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Analytical Study of t
he Fishbone Warehouse Layout
Journal:
International Journal of Logistics
Manuscript ID:
CJOL-2012-0090
Manuscript Type:
Theory Paper
Keywords:
warehousing, warehouse layout, Fishbone designs, non-linear optimization
URL: http:/mc.manuscriptcentral.com/cjol Email: a.e.whiteing@its.leeds.ac.uk
International Journal of Logistics
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Analytical Study of the Fishbone Warehouse Layout
The purpose of this paper is to study the Fishbone Warehouse Layout. Fishbone
Layouts are non-traditional designs for warehouses that have shown to be more
efficient under unit load operations. This paper presents an analytical study of the
design from an optimization point of view, the authors gain insight about the
characteristics of the model and present alternatives for different situations of
interest on industrial environments. Finally, the author’s compare the performance
between the Fishbone designs and the traditional designs; presenting a formal
conception of the expected savings. The author´s focus on finding optimal
conditions for the most important characteristic of the Fishbone design, the slope of
the diagonal cross aisle. The design process is modeled as a non-linear optimization
problem and it is solve with both numerical and exact methods, depending of the
most convenient analysis. In addition, the robustness of the study is revealed for
practical implications.
Keywords: warehousing, warehouse layout, Fishbone designs, non-linear
optimization.
1. Introduction
[Figure 1 near here]
Interest in warehousing has increased in recent decades, given that it is a necessary activity that
does not add value to the product. This interest has been fueled by the improvement in results
and complexity of system analysis and decision-aiding tools.
(Meller & Gue, 2009) proposed new unit-load warehouse layouts that were 20% faster
than traditional rectilinear warehouses. Their proposed design (called a Fishbone Warehouse)
presents two main diagonal aisles that form a “V” and picking aisles that are perpendicular to the
sides of the warehouse, as seen in Figure 1. In this design, the expected travel distance of a single
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picking operation is up to 20% shorter than in a traditional warehouse with the same picking
space. These savings are realized because the diagonal main aisles make the travel between the
pickup and deposit point (P/D point) and the item to be picked closer to the Euclidean distance.
In a traditional rectilinear warehouse, it is always necessary to traverse the full rectilinear in a
picking command.
The Fishbone warehouse is designed for unit-load warehouses with single-command
picking operations. Unit-load warehouses handle standardized types of cargo; a typical example
is a warehouse with selective racks that house palletized products and a customer’s order consists
of an integer number of full pallets. Performing a single-command operation means that the
operator goes with an empty forklift truck to the picking location and brings out a full pallet to
the P/D point. A double command would mean that the operator takes an incoming pallet from
the P/D point, puts it away, and then moves to a picking location to bring out another pallet to
the P/D point. In traditional rectilinear warehouses, (Roodbergen & DeKoster, 2001) showed that
double commands (with the addition of cross-aisles) have the potential to decrease total traveling
(and therefore picking costs). Possible extensions or adaptations of the Fishbone design to other
types of warehouses (double-command, picking warehouses) have been studied; however, the
best fit continues to be for single-command, unit-load warehouses (Section 2). This is why we
have tackled the design issues for a Fishbone warehouse instead of looking for extensions and
alternative applications.
In this paper, we present an analytical study of the central parameter in a Fishbone
design, which is the slope of the diagonal cross aisle. In Section 2, we present a review of work
related to non-traditional warehouse layouts. Section 3 contains the analytical study of the slope
of the diagonal cross aisle. In Section 4, we present the main results of the study. In Section 5 the
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reader will find the main conclusions and takeaways, and in Section 6, future research questions
are presented. We recommend that a reader interested mainly in results and optimal answers
skips Section 3, as it contains the detailed mathematical process to arrive to those solutions. For
such a reader, Sections 4 and 5 may contain what she needs.
2. Background
(Meller & Gue, 2009) proposed the Fishbone layout for unit-load warehouses under a set of
assumptions:
• Single-command operations.
• Random storage: Each sku can be stored anywhere in the warehouse.
• Only one P/D point, located in the lower middle point of the warehouse.
• Two diagonal cross aisles that start at the P/D point.
In their paper, they present an optimization model to find the best slope for the diagonal
cross aisles. The objective function (to be optimized) is the expected value of the travel distance
for a picking command. This expected value is not a linear function, and this is why they decided
to use numerical methods to solve it, without generating an analytical expression for the slope of
the cross aisle. They suggest that the diagonal aisle should start in the P/D point in the middle of
the lower side of the warehouse and go up to a point close to the upper (left or right) corner of
the warehouse.
Due to the great theoretical performance of a fishbone design, (Meller & Gue, 2009,
2011, 2012) have attempted to extend it or adapt it to warehouses with different operating
conditions. Fishbone designs are proposed for dual-command operations, non-random storage
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policies, multiple pick up and deposit points and order-picking operations. Some of the most
relevant results from these papers are mentioned below.
In (Pohl, Meller & Gue, 2009) the fishbone layout was tested for dual-command
operations or task-interleaving. By means of analytical analysis, the paper presents that the
fishbone layout reduces the travel distance by approximately 10-15% compared to the traditional
design. Furthermore, they propose two fishbone design rules for warehouses with single or dual
command operations, (1) the shape of the half-space should be approximately square and, (2) the
diagonal cross-aisle should end in the upper corner of the picking half-space.
In (Pohl, Meller & Gue, 2011) the fishbone layout was tested for non-uniform storage
policies. The results indicate that designs that have good performance under a random policy will
also have a good performance for non-uniform storage policies. Also, they extend the design
rules proposed in Pohl, (Meller & Gue, 2009) for warehouses with turnover-based storage
policies, for single and dual command operations.
In (Gue, Ivanovic, & Meller, 2012) non-traditional aisle designs were tested for
warehouses with multiple P/D points. The results suggest that those designs are better than
traditional ones, but the benefit is not as important as it was in the single P/D point warehouses.
In (Dukic & Opetuk, 2008) the fishbone layout was tested for manual-pick order-picking
systems with different routing policies. They show that with the most common routing polices,
the fishbone layout increases the distances of the travel. The study is not conclusive and future
research is needed to formalize this intuition, but at least with the most common polices the
fishbone layout is not favorable for order picking warehouses.
The operation condition in which the fishbone design shows the best performance is the
unit-load, single-command operation. In other operating conditions, as it is mentioned in the
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previous studies, the fishbone design presents a better performance than a rectangular design but
these improvements are modest.
These previous papers propose some ideas and considerations that should be taken into
account when implementing a fishbone design. Some of these considerations will be used in this
paper.
3. The optimal slope
There are two main types of warehousing projects: greenfield projects (where you will build a
warehouse according to your specifications) and brownfield projects (where the warehouse
building is already in place and you are looking for the best layout for it). The fundamental
difference between these is that in a greenfield project you can define the dimensions of the
warehouse; these dimensions are already known and set for a brownfield project.
The starting assumptions for this paper are:
(1) The warehouse has a rectangular shape.
(2) There is a single P/D point, located in the middle of one side of the warehouse.
(3) The warehouse is symmetrically distributed with respect to the axis of the P/D point.
(4) The inside of the warehouse is considered to be obstacle-free. Structures such as
columns, lifts and walls are ignored.
(5) Congestion issues for material handling equipment are ignored.
(6) Storage spaces are assigned randomly.
(7) In a fishbone layout there are two straight diagonal cross aisles that start at the P/D point.
Due to assumption number 3, it is possible to analyze the picking half-space of the
warehouse without loss of generality (the right hand side was arbitrarily chosen in this case, see
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Figure 2). An indicator of the goodness of the design will be the expected value of the travel
distance for a picking operation; for a symmetrical warehouse every half will have the same
expected value, and given that the pick operation is equally likely to fall on either side of the
warehouse, the expected value of the travel distance will be the same for the whole warehouse as
it would be for the right hand side half of it.
[Figure 2 near here]
In (Meller & Gue, 2009) the authors assume that, for symmetry reasons, we would only
need to consider diagonal aisles that end on the right hand side of the warehouse under the upper
right corner of the building. Having a higher slope implies that the diagonal cross aisle would
end on the upper side of the warehouse, in which case it would be equivalent to flip the sides of
the warehouse. In this paper, each of these cases will be considered independently, and we will
call Case I when the diagonal cross aisle ends below the corner of the warehouse and Case II if
the diagonal aisle ends above the upper right corner of the building (see Figure 3).
[Figure 3 near here]
We are interested in finding a design that minimizes the expected travel distance for a
picking operation. The first development we need is an analytical expression for this expected
travel distance as a function of the dimensions of the warehouse and the slope of the diagonal
aisle (Section 3.1).
With this analytical expression for the expected travel distance we analyze the brownfield
case, where you know the dimensions of the warehouse and the variable of interest is the slope of
the diagonal cross aisle that minimizes the expected travel distance (Section 3.2). Later, we
analyze the greenfield case, in which we want to simultaneously find the dimensions of the
warehouse and the slope of the diagonal aisle. In this case we keep the area of the picking space
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constant and minimize the expected travel distance (Section 3.3). Finally, we make an analytical
comparison between fishbone and rectangular designs (Section 3.4). In Figure 4 we present the
notation that will be used in the following sections:
[Figure 4 near here]
(1) length of the side in which the P/D point is located.
(2) length of the side in which the P/D is not located.
(3) angle of the diagonal cross aisle.
(4) length of the segment opposite to.
(5) the horizontal coordinate of the item to be picked.
(6) the vertical coordinate of the item to be picked.
(7) travel distance.
(8)
travel distance for items located in Zone 1. (Above the diagonal cross aisle)
(9)
travel distance for items located in Zone 2. (Below the diagonal cross aisle)
We are going to use the sub-index to denote variables and parameters for Case I, and to
denote variables and parameters of Case II.
3.1 Expected travel distance
(Meller & Gue, 2009) found an analytical expression for the expected travel distance based on a
discrete representation of the space. This expression is too complex to use for analytical
optimization, this is why we will use a continuous representation of the space. Even though a
continuous representation is less accurate than a discrete representation, we are going to use it
because it allows us to develop an analytical analysis. The purpose of this section is to define an
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analytical expression for the expected travel distance in terms of the dimensions of the picking
half-space and the slope of the diagonal cross aisle. Hence, we are going to define a function for
the travel distance and calculate the expected value for this variable.
Travel distance
The travel distance is defined in terms of the coordinates of the item to be picked, the
zone where the item is located and the angle of the diagonal crossaisle, . Thus, is a
piecewise function defined as follows:
Travel distance in Zone 1
[Figure 5 near here]
Replacing terms,
Organizing terms,
(1)
Travel distance in Zone 2
[Figure 6 near here]
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Replacing terms,
Organizing terms,
(2)
3.1.1 Expected travel distance
Case I
Due to the assumption that storage spaces are assigned randomly, we can define the expected
travel distance as follows:
(3)
Replacing as a piecewise function,
We integrate with Wolfram|Alpha and obtained,
(4)
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We need that be a function in terms of the sizes of the picking half-space and the
slope of the diagonal cross aisle , so we have to rewrite Eq. 4. Note that
, so we are
going to replace by .
(5)
(5 is the expected travel distance
of the picking half-space for Case I that we are going to use for the brownfield and greenfield
problems.
Case II
In the same way of Case I, we can define the expected travel distance as follows:
Replacing as a piecewise function,
We integrate with Wolfram|Alpha and obtained,
(6)
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We need that
be a function of the size of the picking half-space and the slope of the
diagonal cross aisle , so we have to rewrite Eq. 6. Note that
, so we can replace by
.
(7)
Eq. 7¡Error! No se encuentra el origen de la referencia. is the expected travel distance of the
picking half-space for case II that we are going to use for the brownfield and greenfield
problems.
3.2 Brownfield
Given , the fixed, known dimensions of a half picking space, we want to determine the
slope of the diagonal cross aisle
that minimizes the expected value of the travel distance,
. Given that differs for Case I and Case II, each case must be worked independently
and then both solutions will be integrated into a general solution. The following three
subsections present Case I (section 3.2.1), Case II (section 3.2.2) and the general case (section
3.2.3).
3.2.1 Case I
Up to this point,
depends on (the size of the vertical side of the half picking space), on
(the horizontal dimension of the half picking space) and on
(the slope of the diagonal cross
aisle). We will now seek an analytical expression where
is defined by , by the slope of
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the diagonal of the half picking space (
) and the slope of the diagonal cross aisle,
.
Rewriting Equation 5,
Replacing
,
(8)
Label
, then notice that,
Note that
is a constant, therefore Ε
will be minimal only if
is minimal too.
Therefore, it is only necessary to study
to find the parameters that minimize Ε
.
Lastly, note that for the brownfield design case, the dimensions of the warehouse are known and
therefore
is a constant and not an independent variable in
. Thus,
becomes
, a problem with just one variable.
(9)
In this way, the brownfield design problem may be presented as:
• Minimize
(Eq. 9)
• S.t.:
, (this is Case I, where the diagonal cross aisle intersects the right
side of the half picking space).
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The solution strategy involves solving the unrestricted optimization problem and identifying
where a feasible optimal solution is reached and where the optimal solution is unfeasible. Figure
7 shows that the problem may be split into two regions and treated separately. In one region the
constraint
is not binding and the problem is unconstrained. In the other region, the
constraint actually limits the solution space and the problem is constrained. Formally, the first
region is comprised of all points
where
and
(non binding
constraint); the second region contains all the points
where
, but
(the critical point that makes the derivative equal to zero finds itself on a non-feasible
region).
Let us call
the region where the constraint is active and
the region where it is not.
To explicitly define the regions we can use the function of the optimal slope and the constraint.
Function of the optimal slope
,
(10)
Constraint,
(11).
Analyzing the characteristics of the two functions (Eq. 10 and Eq. 11), it is possible to
conclude that in all their domain,
, the functions intersect in only one point (Figure 7),
. To the right of
the function of the optimal slope will be smaller than the constraint, and to
its left the function of the optimal slope is larger than the constraint. In this way,
is defined
for values of
from zero up to
, and
contains values of
greater than
.
[Figure 7 near here]
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To find
, we replace Eq. 11 on Eq. 10 and rewrite the resulting expression:
Organizing terms,
The roots of the equation are
and
. Given that
, we conclude that
.
The two regions might be explicitly defined as
and
. We can define
as the function that describes the optimal slope of the
diagonal cross aisle with respect to
:
Where,
•
•
We will find each term now.
Optimal slope for Region 1,
Problem:
• Minimize
(Eq. 9)
• S.t. :
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This problem was solved using Wolfram|Alpha, obtaining that
Optimal slope for region 2,
Problem:
• Minimize
(Eq. 9)
Given that this is an unconstrained problem, we can find the first derivative of
and make it
equal to zero. In this way, an implicit equation for
and
is found (Eq. 10).
is
obtained finding
from Equation 10. The complete expressions can be found in Appendix A.
From now on, we will denote this expression by
. In this way,
.
Optimal slope
General case (considering
and
)
We can write the function for the optimal slope of the diagonal cross aisle (
) in a half
picking space with fixed, known dimensions as:
(12)
[Figure 8 near here]
3.2.2 Case II
We will follow the same procedure we used in Case I to solve Case II. Due to the similarities in
the procedure, a short version of it will follow. The first step is to obtain the expected travel
distance function
defined by , by the relation
and by the slope of the diagonal
cross aisle,
.
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Label
, then note that,
Given that the dimensions of the warehouse are known,
is a constant in
. Thus,
can be expressed as
:
(13)
In this way, the brownfield design problem becomes:
• Minimize
(Eq. 13)
• S.t. :
, this is Case II where the diagonal cross aisle intersects the upper side of
the half picking space.
We define again
as the region where the constraint is non-binding and
the region where
the constraint is active. To define the regions explicitly we can use the function of the optimal
slope and the constraint.
Runction of the optimal slope:
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(14)
Constraint,
(15)
Analyzing the characteristics of the two functions (Eq. 14 and Eq. 15), it is possible to conclude
that in all their domain,
, the functions intersect in only one point,
. To the right of
the optimal slope function is lower than the constraint, and to the right of
the optimal slope
function is bigger than the constraint. In this way,
is defined from zero to
and
is
defined for values of
greater than
. See Figure 9. To calculate
we must replace Eq. 15
in Eq. 14 and rewrite in the following manner:
Organizing the expression,
[Figure 9 near here]
The roots of the equation are
and
. Given that
,
is the only valid
root, thus,
. In this way, we can explicitly define the two regions as
and
.
Then, we will define
as the function that describes the optimal slope of the diagonal
cross aisle with respect to
as follows:
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Where,
•
•
We will find each term next.
Optimal slope for Region 1
Problem:
• Minimize
(Eq. 13)
Given that this is an unconstrained problem, we can find the first derivative of
and make it
equal to zero. In this way, an implicit equation for
and
is found (Eq. 14).
is
obtained finding
from Equation 14. Appendix A presents the complete expression for finding
in terms of
. From now on, we will denote this expression by
. In this way,
.
Optimal slope for Region 2
Problem
• Minimize
(Eq. 13)
• S.t.
This problem was solved using Wolfram|Alpha, obtaining that
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Optimal slope
. General case (considering
and
)
We can write the function for the optimal slope of the diagonal cross aisle
in a half picking
space with fixed, known dimensions as:
(16)
[Figure 10 near here]
3.2.3 General Case (considering Case I and Case II)
We presented Case I (when the diagonal cross aisle intersects the right side of the half
picking space) in section 3.2.1 and Case II (when the diagonal cross aisle intersects the upper
side of the half picking space) in section 3.2.2. Since it is not known beforehand on which side of
the half picking space should the cross aisle intersect, both cases must be compared and the best
of the two optimal solutions will be the global optimum
.
From Cases I and II we already have the optimal slope function, Equations 12 and 16
respectively. Also, the cost function
for each case is known (Equations 9 and 13).
With these two functions we will build the cost function for the optimal slope in each case. The
cost function will have only one independent variable,
.
Cost function for Case I (
) with the optimal slope of the diagonal cross aisle.
Replacing
in Eq. 9, we get:
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Note that
is a function of
, thus, the cost function with the optimal slope of the diagonal
cross aisle depends only on
. Replacing
(Eq. 12):
(17)
Where,
Cost function for Case II (
) with the optimal slope of the diagonal cross aisle
Replacing
en Eq. 13, we get:
Note that
is a function of
, thus, the cost function with the optimal slope of the diagonal
cross aisle depends only on
. Replacing
(Eq. 16):
(18)
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Where,
Comparison of the cost functions
Let
be the cost function in terms of the relationship between the dimensions of the
half picking space
, then
is defined in terms of
and
. Given the
characteristics of
and
in the domain where
, the global optimal slope
defined as
may be expressed in parts. In Figure 11, the monotony of
both cost functions can be observed, it leads to a single point of intersection between them and
the definition of two regions, one where
and another one where
.
[Figure 11 near here]
Figure 11 shows that the difference between Cases I and II is extremely small, but it is not zero.
The two cost functions intersect in only one point in the domain of interest and this point occurs
when
, in this way,
is defined in terms of
and
by parts as
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follows:
The function of the optimal slope for the diagonal cross aisle (
is defined as:
Replacing
(Eq. 12) and
(Eq. 16),
(19)
Note that for
it is better to have the diagonal cross aisle intersect the upper side of the
half picking space, whereas when
it is better to have the diagonal cross aisle intersect
the right side of the warehouse. In the frontier case when
, the diagonal cross aisle should
go to the upper right corner of the warehouse. Figure 12 presents two examples of
[Figure 12 near here]
[Figure 13 near here]
[Table 1 near here]
Equation 19 presents an optimal value for the slope of the diagonal cross aisle given the
dimensions of the half picking space. Still, it might be preferrable for industrial users to have
simpler design parameters for the warehouses. In an actual application, it is much easier to have
the diagonal cross aisle go to the upper right corner of the warehouse, that is to have
.
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Figure 14 shows the deviation from the optimal cost if the cross aisle always goes to the upper
right corner instead to the optimal
calculated from previous steps. It can be noticed that for
warehouses with
, the deviation from the minimal expected travel distance is less
than 0.6%. Thus, for a practical design it is recommended to always use
.
[Figure 14 near here]
3.3 Greenfield
On a greenfield design situation, we do not have a specific building to organize, so we can
determine the geometry of the warehouse and the slope of the diagonal cross aisle
simultaneously. That is, we wish to determine the values for
and
in such a way that the
expected travel distance is minimal.
Note that Case II has a symmetry relationship with Case I. We can transform Case II into
a Case I through one rotation and one reflection (Figure 15). It can be showed that for any
warehouse that has the diagonal cross aisle intersecting the right side (Case I), it exists a
warehouse with the diagonal cross aisle intercepting the upper side (Case II) that has the same
. Therefore, we need only to consider designs for Case I. Note that this was not feasible for
the brownfield case.
3.3.1 Case I
Problem
• Minimize
• S.t.
Note that
(Eq. 8) does not depend directly on the area, therefore the first thing to do is to
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find an expression for
in terms of , of
and
. We will use the fact that and
that
, therefore
. Replacing
in Eq. 8,
[Figure 15 near here]
Note that to minimize
it is enough to minimize
, where
(20)
The optimal values for
and
do not depend on . This means that these optimal values,
and
, can be found solving the following optimization problem:
Problem
• Minimize
(Eq. 20)
• S.t.
We use MATLAB for the optimization process and found the optimal point at
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Note that the greenfield design situation can be seen as a brownfield problem where
is not a known quantity but a decision variable. Having an analytical expression that describes
in terms of
(
, Appendix A), the problem is then to find
. Figure 16 presents a
chart of
in terms of
.
[Figure 16 near here]
[Figure 17 near here]
Even though the optimal value of the cost function is reached at en
, Figure 16
shows that the expected travel distance is a very robust function. The red region (labeled
“robust region”) is where the expected travel distance does not deviate more than 2% of the
global optimum.
[Figure 18 near here]
To find an ellyptical regression model was used. The details of the equations can be found in
Appendix B.
3.4 Fishbone vs Traditional designs
In this section we compare the expected travel distance of a fishbone design versus a traditional
design. We compare the designs in a half-picking space of area and evaluate the expected
savings of implementing a fishbone design. Let us call to the expected savings, then is
defined as:
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Fishbone expected travel distance,
,
Traditional expected travel distance
In (Meller & Gue, 2009) the rectangular expected travel distance of a half-picking space of area
is defined as:
Replacing
and
,
Rewriting the equation,
Expected savings
Rewriting the equation,
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To make the evaluation in optimal conditions, we replace
,
(21)
In Eq. 21 we can see that the savings depend on
, they are not a constant value. In Figure 18
we can see that we obtain the maximum savings for
, and for
the
savings are greater than 18 %. This agrees with previous works that predict savings around 20 %
(Meller & Gue, 2009).
[Figure 19 near here]
4. Summary
This paper presents an analytical study to find the optimal slope of a Fishbone Warehouse
Design, for both brownfield and greenfield situations. Also, a comparison between Fishbone and
traditional designs was presented.
Brownfield
When the dimensions of the warehouse are known, an analytical expression is presented to
obtain the optimal slope of the diagonal cross aisle,
. To find
, Eq. 19 can be used or the
value may be approximated using Table 1.
A special consideration is given to the case where the diagonal cross aisle goes to the
upper right corner of the warehouse. It was showed that designing the cross aisle this way, only
0.6% of cost penalty would be caused for the most common warehouse geometries.
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Greenfield
When it is desired to build a warehouse and there is freedom to decide on the shape of the
building, the best option is to build a half picking space in the shape of a square and to have the
diagonal cross aisle go to the upper right corner of the warehouse (Section 3.3).
It was also determined that the Fishbone design is very robust, and a region where the
expected travel distance deviated at most 2% from its optimal value was drawn (Section 3.4).
This gives ample flexibility to a warehouse designer, since it gives them the possibility to
consider other factors to include them in the design with the peace of mind that the expected
travel distance will not stray greatly from the optimal values.
Fishbone vs. Traditional designs
An analytical study of the savings for using the fishbone design compared to traditional designs
was presented. For usual warehouse geometries, it was found that the savings on the expected
travel distance are greater than 18%.
5. Conclusions
Studying the slope of the diagonal cross aisle on a Fishbone design with a continuous
representation of space implies some sacrifices in precision, but it presents an easier treatment of
space and its equations. The analytical expression for the optimal slope in the brownfield case
empowers a warehouse designer with decent spreadsheet capabilities to calculate the optimal
slopes on their own. The greenfield case is even simpler, given that the optimal solution is to
design a warehouse that has a square half picking space and the diagonal cross aisle goes from
the P/D point to the upper right corner of the half picking space. No calculations are required.
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It is important to point out that when
it is better to have the diagonal cross aisle
intersect the upper side of the half picking space, which is different than what has been pointed
out in the literature so far. When
it is better to have the diagonal cross aisle intersect the
right side of the half picking space. When
it is simply better to have a diagonal cross
aisle that goes to the upper right corner of the half picking space.
We consider that the most important result of this work is to point out the flexibility that
a warehouse designer has when is implementing a Fishbone layout. This flexibility allows to
consider other practical issues in the design as space utilization and industrial safety, having in
mind that if he needs to deviate from the optimal slope, it won’t increment too much the
operational costs.
6. Future research
The Fishbone warehouse layout was designed for unit-load, single command, single P/D point
and random storage policy. Some extensions have been presented for other types of operations,
but even though Fishbone designs have been shown to perform better than a traditional design,
the biggest savings were realized by unit-load, single command operations. This is why future
research around the Fishbone design should focus on formalizing the design, the study of the
design parameters and the impact they have on the performance of the warehouse. To make the
model practical it is necessary to study discrete representations of space, and to include variables
such as the characteristics of the storage opening and the most efficient way to place aisles and
storage racks.
Even though it is possible to extend the analysis to dual-command warehouses, it is not
recommended. In (Pohl, Meller & Gue, 2009), the authors pose the possibility of generating new
layouts specifically crafted for dual command operations. The Fishbone layout was designed.
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With all the previous work on Fishbone layouts, plus the analytical study presented here,
we contend that the knowledge of this design is mature enough to create software applications
that facilitate the actual design of warehouses in different situations of interest for industry.
Lastly, there are situations that make a traditional design to be more desirable, such as a
high cost per square feet. High picking costs might suggest, on the other hand, that non-
traditional designs might be more attractive. Further research needs to be conducted to build
models that present economically sensible comparisons between traditional and non-traditional
designs to enhance decision making by industry practitioners.
References
Dukic, G., and Opetuk, T., 2008. Analysis of order-picking in warehouse with fishbone layout.
Proceeding Proceeding of International Conference on Industrial Logistics , 197-205.
Gue, K., Ivanovic, G., and Meller, R., 2012. A Unit-Load Warehouse with Multiple Pickup &
Deposit Points and Non-Traditional Aisles. Transportation Research Part E: Logistics
and Transportation Review , 48 (4), 795-806.
Meller, R., and Gue, K., 2009. Aisle configurations for unit-load warehouses. IIE Transactions,
41, 171-182.
Pohl, L., Meller, R., and Gue, K., 2009. Optimizing fishbone aisles for dual command
operations in a warehouse. Naval Research Logistics (NRL) , 56 (5), 389-409.
Pohl, L., Meller, R., and Gue, K., 2011. Turnover-based storage in non-traditional unit-load
warehouse designs. IIE Transactions , 43, 703-720.
Roodbergen, K., and DeKoster, R., 2001. Routing order pickers in a warehouse with a middle
aisle. European Journal of Operational Research , 32-43.
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APPENDIX A
CASE I
This appendix presents an analytical expression for
(Section 3.2.1).
is the
result of solving Eq. 10 for
.
Eq. 10,
Let us define
,
,
and
,
We then define the function
as follows:
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CASE II
This appendix presents an analytic expression for
(Section 3.2.2).
is the
result of solving Eq. 14 for
.
Eq. 14,
Let us define
,
,
,
and
,
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We then define the function
as follows:
APPENDIX B
To find region it was necessary to perform an elliptical regression. The equation of the ellipse
that adjusts the region best is
In this way, may be explicitly defined as
Where
This means that for any
if then the expected travel distance for is at
most 2% higher than the optimal value.
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∗
0,10 0,26
0,20 0,38
0,30 0,48
0,40 0,56
0,50 0,64
0,60 0,72
0,70 0,79
0,80 0,86
0,90 0,93
1,00 1,00
1,10 1,07
1,20 1,13
1,30 1,19
1,40 1,25
1,50 1,33
1,60 1,38
1,70 1,43
1,80 1,48
1,90 1,53
2,00 1,56
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Traditional (rectilinear) and Fishbone warehouse layouts
614x197mm (96 x 96 DPI)
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The right hand side of the warehouse will be analyzed.
1210x333mm (96 x 96 DPI)
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Different end points of the diagonal cross aisle.
188x101mm (96 x 96 DPI)
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Notation
600x349mm (96 x 96 DPI)
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Travel distance Zone 1
553x265mm (96 x 96 DPI)
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Travel distance Zone 2
514x237mm (96 x 96 DPI)
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Unconstrained optimal slope Case I
166x129mm (96 x 96 DPI)
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Optimal slope Case I
178x122mm (96 x 96 DPI)
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Unconstrained Optimal slope Case II
158x118mm (96 x 96 DPI)
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Optimal slope Case II.
165x112mm (96 x 96 DPI)
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Comparison of cost functions.
150x115mm (96 x 96 DPI)
Page 45 of 53
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Optimal slope examples
640x450mm (96 x 96 DPI)
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Global optimal slope
147x106mm (96 x 96 DPI)
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Deviation over the minimun.
123x94mm (96 x 96 DPI)
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Symmetry between case I and case II
793x754mm (96 x 96 DPI)
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Cost function, greenfield situation.
793x643mm (96 x 96 DPI)
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Example, greenfield situation.
806x751mm (96 x 96 DPI)
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Robustness of the cost function for a greenfield situation.
899x793mm (96 x 96 DPI)
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Expected savings.
916x612mm (96 x 96 DPI)
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