(1)
This paper examines the problem of where a storage/retrieval machine should reside, or dwell, when an automated storage and retrieval system (AS/RS) becomes idle to minimize the expected value of the next transaction time. After a review of the relevant literature on AS/RS dwell point strategies, this paper proposes several analytical models of these expected response times of the AS/RS based on the relative locations of the input and output ports of the AS/RS. It uses a continuous rack approximation to provide analytical models of the dwell point location problem. These models provide closed form solutions for the dwell point location in an AS/RS. Extensions are made to consider AS/RS with a variety of configurations including multiple input and output ports. These models not only provide solutions to the dwell point location problem, but they provide considerable insight into the nature of this problem, which is particularly valuable when the requirements facing the AS/RS are uncertain.
In today's manufacturing environments, inventories are maintained at lower levels than in the past. These reduced inventories have led to smaller storage systems, which, in turn, have created the need for quick access to the material being held in storage. Hence, automated storage/retrieval systems used in manufacturing, warehousing, and distribution applications must be designed to provide quick response times to service requests in order to keep the system operating efficiently. One important operational aspect of the AS/RS, which contributes to the system response time, is the dwell point location of the S/R machine. The dwell point is the location where the S/R machine is positioned when the AS/RS is idle and awaiting the next service request. Egbelu and Wu (1993) have shown the choice of the dwell point location can have significant impact on the response time of the AS/RS. Our paper develops an analytical model for the determination of the optimal dwell point location for an S/R machine in an AS/RS system. The main contributions of this paper are (1) the development of a closed form solution for the dwell point location problem under a variety of AS/RS configurations and (2) the considerable insight this model offers for the AS/RS dwell point location problem.
Automated storage/retrieval systems have been the subject of much research over the past several years. Many of the papers deal with performance modeling of the AS/RS. Bozer and White (1984) developed expressions for the expected cycle time of an AS/RS performing single and dual command cycles. They used a continuous rack approximation to develop analytical models of the expected cycle time. In addition, they suggested several static dwell point rules for AS/RS, although they provide no quantitative comparison of their performance. Han et al. (1987) have improved the performance of an AS/RS by sequencing the order of servicing the retrieval requests. In this case, the AS/RS is assumed to be throughput-bound and therefore is never idle. Many other papers have looked at various aspects of AS/RS design and operation (e.g., see Graves et al., 1977; Hausman et al., 1976; Linn and Wysk, 1987).
Egbelu (1991) studied the AS/RS dwell point location problem. He developed formulations for minimizing the expected response time and minimizing the maximum response time for an AS/RS dedicated to a single aisle or shared between two aisles. He then transformed these constrained nonlinear programming formulations into linear programming problems. He discussed a framework for the dynamic selection of the dwell point, where updated information is used to generate a new linear programming problem each time a decision is to be made. The Egbelu formulation requires an estimation of the probability () that the next service request will be a load storage (implying that the probability of a load retrieval request is (1 - )) and the probability (i) that an incoming retrieval request will be for item i.
Hwang and Lim (1993) showed that the formulations from Egbelu (1991) could be transformed to facility location problems in order to improve the solution time. In particular, they transformed the minimize expected response time formulation to a single facility minisum Tchebychev location problem and the minimize the maximum response time formulation to a single facility minimax Tchebychev location problem. These reformulations reduced the required computational times by two orders of magnitude.
Egbelu and Wu (1993) used simulation to compare several dwell point rules. In particular, they compared the two formulations of Egbelu (1991) and the four static rules proposed by Bozer and White (1984). They found that the solution from the minimum expected response time formulation performed well as did the "always dwell at the input point" rule of Bozer and White. These rules dominated the other rules studied, but one was not found to always dominate the other.
We are interested in predicting the expected response time for an S/R machine at rest in an AS/RS to service a request. The response time is the time for the S/R machine to move from the dwell point to the location required to begin the request (e.g., the input point if the request is a load storage). Egbelu (1991) showed that this problem could be solved by summing the expected travel time to each location in the rack from an unknown dwell point and then solving this problem as a linear program. Our approach is to model the AS/RS rack face not as a discrete set of locations as in Egbelu (1991) but as a continuous rack face as in Bozer and White (1984). Many of our results use the expected cycle time equations developed in Bozer and White (1984) as a foundation; therefore, in the next section, we briefly review their assumptions, modeling approach, and results.
The AS/RS under consideration in this paper is a storage system with multiple aisles. Each aisle is served by a dedicated S/R machine that can reach a pick face on each side of the aisle. Hence, the modeling and analysis of the system can treat each aisle individually as long as the storage and retrieval requests to the system are separated by aisle. The S/R machine can carry one unit load at a time and can either store an inbound load in an empty location in the rack or pick up a load from a location in the rack and deliver it to the output point of the system. Each aisle has at least one input point and at least one output point for incoming and outgoing loads, respectively.
Bozer and White (1984) developed expected cycle time expressions for this type of AS/RS. They made the following assumptions about the S/R machines and the system operation, and unless otherwise specified, these assumptions will also hold for our developments in this paper.
Bozer and White (1984) have developed expected cycle time expressions for single and dual command cycles. An illustration of the continuous, normalized rack face is shown in Figure 1. Because randomized storage is used, the expected location for a storage or retrieval is randomly distributed between 0 and 1 in the horizontal direction and 0 and b in the vertical direction for the normalized rack. (Without loss of generality, we assume that the travel time from the I/O point to the farthest rack corner is larger in the horizontal direction than in the vertical direction.) If two random locations are represented by (x1,y1) and (x2,y2), then the normalized travel time between the two locations is given by max(x1-x2,y1-y2) (due to the Tchebychev travel metric). Given this expression, the following equations were developed.
(1)
(2)
(3)
where
Bozer and White (1984) use a statistical approach in deriving these expressions. Since they assume a randomized storage policy and a continuous rack face, a uniform distribution can be used to represent the location of items in the rack. Therefore, the distribution of travel time in the horizontal and vertical directions from a corner of the rack to a random location can be easily developed. The distribution of the travel time between two random points is slightly more complicated. In this case, they recognize that the travel time between two locations can be represented as the difference between the range of two order statistics. Given this realization, previous results for deriving the distribution of the range of order statistics are used to determine the distribution of the travel time in each direction. By the Tchebychev travel metric, the distribution of travel time is the product of the distribution in each direction (horizontal and vertical), assuming the x and y coordinates are independent. Once the distribution of travel times is known, the expected value of the travel time can be easily computed by integrating over the applicable range of the distribution.
Let us consider an AS/RS with the I/O point located at the center of the rack face, which is (0.5, 0.5b) in normalized time units. For convenience, let T = 1. Using the same assumptions and developments as described above, we can easily show that the optimal dwell point location is at the center of the rack face. An intuitive proof of this assertion is provided in the following paragraphs.
If the next service request is a retrieve, then the
server will move from the dwell point to the necessary bin location.
If nothing is known about the next item to be retrieved, then
the expected time to complete this move is 
,
since a randomized storage strategy and continuous rack approximation
are used. This expression is derived from the normalized rack
with the I/O point at the center. With this configuration, the
rack is essentially divided into four racks with the same shape
factor as the original rack and with the I/O point in one corner.
Hence, the results from Bozer and White for a single-command cycle
can be used, but now the normalization factor is 0.5. If the next
service request is a load storage, then the server response time
will be zero, since the dwell point is at the input point to the
AS/RS, and this is where the storage cycle must begin.
If the dwell point is moved to any other location in the rack besides the center, the expected response time will increase. This is clearly true if the next command is a storage, since the S/R machine will have to move from the dwell point location to the input location. If the next command is a retrieval, the expected response time will increase as well. If we assume that nothing is known about the next item to be retrieved, then, with a continuous rack approximation, any location in the rack is equally likely to be chosen. Therefore, the expected time to travel to a random point in the rack will be minimized if the server begins in the center of the rack. So, any dwell point other than the center point we increase the expected response time for a retrieval. Since moving the dwell point from the center of the rack will increase the expected response time for both a storage and a retrieval, the optimal dwell point location is at the center of the rack independent of , the probability that the next service request will be a load storage.
Notice that the location of the output point has no influence on the dwell point location. Once a load has been picked up from the rack, the time to reach the output point is independent of the dwell point location since the S/R machine has already moved from the dwell point location to the rack location containing the requested item. Hence, the determination of the dwell point location does not need to consider the location of the output point.
Consider an aisle of an AS/RS that has a single I/O point located in the lower left corner of the rack face. Let the dwell point for the S/R machine be located at (,b) in the normalized rack. The expected response time of the S/R machine for the next service request is given by E(RT):
E(RT) = () + (1-)E(V) (4)
where E(V) is the expected time to travel from the dwell point to a random location in the rack and is the probability that the next service request is a load storage. The first term in this expression (a(d)) is based on the assumption that the normalization factor (T) is 1. Note that this assumption simplifies the presentation, but does not limit the formulation.
The insights from the previous section also hold in this case. If the next command is a load storage ( = 1), then the optimal dwell point location is at the input point. Alternatively, if the next command is a load retrieval ( = 0), then the optimal dwell point location is at the midpoint of the rack face, which minimizes E(V). Hence, for any probability (0 1), we would expect the optimal dwell point to lie somewhere between the input point and the midpoint of the rack face.
The optimal dwell point location, in general, minimizes E(RT) and depends on the value of . To find this location, we need to develop an expression for E(V). First assume that the rack is square-in-time, i.e., b = 1. We will let T = 1 for ease of presentation. If the dwell point is located at (,), the rack face can be divided into four areas as shown in Figure 2.
Let E(V) represent the expected time to travel
from the corner of the rack face to a random location in the rack.
Then E(V) is one-half of the single command cycle time
shown in equation (1), which is 
. Consequently,
the expected travel time from the dwell point location to a random
rack location in region 1 of Figure 2 is 2/3.
Similarly, the expected travel time from the dwell point location
to a random rack location in region 2 is 2/3(1-).
The expected travel time for region 3 is the same as for region
1, since these regions are both square-in-time with dimension
. The expected travel time from the dwell point location to a
random rack location in region 4 is + bd(1 - 2) = bd. This expression
can be derived as follows. Since region 3 is square-in-time, each
point on the boundary between regions 3 and 4 can be reached in
exactly the same time, namely . Therefore, d
must be less than 0.5 for region 4 to exist. From this boundary,
the S/R machine only moves in one dimension to reach the required
rack location. Therefore, the expected time to reach a random
location will be the time to reach the midpoint, along this travel
dimension, of region 4. Since region 4 is (1 - 2) normalized time
units long in this direction, the expected travel is one-half
of this dimension, giving the expected travel time from the dwell
point to be + ½(1 - 2) = ½. This result shows that,
unlike the other three regions, the expected travel time from
the dwell point to a location in region 4 is a constant value
independent of the value of .
The expression for E(V) is then the weighted average of the expected travel times for the four regions. The weights are given by the relative areas of the regions. The resulting expression for E(V) is shown below.
E(V) = 2(2/3) + (1 - )2(2/3(1-)) + 2 2(2/3) + 2(1 - 2)( + ½(1 - 2))
which simplifies to
E(V) = 4/3 3 - + 2/3 (5)
Note that minimizing E(V) implies that * = ½, which confirms the intuitive argument given above that a dwell point at the center of the rack minimizes travel to a random location in the rack.
Substituting equation (5) for E(V) into the equation for E(RT) yields the following equation.
E(RT) = + (1- )(4/3 3 - + 2/3) (6)
which expands to
E(RT) = 4/3 3 - 4/3 3 + 2 - - 2/3 + 2/3 (7)
To find the optimal dwell point location, we can
take the partial derivative of E(RT) with respect to ,
which yields 
.
(8)
If we set 
= 0, we can find
the local extreme point, *.
(9)
The second derivative of E(RT) with respect
to , 
, is nonnegative since 0 1 and 0
1, and therefore, * is a minimum point. A particular
value of can be substituted into equation (7) and the minimum
value of found as the optimal dwell point location.
There are several subtleties associated with the above results that provide insight into the dwell point location problem. First, notice that equation (9) is only valid if the term under the square root is nonnegative. Since 0 1, the denominator will always be nonnegative. The numerator, however, will only be nonnegative if 0.5. If = 0.5, the numerator will be zero, and * will equal zero. This result implies that if nothing is known about the probability of the next service request (i.e., a load storage or retrieval is equally likely), then the S/R machine should dwell at the input point (0,0).
Now consider the range 0.5 < 1 in equation (7). We see that the expression will take a general form as follows, where K1, K2, and K3 are positive constants.
E(RT) = K1 3 + K2 + K3 (10)
Therefore, for any value of on the range (0.5,1], the optimal dwell point location will be at * = 0, which is at the input point location (0,0). So, if the probability that the next command is a load storage is greater than or equal to the probability that the next command is a load retrieval, the optimal dwell point location is at the input point (0,0); otherwise, the optimal dwell point location is given by equation (9).
Figure 3 shows two plots of the expected response time versus for various values of ranging from 0 to 0.5. Part (a) shows an approximation of the form of the function and part (b) shows a more detailed view of the same function. The interpretation of this plot is that starting from the horizontal axis at a particular value, the first function intersected with a vertical line gives the optimal value of . This plot illustrates the results that at = 0, = 0.5 is optimal and at 0.5, = 0 is optimal.
This insight helps explain the results found by Egbelu and Wu (1993). Their simulation testing showed that the "always dwell at the input point" rule from Bozer and White (1984) was good compared to other static rules. This rule also performed well compared to the location found by Egbelu's (1991) linear programming formulation. Figure 3 shows that the input point is the optimal location for several values of . It also shows that dwelling at the input point is not too far from optimal even if is slightly below 0.5. Only when is relatively small, is there a large penalty for using the input point as opposed to the optimal dwell point location.
In Egbelu's (1991) linear programming formulation, the input point was an alternative optimal dwell point location if = 0.5. In this case, anywhere on the line connecting the input point and the pickup point for the first rack location is optimal. (Note that as the number of rack locations increases to infinity, i.e., a continuous rack approximation, these alternative locations converge to the input point.) Hence, in many cases Egbelu's model may chose a different dwell point location even though the input point is also an optimal location. As Figure 3 shows, the input location is near optimal even if is as small as 0.35 or 0.4, which explains the good performance of the "always dwell at the input point" rule relative to Egbelu's LP formulation.
Further comparisons of the models developed in this paper and Egbelu's (1991) formulation can also be made. In general, the two approaches will give equivalent results if the assumptions made in this paper hold. In particular, the closed form models are more aggregate in that they don't incorporate knowledge about the specific locations of future retrieval requests. If these requests are randomly located throughout the rack, then the results from the closed form models and the results from Egbelu will be equivalent. However, Egbelu's formulation allows specific information about non-uniform retrieval requests to be included in determining the dwell point location.
In general this specific type of data may not be available, but some knowledge of the expected types of operations to be performed might be known. That is, a particular operation might perform mostly receiving and putaway in the morning (or first shift) and perform mostly retrieval and shipping in the afternoon (or second shift). In this case, the value of can be adjusted to account for the different modes of operation. Clearly over the long run, the number of storages must equal the number of retrievals. However, over any specific time period, the storage and retrieval requests may not be balanced and the probability can be dynamically modified to reflect this situation.
The above results are not limited to the I/O point being in the lower left rack corner. In fact, the results hold if the I/O point is located in any corner of the rack. In addition, the output point doesn't have to be collocated with the input point since the output point location does not effect the model. As a result, any output configuration is valid with the above dwell point location results.
This single input point model can be easily extended to other input point locations on the rack face besides a corner. The development procedure outlined above is followed in the same manner. The only change is the determination of the time to move from the dwell point location, represented by , to the input point to start a load storage. The expected time to move from the dwell point location to a random location in the rack is still given by the decomposition approach shown in Figure 2. This yields an equation for E(RT) similar to equation (7). A particular value of can be substituted and the value that minimizes E(RT) can easily be determined.
When the rack is not square-in-time (i.e., 0 b < 1), the results for = 0 and = 1 still hold. That is, if the next command is a load storage ( = 1), then the optimal dwell point location is at the input point. Alternatively, if the next command is a load retrieval ( = 0), then the optimal dwell point location is at the midpoint of the rack face, (1/2,b/2), which minimizes E(V). However, in the non-square-in-time case, if is between 0 and 1, the optimal dwell point does not lie on the line connecting the input point and the midpoint of the rack. Instead, the optimal dwell point will lie on the dotted line shown in Figure 4. This set of dwell point locations is due to the nature of the S/R machine's movement characteristics, i.e., Tchebychev travel. Any dwell point not on this line, e.g., point A in Figure 4, can be moved onto the line and be closer (in terms of travel time) to either the input point or the midpoint of the rack and no farther from the other, since the travel time is the maximum of the travel times in the horizontal and vertical directions. Therefore, the optimal dwell point will lie somewhere on this line and can still be represented by the single parameter, , which is the normalized distance of the dwell point location from the input point in the horizontal direction. Note that if tv, the normalized height of the rack, is greater than th, the normalized length, the rack can essentially be "flipped" on its side and the previous developments hold, so there is no loss of generality in the assumption that th is greater than tv.
In this situation, the decomposition approach, illustrated in Figure 2, is still valid, but the specific regions developed in Figure 2 are not valid. In particular, there are two cases for the decomposition depending on the dwell point location. Case 1 is when the dwell point location is on the 45b0 line originating from the input point, shown in Figure 5a, and case 2 is when the dwell point location is on the horizontal line, as shown in Figure 5b.
For each case, the expected travel time from the dwell point to a random location in the rack can be computed using the procedure developed in the previous section. The difficulty is that the objective function now has two different expressions depending on whether the dwell point is on the "slope" or the "flat." This difficulty requires an adjustment in the procedure for determining the optimal dwell point; however, the basic method outlined above serves as the foundation of a solution methodology for the case of a general AS/RS rack shape.
First consider Case 1 shown in Figure 5a. Using the
procedure described for the square-in-time rack, the expected
travel time from the dwell point to a random location in any of
the seven regions in Figure 5a can be computed. These times are
summarized in Table 1.
The expected response time is still computed from expression (4). (The subscript s is used to denote the case where the dwell point location is on the slope, as in Figure 5a, subscript f will be used for the case in Figure 5b when the dwell point is on the flat.) The expression for E(V)s is then the weighted average of the expected travel times for the seven regions in Table 1. The weights are given by the relative areas of the regions. The resulting expression for E(V)s is shown below.
which simplifies to
With this expression, we can compute the expected response time from expression (4).
(11)
We can then differentiate this expression to find the stationary point and take the second derivative to determine that it is a local minimum point, since the second derivative is always positive for 0 1.
(12)
(13)
From these expressions, we can find the optimal dwell point location, *, that minimizes the expected response time for any value of . In particular, if = 1, i.e., the next command is a load storage, E(RT)s = and, therefore, * = 0. In fact, for 0.5 1, expression (11) becomes K1 2 + K2 + K3, where K1 and K2 are positive constants, and so * = 0. For 0.5, equation (12) can be set to zero and solved for *. However, these expressions are only valid if the dwell point location is on the "slope" as shown in Figure 5a. If * is on the flat, then the case in Figure 5b must be evaluated.
For the case shown in Figure 5b, the decomposition approach yields
the areas and expected travel times for the four regions as shown
in Table 2.
 |  |
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The same development can be used to determine E(RT)f as well as its first and second derivatives.
(14)
(15)
(16)
From these expressions, the optimal dwell point location, *, can be determined for any value of . If = 0, i.e., the next command is a load retrieval, then * = bd. However, these expressions, and the resulting values, are again only valid if the dwell point is on the "flat" as shown in Figure 5b. In general, the region containing the dwell point will not be known, a priori, since that is the decision to be made.
Fortunately, equations (11) and (14) are convex for any given values of b and , since both are expressions in a single variable and the second derivatives, from (13) and (16), respectively, are positive. Therefore, we can exploit the fact that s* and f* are global optimal solutions to equations (11) and (14), respectively, and use the following procedure to determine the optimal dwell point location.
Step 1. For a given and rack shape factor b, solve for s* using equations (11) and (12).
Step 2. If 0 s* < b/2, then STOP, s* is the optimal dwell point location.
Step 3. Otherwise, f*, found using equations (14) and (15), specifies the optimal dwell point location.
This simple algorithm allows the optimal dwell point location to be readily determined. In addition, the two decomposition cases shown in Figure 5 offer considerable insight into the nature of the dwell point location problem. The possible locations for the dwell point were determined. The optimal dwell point was again shown to be at the input point or the midpoint of the rack for the extreme values of , and the input point was shown to be the optimal location if nothing is known about the relative probabilities of a storage or a retrieval for the next S/R machine request. However, given specific values for , the optimal dwell point location can be determined using the algorithm.
The development of the dwell point location models for an AS/RS with a single I/O point provides the foundation for the development of models for more general AS/RS configurations. For example, consider the AS/RS aisle shown in Figure 6, where a square-in-time rack has two I/O points, one in each lower corner.
Figure 6.
AS/RS Example with Multiple I/O PointsAgain, we are interested in determining to minimize E(RT), which is given by
E(RT) = + (1 - )(1 - ) + (1 - )E(V) (17)
where is the probability that I/O point 1, located in the lower left hand corner, is used for a load storage request. In this case, the multiple I/O points do not effect the computation of E(V) - it is still computed using the decomposition approach of Figure 2. Thus, equation (17) expands to
E(RT) = 4/3 3 - 4/3 3 + 2 - - + 1/3 + 2/3 (18)
Clearly, the optimal dwell point location depends on both and . If both and are equal to 1, then equation (18) will take the general form of equation (10) and the optimal dwell point location will be at * = 0, which is at (0,0), input point 1. If = 1 and = 0, then equation (18) reduces to (- + 1). Therefore, the optimal solution is * = 1, and the dwell point location is at (1,1), input point 2. If = 1 and = 0.5, then equation (18) reduces to a constant, and therefore, any value of between 0 and 1 is optimal. If = 0, then for any value of equation (18) reduces to 4/3 3 - + 2/3, which is E(V). The minimum of this expression is found at * = 0.5, implying that the dwell point should be the center of the rack. Finally, if nothing is known about the relative probabilities of a storage or retrieval or about the relative probabilities of which input point will be used (i.e., = 0.5 and = 0.5), then the optimal dwell point location is * = 0.5, implying that the dwell point should be in the center of the rack. (Note that this is a different result than the single I/O point model.)
General conclusions can be drawn from the above results by examining equation (18). We can see that equation (18) will take the general form E(RT) = K1 3 + K2 + K3. K1 will always be a positive constant and K3 will be a constant that doesn't depend on . Therefore, if the constant K2 is nonnegative, * = 0. K2 will be nonnegative if 2 1. Otherwise, the optimal value of can be found by taking the partial derivative of equation (18) with respect to . This yields equation (19), which when set to zero and solved for * yields equation (20). If the appropriate values of and are substituted into equation (20), the optimal dwell point location can be determined.
(19)
(20)
Alternative configurations of AS/RS systems with multiple I/O points can be modeled in the same manner. The development methodology discussed above is used to compute the expected response time. Depending on the location of the I/O points, differing equations will be developed for the first two components. Given these expressions, a model is developed that can be used to directly determine the dwell point location for any given values of and . Clearly, this same approach is applicable if more than two input points are used. The basic equation (17) can be extended to include terms for each input point, with the appropriate probabilities, and then the model development proceeds as illustrated above.
This paper has considered the dwell point location problem in automated storage and retrieval systems. This problem is important for minimizing the expected value of the next transaction time when an AS/RS becomes idle. We use a continuous rack approximation as an alternative modeling approach. Our paper develops analytical models for the determination of the optimal dwell point location for an S/R machine in an AS/RS. The main contributions of this paper are (1) the development of a closed form solution for the dwell point location problem under a variety of AS/RS configurations and (2) the considerable insight this model offers for the AS/RS dwell point location problem. This paper provides valuable insight for designing and operating automated storage and retrieval systems that is not provided by other approaches to the AS/RS dwell point location problem.
Bozer, Y. A. and White, J. A., Travel-Time Models for Automated Storage/Retrieval Systems, IIE Transactions, Vol. 16, No. 4, 1984, pp. 329-338.
Egbelu, P. J., Framework for Dynamic Positioning of Storage/Retrieval Machines in an Automated Storage and Retrieval System, International Journal of Production Research, Vol. 29, No. 1, 1991, pp. 17-37.
Egbelu, P. J. and Wu, C. T., A Comparison of Dwell Point Rules in an Automated Storage/Retrieval System, International Journal of Production Research, Vol. 31, No. 11, 1993, pp. 2515-2530.
Graves, S. C., Hausman, W. H., and Schwarz, L. B., Storage-retrieval Interleaving in Automatic Warehousing Systems, Management Science, Vol. 23, No. 9, 1977, pp. 935-945.
Han, M. H., McGinnis, L. F., Shieh, J. S., and White, J. A., On Sequencing Retrievals in an Automated Storage/Retrieval System, IIE Transactions, Vol. 19, No. 1, 1987, pp. 56-66.
Hausman, W. H., Schwarz, L. B., and Graves, S. C., Optimal Storage Assignment in Automatic Warehousing Systems, Management Science, Vol. 22, No. 6, 1976, pp. 629-638.
Hwang, H. and Lim, J. M., Deriving an Optimal Dwell Point of the Storage/Retrieval Machine in an Automated Storage/Retrieval System, International Journal of Production Research, Vol. 31, No. 11, 1993, pp. 2591-2602.
Linn, R. J. and Wysk, R. A., An Expert System Framework for Automated Storage and Retrieval System Control, Computers and Industrial Engineering, Vol. 18, No. 1, 1990, pp. 37-48.