Problem definition and Research question
Warehouses rely on order pickers moving through aisles to collect items for shop orders. Large retailers already use algorithms to decide how items are grouped onto load carriers (a process referred to as batching) and which route a picker takes in order to minimize travel time. These algorithms do not account for what happens when many pickers execute their routes at the same time, however: workers can end up needing the same aisle at the same moment, forcing them to wait for one another. This congestion adds time on top of what was planned and becomes more pronounced during busy periods.

This thesis treats batching and routing as fixed, since they are already handled by existing systems, and instead asks whether congestion can be reduced through a different lever: when, and to whom, pick orders are assigned. The resulting research question is: how can order-to-picker assignment be optimized to minimize peak aisle congestion in a warehouse?

Methodology
We formalize the problem as a scheduling problem, with pickers as parallel machines and pick orders, referred to as batches, treated as jobs consisting of a fixed sequence of aisle visits. We prove a simplified version, the Aisle Load Balancing Problem (ALBP), to be NP-complete through a reduction from graph edge colouring. An extended version builds on this with realistic constraints: batches visit aisles in sequence with individual processing times and must start within an earliest and latest allowed time drawn from real delivery deadlines.

Since solving this problem’s ILP exactly can take from seconds to hours, we developed two greedy heuristics as faster alternatives. The NeighborhoodN Greedy builds a schedule while actively avoiding congestion increases, using a limited look-ahead among already available batches. MaxM Greedy instead takes a maximum aisle load as a fixed input and schedules as many batches as possible within it.

We tested all methods, plus a benchmark reflecting current practice, on real data from two large distribution centres, across eight representative days each, in the pickzone with the highest realized number of pick orders.

Results
Across all sixteen instances tested, the ILP achieved the lowest maximum aisle load (𝑀 = 2, 3, 4), followed closely by MaxM Greedy (𝑀 between 3 and 6). Both stay well below the operational congestion threshold of six pickers per aisle. Current assignment practice, and its heuristic approximation of real-world practice (NeighborhoodN Greedy (𝑁 = 0)), reach an average close to 𝑀 ≈ 9 to 10, regularly exceeding this threshold.

There are several important caveats. The ILP achieves its result by using the full width of each batch’s allowed time window, which lengthens the schedule, though it never finishes a batch late. MaxM Greedy trades off differently: pushing its bound too low causes batches to finish too late instead. The ILP is also expensive and unpredictable to compute, ranging from 21 seconds to over two hours with no clear relation to instance size, while both heuristics run in under one second on every instance. This pattern—ILP strongest but slowest, MaxM Greedy fast and close behind, and current practice weakest—holds consistently across all eight days and both warehouses.

Conclusions
Aisle congestion can be reduced substantially through scheduling alone, without changing how orders are batched or routed, and this holds regardless of warehouse or workload. Achieving the largest reduction exactly with the ILP requires accepting a longer schedule or an unpredictable amount of computation time, limiting it to a benchmark rather than a daily tool. Achieving it approximately with MaxM Greedy gives up only a small amount of congestion reduction for a schedule produced in under a second and tunable to whatever aisle load a warehouse accepts. Of the methods tested, MaxM Greedy is the most realistic candidate for day-to-day use.

Further work
Several directions remain open. The model’s inputs could be made more realistic through finer norm time estimation, picker breaks, and a safety buffer against real-world delays such as forklift resupply operations. Two assumptions set in this thesis could be relaxed: a constant number of pickers throughout the day, and batching and routing that ignore congestion. The heuristics could also be extended, for instance by allowing a previously raised aisle load bound to decrease again as time advances. Finally, alternative objectives, such as minimizing makespan under a fixed congestion limit or a genuine bi-objective formulation producing a full tradeoff curve, would let a warehouse choose its own balance rather than the discrete points explored here. ... Read more

The magical "Cheney card trick" will be presented in this thesis using a mathematical approach. This trick is performed by a magician and an assistant in the following way: the magician leaves the room, and the assistant lets the audience draw 5 cards from a standard deck of 52 cards. Then, the assistant returns one of the 5 drawn cards to the audience and places the remaining 4 cards on the table. The magician can enter the room once the 4 cards have been placed on the table. The magician looks at the 4 cards on the table and, by these 4 cards, identifies the card that was given to the audience.
This thesis explores the underlying mathematical principles. The trick’s workings will first be explained, after which an algorithm for performing the trick will be given. In the analysis of the trick, we noticed that the size of the deck that we performed the trick with could be expanded. Given that we draw n cards from a deck of size d, we introduce an upper bound. This upper bound on the deck size is: d ≤ n! + n − 1. Furthermore, we introduce Birkhoff-von- Neumann’s theorem and Hall’s marriage theorem. Using these two theorems, we will prove that a convention for the magician and the assistant to perform the trick always exists when we attain the introduced upper bound. We will prove the existence of a convention and provide algorithms to perform the trick with a deck of cards attaining the introduced upper bound. While performing the trick, a specific order of the cards appears more often than others. When analysed, we find that a less preferable order of the cards is only necessary for approximately 10 per cent of the drawn hands. Additionally, a theorem of Gale-Shapley about stable and optimal matches is introduced, adding an extra dimension to Hall’s marriage theorem. Gale-Shapley’s algorithm is then introduced, and the possibility of finding a stable match between hands and messages is explored. Altogether, this thesis aims to find an answer to the question: What mathematical principles and ideas underlie the "Cheney card trick"? ... Read more