The problem considered in this thesis is the box set optimization problem. In
this problem the goal is to find the optimal dimensions of a number of shipping
boxes which have to provide an optimal fit on a set of items. The application
considered is e-commerce, and hence the number of items is relatively large
compared to other applications found in literature.

The literature describes multiple problem formulations where a small num-
ber of optimal shipping boxes are selected from a large set containing candidate
boxes. However, since the instances considered in e-commerce application are
often too large to be solved in this way, we introduce a simulated annealing
heuristic solution method. In this way we are able to generate box sets con-
taining 10 to 20 boxes with centimeter accuracy for data sets containing up to
XX single item orders.

We pay special attention to an algorithm to very efficiently calculate the
objective value associated to the fit of a selection of candidate boxes on a set of
items. This is done by assigning items to shipping boxes using a pre-calculation
table. Because the heuristic can process a larger set of candidate boxes than
the exact solution method, ultimately results of the heuristic are better than
those found by the exact solution method.

Finally we generate multiple box sets based on the input data provided by
the industry partner bol.com, and compare the results. We conclude that on
almost all tested instances, the total amount of carton used can be reduced by
more than XX in combination with a total send volume reduction of over XX
in comparison with the box set that is currently used. Furthermore, we answer
additional questions. We find that minimizing the total amount of carton used,
instead of minimizing the total volume send, can lead to a result that is most
desirable in practice. We discover that pack-line specific, warehouse specific or
seasonal specific box sets provide none to only minor additional improvements.

Het plankprobleem gaat over het overdekken van convexe vormen met hypervlakken. De gegeven stellingen en bewijzen in het artikel \The plank problem for symmetric bodies" van Keith Ball zijn lastig te begrijpen voor bacholor wiskunde studenten. In dit verslag wordt verduidelijking gegeven van dit artikel zodat deze begrijpelijk en toegankelijk wordt voor anderen. Hierbij speelt de trace-class een belangrijke rol en komt er veel lineaire algebra over symmetrische matrices voorbij.