symbolic expression
alpar@9	2115 used to compute a value assigned to the parameter or its member
alpar@9	2116 whenever no appropriate data are available in the data section. If
alpar@9	2117 neither assign nor {\tt default} attribute is specified, missing data
alpar@9	2118 will cause an error.
alpar@9	2119
alpar@9	2120 \subsection{Variable statement}
alpar@9	2121
alpar@9	2122 \medskip
alpar@9	2123
alpar@9	2124 \framebox[345pt][l]{
alpar@9	2125 \parbox[c][24pt]{345pt}{
alpar@9	2126 \hspace{6pt} {\tt var} {\it name} {\it alias} {\it domain} {\tt,}
alpar@9	2127 {\it attrib} {\tt,} \dots {\tt,} {\it attrib} {\tt;}
alpar@9	2128 }}
alpar@9	2129
alpar@9	2130 \setlength{\leftmargini}{60pt}
alpar@9	2131
alpar@9	2132 \begin{description}
alpar@9	2133 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	2134 variable;
alpar@9	2135 \item[\hspace*{54pt}] {\it alias} is an optional string literal, which
alpar@9	2136 specifies an alias of the variable;
alpar@9	2137 \item[\hspace*{54pt}] {\it domain} is an optional indexing expression,
alpar@9	2138 which specifies a subscript domain of the variable;
alpar@9	2139 \item[\hspace*{54pt}] {\it attrib}, \dots, {\it attrib} are optional
alpar@9	2140 attributes of the variable. (Commae preceding attributes may be
alpar@9	2141 omitted.)
alpar@9	2142 \end{description}
alpar@9	2143
alpar@9	2144 \noindent Optional attributes:
alpar@9	2145
alpar@9	2146 \begin{description}
alpar@9	2147 \item[{\tt integer}\hspace*{18.5pt}] restricts the variable to be
alpar@9	2148 integer;
alpar@9	2149 \item[{\tt binary}\hspace*{24pt}] restricts the variable to be binary;
alpar@9	2150 \item[{\tt>=} {\it expression}]\hspace*{0pt}\\
alpar@9	2151 specifies an lower bound of the variable;
alpar@9	2152 \item[{\tt<=} {\it expression}]\hspace*{0pt}\\
alpar@9	2153 specifies an upper bound of the variable;
alpar@9	2154 \item[{\tt=} {\it expression}]\hspace*{0pt}\\
alpar@9	2155 specifies a fixed value of the variable;
alpar@9	2156 \end{description}
alpar@9	2157
alpar@9	2158 \noindent{\bf Examples}
alpar@9	2159
alpar@9	2160 \begin{verbatim}
alpar@9	2161 var x >= 0;
alpar@9	2162 var y{I,J};
alpar@9	2163 var make{p in prd}, integer, >= commit[p], <= market[p];
alpar@9	2164 var store{raw, 1..T+1} >= 0;
alpar@9	2165 var z{i in I, j in J} >= i+j;
alpar@9	2166 \end{verbatim}
alpar@9	2167
alpar@9	2168 The variable statement declares a variable. If a subscript domain is
alpar@9	2169 not specified, the variable is a simple (scalar) variable, otherwise it
alpar@9	2170 is a $n$-dimensional array of elemental variables.
alpar@9	2171
alpar@9	2172 Elemental variable(s) associated with the model variable (if it is a
alpar@9	2173 simple variable) or its members (if it is an array) correspond to the
alpar@9	2174 variables in the LP/MIP problem formulation (see Subsection
alpar@9	2175 \ref{problem}, page \pageref{problem}). Note that only elemental
alpar@9	2176 variables actually referenced in some constraints and/or objectives are
alpar@9	2177 included in the LP/MIP problem instance to be generated.
alpar@9	2178
alpar@9	2179 The type attributes {\tt integer} and {\tt binary} restrict the
alpar@9	2180 variable to be integer or binary, respectively. If no type attribute is
alpar@9	2181 specified, the variable is continuous. If all variables in the model
alpar@9	2182 are continuous, the corresponding problem is of LP class. If there is
alpar@9	2183 at least one integer or binary variable, the problem is of MIP class.
alpar@9	2184
alpar@9	2185 The lower bound ({\tt>=}) attribute specifies a numeric expression for
alpar@9	2186 computing an lower bound of the variable. At most one lower bound can
alpar@9	2187 be specified. By default all variables (except binary ones) have no
alpar@9	2188 lower bound, so if a variable is required to be non-negative, its zero
alpar@9	2189 lower bound should be explicitly specified.
alpar@9	2190
alpar@9	2191 The upper bound ({\tt<=}) attribute specifies a numeric expression for
alpar@9	2192 computing an upper bound of the variable. At most one upper bound
alpar@9	2193 attribute can be specified.
alpar@9	2194
alpar@9	2195 The fixed value ({\tt=}) attribute specifies a numeric expression for
alpar@9	2196 computing a value, at which the variable is fixed. This attribute
alpar@9	2197 cannot be specified along with the bound attributes.
alpar@9	2198
alpar@9	2199 \subsection{Constraint statement}
alpar@9	2200
alpar@9	2201 \medskip
alpar@9	2202
alpar@9	2203 \framebox[345pt][l]{
alpar@9	2204 \parbox[c][96pt]{345pt}{
alpar@9	2205 \hspace{6pt} {\tt s.t.} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2206 {\it expression} {\tt,} {\tt=} {\it expression} {\tt;}
alpar@9	2207
alpar@9	2208 \medskip
alpar@9	2209
alpar@9	2210 \hspace{6pt} {\tt s.t.} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2211 {\it expression} {\tt,} {\tt<=} {\it expression} {\tt;}
alpar@9	2212
alpar@9	2213 \medskip
alpar@9	2214
alpar@9	2215 \hspace{6pt} {\tt s.t.} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2216 {\it expression} {\tt,} {\tt>=} {\it expression} {\tt;}
alpar@9	2217
alpar@9	2218 \medskip
alpar@9	2219
alpar@9	2220 \hspace{6pt} {\tt s.t.} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2221 {\it expression} {\tt,} {\tt<=} {\it expression} {\tt,} {\tt<=}
alpar@9	2222 {\it expression} {\tt;}
alpar@9	2223
alpar@9	2224 \medskip
alpar@9	2225
alpar@9	2226 \hspace{6pt} {\tt s.t.} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2227 {\it expression} {\tt,} {\tt>=} {\it expression} {\tt,} {\tt>=}
alpar@9	2228 {\it expression} {\tt;}
alpar@9	2229 }}
alpar@9	2230
alpar@9	2231 \setlength{\leftmargini}{60pt}
alpar@9	2232
alpar@9	2233 \begin{description}
alpar@9	2234 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	2235 constraint;
alpar@9	2236 \item[\hspace*{54pt}] {\it alias} is an optional string literal, which
alpar@9	2237 specifies an alias of the constraint;
alpar@9	2238 \item[\hspace*{54pt}] {\it domain} is an optional indexing expression,
alpar@9	2239 which specifies a subscript domain of the constraint;
alpar@9	2240 \item[\hspace*{54pt}] {\it expression} is a linear expression used to
alpar@9	2241 compute a component of the constraint. (Commae following expressions
alpar@9	2242 may be omitted.)
alpar@9	2243 \end{description}
alpar@9	2244
alpar@9	2245 \begin{description}
alpar@9	2246 \item[{\rm Note:}\hspace*{31pt}] The keyword {\tt s.t.} may be written
alpar@9	2247 as {\tt subject to} or as {\tt subj to}, or may be omitted at all.
alpar@9	2248 \end{description}
alpar@9	2249
alpar@9	2250 \noindent{\bf Examples}
alpar@9	2251
alpar@9	2252 \begin{verbatim}
alpar@9	2253 s.t. r: x + y + z, >= 0, <= 1;
alpar@9	2254 limit{t in 1..T}: sum{j in prd} make[j,t] <= max_prd;
alpar@9	2255 subject to balance{i in raw, t in 1..T}: store[i,t+1] -
alpar@9	2256 store[i,t] - sum{j in prd} units[i,j] * make[j,t];
alpar@9	2257 subject to rlim 'regular-time limit' {t in time}:
alpar@9	2258 sum{p in prd} pt[p] * rprd[p,t] <= 1.3 * dpp[t] * crews[t];
alpar@9	2259 \end{verbatim}
alpar@9	2260
alpar@9	2261 The constraint statement declares a constraint. If a subscript domain
alpar@9	2262 is not specified, the constraint is a simple (scalar) constraint,
alpar@9	2263 otherwise it is a $n$-dimensional array of elemental constraints.
alpar@9	2264
alpar@9	2265 Elemental constraint(s) associated with the model constraint (if it is
alpar@9	2266 a simple constraint) or its members (if it is an array) correspond to
alpar@9	2267 the linear constraints in the LP/MIP problem formulation (see
alpar@9	2268 Subsection \ref{problem}, page \pageref{problem}).
alpar@9	2269
alpar@9	2270 If the constraint has the form of equality or single inequality, i.e.
alpar@9	2271 includes two expressions, one of which follows the colon and other
alpar@9	2272 follows the relation sign {\tt=}, {\tt<=}, or {\tt>=}, both expressions
alpar@9	2273 in the statement can be linear expressions. If the constraint has the
alpar@9	2274 form of double inequality, i.e. includes three expressions, the middle
alpar@9	2275 expression can be a linear expression while the leftmost and rightmost
alpar@9	2276 ones can be only numeric expressions.
alpar@9	2277
alpar@9	2278 Generating the model is, roughly speaking, generating its constraints,
alpar@9	2279 which are always evaluated for the entire subscript domain. Evaluation
alpar@9	2280 of the constraints leads, in turn, to evaluation of other model objects
alpar@9	2281 such as sets, parameters, and variables.
alpar@9	2282
alpar@9	2283 Constructing an actual linear constraint included in the problem
alpar@9	2284 instance, which (constraint) corresponds to a particular elemental
alpar@9	2285 constraint, is performed as follows.
alpar@9	2286
alpar@9	2287 If the constraint has the form of equality or single inequality,
alpar@9	2288 evaluation of both linear expressions gives two resultant linear forms:
alpar@9	2289 $$\begin{array}{r@{\ }c@{\ }r@{\ }c@{\ }r@{\ }c@{\ }r@{\ }c@{\ }r}
alpar@9	2290 f&=&a_1x_1&+&a_2x_2&+\dots+&a_nx_n&+&a_0,\\
alpar@9	2291 g&=&b_1x_1&+&a_2x_2&+\dots+&a_nx_n&+&b_0,\\
alpar@9	2292 \end{array}$$
alpar@9	2293 where $x_1$, $x_2$, \dots, $x_n$ are elemental variables; $a_1$, $a_2$,
alpar@9	2294 \dots, $a_n$, $b_1$, $b_2$, \dots, $b_n$ are numeric coefficients;
alpar@9	2295 $a_0$ and $b_0$ are constant terms. Then all linear terms of $f$ and
alpar@9	2296 $g$ are carried to the left-hand side, and the constant terms are
alpar@9	2297 carried to the right-hand side, that gives the final elemental
alpar@9	2298 constraint in the standard form:
alpar@9	2299 $$(a_1-b_1)x_1+(a_2-b_2)x_2+\dots+(a_n-b_n)x_n\left\{
alpar@9	2300 \begin{array}{@{}c@{}}=\\\leq\\\geq\\\end{array}\right\}b_0-a_0.$$
alpar@9	2301
alpar@9	2302 If the constraint has the form of double inequality, evaluation of the
alpar@9	2303 middle linear expression gives the resultant linear form:
alpar@9	2304 $$f=a_1x_1+a_2x_2+\dots+a_nx_n+a_0,$$
alpar@9	2305 and evaluation of the leftmost and rightmost numeric expressions gives
alpar@9	2306 two numeric values $l$ and $u$, respectively. Then the constant term of
alpar@9	2307 the linear form is carried to both left-hand and right-handsides that
alpar@9	2308 gives the final elemental constraint in the standard form:
alpar@9	2309 $$l-a_0\leq a_1x_1+a_2x_2+\dots+a_nx_n\leq u-a_0.$$
alpar@9	2310
alpar@9	2311 \subsection{Objective statement}
alpar@9	2312
alpar@9	2313 \medskip
alpar@9	2314
alpar@9	2315 \framebox[345pt][l]{
alpar@9	2316 \parbox[c][44pt]{345pt}{
alpar@9	2317 \hspace{6pt} {\tt minimize} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2318 {\it expression} {\tt;}
alpar@9	2319
alpar@9	2320 \medskip
alpar@9	2321
alpar@9	2322 \hspace{6pt} {\tt maximize} {\it name} {\it alias} {\it domain} {\tt:}
alpar@9	2323 {\it expression} {\tt;}
alpar@9	2324 }}
alpar@9	2325
alpar@9	2326 \setlength{\leftmargini}{60pt}
alpar@9	2327
alpar@9	2328 \begin{description}
alpar@9	2329 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	2330 objective;
alpar@9	2331 \item[\hspace*{54pt}] {\it alias} is an optional string literal, which
alpar@9	2332 specifies an alias of the objective;
alpar@9	2333 \item[\hspace*{54pt}] {\it domain} is an optional indexing expression,
alpar@9	2334 which specifies a subscript domain of the objective;
alpar@9	2335 \item[\hspace*{54pt}] {\it expression} is a linear expression used to
alpar@9	2336 compute the linear form of the objective.
alpar@9	2337 \end{description}
alpar@9	2338
alpar@9	2339 \noindent{\bf Examples}
alpar@9	2340
alpar@9	2341 \begin{verbatim}
alpar@9	2342 minimize obj: x + 1.5 * (y + z);
alpar@9	2343 maximize total_profit: sum{p in prd} profit[p] * make[p];
alpar@9	2344 \end{verbatim}
alpar@9	2345
alpar@9	2346 The objective statement declares an objective. If a subscript domain is
alpar@9	2347 not specified, the objective is a simple (scalar) objective. Otherwise
alpar@9	2348 it is a $n$-dimensional array of elemental objectives.
alpar@9	2349
alpar@9	2350 Elemental objective(s) associated with the model objective (if it is a
alpar@9	2351 simple objective) or its members (if it is an array) correspond to
alpar@9	2352 general linear constraints in the LP/MIP problem formulation (see
alpar@9	2353 Subsection \ref{problem}, page \pageref{problem}). However, unlike
alpar@9	2354 constraints the corresponding linear forms are free (unbounded).
alpar@9	2355
alpar@9	2356 Constructing an actual linear constraint included in the problem
alpar@9	2357 instance, which (constraint) corresponds to a particular elemental
alpar@9	2358 constraint, is performed as follows. The linear expression specified in
alpar@9	2359 the objective statement is evaluated that, gives the resultant linear
alpar@9	2360 form:
alpar@9	2361 $$f=a_1x_1+a_2x_2+\dots+a_nx_n+a_0,$$
alpar@9	2362 where $x_1$, $x_2$, \dots, $x_n$ are elemental variables; $a_1$, $a_2$,
alpar@9	2363 \dots, $a_n$ are numeric coefficients; $a_0$ is the constant term. Then
alpar@9	2364 the linear form is used to construct the final elemental constraint in
alpar@9	2365 the standard form:
alpar@9	2366 $$-\infty<a_1x_1+a_2x_2+\dots+a_nx_n+a_0<+\infty.$$
alpar@9	2367
alpar@9	2368 As a rule the model description contains only one objective statement
alpar@9	2369 that defines the objective function used in the problem instance.
alpar@9	2370 However, it is allowed to declare arbitrary number of objectives, in
alpar@9	2371 which case the actual objective function is the first objective
alpar@9	2372 encountered in the model description. Other objectives are also
alpar@9	2373 included in the problem instance, but they do not affect the objective
alpar@9	2374 function.
alpar@9	2375
alpar@9	2376 \subsection{Solve statement}
alpar@9	2377
alpar@9	2378 \medskip
alpar@9	2379
alpar@9	2380 \framebox[345pt][l]{
alpar@9	2381 \parbox[c][24pt]{345pt}{
alpar@9	2382 \hspace{6pt} {\tt solve} {\tt;}
alpar@9	2383 }}
alpar@9	2384
alpar@9	2385 \setlength{\leftmargini}{60pt}
alpar@9	2386
alpar@9	2387 \begin{description}
alpar@9	2388 \item[{\rm Note:}\hspace*{31pt}] The solve statement is optional and
alpar@9	2389 can be used only once. If no solve statement is used, one is assumed at
alpar@9	2390 the end of the model section.
alpar@9	2391 \end{description}
alpar@9	2392
alpar@9	2393 The solve statement causes the model to be solved, that means computing
alpar@9	2394 numeric values of all model variables. This allows using variables in
alpar@9	2395 statements below the solve statement in the same way as if they were
alpar@9	2396 numeric parameters.
alpar@9	2397
alpar@9	2398 Note that the variable, constraint, and objective statements cannot be
alpar@9	2399 used below the solve statement, i.e. all principal components of the
alpar@9	2400 model must be declared above the solve statement.
alpar@9	2401
alpar@9	2402 \subsection{Check statement}
alpar@9	2403
alpar@9	2404 \medskip
alpar@9	2405
alpar@9	2406 \framebox[345pt][l]{
alpar@9	2407 \parbox[c][24pt]{345pt}{
alpar@9	2408 \hspace{6pt} {\tt check} {\it domain} {\tt:} {\it expression} {\tt;}
alpar@9	2409 }}
alpar@9	2410
alpar@9	2411 \setlength{\leftmargini}{60pt}
alpar@9	2412
alpar@9	2413 \begin{description}
alpar@9	2414 \item[{\rm Where:}\hspace*{23pt}] {\it domain} is an optional indexing
alpar@9	2415 expression, which specifies a subscript domain of the check statement;
alpar@9	2416 \item[\hspace*{54pt}] {\it expression} is an logical expression which
alpar@9	2417 specifies the logical condition to be checked. (The colon preceding
alpar@9	2418 {\it expression} may be omitted.)
alpar@9	2419 \end{description}
alpar@9	2420
alpar@9	2421 \noindent{\bf Examples}
alpar@9	2422
alpar@9	2423 \begin{verbatim}
alpar@9	2424 check: x + y <= 1 and x >= 0 and y >= 0;
alpar@9	2425 check sum{i in ORIG} supply[i] = sum{j in DEST} demand[j];
alpar@9	2426 check{i in I, j in 1..10}: S[i,j] in U[i] union V[j];
alpar@9	2427 \end{verbatim}
alpar@9	2428
alpar@9	2429 The check statement allows checking the resultant value of an logical
alpar@9	2430 expression specified in the statement. If the value is {\it false}, an
alpar@9	2431 error is reported.
alpar@9	2432
alpar@9	2433 If the subscript domain is not specified, the check is performed only
alpar@9	2434 once. Specifying the subscript domain allows performing multiple checks
alpar@9	2435 for every\linebreak $n$-tuple in the domain set. In the latter case the
alpar@9	2436 logical expression may include dummy indices introduced in
alpar@9	2437 corresponding indexing expression.
alpar@9	2438
alpar@9	2439 \subsection{Display statement}
alpar@9	2440
alpar@9	2441 \medskip
alpar@9	2442
alpar@9	2443 \framebox[345pt][l]{
alpar@9	2444 \parbox[c][24pt]{345pt}{
alpar@9	2445 \hspace{6pt} {\tt display} {\it domain} {\tt:} {\it item} {\tt,}
alpar@9	2446 \dots {\tt,} {\it item} {\tt;}
alpar@9	2447 }}
alpar@9	2448
alpar@9	2449 \setlength{\leftmargini}{60pt}
alpar@9	2450
alpar@9	2451 \begin{description}
alpar@9	2452 \item[{\rm Where:}\hspace*{23pt}] {\it domain} is an optional indexing
alpar@9	2453 expression, which specifies a subscript domain of the check statement;
alpar@9	2454 \item[\hspace*{54pt}] {\it item}, \dots, {\it item} are items to be
alpar@9	2455 displayed. (The colon preceding the first item may be omitted.)
alpar@9	2456 \end{description}
alpar@9	2457
alpar@9	2458 \noindent{\bf Examples}
alpar@9	2459
alpar@9	2460 \begin{verbatim}
alpar@9	2461 display: 'x =', x, 'y =', y, 'z =', z;
alpar@9	2462 display sqrt(x 2 + y 2 + z ** 2);
alpar@9	2463 display{i in I, j in J}: i, j, a[i,j], b[i,j];
alpar@9	2464 \end{verbatim}
alpar@9	2465
alpar@9	2466 \newpage
alpar@9	2467
alpar@9	2468 The display statement evaluates all items specified in the statement
alpar@9	2469 and writes their values to the terminal in plain text format.
alpar@9	2470
alpar@9	2471 If a subscript domain is not specified, items are evaluated and then
alpar@9	2472 displayed only once. Specifying the subscript domain causes items to be
alpar@9	2473 evaluated and displayed for every $n$-tuple in the domain set. In the
alpar@9	2474 latter case items may include dummy indices introduced in corresponding
alpar@9	2475 indexing expression.
alpar@9	2476
alpar@9	2477 An item to be displayed can be a model object (set, parameter, variable,
alpar@9	2478 constraint, objective) or an expression.
alpar@9	2479
alpar@9	2480 If the item is a computable object (i.e. a set or parameter provided
alpar@9	2481 with the assign attribute), the object is evaluated over the entire
alpar@9	2482 domain and then its content (i.e. the content of the object array) is
alpar@9	2483 displayed. Otherwise, if the item is not a computable object, only its
alpar@9	2484 current content (i.e. members actually generated during the model
alpar@9	2485 evaluation) is displayed.
alpar@9	2486
alpar@9	2487 If the item is an expression, the expression is evaluated and its
alpar@9	2488 resultant value is displayed.
alpar@9	2489
alpar@9	2490 \subsection{Printf statement}
alpar@9	2491
alpar@9	2492 \medskip
alpar@9	2493
alpar@9	2494 \framebox[345pt][l]{
alpar@9	2495 \parbox[c][60pt]{345pt}{
alpar@9	2496 \hspace{6pt} {\tt printf} {\it domain} {\tt:} {\it format} {\tt,}
alpar@9	2497 {\it expression} {\tt,} \dots {\tt,} {\it expression} {\tt;}
alpar@9	2498
alpar@9	2499 \medskip
alpar@9	2500
alpar@9	2501 \hspace{6pt} {\tt printf} {\it domain} {\tt:} {\it format} {\tt,}
alpar@9	2502 {\it expression} {\tt,} \dots {\tt,} {\it expression} {\tt>}
alpar@9	2503 {\it filename} {\tt;}
alpar@9	2504
alpar@9	2505 \medskip
alpar@9	2506
alpar@9	2507 \hspace{6pt} {\tt printf} {\it domain} {\tt:} {\it format} {\tt,}
alpar@9	2508 {\it expression} {\tt,} \dots {\tt,} {\it expression} {\tt>>}
alpar@9	2509 {\it filename} {\tt;}
alpar@9	2510 }}
alpar@9	2511
alpar@9	2512 \setlength{\leftmargini}{60pt}
alpar@9	2513
alpar@9	2514 \begin{description}
alpar@9	2515 \item[{\rm Where:}\hspace*{23pt}] {\it domain} is an optional indexing
alpar@9	2516 expression, which specifies a subscript domain of the printf statement;
alpar@9	2517 \item[\hspace*{54pt}] {\it format} is a symbolic expression whose value
alpar@9	2518 specifies a format control string. (The colon preceding the format
alpar@9	2519 expression may be omitted.)
alpar@9	2520 \item[\hspace*{54pt}] {\it expression}, \dots, {\it expression} are
alpar@9	2521 zero or more expressions whose values have to be formatted and printed.
alpar@9	2522 Each expression must be of numeric, symbolic, or logical type.
alpar@9	2523 \item[\hspace*{54pt}] {\it filename} is a symbolic expression whose
alpar@9	2524 value specifies a name of a text file, to which the output is
alpar@9	2525 redirected. The flag {\tt>} means creating a new empty file while the
alpar@9	2526 flag {\tt>>} means appending the output to an existing file. If no file
alpar@9	2527 name is specified, the output is written to the terminal.
alpar@9	2528 \end{description}
alpar@9	2529
alpar@9	2530 \noindent{\bf Examples}
alpar@9	2531
alpar@9	2532 \begin{verbatim}
alpar@9	2533 printf 'Hello, world!\n';
alpar@9	2534 printf: "x = %.3f; y = %.3f; z = %.3f\n",
alpar@9	2535 x, y, z > "result.txt";
alpar@9	2536 printf{i in I, j in J}: "flow from %s to %s is %d\n",
alpar@9	2537 i, j, x[i,j] >> result_file & ".txt";
alpar@9	2538 \end{verbatim}
alpar@9	2539
alpar@9	2540 \newpage
alpar@9	2541
alpar@9	2542 \begin{verbatim}
alpar@9	2543 printf{i in I} 'total flow from %s is %g\n',
alpar@9	2544 i, sum{j in J} x[i,j];
alpar@9	2545 printf{k in K} "x[%s] = " & (if x[k] < 0 then "?" else "%g"),
alpar@9	2546 k, x[k];
alpar@9	2547 \end{verbatim}
alpar@9	2548
alpar@9	2549 The printf statement is similar to the display statement, however, it
alpar@9	2550 allows formatting data to be written.
alpar@9	2551
alpar@9	2552 If a subscript domain is not specified, the printf statement is
alpar@9	2553 executed only once. Specifying a subscript domain causes executing the
alpar@9	2554 printf statement for every $n$-tuple in the domain set. In the latter
alpar@9	2555 case the format and expression may include dummy indices introduced in
alpar@9	2556 corresponding indexing expression.
alpar@9	2557
alpar@9	2558 The format control string is a value of the symbolic expression
alpar@9	2559 {\it format} specified in the printf statement. It is composed of zero
alpar@9	2560 or more directives as follows: ordinary characters (not {\tt\%}), which
alpar@9	2561 are copied unchanged to the output stream, and conversion
alpar@9	2562 specifications, each of which causes evaluating corresponding
alpar@9	2563 expression specified in the printf statement, formatting it, and
alpar@9	2564 writing its resultant value to the output stream.
alpar@9	2565
alpar@9	2566 Conversion specifications that may be used in the format control string
alpar@9	2567 are the following: {\tt d}, {\tt i}, {\tt f}, {\tt F}, {\tt e}, {\tt E},
alpar@9	2568 {\tt g}, {\tt G}, and {\tt s}. These specifications have the same
alpar@9	2569 syntax and semantics as in the C programming language.
alpar@9	2570
alpar@9	2571 \subsection{For statement}
alpar@9	2572
alpar@9	2573 \medskip
alpar@9	2574
alpar@9	2575 \framebox[345pt][l]{
alpar@9	2576 \parbox[c][44pt]{345pt}{
alpar@9	2577 \hspace{6pt} {\tt for} {\it domain} {\tt:} {\it statement} {\tt;}
alpar@9	2578
alpar@9	2579 \medskip
alpar@9	2580
alpar@9	2581 \hspace{6pt} {\tt for} {\it domain} {\tt:} {\tt\{} {\it statement}
alpar@9	2582 \dots {\it statement} {\tt\}} {\tt;}
alpar@9	2583 }}
alpar@9	2584
alpar@9	2585 \setlength{\leftmargini}{60pt}
alpar@9	2586
alpar@9	2587 \begin{description}
alpar@9	2588 \item[{\rm Where:}\hspace*{23pt}] {\it domain} is an indexing
alpar@9	2589 expression which specifies a subscript domain of the for statement.
alpar@9	2590 (The colon following the indexing expression may be omitted.)
alpar@9	2591 \item[\hspace*{54pt}] {\it statement} is a statement, which should be
alpar@9	2592 executed under control of the for statement;
alpar@9	2593 \item[\hspace*{54pt}] {\it statement}, \dots, {\it statement} is a
alpar@9	2594 sequence of statements (enclosed in curly braces), which should be
alpar@9	2595 executed under control of the for statement.
alpar@9	2596 \end{description}
alpar@9	2597
alpar@9	2598 \begin{description}
alpar@9	2599 \item[{\rm Note:}\hspace*{31pt}] Only the following statements can be
alpar@9	2600 used within the for statement: check, display, printf, and another for.
alpar@9	2601 \end{description}
alpar@9	2602
alpar@9	2603 \noindent{\bf Examples}
alpar@9	2604
alpar@9	2605 \begin{verbatim}
alpar@9	2606 for {(i,j) in E: i != j}
alpar@9	2607 { printf "flow from %s to %s is %g\n", i, j, x[i,j];
alpar@9	2608 check x[i,j] >= 0;
alpar@9	2609 }
alpar@9	2610 \end{verbatim}
alpar@9	2611
alpar@9	2612 \newpage
alpar@9	2613
alpar@9	2614 \begin{verbatim}
alpar@9	2615 for {i in 1..n}
alpar@9	2616 { for {j in 1..n} printf " %s", if x[i,j] then "Q" else ".";
alpar@9	2617 printf("\n");
alpar@9	2618 }
alpar@9	2619 for {1..72} printf("*");
alpar@9	2620 \end{verbatim}
alpar@9	2621
alpar@9	2622 The for statement causes a statement or a sequence of statements
alpar@9	2623 specified as part of the for statement to be executed for every
alpar@9	2624 $n$-tuple in the domain set. Thus, statements within the for statement
alpar@9	2625 may include dummy indices introduced in corresponding indexing
alpar@9	2626 expression.
alpar@9	2627
alpar@9	2628 \subsection{Table statement}
alpar@9	2629
alpar@9	2630 \medskip
alpar@9	2631
alpar@9	2632 \framebox[345pt][l]{
alpar@9	2633 \parbox[c][68pt]{345pt}{
alpar@9	2634 \hspace{6pt} {\tt table} {\it name} {\it alias} {\tt IN} {\it driver}
alpar@9	2635 {\it arg} \dots {\it arg} {\tt:}
alpar@9	2636
alpar@9	2637 \hspace{6pt} {\tt\ \ \ \ \ } {\it set} {\tt<-} {\tt[} {\it fld} {\tt,}
alpar@9	2638 \dots {\tt,} {\it fld} {\tt]} {\tt,} {\it par} {\tt\textasciitilde}
alpar@9	2639 {\it fld} {\tt,} \dots {\tt,} {\it par} {\tt\textasciitilde} {\it fld}
alpar@9	2640 {\tt;}
alpar@9	2641
alpar@9	2642 \medskip
alpar@9	2643
alpar@9	2644 \hspace{6pt} {\tt table} {\it name} {\it alias} {\it domain} {\tt OUT}
alpar@9	2645 {\it driver} {\it arg} \dots {\it arg} {\tt:}
alpar@9	2646
alpar@9	2647 \hspace{6pt} {\tt\ \ \ \ \ } {\it expr} {\tt\textasciitilde} {\it fld}
alpar@9	2648 {\tt,} \dots {\tt,} {\it expr} {\tt\textasciitilde} {\it fld} {\tt;}
alpar@9	2649 }}
alpar@9	2650
alpar@9	2651 \setlength{\leftmargini}{60pt}
alpar@9	2652
alpar@9	2653 \begin{description}
alpar@9	2654 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	2655 table;
alpar@9	2656 \item[\hspace*{54pt}] {\it alias} is an optional string literal, which
alpar@9	2657 specifies an alias of the table;
alpar@9	2658 \item[\hspace*{54pt}] {\it domain} is an indexing expression, which
alpar@9	2659 specifies a subscript domain of the (output) table;
alpar@9	2660 \item[\hspace*{54pt}] {\tt IN} means reading data from the input table;
alpar@9	2661 \item[\hspace*{54pt}] {\tt OUT} means writing data to the output table;
alpar@9	2662 \item[\hspace*{54pt}] {\it driver} is a symbolic expression, which
alpar@9	2663 specifies the driver used to access the table (for details see Section
alpar@9	2664 \ref{drivers}, page \pageref{drivers});
alpar@9	2665 \item[\hspace*{54pt}] {\it arg} is an optional symbolic expression,
alpar@9	2666 which is an argument pass\-ed to the table driver. This symbolic
alpar@9	2667 expression must not include dummy indices specified in the domain;
alpar@9	2668 \item[\hspace*{54pt}] {\it set} is the name of an optional simple set
alpar@9	2669 called {\it control set}. It can be omitted along with the delimiter
alpar@9	2670 {\tt<-};
alpar@9	2671 \item[\hspace*{54pt}] {\it fld} is a field name. Within square brackets
alpar@9	2672 at least one field should be specified. The field name following
alpar@9	2673 a parameter name or expression is optional and can be omitted along
alpar@9	2674 with the delimiter {\tt\textasciitilde}, in which case the name of
alpar@9	2675 corresponding model object is used as the field name;
alpar@9	2676 \item[\hspace*{54pt}] {\it par} is a symbolic name of a model parameter;
alpar@9	2677 \item[\hspace*{54pt}] {\it expr} is a numeric or symbolic expression.
alpar@9	2678 \end{description}
alpar@9	2679
alpar@9	2680 \newpage
alpar@9	2681
alpar@9	2682 \noindent{\bf Examples}
alpar@9	2683
alpar@9	2684 \begin{verbatim}
alpar@9	2685 table data IN "CSV" "data.csv":
alpar@9	2686 S <- [FROM,TO], d~DISTANCE, c~COST;
alpar@9	2687 table result{(f,t) in S} OUT "CSV" "result.csv":
alpar@9	2688 f~FROM, t~TO, x[f,t]~FLOW;
alpar@9	2689 \end{verbatim}
alpar@9	2690
alpar@9	2691 The table statement allows reading data from a table into model
alpar@9	2692 objects such as sets and (non-scalar) parameters as well as writing
alpar@9	2693 data from the model to a table.
alpar@9	2694
alpar@9	2695 \subsubsection{Table structure}
alpar@9	2696
alpar@9	2697 A {\it data table} is an (unordered) set of {\it records}, where each
alpar@9	2698 record consists of the same number of {\it fields}, and each field is
alpar@9	2699 provided with a unique symbolic name called the {\it field name}. For
alpar@9	2700 example:
alpar@9	2701
alpar@9	2702 \bigskip
alpar@9	2703
alpar@9	2704 \begin{tabular}{@{\hspace{38mm}}c@{\hspace{11mm}}c@{\hspace*{10mm}}c
alpar@9	2705 @{\hspace*{9mm}}c}
alpar@9	2706 First&Second&&Last\\
alpar@9	2707 field&field&.\ \ .\ \ .&field\\
alpar@9	2708 $\downarrow$&$\downarrow$&&$\downarrow$\\
alpar@9	2709 \end{tabular}
alpar@9	2710
alpar@9	2711 \begin{tabular}{ll@{}}
alpar@9	2712 Table header&$\rightarrow$\\
alpar@9	2713 First record&$\rightarrow$\\
alpar@9	2714 Second record&$\rightarrow$\\
alpar@9	2715 \\
alpar@9	2716 \hfil .\ \ .\ \ .\\
alpar@9	2717 \\
alpar@9	2718 Last record&$\rightarrow$\\
alpar@9	2719 \end{tabular}
alpar@9	2720 \begin{tabular}{\|l\|l\|c\|c\|}
alpar@9	2721 \hline
alpar@9	2722 {\tt FROM}&{\tt TO}&{\tt DISTANCE}&{\tt COST}\\
alpar@9	2723 \hline
alpar@9	2724 {\tt Seattle} &{\tt New-York}&{\tt 2.5}&{\tt 0.12}\\
alpar@9	2725 {\tt Seattle} &{\tt Chicago} &{\tt 1.7}&{\tt 0.08}\\
alpar@9	2726 {\tt Seattle} &{\tt Topeka} &{\tt 1.8}&{\tt 0.09}\\
alpar@9	2727 {\tt San-Diego}&{\tt New-York}&{\tt 2.5}&{\tt 0.15}\\
alpar@9	2728 {\tt San-Diego}&{\tt Chicago} &{\tt 1.8}&{\tt 0.10}\\
alpar@9	2729 {\tt San-Diego}&{\tt Topeka} &{\tt 1.4}&{\tt 0.07}\\
alpar@9	2730 \hline
alpar@9	2731 \end{tabular}
alpar@9	2732
alpar@9	2733 \subsubsection{Reading data from input table}
alpar@9	2734
alpar@9	2735 The input table statement causes reading data from the specified table
alpar@9	2736 record by record.
alpar@9	2737
alpar@9	2738 Once a next record has been read, numeric or symbolic values of fields,
alpar@9	2739 whose names are enclosed in square brackets in the table statement, are
alpar@9	2740 gathered into $n$-tuple, and if the control set is specified in the
alpar@9	2741 table statement, this $n$-tuple is added to it. Besides, a numeric or
alpar@9	2742 symbolic value of each field associated with a model parameter is
alpar@9	2743 assigned to the parameter member identified by subscripts, which are
alpar@9	2744 components of the $n$-tuple just read.
alpar@9	2745
alpar@9	2746 For example, the following input table statement:
alpar@9	2747
alpar@9	2748 \medskip
alpar@9	2749
alpar@9	2750 \noindent\hfil
alpar@9	2751 \verb\|table data IN "...": S <- [FROM,TO], d~DISTANCE, c~COST;\|
alpar@9	2752
alpar@9	2753 \medskip
alpar@9	2754
alpar@9	2755 \noindent
alpar@9	2756 causes reading values of four fields named {\tt FROM}, {\tt TO},
alpar@9	2757 {\tt DISTANCE}, and {\tt COST} from each record of the specified table.
alpar@9	2758 Values of fields {\tt FROM} and {\tt TO} give a pair $(f,t)$, which is
alpar@9	2759 added to the control set {\tt S}. The value of field {\tt DISTANCE} is
alpar@9	2760 assigned to parameter member ${\tt d}[f,t]$, and the value of field
alpar@9	2761 {\tt COST} is assigned to parameter member ${\tt c}[f,t]$.
alpar@9	2762
alpar@9	2763 Note that the input table may contain extra fields whose names are not
alpar@9	2764 specified in the table statement, in which case values of these fields
alpar@9	2765 on reading the table are ignored.
alpar@9	2766
alpar@9	2767 \subsubsection{Writing data to output table}
alpar@9	2768
alpar@9	2769 The output table statement causes writing data to the specified table.
alpar@9	2770 Note that some drivers (namely, CSV and xBASE) destroy the output table
alpar@9	2771 before writing data, i.e. delete all its existing records.
alpar@9	2772
alpar@9	2773 Each $n$-tuple in the specified domain set generates one record written
alpar@9	2774 to the output table. Values of fields are numeric or symbolic values of
alpar@9	2775 corresponding expressions specified in the table statement. These
alpar@9	2776 expressions are evaluated for each $n$-tuple in the domain set and,
alpar@9	2777 thus, may include dummy indices introduced in the corresponding indexing
alpar@9	2778 expression.
alpar@9	2779
alpar@9	2780 For example, the following output table statement:
alpar@9	2781
alpar@9	2782 \medskip
alpar@9	2783
alpar@9	2784 \noindent
alpar@9	2785 \verb\| table result{(f,t) in S} OUT "...": f~FROM, t~TO, x[f,t]~FLOW;\|
alpar@9	2786
alpar@9	2787 \medskip
alpar@9	2788
alpar@9	2789 \noindent
alpar@9	2790 causes writing records, by one record for each pair $(f,t)$ in set
alpar@9	2791 {\tt S}, to the output table, where each record consists of three
alpar@9	2792 fields named {\tt FROM}, {\tt TO}, and {\tt FLOW}. The values written
alpar@9	2793 to fields {\tt FROM} and {\tt TO} are current values of dummy indices
alpar@9	2794 {\tt f} and {\tt t}, and the value written to field {\tt FLOW} is
alpar@9	2795 a value of member ${\tt x}[f,t]$ of corresponding subscripted parameter
alpar@9	2796 or variable.
alpar@9	2797
alpar@9	2798 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	2799
alpar@9	2800 \newpage
alpar@9	2801
alpar@9	2802 \section{Model data}
alpar@9	2803
alpar@9	2804 {\it Model data} include elemental sets, which are ``values'' of model
alpar@9	2805 sets, and numeric and symbolic values of model parameters.
alpar@9	2806
alpar@9	2807 In MathProg there are two different ways to saturate model sets and
alpar@9	2808 parameters with data. One way is simply providing necessary data using
alpar@9	2809 the assign attribute. However, in many cases it is more practical to
alpar@9	2810 separate the model itself and particular data needed for the model. For
alpar@9	2811 the latter reason in MathProg there is another way, when the model
alpar@9	2812 description is divided into two parts: model section and data section.
alpar@9	2813
alpar@9	2814 A {\it model section} is a main part of the model description that
alpar@9	2815 contains declarations of all model objects and is common for all
alpar@9	2816 problems based on that model.
alpar@9	2817
alpar@9	2818 A {\it data section} is an optional part of the model description that
alpar@9	2819 contains model data specific for a particular problem.
alpar@9	2820
alpar@9	2821 In MathProg model and data sections can be placed either in one text
alpar@9	2822 file or in two separate text files.
alpar@9	2823
alpar@9	2824 1. If both model and data sections are placed in one file, the file is
alpar@9	2825 composed as follows:
alpar@9	2826
alpar@9	2827 \bigskip
alpar@9	2828
alpar@9	2829 \noindent\hfil
alpar@9	2830 \framebox{\begin{tabular}{l}
alpar@9	2831 {\it statement}{\tt;}\\
alpar@9	2832 {\it statement}{\tt;}\\
alpar@9	2833 \hfil.\ \ .\ \ .\\
alpar@9	2834 {\it statement}{\tt;}\\
alpar@9	2835 {\tt data;}\\
alpar@9	2836 {\it data block}{\tt;}\\
alpar@9	2837 {\it data block}{\tt;}\\
alpar@9	2838 \hfil.\ \ .\ \ .\\
alpar@9	2839 {\it data block}{\tt;}\\
alpar@9	2840 {\tt end;}
alpar@9	2841 \end{tabular}}
alpar@9	2842
alpar@9	2843 \bigskip
alpar@9	2844
alpar@9	2845 2. If the model and data sections are placed in two separate files, the
alpar@9	2846 files are composed as follows:
alpar@9	2847
alpar@9	2848 \bigskip
alpar@9	2849
alpar@9	2850 \noindent\hfil
alpar@9	2851 \begin{tabular}{@{}c@{}}
alpar@9	2852 \framebox{\begin{tabular}{l}
alpar@9	2853 {\it statement}{\tt;}\\
alpar@9	2854 {\it statement}{\tt;}\\
alpar@9	2855 \hfil.\ \ .\ \ .\\
alpar@9	2856 {\it statement}{\tt;}\\
alpar@9	2857 {\tt end;}\\
alpar@9	2858 \end{tabular}}\\
alpar@9	2859 \\\\Model file\\
alpar@9	2860 \end{tabular}
alpar@9	2861 \hspace{32pt}
alpar@9	2862 \begin{tabular}{@{}c@{}}
alpar@9	2863 \framebox{\begin{tabular}{l}
alpar@9	2864 {\tt data;}\\
alpar@9	2865 {\it data block}{\tt;}\\
alpar@9	2866 {\it data block}{\tt;}\\
alpar@9	2867 \hfil.\ \ .\ \ .\\
alpar@9	2868 {\it data block}{\tt;}\\
alpar@9	2869 {\tt end;}\\
alpar@9	2870 \end{tabular}}\\
alpar@9	2871 \\Data file\\
alpar@9	2872 \end{tabular}
alpar@9	2873
alpar@9	2874 \bigskip
alpar@9	2875
alpar@9	2876 \begin{description}
alpar@9	2877 \item[{\rm Note:}\hspace*{31pt}] If the data section is placed in a
alpar@9	2878 separate file, the keyword {\tt data} is optional and may be omitted
alpar@9	2879 along with the semicolon that follows it.
alpar@9	2880 \end{description}
alpar@9	2881
alpar@9	2882 \subsection{Coding data section}
alpar@9	2883
alpar@9	2884 The {\it data section} is a sequence of data blocks in various formats,
alpar@9	2885 which are discussed in following subsections. The order, in which data
alpar@9	2886 blocks follow in the data section, may be arbitrary, not necessarily
alpar@9	2887 the same, in which corresponding model objects follow in the model
alpar@9	2888 section.
alpar@9	2889
alpar@9	2890 The rules of coding the data section are commonly the same as the rules
alpar@9	2891 of coding the model description (see Subsection \ref{coding}, page
alpar@9	2892 \pageref{coding}), i.e. data blocks are composed from basic lexical
alpar@9	2893 units such as symbolic names, numeric and string literals, keywords,
alpar@9	2894 delimiters, and comments. However, for the sake of convenience and
alpar@9	2895 improving readability there is one deviation from the common rule: if
alpar@9	2896 a string literal consists of only alphanumeric characters (including
alpar@9	2897 the underscore character), the signs {\tt+} and {\tt-}, and/or the
alpar@9	2898 decimal point, it may be coded without bordering by (single or double)
alpar@9	2899 quotes.
alpar@9	2900
alpar@9	2901 All numeric and symbolic material provided in the data section is coded
alpar@9	2902 in the form of numbers and symbols, i.e. unlike the model section
alpar@9	2903 no expressions are allowed in the data section. Nevertheless, the signs
alpar@9	2904 {\tt+} and {\tt-} can precede numeric literals to allow coding signed
alpar@9	2905 numeric quantities, in which case there must be no white-space
alpar@9	2906 characters between the sign and following numeric literal (if there is
alpar@9	2907 at least one white-space, the sign and following numeric literal are
alpar@9	2908 recognized as two different lexical units).
alpar@9	2909
alpar@9	2910 \subsection{Set data block}
alpar@9	2911
alpar@9	2912 \medskip
alpar@9	2913
alpar@9	2914 \framebox[345pt][l]{
alpar@9	2915 \parbox[c][44pt]{345pt}{
alpar@9	2916 \hspace{6pt} {\tt set} {\it name} {\tt,} {\it record} {\tt,} \dots
alpar@9	2917 {\tt,} {\it record} {\tt;}
alpar@9	2918
alpar@9	2919 \medskip
alpar@9	2920
alpar@9	2921 \hspace{6pt} {\tt set} {\it name} {\tt[} {\it symbol} {\tt,} \dots
alpar@9	2922 {\tt,} {\it symbol} {\tt]} {\tt,} {\it record} {\tt,} \dots {\tt,}
alpar@9	2923 {\it record} {\tt;}
alpar@9	2924 }}
alpar@9	2925
alpar@9	2926 \setlength{\leftmargini}{60pt}
alpar@9	2927
alpar@9	2928 \begin{description}
alpar@9	2929 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	2930 set;
alpar@9	2931 \item[\hspace*{54pt}] {\it symbol}, \dots, {\it symbol} are subscripts,
alpar@9	2932 which specify a particular member of the set (if the set is an array,
alpar@9	2933 i.e. a set of sets);
alpar@9	2934 \item[\hspace*{54pt}] {\it record}, \dots, {\it record} are data
alpar@9	2935 records.
alpar@9	2936 \end{description}
alpar@9	2937
alpar@9	2938 \begin{description}
alpar@9	2939 \item[{\rm Note:}\hspace*{31pt}] Commae preceding data records may be
alpar@9	2940 omitted.
alpar@9	2941 \end{description}
alpar@9	2942
alpar@9	2943 \noindent Data records:
alpar@9	2944
alpar@9	2945 \begin{description}
alpar@9	2946 \item[{\tt :=}\hspace*{45pt}] is a non-significant data record, which
alpar@9	2947 may be used freely to improve readability;
alpar@9	2948 \item[{\tt(} {\it slice} {\tt)}\hspace*{18.5pt}] specifies a slice;
alpar@9	2949 \item[{\it simple-data}\hspace*{5.5pt}] specifies set data in the
alpar@9	2950 simple format;
alpar@9	2951 \item[{\tt:} {\it matrix-data}]\hspace*{0pt}\\
alpar@9	2952 specifies set data in the matrix format;
alpar@9	2953 \item[{\tt(tr)} {\tt:} {\it matrix-data}]\hspace*{0pt}\\
alpar@9	2954 specifies set data in the transposed matrix format. (In this case the
alpar@9	2955 colon following the keyword {\tt(tr)} may be omitted.)
alpar@9	2956 \end{description}
alpar@9	2957
alpar@9	2958 \noindent{\bf Examples}
alpar@9	2959
alpar@9	2960 \begin{verbatim}
alpar@9	2961 set month := Jan Feb Mar Apr May Jun;
alpar@9	2962 set month "Jan", "Feb", "Mar", "Apr", "May", "Jun";
alpar@9	2963 set A[3,Mar] := (1,2) (2,3) (4,2) (3,1) (2,2) (4,4) (3,4);
alpar@9	2964 set A[3,'Mar'] := 1 2 2 3 4 2 3 1 2 2 4 4 2 4;
alpar@9	2965 set A[3,'Mar'] : 1 2 3 4 :=
alpar@9	2966 1 - + - -
alpar@9	2967 2 - + + -
alpar@9	2968 3 + - - +
alpar@9	2969 4 - + - + ;
alpar@9	2970 set B := (1,2,3) (1,3,2) (2,3,1) (2,1,3) (1,2,2) (1,1,1) (2,1,1);
alpar@9	2971 set B := (,,*) 1 2 3, 1 3 2, 2 3 1, 2 1 3, 1 2 2, 1 1 1, 2 1 1;
alpar@9	2972 set B := (1,,2) 3 2 (2,,1) 3 1 (1,2,3) (2,1,3) (1,1,1);
alpar@9	2973 set B := (1,,) : 1 2 3 :=
alpar@9	2974 1 + - -
alpar@9	2975 2 - + +
alpar@9	2976 3 - + -
alpar@9	2977 (2,,) : 1 2 3 :=
alpar@9	2978 1 + - +
alpar@9	2979 2 - - -
alpar@9	2980 3 + - - ;
alpar@9	2981 \end{verbatim}
alpar@9	2982
alpar@9	2983 \noindent(In these examples {\tt month} is a simple set of singlets,
alpar@9	2984 {\tt A} is a 2-dimensional array of doublets, and {\tt B} is a simple
alpar@9	2985 set of triplets. Data blocks for the same set are equivalent in the
alpar@9	2986 sense that they specify the same data in different formats.)
alpar@9	2987
alpar@9	2988 \medskip
alpar@9	2989
alpar@9	2990 The {\it set data block} is used to specify a complete elemental set,
alpar@9	2991 which is assigned to a set (if it is a simple set) or one of its
alpar@9	2992 members (if the set is an array of sets).\footnote{There is another way
alpar@9	2993 to specify data for a simple set along with data for parameters. This
alpar@9	2994 feature is discussed in the next subsection.}
alpar@9	2995
alpar@9	2996 Data blocks can be specified only for non-computable sets, i.e. for
alpar@9	2997 sets, which have no assign ({\tt:=}) attribute in the corresponding set
alpar@9	2998 statements.
alpar@9	2999
alpar@9	3000 If the set is a simple set, only its symbolic name should be specified
alpar@9	3001 in the header of the data block. Otherwise, if the set is a
alpar@9	3002 $n$-dimensional array, its symbolic name should be provided with a
alpar@9	3003 complete list of subscripts separated by commae and enclosed in square
alpar@9	3004 brackets to specify a particular member of the set array. The number of
alpar@9	3005 subscripts must be the same as the dimension of the set array, where
alpar@9	3006 each subscript must be a number or symbol.
alpar@9	3007
alpar@9	3008 An elemental set defined in the set data block is coded as a sequence
alpar@9	3009 of data records described below.\footnote{{\it Data record} is simply a
alpar@9	3010 technical term. It does not mean that data records have any special
alpar@9	3011 formatting.}
alpar@9	3012
alpar@9	3013 \newpage
alpar@9	3014
alpar@9	3015 \subsubsection{Assign data record}
alpar@9	3016
alpar@9	3017 The {\it assign} ({\tt:=}) {\it data record} is a non-signficant
alpar@9	3018 element. It may be used for improving readability of data blocks.
alpar@9	3019
alpar@9	3020 \subsubsection{Slice data record}
alpar@9	3021
alpar@9	3022 The {\it slice data record} is a control record, which specifies a
alpar@9	3023 {\it slice} of the elemental set defined in the data block. It has the
alpar@9	3024 following syntactic form:
alpar@9	3025
alpar@9	3026 \medskip
alpar@9	3027
alpar@9	3028 \noindent\hfil
alpar@9	3029 {\tt(} $s_1$ {\tt,} $s_2$ {\tt,} \dots {\tt,} $s_n$ {\tt)}
alpar@9	3030
alpar@9	3031 \medskip
alpar@9	3032
alpar@9	3033 \noindent where $s_1$, $s_2$, \dots, $s_n$ are components of the slice.
alpar@9	3034
alpar@9	3035 Each component of the slice can be a number or symbol or the asterisk
alpar@9	3036 ({\tt*}). The number of components in the slice must be the same as the
alpar@9	3037 dimension of $n$-tuples in the elemental set to be defined. For
alpar@9	3038 instance, if the elemental set contains 4-tuples (quadruplets), the
alpar@9	3039 slice must have four components. The number of asterisks in the slice
alpar@9	3040 is called the {\it slice dimension}.
alpar@9	3041
alpar@9	3042 The effect of using slices is the following. If a $m$-dimensional slice
alpar@9	3043 (i.e. a slice having $m$ asterisks) is specified in the data block, all
alpar@9	3044 subsequent data records must specify tuples of the dimension $m$.
alpar@9	3045 Whenever a $m$-tuple is encountered, each asterisk in the slice is
alpar@9	3046 replaced by corresponding components of the $m$-tuple that gives the
alpar@9	3047 resultant $n$-tuple, which is included in the elemental set to be
alpar@9	3048 defined. For example, if the slice $(a,,1,2,)$ is in effect, and
alpar@9	3049 2-tuple $(3,b)$ is encountered in a subsequent data record, the
alpar@9	3050 resultant 5-tuple included in the elemental set is $(a,3,1,2,b)$.
alpar@9	3051
alpar@9	3052 The slice having no asterisks itself defines a complete $n$-tuple,
alpar@9	3053 which is included in the elemental set.
alpar@9	3054
alpar@9	3055 Being once specified the slice effects until either a new slice or the
alpar@9	3056 end of data block is encountered. Note that if no slice is specified in
alpar@9	3057 the data block, one, components of which are all asterisks, is assumed.
alpar@9	3058
alpar@9	3059 \subsubsection{Simple data record}
alpar@9	3060
alpar@9	3061 The {\it simple data record} defines one $n$-tuple in a simple format
alpar@9	3062 and has the following syntactic form:
alpar@9	3063
alpar@9	3064 \medskip
alpar@9	3065
alpar@9	3066 \noindent\hfil
alpar@9	3067 $t_1$ {\tt,} $t_2$ {\tt,} \dots {\tt,} $t_n$
alpar@9	3068
alpar@9	3069 \medskip
alpar@9	3070
alpar@9	3071 \noindent where $t_1$, $t_2$, \dots, $t_n$ are components of the
alpar@9	3072 $n$-tuple. Each component can be a number or symbol. Commae between
alpar@9	3073 components are optional and may be omitted.
alpar@9	3074
alpar@9	3075 \subsubsection{Matrix data record}
alpar@9	3076
alpar@9	3077 The {\it matrix data record} defines several 2-tuples (doublets) in
alpar@9	3078 a matrix format and has the following syntactic form:
alpar@9	3079
alpar@9	3080 \newpage
alpar@9	3081
alpar@9	3082 $$\begin{array}{cccccc}
alpar@9	3083 \mbox{{\tt:}}&c_1&c_2&\dots&c_n&\mbox{{\tt:=}}\\
alpar@9	3084 r_1&a_{11}&a_{12}&\dots&a_{1n}&\\
alpar@9	3085 r_2&a_{21}&a_{22}&\dots&a_{2n}&\\
alpar@9	3086 \multicolumn{5}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}&\\
alpar@9	3087 r_m&a_{m1}&a_{m2}&\dots&a_{mn}&\\
alpar@9	3088 \end{array}$$
alpar@9	3089 where $r_1$, $r_2$, \dots, $r_m$ are numbers and/or symbols
alpar@9	3090 corresponding to rows of the matrix; $c_1$, $c_2$, \dots, $c_n$ are
alpar@9	3091 numbers and/or symbols corresponding to columns of the matrix, $a_{11}$,
alpar@9	3092 $a_{12}$, \dots, $a_{mn}$ are matrix elements, which can be either
alpar@9	3093 {\tt+} or {\tt-}. (In this data record the delimiter {\tt:} preceding
alpar@9	3094 the column list and the delimiter {\tt:=} following the column list
alpar@9	3095 cannot be omitted.)
alpar@9	3096
alpar@9	3097 Each element $a_{ij}$ of the matrix data block (where $1\leq i\leq m$,
alpar@9	3098 $1\leq j\leq n$) corresponds to 2-tuple $(r_i,c_j)$. If $a_{ij}$ is the
alpar@9	3099 plus sign ({\tt+}), that 2-tuple (or a longer $n$-tuple, if a slice is
alpar@9	3100 used) is included in the elemental set. Otherwise, if $a_{ij}$ is the
alpar@9	3101 minus sign ({\tt-}), that 2-tuple is not included in the elemental set.
alpar@9	3102
alpar@9	3103 Since the matrix data record defines 2-tuples, either the elemental set
alpar@9	3104 must consist of 2-tuples or the slice currently used must be
alpar@9	3105 2-dimensional.
alpar@9	3106
alpar@9	3107 \subsubsection{Transposed matrix data record}
alpar@9	3108
alpar@9	3109 The {\it transposed matrix data record} has the following syntactic
alpar@9	3110 form:
alpar@9	3111 $$\begin{array}{cccccc}
alpar@9	3112 \mbox{{\tt(tr) :}}&c_1&c_2&\dots&c_n&\mbox{{\tt:=}}\\
alpar@9	3113 r_1&a_{11}&a_{12}&\dots&a_{1n}&\\
alpar@9	3114 r_2&a_{21}&a_{22}&\dots&a_{2n}&\\
alpar@9	3115 \multicolumn{5}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}&\\
alpar@9	3116 r_m&a_{m1}&a_{m2}&\dots&a_{mn}&\\
alpar@9	3117 \end{array}$$
alpar@9	3118 (In this case the delimiter {\tt:} following the keyword {\tt(tr)} is
alpar@9	3119 optional and may be omitted.)
alpar@9	3120
alpar@9	3121 This data record is completely analogous to the matrix data record (see
alpar@9	3122 above) with only exception that in this case each element $a_{ij}$ of
alpar@9	3123 the matrix corresponds to 2-tuple $(c_j,r_i)$ rather than $(r_i,c_j)$.
alpar@9	3124
alpar@9	3125 Being once specified the {\tt(tr)} indicator affects all subsequent
alpar@9	3126 data records until either a slice or the end of data block is
alpar@9	3127 encountered.
alpar@9	3128
alpar@9	3129 \subsection{Parameter data block}
alpar@9	3130
alpar@9	3131 \medskip
alpar@9	3132
alpar@9	3133 \framebox[345pt][l]{
alpar@9	3134 \parbox[c][80pt]{345pt}{
alpar@9	3135 \hspace{6pt} {\tt param} {\it name} {\tt,} {\it record} {\tt,} \dots
alpar@9	3136 {\tt,} {\it record} {\tt;}
alpar@9	3137
alpar@9	3138 \medskip
alpar@9	3139
alpar@9	3140 \hspace{6pt} {\tt param} {\it name} {\tt default} {\it value} {\tt,}
alpar@9	3141 {\it record} {\tt,} \dots {\tt,} {\it record} {\tt;}
alpar@9	3142
alpar@9	3143 \medskip
alpar@9	3144
alpar@9	3145 \hspace{6pt} {\tt param} {\tt:} {\it tabbing-data} {\tt;}
alpar@9	3146
alpar@9	3147 \medskip
alpar@9	3148
alpar@9	3149 \hspace{6pt} {\tt param} {\tt default} {\it value} {\tt:}
alpar@9	3150 {\it tabbing-data} {\tt;}
alpar@9	3151 }}
alpar@9	3152
alpar@9	3153 \newpage
alpar@9	3154
alpar@9	3155 \setlength{\leftmargini}{60pt}
alpar@9	3156
alpar@9	3157 \begin{description}
alpar@9	3158 \item[{\rm Where:}\hspace*{23pt}] {\it name} is a symbolic name of the
alpar@9	3159 parameter;
alpar@9	3160 \item[\hspace*{54pt}] {\it value} is an optional default value of the
alpar@9	3161 parameter;
alpar@9	3162 \item[\hspace*{54pt}] {\it record}, \dots, {\it record} are data
alpar@9	3163 records;
alpar@9	3164 \item[\hspace*{54pt}] {\it tabbing-data} specifies parameter data in
alpar@9	3165 the tabbing format.
alpar@9	3166 \end{description}
alpar@9	3167
alpar@9	3168 \begin{description}
alpar@9	3169 \item[{\rm Note:}\hspace*{31pt}] Commae preceding data records may be
alpar@9	3170 omitted.
alpar@9	3171 \end{description}
alpar@9	3172
alpar@9	3173 \noindent Data records:
alpar@9	3174
alpar@9	3175 \begin{description}
alpar@9	3176 \item[{\tt :=}\hspace*{45pt}] is a non-significant data record, which
alpar@9	3177 may be used freely to improve readability;
alpar@9	3178 \item[{\tt[} {\it slice} {\tt]}\hspace*{18.5pt}] specifies a slice;
alpar@9	3179 \item[{\it plain-data}\hspace*{11pt}] specifies parameter data in the
alpar@9	3180 plain format;
alpar@9	3181 \item[{\tt:} {\it tabular-data}]\hspace*{0pt}\\
alpar@9	3182 specifies parameter data in the tabular format;
alpar@9	3183 \item[{\tt(tr)} {\tt:} {\it tabular-data}]\hspace*{0pt}\\
alpar@9	3184 specifies set data in the transposed tabular format. (In this case the
alpar@9	3185 colon following the keyword {\tt(tr)} may be omitted.)
alpar@9	3186 \end{description}
alpar@9	3187
alpar@9	3188 \noindent{\bf Examples}
alpar@9	3189
alpar@9	3190 \begin{verbatim}
alpar@9	3191 param T := 4;
alpar@9	3192 param month := 1 'Jan' 2 'Feb' 3 'Mar' 4 'Apr' 5 'May';
alpar@9	3193 param month := [1] Jan, [2] Feb, [3] Mar, [4] Apr, [5] May;
alpar@9	3194 param day := [Sun] 0, [Mon] 1, [Tue] 2, [Wed] 3, [Thu] 4,
alpar@9	3195 [Fri] 5, [Sat] 6;
alpar@9	3196 param init_stock := iron 7.32 nickel 35.8;
alpar@9	3197 param init_stock [*] iron 7.32, nickel 35.8;
alpar@9	3198 param cost [iron] .025 [nickel] .03;
alpar@9	3199 param value := iron -.1, nickel .02;
alpar@9	3200 param : init_stock cost value :=
alpar@9	3201 iron 7.32 .025 -.1
alpar@9	3202 nickel 35.8 .03 .02 ;
alpar@9	3203 param : raw : init_stock cost value :=
alpar@9	3204 iron 7.32 .025 -.1
alpar@9	3205 nickel 35.8 .03 .02 ;
alpar@9	3206 param demand default 0 (tr)
alpar@9	3207 : FRA DET LAN WIN STL FRE LAF :=
alpar@9	3208 bands 300 . 100 75 . 225 250
alpar@9	3209 coils 500 750 400 250 . 850 500
alpar@9	3210 plate 100 . . 50 200 . 250 ;
alpar@9	3211 \end{verbatim}
alpar@9	3212
alpar@9	3213 \newpage
alpar@9	3214
alpar@9	3215 \begin{verbatim}
alpar@9	3216 param trans_cost :=
alpar@9	3217 [,,bands]: FRA DET LAN WIN STL FRE LAF :=
alpar@9	3218 GARY 30 10 8 10 11 71 6
alpar@9	3219 CLEV 22 7 10 7 21 82 13
alpar@9	3220 PITT 19 11 12 10 25 83 15
alpar@9	3221 [,,coils]: FRA DET LAN WIN STL FRE LAF :=
alpar@9	3222 GARY 39 14 11 14 16 82 8
alpar@9	3223 CLEV 27 9 12 9 26 95 17
alpar@9	3224 PITT 24 14 17 13 28 99 20
alpar@9	3225 [,,plate]: FRA DET LAN WIN STL FRE LAF :=
alpar@9	3226 GARY 41 15 12 16 17 86 8
alpar@9	3227 CLEV 29 9 13 9 28 99 18
alpar@9	3228 PITT 26 14 17 13 31 104 20 ;
alpar@9	3229 \end{verbatim}
alpar@9	3230
alpar@9	3231 The {\it parameter data block} is used to specify complete data for a
alpar@9	3232 parameter (or parameters, if data are specified in the tabbing format).
alpar@9	3233
alpar@9	3234 Data blocks can be specified only for non-computable parameters, i.e.
alpar@9	3235 for parameters, which have no assign ({\tt:=}) attribute in the
alpar@9	3236 corresponding parameter statements.
alpar@9	3237
alpar@9	3238 Data defined in the parameter data block are coded as a sequence of
alpar@9	3239 data records described below. Additionally the data block can be
alpar@9	3240 provided with the optional {\tt default} attribute, which specifies a
alpar@9	3241 default numeric or symbolic value of the parameter (parameters). This
alpar@9	3242 default value is assigned to the parameter or its members, if
alpar@9	3243 no appropriate value is defined in the parameter data block. The
alpar@9	3244 {\tt default} attribute cannot be used, if it is already specified in
alpar@9	3245 the corresponding parameter statement.
alpar@9	3246
alpar@9	3247 \subsubsection{Assign data record}
alpar@9	3248
alpar@9	3249 The {\it assign} ({\tt:=}) {\it data record} is a non-signficant
alpar@9	3250 element. It may be used for improving readability of data blocks.
alpar@9	3251
alpar@9	3252 \subsubsection{Slice data record}
alpar@9	3253
alpar@9	3254 The {\it slice data record} is a control record, which specifies a
alpar@9	3255 {\it slice} of the parameter array. It has the following syntactic form:
alpar@9	3256
alpar@9	3257 \medskip
alpar@9	3258
alpar@9	3259 \noindent\hfil
alpar@9	3260 {\tt[} $s_1$ {\tt,} $s_2$ {\tt,} \dots {\tt,} $s_n$ {\tt]}
alpar@9	3261
alpar@9	3262 \medskip
alpar@9	3263
alpar@9	3264 \noindent where $s_1$, $s_2$, \dots, $s_n$ are components of the slice.
alpar@9	3265
alpar@9	3266 Each component of the slice can be a number or symbol or the asterisk
alpar@9	3267 ({\tt*}). The number of components in the slice must be the same as the
alpar@9	3268 dimension of the parameter. For instance, if the parameter is a
alpar@9	3269 4-dimensional array, the slice must have four components. The number of
alpar@9	3270 asterisks in the slice is called the {\it slice dimension}.
alpar@9	3271
alpar@9	3272 The effect of using slices is the following. If a $m$-dimensional slice
alpar@9	3273 (i.e. a slice having $m$ asterisks) is specified in the data block, all
alpar@9	3274 subsequent data records must specify subscripts of the parameter
alpar@9	3275 members as if the parameter were $m$-dimensional, not $n$-dimensional.
alpar@9	3276
alpar@9	3277 Whenever $m$ subscripts are encountered, each asterisk in the slice is
alpar@9	3278 replaced by corresponding subscript that gives $n$ subscripts, which
alpar@9	3279 define the actual parameter member. For example, if the slice
alpar@9	3280 $[a,,1,2,]$ is in effect, and subscripts 3 and $b$ are encountered in
alpar@9	3281 a subsequent data record, the complete subscript list used to choose a
alpar@9	3282 parameter member is $[a,3,1,2,b]$.
alpar@9	3283
alpar@9	3284 It is allowed to specify a slice having no asterisks. Such slice itself
alpar@9	3285 defines a complete subscript list, in which case the next data record
alpar@9	3286 should define only a single value of corresponding parameter member.
alpar@9	3287
alpar@9	3288 Being once specified the slice effects until either a new slice or the
alpar@9	3289 end of data block is encountered. Note that if no slice is specified in
alpar@9	3290 the data block, one, components of which are all asterisks, is assumed.
alpar@9	3291
alpar@9	3292 \subsubsection{Plain data record}
alpar@9	3293
alpar@9	3294 The {\it plain data record} defines a subscript list and a single value
alpar@9	3295 in the plain format. This record has the following syntactic form:
alpar@9	3296
alpar@9	3297 \medskip
alpar@9	3298
alpar@9	3299 \noindent\hfil
alpar@9	3300 $t_1$ {\tt,} $t_2$ {\tt,} \dots {\tt,} $t_n$ {\tt,} $v$
alpar@9	3301
alpar@9	3302 \medskip
alpar@9	3303
alpar@9	3304 \noindent where $t_1$, $t_2$, \dots, $t_n$ are subscripts, and $v$ is a
alpar@9	3305 value. Each subscript as well as the value can be a number or symbol.
alpar@9	3306 Commae following subscripts are optional and may be omitted.
alpar@9	3307
alpar@9	3308 In case of 0-dimensional parameter or slice the plain data record has
alpar@9	3309 no subscripts and consists of a single value only.
alpar@9	3310
alpar@9	3311 \subsubsection{Tabular data record}
alpar@9	3312
alpar@9	3313 The {\it tabular data record} defines several values, where each value
alpar@9	3314 is provided with two subscripts. This record has the following
alpar@9	3315 syntactic form:
alpar@9	3316 $$\begin{array}{cccccc}
alpar@9	3317 \mbox{{\tt:}}&c_1&c_2&\dots&c_n&\mbox{{\tt:=}}\\
alpar@9	3318 r_1&a_{11}&a_{12}&\dots&a_{1n}&\\
alpar@9	3319 r_2&a_{21}&a_{22}&\dots&a_{2n}&\\
alpar@9	3320 \multicolumn{5}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}&\\
alpar@9	3321 r_m&a_{m1}&a_{m2}&\dots&a_{mn}&\\
alpar@9	3322 \end{array}$$
alpar@9	3323 where $r_1$, $r_2$, \dots, $r_m$ are numbers and/or symbols
alpar@9	3324 corresponding to rows of the table; $c_1$, $c_2$, \dots, $c_n$ are
alpar@9	3325 numbers and/or symbols corresponding to columns of the table, $a_{11}$,
alpar@9	3326 $a_{12}$, \dots, $a_{mn}$ are table elements. Each element can be a
alpar@9	3327 number or symbol or the single decimal point ({\tt.}). (In this data
alpar@9	3328 record the delimiter {\tt:} preceding the column list and the delimiter
alpar@9	3329 {\tt:=} following the column list cannot be omitted.)
alpar@9	3330
alpar@9	3331 Each element $a_{ij}$ of the tabular data block ($1\leq i\leq m$,
alpar@9	3332 $1\leq j\leq n$) defines two subscripts, where the first subscript is
alpar@9	3333 $r_i$, and the second one is $c_j$. These subscripts are used in
alpar@9	3334 conjunction with the current slice to form the complete subscript list
alpar@9	3335 that identifies a particular member of the parameter array. If $a_{ij}$
alpar@9	3336 is a number or symbol, this value is assigned to the parameter member.
alpar@9	3337 However, if $a_{ij}$ is the single decimal point, the member is
alpar@9	3338 assigned a default value specified either in the parameter data block
alpar@9	3339 or in the parameter statement, or, if no default value is specified,
alpar@9	3340 the member remains undefined.
alpar@9	3341
alpar@9	3342 Since the tabular data record provides two subscripts for each value,
alpar@9	3343 either the parameter or the slice currently used must be 2-dimensional.
alpar@9	3344
alpar@9	3345 \subsubsection{Transposed tabular data record}
alpar@9	3346
alpar@9	3347 The {\it transposed tabular data record} has the following syntactic
alpar@9	3348 form:
alpar@9	3349 $$\begin{array}{cccccc}
alpar@9	3350 \mbox{{\tt(tr) :}}&c_1&c_2&\dots&c_n&\mbox{{\tt:=}}\\
alpar@9	3351 r_1&a_{11}&a_{12}&\dots&a_{1n}&\\
alpar@9	3352 r_2&a_{21}&a_{22}&\dots&a_{2n}&\\
alpar@9	3353 \multicolumn{5}{c}{.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}&\\
alpar@9	3354 r_m&a_{m1}&a_{m2}&\dots&a_{mn}&\\
alpar@9	3355 \end{array}$$
alpar@9	3356 (In this case the delimiter {\tt:} following the keyword {\tt(tr)} is
alpar@9	3357 optional and may be omitted.)
alpar@9	3358
alpar@9	3359 This data record is completely analogous to the tabular data record
alpar@9	3360 (see above) with only exception that the first subscript defined by
alpar@9	3361 element $a_{ij}$ is $c_j$ while the second one is $r_i$.
alpar@9	3362
alpar@9	3363 Being once specified the {\tt(tr)} indicator affects all subsequent
alpar@9	3364 data records until either a slice or the end of data block is
alpar@9	3365 encountered.
alpar@9	3366
alpar@9	3367 \subsubsection{Tabbing data format}
alpar@9	3368
alpar@9	3369 The parameter data block in the {\it tabbing format} has the following
alpar@9	3370 syntactic form:
alpar@9	3371 $$\begin{array}{p{12pt}@{\ }l@{\ }c@{\ }l@{\ }c@{\ }l@{\ }r@{\ }l@{\ }c
alpar@9	3372 @{\ }l@{\ }c@{\ }l@{\ }l}
alpar@9	3373 \multicolumn{7}{@{}c@{}}{\mbox{\tt param}\ \mbox{\tt default}\ \mbox
alpar@9	3374 {\it value}\ \mbox{\tt:}\ \mbox{\it s}\ \mbox{\tt:}}&
alpar@9	3375 p_1&\mbox{\tt,}&p_2&\mbox{\tt,} \dots \mbox{\tt,}&p_k&\mbox{\tt:=}\\
alpar@9	3376 &t_{11}&\mbox{\tt,}&t_{12}&\mbox{\tt,} \dots \mbox{\tt,}&t_{1n}&
alpar@9	3377 \mbox{\tt,}&a_{11}&\mbox{\tt,}&a_{12}&\mbox{\tt,} \dots \mbox{\tt,}&
alpar@9	3378 a_{1k}\\
alpar@9	3379 &t_{21}&\mbox{\tt,}&t_{22}&\mbox{\tt,} \dots \mbox{\tt,}&t_{2n}&
alpar@9	3380 \mbox{\tt,}&a_{21}&\mbox{\tt,}&a_{22}&\mbox{\tt,} \dots \mbox{\tt,}&
alpar@9	3381 a_{2k}\\
alpar@9	3382 \multicolumn{13}{c}
alpar@9	3383 {.\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .\ \ .}\\
alpar@9	3384 &t_{m1}&\mbox{\tt,}&t_{m2}&\mbox{\tt,} \dots \mbox{\tt,}&t_{mn}&
alpar@9	3385 \mbox{\tt,}&a_{m1}&\mbox{\tt,}&a_{m2}&\mbox{\tt,} \dots \mbox{\tt,}&
alpar@9	3386 a_{mk}&\mbox{\tt;}\\
alpar@9	3387 \end{array}$$
alpar@9	3388
alpar@9	3389 {\it Notes:}
alpar@9	3390
alpar@9	3391 1. The keyword {\tt default} may be omitted along with a value
alpar@9	3392 following it.
alpar@9	3393
alpar@9	3394 2. Symbolic name {\tt s} may be omitted along with the colon following
alpar@9	3395 it.
alpar@9	3396
alpar@9	3397 3. All comae are optional and may be omitted.
alpar@9	3398
alpar@9	3399 \medskip
alpar@9	3400
alpar@9	3401 The data block in the tabbing format shown above is exactly equivalent
alpar@9	3402 to the following data blocks for $j=1,2,\dots,k$:
alpar@9	3403
alpar@9	3404 \medskip
alpar@9	3405
alpar@9	3406 {\tt set} {\it s} {\tt:=}
alpar@9	3407 {\tt(}$t_{11}${\tt,}$t_{12}${\tt,}\dots{\tt,}$t_{1n}${\tt)}
alpar@9	3408 {\tt(}$t_{21}${\tt,}$t_{22}${\tt,}\dots{\tt,}$t_{2n}${\tt)} \dots
alpar@9	3409 {\tt(}$t_{m1}${\tt,}$t_{m2}${\tt,}\dots{\tt,}$t_{mn}${\tt)} {\tt;}
alpar@9	3410
alpar@9	3411 {\tt param} $p_j$ {\tt default} {\it value} {\tt:=}
alpar@9	3412
alpar@9	3413 $\!${\tt[}$t_{11}${\tt,}$t_{12}${\tt,}\dots{\tt,}$t_{1n}${\tt]}
alpar@9	3414 $a_{1j}$
alpar@9	3415 {\tt[}$t_{21}${\tt,}$t_{22}${\tt,}\dots{\tt,}$t_{2n}${\tt]} $a_{2j}$
alpar@9	3416 \dots
alpar@9	3417 {\tt[}$t_{m1}${\tt,}$t_{m2}${\tt,}\dots{\tt,}$t_{mn}${\tt]} $a_{mj}$
alpar@9	3418 {\tt;}
alpar@9	3419
alpar@9	3420 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	3421
alpar@9	3422 \appendix
alpar@9	3423
alpar@9	3424 \newpage
alpar@9	3425
alpar@9	3426 \section{Using suffixes}
alpar@9	3427
alpar@9	3428 Suffixes can be used to retrieve additional values associated with
alpar@9	3429 model variables, constraints, and objectives.
alpar@9	3430
alpar@9	3431 A {\it suffix} consists of a period ({\tt.}) followed by a non-reserved
alpar@9	3432 keyword. For example, if {\tt x} is a two-dimensional variable,
alpar@9	3433 {\tt x[i,j].lb} is a numeric value equal to the lower bound of
alpar@9	3434 elemental variable {\tt x[i,j]}, which (value) can be used everywhere
alpar@9	3435 in expressions like a numeric parameter.
alpar@9	3436
alpar@9	3437 For model variables suffixes have the following meaning:
alpar@9	3438
alpar@9	3439 \medskip
alpar@9	3440
alpar@9	3441 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	3442 {\tt.lb}&lower bound\\
alpar@9	3443 {\tt.ub}&upper bound\\
alpar@9	3444 {\tt.status}&status in the solution:\\
alpar@9	3445 &0 --- undefined\\
alpar@9	3446 &1 --- basic\\
alpar@9	3447 &2 --- non-basic on lower bound\\
alpar@9	3448 &3 --- non-basic on upper bound\\
alpar@9	3449 &4 --- non-basic free (unbounded) variable\\
alpar@9	3450 &5 --- non-basic fixed variable\\
alpar@9	3451 {\tt.val}&primal value in the solution\\
alpar@9	3452 {\tt.dual}&dual value (reduced cost) in the solution\\
alpar@9	3453 \end{tabular}
alpar@9	3454
alpar@9	3455 \medskip
alpar@9	3456
alpar@9	3457 For model constraints and objectives suffixes have the following
alpar@9	3458 meaning:
alpar@9	3459
alpar@9	3460 \medskip
alpar@9	3461
alpar@9	3462 \begin{tabular}{@{}p{96pt}p{222pt}@{}}
alpar@9	3463 {\tt.lb}&lower bound of the linear form\\
alpar@9	3464 {\tt.ub}&upper bound of the linear form\\
alpar@9	3465 {\tt.status}&status in the solution:\\
alpar@9	3466 &0 --- undefined\\
alpar@9	3467 &1 --- non-active\\
alpar@9	3468 &2 --- active on lower bound\\
alpar@9	3469 &3 --- active on upper bound\\
alpar@9	3470 &4 --- active free (unbounded) row\\
alpar@9	3471 &5 --- active equality constraint\\
alpar@9	3472 {\tt.val}&primal value of the linear form in the solution\\
alpar@9	3473 {\tt.dual}&dual value (reduced cost) of the linear form in the
alpar@9	3474 solution\\
alpar@9	3475 \end{tabular}
alpar@9	3476
alpar@9	3477 \medskip
alpar@9	3478
alpar@9	3479 Note that suffixes {\tt.status}, {\tt.val}, and {\tt.dual} can be used
alpar@9	3480 only below the solve statement.
alpar@9	3481
alpar@9	3482 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	3483
alpar@9	3484 \newpage
alpar@9	3485
alpar@9	3486 \section{Date and time functions}
alpar@9	3487
alpar@9	3488 \noindent\hfil
alpar@9	3489 by Andrew Makhorin \verb\|<mao@gnu.org>\|
alpar@9	3490
alpar@9	3491 \noindent\hfil
alpar@9	3492 and Heinrich Schuchardt \verb\|<heinrich.schuchardt@gmx.de>\|
alpar@9	3493
alpar@9	3494 \subsection{Obtaining current calendar time}
alpar@9	3495 \label{gmtime}
alpar@9	3496
alpar@9	3497 To obtain the current calendar time in MathProg there exists the
alpar@9	3498 function {\tt gmtime}. It has no arguments and returns the number of
alpar@9	3499 seconds elapsed since 00:00:00 on January 1, 1970, Coordinated
alpar@9	3500 Universal Time (UTC). For example:
alpar@9	3501
alpar@9	3502 \medskip
alpar@9	3503
alpar@9	3504 \verb\| param utc := gmtime();\|
alpar@9	3505
alpar@9	3506 \medskip
alpar@9	3507
alpar@9	3508 MathProg has no function to convert UTC time returned by the function
alpar@9	3509 {\tt gmtime} to {\it local} calendar times. Thus, if you need to
alpar@9	3510 determine the current local calendar time, you have to add to the UTC
alpar@9	3511 time returned the time offset from UTC expressed in seconds. For
alpar@9	3512 example, the time in Berlin during the winter is one hour ahead of UTC
alpar@9	3513 that corresponds to the time offset +1 hour = +3600 secs, so the
alpar@9	3514 current winter calendar time in Berlin may be determined as follows:
alpar@9	3515
alpar@9	3516 \medskip
alpar@9	3517
alpar@9	3518 \verb\| param now := gmtime() + 3600;\|
alpar@9	3519
alpar@9	3520 \medskip
alpar@9	3521
alpar@9	3522 \noindent Similarly, the summer time in Chicago (Central Daylight Time)
alpar@9	3523 is five hours behind UTC, so the corresponding current local calendar
alpar@9	3524 time may be determined as follows:
alpar@9	3525
alpar@9	3526 \medskip
alpar@9	3527
alpar@9	3528 \verb\| param now := gmtime() - 5 * 3600;\|
alpar@9	3529
alpar@9	3530 \medskip
alpar@9	3531
alpar@9	3532 Note that the value returned by {\tt gmtime} is volatile, i.e. being
alpar@9	3533 called several times this function may return different values.
alpar@9	3534
alpar@9	3535 \subsection{Converting character string to calendar time}
alpar@9	3536 \label{str2time}
alpar@9	3537
alpar@9	3538 The function {\tt str2time(}{\it s}{\tt,} {\it f}{\tt)} converts a
alpar@9	3539 character string (timestamp) specified by its first argument {\it s},
alpar@9	3540 which must be a symbolic expression, to the calendar time suitable for
alpar@9	3541 arithmetic calculations. The conversion is controlled by the specified
alpar@9	3542 format string {\it f} (the second argument), which also must be a
alpar@9	3543 symbolic expression.
alpar@9	3544
alpar@9	3545 The result of conversion returned by {\tt str2time} has the same
alpar@9	3546 meaning as values returned by the function {\tt gmtime} (see Subsection
alpar@9	3547 \ref{gmtime}, page \pageref{gmtime}). Note that {\tt str2time} does
alpar@9	3548 {\tt not} correct the calendar time returned for the local timezone,
alpar@9	3549 i.e. being applied to 00:00:00 on January 1, 1970 it always returns 0.
alpar@9	3550
alpar@9	3551 For example, the model statements:
alpar@9	3552
alpar@9	3553 \medskip
alpar@9	3554
alpar@9	3555 \verb\| param s, symbolic, := "07/14/98 13:47";\|
alpar@9	3556
alpar@9	3557 \verb\| param t := str2time(s, "%m/%d/%y %H:%M");\|
alpar@9	3558
alpar@9	3559 \verb\| display t;\|
alpar@9	3560
alpar@9	3561 \medskip
alpar@9	3562
alpar@9	3563 \noindent produce the following printout:
alpar@9	3564
alpar@9	3565 \medskip
alpar@9	3566
alpar@9	3567 \verb\| t = 900424020\|
alpar@9	3568
alpar@9	3569 \medskip
alpar@9	3570
alpar@9	3571 \noindent where the calendar time printed corresponds to 13:47:00 on
alpar@9	3572 July 14, 1998.
alpar@9	3573
alpar@9	3574 \newpage
alpar@9	3575
alpar@9	3576 The format string passed to the function {\tt str2time} consists of
alpar@9	3577 conversion specifiers and ordinary characters. Each conversion
alpar@9	3578 specifier begins with a percent ({\tt\%}) character followed by a
alpar@9	3579 letter.
alpar@9	3580
alpar@9	3581 The following conversion specifiers may be used in the format string:
alpar@9	3582
alpar@9	3583 \medskip
alpar@9	3584
alpar@9	3585 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3586 {\tt\%b}&The abbreviated month name (case insensitive). At least three
alpar@9	3587 first letters of the month name must appear in the input string.\\
alpar@9	3588 \end{tabular}
alpar@9	3589
alpar@9	3590 \medskip
alpar@9	3591
alpar@9	3592 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3593 {\tt\%d}&The day of the month as a decimal number (range 1 to 31).
alpar@9	3594 Leading zero is permitted, but not required.\\
alpar@9	3595 \end{tabular}
alpar@9	3596
alpar@9	3597 \medskip
alpar@9	3598
alpar@9	3599 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3600 {\tt\%h}&The same as {\tt\%b}.\\
alpar@9	3601 \end{tabular}
alpar@9	3602
alpar@9	3603 \medskip
alpar@9	3604
alpar@9	3605 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3606 {\tt\%H}&The hour as a decimal number, using a 24-hour clock (range 0
alpar@9	3607 to 23). Leading zero is permitted, but not required.\\
alpar@9	3608 \end{tabular}
alpar@9	3609
alpar@9	3610 \medskip
alpar@9	3611
alpar@9	3612 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3613 {\tt\%m}&The month as a decimal number (range 1 to 12). Leading zero is
alpar@9	3614 permitted, but not required.\\
alpar@9	3615 \end{tabular}
alpar@9	3616
alpar@9	3617 \medskip
alpar@9	3618
alpar@9	3619 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3620 {\tt\%M}&The minute as a decimal number (range 0 to 59). Leading zero
alpar@9	3621 is permitted, but not required.\\
alpar@9	3622 \end{tabular}
alpar@9	3623
alpar@9	3624 \medskip
alpar@9	3625
alpar@9	3626 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3627 {\tt\%S}&The second as a decimal number (range 0 to 60). Leading zero
alpar@9	3628 is permitted, but not required.\\
alpar@9	3629 \end{tabular}
alpar@9	3630
alpar@9	3631 \medskip
alpar@9	3632
alpar@9	3633 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3634 {\tt\%y}&The year without a century as a decimal number (range 0 to 99).
alpar@9	3635 Leading zero is permitted, but not required. Input values in the range
alpar@9	3636 0 to 68 are considered as the years 2000 to 2068 while the values 69 to
alpar@9	3637 99 as the years 1969 to 1999.\\
alpar@9	3638 \end{tabular}
alpar@9	3639
alpar@9	3640 \medskip
alpar@9	3641
alpar@9	3642 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3643 {\tt\%z}&The offset from GMT in ISO 8601 format.\\
alpar@9	3644 \end{tabular}
alpar@9	3645
alpar@9	3646 \medskip
alpar@9	3647
alpar@9	3648 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3649 {\tt\%\%}&A literal {\tt\%} character.\\
alpar@9	3650 \end{tabular}
alpar@9	3651
alpar@9	3652 \medskip
alpar@9	3653
alpar@9	3654 All other (ordinary) characters in the format string must have a
alpar@9	3655 matching character in the input string to be converted. Exceptions are
alpar@9	3656 spaces in the input string which can match zero or more space
alpar@9	3657 characters in the format string.
alpar@9	3658
alpar@9	3659 If some date and/or time component(s) are missing in the format and,
alpar@9	3660 therefore, in the input string, the function {\tt str2time} uses their
alpar@9	3661 default values corresponding to 00:00:00 on January 1, 1970, that is,
alpar@9	3662 the default value of the year is 1970, the default value of the month
alpar@9	3663 is January, etc.
alpar@9	3664
alpar@9	3665 The function {\tt str2time} is applicable to all calendar times in the
alpar@9	3666 range 00:00:00 on January 1, 0001 to 23:59:59 on December 31, 4000 of
alpar@9	3667 the Gregorian calendar.
alpar@9	3668
alpar@9	3669 \subsection{Converting calendar time to character string}
alpar@9	3670 \label{time2str}
alpar@9	3671
alpar@9	3672 The function {\tt time2str(}{\it t}{\tt,} {\it f}{\tt)} converts the
alpar@9	3673 calendar time specified by its first argument {\it t}, which must be a
alpar@9	3674 numeric expression, to a character string (symbolic value). The
alpar@9	3675 conversion is controlled by the specified format string {\it f} (the
alpar@9	3676 second argument), which must be a symbolic expression.
alpar@9	3677
alpar@9	3678 The calendar time passed to {\tt time2str} has the same meaning as
alpar@9	3679 values returned by the function {\tt gmtime} (see Subsection
alpar@9	3680 \ref{gmtime}, page \pageref{gmtime}). Note that {\tt time2str} does
alpar@9	3681 {\it not} correct the specified calendar time for the local timezone,
alpar@9	3682 i.e. the calendar time 0 always corresponds to 00:00:00 on January 1,
alpar@9	3683 1970.
alpar@9	3684
alpar@9	3685 For example, the model statements:
alpar@9	3686
alpar@9	3687 \medskip
alpar@9	3688
alpar@9	3689 \verb\| param s, symbolic, := time2str(gmtime(), "%FT%TZ");\|
alpar@9	3690
alpar@9	3691 \verb\| display s;\|
alpar@9	3692
alpar@9	3693 \medskip
alpar@9	3694
alpar@9	3695 \noindent may produce the following printout:
alpar@9	3696
alpar@9	3697 \medskip
alpar@9	3698
alpar@9	3699 \verb\| s = '2008-12-04T00:23:45Z'\|
alpar@9	3700
alpar@9	3701 \medskip
alpar@9	3702
alpar@9	3703 \noindent which is a timestamp in the ISO format.
alpar@9	3704
alpar@9	3705 The format string passed to the function {\tt time2str} consists of
alpar@9	3706 conversion specifiers and ordinary characters. Each conversion
alpar@9	3707 specifier begins with a percent ({\tt\%}) character followed by a
alpar@9	3708 letter.
alpar@9	3709
alpar@9	3710 The following conversion specifiers may be used in the format string:
alpar@9	3711
alpar@9	3712 \medskip
alpar@9	3713
alpar@9	3714 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3715 {\tt\%a}&The abbreviated (2-character) weekday name.\\
alpar@9	3716 \end{tabular}
alpar@9	3717
alpar@9	3718 \medskip
alpar@9	3719
alpar@9	3720 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3721 {\tt\%A}&The full weekday name.\\
alpar@9	3722 \end{tabular}
alpar@9	3723
alpar@9	3724 \medskip
alpar@9	3725
alpar@9	3726 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3727 {\tt\%b}&The abbreviated (3-character) month name.\\
alpar@9	3728 \end{tabular}
alpar@9	3729
alpar@9	3730 \medskip
alpar@9	3731
alpar@9	3732 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3733 {\tt\%B}&The full month name.\\
alpar@9	3734 \end{tabular}
alpar@9	3735
alpar@9	3736 \medskip
alpar@9	3737
alpar@9	3738 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3739 {\tt\%C}&The century of the year, that is the greatest integer not
alpar@9	3740 greater than the year divided by 100.\\
alpar@9	3741 \end{tabular}
alpar@9	3742
alpar@9	3743 \medskip
alpar@9	3744
alpar@9	3745 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3746 {\tt\%d}&The day of the month as a decimal number (range 01 to 31).\\
alpar@9	3747 \end{tabular}
alpar@9	3748
alpar@9	3749 \medskip
alpar@9	3750
alpar@9	3751 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3752 {\tt\%D}&The date using the format \verb\|%m/%d/%y\|.\\
alpar@9	3753 \end{tabular}
alpar@9	3754
alpar@9	3755 \medskip
alpar@9	3756
alpar@9	3757 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3758 {\tt\%e}&The day of the month like with \verb\|%d\|, but padded with
alpar@9	3759 blank rather than zero.\\
alpar@9	3760 \end{tabular}
alpar@9	3761
alpar@9	3762 \medskip
alpar@9	3763
alpar@9	3764 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3765 {\tt\%F}&The date using the format \verb\|%Y-%m-%d\|.\\
alpar@9	3766 \end{tabular}
alpar@9	3767
alpar@9	3768 \medskip
alpar@9	3769
alpar@9	3770 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3771 {\tt\%g}&The year corresponding to the ISO week number, but without the
alpar@9	3772 century (range 00 to 99). This has the same format and value as
alpar@9	3773 \verb\|%y\|, except that if the ISO week number (see \verb\|%V\|) belongs
alpar@9	3774 to the previous or next year, that year is used instead.\\
alpar@9	3775 \end{tabular}
alpar@9	3776
alpar@9	3777 \medskip
alpar@9	3778
alpar@9	3779 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3780 {\tt\%G}&The year corresponding to the ISO week number. This has the
alpar@9	3781 same format and value as \verb\|%Y\|, except that if the ISO week number
alpar@9	3782 (see \verb\|%V\|) belongs to the previous or next year, that year is used
alpar@9	3783 instead.
alpar@9	3784 \end{tabular}
alpar@9	3785
alpar@9	3786 \medskip
alpar@9	3787
alpar@9	3788 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3789 {\tt\%h}&The same as \verb\|%b\|.\\
alpar@9	3790 \end{tabular}
alpar@9	3791
alpar@9	3792 \medskip
alpar@9	3793
alpar@9	3794 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3795 {\tt\%H}&The hour as a decimal number, using a 24-hour clock (range 00
alpar@9	3796 to 23).\\
alpar@9	3797 \end{tabular}
alpar@9	3798
alpar@9	3799 \medskip
alpar@9	3800
alpar@9	3801 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3802 {\tt\%I}&The hour as a decimal number, using a 12-hour clock (range 01
alpar@9	3803 to 12).\\
alpar@9	3804 \end{tabular}
alpar@9	3805
alpar@9	3806 \medskip
alpar@9	3807
alpar@9	3808 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3809 {\tt\%j}&The day of the year as a decimal number (range 001 to 366).\\
alpar@9	3810 \end{tabular}
alpar@9	3811
alpar@9	3812 \medskip
alpar@9	3813
alpar@9	3814 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3815 {\tt\%k}&The hour as a decimal number, using a 24-hour clock like
alpar@9	3816 \verb\|%H\|, but padded with blank rather than zero.\\
alpar@9	3817 \end{tabular}
alpar@9	3818
alpar@9	3819 \medskip
alpar@9	3820
alpar@9	3821 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3822 {\tt\%l}&The hour as a decimal number, using a 12-hour clock like
alpar@9	3823 \verb\|%I\|, but padded with blank rather than zero.
alpar@9	3824 \end{tabular}
alpar@9	3825
alpar@9	3826 \medskip
alpar@9	3827
alpar@9	3828 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3829 {\tt\%m}&The month as a decimal number (range 01 to 12).\\
alpar@9	3830 \end{tabular}
alpar@9	3831
alpar@9	3832 \medskip
alpar@9	3833
alpar@9	3834 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3835 {\tt\%M}&The minute as a decimal number (range 00 to 59).\\
alpar@9	3836 \end{tabular}
alpar@9	3837
alpar@9	3838 \medskip
alpar@9	3839
alpar@9	3840 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3841 {\tt\%p}&Either {\tt AM} or {\tt PM}, according to the given time value.
alpar@9	3842 Midnight is treated as {\tt AM} and noon as {\tt PM}.\\
alpar@9	3843 \end{tabular}
alpar@9	3844
alpar@9	3845 \medskip
alpar@9	3846
alpar@9	3847 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3848 {\tt\%P}&Either {\tt am} or {\tt pm}, according to the given time value.
alpar@9	3849 Midnight is treated as {\tt am} and noon as {\tt pm}.\\
alpar@9	3850 \end{tabular}
alpar@9	3851
alpar@9	3852 \medskip
alpar@9	3853
alpar@9	3854 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3855 {\tt\%R}&The hour and minute in decimal numbers using the format
alpar@9	3856 \verb\|%H:%M\|.\\
alpar@9	3857 \end{tabular}
alpar@9	3858
alpar@9	3859 \medskip
alpar@9	3860
alpar@9	3861 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3862 {\tt\%S}&The second as a decimal number (range 00 to 59).\\
alpar@9	3863 \end{tabular}
alpar@9	3864
alpar@9	3865 \medskip
alpar@9	3866
alpar@9	3867 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3868 {\tt\%T}&The time of day in decimal numbers using the format
alpar@9	3869 \verb\|%H:%M:%S\|.\\
alpar@9	3870 \end{tabular}
alpar@9	3871
alpar@9	3872 \medskip
alpar@9	3873
alpar@9	3874 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3875 {\tt\%u}&The day of the week as a decimal number (range 1 to 7), Monday
alpar@9	3876 being 1.\\
alpar@9	3877 \end{tabular}
alpar@9	3878
alpar@9	3879 \medskip
alpar@9	3880
alpar@9	3881 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3882 {\tt\%U}&The week number of the current year as a decimal number (range
alpar@9	3883 00 to 53), starting with the first Sunday as the first day of the first
alpar@9	3884 week. Days preceding the first Sunday in the year are considered to be
alpar@9	3885 in week 00.
alpar@9	3886 \end{tabular}
alpar@9	3887
alpar@9	3888 \medskip
alpar@9	3889
alpar@9	3890 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3891 {\tt\%V}&The ISO week number as a decimal number (range 01 to 53). ISO
alpar@9	3892 weeks start with Monday and end with Sunday. Week 01 of a year is the
alpar@9	3893 first week which has the majority of its days in that year; this is
alpar@9	3894 equivalent to the week containing January 4. Week 01 of a year can
alpar@9	3895 contain days from the previous year. The week before week 01 of a year
alpar@9	3896 is the last week (52 or 53) of the previous year even if it contains
alpar@9	3897 days from the new year. In other word, if 1 January is Monday, Tuesday,
alpar@9	3898 Wednesday or Thursday, it is in week 01; if 1 January is Friday,
alpar@9	3899 Saturday or Sunday, it is in week 52 or 53 of the previous year.\\
alpar@9	3900 \end{tabular}
alpar@9	3901
alpar@9	3902 \medskip
alpar@9	3903
alpar@9	3904 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3905 {\tt\%w}&The day of the week as a decimal number (range 0 to 6), Sunday
alpar@9	3906 being 0.\\
alpar@9	3907 \end{tabular}
alpar@9	3908
alpar@9	3909 \medskip
alpar@9	3910
alpar@9	3911 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3912 {\tt\%W}&The week number of the current year as a decimal number (range
alpar@9	3913 00 to 53), starting with the first Monday as the first day of the first
alpar@9	3914 week. Days preceding the first Monday in the year are considered to be
alpar@9	3915 in week 00.\\
alpar@9	3916 \end{tabular}
alpar@9	3917
alpar@9	3918 \medskip
alpar@9	3919
alpar@9	3920 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3921 {\tt\%y}&The year without a century as a decimal number (range 00 to
alpar@9	3922 99), that is the year modulo 100.\\
alpar@9	3923 \end{tabular}
alpar@9	3924
alpar@9	3925 \medskip
alpar@9	3926
alpar@9	3927 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3928 {\tt\%Y}&The year as a decimal number, using the Gregorian calendar.\\
alpar@9	3929 \end{tabular}
alpar@9	3930
alpar@9	3931 \medskip
alpar@9	3932
alpar@9	3933 \begin{tabular}{@{}p{20pt}p{298pt}@{}}
alpar@9	3934 {\tt\%\%}&A literal \verb\|%\| character.\\
alpar@9	3935 \end{tabular}
alpar@9	3936
alpar@9	3937 \medskip
alpar@9	3938
alpar@9	3939 All other (ordinary) characters in the format string are simply copied
alpar@9	3940 to the resultant string.
alpar@9	3941
alpar@9	3942 The first argument (calendar time) passed to the function {\tt time2str}
alpar@9	3943 must be in the range from $-62135596800$ to $+64092211199$ that
alpar@9	3944 corresponds to the period from 00:00:00 on January 1, 0001 to 23:59:59
alpar@9	3945 on December 31, 4000 of the Gregorian calendar.
alpar@9	3946
alpar@9	3947 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	3948
alpar@9	3949 \newpage
alpar@9	3950
alpar@9	3951 \section{Table drivers}
alpar@9	3952 \label{drivers}
alpar@9	3953
alpar@9	3954 \noindent\hfil
alpar@9	3955 by Andrew Makhorin \verb\|<mao@gnu.org>\|
alpar@9	3956
alpar@9	3957 \noindent\hfil
alpar@9	3958 and Heinrich Schuchardt \verb\|<heinrich.schuchardt@gmx.de>\|
alpar@9	3959
alpar@9	3960 \bigskip\bigskip
alpar@9	3961
alpar@9	3962 The {\it table driver} is a program module which provides transmitting
alpar@9	3963 data between MathProg model objects and data tables.
alpar@9	3964
alpar@9	3965 Currently the GLPK package has four table drivers:
alpar@9	3966
alpar@9	3967 \setlength{\leftmargini}{2.5em}
alpar@9	3968
alpar@9	3969 \begin{itemize}
alpar@9	3970 \item built-in CSV table driver;
alpar@9	3971 \item built-in xBASE table driver;
alpar@9	3972 \item ODBC table driver;
alpar@9	3973 \item MySQL table driver.
alpar@9	3974 \end{itemize}
alpar@9	3975
alpar@9	3976 \subsection{CSV table driver}
alpar@9	3977
alpar@9	3978 The CSV table driver assumes that the data table is represented in the
alpar@9	3979 form of a plain text file in the CSV (comma-separated values) file
alpar@9	3980 format as described below.
alpar@9	3981
alpar@9	3982 To choose the CSV table driver its name in the table statement should
alpar@9	3983 be specified as \verb\|"CSV"\|, and the only argument should specify the
alpar@9	3984 name of a plain text file containing the table. For example:
alpar@9	3985
alpar@9	3986 \medskip
alpar@9	3987
alpar@9	3988 \verb\| table data IN "CSV" "data.csv": ... ;\|
alpar@9	3989
alpar@9	3990 \medskip
alpar@9	3991
alpar@9	3992 The filename suffix may be arbitrary, however, it is recommended to use
alpar@9	3993 the suffix `\verb\|.csv\|'.
alpar@9	3994
alpar@9	3995 On reading input tables the CSV table driver provides an implicit field
alpar@9	3996 named \verb\|RECNO\|, which contains the current record number. This
alpar@9	3997 field can be specified in the input table statement as if there were
alpar@9	3998 the actual field having the name \verb\|RECNO\| in the CSV file. For
alpar@9	3999 example:
alpar@9	4000
alpar@9	4001 \medskip
alpar@9	4002
alpar@9	4003 \verb\| table list IN "CSV" "list.csv": num <- [RECNO], ... ;\|
alpar@9	4004
alpar@9	4005 \subsubsection*{CSV format\footnote{This material is based on the RFC
alpar@9	4006 document 4180.}}
alpar@9	4007
alpar@9	4008 The CSV (comma-separated values) format is a plain text file format
alpar@9	4009 defined as follows.
alpar@9	4010
alpar@9	4011 1. Each record is located on a separate line, delimited by a line
alpar@9	4012 break. For example:
alpar@9	4013
alpar@9	4014 \medskip
alpar@9	4015
alpar@9	4016 \verb\| aaa,bbb,ccc\n\|
alpar@9	4017
alpar@9	4018 \verb\| xxx,yyy,zzz\n\|
alpar@9	4019
alpar@9	4020 \medskip
alpar@9	4021
alpar@9	4022 \noindent
alpar@9	4023 where \verb\|\n\| means the control character \verb\|LF\| ({\tt 0x0A}).
alpar@9	4024
alpar@9	4025 \newpage
alpar@9	4026
alpar@9	4027 2. The last record in the file may or may not have an ending line
alpar@9	4028 break. For example:
alpar@9	4029
alpar@9	4030 \medskip
alpar@9	4031
alpar@9	4032 \verb\| aaa,bbb,ccc\n\|
alpar@9	4033
alpar@9	4034 \verb\| xxx,yyy,zzz\|
alpar@9	4035
alpar@9	4036 \medskip
alpar@9	4037
alpar@9	4038 3. There should be a header line appearing as the first line of the
alpar@9	4039 file in the same format as normal record lines. This header should
alpar@9	4040 contain names corresponding to the fields in the file. The number of
alpar@9	4041 field names in the header line should be the same as the number of
alpar@9	4042 fields in the records of the file. For example:
alpar@9	4043
alpar@9	4044 \medskip
alpar@9	4045
alpar@9	4046 \verb\| name1,name2,name3\n\|
alpar@9	4047
alpar@9	4048 \verb\| aaa,bbb,ccc\n\|
alpar@9	4049
alpar@9	4050 \verb\| xxx,yyy,zzz\n\|
alpar@9	4051
alpar@9	4052 \medskip
alpar@9	4053
alpar@9	4054 4. Within the header and each record there may be one or more fields
alpar@9	4055 separated by commas. Each line should contain the same number of fields
alpar@9	4056 throughout the file. Spaces are considered as part of a field and
alpar@9	4057 therefore not ignored. The last field in the record should not be
alpar@9	4058 followed by a comma. For example:
alpar@9	4059
alpar@9	4060 \medskip
alpar@9	4061
alpar@9	4062 \verb\| aaa,bbb,ccc\n\|
alpar@9	4063
alpar@9	4064 \medskip
alpar@9	4065
alpar@9	4066 5. Fields may or may not be enclosed in double quotes. For example:
alpar@9	4067
alpar@9	4068 \medskip
alpar@9	4069
alpar@9	4070 \verb\| "aaa","bbb","ccc"\n\|
alpar@9	4071
alpar@9	4072 \verb\| zzz,yyy,xxx\n\|
alpar@9	4073
alpar@9	4074 \medskip
alpar@9	4075
alpar@9	4076 6. If a field is enclosed in double quotes, each double quote which is
alpar@9	4077 part of the field should be coded twice. For example:
alpar@9	4078
alpar@9	4079 \medskip
alpar@9	4080
alpar@9	4081 \verb\| "aaa","b""bb","ccc"\n\|
alpar@9	4082
alpar@9	4083 \medskip
alpar@9	4084
alpar@9	4085 \noindent{\bf Example}
alpar@9	4086
alpar@9	4087 \begin{verbatim}
alpar@9	4088 FROM,TO,DISTANCE,COST
alpar@9	4089 Seattle,New-York,2.5,0.12
alpar@9	4090 Seattle,Chicago,1.7,0.08
alpar@9	4091 Seattle,Topeka,1.8,0.09
alpar@9	4092 San-Diego,New-York,2.5,0.15
alpar@9	4093 San-Diego,Chicago,1.8,0.10
alpar@9	4094 San-Diego,Topeka,1.4,0.07
alpar@9	4095 \end{verbatim}
alpar@9	4096
alpar@9	4097 \subsection{xBASE table driver}
alpar@9	4098
alpar@9	4099 The xBASE table driver assumes that the data table is stored in the
alpar@9	4100 .dbf file format.
alpar@9	4101
alpar@9	4102 To choose the xBASE table driver its name in the table statement should
alpar@9	4103 be specified as \verb\|"xBASE"\|, and the first argument should specify
alpar@9	4104 the name of a .dbf file containing the table. For the output table there
alpar@9	4105 should be the second argument defining the table format in the form
alpar@9	4106 \verb\|"FF...F"\|, where \verb\|F\| is either {\tt C({\it n})},
alpar@9	4107 which specifies a character field of length $n$, or
alpar@9	4108 {\tt N({\it n}{\rm [},{\it p}{\rm ]})}, which specifies a numeric field
alpar@9	4109 of length $n$ and precision $p$ (by default $p$ is 0).
alpar@9	4110
alpar@9	4111 The following is a simple example which illustrates creating and
alpar@9	4112 reading a .dbf file:
alpar@9	4113
alpar@9	4114 \begin{verbatim}
alpar@9	4115 table tab1{i in 1..10} OUT "xBASE" "foo.dbf"
alpar@9	4116 "N(5)N(10,4)C(1)C(10)": 2*i+1 ~ B, Uniform(-20,+20) ~ A,
alpar@9	4117 "?" ~ FOO, "[" & i & "]" ~ C;
alpar@9	4118 set S, dimen 4;
alpar@9	4119 table tab2 IN "xBASE" "foo.dbf": S <- [B, C, RECNO, A];
alpar@9	4120 display S;
alpar@9	4121 end;
alpar@9	4122 \end{verbatim}
alpar@9	4123
alpar@9	4124 \subsection{ODBC table driver}
alpar@9	4125
alpar@9	4126 The ODBC table driver allows connecting to SQL databases using an
alpar@9	4127 implementation of the ODBC interface based on the Call Level Interface
alpar@9	4128 (CLI).\footnote{The corresponding software standard is defined in
alpar@9	4129 ISO/IEC 9075-3:2003.}
alpar@9	4130
alpar@9	4131 \paragraph{Debian GNU/Linux.}
alpar@9	4132 Under Debian GNU/Linux the ODBC table driver uses the iODBC
alpar@9	4133 package,\footnote{See {\tt<http://www.iodbc.org/>}.} which should be
alpar@9	4134 installed before building the GLPK package. The installation can be
alpar@9	4135 effected with the following command:
alpar@9	4136
alpar@9	4137 \begin{verbatim}
alpar@9	4138 sudo apt-get install libiodbc2-dev
alpar@9	4139 \end{verbatim}
alpar@9	4140
alpar@9	4141 Note that on configuring the GLPK package to enable using the iODBC
alpar@9	4142 library the option `\verb\|--enable-odbc\|' should be passed to the
alpar@9	4143 configure script.
alpar@9	4144
alpar@9	4145 The individual databases must be entered for systemwide usage in
alpar@9	4146 \linebreak \verb\|/etc/odbc.ini\| and \verb\|/etc/odbcinst.ini\|. Database
alpar@9	4147 connections to be used by a single user are specified by files in the
alpar@9	4148 home directory (\verb\|.odbc.ini\| and \verb\|.odbcinst.ini\|).
alpar@9	4149
alpar@9	4150 \paragraph{Microsoft Windows.}
alpar@9	4151 Under Microsoft Windows the ODBC table driver uses the Microsoft ODBC
alpar@9	4152 library. To enable this feature the symbol:
alpar@9	4153
alpar@9	4154 \begin{verbatim}
alpar@9	4155 #define ODBC_DLNAME "odbc32.dll"
alpar@9	4156 \end{verbatim}
alpar@9	4157
alpar@9	4158 \noindent
alpar@9	4159 should be defined in the GLPK configuration file `\verb\|config.h\|'.
alpar@9	4160
alpar@9	4161 Data sources can be created via the Administrative Tools from the
alpar@9	4162 Control Panel.
alpar@9	4163
alpar@9	4164 \bigskip
alpar@9	4165
alpar@9	4166 To choose the ODBC table driver its name in the table statement should
alpar@9	4167 be specified as \verb\|'ODBC'\| or \verb\|'iODBC'\|.
alpar@9	4168
alpar@9	4169 The argument list is specified as follows.
alpar@9	4170
alpar@9	4171 The first argument is the connection string passed to the ODBC library,
alpar@9	4172 for example:
alpar@9	4173
alpar@9	4174 \verb\|'DSN=glpk;UID=user;PWD=password'\|, or
alpar@9	4175
alpar@9	4176 \verb\|'DRIVER=MySQL;DATABASE=glpkdb;UID=user;PWD=password'\|.
alpar@9	4177
alpar@9	4178 Different parts of the string are separated by semicolons. Each part
alpar@9	4179 consists of a pair {\it fieldname} and {\it value} separated by the
alpar@9	4180 equal sign. Allowable fieldnames depend on the ODBC library. Typically
alpar@9	4181 the following fieldnames are allowed:
alpar@9	4182
alpar@9	4183 \verb\|DATABASE \| database;
alpar@9	4184
alpar@9	4185 \verb\|DRIVER \| ODBC driver;
alpar@9	4186
alpar@9	4187 \verb\|DSN \| name of a data source;
alpar@9	4188
alpar@9	4189 \verb\|FILEDSN \| name of a file data source;
alpar@9	4190
alpar@9	4191 \verb\|PWD \| user password;
alpar@9	4192
alpar@9	4193 \verb\|SERVER \| database;
alpar@9	4194
alpar@9	4195 \verb\|UID \| user name.
alpar@9	4196
alpar@9	4197 The second argument and all following are considered to be SQL
alpar@9	4198 statements
alpar@9	4199
alpar@9	4200 SQL statements may be spread over multiple arguments. If the last
alpar@9	4201 character of an argument is a semicolon this indicates the end of
alpar@9	4202 a SQL statement.
alpar@9	4203
alpar@9	4204 The arguments of a SQL statement are concatenated separated by space.
alpar@9	4205 The eventual trailing semicolon will be removed.
alpar@9	4206
alpar@9	4207 All but the last SQL statement will be executed directly.
alpar@9	4208
alpar@9	4209 For IN-table the last SQL statement can be a SELECT command starting
alpar@9	4210 with the capitalized letters \verb\|'SELECT '\|. If the string does not
alpar@9	4211 start with \verb\|'SELECT '\| it is considered to be a table name and a
alpar@9	4212 SELECT statement is automatically generated.
alpar@9	4213
alpar@9	4214 For OUT-table the last SQL statement can contain one or multiple
alpar@9	4215 question marks. If it contains a question mark it is considered a
alpar@9	4216 template for the write routine. Otherwise the string is considered a
alpar@9	4217 table name and an INSERT template is automatically generated.
alpar@9	4218
alpar@9	4219 The writing routine uses the template with the question marks and
alpar@9	4220 replaces the first question mark by the first output parameter, the
alpar@9	4221 second question mark by the second output parameter and so forth. Then
alpar@9	4222 the SQL command is issued.
alpar@9	4223
alpar@9	4224 The following is an example of the output table statement:
alpar@9	4225
alpar@9	4226 \begin{small}
alpar@9	4227 \begin{verbatim}
alpar@9	4228 table ta { l in LOCATIONS } OUT
alpar@9	4229 'ODBC'
alpar@9	4230 'DSN=glpkdb;UID=glpkuser;PWD=glpkpassword'
alpar@9	4231 'DROP TABLE IF EXISTS result;'
alpar@9	4232 'CREATE TABLE result ( ID INT, LOC VARCHAR(255), QUAN DOUBLE );'
alpar@9	4233 'INSERT INTO result 'VALUES ( 4, ?, ? )' :
alpar@9	4234 l ~ LOC, quantity[l] ~ QUAN;
alpar@9	4235 \end{verbatim}
alpar@9	4236 \end{small}
alpar@9	4237
alpar@9	4238 \noindent
alpar@9	4239 Alternatively it could be written as follows:
alpar@9	4240
alpar@9	4241 \begin{small}
alpar@9	4242 \begin{verbatim}
alpar@9	4243 table ta { l in LOCATIONS } OUT
alpar@9	4244 'ODBC'
alpar@9	4245 'DSN=glpkdb;UID=glpkuser;PWD=glpkpassword'
alpar@9	4246 'DROP TABLE IF EXISTS result;'
alpar@9	4247 'CREATE TABLE result ( ID INT, LOC VARCHAR(255), QUAN DOUBLE );'
alpar@9	4248 'result' :
alpar@9	4249 l ~ LOC, quantity[l] ~ QUAN, 4 ~ ID;
alpar@9	4250 \end{verbatim}
alpar@9	4251 \end{small}
alpar@9	4252
alpar@9	4253 Using templates with `\verb\|?\|' supports not only INSERT, but also
alpar@9	4254 UPDATE, DELETE, etc. For example:
alpar@9	4255
alpar@9	4256 \begin{small}
alpar@9	4257 \begin{verbatim}
alpar@9	4258 table ta { l in LOCATIONS } OUT
alpar@9	4259 'ODBC'
alpar@9	4260 'DSN=glpkdb;UID=glpkuser;PWD=glpkpassword'
alpar@9	4261 'UPDATE result SET DATE = ' & date & ' WHERE ID = 4;'
alpar@9	4262 'UPDATE result SET QUAN = ? WHERE LOC = ? AND ID = 4' :
alpar@9	4263 quantity[l], l;
alpar@9	4264 \end{verbatim}
alpar@9	4265 \end{small}
alpar@9	4266
alpar@9	4267 \subsection{MySQL table driver}
alpar@9	4268
alpar@9	4269 The MySQL table driver allows connecting to MySQL databases.
alpar@9	4270
alpar@9	4271 \paragraph{Debian GNU/Linux.}
alpar@9	4272 Under Debian GNU/Linux the MySQL table\linebreak driver uses the MySQL
alpar@9	4273 package,\footnote{For download development files see
alpar@9	4274 {\tt<http://dev.mysql.com/downloads/mysql/>}.} which should be installed
alpar@9	4275 before building the GLPK package. The installation can be effected with
alpar@9	4276 the following command:
alpar@9	4277
alpar@9	4278 \begin{verbatim}
alpar@9	4279 sudo apt-get install libmysqlclient15-dev
alpar@9	4280 \end{verbatim}
alpar@9	4281
alpar@9	4282 Note that on configuring the GLPK package to enable using the MySQL
alpar@9	4283 library the option `\verb\|--enable-mysql\|' should be passed to the
alpar@9	4284 configure script.
alpar@9	4285
alpar@9	4286 \paragraph{Microsoft Windows.}
alpar@9	4287 Under Microsoft Windows the MySQL table driver also uses the MySQL
alpar@9	4288 library. To enable this feature the symbol:
alpar@9	4289
alpar@9	4290 \begin{verbatim}
alpar@9	4291 #define MYSQL_DLNAME "libmysql.dll"
alpar@9	4292 \end{verbatim}
alpar@9	4293
alpar@9	4294 \noindent
alpar@9	4295 should be defined in the GLPK configuration file `\verb\|config.h\|'.
alpar@9	4296
alpar@9	4297 \bigskip
alpar@9	4298
alpar@9	4299 To choose the MySQL table driver its name in the table statement should
alpar@9	4300 be specified as \verb\|'MySQL'\|.
alpar@9	4301
alpar@9	4302 The argument list is specified as follows.
alpar@9	4303
alpar@9	4304 The first argument specifies how to connect the data base in the DSN
alpar@9	4305 style, for example:
alpar@9	4306
alpar@9	4307 \verb\|'Database=glpk;UID=glpk;PWD=gnu'\|.
alpar@9	4308
alpar@9	4309 Different parts of the string are separated by semicolons. Each part
alpar@9	4310 consists of a pair {\it fieldname} and {\it value} separated by the
alpar@9	4311 equal sign. The following fieldnames are allowed:
alpar@9	4312
alpar@9	4313 \verb\|Server \| server running the database (defaulting to localhost);
alpar@9	4314
alpar@9	4315 \verb\|Database \| name of the database;
alpar@9	4316
alpar@9	4317 \verb\|UID \| user name;
alpar@9	4318
alpar@9	4319 \verb\|PWD \| user password;
alpar@9	4320
alpar@9	4321 \verb\|Port \| port used by the server (defaulting to 3306).
alpar@9	4322
alpar@9	4323 The second argument and all following are considered to be SQL
alpar@9	4324 statements
alpar@9	4325
alpar@9	4326 SQL statements may be spread over multiple arguments. If the last
alpar@9	4327 character of an argument is a semicolon this indicates the end of
alpar@9	4328 a SQL statement.
alpar@9	4329
alpar@9	4330 The arguments of a SQL statement are concatenated separated by space.
alpar@9	4331 The eventual trailing semicolon will be removed.
alpar@9	4332
alpar@9	4333 All but the last SQL statement will be executed directly.
alpar@9	4334
alpar@9	4335 For IN-table the last SQL statement can be a SELECT command starting
alpar@9	4336 with the capitalized letters \verb\|'SELECT '\|. If the string does not
alpar@9	4337 start with \verb\|'SELECT '\| it is considered to be a table name and a
alpar@9	4338 SELECT statement is automatically generated.
alpar@9	4339
alpar@9	4340 For OUT-table the last SQL statement can contain one or multiple
alpar@9	4341 question marks. If it contains a question mark it is considered a
alpar@9	4342 template for the write routine. Otherwise the string is considered a
alpar@9	4343 table name and an INSERT template is automatically generated.
alpar@9	4344
alpar@9	4345 The writing routine uses the template with the question marks and
alpar@9	4346 replaces the first question mark by the first output parameter, the
alpar@9	4347 second question mark by the second output parameter and so forth. Then
alpar@9	4348 the SQL command is issued.
alpar@9	4349
alpar@9	4350 The following is an example of the output table statement:
alpar@9	4351
alpar@9	4352 \begin{small}
alpar@9	4353 \begin{verbatim}
alpar@9	4354 table ta { l in LOCATIONS } OUT
alpar@9	4355 'MySQL'
alpar@9	4356 'Database=glpkdb;UID=glpkuser;PWD=glpkpassword'
alpar@9	4357 'DROP TABLE IF EXISTS result;'
alpar@9	4358 'CREATE TABLE result ( ID INT, LOC VARCHAR(255), QUAN DOUBLE );'
alpar@9	4359 'INSERT INTO result VALUES ( 4, ?, ? )' :
alpar@9	4360 l ~ LOC, quantity[l] ~ QUAN;
alpar@9	4361 \end{verbatim}
alpar@9	4362 \end{small}
alpar@9	4363
alpar@9	4364 \noindent
alpar@9	4365 Alternatively it could be written as follows:
alpar@9	4366
alpar@9	4367 \begin{small}
alpar@9	4368 \begin{verbatim}
alpar@9	4369 table ta { l in LOCATIONS } OUT
alpar@9	4370 'MySQL'
alpar@9	4371 'Database=glpkdb;UID=glpkuser;PWD=glpkpassword'
alpar@9	4372 'DROP TABLE IF EXISTS result;'
alpar@9	4373 'CREATE TABLE result ( ID INT, LOC VARCHAR(255), QUAN DOUBLE );'
alpar@9	4374 'result' :
alpar@9	4375 l ~ LOC, quantity[l] ~ QUAN, 4 ~ ID;
alpar@9	4376 \end{verbatim}
alpar@9	4377 \end{small}
alpar@9	4378
alpar@9	4379 Using templates with `\verb\|?\|' supports not only INSERT, but also
alpar@9	4380 UPDATE, DELETE, etc. For example:
alpar@9	4381
alpar@9	4382 \begin{small}
alpar@9	4383 \begin{verbatim}
alpar@9	4384 table ta { l in LOCATIONS } OUT
alpar@9	4385 'MySQL'
alpar@9	4386 'Database=glpkdb;UID=glpkuser;PWD=glpkpassword'
alpar@9	4387 'UPDATE result SET DATE = ' & date & ' WHERE ID = 4;'
alpar@9	4388 'UPDATE result SET QUAN = ? WHERE LOC = ? AND ID = 4' :
alpar@9	4389 quantity[l], l;
alpar@9	4390 \end{verbatim}
alpar@9	4391 \end{small}
alpar@9	4392
alpar@9	4393 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	4394
alpar@9	4395 \newpage
alpar@9	4396
alpar@9	4397 \section{Solving models with glpsol}
alpar@9	4398
alpar@9	4399 The GLPK package\footnote{{\tt http://www.gnu.org/software/glpk/}}
alpar@9	4400 includes the program {\tt glpsol}, which is a stand-alone LP/MIP solver.
alpar@9	4401 This program can be launched from the command line or from the shell to
alpar@9	4402 solve models written in the GNU MathProg modeling language.
alpar@9	4403
alpar@9	4404 In order to tell the solver that the input file contains a model
alpar@9	4405 description, you need to specify the option \verb\|--model\| in the
alpar@9	4406 command line. For example:
alpar@9	4407
alpar@9	4408 \medskip
alpar@9	4409
alpar@9	4410 \verb\| glpsol --model foo.mod\|
alpar@9	4411
alpar@9	4412 \medskip
alpar@9	4413
alpar@9	4414 Sometimes it is necessary to use the data section placed in a separate
alpar@9	4415 file, in which case you may use the following command:
alpar@9	4416
alpar@9	4417 \medskip
alpar@9	4418
alpar@9	4419 \verb\| glpsol --model foo.mod --data foo.dat\|
alpar@9	4420
alpar@9	4421 \medskip
alpar@9	4422
alpar@9	4423 \noindent Note that if the model file also contains the data section,
alpar@9	4424 that section is ignored.
alpar@9	4425
alpar@9	4426 If the model description contains some display and/or printf statements,
alpar@9	4427 by default the output is sent to the terminal. In order to redirect the
alpar@9	4428 output to a file you may use the following command:
alpar@9	4429
alpar@9	4430 \medskip
alpar@9	4431
alpar@9	4432 \verb\| glpsol --model foo.mod --display foo.out\|
alpar@9	4433
alpar@9	4434 \medskip
alpar@9	4435
alpar@9	4436 If you need to look at the problem, which has been generated by the
alpar@9	4437 model translator, you may use the option \verb\|--wlp\| as follows:
alpar@9	4438
alpar@9	4439 \medskip
alpar@9	4440
alpar@9	4441 \verb\| glpsol --model foo.mod --wlp foo.lp\|
alpar@9	4442
alpar@9	4443 \medskip
alpar@9	4444
alpar@9	4445 \noindent in which case the problem data is written to file
alpar@9	4446 \verb\|foo.lp\| in CPLEX LP format suitable for visual analysis.
alpar@9	4447
alpar@9	4448 Sometimes it is needed merely to check the model description not
alpar@9	4449 solving the generated problem instance. In this case you may specify
alpar@9	4450 the option \verb\|--check\|, for example:
alpar@9	4451
alpar@9	4452 \medskip
alpar@9	4453
alpar@9	4454 \verb\| glpsol --check --model foo.mod --wlp foo.lp\|
alpar@9	4455
alpar@9	4456 \medskip
alpar@9	4457
alpar@9	4458 In order to write a numeric solution obtained by the solver you may use
alpar@9	4459 the following command:
alpar@9	4460
alpar@9	4461 \medskip
alpar@9	4462
alpar@9	4463 \verb\| glpsol --model foo.mod --output foo.sol\|
alpar@9	4464
alpar@9	4465 \medskip
alpar@9	4466
alpar@9	4467 \noindent in which case the solution is written to file \verb\|foo.sol\|
alpar@9	4468 in a plain text format.
alpar@9	4469
alpar@9	4470 The complete list of the \verb\|glpsol\| options can be found in the
alpar@9	4471 reference manual included in the GLPK distribution.
alpar@9	4472
alpar@9	4473 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	4474
alpar@9	4475 \newpage
alpar@9	4476
alpar@9	4477 \section{Example model description}
alpar@9	4478
alpar@9	4479 \subsection{Model description written in MathProg}
alpar@9	4480
alpar@9	4481 Below here is a complete example of the model description written in
alpar@9	4482 the GNU MathProg modeling language.
alpar@9	4483
alpar@9	4484 \begin{small}
alpar@9	4485 \begin{verbatim}
alpar@9	4486 # A TRANSPORTATION PROBLEM
alpar@9	4487 #
alpar@9	4488 # This problem finds a least cost shipping schedule that meets
alpar@9	4489 # requirements at markets and supplies at factories.
alpar@9	4490 #
alpar@9	4491 # References:
alpar@9	4492 # Dantzig G B, "Linear Programming and Extensions."
alpar@9	4493 # Princeton University Press, Princeton, New Jersey, 1963,
alpar@9	4494 # Chapter 3-3.
alpar@9	4495
alpar@9	4496 set I;
alpar@9	4497 /* canning plants */
alpar@9	4498
alpar@9	4499 set J;
alpar@9	4500 /* markets */
alpar@9	4501
alpar@9	4502 param a{i in I};
alpar@9	4503 /* capacity of plant i in cases */
alpar@9	4504
alpar@9	4505 param b{j in J};
alpar@9	4506 /* demand at market j in cases */
alpar@9	4507
alpar@9	4508 param d{i in I, j in J};
alpar@9	4509 /* distance in thousands of miles */
alpar@9	4510
alpar@9	4511 param f;
alpar@9	4512 /* freight in dollars per case per thousand miles */
alpar@9	4513
alpar@9	4514 param c{i in I, j in J} := f * d[i,j] / 1000;
alpar@9	4515 /* transport cost in thousands of dollars per case */
alpar@9	4516
alpar@9	4517 var x{i in I, j in J} >= 0;
alpar@9	4518 /* shipment quantities in cases */
alpar@9	4519
alpar@9	4520 minimize cost: sum{i in I, j in J} c[i,j] * x[i,j];
alpar@9	4521 /* total transportation costs in thousands of dollars */
alpar@9	4522
alpar@9	4523 s.t. supply{i in I}: sum{j in J} x[i,j] <= a[i];
alpar@9	4524 /* observe supply limit at plant i */
alpar@9	4525
alpar@9	4526 s.t. demand{j in J}: sum{i in I} x[i,j] >= b[j];
alpar@9	4527 /* satisfy demand at market j */
alpar@9	4528
alpar@9	4529 data;
alpar@9	4530
alpar@9	4531 set I := Seattle San-Diego;
alpar@9	4532
alpar@9	4533 set J := New-York Chicago Topeka;
alpar@9	4534
alpar@9	4535 param a := Seattle 350
alpar@9	4536 San-Diego 600;
alpar@9	4537
alpar@9	4538 param b := New-York 325
alpar@9	4539 Chicago 300
alpar@9	4540 Topeka 275;
alpar@9	4541
alpar@9	4542 param d : New-York Chicago Topeka :=
alpar@9	4543 Seattle 2.5 1.7 1.8
alpar@9	4544 San-Diego 2.5 1.8 1.4 ;
alpar@9	4545
alpar@9	4546 param f := 90;
alpar@9	4547
alpar@9	4548 end;
alpar@9	4549 \end{verbatim}
alpar@9	4550 \end{small}
alpar@9	4551
alpar@9	4552 \subsection{Generated LP problem instance}
alpar@9	4553
alpar@9	4554 Below here is the result of the translation of the example model
alpar@9	4555 produced by the solver \verb\|glpsol\| and written in CPLEX LP format
alpar@9	4556 with the option \verb\|--wlp\|.
alpar@9	4557
alpar@9	4558 \begin{small}
alpar@9	4559 \begin{verbatim}
alpar@9	4560 \* Problem: transp *\
alpar@9	4561
alpar@9	4562 Minimize
alpar@9	4563 cost: + 0.225 x(Seattle,New~York) + 0.153 x(Seattle,Chicago)
alpar@9	4564 + 0.162 x(Seattle,Topeka) + 0.225 x(San~Diego,New~York)
alpar@9	4565 + 0.162 x(San~Diego,Chicago) + 0.126 x(San~Diego,Topeka)
alpar@9	4566
alpar@9	4567 Subject To
alpar@9	4568 supply(Seattle): + x(Seattle,New~York) + x(Seattle,Chicago)
alpar@9	4569 + x(Seattle,Topeka) <= 350
alpar@9	4570 supply(San~Diego): + x(San~Diego,New~York) + x(San~Diego,Chicago)
alpar@9	4571 + x(San~Diego,Topeka) <= 600
alpar@9	4572 demand(New~York): + x(Seattle,New~York) + x(San~Diego,New~York) >= 325
alpar@9	4573 demand(Chicago): + x(Seattle,Chicago) + x(San~Diego,Chicago) >= 300
alpar@9	4574 demand(Topeka): + x(Seattle,Topeka) + x(San~Diego,Topeka) >= 275
alpar@9	4575
alpar@9	4576 End
alpar@9	4577 \end{verbatim}
alpar@9	4578 \end{small}
alpar@9	4579
alpar@9	4580 \subsection{Optimal LP solution}
alpar@9	4581
alpar@9	4582 Below here is the optimal solution of the generated LP problem instance
alpar@9	4583 found by the solver \verb\|glpsol\| and written in plain text format
alpar@9	4584 with the option \verb\|--output\|.
alpar@9	4585
alpar@9	4586 \newpage
alpar@9	4587
alpar@9	4588 \begin{small}
alpar@9	4589 \begin{verbatim}
alpar@9	4590 Problem: transp
alpar@9	4591 Rows: 6
alpar@9	4592 Columns: 6
alpar@9	4593 Non-zeros: 18
alpar@9	4594 Status: OPTIMAL
alpar@9	4595 Objective: cost = 153.675 (MINimum)
alpar@9	4596
alpar@9	4597 No. Row name St Activity Lower bound Upper bound Marginal
alpar@9	4598 --- ------------ -- ------------ ------------ ------------ ------------
alpar@9	4599 1 cost B 153.675
alpar@9	4600 2 supply[Seattle]
alpar@9	4601 B 300 350
alpar@9	4602 3 supply[San-Diego]
alpar@9	4603 NU 600 600 < eps
alpar@9	4604 4 demand[New-York]
alpar@9	4605 NL 325 325 0.225
alpar@9	4606 5 demand[Chicago]
alpar@9	4607 NL 300 300 0.153
alpar@9	4608 6 demand[Topeka]
alpar@9	4609 NL 275 275 0.126
alpar@9	4610
alpar@9	4611 No. Column name St Activity Lower bound Upper bound Marginal
alpar@9	4612 --- ------------ -- ------------ ------------ ------------ ------------
alpar@9	4613 1 x[Seattle,New-York]
alpar@9	4614 B 0 0
alpar@9	4615 2 x[Seattle,Chicago]
alpar@9	4616 B 300 0
alpar@9	4617 3 x[Seattle,Topeka]
alpar@9	4618 NL 0 0 0.036
alpar@9	4619 4 x[San-Diego,New-York]
alpar@9	4620 B 325 0
alpar@9	4621 5 x[San-Diego,Chicago]
alpar@9	4622 NL 0 0 0.009
alpar@9	4623 6 x[San-Diego,Topeka]
alpar@9	4624 B 275 0
alpar@9	4625
alpar@9	4626 End of output
alpar@9	4627 \end{verbatim}
alpar@9	4628 \end{small}
alpar@9	4629
alpar@9	4630 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alpar@9	4631
alpar@9	4632 \newpage
alpar@9	4633
alpar@9	4634 \setcounter{secnumdepth}{-1}
alpar@9	4635
alpar@9	4636 \section{Acknowledgment}
alpar@9	4637
alpar@9	4638 The authors would like to thank the following people, who kindly read,
alpar@9	4639 commented, and corrected the draft of this document:
alpar@9	4640
alpar@9	4641 \medskip
alpar@9	4642
alpar@9	4643 \noindent Juan Carlos Borras \verb\|<borras@cs.helsinki.fi>\|
alpar@9	4644
alpar@9	4645 \medskip
alpar@9	4646
alpar@9	4647 \noindent Harley Mackenzie \verb\|<hjm@bigpond.com>\|
alpar@9	4648
alpar@9	4649 \medskip
alpar@9	4650
alpar@9	4651 \noindent Robbie Morrison \verb\|<robbie@actrix.co.nz>\|
alpar@9	4652
alpar@9	4653 \end{document}