Added a section on lower bounds to tutorial

Author: cicirello

dist/interactive-bin-packing-3.0.jar

@@ -34,13 +34,14 @@ <h2>Introduction</h2>
 <h2><a name="ToC"></a>Table of Contents</h2> 
 The remainder of this tutorial is organized as follows:
 <ul>
-<li><a href="#copt">Combinatorial optimization</a></li>
-<li><a href="#bin">Bin packing</a></li>
-<li><a href="#heur">Constructive heuristics</a></li>
-<li><a href="#ff">First-fit</a></li>
-<li><a href="#ffd">First-fit decreasing</a></li>
-<li><a href="#bf">Best-fit</a></li>
-<li><a href="#bfd">Best-fit decreasing</a></li>
+<li><a href="#copt">Combinatorial Optimization</a></li>
+<li><a href="#bin">Bin Packing</a></li>
+<li><a href="#bound">Lower Bounds</a></li>
+<li><a href="#heur">Constructive Heuristics</a></li>
+<li><a href="#ff">First-Fit</a></li>
+<li><a href="#ffd">First-Fit Decreasing</a></li>
+<li><a href="#bf">Best-Fit</a></li>
+<li><a href="#bfd">Best-Fit Decreasing</a></li>
 </ul>
 
 <hr>
@@ -170,6 +171,38 @@ <h2><a name="bin"></a>Bin Packing</h2>
 
 <hr>
 
+<h2><a name="bound"></a>Lower Bounds</h2>
+
+When solving a combinatorial optimization problem, it is sometimes useful
+to compute a lower bound for the objective function if the problem is a
+minimization problem, and similarly an upper bound if the problem is a
+maximization problem. Since Bin Packing is a minimization problem, we will
+explain the concept with a lower bound. A lower bound is a value that is 
+definitely less than or equal to the optimal value. It is often possible to
+compute a lower bound much easier than it is to compute the actual optimal
+solution. Although there is no guarantee that a solution exists whose value
+is that of the lower bound, if you are able to find a solution whose value is
+the lower bound, then you know there is no reason to search any further 
+for a better solution.
+
+<p>There is a very easy way to compute a lower bound for a Bin Packing instance.
+Simply sum the sizes of the items. And then compute the ceiling of that sum
+divided by the bin capacity. This lower bound makes the very naive assumption
+that it is possible to pack the items in bins so that there is no wasted space.
+You clearly can't do any better than this, although it is rarely possible to
+actually pack the bins in this way.</p>
+
+<p>In the Operations Menu of the application, there is a command "Compute
+Lower Bound" that computes a lower bound for the current instance. Use that
+command to compute the lower bound for the current Bin Packing instance.
+If you are still on the default instance, you will find that the lower
+bound is 5 bins. In this case, it turns out that the optimal solution is
+also 5 bins, but you have no way of knowing for sure at this point.</p>
+
+<p>Return to <a href="#TOP">Top</a> or <a href="#ToC">Table of Contents</a>.</p>
+
+<hr>
+
 <h2><a name="heur"></a>Constructive Heuristics</h2>
 
 Recall that the Bin Packing problem, as well as many other combinatorial optimization