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Weekly Updates1. Introduction
A Distribution Center (DC) is an integral element of a logistics system. A DC is responsible for obtaining materials from different suppliers, performing value-added activities and assembling (or sorting) them to fulfill a number of different customer orders. Some of the major activities within a DC include the receipt of items and customer orders, storing items, order picking, shipping, customer service and reclamation, and control. Amongst these activities, order picking has been identified as the highest-priority activity in a DC for productivity improvements due to its relatively high (about 50%) contribution to the total DC operating cost (Tompkins et al., 2003). Order picking refers to an operation through which the items are retrieved from a storage location to fulfill customer orders. Some of the order picking work elements include traveling to the item, searching for the item, reaching and extracting the item from the storage, documenting the pick, sorting items, etc.
While designing an Order Picking System (OPS), a designer is always posed with the question: which OPS best meets a given set of objectives? Some of the objectives the designer is required to optimize include maximizing throughput or minimizing cost, space, response time and error rate. Optimization of one or more of these objectives requires a complex task of designing (or selecting) several parameters; e.g., forward-reserve area allocation, number and width of aisles, storage system configuration, storage policy (randomized, class based, volume based, family-based, Cube-per-Order Index (COI) based, etc.), picking strategy (discrete, batch or zone), picking method (manual, semi-automated or automated) and material handling equipment (order picker trucks, carousels, Verticle Lift Modules (VLMs), etc.).
In this paper we focus on picker blocking, a common factor that affects several OPS design parameters (e.g., width of aisles, storage system and picking strategy). Blocking can be prominent in a high-throughput system that requires a large number of pickers in the picking area. Blocking can lead to increased picker idle time and reduce their productivity.
We develop analytical models to estimate picker blocking in OPSs. In particular, our models concern a wide-aisle OPS, which is an OPS in which the aisles are wide enough to allow pickers to pass each other in the aisle. In such OPSs pickers can experience blocking at a pick-face when two or more pickers need to pick at the same pick face--we refer to this form of blocking as pick-face blocking. On the other hand, a narrow-aisle OPS is one in which aisles are too narrow to allow pickers to pass each other in the aisle. For such OPSs, pickers not only experience pick-face blocking, but also experience blocking due to their inability to pass other pickers in the aisle. This form of blocking is referred to as in-the-aisle blocking, models to estimate which have been developed by Gue et al. (2006). Figure 1 illustrates pick face blocking in wide-aisle OPSs and in-the-aisle blocking in narrow-aisle OPSs.
[FIGURE 1 OMITTED]
The remainder of this paper is organized as follows. In Section 2 we briefly review literature in the area of design of DCs and OPSs. In Section 3 we present the assumptions that we make in developing the blocking models. It is important to note that the ratio of the time to pick at a location to the time to walk past a location--referred to as pick:walk time ratio--is critical in modeling blocking since it defines how much the distance between pickers can change while one is picking and one is walking. In Section 4 we develop analytical models to estimate pick-face blocking, considering two extreme values of the pick: walk time ratios, for the case when pickers pick at most one Stock Keeping Unit (SKU) at a pick face. (SKU is a common term in industry, uniquely defining each product or item that is stocked in a DC.) Such an assumption of picking at most one SKU leads to no variance in the time to pick at a pick-face for each picker. Alternatively, in Section 5 we develop analytical models to estimate pick face blocking, again considering two extreme values of pick:walk time ratios, for the case when pickers may pick more than one SKU at a pick face. The possibility of picking more than one SKU leads to a higher value of variance in the time to pick at a pick-face for each picker. In Section 6 we summarize our understanding of the blocking phenomenon.
2. Literature review
In the area of overall DC (or a warehouse) design, Gray et al. (1992) propose a multi-stage hierarchical decision approach to model the composite design and operating problems for a typical order-consolidation warehouse. Their hierarchical approach utilizes a sequence of coordinated mathematical models to evaluate the major economic trade-offs and to prune the decision space to a few superior alternatives. Rouwenhorst et al. (2000) present a reference framework and a classification of warehouse design and control problems. They define a warehouse design problem as a "structured approach of decision making at a strategic, tactical, and operational level in an attempt to meet a number of well-defined performance criteria."
Specific to the design of an OPS in a DC, Brynzer et al. (1994) present a zero-based analysis methodology for the evaluation of OPSs as a base for system design and managerial decisions. Yoon and Sharp (1996) propose a structured procedure for the analysis and design of an OPS that considers the inter-dependent relationships between different functional areas (e.g., receiving, picking, sorting, etc.). The survey papers by Van den Berg (1999), Rouwenhorst et al. (2000) and Gu et al. (2007) provide a good review of various contributions in the area of DC and OPS design.
Ruben and Jacobs (1999) explore the relationship between five order-batching heuristics and three storage policies (randomized, class-based and family-based). Through simulation studies, they conclude that blocking plays a vital role in the simultaneous selection of order batching heuristics and storage policies.
To understand the effect of pick density (the probability that a picker will pick at a pick face) on blocking in narrowaisle OPSs, Gue et al. (2006) develop analytical models to estimate in-the-aisle picker blocking in such systems. They use discrete-time Markov chains to estimate bounds on the percentage of time each picker is blocked. The critical assumptions in their work include: the order picking aisle may be represented as a circle, the pick: walk time ratio is 1:1 or [infinity]:1, and pickers can make only one pick at a pick face. They perform simulation studies to estimate blocking for cases when the pick:walk time ratios are 5:1, 10:1 and 20:1. Through their results, the authors suggest that blocking increases with an increase in the number of pickers in narrow-aisle OPSs, but decreases with an increase in the picking area (for the same number of pickers). Skufca (2005) derives an analytical expression to estimate blocking in a circular warehouse with k workers walking at infinite speed for the case when the aisle is too narrow to allow passing and pickers pick at most one SKU at a pick face.
From our review of the OPS literature, we conclude that the only two contributions that develop analytical models to estimate blocking in an OPS are those by Skufca (2005) and Gue et al. (2006). However, both these contributions consider only a narrow-aisle OPS, in which pickers are unable to pass within the aisle. Many OPSs in the real-world have aisles wide enough to allow pickers to pass. Models to estimate blocking in such OPSs will aid a designer in deciding the width of the aisles and the possible implication of this decision on space utilization, storage system configuration and picking strategy selection. Furthermore, pickers may pick move than one SKU at a pick-face--a situation not considered in Skufca (2005) and Gue et al. (2006).
With this perspective, we next develop blocking models to estimate pick-face blocking in wide-aisle OPSs in which pickers pick one SKU (i.e., there is no variance in the time to pick) or more than one SKU (i.e., there is high variance in the time to pick) at a pick-face. For ease of presentation, a picker will be referred to as he as we develop our models.
3. Blocking models for wide-aisle OPSs
In developing the blocking models we make the following assumptions:
1. The order picking area consists of n pick-faces. A pick face might represent a column of pallet rack, a bay of flow-rack or a section of bin-shelving. The place at which a picker stops is considered the pick-face (however, in reality each pick-face might consist of several storage locations).
2. Pickers follow a traversal or S-shape routing policy, which means that they visit each aisle and travel through that aisle in only one direction. In practice, pick-faces are located on both sides of the aisle. However, for ease of modeling, we assume that the pick-faces are staggered and pickers travel from the left pick-face to right pickface in succession. Consequently, we assume the layout of the picking area may be represented as a (one-sided) circle.
3. If a picker picks I (0 < I < n) SKUs on average, then the probability with which a picker stops at a pick-face, p, can be calculated as p = I/n. Note that typically there will be multiple SKUs stored, at each pick location, each with a different activity level, the effect of which under this assumption is that the pick-density for each picking location is nearly equivalent. Note that this assumption may or may not hold for a particular order picking system.
4. We assume deterministic (average) times to pick ([t.sub.p]) and walk from one pick-face to another ([t.sub.w]). The time to pick ([t.sub.p]) represents the average time the picker is stopped; which may include the time spent picking multiple items (copies) of the same SKU. The time to walk ([t.sub.w]) represents the average time to walk past a location in our model (which is a one-sided circle) and so represents one-half the time to walk past one physical location in the two-sided real system.
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