## Multiple Slot Heights in Unit-Load Warehouses

In a unit-load warehouse, products are stored and retrieved in pallet quantities. Examples of unit-load warehouses are retail distribution centers (DC), 3rd-party DCs, and transshipment hubs in freight transportation. Operations are commonly divided into three main functions: receiving, bulk storage, and shipping. The flow of products typically follows a well-defined path. Pallets arrive in truck loads at the receiving docks. From there, lift truck operators unload the pallets and transfer them to the receiving staging area. Then, the pallets are processed and stored in the bulk storage area, which consists of pallet racks arranged to form parallel picking aisles. When customer orders arrive, lift truck operators pick the pallets from the bulk storage area and move them to the shipping staging area. Finally, the pallets are prepared for shipping and loaded into the trucks.

From the areas of the warehouse, bulk storage generally represents most of its footprint and space related costs are one of the most critical resources to administer in unit-load warehouses. Hence, maximizing space utilization — cubic feet of product divided by the total cubic feet available — is of the utmost importance.

Luis F. Cardona and Kevin Gue, recently published a research paper that takes aim at a universal but often overlooked source of waste — the space between the top of a pallet and the rack above it. Pallet heights are variable, and they change through time. However, most warehouses set up their racks with all slots having the same height. This creates unused vertical space between the top of the pallet and the bottom of the slot above.

#### Model

We propose a method to generate the slot profile of a storage rack using multiple slot heights. A slot profile is the composition of slot heights and corresponding quantities of the racks — e.g., 100 slots 4000 high and 250 slots 7200 high. Our method analyzes historical inventory levels and pallet heights to determine a slot profile that maximizes space utilization and guarantees a desired storage service level.

The design process is defined by three questions:

- What are the best heights for different slots?
- How many different slot types should the rack system have? and
- How many slots of each type?

The height of the slots will be determined by the distribution of pallet heights and how this changes over time. The graph below shows the distribution of pallet heights at an instance in time from a warehouse we visited.

Pallet heights vary from 6 to 72 inches, while most pallets were between 30 and 50 inches. If the warehouse were to have only one slot type, it would have to be 72 inches, leaving unused vertical space when storing smaller pallets. We formulated a mathematical model that determines the optimal set of heights to minimize the loss of vertical space. Our model considers that selective racks in industry can only be adjusted in increments of two inches, as you can see below.

For each potential value of slot heights, our method calculates the minimum slot quantities that will provide the desired service level. In practice, service level in a unit-load warehouse is the ability to store an arriving pallet. The fundamental principle is that a pallet can be stored in any slot of height greater than its own, but cannot be stored in a slot of lesser height. In our model, service level is the probability that all pallets present during an operational shift can be stored, which is similar to service level Type I in inventory theory. We assume discrete time periods during which a storage requirement is known or forecast. We call this period an operational shift,” which could be a day, an 8-hour shift, or a 10-minute interval. All pallets in the warehouse during an operational shift are served if they can be assigned to slots; otherwise, there is a failure to serve or, in inventory terms, a stockout.

In a warehouse with one slot type, service level is calculated as the probability that the number of pallets to be stored in the warehouse in an operational shift is less than or equal to the number of slots. In a warehouse with multiple slot types, we need to consider each category of slots first. For example, the service level of slots 50 inches high is the probability that the number of pallets with height less than or equal to 50 inches to be stored in the warehouse in an operational shift is less than or equal to the number of slots of 50 inches; and so on, for all slot heights. Then, the service level of the warehouse is the probability there will not be a stockout of slots of any height in an operational shift. In this way, we build a mathematical model for the service level of a warehouse with multiple slot heights and use it to calculate the slot quantities. The graph below shows an example of this dynamic.

#### Benefits

To illustrate the potential benefits of using multiple slot heights, we simulated 3.600 random scenarios based on a data set of five distribution centers. We define benefit of using multiple slot heights as 100% minus ratio of the space requirements of a slot profile with multiple slot heights to that of a slot profile with one slot height — we chose six slot types to be the maximum number of slot heights tested because in preliminary studies, we noticed that for more slot types the expected benefit flattens in most cases. The graph below presents the results. For example, a slot profile with five slot types requires 38.7% less space on average than a slot profile with one slot height.

#### Use it

We implemented the model in a user friendly webtool tailored for practitioners.

##### https://www.warehousingtools.com/

Our experiments indicate significant potential benefits of using multiple slot heights; however, there are situations in which they may not be ideal. If the range of pallet heights is small, then using multiple slot heights is unlikely to bring much benefit. A warehouse with very few skus is not a good candidate for our models because the Gaussian approximation of aggregate inventory levels could be far from what happens in practice. Finally, distribution environments with unpredictable mixes of skus are also not good candidates, including warehouses with highly variable inventory levels, or many skus with strongly non-stationary or seasonal demands.

Multiple slot heights require additional operational procedures to guarantee the pallets are assigned to feasible slots and are retrieved from the most convenient locations. This implies modifications in the warehouse management system and additional administrative effort from warehouse associates. The best number of slot types will be determined by the trade-off between better space utilization and additional administrative effort. Despite these limitations, multiple slot profiles offer significant benefits to warehouses that meet, to a reasonable extent, the assumptions of the model.

If you want to understand all the details and implications, this is the full paper.

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