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Weekly Updates1. Warehouse design
A distribution center consists of several component subsystems, including receiving, storage, order picking and shipping. Perhaps the most common building block in these systems is the pallet storage area, which consists of storage racks, aisles between them and one or more pickup and deposit (P&D) points. In the academic literature this area is commonly called a "warehouse," and we adopt that terminology here. Because almost all products are received and stored in pallet quantities, pallet warehouses tend to consume the majority of space within a distribution center.
Warehouses in industry are typically comprised of single-or double-deep pallet racks arranged in parallel picking aisles, as in Fig. 1. In order picking warehouses, workers travel through aisles with picking carts (or perhaps ride forklifts, with empty pallets) and build orders by picking items or cases from the stored pallets. Large order picking warehouses usually have one or more cross aisles (Fig. 1, right), which tend to reduce the travel distance between successive picks in a tour (Roodbergen and De Koster, 2001).
[FIGURE 1 OMITTED]
In unit-load warehouses, which are the subject of our work, items are stowed and retrieved in pallet quantities, and each stow or pick is for a single pallet. Unit-load warehouses are used in at least two ways in a distribution center: (i) as order picking areas, where products are received and shipped in pallet quantities (distributors of groceries or appliances are two examples); and (ii) as reserve areas that replenish fast-pick areas (Bartholdi and Hackman, 2008). For example, a common arrangement is to have a fast-pick "module," consisting of a gravity-fed, carton flow rack, replenished from a pallet reserve area. Order pickers build detailed orders from the flow rack, while workers on lift trucks replenish product into the flow rack from the reserve pallet area. Unit-load operations are also common in crossdocking, where pallets are stored briefly before being loaded onto outbound trucks.
Unit-load warehouses may use single-command cycles, dual-command cycles, or both. In a single-command cycle, a worker accomplishes either a stow or a pick in each trip into the warehouse. Therefore, travel is to and from a single storage location. In a dual-command cycle, often referred to as "task interleaving," a worker visits two storage locations per trip into the space--first for a stow, then for a pick. Dual commands are more efficient with respect to travel, but they require concurrent receiving (putaway) and shipping (picking) operations and advanced IT systems to direct workers to picks. As a result, single-command operations are common.
Whether a unit-load warehouse uses single or dual commands determines, to some extent, the design of the storage space. If operations are exclusively single command, then a traditional middle aisle (see Fig. 1) confers no benefit. It is easy to see why: if the P&D point is along the lower boundary of the picking space, inserting a cross aisle moves approximately half the locations farther away. There is no corresponding benefit because the travel distance to every location is still the rectilinear distance from the P&D point. If operations are exclusively dual command, then a middle aisle can be beneficial, depending on the size of the warehouse (Roodbergen and De Koster, 2001). Our work addresses only single-command operations in unit-load warehouses.
We believe that all, or nearly all, unit-load warehouses in industry conform to two Unspoken Design Rules.
1. Picking aisles must be straight, and parallel to one another.
2. If present, cross aisles must be straight, and they must meet picking aisles at right angles.
Given these two rules and a rectangular warehouse space, the designs in Fig. 1 are probably the only options (large warehouses may have more than one middle aisle). However, both designs tend to limit productivity in a single-command unit-load warehouse. This leads us to ask
How should cross aisles and picking aisles be arranged to minimize the expected distance to pick in a single-command unit-load warehouse?
We answer this question with two design models, which produce designs that, to our knowledge, were not found in industry before our work, but that offer significant reductions in expected travel distances over traditional designs.
2. Literature review
The aisle design problem is the first of three related problems in warehouse design. The second is product allocation, which seeks to put products in the right locations. The third is the order picker routing problem (and order batching problem, if appropriate), which determines the best sequence of locations for a worker to visit when building orders.
Fundamental relationships for the length and width of a rectangular warehouse are in Tompkins et al. (2003) and Heragu (2006). Francis (1967) investigated rectangular warehouse shapes to minimize picking and construction costs. He assumed rectilinear travel paths, which "presupposes that there is an orthogonal network of aisles running parallel to the x and y axes." Bassan et al. (1980) developed models to determine when it is best to align picking aisles horizontally or vertically in a warehouse, but they assume the traditional structure of Fig. 1, with all picking aisles parallel. Berry (1968) noticed that floor-stored pallets should be arranged in lanes with different depths, based on demand characteristics for the Stock Keeping Unit (SKU), and that different lane depths can be arranged to form "diagonal gangways" in the storage space. He did not explore the implications of this observation.
Especially relevant to our work is White (1972), which proposes non-rectangular warehouses with two or more "radial aisles" projecting away from a single P&D point. His goal was to "approximate Euclidean efficiencies." Radial aisles are similar to the "fish bone" designs we propose below, however, our work is different than White's in a number of ways.
1. White's model is descriptive, taking as input a particular design and producing as output expected travel distance. Our models are prescriptive: they take as input some system characteristics, such as the distance between aisles and the length and width of the space, and they produce as output an aisle design that minimizes expected travel distance.
2. Our designs adhere to the industry standard of rectangular picking spaces, which we believe makes them more likely to be implemented in practice.
3. We model the picking space as a set of discrete aisles, whereas White models it as a continuous space. This makes our model slightly more accurate.
4. Our models account for the width of the cross aisle and its effects on travel distances; White's model assumes aisles have zero width. This is an important difference because, in our experience, the first objection of managers to new aisle designs is the effect of cross aisles on storage density.
We should also add that White does not address the best shape for a single cross aisle when picking aisles are parallel, which we describe below.
The warehouse aisle design problem is similar in principle to street design in an urban area. Arlinghaus and Nystuen (1991) mention the effect of a diagonal link in an otherwise rectangular grid network, but they do not offer a model. Their concern is the interaction between pedestrians and automobiles. A famous example of "diagonal travel" in an urban setting is Broadway in Manhattan, which affords a benefit over traveling streets and avenues exclusively.
Product allocation problems in warehouses are of two main types: allocating products among areas in a warehouse, and allocating products to locations within those areas. The first type includes work on the forward reserve problem (e.g, Hackman and Rosenblatt (1990) and Bartholdi and Hackman (2008) and more general product allocation models (Heragu et al., 2005). Product allocation problems of the second type are based on the well-known cube-per-order index rule (Heskett, 1963; Kallina and Lynn, 1976), which assigns products with the highest activity per location to the best locations. In a warehouse similar to the one we study, this leads to storing the "fast movers" in a triangular pattern around the P&D point (Francis et al., 1992; Tompkins et al., 2003).
Operationally, product allocation models assume some form of dedicated storage, which reserves particular locations for particular SKUs. Dedicated storage is common in order picking warehouses because such a policy tends to reduce labor costs, which is the primary design concern. In unit-load warehouses, dedicated storage is less common because it tends to reduce storage density, which is a primary design concern. Many unit-load warehouses--most, in our experience--use a storage policy in which any product may occupy any location, depending on availability at the time of storage (e.g., "closest-open location" or randomized policies). Our models assume randomized storage, which was shown by Schwarz et al. (1978) to approximate closest-open location.
The seminal work in routing order pickers is by Ratliff and Rosenthal (1983), who showed that routing a worker to pick several items from a rectangular order picking warehouse (Fig. 1, left) is a solvable case of the Traveling Salesman Problem. Roodbergen and De Koster (2001) extended Ratliff and Rosenthal's results to the case of rectangular picking areas with one or more cross aisles (Fig. 1, right) in the middle of the picking space. They found that having a "middle aisle" is not beneficial when retrieving a single item, for reasons we have already discussed. For "reasonably sized" pick lists, a cross aisle allows shorter tours. For very large pick lists, which are not uncommon in industry, the cross aisle again confers a dis-advantage because nearly every aisle is traversed in its entirety, and the cross aisle effectively makes the picking aisles longer. Vaughan and Petersen (1999) used heuristic routing techniques to determine the number of cross aisles in a picking area.
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