SCIP Doxygen Documentation: examples/Binpacking/doc/xternal

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 /*         SCIP --- Solving Constraint Integer Programs                      */
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 /**@file   xternal_binpacking.c
  * @brief  The bin packing example of SCIP
  * @author Timo Berthold
  * @author Stefan Heinz
  */
 
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 /**@page BINPACKING_MAIN Overview
  * @author Timo Berthold
  * @author Stefan Heinz
  *
  * This example contains a branch-and-price approach for the binpacking problem which is realized with the framework
  * \SCIP. Therefore, the following plugins are implemented:
  *
  * - a \ref reader_bpa.c "problem reader" which parses the problem out of file and creates the corresponding problem within \SCIP
  * - a \ref probdata_binpacking.c "(global) problem data structure" which contains all necessary information
  * - a \ref pricer_binpacking.c "pricer" which generates new variables/columns during the search
  * - the \ref branch_ryanfoster.c "Ryan/Foster branching rule"
  * - a \ref cons_samediff.c "constraint handler" which handles the branching decisions of the Ryan/Foster branching
  * - a \ref vardata_binpacking.c "variable data structure" which stores information for each variable and is needed to perform the Ryan/Foster
  *   branching
  *
  * In the following we introduce the problem, explain the use of the reader plugin and pricer plugin. Finally, we
  * introduce the Ryan/Foster branching rule and briefly discuss how that specific branching rule is realized within
  * the framework \SCIP.
  *
  * -# \ref BINPACKING_PROBLEM "Problem description"
  * -# \ref BINPACKING_READER "Parsing the input format and creating the problem"
  * -# \ref BINPACKING_PROBLEMDATA "Main problem data"
  * -# \ref BINPACKING_PRICER "Pricing new variables"
  * -# \ref BINPACKING_BRANCHING "Ryan/Foster branching"
  * -# \ref BINPACKING_MAKEFILE "The Makefile"
  *
  */
 
 /**@page BINPACKING_PROBLEM Problem description
  *
  * The binpacking problem consists of the task to distribute a given set of items \f$ [n] := \{1, \dots, n\}\f$ with
  * nonnegative size \f$s_i\f$ to a minimal number of bins, all of the same capacity \f$\kappa\f$.  As example consider 9
  * items with sizes: 2, 1, 2, 1, 1, 2, 3, 2, and 1 and a bin capacity of \f$\kappa\f$ of 4. The following pictures show
  * a feasible solution which needs 5 bins. The minimum number of bins needed for that example is 3.
  *
  * <CENTER>
  * \image html binpacking.png
  * </CENTER>
  *
  * This problem can be formulated as a set covering problem as discussed by Gilmore and Gomory in the following two classical papers:
  * - Gilmore P. C., R. E. Gomory: A linear programming approach to the cutting-stock problem. Operations Research 9 (1961): 849-859.
  * - Gilmore P. C., R. E. Gomory: A linear programming approach to the cutting-stock problem - Part II. Operations Research 11 (1963): 863-888
  *
  * We introduce a binary variable \f$x_{S}\f$ for each feasible packing \f$S\f$. A <b>packing</b> \f$S\f$ is an
  * assignment vector \f$ \lambda_{S}\in\{0,1\}^n \f$ which states the items belonging to that packing. It is
  * <b>feasible</b>, if and only if the total size of the items contained in this assignment is not greater than the
  * given capacity \f$\kappa\f$. Let \f$\mathcal{S}\f$ be the set of all feasible packing, this measns:
  *
  * \f[
  *    \mathcal{S} := \{S\subseteq [n] \mid \sum_{i:i\in S} s_{i} \leq \kappa \}
  * \f]
  *
  * An integer program can be formulated as follows:
  *
  * \f[
  *  \begin{array}[t]{rll}
  *    \min & \displaystyle \sum_{S \in \mathcal{S}} x_{S} \\
  *         & \\
  *    subject \ to & \displaystyle \sum_{S \in \mathcal{S}} (\lambda_{S})_{i}x_{S} \ge 1 & \quad \forall i \in \{1,\dots,n\} \\
  *         & \\
  *         & x_{S} \in \{0,1\} & \quad \forall S \in \mathcal{S} \\
  *  \end{array}
  * \f]
  *
  * This means we are searching for a set of packings such that each item is contained in at least one of the selected
  * packings. Since the objective is to minimize the number of used packings, each optimal solution to the above problem
  * can be transformed into a solution where each item is packed exactly once with the same cost.
  *
  *
  * Since \f$\mathcal{S}\f$ can be of exponential size, we are using a column generation approach to solve this
  * problem. We initialize the (master) problem with a set of \f$ n \f$ variables representing packings of a single item
  * per bin.  Now, we have to iteratively search for variables representing "better" packings, i.e., a packing pattern
  * which reduces the overall cost. For a given solution \f$y^\star\f$ of the (restricted) dual linear program, we have
  * to ﬁnd a variable/packing \f$ \lambda_{S} \f$ for which the reduced costs is negative. This means:
  *
  * \f[
  *     c_{S} - \sum_{i=0}^n (\lambda_S)_i y_i^\star < 0.
  * \f]
  *
  * Since all variables \f$ \lambda_{S} \f$ have an objective coefficient \f$ c_{S} = 1 \f$ the above condition is
  * equivalent to
  *
  * \f[
  *     \sum_{i=0}^n  (\lambda_S)_i y_i^\star > 1.
  * \f]
  *
  *
  * To find such a variable/packing we solve the following integer program:
  *
  *  \f[
  *  \begin{array}[t]{rll}
  *       \max & \displaystyle \sum_{i=1}^n (\lambda_S)_i y^\star_i\\
  *        & \\
  *        subject \ to & \displaystyle \sum_{i=0}^n (\lambda_S)_i s_i \leq \kappa \\
  *        & \\
  *        & (\lambda_S)_i \in \{0,1\} & \quad \forall i \in \{ 1, \dots , n \} \\
  *  \end{array}
  * \f]
  *
  * where \f$ (\lambda_S)_i \f$ for \f$i\in\{1,\dots,n\}\f$ are binary variables and \f$y^\star_i\f$ given by the dual
  * solution of the restricted master problem.
  *
  * The above problem is a knapsack problem which can be solved via dynamic programming or by solving the above integer
  * program. In this example we implemented a pricer which solves the integer program.
  *
  */
 
 
 
 /**@page BINPACKING_MAKEFILE The Makefile
  *
  * The Makefile is based on the main \SCIP Makefile. This means that all compiling options which are
  * available for \SCIP are also available for the binpacking project. Below, you find a list
  * of the most important compiling flags, the values they can take, and a short description. The
  * values in bold face are the default values.
  *
  * - <code>LPS={clp | cpx | none | <b>spx</b>}</code>
  *   <br>
  *   Defines the linear program solver to use:
  *   - <code>clp</code> use COIN-OR CLP
  *   - <code>cpx</code> use IBM CPLEX
  *   - <code>none</code> no LP solver
  *   - <code><b>spx</b></code> use SoPlex
  *
  * - <code>OPT={dbg | <b>opt</b>}</code>
  *   <br>
  *   Defines if the projects gets compiled in debug (<code>dbg</code>) mode or
  *   optimized (<code><b>opt</b></code>) mode. In the debug mode all assertions are checked.
  *
  * - <code>ZIMPL={false | <b>true</b>}</code>
  *   <br>
  *   Defines if the modeling language ZIMPL should be linked to binary or not.
  *
  * In the following we explain the all <b>Makefile targets</b>.
  *
  * - <b>lint</b>
  *   <br>
  *   Statically checks the code for uninitialized variables and many other possible problems,
  *   which even do not lead to compiler errors. For this,
  *   the external tool flexelint is needed. The call produces the file <code>lint.out</code>
  *   which contains all the detected warnings. From the experience when developing \SCIP, we strongly
  *   recommend to use such a code checker. It is always a surprising the stuff such tools detect.
  *   <br>
  *
  * - <b>clean</b>
  *   <br>
  *   Remove all objective files, libraries, and binaries.
  *   <br>
  *
  * - <b>test</b>
  *   <br>
  *   Starts an automated test run based on the SCIP test runs (see \ref TEST "How to run automated tests with SCIP").
  *   <br>
  *
  * - <b>tags</b>
  *   <br>
  *   Generates tags which can be used in the editor <b>emacs</b> and <b>xemacs</b>.
  *   <br>
  *
  * - <b>depend</b>
  *   <br>
  *   Generates the dependencies for the compiling process.
  */