Travel Models for Warehouses with Task Interleaving

Transcription

1 Proceedings of the 2008 Industrial Engineering Research Conference J. Fowler and S. Mason, eds. Travel Models for Warehouses with Task Interleaving Letitia M. Pohl and Russell D. Meller Department of Industrial Engineering University of Arkansas, Fayetteville, Arkansas 72701, USA Kevin R. Gue Department of Industrial & Systems Engineering Auburn, Alabama 36849, USA Abstract Operations in unit-load warehouses include single-command cycles and dual-command cycles, where storage and retrieval operations are interleaved. We model dual-command travel in two warehouse layouts that occur commonly in industry, and compare the efficiency of the two designs. General guidelines for optimizing the two aisle layouts are provided. Keywords Aisle design, warehouses, operations research 1 Introduction Unit-load warehouses receive and ship material or products in single discrete units, usually as pallet loads. They are commonly found in industry; examples include third-party transshipment warehouses, grocery and beverage distributors, appliance manufacturers and import distribution centers. Many break-bulk facilities, which break down a pallet into smaller volume packages for shipment, also have a portion of their activity dedicated to unit-loads, such as the reserve area that is used to receive and store pallets until they are needed to replenish a forward, or fast-pick area [1]. Travel from a common pickup/deposit () point to a single pallet location and back again to execute a storage or retrieval request is referred to as a single-command cycle. Since interleaving storage and retrieval requests (referred to as task interleaving or a dual-command cycle when the cycle includes one storage and one retrieval request) makes more efficient use of time and resources, and contemporary unit-load warehouses utilize warehouse management systems to schedule activities, the use of dual-command operations is more common than in the past. Even with the more efficient dual-command cycles, storage and retrieval operations are labor-intensive and account for a significant portion of warehouse operating costs, particularly in the very large unit-load warehouses of today. For this reason, we focus on optimizing the design of unit-load warehouses by minimizing expected travel distances. Figure 1 shows three conventional warehouse designs, with parallel picking aisles and orthogonal cross aisles. Layout A has two cross aisles, while Layouts B and C have three cross aisles. (a) Layout A (b) (c) Figure 1: Conventional Warehouses with Parallel Aisles is simply Layout A with the addition of a middle cross aisle. There is a penalty to pay for adding a middle aisle, since it reduces floor space that could otherwise be allocated to storage, or requires a slightly larger 869

2 facility to maintain the same amount of storage. For a warehouse that performs strictly single-command cycles, Layout B is actually less efficient than Layout A. Roodbergen and de Koster [6] point out that for a given number of picking locations, adding a middle cross aisle increases the expected travel to a single pick, since half of the locations are now farther from the point. However, they show that for practically-sized picklists, a middle cross aisle will provide savings in travel distance because it creates more possible routes for a pick tour. Because of this, Layouts B and C are potentially more efficient for the travel between pick locations than Layout A, and are therefore potentially better choices for facilities that perform dual-command operations. In this paper, we investigate Layouts B and C by developing expressions for dual-command travel in each and providing some guidelines for determining when each layout is preferable. Early researchers have modeled single-command travel in Layout A [3, 2] and [2] and have presented some well-known results on optimal warehouse shape and location. The few papers that model dual-command travel in Layout A [5, 4], do not apply their results to optimizing warehouse design. We are unaware of any published analytical models for optimal travel paths in. The papers that consider focus on orderpicking and use simulation to estimate expected travel [6, 8, 10] or model heuristic travel paths [9]. The contribution of this research lies in the development of dual-command travel models for optimal paths in Layouts B and C, and the resulting insight we gain from the analysis of our models. Our results provide practical tools for optimizing the design of unit-load warehouses, particularly those that perform dual-command operations, or a combination of single and dual-command operations. In Section 2 we model expected single and dual-command travel in two warehouse types (Layouts B and C), assuming a random storage policy and optimal travel paths. These analytical expressions allow us to compare the efficiency of the two designs for dual-command travel in Section 3. Section 4 presents our conclusions and thoughts on future research. 2 Expected Dual-Command Travel 2.1 : Picking Aisles Perpendicular to the Front Wall The warehouse in Figure 2(a) has parallel picking aisles that are perpendicular to the front wall (where the point is located). There are cross aisles at the top and bottom of the warehouse, and a middle cross aisle of width 2v, half-way between the top and bottom cross aisles. The travel paths along the aisles are indicated by solid black lines, where the distance between picking aisles is a. We assume that aisles are sized such that the racks on each side of the picking aisle can be accessed, but the lateral travel within the picking aisle is negligible. Workers can travel in either direction in an aisle and change directions within an aisle. We are interested only in travel distance, therefore the time to insert or extract pallets is considered to be a fixed cost. The warehouse is modeled as a set of discrete picking aisles, with continuous picking activity in each aisle, and picking uniformly distributed within and among all aisles. The storage and retrieval requests are assumed to be independent, and processed on a FCFS (first come, first served) basis. Two pick locations are indicated in black in Figure 2(a), where the pick on the left is a distance x from the bottom of aisle i, and the pick on the right is a distance y + 2v from the bottom of aisle j. The total picking length of each aisle is L and the distance that must be traveled to enter a picking aisle from either the bottom or top cross aisle is v, or half the width of the cross aisle. The expected dual-command travel can be expressed as the sum of the expected single-command travel and the expected travel-between, E[DC] = E[SC] + E[T B]. To determine E[SC], we consider two components of travel: cross aisle travel, and picking aisle travel (including the distance v to enter/exit the picking aisles, if necessary). Since we assume uniform picking activity, the expected picking aisle travel in a single-command cycle is L + 4v. In order to determine the expected cross aisle travel we must make an assumption for the location and whether there is an odd or even number of aisles, where n represents the number of aisles. We assume the point is optimally located in the middle of the bottom cross aisle [7], as shown in Figure 2(a), and that there are an odd number of aisles. We derive the expected cross aisle travel as a(n 2 1)/2n. (The derivation for an even number of aisles is similar.) Thus, the expected single-command travel for with an odd number of aisles is E[SC B ] = L + 4v + a(n2 1). 2n To determine the expected travel between two random picks, or E[T B], we again consider two components: cross 870

3 a v x 2v aisle i aisle j aisle i 2v x y+2v aisle j v y+2v (a) (b) Figure 2: (a) : Picking Aisles Perpendicular to Front Wall (b) : Picking Aisles Parallel to Front Wall aisle travel, and picking aisle travel. The expected cross aisle travel is and the expected picking aisle travel is a(n 2 1), (1) ( ) 1 L n 3 + v + n 1 ( ) 5 n 12 L + 2v. (2) The derivations of (1) and (2) are found in the appendix. The total expected travel between two picks is then E[T B B ] = 1 [ ( )] L 5 + v + (n 1) n 3 12 L + 2v + a(n2 1), (3) and the expected dual-command travel for is ( ) ( ) 17n 1 6n 1 E[DC B ] = L + v + a 12n n ( 5n 2 5 6n ). (4) 2.2 : Picking Aisles Parallel to Front Wall, represented in Figure 2(b), is similar to except that the picking aisles are parallel to the front wall of the warehouse where the point is located, and the cross aisles are perpendicular to the front wall. The two layouts can actually be viewed as the same general design with the point moved to another position. As we will see, however, this change in the position of the point has a significant impact on the optimal shape of the warehouse, therefore it is beneficial to our comparison to view Layouts B and C as two distinct designs. As before, the warehouse has n picking aisles, each with picking length L, and cross aisles of width 2v. Given a point at the bottom center of the warehouse, as shown in Figure 2(b), the expected single-command travel for is E[SC C ] = L + 2v + an. 2 The expected travel between two picks in is the same as the expected travel-between in, where 871

4 E[T B C ] is given by (3). The resulting dual-command travel for this warehouse is then ( ) ( ) ( 11n 1 2n 1 4n 2 ) 1 E[DC C ] = L + v + a. (5) 12n n 3 Comparison of Layouts B and C To compare Layouts B and C, we evaluate their performance for fixed values of total picking length (a surrogate for warehouse storage capacity), which we define as T = nl. For each value of T, we vary the number of aisles n and evaluate E[SC] and E[DC]. We find that for single-command travel the optimal shape for both Layouts B and C is a warehouse that is approximately twice as wide as it is tall, or has a shape factor of height/width 0.5. This is a well-known result discussed by several authors (e.g. [3, 2]). Figure 3 shows the value of E[DC] for both designs over a range of shape factors for three different values of T. We note in Figure 3 that E[DC] appears to be a convex function with respect to the shape factor, and that the performance of Layouts B and C are very similar. For each of the three values of T, the optimal shape factor for both designs is approximately 0.6. While the shape of the two optimal warehouses is similar, will have considerably fewer, but longer aisles than Layout A. E [DC ] T = Shape Factor - height/width (a) E [DC ] 200 T = Shape Factor - height/width (b) E [DC ] 320 T = Shape Factor - height/width (c) Figure 3: Comparison of E[DC] for Layouts B and C: (a) T = 300 (b) T = 1000 (c) T = 3000 Figure 4 compares E[SC], E[T B] and E[DC] for Layouts B and C, over a range of values of T, where the warehouse shapes evaluated have all been optimized for E[DC]. The performance of the two designs is similar, therefore in order to discriminate between the two, we plot the performance ratio of each design, which as an example, is defined for E[SC B ] as E[SC B ]/ min(e[sc B ], E[SC C ]). Note in Figure 4(a) that is preferable for singlecommand travel, and in Figure 4(b) that is more efficient for travel-between. The resulting difference for dual-command travel is shown in Figure 4(c), where is preferred for small warehouses (T < 1500), while is preferred for large warehouses (T > 1500). The difference between the two designs is small for dualcommand travel, being within 1% of each other for values of T > Conclusions and Future Research This paper presents dual-command travel models for two aisle layouts that are commonly found in industry. We determine for each layout the shape that minimizes the expected dual-command travel and compare the performance of optimized designs over a range of warehouse sizes. While is preferred for small warehouses and for larger warehouses, we note that the dual-command performance and optimal warehouse shapes are very similar for the two layouts, thereby making them equally-viable design alternatives for unit-load warehouses. Future work in this area could include a more thorough investigation into the optimal design of the aisle structure, such as defining the optimal location of the middle cross aisle in, determining when is preferable to Layout A, and deriving expressions for the optimal number of aisles. 872

5 E [SC ] PR Total Picking Length (a) E [TB ] PR Total Picking Length (b) E [DC ] PR Total Picking Length (c) Figure 4: Performance Ratio (PR) Comparisons of Layouts B and C for: (a) Single-Command Travel (b) Travel- Between and (c) Dual-Command Travel Appendix Since the aisles are equivalent in length, all aisles are equally likely to contain one of the two picks. If the number of aisles is n, there are n 2 possible combinations of i and j, all equally likely. For instance, there are n possible ways that i = j, for which there is no cross aisle travel required. Therefore, the probability that i j = 0 is n/n 2. Likewise, there are 2(n 1) ways that i j = 1, so P r( i j = 1) = 2(n 1)/n 2. The expected number of aisle widths between i and j is n 1 n 1 2(n k) E[ i j ] = k P r( i j = k) = k n 2 = (n2 1). k=0 The expected cross aisle travel is then a(n 2 1). (1) For picking aisle travel, X and Y are random variables, such that X U(0, L) and Y U(0, L). From order statistics, we know the expected distance between two picks on the same aisle of length L is E[ x y ] = L/3. When the two picks are in the same picking aisle, we expect to traverse the middle cross aisle half the time, therefore, the expected travel is L/3 + 2v(1/2) = L/3 + v. For the purpose of this derivation, we assume for a moment that there are only two cross aisles at the top and bottom of the warehouse. For picks that are on different picking aisles, there are therefore only two alternative paths, with the optimal path equal to min[x + y, 2L (x + y)] + 2v. We let Z = X + Y. The convolution of two identical uniform density functions is triangular. Using the probability density function of Z, f Z (z), we find that the expected value of min[x + y, 2L (x + y)] is then E[min(z, 2L z)] = 2L 0 min(z, 2L z)f Z (z)dz = k=1 L 0 zf Z (z)dz + 2L L (2L z)f Z (z)dz = 2 3 L. Now we consider again, with three cross aisles. To evaluate picking aisle travel when the picks are in different aisles, we condition on whether the picks are above or below the middle cross aisle and consider four mutually exclusive cases: (1) x L 2, y L 2 (2) x > L 2, y > L 2 (3) x L 2, y > L 2 (4) x > L 2, y L 2. In case 1, both picks are below the middle cross aisle, therefore the shortest path uses either the bottom cross aisle or the middle cross aisle. The picking aisle travel is this case is min[x + y, L (x + y)] + 2v. Applying the results above for E[min(z, 2L z)] (with an aisle length of ) yields expected picking aisle travel of L/3 + 2v for case 1. In case 2, both picks are above the middle cross aisle, and by similarity with case 1, the expected picking aisle travel 873

6 is also L/3 + 2v. In cases 3 and 4, we have one pick above the middle cross aisle and one pick below. In these cases it is always optimal to use the middle cross aisle, therefore the expected picking aisle travel is x y + 2v. For case 3, E [ Y X X L 2, Y > L 2 ] = E [ Y Y > L 2 ] E [ X X L 2 ] = 3L 4 L 4 = L 2, therefore the expected picking aisle travel is + 2v. By similarity, this applies for case 4 as well. Since cases 1 4 are equally likely, the expected picking aisle travel when the picks are on different aisles is [ ( ) L 1 + L ( ) 1 + L ( ) 1 + L ( )] 1 + 2v = L + 2v. The probability the second pick will be in the same aisle as the first pick is 1/n, and the probability the second pick will be in a different aisle is (n 1)/n. The expected picking aisle travel is then ( ) 1 L n 3 + v + n 1 ( ) 5 n 12 L + 2v, (2) and the total expected travel between two picks in the warehouse of Figure 2(a) is E[T B] = 1 [ ( )] L 5 + v + (n 1) n 3 12 L + 2v a(n + 1)(n 1) +. (3) Acknowledgements This research was supported in part by the National Science Foundation under Grants DMI and DMI References [1] Bartholdi, III, J. J., and Hackman, S. T., 2007, Warehouse & Distribution Science, Release 0.85, www. warehouse-science.com. [2] Bassan, Y., Roll, Y., and Rosenblatt, M. J., 1980, Internal Layout Design of a Warehouse, AIIE Transactions, 12, 4, [3] Francis, R. L., 1967, On Some Problems of Rectangular Warehouse Design and Layout, Journal of Industrial Engineering, 18, 10, [4] Malmborg, C. J., and Krishnakumar, B., 1987, On the optimality of the cube per order index for conventional warehouses with dual command cycles, Material Flow, 4, [5] Mayer, Jr., H. E., 1961, Storage and retrieval of material, The Western Electric Engineer, 5, 1, [6] Roodbergen, K. J., and de Koster, R., 2001, Routing Order Pickers in a Warehouse with a Middle Aisle, European Journal of Operational Research, 133, 1, [7] Roodbergen, K. J., and Vis, I., 2006, A Model for Warehouse Layout, IIE Transactions, 38, 10, [8] Roodbergen, K. J., and de Koster, R., 2001, Routing Methods for Warehouses with Multiple Cross Aisles, International Journal of Production Research, 39, 9, [9] Roodbergen, K. J., Sharp, G. P., and Vis, I. F., 2007, Designing the Structure of Manual Order-Picking Areas in Warehouses, IIE Transactions, to appear. [10] Vaughan, T. S., and Petersen, C. G., 1999, The Effect of Warehouse Cross Aisles on Order Picking Efficiency, International Journal of Production Research, 37, 4,