Placement of warehouse cross-aisles
Paul G. Berglund1 and Rajan Batta2
1
Department of Industrial and Systems Engineering, University at Buffalo, 438 Bell Hall,
Buffalo, NY 14260-2050, USA
Email: [email protected]
2
Department of Industrial and Systems Engineering, University at Buffalo, 438 Bell Hall,
Buffalo, NY 14260-2050, USA
Email: [email protected]
Abstract
Given an order-consolidation warehouse having a simple rectilinear aisle arrangement
with north-south storage aisles and east-west travel aisles (or “cross aisles”), this paper
investigates the optimal placement of the cross-aisles as a consequence of the probability
density function of the order pick locations, as determined by the storage policy. That is,
given some storage policy, what placement of the cross aisles will result in a minimal
expected path length for the picker?
An analytical solution procedure is developed for
the optimal placement of a single cross-aisle given for a given storage policy. Simplifying
assumptions are made as regards to the nature of this policy, but in particular the picks
need not be uniformly distributed. The solution procedure is generalized to a method for
multiple cross-aisles. The result obtained is checked by a simulation study.
Keywords: Warehousing, aisle design, order-picking, material handling
1
1. Introduction
A significant portion of the time (and therefore money) spent in order-picking in
an order-consolidation warehouse is the time spent traveling from pick location to pick
location. There are a number of possible approaches to reduce the costs associated with
picker travel: use an efficient routing policy (have pickers take the shortest possible path
that visits all pick locations), store items in the warehouse so that frequently accessed
items are nearby, and so on.
A key factor (but one that is more often assumed as a fait accompli than made the
object of analysis) is the layout of the warehouse. Travel through a warehouse
necessarily involves going around obstacles: the picker travels from pick point to pick
point, going through aisles. As the picker is constrained to move within these aisles, their
layout (number and positioning) will naturally have an effect on travel distances (and
therefore times). The number and width of aisles will also impact the effective storage
density of the warehouse, resulting in a trade-off between space and time efficiency: as
the number of aisles increases, picking travel times will decrease, but so will storage
density.
It should be noted at the outset that in practice travel-aisle configurations may be
changed without incurring prohibitive costs. Product is typically stored on shelving that
consists of a number of modular units bolted together. Thus they may be reconfigured
fairly easily, the greatest cost being the effort of unloading the shelves and then reloading
them once the shelves have been re-assembled in their new configuration. Thus the
optimum aisle configuration is potentially valuable information, due to the practical
possibility of acting on it.
2
The optimal positioning of aisles is conceptually simple: a cross aisle will provide
a greater benefit if it is close to those locations where the most picks are made; to add an
aisle in a seldom-visited area of a warehouse would be to trade storage capacity for only a
small benefit in picking efficiency. Therefore a proper analysis of optimal aisle
placement should take into account pick densities (storage policies). This paper therefore
presents a solution method for the problem of where cross aisles should best be
positioned to optimize picker travel distance given some storage policy.
2. Literature Review
In order to minimize picking costs, a number of strategies may be employed,
including order batching, optimal routing algorithms, strategies to minimize congestion
(and therefore blocking), storage policies and facility layouts. All of these factors
interact and to consider all of these factors at once would be an overwhelming task, so
studies typically make some simple assumptions about those factors which are not their
main focus. Therefore studies might be characterized by which factors they ignore (or
essentially remove from the problem via simplifying assumptions) and which they focus
on.
Although a very substantial literature exists on routing algorithms, storage
policies and facility layouts, less has been written specifically on the impact of aisle
layouts on picker travel distances or times. Vaughn and Petersen (1999) used a
simulation study to calculate the optimal number of evenly-spaced cross-aisles in a
warehouse, assuming a certain routing policy and a uniform (random) storage policy.
Thalayan (2008) used simulation to compare the effects on travel time of a number of
3
different factors, including the number of cross-aisles, storage policy and routing policy.
Gue and Meller (2009) used an analytical approach to derive an unusual but efficient
aisle configuration for a unit-load warehouse assuming uniform pick density. Pohl et. al.
(2009) use an analytical approach to evaluate the efficiency of three different aisle
configurations for dual-command operation and a uniform pick density. They also prove
that, for this case, the optimal position of a single east-west cross aisle in a warehouse
with north-south storage aisles must be between the center of the warehouse and the top
cross aisle. It should be noted that although these authors consider a variety of different
ways in which warehouse configurations might be varied, such as shape, non-rectilinear
aisle configurations, etc., this paper addresses only the optimal positioning of east-west
travel aisles in a warehouse with north-south storage aisles.
This paper focuses on two factors: storage policy (represented here in the form of
the distribution function of pick locations) and facility layout (more specifically the
question of cross-aisle position), and their effect on picker travel distances. Given some
storage policy, an optimal cross-aisle position is calculated. An assumption is made that
pickers will be routed by a simple heuristic that will not always generate the shortest
possible route. Possible congestion effects are not considered.
The remainder of this paper is organized as follows: the model is described and an
analytical solution for the case of positioning a single cross-aisle is presented (including
an illustrative example). The solution is then extended to an arbitrary number of crossaisles. Some examples of the possible applications of the procedure are given.
Simulations are performed to estimate the effect of the simple routing heuristic used on
picker path lengths and on the resulting optimal cross-aisle positions.
4
3. Model
Consider an order-picking warehouse with M vertical (or “north-south”) storage aisles.
Each storage aisle has B discrete pick locations of uniform size (e.g. one pallet width),
numbered 1,2…B with the 1st location being the “southernmost” and the Bth location
being the “northernmost”. There are three lateral (or “east-west”) cross aisles, one at y=0
(i.e. “south” of all pick locations), one at y=B (“north” of all pick locations) and one at
y=h, where h is to be determined (h being the number of pick locations “south” of the
middle cross-aisle, where 0<h<B). There is a single I/O point at y=0, at some x
coordinate. For our purposes the position of the I/O point may be disregarded because it
has no effect on the amount of north-south travel (i.e. travel in the storage aisles) and
therefore no effect on the optimal location of the cross-aisle. The position of the I/O
point will of course have an impact on the amount of east-west travel (i.e. travel in the
cross-aisles), but we will neglect this question.
We will begin by assuming the same routing model as Vaughan and Petersen: the
picker begins at the leftmost storage aisle from which items must be picked and picks all
items in that aisle, then proceeds to the nearest aisle to the right that has any items to be
picked, picks all items in that aisle, and so on until all items have been picked. (Pickers
are able to turn around in storage aisles and to traverse them in either direction, but
always move east-to-west in travel aisles, except before making the first pick or after
making the last.) The shortest path using this routing is easier to calculate than in the
general case, however the necessary computations are still complex, an analytical
solution will be correspondingly difficult to obtain.
5
In order to simplify this computation sufficiently to allow us to develop an
analytical solution, we will make the additional simplifying “naïve routing assumption”
that after making the final pick in a given storage aisle, the picker then departs via the
closest cross aisle to his current location, without considering the picking locations to be
visited in subsequent aisles. If equidistant from two cross aisles, the picker will choose
the cross aisle closest to the I/O point. Note that this will not always result in the picker
choosing the optimal route (see figure 1). The penalty for this simplified routing strategy
will vary depending on the pick density, the number of cross-aisles and the distribution
function of the pick points, but for the 80-20 distribution outlined below and a moderate
pick density the penalty is about 8 ½ per cent. See the appendix for a more detailed
discussion of the effects of the simplified routing assumption.
We will use the following terms (after Vaughan and Petersen):
N
the number of pick locations to be visited in a trip
M
the number of vertical (storage) aisles
Km
the number of pick locations to be visited in storage aisle m
Xm(t) the tth pick location in aisle m
Xm+
the largest pick location in aisle m
Xm-
the smallest pick location in aisle m
Assume that the pick locations to be visited are equally likely to be in any of the
M storage aisles and that for all m, Xm(1), Xm(2),…, Xm(Km) are independent and
identically distributed with probability mass function fX(x). The number of pick locations
to be visited in any given aisle will be binomially distributed with pmf
6
x
g x ( x) 
n!
1 
 1  
  1  
(n  x)! x!  M   M 
n x
Note that Xm+ and Xm- are respectively the Kmth and 1st order statistics for a discrete sample
of Km items, and will thus have probability mass functions, given by Siotani (1956), of
the form:
f X ( x)  ( FX ( x)) K m  ( FX ( x  1)) K m
and
f X ( x)  (1  FX ( x  1)) K m  (1  FX ( x)) K m
and their joint pmf is
f ( X m , X m )  [ F ( X m )  F ( X m  1)] K m  [ F ( X m )  F ( X m )] K m
 [ F ( X m  1)  F ( X m  1)] K m  [ F ( X m  1)  F ( X m )] K m
Optimal placement for a single cross-aisle will result in minimal expected “northsouth” travel (that is, travel in storage aisles), given the assumption that pickers will
employ our simplified routing strategy. Note that the expected amount of lateral or “eastwest” travel (that is, travel in the cross-aisles) will be the same regardless of the value of
h.
4. Algorithm
Define Pm as the north-south travel distance in aisle m, and P as the total northsouth travel distance. In order to find the optimal value of h, we must first find a formula
for the expected value of P as a function of h. Our N picks must be distributed among
our M aisles in some way. That is, we have some ordered set K= {K1,K2,…,KM} such
7
M
that
K
m 1
m
 N . If we know all the different possible distributions of picks among our
aisles, and the probability of each distribution, then we can calculate E(P) as follows:
E ( P) 
 Pr( K ) * E ( P | K )
K K *
where K* is the set of all possible distributions of picks among our aisles. The
calculation of the different possible distributions and their respective probabilities is
relatively straightforward.
To calculate E(P|K), we sum the expected path lengths in all aisles:
M
M
m 1
m 1
P   Pm , so E ( P | K )   E ( Pm | K )
The formula for the path length in a given aisle may be found by decomposing the
problem into the differing cases:
Case 1: aisle m is entered at y=B
Case 2: aisle m is entered at y=0
Case 3: aisle m is entered at y=h
E ( Pm | K ) 
Pr(Case 1)*E(Pm|K and Case 1) +
Pr(Case 2)*E(Pm|K and Case 2) +
Pr(Case 3)*E(Pm|K and Case 3)
The expected path lengths for the different cases are calculated as follows. Cases 1 and 2
are straightforward; the picker will proceed either up (if entering at y=0) or down (if
entering at y=B) the storage aisle until all picks have been made and will then exit via the
closest cross-aisle to the final pick location, as shown in Fig. 2.
8
Case 1: storage aisle m is entered at y=B. We must travel down the storage aisle far
enough to make all picks (which amounts to traveling down to Xm-), whereupon we then
leave by the closest exit point (0, 1 or h). Therefore the length of the optimal path
depends only on the value of Xm-. There are four possible sub-cases:
h
, the closest exit point to Xm- is at y=0. Then the shortest possible
2
1. If 0  X m 
path length is B*wb+2*wa, where wa is the width of a cross-aisle and wb is the
width of a pick location.
2. If
h
 X m  h , the closest exit point to Xm- is at y=h, and at least one pick is made
2
at some location ≤ h. Then the shortest possible path length is
B  X   h  X   1* w

m

m
3. If h  X m 
b
 2 * wa .
Bh
, the closest exit point to Xm- is at y=h, and no picks are made at
2
any locations ≤ h. Then the shortest possible path length is
(B- h)*wb + wa.
4. If
Bh
 X m  B , the closest exit point to Xm- is at y=B and the shortest possible
2
path length is (2*(B- Xm-)+1)*wb+wa
From this we can derive an expression for the expected path length in case 1:
E ( P | Case1) 
f
1 x  h / 2

f

X

X
h x ( B  h ) / 2
( x)B * wb  2 * wa  
f
h / 2 x  h
( x)B  h  * wb  wa  
9

X
( x)B  x   h  x   1 * wb  2 * wa 
f

X
( b  h ) / 2 x  B
( x)2 * B  x   1 * wb  wa 
Case 2: storage aisle m is entered at y=0. Then, analogously to case 1, the length of the
optimal path depends only on the value of Xm+, and is given by
E ( P | Case2) 
f
1 x  h / 2

f

X

X
h x ( B  h ) / 2
( x)2 * x  1 * wb  wa  
f
h / 2 x  h

X
( x)h * wb  wa 
( x)h  2 * ( x  h)  1 * wb  2 * wa  
f

X
( b  h ) / 2 x  B
( x)B * wb  2 * wa 
Case 3: storage aisle m is entered at y=h. In this case, the values of both Xm+ and Xm- are
relevant, and to find the expected minimum path length we must sum over the domain of
the joint pmf of Xm+ and Xm-. Because we may have some picks above h and some below,
we must calculate the path lengths of the two possible routes (first making all picks above
h and then picks all below h, or the reverse), and take the minimum of the two.
For all i,j , 0  i  j  B , define Pm* (i, j ) as the minimum path length to pick all items
in aisle m given that X m  i and X m  j . For any given i,j Pm* (i, j ) is straightforwardly
calculated as follows:
1. Let Pm1 (i, j ) =
(the distance from the cross-aisle at y=h to i)+
(the distance from i to j)+
(the distance from j to the closest cross-aisle to j)
2. Let Pm2 (i, j ) =
(the distance from the cross-aisle at y=h to j)+
(the distance from j to i)+
(the distance from i to the closest cross-aisle to i)
10
Then Pm* (i, j )  Min Pm1 (i, j ), Pm2 (i, j ), and
B
B
E ( P | Case3)   f (i, j ) Pm* (i, j )
i 1 j i
Once we have computed the values of the above conditional expectations, to be
able to compute E(P), the expected optimal path length, as a function of h, we know only
the probabilities of entering the aisle at y=0, y=B and y=h (i.e. P(Case1), P(Case2) and
P(Case3)). We start with the boundary condition that in the first aisle with picks, the
aisle will always be entered at y=0. Likewise, the rightmost aisle with picks will always
be exited at y=0. Furthermore, if for a given aisle we know the probabilities of the three
cases and the number of picks in the aisle, we can calculate both the expected travel
distance within the aisle and the respective probabilities of leaving the aisle at y=0, y=B
or y=h. Thus if we know the number of picks in each aisle, we may iterate over all aisles
to compute the total expected travel distance for making all picks. Therefore, if we
compute the probabilities of each combination of number of picks per aisle, we can
iterate over all of these combinations to calculate, for given values of h, N, M, and fX(x),
the expected travel distance needed to make all picks.
5. Computation of the optimal aisle position
As we are able to compute the expected path length for each pattern, and we know
the probability of each pattern, we are thus able to compute the expected path length for a
given value of h. Once we know how to compute the expected path length for a given
value of h, the next step is to compute that value of h for which the picker’s expected
travel distance is minimized. We can do this by evaluating E(P) for various values of h.
11
As an example, we do this for the “80-20” distribution function, so-called because 80%
of picks are in the 20% of the aisle closest to the I/O point:
 0.08 1  x  8

f X ( x)  0.005 9  x  50
 0
otherwise

Assume 50 pick locations per aisle, 20 storage aisles and 5 picks per trip, wa=10
and wb=5. For this case we obtain the results shown in Figure 3: the expected path length
E(P) as a function of h, the position of the intermediate cross-aisle. The minimal value of
E(P) is 454.455, achieved when h=8.
6. Extension for additional cross-aisles
The foregoing may be straightforwardly extended to finding the optimal positions
for two or more “floating” cross-aisles, at y=h1, y=h2, etc. Calculations for the fixed
aisles at y=B and y=1 may be computed in the same way same as for cases 1 and 2 above,
whereas calculations involving the movable interior aisles are done as in case 3. It should
be noted that as the number of cross-aisles increases so too does the cost of evaluating the
objective function, due to the number of possible paths through the different aisles, the
probabilities of all of which must be computed. Furthermore, the number of objective
function evaluations necessary to find the optimal solution increases as well. However as
a single objective function evaluation remains sufficiently inexpensive that it is still
feasible to use full enumeration to find the optimal solution. For example, the optimal
solution for the 80-20 distribution with five cross-aisles was found by full enumeration in
136 seconds. See table 1 for detailed performance data for objective function evaluation.
7. Examples
12
The procedure described above could be applied in a number of ways. For
example, once it has been used to identify an efficient aisle configuration for a given
storage policy, that aisle configuration may be compared to some baseline configuration
to measure the savings that it would produce. Table 2 shows such a comparison: the
predicted optimal aisle configuration is shown in simulation to be between 6 and 13.9
percent more efficient than a configuration with equally-spaced cross aisles, depending
on the pick density.
Another example would be to show the how the optimal aisle configurations vary
for different storage policies. The procedure was used to analyze three different storage
allocation policies. For each, the optimal aisle configuration was computed. In all cases,
B=50, A=5, N=5 and M=20.
Case 1: the “80-20” distribution, as described above
Optimal Aisle positions: 0, 6, 29, 44, 50
Objective Function Value: 426
Case 2: the “60-30-10” distribution:
 3 / 25 if

f ( x)   1 / 50 if
1 / 300 if

1 x  5
6  x  20
21  x  50
Optimal Aisle positions: 0, 4, 15, 38, 50
Objective Function Value: 399
Case 3: the “offset 80-20” distribution:
1 / 200 if

f ( x)   2 / 25 if
1 / 200 if

1  x  10
11  x  20
21  x  50
Optimal Aisle positions: 0, 16, 35, 48, 50
13
Objective Function Value: 449
As we can see, each storage policy has a different optimal aisle configuration. As
compared to the “80-20” distribution, the “60-30-10” distribution has its first movable
cross aisle closer to the I/O point, at h=4, whereas the “offset 80-20” distribution has its
first movable cross aisle farther away, at h=16. This makes intuitive sense as the “60-3010” distribution bunches its picks closer to the I/O point (90 % of them are at x≤20), as
contrasted with the “offset 80-20” distribution where they are pushed father away.
Finally, we can corroborate the theorem proven by Pohl et. al., that the optimal
position of a single movable cross-aisle with dual command-picking will be between the
center of the warehouse and the top cross aisle (their Proposition 1). The results, as
shown in table 3, agree with the proof by Pohl et. al. that the optimal aisle position will
be beyond the midpoint of the warehouse.
8. Conclusions
As stated earlier, there are many examples of warehouses that, in order to increase
efficiency, do not store goods using random storage policies (i.e. with uniform
distributions). In such cases, the most efficient cross-aisle positions will not be equallyspaced. Furthermore, as the cost of adjusting cross-aisle positions is not prohibitive,
practitioners will be able to benefit from knowing the maximally efficient positions for
cross-aisles corresponding to their storage policies.
We have presented a method for calculating maximally efficient cross-aisle
positions for order picking warehouses, subject to a certain set of assumptions and
constraints. This work may be developed into a tool that could be used to calculate
14
optimal cross-aisle positions. At some difficulty, various assumptions of the current
work may be relaxed so as to make the results applicable to a wider set of circumstances.
15
Appendix
Simulation was used to validate the results of the analytical study in two different
ways. An estimate was made of the increase in average path length due to the naïve
routing assumption. The path length resulting from the naïve routing assumption was
compared to that resulting from the routing assumption used by Vaughan and Petersen,
where storage aisles are visited in a strict left-to-right sequence, but the shortest possible
path subject to that restriction was found (using a dynamic programming algorithm). For
the sample problem with the 80-20 distribution, cross-aisles at 0, 6, 29, 44 and 50 , 50
pick locations per storage aisle, 20 storage aisles and 5 picks per trip, the results were that
the average penalty of the naïve routing assumption was 8.64%. This is based on a
sample of 100,000 picking trips. For the three-aisle case, the discrepancy for the optimal
configuration (cross-aisles at 0, 8 and 50) was 13.49%. It makes intuitive sense that the
discrepancy should be larger in this case: when the cross-aisles are fewer and therefore
farther apart, the penalty for using the wrong cross-aisle will be larger.
But does this larger discrepancy between the expected path lengths when using
the different routing assumptions cause us to find the wrong optimal solution?
Simulation was used to estimate the expected path length using Vaughan and Petersen’s
routing assumption for a number of different candidate solutions to the sample problem
of the 80-20 distribution, with N=5 and M=20, 50 pick locations per aisle and 3 crossaisles, and the position of the middle aisle varying between 1 and 49. Figure 4 plots these
simulated path lengths against the values calculated analytically using the naïve routing
assumption. Although Vaughan and Petersen’s routing assumption resulted in
16
significantly shorter path lengths, the two curves are quite similar in shape and have the
same minimum point (at h=8).
17
References
Gray, A.E., Karmarkar, U.S., and Seidmann, A. (1992) Design and operation of an orderconsolidation warehouse: models and application. European Journal of Operational
Research, 58(1), 3-13.
Gue K.R., Meller, R.D. and Skufca, J.D. (2006) The effects of pick density on order
picking areas with narrow aisles. IIE Transactions, 38(10), 859-868.
Gue, K.R. and Meller, R. D. (2009) Aisle configurations for unit-load warehouses, IIE
Transactions, 41(3), 171-182.
Jarvis, J.M. and McDowell, E.D. (1991) Optimal product layout in an order-picking
warehouse. IIE Transactions, 23(1), 93-102.
Pohl, L.M., Meller, R.D. and Gue, K.R. (2009) An analysis of dual-command operations
in common warehouse designs, Transportation Research Part E, 45 (3), 367–379.
Siotani, M. (1956) Order statistics for discrete case with a numerical application to the
binomial distribution, Annals of the Institute of Statistical Mathematics, 8(1), 95-104.
Thalayan, P. (2008) Comparative study of item storage policies, vehicle routing strategies
and warehouse layouts under congestion, Unpublished M.S. thesis, State University of
New York at Buffalo, Buffalo NY.
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), 881-897.
18
Figure 1 In the above example, the “naïve” routing method (shown on the left) results in a longer
path length than the optimal routing (shown on the right).
19
Figure 2 In the case where we enter an aisle at y=0 or y=B, make all picks and then exit via the closest
cross-aisle to our last pick location.
20
Optimal aisle position
540
530
520
510
500
E[P]
490
E(P)
480
470
460
450
440
0
10
20
30
40
50
60
h
Figure 3 – Expected vertical path length as a function of cross-aisle location (h) for the “80-20”
distribution, with 50 pick locations per aisle, 20 storage aisles and 5 picks per trip, wa=10 and wb=5.
21
Effect of Routing Assumption
520
500
Path Length
480
460
E(P)
sim
440
420
400
380
0
10
20
30
40
50
60
cross-aisle position
Figure 4: simulation results for the 80-20 distribution, with N=5 and M=20, 50 pick locations per aisle
and 3 cross-aisles, with the position of the middle aisle varying between 1 and 49. The blue line
shows the analytical result for E[P] given the naïve routing assumption, and the pink line shows the
simulation result for the same aisle configurations, given the routing assumption used by Vaughan
and Petersen. Although the naïve routing assumption results in a significant penalty (longer trip
lengths), the optimal aisle position is the same in both cases (h=8).
22
N=2
N=3
N=4
N=5
N=6
N=7
N=8
N=9 N=10
A=3,B=25
A=4,B=25
A=5,B=25
1
1
2
1
1
2
1
1
2
1
2
2
1
2
3
1
2
3
1
2
3
2
3
3
2
3
4
A=3,B=50
A=4,B=50
A=5,B=50
2
4
6
3
5
7
3
5
7
4
6
8
4
7
9
5
7
10
5
8
11
6
9
12
7
10
13
A=3,B=100
A=4,B=100
A=5,B=100
10
16
23
12
18
25
13
20
28
15
23
31
17
25
34
19
28
35
21
30
38
23
32
41
25
35
44
Table 1 – function evaluation speed (mS)
N – number of picks per trip
A – number of cross aisles
B – number of pick locations per storage aisle
The results are averaged over various values of M (the number of storage aisles), though in practice
the value of M affects performance little if at all. All performance numbers were obtained running
64-bit Windows Vista on a 2.10 GHz machine with 4.00 GB of RAM.
23
N=2
N=3
N=4
N=5
N=6
N=7
N=8
N=9
N=10
Path
length,
h=8
189.24
259.21
328.42
397.31
465.11
531.75
597.70
662.80
726.66
Path
length,
h=25
201.33
287.48
371.83
454.89
536.10
615.42
693.45
769.70
843.99
Difference
12.09
28.27
43.41
57.58
70.99
83.67
95.75
106.90
117.33
Percent
savings
6.01
9.83
11.67
12.66
13.24
13.60
13.81
13.89
13.90
Table 2 - Savings of optimal cross-aisle position compared to centered cross-aisle for the 80-20
distribution, with M=25, N=2 thru 10, based on 1,000,000 simulation runs per value of N. The
optimal aisle position of h=8 saved between 6 and 13.9 percent as compared to the centered cross
aisle at h=25.
24
N
M
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Z*
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
h*
382.00
390.45
394.63
397.14
398.81
400.00
400.90
401.59
402.15
402.61
402.99
403.31
403.58
403.82
404.02
404.20
404.37
404.51
404.64
404.76
404.87
404.97
405.06
405.14
405.22
405.29
405.35
405.41
405.47
30
28
28
28
28
28
28
28
28
28
28
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
Table 3 - Optimal cross-aisle position for dual-command operation for the uniform distribution, with
B=50 and M between 2 and 30. The value of h* is always greater than 25, which agrees with Pohl et
al’s proposition 1.
25