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Order batching procedures
European Journal of Operational Research 58 ( 1 9 9 2 ) 5 7 - 6 7 North-Holland
57
Order batching procedures D a v i d R. G i b s o n and G u n t e r P. Sharp
Georgia Institute of Technology, School of Industrial and Systems Engineering, Atlanta, GA 30332-0205, USA Received June 1990; revised March 1991
Abstract: Computer simulation is used to compare two new procedures for batching orders in an order retrieval system against a baseline procedure. The factors considered are the travel metric, warehouse representation, item location assignments, number of items per order, and the total number of orders considered. The results indicate that the two new procedures, in combination with skewed (ABC) item location assignments can reduce batch tour lengths by up to 44%. Keywords: Order retrieval, order batching, clustering, warehousing
1. Introduction
systems. For example, we might store pallets (unit loads) in a warehouse and retrieve individual cartons (items) in response to a specific request or order. It is often better to group several small orders into a larger amount of work, called a batch, to realize an increase in labor efficiency.
1.1. Background The retrieval of items that are smaller than a unit-load is an important function in warehouse 1.0
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© 1992 -
Elsevier Science Publishers B.V. All rights reserved92
58
D.R. Gibson, G.P. Sharp /Order batchingprocedures
Figure 1 is a schematic of an order retrieval system with horizontal travel. If each order is retrieved separately, called single-order picking, the order-picker fills the order by leaving the i n p u t / o u t p u t ( I / O ) point, making a tour through the storage region, retrieving the items specified in the order, and returning to the i n p u t / o u t p u t point. If a batch of orders is grouped for simultaneous retrieval, the order-picker makes an efficient tour through the storage system to retrieve the items specified in the batch. The length of the tour necessary to fill a batch of orders is usually shorter than the sum of the tour lengths if they are filled individually. In Figure 1, order 1 requires visiting 5 aisles, and order 2 requires visiting 7 aisles; the simultaneous retrieval of both orders requires visiting 8 aisles. Batching may require a different type of vehicle, the travel speed may be different, and the order picker or someone else may have to perform extra functions such as separating the batch into the individual orders. Batching also increases the response time for filling an order. A size limit on a batch is most often determined by the capacity of the vehicle and in some situations by an upper limit on response time. 1.2. Purpose of research There are two objectives of this research. First, we want to identify batching procedures which perform well in a typical horizontal travel system. Performance will be measured by improvement in labor efficiency as well as computational effort to execute the procedure. Second, we want to identify the conditions under which different batching procedures perform well. These conditions include the travel metric, the method for assigning items to storage locations, the representation of the storage system, fixed or variable number of items per order, and the number of orders to be considered. 1.3. Literature review Several authors present hierarchical methods for batching orders. These methods basically follow three steps: 1) a method of initiating batches, 2) a method of allocating orders to batches, 3) a stopping rule to determine when a batch has been completed. For example, Elsayed and Stern
[1983] consider a square, Euclidean region with 30 randomly generated points as possible item locations. These points are fixed for each of the experiments, thus giving a discrete representation of the storage region. The number of orders considered in each experiment is an integer, uniform random variable over 9 and 30. The number of items in each order is an integer, uniform random variable over 1 and 12. Similar methods are reported by Cunningham and Ogilvie [1971], Hartigan [1975], and Lance and Williams [1967]. Vinod [1969] presents two integer programming formulations of the batching problem, one with a linear objective, and the other with a quadratic objective. Th objective function coefficients are derived from assigning a value to each order. There is no discussion of how to determine the value of an order. One example is given where seven batches are created from 14 orders. Kusiak [1986] presents an integer, quadratic programming formulation of the batching problem. An eigenvector based approach is described for finding an approximate solution. This formulation uses distances between orders as coefficients in the objective function. No definition is given for the distance between two orders. One example is given where two batches are created from eight orders. Bartholdi and Platzman [1988] present a method for batching orders based on Spacefilling Curves (SFC). A rectangular range is identified for each order and is summarized as a scalar by a 4-dimensional SFC transformation. These scalars are then used to make batching decisions. No numerical results are given. (Their procedure is described in Section 2.2.) 1.4. Problem statement Give a set of orders, each consisting of several items to be retrieved from known locations in a storage region with horizontal travel, we wish to group these orders to minimize the total travel distance for retrieving all orders. Each order must be contained entirely in one batch, and each batch should have approximately 50 items. We assume that labor efficiency is proportional to travel distance, and that the time to extract items from storage locations is not affected by the method of batching. Splitting an order among two or more batches is not allowed,
D.R. Gibson, G.P. Sharp / Order batching procedures
59 0.5
so that we avoid downstream sorting. We envision a pick vehicle with a capacity of approximately 50 items. The retrieval quantity per item (product) in our system is one.
UNIT CIRCLE 2. Heuristics for the batching problem In this section we briefly describe several heuristic algorithms for grouping orders into batches. Besides the first-come, first-served (FCFS) heuristic, we present 'intelligent' or 'look-ahead' procedures.
1.0
2.1. First-Come, First-Served batching heuristic The FCFS batching heuristic is straight-forward. We group the first n orders from the input order list so that the batch size is as close to the desired size (50) as possible. Then we group the next m orders so that the batch size is as close to the desired size as possible. We keep doing this until all the orders are batched. This method can be described as 'naive' batching. The performance of the FCFS heuristic will be used as a baseline for comparison with the other heuristics.
UNIT SQUARE
0.0
I 0.0
4.0
Figure 2. Inverse spacefilling curve mapping
2.2. Spacefilling curve batching heuristic 2.2.1. 2-Dimensional spacefilling curve 2.2.1.1. Background. First, we consider the 2-dimensional spacefilling curve (SFC), which is a continuous mapping from a point, theta, on the unit circle onto the unit square. Theta ranges in value from 0 to 1 measuring the revolutions removed, clockwise from a fixed reference point. A theta value of 1.0 corresponds to 360 °. As theta ranges from 0 to 1, the SFC mapping traces out a tour of all the points in the unit square [Bartholdi and Platzman, 1982]. An example SFC is shown in Figure 2. Given a set of points in the unit square to be visited, a reasonable strategy is to sequence them according to where they fall along the SFC. The inverse 2-dimensional SFC mapping, which takes points in the unit square and transforms them to theta values on the unit circle, gives precisely this location along the SFC [Bartholdi and Platzman, 1982]. Figure 2 shows the transformation of sev-
eral points in the unit square to theta values, which provide the sequence for visiting the points. The actual tour would depend upon the travel metric that applies in the situation, such as straight-line, rectilinear, and so forth. The SFC method has been used successfully in vehicle routing [Bartholdi, Platzman, Collins and Warden, 1983]. The idea of using the SFC for batching is that items that are close to one another in the storage system will, hopefully, generate theta values which are close in magnitude. In Figure 2 the points (0.725, 0.78) and (0.625, 0.90) are close together in the unit square and generate theta values of 0.47 and 0.48, respectively [Bartholdi and Platzman, 1982]. This suggests that we can make relative proximity assessments of a group of items by examining their corresponding theta values. Thus, we have transformed a 2-dimensional problem to a problem on the unit circle, which is much easier to solve; however, the representation on the unit circle is an approximate one.
D.R. Gibson, G.P. Sharp /Order batchingprocedures
60
Table la Example of order batching using two-dimensional spacefilling curve mapping Order
Item
Location
Item theta
Ordertheta
1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
(0.32, (0.90, (0.51, (0.58, (0.04, (0.49, (0.78, (0.09, (0.19, (0.71, (0.22, (0.11, (0.94, (0.82, (0.39, (0.68, (0.14, (0.31, (0.70, (0.54,
0.23 0.54 0.44 0.42 0.07 0.88 0.59 0.04 0.03 0.65 0.16 0.23 0.55 0.78 0.37 0.85 0.04 0.30 0.79 0.44
0.19
2
3
4
0.78) 0.67) 0.96) 0.82) 0.48) 0.49) 0.66) 0.26) 0.24) 0.36) 0.63) 0.84) 0.65) 0.17) 0.62) 0.29) 0,28) 1.00) 0.06) 0.93)
max
rain max
2.2.2. 4-Dimensional spacefilling curue
0.25
rain min
0.22
max max min
0.24
2.2.1.2. Procedure. This was the first SFC batching heuristic developed; however, it proved inferior during preliminary investigations to the 4-dimensional variation presented in Section 2.2.2.2. The steps of the algorithm are as follows: The heuristic calculates theta values for all items, called item-theta values. Next, for each order, we identify the minimum and maximum item-theta values. These two values for each order are mapped, in the same way an item location is mapped to an item-theta value, to what we call an order-theta value. The third step is to sort the orders according to ascending order-theta values. The last step is to group orders from the sorted order list to form batches, using a FCFS method, close to the desired size. Table 1 shows an example with four orders and five items per order with two batches produced.
Table lb Batch
Order
Order-theta
1
1
2
3 4 2
0.19 0.22 0.24 0.25
We are actually applying the 2-dimensional SFC heuristic twice, once to the 2-dimensional coordinates of each item location, and the second time to the pair of theta values that are minimum and maximum item-theta values for each order.
2.2.2.1. Background. The idea of the 4-dimensional SFC heuristic is similar to the 2-dimensional case, except that the inverse mapping takes points in 4-dimensional space directly in one step onto the unit circle [Bartholdi and Platzman, 1988]. To determine an order-theta value using this heuristic, we first identify the minimum and maximum x coordinates and minimum and maximum y coordinates for all the items in a particular order. The motivation for using this particular vector of numbers is that the points (Xmin, Ymin) and (x . . . . Ymax) define the smallest rectangle (orthogonal to the storage system) which encloses all the items in an order. Next, we map the four numbers (Xmin, Ymin, X. . . . Ymax) onto the unit circle using the inverse 4-dimensional SFC mapping. The resulting scalar, which summarizes the defining coordinates of this rectangle, is the order-theta value. 2.2.2.2. Procedure. The first step in the 4-dimensional SFC heuristic is to identify, for each order, the minimum and maximum x and y coordinates and map these four values to an order-theta value as described above. Then the orders are sorted according to ascending order-theta values. The third step is to batch the orders, by considering the sorted order list in a FCFS manner. Table 2 shows an example of the 4-dimensional SFC batching heuristic. The input data is the same as in the example in Table 1, but the batching results are different. 2.3. Sequential minimum distance batching heuristic 2.3.1. Background With this heuristic we batch orders by using the distances between orders. The distance between two orders is a measure of the travel distance between the item locations of one order to those of another. The specific distance measures are defined in Section 3.1.2.
D.R. Gibson, G.P. Sharp / Order batching procedures Table 2a Example of order batching using four-dimensional spacefilling curve mapping Order
Item
Location
Ordertheta
1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
(0.32, 0.78) max x(0.90, 0.67) (0.51, 0.96) max y (0.58, 0.82) min x(0.04, 0.48) rain y (0.49, 0.49) max x (0.78, 0.66) max y rain x(0.09, 0.26) (0.19, 0.24) min y (0.71, 0.36) (0.22, 0.63) min x(0.11, 0.84) max y max x(0.94, 0.65) (0.82, 0.17) min y (0.39, 0.62) max x(0.68, 0.29) rain x(0.14, 0.28) (0.31, 1.00) max y (0.70, 0.06) rain y (0.54, 0.93)
0.137
2
3
4
0.130
0.125
0.120
2.3.2. Procedure The Sequential Minimal Distance (SMD) batching heuristic takes the first order from the given set of orders and puts it as a seed in batch 1. Next, we calculate the distance from order 1 to each of the other orders. We choose the n orders which are closest to order 1 and include them in batch 1, n being chosen so that the batch size is as close to the desired size (50) as possible. When batch 1 is completed, it is set aside and one of the remaining orders is selected as a seed for batch 2, and so forth. To illustrate this greedy algorithm, consider an example with six orders, where order 1 is selected as the seed for batch 1. Let the distance from order 1 to each other order be: Order: 2 3 4 5 Distance from 1 : 5 6 2 7 4
Table 2b Batch
Order
Order-theta
1
4 3 2 1
0.120 0.125 0.130 0.137
2
61
Order 4 is closest so it is placed in batch 1. We assume batch 1 is close to the desired size and that we set it aside. Then we select order 2 as the seed for batch 2 and compute the distances from order 2 to each remaining order: Order: 3 5 6 Distance from 2 : 4 5 1 Order 6 is closest so it is placed in batch 2. Assume batch 2 is close to the desired size and that we set it aside. If the remaining two orders, 3 and 5, can be accommodated in a batch, the overall result is batch 1 : orders 1 and 4, batch 2 : orders 2 and 6, batch 3 : orders 3 and 5.
3. Experimental factors There are seven experimental factors considered in the evaluation of batching: 1) Batching heuristic. 2) Travel metric. 3) Storage region representation. 4) Item location assignment. 5) Fixed or variable number of items per order. 6) Number of orders. 7) Number of replications to minimize random bias. The batching algorithms are FCFS, 4-dimensional SFC, and SMD, as described in the previous section. The other factors are described below.
3.1. Travel metrics 3.1.1. Types of metrics Four travel metrics are considered: Euclidean, rectilinear, Chebyshev, and aisle. Euclidean distances are measured by using the formula d E = [(Xl - - X 2 )2 + (Yl -- Y21~.210.5 ] , where (x 1, yj) and (x2, y2 ) indicate the coordinate locations of two points in the storage region. The Euclidean metric rarely applies in warehousing systems where order picking occurs. It is included here so that we can compare our results with those of other researchers. Rectilinear distances are measured using the formula d R = I X 1 - - X 2 1 + I Yl -Y21. This metric
D.R. Gibson, G.P. Sharp /Order batching procedures
62
applies to open-yard storage systems, such as those found one some military bases. Chebyshev distances are measured using the formula d c = m a x { I x l - x 21, l Y l - Y z l } . The Chebyshev metric applies to aisle-captive, person-aboard storage/retrieval systems. The aisle travel metric characterizes the distances along the cross aisle in a typical warehouse with an aisle structure as shown in Figure 1. We assume that an order picker may select which end of an aisle to enter, but must then traverse its entire length and exit at the other end (selective, one-way routing). Thus, the distance between an item in aisle 1 and another item in aisle 3 is the distance, measure along the cross aisle, between the centerlines of aisles 1 and 3. The distance between two items in the same aisle is by definition zero. The reason for this definition of the travel metric is to force the batching of orders with items in the same aisles.
3.1.2. Order distances The SMD heuristic uses the concept of the distance between two orders, which is defined as the sum of the distances from each item in the first order to the closest item in the second order. For the Euclidean metric, let (Xim, Yim) be the coordinates of item i in order m. Define 210.5
gimn= m!n[(Xim-Xjn)2+(Yim-Yjn) 3
J
•
This Lim n is the distance between item i in order m and the nearest item in order n. We then sum the Li.,n to obtain the distance d(m, n) between orders m and n:
d(m, n)= •Lim,,. i
In effect, each item in order m identifies a closest item in order n, and the corresponding distances are added. This pairing need not be unique. Further, the distance measure is not symmetric. That is, in general
d(m, n) 4~d(n, m). For the rectilinear and Chebyshev metrics, the Lim n a r e defined using the appropriate analog of d R and d c, respectively. For the aisle metric, let aim be the number of the aisle that contains item
i of order m. Then, for equally spaced and sequentially numbered aisles: Lira n
=
mini aim -- ajn 1.
J
The d(m, n) are defined in the same way for all metrics.
3.1.3. Tour sequencing The sequencing of retrievals of the items in a batch is an example of the Traveling Salesman Problem, which is NP-complete [Papadimitriou, 1977]. Under the Euclidean, rectilinear, and Chebyshev travel metrics, all the items in a batch are sorted according to their item-theta values, and the tour simply follows this list in sequence. The distances between items are measured using the appropriate metric. Under the aisle travel metric, with selective, one-way routing, tour length calculation is based on the particular set of aisles which must be visited in order to complete the tour effectively. If the number of aisles to be visited is odd, an extra aisle traversal is needed to return to the I / O point.
3.2. Storage region representation The storage region, which occupies a unit square, is represented in two ways. The first is the continuous representation, where any point inside or on the unit square is a possible item location. This representation, like the Euclidean travel metric, was included so that we may compare our results with those of others. We are also interested in any possible difference between it and the second representation. The second representation is the discrete representation, where there are only a finite number of item locations (800) in the storage region, as shown in Figure 3. This corresponds, in the context of a horizontal travel system, to 10 aisles with 40 locations along each side of an aisle. In the context of a person-aboard storage system this means 40 horizontal positions along each of 20 vertical levels.
3.3. Item location assignement Uniform item location assignment means that the probabilities of storage locations being item
D.R. Gibson, G.P. Sharp /Order batchingprocedures
locations are independent and equally likely. For example, under the continuous representation, the density function corresponding to item locations in the unit square is the uniform distribution: f ( x , y) = 1, 0 ~
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Figure 4. Skewed item location assignment
use 5 items per order. For the variable case we assume the number of items follows a positive geometric distribution with a mean of 5.
3.5. Number of orders considered in batching process
3.4. Number of items per order We consider two cases, fixed and variable number of items per order. For the fixed case we
1140 1/41 1/41
1/20
1120
We consider 12 situations: 100 order, 200 orders, and so on up to 1200 orders. This corresponds to generating 500, 1000, 1500 . . . . . 6000
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D.R. Gibson, G.P. Sharp /Order batchingprocedures
item locations, respectively. With a fixed number of items per order, and exactly 10 orders per batch, we generate 10, 20, 30 . . . . . 120 batches of size 50, respectively. When the number of items per order is variable, we generate exactly the same number of items as in the fixed situation; however, the number of orders is not necessarily 100, 200 . . . . . 1200. For the variable situation we do, however, create exactly the same number of batches as in the fixed Case, making them as close to size 50 as possible.
3.6. Framework for experiment, replications Simulation runs were performed for each of the three heuristics under 24 identical sets of experimental factors, for 72 combinations (shown in Table 4). Not all metrics were paired with both storage region representations: with Euclidean only a continuous representation was used, and with aisle only a discrete representation was used. For each of the 72 combinations, the number of orders varied from 100 to 1200 in steps of 100. A different random number seed was used to generate the item locations for each of the 12 values. Also, each of the 12 values was replicated 10 times using different random number seeds. Thus, a total of 72 * 12 * 10 = 8640 simulations were performed, with some exceptions. The batching heuristics were coded in Turbo Pascal, Version 5.0, and the simulations were performed on an IBM P S / 2 Model 80, running at 17.6 MHz.
4.2. Item location assignment Skewed item location assignment plays a significant role in the reduction of tour lengths. FCFS batching experiences reductions of 15-19% when item locations are skewed, as shown in Figure 5, by comparing the curves for baseline (FADUV) and baseline, skewed (FADSF). The most dramatic reductions in tour length occur when intelligent batching, either SFC or SMD, is performed and item locations are skewed, as shown in Table 3. The numbers are the range of average batch tour length reductions from the baseline case, as the number of orders ranges from 100 to 1200. As an example, consider SRDSV, in the fourth column. The average batch tour lengths produced by SFC batching under rectilinear travel, discrete representation, skewed item locations, and variable number of items per order ranges from 26 to 32% shorter than tours produced by FCFS batching under identical experimental conditions, ex-
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4. Analysis of experiment 4.1. General obseruations As expected, FCFS batching produces no decrease in average batch tour length regardless of the number of orders and experimental factors considered. The FCFS heuristic used in conjunction with the aisle travel metric produces the longest tour with an average batch tour length of 11.8, followed by the rectilinear metric at 8.6, Euclidean at 7.0, and Chebyshev at 6.3. These values were obtained under the baseline conditions where item location assignment is uniform, variable number of items per order, and the number of orders range from 100 to 1200.
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4
Batching, Skewed, MADSV
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300
i
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500
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too
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9oo
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11100
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Number of Orders FADUV - FCFS betC.hing, aisle metric, discrete representation, uniform ~arn location, v~ixble order size •
FADSV - FCFS betching, aisle metric, discrete representation, skewed item location, vaZiable order size MADSV. SMD heuristic, aJsJemetric, discrete representation, skewed ~tam location, varlablo order size
Figure 5. Results for skewed item location assignment
D.R. Gibson, G.P. Sharp / Order batching procedures Table 3 Selected tour length comparisons Baseline
Skewed item locations, savings over baseline
Intelligent batching, savings over baseline
Skewed item locations and intelligent/ batching savings o v e r / baseline
FADUV a
FADSV 18%
SADUV 1-10% MADUV
FRDUV
FRDSV 18%
SRDUV 2-14% MRDUV 1-10% SECUV 2-8% MECUV 2-8% SCDUV 1-8% MCDUV
SADSV 14-21% MADSV 27-44% SRDSV 26-32% MRDSV 21-27% SECSV 23-29% MECSV 20-28% SCDSV 24-29% MCDSV 20-26%
1-10%
FECUV
FECSV 17%
FCDUV
FCDSV 17%
1-8% a Mnemonic explanation:
I st character: Heuristic. F - First-come, First-Served. S - Spacefilling Curve. M - Sequential Minimum Distance. 2 na character: Travel metric. A - Aisle. R - Rectilinear. E - Euclidean. C - Chebyshev. 3 ra character: Warehouse Representation. C - Continuous. D - Discrete. 4 th character: Item location assignment. U - Uniform. S - Skewed. 5 th character: Nu mb er of items per order. F - Fixed. V - Variable.
cept that item locations are uniform, as the number of orders ranges from 100 to 1200. These percentage reductions reflect the combined effect of intelligent batching and skewed item locations. SFC batching, under the rectilinear, Euclidean, and Chebyshev travel metrics, results in reduction of 23-29%, and under the aisle travel metric, reductions of 14-21% below the values obtained from FCFS batching with uniform item locations. Run times for the SFC heuristic are about 2 minutes to batch 1200 orders. The SMD heuristic with skewed item locations results in reductions of 20-26% under the recti-
65
linear, Euclidean, and Chebyshev travel metrics, and reductions of 27-44% under the aisle metric. The entry 27-44%, under MADSV in Table 3, is shown graphically in Figure 5. The curve labelled intelligent batching, skewed (MADSV) is 27-44% below the curve labelled baseline (FADUV). Run times of this heuristic under the aisle metric average about 4 minutes to batch 1200 orders, 10 minutes under the rectilinear and Chebyshev metrics, and 69 minutes for the Euclidean metric. 4.3. Storage region representation A discrete representation of the storage region always produces shorter average tour lengths than a continuous representation. These reductions range from 2-9% over all experimental conditions. A typical example is shown in Figure 6 by the two curves labelled continuous (FRCUF) and discrete (FRDUF). In general, however, the storage region representation does not cause any other experimental factor to behave much differently. For example, if a particular heuristic performs better than another under the continuous representation, the same is true under the discrete representation. Thus, the storage region representation should not be considered in future research. 4.4. Number of items per order Using the FCFS heuristic, shorter tours result with variable order size; however, these reductions rarely exceed 1%. The two other heuristics consistently produce shorter tours with fixed order size. With SFC and SMD batching under the rectilinear Chebyshev metrics, reductions are 710%. The SMD heuristic under the aisle metric has reductions of 7-18%, as shown in Figure 6 by the curves labelled fixed (MADUF) and variable (MADUV). These values point to the danger of under-estimating tour lengths when a fixed order size is assumed. 4.5. Travel metric The aisle travel metric, the most realistic, produces the longest tours and therefore has room for the greatest tour length reduction. With aisle travel the SMD heuristic outperforms the SFC heuristic under all conditions. With the rectilinear and Chebyshev metrics, the SFC heuristic
D.R. Gibson, G.P. Sharp / Order batchingprocedures
66 Table 4 Summary of results Exptl. factors
Table 4 (continued)
Number of Orders 100
300
600
1200
SADUF SADUV SADSF SADSV SRCUF SRCUV SRCSF SRCSV SRDUF SRDUV SRDSF SRDSV SECUF SECSF SECSV SCCUF SCCUV SCCSF SCCSV SCDUF SCDSF SCDSV
11.503 11.768 0,748 10.131 8,529 8,726 6,399 6.781 8.160 8.413 5.859 6.340 6.723 5.053 5.356 6.066 6.181 4.568 4.842 5.876 4.319 4.647
11.096 11,402 9.235 9.878 8.118 8.436 5.999 6.597 7.775 8.118 5.706 6.207 6.398 4.766 5.254 5.769 6.004 4.330 4.777 5.605 4.222 4.582
10.790 11.268 8.738 9.632 7.875 8.254 5.852 6.431 7.542 7.941 5.434 6.035 6.215 4,663 5.132 5.608 5.863 4.243 4.673 5.462 4.055 4.499
10.554 11.016 8.430 9.326 7.771 8.134 5.691 6.231 7.368 7.792 5.251 5,847 6.101 4.549 4.981 5.509 5.785 4.147 4.540 5.345 3.944 4.372
FADUF FADUV FADSF FADSV FRCUF FRCUV FRCSF FRCSV FRDUF FRDUV FRDSF FRDSV
1.778 11.778 9.678 9.789 8.918 8.898 7.404 7.310 8.542 8.586 6.950 6.885
1.778 1.777 9.613 9.547 8.881 8.893 7.354 7.379 8.586 8.627 7.092 7.027
1.774 1.778 9.645 9.624 8.838 8.894 7.421 7.414 8,577 8.594 7.022 7.004
1.775 1.772 9.589 9.635 8.882 8.865 7.392 7.370 8.607 8.595 7.092 7.019
MADUF MADUV MADSF MADSV MRCUF MRCUV MRCSF MRCSV MRDUF MRDUV MRDSF MRDSV MECUF MECUV MECSF MECSV MCCUF MCCUV MCCSF MCCSV
11.381 11,691 7.969 8.634 8.517 8.715 6.937 7.124 8.418 8.517 6.428 6.644 6.666 6.864 5.433 5.574 5.985 6.162 4.942 5.061
10,398 11,207 6,535 7,892 8,235 8,501 6.681 6.970 8.198 8.235 6.343 6.598 N/A N/A N/A N/A 5.835 6.020 4.711 4.956
9.767 10.773 5.851 7.187 8.052 8.360 6.521 6.866 8.068 8.052 6.030 6.432 6.346 6,566 5.083 5.375 5.724 5.936 4.598 4.857
9.031 10.331 5.426 6.547 7.967 8.250 6.327 6.684 7.933 7.967 5.877 6.296 6.248 6.461 4.919 5.249 5.619 5.819 4.454 4.717
Exptl. factors a
Number of Orders 100
300
600
1200
MCDUF MCDUV MCDSF MCDSV
5,882 6.044 4.510 4.919
5.722 5.876 4.280 4.813
5.577 5.786 4.188 4.680
5.435 5.686 4.062 4.545
FECUF FECUV FECSF FECSV FCCUF FCCUV FCCSF FCDUF FCDUV FCDSF FCDSV
7.021 6.987 5.840 5.765 6.334 6.293 5.280 6.168 6.135 5.067 5.004
6.995 7.002 5.818 5.839 6.308 6.306 5.270 6.180 6.165 5,138 5.112
6.958 6.998 5.872 5,854 6.271 6,307 5,318 6,151 6.133 5.117 5.102
6,988 6.983 5.840 5.824 6.297 6.298 5.218 6.156 6.162 5.113 5.107
a Experimental factors are defined by the same mnemonics as in Table 3.
12
Continuous, FRCUF
I0
1
e-
3 ~8
=-~=
=
"
=
=
= l=
=
:
"
Fixed, MADUF
I 100
I
I
300
I
I
500
I
I
I
700
I
I
900
I
I
1100 1200
Number of Orders •
FRCUF - FCFS hatching, reclJlinNr metric, continuoul representation, uniform item location, fixed order SiZe
•
FRDUF - FCFS bitching, rectilinear metric, dilcrete reprelentation, uniform ~
[]
Io¢~ltJon, fixed or~o," liZe
MADUV - SMD heuristic, l i l l e metric, dilcreta repreltantation, uniform item location, variable Order IJzO
©
MADUF - SMD heuristic, l i l l e metric, dil~rete representation, uniform item location, fixed order size
Figure 6. R~sults for warehouse representation and number of items per order
D.R. Gibson, G.P. Sharp /Order batching procedures
almost always outperforms the SMD heuristic. With the Euclidean metric, the SFC heuristic is again superior to the SMD heuristic, but only slightly. Table 4 contains a summary of results obtained under various experimental conditions.
67
at the Georgia Institute of Technology and by the National Science Foundation under grant ECS8351313. The authors gratefully thank two anonymous reviewers for their helpful comments.
5. Conclusions and recommendations References
Skewed item location assignment used in conjunction with intelligent batching yields significant reductions in average batch tour lengths. In some instances these reductions are 44%. In situations where the Euclidean, rectilinear, or Chebyshev metric is appropriate, it is recommended that the SFC heuristic be used because tours are generally shorter and the execution time is much quicker than the SMD heuristic. In a typical warehouse with an aisle metric, it is recommended that the SMD heuristic be used, because the tours are significantly shorter than the ones obtained by the SFC heuristic and the execution time of the SMD heuristic is more reasonable. A discrete storage region representation is preferred because it is more realistic than the continuous case. However, the relative performance of batching algorithms is not affected much by the representation. There is a danger of under-estimating tour lengths when a fixed order size is assumed.
Acknowledgement
This research was supported by US industry through the Material Handling Research Center
Bartholdi, J.J., and Platzman, L.K. (1982), "An O(N log N) planar travelling salesman heuristic based on spacefilling curves". Operations Research Letters 1/4, 121-125. Bartholdi, J.J., Platzman, L.K., Collins, R.L., and Warden, W.H. (1983), "How to implement a simple routing system", Technical Report PDRC 83-04, Production and Distribution Research Center, Georgia Institute of Technology, Atlanta, GA. Bartholdi, J.J., and Platzman, L.K. (1988), "Heuristics based on spacefilling curves for combinatorial problems in euclidean space", Management Science 34/3, 291-305. Cunningham, K.M., and Ogilvie, J.C. (1972), "Evaluation of hierarchical grouping techniques: A preliminary study", Computer Journal 15/3, 209-213. Elsayed, E.A., and Stern, R.G. (1983), "Computerized algorithms for order processing in automated warehousing systems"International Journal of Production Research 21/4, 579-585. Hartigan, J.A. (1975), ClusteringAlgorithms, John Wiley, New York, 19.74. Kusiak, A., Vanelli, A., and Kumar, K.R. (1986), "Clustering analysis: Models and algorithms", Control and Cybernetics 15/2, 139-152. Lance, G.N., and Williams, W.T. (1967), "Computer programs for hierarchical polythetic classification ('similarity analysis')", Computer Journal 9, 373-380. Papadimitriou, C.H., and Steiglitz, K. (1982), Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ. Vinod, H.D. (1969), "Integer programming and the theory of grouping", Journal of the American Statistical Association 64, 506-515.
57
Order batching procedures D a v i d R. G i b s o n and G u n t e r P. Sharp
Georgia Institute of Technology, School of Industrial and Systems Engineering, Atlanta, GA 30332-0205, USA Received June 1990; revised March 1991
Abstract: Computer simulation is used to compare two new procedures for batching orders in an order retrieval system against a baseline procedure. The factors considered are the travel metric, warehouse representation, item location assignments, number of items per order, and the total number of orders considered. The results indicate that the two new procedures, in combination with skewed (ABC) item location assignments can reduce batch tour lengths by up to 44%. Keywords: Order retrieval, order batching, clustering, warehousing
1. Introduction
systems. For example, we might store pallets (unit loads) in a warehouse and retrieve individual cartons (items) in response to a specific request or order. It is often better to group several small orders into a larger amount of work, called a batch, to realize an increase in labor efficiency.
1.1. Background The retrieval of items that are smaller than a unit-load is an important function in warehouse 1.0
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I/0 Figure 1. Order retrieval system with horizontal travel. 1: location of item in order 1; 2: location of item in order 2 0377-2217/92/$05.00
© 1992 -
Elsevier Science Publishers B.V. All rights reserved92
58
D.R. Gibson, G.P. Sharp /Order batchingprocedures
Figure 1 is a schematic of an order retrieval system with horizontal travel. If each order is retrieved separately, called single-order picking, the order-picker fills the order by leaving the i n p u t / o u t p u t ( I / O ) point, making a tour through the storage region, retrieving the items specified in the order, and returning to the i n p u t / o u t p u t point. If a batch of orders is grouped for simultaneous retrieval, the order-picker makes an efficient tour through the storage system to retrieve the items specified in the batch. The length of the tour necessary to fill a batch of orders is usually shorter than the sum of the tour lengths if they are filled individually. In Figure 1, order 1 requires visiting 5 aisles, and order 2 requires visiting 7 aisles; the simultaneous retrieval of both orders requires visiting 8 aisles. Batching may require a different type of vehicle, the travel speed may be different, and the order picker or someone else may have to perform extra functions such as separating the batch into the individual orders. Batching also increases the response time for filling an order. A size limit on a batch is most often determined by the capacity of the vehicle and in some situations by an upper limit on response time. 1.2. Purpose of research There are two objectives of this research. First, we want to identify batching procedures which perform well in a typical horizontal travel system. Performance will be measured by improvement in labor efficiency as well as computational effort to execute the procedure. Second, we want to identify the conditions under which different batching procedures perform well. These conditions include the travel metric, the method for assigning items to storage locations, the representation of the storage system, fixed or variable number of items per order, and the number of orders to be considered. 1.3. Literature review Several authors present hierarchical methods for batching orders. These methods basically follow three steps: 1) a method of initiating batches, 2) a method of allocating orders to batches, 3) a stopping rule to determine when a batch has been completed. For example, Elsayed and Stern
[1983] consider a square, Euclidean region with 30 randomly generated points as possible item locations. These points are fixed for each of the experiments, thus giving a discrete representation of the storage region. The number of orders considered in each experiment is an integer, uniform random variable over 9 and 30. The number of items in each order is an integer, uniform random variable over 1 and 12. Similar methods are reported by Cunningham and Ogilvie [1971], Hartigan [1975], and Lance and Williams [1967]. Vinod [1969] presents two integer programming formulations of the batching problem, one with a linear objective, and the other with a quadratic objective. Th objective function coefficients are derived from assigning a value to each order. There is no discussion of how to determine the value of an order. One example is given where seven batches are created from 14 orders. Kusiak [1986] presents an integer, quadratic programming formulation of the batching problem. An eigenvector based approach is described for finding an approximate solution. This formulation uses distances between orders as coefficients in the objective function. No definition is given for the distance between two orders. One example is given where two batches are created from eight orders. Bartholdi and Platzman [1988] present a method for batching orders based on Spacefilling Curves (SFC). A rectangular range is identified for each order and is summarized as a scalar by a 4-dimensional SFC transformation. These scalars are then used to make batching decisions. No numerical results are given. (Their procedure is described in Section 2.2.) 1.4. Problem statement Give a set of orders, each consisting of several items to be retrieved from known locations in a storage region with horizontal travel, we wish to group these orders to minimize the total travel distance for retrieving all orders. Each order must be contained entirely in one batch, and each batch should have approximately 50 items. We assume that labor efficiency is proportional to travel distance, and that the time to extract items from storage locations is not affected by the method of batching. Splitting an order among two or more batches is not allowed,
D.R. Gibson, G.P. Sharp / Order batching procedures
59 0.5
so that we avoid downstream sorting. We envision a pick vehicle with a capacity of approximately 50 items. The retrieval quantity per item (product) in our system is one.
UNIT CIRCLE 2. Heuristics for the batching problem In this section we briefly describe several heuristic algorithms for grouping orders into batches. Besides the first-come, first-served (FCFS) heuristic, we present 'intelligent' or 'look-ahead' procedures.
1.0
2.1. First-Come, First-Served batching heuristic The FCFS batching heuristic is straight-forward. We group the first n orders from the input order list so that the batch size is as close to the desired size (50) as possible. Then we group the next m orders so that the batch size is as close to the desired size as possible. We keep doing this until all the orders are batched. This method can be described as 'naive' batching. The performance of the FCFS heuristic will be used as a baseline for comparison with the other heuristics.
UNIT SQUARE
0.0
I 0.0
4.0
Figure 2. Inverse spacefilling curve mapping
2.2. Spacefilling curve batching heuristic 2.2.1. 2-Dimensional spacefilling curve 2.2.1.1. Background. First, we consider the 2-dimensional spacefilling curve (SFC), which is a continuous mapping from a point, theta, on the unit circle onto the unit square. Theta ranges in value from 0 to 1 measuring the revolutions removed, clockwise from a fixed reference point. A theta value of 1.0 corresponds to 360 °. As theta ranges from 0 to 1, the SFC mapping traces out a tour of all the points in the unit square [Bartholdi and Platzman, 1982]. An example SFC is shown in Figure 2. Given a set of points in the unit square to be visited, a reasonable strategy is to sequence them according to where they fall along the SFC. The inverse 2-dimensional SFC mapping, which takes points in the unit square and transforms them to theta values on the unit circle, gives precisely this location along the SFC [Bartholdi and Platzman, 1982]. Figure 2 shows the transformation of sev-
eral points in the unit square to theta values, which provide the sequence for visiting the points. The actual tour would depend upon the travel metric that applies in the situation, such as straight-line, rectilinear, and so forth. The SFC method has been used successfully in vehicle routing [Bartholdi, Platzman, Collins and Warden, 1983]. The idea of using the SFC for batching is that items that are close to one another in the storage system will, hopefully, generate theta values which are close in magnitude. In Figure 2 the points (0.725, 0.78) and (0.625, 0.90) are close together in the unit square and generate theta values of 0.47 and 0.48, respectively [Bartholdi and Platzman, 1982]. This suggests that we can make relative proximity assessments of a group of items by examining their corresponding theta values. Thus, we have transformed a 2-dimensional problem to a problem on the unit circle, which is much easier to solve; however, the representation on the unit circle is an approximate one.
D.R. Gibson, G.P. Sharp /Order batchingprocedures
60
Table la Example of order batching using two-dimensional spacefilling curve mapping Order
Item
Location
Item theta
Ordertheta
1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
(0.32, (0.90, (0.51, (0.58, (0.04, (0.49, (0.78, (0.09, (0.19, (0.71, (0.22, (0.11, (0.94, (0.82, (0.39, (0.68, (0.14, (0.31, (0.70, (0.54,
0.23 0.54 0.44 0.42 0.07 0.88 0.59 0.04 0.03 0.65 0.16 0.23 0.55 0.78 0.37 0.85 0.04 0.30 0.79 0.44
0.19
2
3
4
0.78) 0.67) 0.96) 0.82) 0.48) 0.49) 0.66) 0.26) 0.24) 0.36) 0.63) 0.84) 0.65) 0.17) 0.62) 0.29) 0,28) 1.00) 0.06) 0.93)
max
rain max
2.2.2. 4-Dimensional spacefilling curue
0.25
rain min
0.22
max max min
0.24
2.2.1.2. Procedure. This was the first SFC batching heuristic developed; however, it proved inferior during preliminary investigations to the 4-dimensional variation presented in Section 2.2.2.2. The steps of the algorithm are as follows: The heuristic calculates theta values for all items, called item-theta values. Next, for each order, we identify the minimum and maximum item-theta values. These two values for each order are mapped, in the same way an item location is mapped to an item-theta value, to what we call an order-theta value. The third step is to sort the orders according to ascending order-theta values. The last step is to group orders from the sorted order list to form batches, using a FCFS method, close to the desired size. Table 1 shows an example with four orders and five items per order with two batches produced.
Table lb Batch
Order
Order-theta
1
1
2
3 4 2
0.19 0.22 0.24 0.25
We are actually applying the 2-dimensional SFC heuristic twice, once to the 2-dimensional coordinates of each item location, and the second time to the pair of theta values that are minimum and maximum item-theta values for each order.
2.2.2.1. Background. The idea of the 4-dimensional SFC heuristic is similar to the 2-dimensional case, except that the inverse mapping takes points in 4-dimensional space directly in one step onto the unit circle [Bartholdi and Platzman, 1988]. To determine an order-theta value using this heuristic, we first identify the minimum and maximum x coordinates and minimum and maximum y coordinates for all the items in a particular order. The motivation for using this particular vector of numbers is that the points (Xmin, Ymin) and (x . . . . Ymax) define the smallest rectangle (orthogonal to the storage system) which encloses all the items in an order. Next, we map the four numbers (Xmin, Ymin, X. . . . Ymax) onto the unit circle using the inverse 4-dimensional SFC mapping. The resulting scalar, which summarizes the defining coordinates of this rectangle, is the order-theta value. 2.2.2.2. Procedure. The first step in the 4-dimensional SFC heuristic is to identify, for each order, the minimum and maximum x and y coordinates and map these four values to an order-theta value as described above. Then the orders are sorted according to ascending order-theta values. The third step is to batch the orders, by considering the sorted order list in a FCFS manner. Table 2 shows an example of the 4-dimensional SFC batching heuristic. The input data is the same as in the example in Table 1, but the batching results are different. 2.3. Sequential minimum distance batching heuristic 2.3.1. Background With this heuristic we batch orders by using the distances between orders. The distance between two orders is a measure of the travel distance between the item locations of one order to those of another. The specific distance measures are defined in Section 3.1.2.
D.R. Gibson, G.P. Sharp / Order batching procedures Table 2a Example of order batching using four-dimensional spacefilling curve mapping Order
Item
Location
Ordertheta
1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
(0.32, 0.78) max x(0.90, 0.67) (0.51, 0.96) max y (0.58, 0.82) min x(0.04, 0.48) rain y (0.49, 0.49) max x (0.78, 0.66) max y rain x(0.09, 0.26) (0.19, 0.24) min y (0.71, 0.36) (0.22, 0.63) min x(0.11, 0.84) max y max x(0.94, 0.65) (0.82, 0.17) min y (0.39, 0.62) max x(0.68, 0.29) rain x(0.14, 0.28) (0.31, 1.00) max y (0.70, 0.06) rain y (0.54, 0.93)
0.137
2
3
4
0.130
0.125
0.120
2.3.2. Procedure The Sequential Minimal Distance (SMD) batching heuristic takes the first order from the given set of orders and puts it as a seed in batch 1. Next, we calculate the distance from order 1 to each of the other orders. We choose the n orders which are closest to order 1 and include them in batch 1, n being chosen so that the batch size is as close to the desired size (50) as possible. When batch 1 is completed, it is set aside and one of the remaining orders is selected as a seed for batch 2, and so forth. To illustrate this greedy algorithm, consider an example with six orders, where order 1 is selected as the seed for batch 1. Let the distance from order 1 to each other order be: Order: 2 3 4 5 Distance from 1 : 5 6 2 7 4
Table 2b Batch
Order
Order-theta
1
4 3 2 1
0.120 0.125 0.130 0.137
2
61
Order 4 is closest so it is placed in batch 1. We assume batch 1 is close to the desired size and that we set it aside. Then we select order 2 as the seed for batch 2 and compute the distances from order 2 to each remaining order: Order: 3 5 6 Distance from 2 : 4 5 1 Order 6 is closest so it is placed in batch 2. Assume batch 2 is close to the desired size and that we set it aside. If the remaining two orders, 3 and 5, can be accommodated in a batch, the overall result is batch 1 : orders 1 and 4, batch 2 : orders 2 and 6, batch 3 : orders 3 and 5.
3. Experimental factors There are seven experimental factors considered in the evaluation of batching: 1) Batching heuristic. 2) Travel metric. 3) Storage region representation. 4) Item location assignment. 5) Fixed or variable number of items per order. 6) Number of orders. 7) Number of replications to minimize random bias. The batching algorithms are FCFS, 4-dimensional SFC, and SMD, as described in the previous section. The other factors are described below.
3.1. Travel metrics 3.1.1. Types of metrics Four travel metrics are considered: Euclidean, rectilinear, Chebyshev, and aisle. Euclidean distances are measured by using the formula d E = [(Xl - - X 2 )2 + (Yl -- Y21~.210.5 ] , where (x 1, yj) and (x2, y2 ) indicate the coordinate locations of two points in the storage region. The Euclidean metric rarely applies in warehousing systems where order picking occurs. It is included here so that we can compare our results with those of other researchers. Rectilinear distances are measured using the formula d R = I X 1 - - X 2 1 + I Yl -Y21. This metric
D.R. Gibson, G.P. Sharp /Order batching procedures
62
applies to open-yard storage systems, such as those found one some military bases. Chebyshev distances are measured using the formula d c = m a x { I x l - x 21, l Y l - Y z l } . The Chebyshev metric applies to aisle-captive, person-aboard storage/retrieval systems. The aisle travel metric characterizes the distances along the cross aisle in a typical warehouse with an aisle structure as shown in Figure 1. We assume that an order picker may select which end of an aisle to enter, but must then traverse its entire length and exit at the other end (selective, one-way routing). Thus, the distance between an item in aisle 1 and another item in aisle 3 is the distance, measure along the cross aisle, between the centerlines of aisles 1 and 3. The distance between two items in the same aisle is by definition zero. The reason for this definition of the travel metric is to force the batching of orders with items in the same aisles.
3.1.2. Order distances The SMD heuristic uses the concept of the distance between two orders, which is defined as the sum of the distances from each item in the first order to the closest item in the second order. For the Euclidean metric, let (Xim, Yim) be the coordinates of item i in order m. Define 210.5
gimn= m!n[(Xim-Xjn)2+(Yim-Yjn) 3
J
•
This Lim n is the distance between item i in order m and the nearest item in order n. We then sum the Li.,n to obtain the distance d(m, n) between orders m and n:
d(m, n)= •Lim,,. i
In effect, each item in order m identifies a closest item in order n, and the corresponding distances are added. This pairing need not be unique. Further, the distance measure is not symmetric. That is, in general
d(m, n) 4~d(n, m). For the rectilinear and Chebyshev metrics, the Lim n a r e defined using the appropriate analog of d R and d c, respectively. For the aisle metric, let aim be the number of the aisle that contains item
i of order m. Then, for equally spaced and sequentially numbered aisles: Lira n
=
mini aim -- ajn 1.
J
The d(m, n) are defined in the same way for all metrics.
3.1.3. Tour sequencing The sequencing of retrievals of the items in a batch is an example of the Traveling Salesman Problem, which is NP-complete [Papadimitriou, 1977]. Under the Euclidean, rectilinear, and Chebyshev travel metrics, all the items in a batch are sorted according to their item-theta values, and the tour simply follows this list in sequence. The distances between items are measured using the appropriate metric. Under the aisle travel metric, with selective, one-way routing, tour length calculation is based on the particular set of aisles which must be visited in order to complete the tour effectively. If the number of aisles to be visited is odd, an extra aisle traversal is needed to return to the I / O point.
3.2. Storage region representation The storage region, which occupies a unit square, is represented in two ways. The first is the continuous representation, where any point inside or on the unit square is a possible item location. This representation, like the Euclidean travel metric, was included so that we may compare our results with those of others. We are also interested in any possible difference between it and the second representation. The second representation is the discrete representation, where there are only a finite number of item locations (800) in the storage region, as shown in Figure 3. This corresponds, in the context of a horizontal travel system, to 10 aisles with 40 locations along each side of an aisle. In the context of a person-aboard storage system this means 40 horizontal positions along each of 20 vertical levels.
3.3. Item location assignement Uniform item location assignment means that the probabilities of storage locations being item
D.R. Gibson, G.P. Sharp /Order batchingprocedures
locations are independent and equally likely. For example, under the continuous representation, the density function corresponding to item locations in the unit square is the uniform distribution: f ( x , y) = 1, 0 ~
63
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Figure 4. Skewed item location assignment
use 5 items per order. For the variable case we assume the number of items follows a positive geometric distribution with a mean of 5.
3.5. Number of orders considered in batching process
3.4. Number of items per order We consider two cases, fixed and variable number of items per order. For the fixed case we
1140 1/41 1/41
1/20
1120
We consider 12 situations: 100 order, 200 orders, and so on up to 1200 orders. This corresponds to generating 500, 1000, 1500 . . . . . 6000
I120
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,,oi
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64
D.R. Gibson, G.P. Sharp /Order batchingprocedures
item locations, respectively. With a fixed number of items per order, and exactly 10 orders per batch, we generate 10, 20, 30 . . . . . 120 batches of size 50, respectively. When the number of items per order is variable, we generate exactly the same number of items as in the fixed situation; however, the number of orders is not necessarily 100, 200 . . . . . 1200. For the variable situation we do, however, create exactly the same number of batches as in the fixed Case, making them as close to size 50 as possible.
3.6. Framework for experiment, replications Simulation runs were performed for each of the three heuristics under 24 identical sets of experimental factors, for 72 combinations (shown in Table 4). Not all metrics were paired with both storage region representations: with Euclidean only a continuous representation was used, and with aisle only a discrete representation was used. For each of the 72 combinations, the number of orders varied from 100 to 1200 in steps of 100. A different random number seed was used to generate the item locations for each of the 12 values. Also, each of the 12 values was replicated 10 times using different random number seeds. Thus, a total of 72 * 12 * 10 = 8640 simulations were performed, with some exceptions. The batching heuristics were coded in Turbo Pascal, Version 5.0, and the simulations were performed on an IBM P S / 2 Model 80, running at 17.6 MHz.
4.2. Item location assignment Skewed item location assignment plays a significant role in the reduction of tour lengths. FCFS batching experiences reductions of 15-19% when item locations are skewed, as shown in Figure 5, by comparing the curves for baseline (FADUV) and baseline, skewed (FADSF). The most dramatic reductions in tour length occur when intelligent batching, either SFC or SMD, is performed and item locations are skewed, as shown in Table 3. The numbers are the range of average batch tour length reductions from the baseline case, as the number of orders ranges from 100 to 1200. As an example, consider SRDSV, in the fourth column. The average batch tour lengths produced by SFC batching under rectilinear travel, discrete representation, skewed item locations, and variable number of items per order ranges from 26 to 32% shorter than tours produced by FCFS batching under identical experimental conditions, ex-
12 O
¢
¢
¢
¢
¢
¢
.
.
.
.
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Baseline, F A D U V
lO
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4. Analysis of experiment 4.1. General obseruations As expected, FCFS batching produces no decrease in average batch tour length regardless of the number of orders and experimental factors considered. The FCFS heuristic used in conjunction with the aisle travel metric produces the longest tour with an average batch tour length of 11.8, followed by the rectilinear metric at 8.6, Euclidean at 7.0, and Chebyshev at 6.3. These values were obtained under the baseline conditions where item location assignment is uniform, variable number of items per order, and the number of orders range from 100 to 1200.
s
4
Batching, Skewed, MADSV
I
loo
I
i
300
i
i
500
I
I
too
I
I
9oo
I
11100
I
12oo
Number of Orders FADUV - FCFS betC.hing, aisle metric, discrete representation, uniform ~arn location, v~ixble order size •
FADSV - FCFS betching, aisle metric, discrete representation, skewed item location, vaZiable order size MADSV. SMD heuristic, aJsJemetric, discrete representation, skewed ~tam location, varlablo order size
Figure 5. Results for skewed item location assignment
D.R. Gibson, G.P. Sharp / Order batching procedures Table 3 Selected tour length comparisons Baseline
Skewed item locations, savings over baseline
Intelligent batching, savings over baseline
Skewed item locations and intelligent/ batching savings o v e r / baseline
FADUV a
FADSV 18%
SADUV 1-10% MADUV
FRDUV
FRDSV 18%
SRDUV 2-14% MRDUV 1-10% SECUV 2-8% MECUV 2-8% SCDUV 1-8% MCDUV
SADSV 14-21% MADSV 27-44% SRDSV 26-32% MRDSV 21-27% SECSV 23-29% MECSV 20-28% SCDSV 24-29% MCDSV 20-26%
1-10%
FECUV
FECSV 17%
FCDUV
FCDSV 17%
1-8% a Mnemonic explanation:
I st character: Heuristic. F - First-come, First-Served. S - Spacefilling Curve. M - Sequential Minimum Distance. 2 na character: Travel metric. A - Aisle. R - Rectilinear. E - Euclidean. C - Chebyshev. 3 ra character: Warehouse Representation. C - Continuous. D - Discrete. 4 th character: Item location assignment. U - Uniform. S - Skewed. 5 th character: Nu mb er of items per order. F - Fixed. V - Variable.
cept that item locations are uniform, as the number of orders ranges from 100 to 1200. These percentage reductions reflect the combined effect of intelligent batching and skewed item locations. SFC batching, under the rectilinear, Euclidean, and Chebyshev travel metrics, results in reduction of 23-29%, and under the aisle travel metric, reductions of 14-21% below the values obtained from FCFS batching with uniform item locations. Run times for the SFC heuristic are about 2 minutes to batch 1200 orders. The SMD heuristic with skewed item locations results in reductions of 20-26% under the recti-
65
linear, Euclidean, and Chebyshev travel metrics, and reductions of 27-44% under the aisle metric. The entry 27-44%, under MADSV in Table 3, is shown graphically in Figure 5. The curve labelled intelligent batching, skewed (MADSV) is 27-44% below the curve labelled baseline (FADUV). Run times of this heuristic under the aisle metric average about 4 minutes to batch 1200 orders, 10 minutes under the rectilinear and Chebyshev metrics, and 69 minutes for the Euclidean metric. 4.3. Storage region representation A discrete representation of the storage region always produces shorter average tour lengths than a continuous representation. These reductions range from 2-9% over all experimental conditions. A typical example is shown in Figure 6 by the two curves labelled continuous (FRCUF) and discrete (FRDUF). In general, however, the storage region representation does not cause any other experimental factor to behave much differently. For example, if a particular heuristic performs better than another under the continuous representation, the same is true under the discrete representation. Thus, the storage region representation should not be considered in future research. 4.4. Number of items per order Using the FCFS heuristic, shorter tours result with variable order size; however, these reductions rarely exceed 1%. The two other heuristics consistently produce shorter tours with fixed order size. With SFC and SMD batching under the rectilinear Chebyshev metrics, reductions are 710%. The SMD heuristic under the aisle metric has reductions of 7-18%, as shown in Figure 6 by the curves labelled fixed (MADUF) and variable (MADUV). These values point to the danger of under-estimating tour lengths when a fixed order size is assumed. 4.5. Travel metric The aisle travel metric, the most realistic, produces the longest tours and therefore has room for the greatest tour length reduction. With aisle travel the SMD heuristic outperforms the SFC heuristic under all conditions. With the rectilinear and Chebyshev metrics, the SFC heuristic
D.R. Gibson, G.P. Sharp / Order batchingprocedures
66 Table 4 Summary of results Exptl. factors
Table 4 (continued)
Number of Orders 100
300
600
1200
SADUF SADUV SADSF SADSV SRCUF SRCUV SRCSF SRCSV SRDUF SRDUV SRDSF SRDSV SECUF SECSF SECSV SCCUF SCCUV SCCSF SCCSV SCDUF SCDSF SCDSV
11.503 11.768 0,748 10.131 8,529 8,726 6,399 6.781 8.160 8.413 5.859 6.340 6.723 5.053 5.356 6.066 6.181 4.568 4.842 5.876 4.319 4.647
11.096 11,402 9.235 9.878 8.118 8.436 5.999 6.597 7.775 8.118 5.706 6.207 6.398 4.766 5.254 5.769 6.004 4.330 4.777 5.605 4.222 4.582
10.790 11.268 8.738 9.632 7.875 8.254 5.852 6.431 7.542 7.941 5.434 6.035 6.215 4,663 5.132 5.608 5.863 4.243 4.673 5.462 4.055 4.499
10.554 11.016 8.430 9.326 7.771 8.134 5.691 6.231 7.368 7.792 5.251 5,847 6.101 4.549 4.981 5.509 5.785 4.147 4.540 5.345 3.944 4.372
FADUF FADUV FADSF FADSV FRCUF FRCUV FRCSF FRCSV FRDUF FRDUV FRDSF FRDSV
1.778 11.778 9.678 9.789 8.918 8.898 7.404 7.310 8.542 8.586 6.950 6.885
1.778 1.777 9.613 9.547 8.881 8.893 7.354 7.379 8.586 8.627 7.092 7.027
1.774 1.778 9.645 9.624 8.838 8.894 7.421 7.414 8,577 8.594 7.022 7.004
1.775 1.772 9.589 9.635 8.882 8.865 7.392 7.370 8.607 8.595 7.092 7.019
MADUF MADUV MADSF MADSV MRCUF MRCUV MRCSF MRCSV MRDUF MRDUV MRDSF MRDSV MECUF MECUV MECSF MECSV MCCUF MCCUV MCCSF MCCSV
11.381 11,691 7.969 8.634 8.517 8.715 6.937 7.124 8.418 8.517 6.428 6.644 6.666 6.864 5.433 5.574 5.985 6.162 4.942 5.061
10,398 11,207 6,535 7,892 8,235 8,501 6.681 6.970 8.198 8.235 6.343 6.598 N/A N/A N/A N/A 5.835 6.020 4.711 4.956
9.767 10.773 5.851 7.187 8.052 8.360 6.521 6.866 8.068 8.052 6.030 6.432 6.346 6,566 5.083 5.375 5.724 5.936 4.598 4.857
9.031 10.331 5.426 6.547 7.967 8.250 6.327 6.684 7.933 7.967 5.877 6.296 6.248 6.461 4.919 5.249 5.619 5.819 4.454 4.717
Exptl. factors a
Number of Orders 100
300
600
1200
MCDUF MCDUV MCDSF MCDSV
5,882 6.044 4.510 4.919
5.722 5.876 4.280 4.813
5.577 5.786 4.188 4.680
5.435 5.686 4.062 4.545
FECUF FECUV FECSF FECSV FCCUF FCCUV FCCSF FCDUF FCDUV FCDSF FCDSV
7.021 6.987 5.840 5.765 6.334 6.293 5.280 6.168 6.135 5.067 5.004
6.995 7.002 5.818 5.839 6.308 6.306 5.270 6.180 6.165 5,138 5.112
6.958 6.998 5.872 5,854 6.271 6,307 5,318 6,151 6.133 5.117 5.102
6,988 6.983 5.840 5.824 6.297 6.298 5.218 6.156 6.162 5.113 5.107
a Experimental factors are defined by the same mnemonics as in Table 3.
12
Continuous, FRCUF
I0
1
e-
3 ~8
=-~=
=
"
=
=
= l=
=
:
"
Fixed, MADUF
I 100
I
I
300
I
I
500
I
I
I
700
I
I
900
I
I
1100 1200
Number of Orders •
FRCUF - FCFS hatching, reclJlinNr metric, continuoul representation, uniform item location, fixed order SiZe
•
FRDUF - FCFS bitching, rectilinear metric, dilcrete reprelentation, uniform ~
[]
Io¢~ltJon, fixed or~o," liZe
MADUV - SMD heuristic, l i l l e metric, dilcreta repreltantation, uniform item location, variable Order IJzO
©
MADUF - SMD heuristic, l i l l e metric, dil~rete representation, uniform item location, fixed order size
Figure 6. R~sults for warehouse representation and number of items per order
D.R. Gibson, G.P. Sharp /Order batching procedures
almost always outperforms the SMD heuristic. With the Euclidean metric, the SFC heuristic is again superior to the SMD heuristic, but only slightly. Table 4 contains a summary of results obtained under various experimental conditions.
67
at the Georgia Institute of Technology and by the National Science Foundation under grant ECS8351313. The authors gratefully thank two anonymous reviewers for their helpful comments.
5. Conclusions and recommendations References
Skewed item location assignment used in conjunction with intelligent batching yields significant reductions in average batch tour lengths. In some instances these reductions are 44%. In situations where the Euclidean, rectilinear, or Chebyshev metric is appropriate, it is recommended that the SFC heuristic be used because tours are generally shorter and the execution time is much quicker than the SMD heuristic. In a typical warehouse with an aisle metric, it is recommended that the SMD heuristic be used, because the tours are significantly shorter than the ones obtained by the SFC heuristic and the execution time of the SMD heuristic is more reasonable. A discrete storage region representation is preferred because it is more realistic than the continuous case. However, the relative performance of batching algorithms is not affected much by the representation. There is a danger of under-estimating tour lengths when a fixed order size is assumed.
Acknowledgement
This research was supported by US industry through the Material Handling Research Center
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