source: trunk/Cbc/doc/cbcmodelclass.xml @ 2188

Last change on this file since 2188 was 120, checked in by rlh, 14 years ago

Rennovated Cbc user guide for INFORMS Tutorial book chapter

  • Property svn:eol-style set to native
  • Property svn:keywords set to Author Date Id Revision
File size: 24.6 KB
Line 
1<?xml version="1.0" encoding="ISO-8859-1"?>
2  <chapter id="cbcmodelclass">
3  <title>
4   The CBC Model Class
5  </title>
6  <section id="hierarchy">
7  <title>
8  Overview
9  </title>
10  <para>
11  The main class in CBC is <classname>CbcModel</classname>.  The <classname>CbcModel</classname> class is where most
12  of the parameter setting is done. The absolute minimum number of actions taken with <classname>CbcModel</classname> is two,
13    <itemizedlist>
14    <listitem>
15    <simpara>
16    <function>CbcModel(OsiSolverInterface &amp; linearSolver)</function> as constructor, and
17    </simpara>
18    </listitem>
19    <listitem>
20    <simpara>
21    <function>branchAndBound()</function> for solving the problem.   
22    </simpara>
23    </listitem>
24    </itemizedlist>
25  </para>
26  </section>
27
28  <section id="firstexample">
29  <title>
30  Simple Branch-and-Bound Example
31  </title>
32  <para>
33  The first sample program shows how to perform simple branch-and-bound with CBC.  This program is short enough to present in full.  Most of the remaining examples will take the form of small code fragments.
34  The complete code for all the examples in this Guide can be found in the CBC Samples directory, <filename>COIN/Cbc/Samples</filename>.
35
36  </para>
37  <example id="minimum.cpp">
38  <title>minimum.cpp</title>
39  <programlisting>
40  <![CDATA[ 
41// Copyright (C) 2005, International Business Machines
42// Corporation and others.  All Rights Reserved.
43
44#include "CbcModel.hpp"
45
46// Using CLP as the solver
47#include "OsiClpSolverInterface.hpp"
48
49int main (int argc, const char *argv[])
50{
51  OsiClpSolverInterface solver1;
52
53  // Read in example model in MPS file format
54  // and assert that it is a clean model
55  int numMpsReadErrors = solver1.readMps("../../Mps/Sample/p0033.mps","");
56  assert(numMpsReadErrors==0);
57
58  // Pass the solver with the problem to be solved to CbcModel
59  CbcModel model(solver1);
60
61  // Do complete search
62  model.branchAndBound();
63
64  /* Print the solution.  CbcModel clones the solver so we
65     need to get current copy from the CbcModel */
66  int numberColumns = model.solver()->getNumCols();
67   
68  const double * solution = model.solver()->getColSolution();
69   
70  for (int iColumn=0;iColumn<numberColumns;iColumn++) {
71    double value=solution[iColumn];
72    if (fabs(value)>1.0e-7&&model.solver()->isInteger(iColumn))
73      printf("%d has value %g\n",iColumn,value);
74   }
75  return 0;
76}   
77  ]]>   
78  </programlisting>
79  </example>
80  <!-- Clone comment  needs more explanation. What info is safe? When do you need to do a refresh?? -->
81  <!-- printf vs cout  -->
82
83  <para>
84  The program in <xref linkend="minimum.cpp"/> creates a <classname>OsiClpSolverInterface</classname> solver interface (i.e., <varname>solver1</varname>), and reads an MPS file. If there are no errors, the program passes the problem to <classname>CbcModel</classname> which solves the problem using the branch-and-bound algorithm. The part of the program which solves the problem is very small (one line!) but before that one line, the LP solver (i.e., <varname>solver1</varname>) had to be created and populated with the problem. After that one line, the results were printed out.
85 </para>
86</section>
87
88<section id="osiAndCbc">
89<title>
90The Relationship Between OSI and CBC
91</title>
92<para> 
93The program in <xref linkend="minimum.cpp"/> illustrates the dependency of CBC on 
94  the <classname>OsiSolverInterface</classname> class. The constructor of <classname>CbcModel</classname> takes a pointer to an <classname>OsiSolverInterface</classname> (i.e., a solver). The <classname>CbcModel</classname> clones the solver, and uses its own instance of the solver. The <classname>CbcModel</classname>'s solver and the original solver (e.g., <varname>solver1</varname>) are not in sync unless the user synchronizes them. The user can always access the <classname>CbcModel</classname>'s solver through the <function>model()</function> method.  To synchronize the two solvers, explicitly refreshing the original, e.g., 
95 <programlisting>
96  solver1 = model.solver();
97</programlisting>
98<classname>CbcModel</classname>'s method <function>solve()</function> returns a pointer to CBC's cloned solver.
99</para>
100
101<para>
102For convenience, many of the OSI methods to access problem data have identical method names in  <classname>CbcModel</classname>. (It's just more convenient to type <function>model.getNumCols()</function> rather than <function>model.solver()->getNumCols()</function>). The <classname>CbcModel</classname> refreshes its solver at certain logical points during the algorithm. At these points, the information from the <classname>CbcModel</classname> <varname>model</varname> will match the information from the <function>model.solver()</function>. Elsewhere, the information may vary. For instance, the OSI method <function>getColSolution()</function> will contain the best solution so far, while the <classname>CbcModel</classname> method may not. In this case, it is safer to use <function>CbcModel::bestSolution()</function>.
103</para>
104<para>
105While all the OSI methods used in <filename>minimum.cpp</filename> have equivalent methods in <classname>CbcModel</classname>, there are some OSI methods which do not. For example, if  the program produced a lot of undesired output, one might add the line
106</para>
107<programlisting>
108  model.solver()->setHintParam(OsiDoReducePrint,true,OsiHintTry);
109</programlisting>
110<para> 
111  <!-- model.solver() returns an OSISolverInterface? -->
112  to reduce the output. There is no <function>setHintParam()</function> method in <classname>CbcModel</classname>.
113  </para>
114  </section>
115
116  <section id="gettingsolution">
117  <title>
118  Getting Solution Information
119  </title>
120  <para>
121  Optimality can be checked through a call to <function>model.isProvenOptimal()</function>.  Also
122  available are <function>isProvenInfeasible()</function>,
123  <function>isSolutionLimitReached()</function>,
124  <function>isNodeLimitReached()</function> or the feared
125  <function>isAbandoned()</function>. There is also
126  <function>int&nbsp;status()</function> which returns 0 if finished (which includes the case when the algorithm is finished because it has been proved infeasible), 1 if stopped by user, and 2 if difficulties arose.
127  </para>
128  <para>
129  In addition to these <classname>CbcModel</classname> methods, solution values can be accessed via OSI methods.  The OSI methods pick up the current solution in the <classname>CBCModel</classname>.  The current solution will match the best solution found so far if called after <function>branchAndBound()</function> and a solution was found.
130  </para>
131  <table frame="none">
132  <title>
133  Methods for Getting Solution Information from OSI
134  </title>
135  <tgroup cols="2">
136  <thead>
137    <row>
138      <entry>
139      Purpose
140      </entry>
141      <entry>
142      Name
143      </entry>
144      <entry>
145      Notes
146      </entry>
147    </row>
148  </thead>
149  <tbody>
150    <row>
151      <entry align="left" valign="top">
152      Primal column solution
153      </entry>
154      <entry align="left" valign="top">
155      <function>const double * getColSolution()</function>
156      </entry>
157      <entry align="left" valign="top">
158      The OSI method will return the best solution found thus far, unless none has been found. It is safer to use <classname>CbcModel</classname> version, <function>CbcModel::bestSolution()</function>
159      </entry>
160    </row>
161    <row>
162      <entry align="left" valign="top">
163      Dual row solution
164      </entry>
165      <entry align="left" valign="top">
166      <function>const double * getRowPrice()</function>
167      </entry>
168      <entry align="left" valign="top">
169      Identical <classname>CbcModel</classname> version available, <function>CbcModel::getRowPrice()</function>.
170      </entry>
171    </row>
172    <row>
173      <entry align="left" valign="top">
174      Primal row solution
175      </entry>
176      <entry align="left" valign="top">
177      <function>const double * getRowActivity()</function>
178      </entry>
179      <entry align="left" valign="top">
180      Identical <classname>CbcModel</classname> version available, <function>CbcModel::getRowActivity()</function>.
181      </entry>
182    </row>
183    <row>
184      <entry align="left" valign="top">
185      Dual column solution
186      </entry>
187      <entry align="left" valign="top">
188      <function>const double * getReducedCost()</function>
189      </entry>
190     <entry align="left" valign="top">
191      Identical <classname>CbcModel</classname> version available, <function>CbcModel::gtReducedCost()</function>.
192      </entry>
193    </row>
194    <row>
195      <entry align="left" valign="top">
196      Number of rows in model
197      </entry>
198      <entry align="left" valign="top">
199      <function>int getNumRows()</function>
200      </entry>
201      <entry align="left" valign="top">
202      Identical <classname>CbcModel</classname> version available, <function>CbcModel::getNumRows()</function>. Note: the number of rows can change due to cuts.
203      </entry>
204    </row>
205    <row>
206      <entry align="left" valign="top">
207      Number of columns in model
208      </entry>
209      <entry align="left" valign="top">
210      <function>int getNumCols()</function>
211      </entry>
212      <entry align="left" valign="top">
213      Identical <classname>CbcModel</classname> version available, <function>CbcModel::getNumCols()</function>.
214      </entry>
215    </row>
216  </tbody>
217  </tgroup>
218  </table>
219  </section>
220
221  <section id="setsandgets">
222  <title>
223   Useful Set and Get Methods in <classname>CbcModel</classname>
224  </title>
225<para>
226Most of the parameter setting in CBC is done through <classname>CbcModel</classname> methods. The most commonly used set and get methods are listed in <xref linkend="setGet"/>.
227</para>
228  <table frame="none" id="setGet">
229  <title>Useful Set and Get Methods in <classname>CbcModel</classname></title>
230    <tgroup cols="2">
231    <thead>
232    <row>
233    <entry>
234    Method(s)
235    </entry>
236    <entry>
237    Description
238    </entry>
239    </row>
240    </thead>
241    <tbody>
242    <row>
243      <entry align="left" valign="top">
244      <function>bool&nbsp;setMaximumNodes(int value)</function><sbr/>
245      <function>int&nbsp;getMaximumNodes() const</function><sbr/>
246      <function>bool&nbsp;setMaximumSeconds(double value)</function><sbr/>
247
248      <function>double&nbsp;getMaximumSeconds()</function><sbr/>
249      <function>bool&nbsp;setMaximumSolutions(double value)</function><sbr/>
250      <function>double&nbsp;getMaximumSolutions() const</function>
251      </entry>
252      <entry align="left" valign="top">
253      These set methods tell CBC to stop after a given number of nodes,
254      seconds, or solutions is reached. The get methods return the corresponding values.
255      </entry>
256    </row>
257    <row>
258      <entry align="left" valign="top">
259      <function>bool&nbsp;setIntegerTolerance(double value) const</function><sbr/>
260      <function>double&nbsp;getIntegerTolerance() const</function>
261      </entry>
262      <entry align="left" valign="top">
263      An integer variable is deemed to be at an integral value if it is no further than this <parameter>value</parameter> (tolerance) away.
264      </entry>
265    </row>
266    <row>
267      <entry align="left" valign="top">
268      <function>bool&nbsp;setAllowableGap(double value)</function><sbr/>
269      <function>double&nbsp;getAllowableGap() const</function><sbr/>
270      <function>bool&nbsp;setAllowablePercentageGap(double value)</function><sbr/>
271      <function>double&nbsp;getAllowablePercentageGap() const</function><sbr/>
272      <function>bool&nbsp;setAllowableFractionGap(double value)</function><sbr/>
273      <function>double&nbsp;getAllowableFractionGap() const</function><sbr/>
274      </entry>
275      <entry align="left" valign="top">
276      <classname>CbcModel</classname> returns if the gap between the best known solution and the best
277      possible solution is less than this <parameter>value</parameter>, or as a percentage, or a fraction.
278      </entry>
279    </row>
280    <row>
281      <entry align="left" valign="top">
282      <function>void&nbsp;setNumberStrong(double value) </function><sbr/>
283      <function>int&nbsp;numberStrong() const </function> <!-- should be a "get" -->
284      </entry>
285      <entry align="left" valign="top">
286      These methods set or get the maximum number of candidates at a node to
287      be evaluated for strong branching.
288      </entry>
289    </row>
290    <row>
291      <entry align="left" valign="top">
292      <function>void&nbsp;setPrintFrequency(int value) </function><sbr/>
293      <function>int&nbsp;printFrequency() const</function>
294      </entry>
295      <entry align="left" valign="top">
296      Controls the number of nodes evaluated between status prints.
297      Print frequency has a very slight overhead, if <parameter>value</parameter> is small.
298      </entry>
299    </row>
300    <row>
301      <entry align="left" valign="top">
302      <function>int&nbsp;getNodeCount() const</function>
303      </entry>
304      <entry align="left" valign="top">
305      Returns number of nodes evaluated in the search.
306      </entry>
307    </row>
308    <row>
309      <entry align="left" valign="top">
310      <function>int&nbsp;numberRowsAtContinuous() const</function>
311      </entry>
312      <entry align="left" valign="top">
313      Returns number of rows at continuous <!-- rlh: huh? -->
314      </entry>
315    </row>
316    <row>
317      <entry align="left" valign="top">
318      <function>int &nbsp;numberIntegers() const</function><sbr/>
319      <function>const int&nbsp;*&nbsp;integerVariable() const</function>
320      </entry>
321      <entry align="left" valign="top">
322      Returns number of integer variables and an array specifying them.
323      </entry>
324    </row>
325    <row>
326      <entry align="left" valign="top">
327      <function>bool&nbsp;isBinary(int colIndex) const</function><sbr/>
328      <function>bool&nbsp;isContinuous(int colIndex) const</function><sbr/>
329      <function>bool&nbsp;isInteger(int colIndex) const</function>
330      </entry>
331      <entry align="left" valign="top">
332      Returns information on variable <parameter>colIndex</parameter>. OSI methods
333      can be used to set these attributes (before handing the model to <classname>CbcModel</classname>).
334      </entry>
335    </row>
336    <row>
337      <entry align="left" valign="top">
338      <function>double&nbsp;getObjValue() const</function>
339      </entry>
340      <entry align="left" valign="top">
341      This method returns the best objective value so far.
342      </entry>
343    </row>
344    <row>
345      <entry align="left" valign="top">
346      <function>double&nbsp;getCurrentObjValue() const</function>
347      </entry>
348      <entry align="left" valign="top">
349      This method returns the current objective value.
350      </entry>
351    </row>
352    <row>
353      <entry align="left" valign="top">
354      <function>const&nbsp;double&nbsp;*&nbsp;getObjCoefficients() const</function><sbr/>
355       </entry>
356      <entry align="left" valign="top">
357      This method return the objective coefficients.
358      </entry>
359    </row>
360    <row>
361      <entry align="left" valign="top">
362      <function>const&nbsp;double&nbsp;*&nbsp;getRowLower() const</function><sbr/>
363       <function>const&nbsp;double&nbsp;*&nbsp;getRowUpper() const</function><sbr/>
364       <function>const&nbsp;double&nbsp;*&nbsp;getColLower() const</function><sbr/>
365       <function>const&nbsp;double&nbsp;*&nbsp;getColUpper() const</function><sbr/>
366       </entry>
367      <entry align="left" valign="top">
368      These methods return the lower and upper bounds on row and column activities.
369      </entry>
370    </row>
371    <row>
372      <entry align="left" valign="top">
373      <function>const&nbsp;CoinPackMatrix&nbsp;*&nbsp;getMatrixByRow() const</function>
374      </entry>
375      <entry align="left" valign="top">
376      This method returns a pointer to a row copy of matrix stored as a
377      <classname>CoinPackedMatrix</classname> which can be further examined.
378      </entry>
379    </row>
380    <row>
381      <entry align="left" valign="top">
382      <function>const CoinPackMatrix * getMatrixByCol() const</function>
383      </entry>
384      <entry align="left" valign="top">
385      This method returns a pointer to a column copy of matrix stored as a
386      <classname>CoinPackedMatrix</classname> which can be further examined.
387      </entry>
388    </row>
389    <row>
390      <entry align="left" valign="top">
391    <function>CoinBigIndex&nbsp;getNumElements() const</function> 
392     <footnote> 
393     <para> 
394        <type>CoinBigIndex</type> is a <function>typedef</function> which in
395        most cases is the same as <type>int</type>.
396        </para> 
397    </footnote> 
398      </entry>
399      <entry align="left" valign="top">
400      Returns the number of nonzero elements in the problem matrix.
401      </entry>
402    </row>
403    <row>
404      <entry align="left" valign="top">
405      <function>void setObjSense(double&nbsp;value)</function><sbr/>
406      <function>double getObjSense() const</function> 
407      </entry>
408      <entry align="left" valign="top">
409      These methods set and get the objective sense.  The parameter
410      <parameter>value</parameter> should be +1 to minimize and -1 to maximize.
411      </entry>
412    </row>
413    </tbody>
414  </tgroup>
415  </table>
416  </section>
417  <section id="majormethods">
418  <title>
419  Impacting the Solution Process
420  </title>
421<para>
422<classname>CbcModel</classname> is extremely flexible and customizable. The class structure of CBC is designed to make the most commonly desired customizations of branch-and-cut possible. These include:
423    <itemizedlist>
424    <listitem>
425    <simpara>
426     selecting the next node to consider in the search tree,
427    </simpara>
428    </listitem>
429    <listitem>
430    <simpara>
431    determining which variable to branch on,
432    </simpara>
433    </listitem>
434   <listitem>
435    <simpara>
436    using heuristics to generate MIP-feasible solutions quickly,
437    </simpara>
438    </listitem>
439   <listitem>
440    <simpara>
441    including cut generation when solving the LP-relaxations, and
442    </simpara>
443    </listitem>
444  <listitem>
445    <simpara>
446    invoking customized subproblem solvers.
447    </simpara>
448    </listitem>
449    </itemizedlist>
450
451
452To enable this flexibility,  <classname>CbcModel</classname> uses other classes in CBC (some of which are virtual and may have multiple instances). Not all classes are created equal. The two tables below list in alphabetical order the classes used by <classname>CbcModel</classname> that are of most interest and of least interest.
453</para>
454    <table frame="none">
455  <title>Classes Used by CbcModel - Most Useful</title>
456    <tgroup cols="3">
457    <thead>
458    <row>
459    <entry>
460    Class name
461    </entry>
462    <entry>
463    Description
464    </entry>
465    <entry>
466    Notes
467    </entry>
468    </row>
469    </thead>
470    <tbody>
471    <row>
472      <entry align="left" valign="top">
473      <classname>CbcCompareBase</classname>
474      </entry>
475      <entry align="left" valign="top">
476      Controls which node on the tree is selected.
477      </entry>
478      <entry align="left" valign="top">
479      The default is <classname>CbcCompareDefault</classname>. Other comparison classes in <filename>CbcCompareActual.hpp</filename> include <classname>CbcCompareDepth</classname> and <classname>CbcCompareObjective</classname>. Experimenting with these classes and creating new compare classes is easy.
480      </entry>
481    </row>
482    <row>
483      <entry align="left" valign="top">
484      <classname>CbcCutGenerator</classname>
485      </entry>
486      <entry align="left" valign="top">
487      A wrapper for <classname>CglCutGenerator</classname> with additional data to control when the cut generator is invoked during the tree search.
488      </entry>
489      <entry align="left" valign="top">
490      Other than knowing how to add a cut generator to <classname>CbcModel</classname>, there is not much the average user needs to know about this class. However, sophisticated users can implement their own cut generators. <!-- what's the default? -->
491      </entry>
492    </row>
493    <row>
494      <entry align="left" valign="top">
495      <classname>CbcHeuristic</classname>
496      </entry>
497      <entry align="left" valign="top">
498      Heuristic that attempts to generate valid MIP-solutions leading to good upper bounds.
499      </entry>
500      <entry align="left" valign="top">
501      Specialized heuristics can dramatically improve branch-and-cut performance. As many different heuristics as desired can be used in CBC. Advanced users should consider implementing custom heuristics when tackling difficult problems. <!-- What's the default? -->
502      </entry>
503    </row>
504    <row>
505      <entry align="left" valign="top">
506      <classname>CbcObject</classname>
507      </entry>
508      <entry align="left" valign="top">
509      Defines what it means for a variable to be satisfied. Used in branching.
510      </entry>
511      <entry align="left" valign="top">
512      Virtual class. CBC's concept of branching is based on the idea of an "object". An object has (i) a feasible region, (ii) can be evaluated for infeasibility, (iii) can be branched on, e.g., a method of generating a branching object, which defines an up branch and a down branch, and (iv) allows comparsion of the effect of branching. Instances of objects include <classname>CbcSimpleInteger</classname>, <classname>CbcSimpleIntegerPseudoCosts</classname>, <classname>CbcClique</classname>, <classname>CbcSOS</classname> (type 1 and 2), <classname>CbcFollowOn</classname>, and <classname>CbcLotsize</classname>.
513      </entry>
514    </row>
515    <row>
516      <entry align="left" valign="top">
517      <classname>OsiSolverInterface</classname>
518      </entry>
519      <entry align="left" valign="top">
520      Defines the LP solver being used and the LP model. Normally
521      a pointer to the desired <classname>OsiSolverInteface</classname> is passed to <classname>CbcModel</classname> before branch and cut.
522      </entry>
523      <entry align="left" valign="top">
524      Virtual class. The user instantiates the solver interface of their choice, e.g.,
525      <classname>OsiClpSolverInterface</classname>.
526      </entry>
527    </row>
528    </tbody>
529  </tgroup>
530  </table>
531<para>
532There is not much about the classes listed in <xref linkend="least"/> that the average user needs to know about.
533</para>
534    <table frame="none" id="least">
535  <title>Classes Used by CbcModel - Least Useful</title>
536    <tgroup cols="3">
537    <thead>
538    <row>
539    <entry>
540    Class name
541    </entry>
542    <entry>
543    Description
544    </entry>
545    <entry>
546    Notes
547    </entry>
548    </row>
549    </thead>
550    <tbody>
551    <row>
552      <entry align="left" valign="top">
553      <classname>CbcBranchDecision</classname>
554      </entry>
555      <entry align="left" valign="top">
556      Used in choosing which variable to branch on, however, most of
557      the work is done by the definitions in <classname>CbcObject</classname>.
558      </entry>
559      <entry align="left" valign="top">
560      Defaults to <classname>CbcBranchDefaultDecision</classname>.
561      </entry>
562    </row>
563    <row>
564      <entry align="left" valign="top">
565      <classname>CbcCountRowCut</classname>
566      </entry>
567      <entry align="left" valign="top">
568      Interface to <classname>OsiRowCut</classname>. It counts the usage so cuts can gracefully vanish.
569      </entry>
570      <entry align="left" valign="top">
571      See <classname>OsiRowCut</classname> for more details. <!-- Default? -->
572      </entry>
573    </row>
574    <row>
575      <entry align="left" valign="top">
576      <classname>CbcNode</classname>
577      </entry>
578      <entry align="left" valign="top">
579      Controls which variable/entity is selected to be branch on.
580      </entry>
581      <entry align="left" valign="top">
582      Controlled via <classname>CbcModel</classname> parameters. Information from <classname>CbcNode</classname> can be useful in creating customized node selection rules.  <!-- Default? -->
583      </entry>
584    </row>
585    <row>
586      <entry align="left" valign="top">
587      <classname>CbcNodeInfo</classname>
588      </entry>
589      <entry align="left" valign="top">
590      Contains data on bounds, basis, etc. for one node of the search tree.
591      </entry>
592      <entry align="left" valign="top">
593      Header is located in <filename>CbcNode.hpp</filename>. <!-- Defaults? -->
594      </entry>
595    </row>
596    <row>
597      <entry align="left" valign="top">
598      <classname>CbcTree</classname>
599      </entry>
600      <entry align="left" valign="top">
601      Defines how the search tree is stored.
602      </entry>
603      <entry align="left" valign="top">
604      This class can be changed but it is not likely to be modified.<!-- Defaults? -->
605      </entry>
606    </row>
607    <row>
608      <entry align="left" valign="top">
609      <classname>CoinMessageHandler</classname>
610      </entry>
611      <entry align="left" valign="top">
612      Deals with message handling
613      </entry>
614      <entry align="left" valign="top">
615      The user can inherit from <classname>CoinMessageHandler</classname> to specialize message handling. 
616      <!-- Defaults? -->
617      </entry>
618    </row>
619    <row>
620      <entry align="left" valign="top">
621      <classname>CoinWarmStartBasis</classname>
622      </entry>
623      <entry align="left" valign="top">
624      Basis representation to be used by solver
625      </entry>
626      <entry align="left" valign="top">
627      <!-- Defaults? -->
628      </entry>
629    </row>
630    </tbody>
631  </tgroup>
632  </table>
633</section>
634
635  </chapter>
Note: See TracBrowser for help on using the repository browser.