Detailed design of fishbone warehouse layouts with vertical travel

Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Detailed design of fishbone warehouse layouts with vertical travel Luis F. Cardona a,n, Diego F. Soto b, Leonardo Rivera c, Hector J. Martínez d a

Department of Industrial Engineering, Faculty of Engineering, Universidad Icesi, Street 18 No. 122-135 Pance, Cali, Colombia Department of Operations Planning, Fanalca, Acopi Yumbo, Colombia School of Industrial Engineering, Faculty of Engineering, Universidad del Valle, Cali, Colombia d Department of Mathematics, Faculty of Science, Universidad del Valle, Cali, Colombia b c

art ic l e i nf o

a b s t r a c t

Article history: Received 31 December 2013 Accepted 9 March 2015

In this paper, we provide a method to generate a three-dimensional detailed design of fishbone layouts. This method takes the desired storage capacity and returns the location ðx; y; zÞ of each opening of the warehouse in such a way that the total operational cost – area cost and material handling cost – of the warehouse is minimal. We model the arrangement of the openings using mathematical finite sequences and represent a fishbone layout in terms of four primary characteristics. Next, we develop an algorithm that generates a detailed design of a fishbone layout given values of its four primary characteristics. Then, we present an optimization model that finds the values for the four primary characteristics that minimize the total operational cost of the warehouse. Finally, we solve the optimization model using a genetic algorithm. Our results suggest that in 91.74% of the cases, our optimization procedure reaches a near optimum point – deviated only by 0.587% – in a reasonable computational time (maximum 4.5 min). This paper aims to diminish dependence upon experts and human decision making in the process of implementing a fishbone layout on greenfield projects, and fulfills an identified need of warehouse practitioners by integrating the most recent advances on non-traditional layouts and detailed warehouse design. & 2015 Elsevier B.V. All rights reserved.

Keywords: Warehouse layout Fishbone layout Warehouse design Aisle configuration

1. Introduction Warehouses around the world have the same layout, called the traditional layout (Caron et al., 2000), which is characterized by the arrangement of parallel aisles, orthogonal to the walls. Besides traditional layouts, there have been some proposals that hold divergent ideas (White, 1972; Berry, 1968), but only in Gue and Meller (2009) appeared a non-traditional layout, called the fishbone layout, that caught the attention of warehouse academics. A fishbone layout has two diagonal cross aisles and the aisles in the lower zones are perpendicular to the aisles in the upper zones (Fig. 1). The improvement in the performance of the layout is accomplished by making travel distances closer to be “Euclidean” (Fig. 1), rather than the rectilinear paths that have to be traveled in traditional layouts. Gue and Meller (2009) found that the expected travel distance of the fishbone layout is up to 20% lower compared with traditional layouts for unit load warehouses with single command operations.

The fishbone layout has also caught the attention of warehouse practitioners, who have implemented it in some warehouses and distribution centers, mostly in the United States (Meller and Gue, 2010, 2009). These implementations are mostly supported by human decision making and expert guidance, given that there is not a scientific method that supports specific concerns as how many picking aisles should have the layout? How many openings in each picking aisle? How many tiers? In the fishbone layout these questions become important, because obtaining a regular pattern for the arrangement of the openings while satisfying a desired storage capacity is a highly complex combinatorial problem. In this paper, we close this gap by providing a method that generates the detailed three-dimensional design1 of a fishbone layout in a greenfield warehouse project2 with single command operations. We assume a random storage policy, all racks of the warehouses are of the same type and size, and the material handling system is constituted only by forklifts. Our method considers the dimensions of the openings, the aisle width, the speeds of the forklifts, the cost of leasing and maintaining

n

Corresponding author. Tel.: þ 57 25552334. E-mail addresses: [email protected] (L.F. Cardona), [email protected] (D.F. Soto), [email protected] (L. Rivera), [email protected] (H.J. Martínez).

1 A detailed design provides the location of each opening of the warehouse, where an opening is the elemental storage position of a rack. 2 A project that starts from scratch without limitations of previous infrastructure.

http://dx.doi.org/10.1016/j.ijpe.2015.03.006 0925-5273/& 2015 Elsevier B.V. All rights reserved.

Please cite this article as: Cardona, L.F., et al., Detailed design of fishbone warehouse layouts with vertical travel. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.03.006i

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Fig. 1. Detailed three-dimensional fishbone layout.

Fig. 2. Three primary characteristics of the fishbone layout.

the warehouse area and the material handling cost, to generate a detailed design of a fishbone layout that satisfies a desired storage capacity. First, we defined four primary characteristics for the fishbone layout (see Fig. 2): the number of openings of the first line (s), the increment between lines (I2), the number of lines (η) and the number of tiers of the three-dimensional design (τ). Second, we developed an algorithm that generates a detailed design depending on these four primary characteristics using mathematical finite sequences to represent the arrangement of the openings. Next, we presented an optimization model that determines these four primary characteristics while minimizing the total operational cost. Finally, we provided an optimization procedure based on a genetic algorithm to solve the optimization model. The rest of this paper is organized as follows: Section 2 presents a review of the literature related to the fishbone layout and detailed warehouse design. Section 3.1 presents an algorithm that generates a fishbone layout given values of the four primary characteristics. Section 3.2 shows the optimization model. Section 3.3 presents the optimization procedure. Section 4 illustrates the method through a hypothetical numerical example. Finally, Section 5 presents the conclusions and future research.

2. Literature review Since 2009, when Gue and Meller (2009) introduced the fishbone layout, warehouse academics have studied the most important characteristics of this layout (the dimensions of the warehouse and the slope for the diagonal cross aisles) aiming to maximize its operational efficiency. Gue and Meller (2009) studied the optimal configuration of a fishbone layout with single command operations and found that the width of a fishbone layout should be twice its length (considering width the side of the warehouse where the pick up and deposit point is located) and the diagonal cross aisles should be extended to the upper corners of the warehouse. Pohl et al. (2009) tested the fishbone layout for dual command operations (or task interleaving) and they concluded that the fishbone layout reduces the expected travel distance by 10–15% compared to traditional layouts. Pohl et al. (2011) tested the fishbone layout with non-uniform storage policies for single and dual command operations; they found that the benefits shown by

the fishbone layout under a random storage policy can also be found under non-uniform storage policies. Afterwards, Cardona et al. (2012) presented a non-linear optimization model to determine the best slope of diagonal cross aisle and the best aspect ratio for fishbone designs. In this model they minimize the expected travel time using a continuous representation of the space and assuming uniform picking density in the warehouse. In other words, they simplified the problem ignoring the internal arrangement of the openings in the warehouse. They found that the results of Gue and Meller (2009) are valid only for greenfield projects and they provided an analytical function to find the optimal slope of the diagonal aisle in brownfield projects.3 However, the variety in rack geometries and aisles sizes in the industry does not guarantee the possibility to design a warehouse with the optimal shape presented in Cardona et al. (2012). Operational cost for warehouses differs around the world. Some countries have low land prices and high material handling cost while others have high land prices and low material handling cost. In consequence, two optimal warehouses with the same storage capacity and the same rack dimensions can have a different layout considering the costs involved in their particular operation. For regions where the land price is low, the optimal layout can be long and wide but low, while for regions with high land prices, the optimal layout can be high but cover a small area. Thus, in this paper we aim to provide a methodology such that warehouse practitioners can generate a three-dimensional design for their specific needs, considering racks geometries, aisles size, storage capacity, and area and material handling costs. Dukic and Opetuk (2012) and Çelik and Süral (2014) tested the fishbone layout for order-picking systems with different routing policies. They found that the fishbone layout “can perform as high as around 30% worse than an equivalent parallel-aisle layout” under a random storage policy and uniform demand. They also found that for dedicated storage with non-uniform demand, the fishbone layout can outperform a parallel-aisle layout depending on how skewed the demand is. Öztürkoğlu et al. (2012) introduced three new non-traditional optimal designs for unit load warehouses with one, two, and three diagonal cross aisles (chevron, leaf, and butterfly). They presented a continuous optimization model for the designs and used a discrete model to discuss practical implications of them. They recommend the chevron design for industrial applications comparing the area and the expected travel time of each design. They also discuss some implementation issues of non-traditional designs such as safety and maneuverability of the forklifts. Authors do not compare the fishbone layout with their new proposals, but it is worth noting that the recommended layout (chevron) has the same expected travel time as the fishbone layout. Finally, Clark and Meller (2013) studied the fishbone layout considering the vertical travel of the forklifts (for unit load and random storage as well as Gue and Meller, 2009). Their model incorporates Chebychev travel within the picking aisles, i.e. the forklift raises or lowers its fork while driving within the picking aisles but not while driving within the diagonal cross aisles. They found that the optimal shape of the fishbone layout proposed by Gue and Meller (2009) is still valid regardless of the height of the rack and they stated the need for further research to guide implementations of nontraditional designs on warehouse projects. In all unit-load operational conditions, the fishbone layout is more efficient than traditional layouts. But out of all operating conditions, the fishbone layout showed the best performance in a greenfield project of unit load warehouses with random storage. In our methodology, we assume this operational condition aiming to

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A project that uses a previously existing building.

Please cite this article as: Cardona, L.F., et al., Detailed design of fishbone warehouse layouts with vertical travel. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.03.006i

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provide a platform for the most beneficial proven aspects of the fishbone layout. Industries have also tested the fishbone layout (Meller and Gue, 2010, 2009). The benefits found in the implementations of fishbone layouts were greater than expected, and not only on the expected travel distance. For instance, forklift operators point out that the 451 turns of fishbone layouts are easier to make than the 901 turns of traditional layout. The literature of fishbone layout lacks methods for designing the specifics of a fishbone layout in three dimensions, as they exist for traditional layouts (Rouwenhorst et al., 2000). Francis (1967) and Bassan et al. (1980) started the groundwork in detailed design of traditional layouts. Park and Webster (1989), Zhang et al. (2002) and Onut et al. (2008) studied the effect of the vertical and horizontal transportation costs to define the dimensions of the warehouse. Parikh and Meller (2010) presented an optimization model to consider the height of the warehouse. We will include these formulations to model the vertical travel in the fishbone layout. Bartholdi and Hackman (2011) and Roodbergen and Vis (2006) presented optimization approaches for defining the number of P&D points and their optimal locations. Roodbergen et al. (2008) and Hsieh and Tsai (2006) studied detailed designs for minimizing travel distances in picking systems. We will follow these formulations to model the expected travel distance of the warehouse. In this paper, we provide a tool for warehouse designers that aims to integrate the advances on fishbone layouts and the literature of detailed design of traditional layouts.

3. Detailed design of fishbone layouts To generate a fishbone detailed design, we assume a greenfield project of a warehouse that will operate under the following conditions:

      

 

Unit load, single command operation. Random storage policy. All openings of the warehouse are equal. All racks have the same height (same number of tiers). All forklifts have similar technical characteristics and their speeds – linear and lifting – can be modelled as deterministic parameters. All forklifts transport unit loads to every opening of the layout regardless of the product they are carrying. The forklifts hold their fork on its resting position while they are moving within aisles and only when they arrive to the desired opening location, they lift the forks to reach the desired opening. We consider a maximum height for the warehouse caused by construction limits or by the maximum height that the forklifts can reach. All aisles have the same width (ap) and are wide enough for the forklift to turn into them.

With these assumptions, we developed a method with three components: 1. An algorithm that generates a detailed three-dimensional design of a fishbone layout depending on the four primary characteristics of the layout (Section 3.1). 2. An optimization model that finds optimal values for the four primary characteristics of the layout while minimizing the total operational cost (Section 3.2). 3. An optimization procedure that solves the optimization model (Section 3.3).

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3.1. Detailed design algorithm The detailed design algorithm takes the dimensions of the openings, the aisle width, and the four primary characteristics of the layout, to generate a detailed three-dimensional design of a fishbone layout. First, we will define a standard geometry for the fishbone layout as follows (Fig. 3): 1. The layout is constituted by four triangular areas, which we will call Zones and we denote them as Zone 1, Zone 2, Zone 3 and Zone 4. 2. The four triangular zones are equal. 3. The line segment between p1 and p2 is perpendicular to the line segment between p1 and p3. 4. There is a central aisle between Zone 2 and Zone 3. 5. All aisles have the same width Fig. 3 uses the following notation:

α β ap

θ

length of the base of triangle of Zone 4 (or Zone 1) length of the height of triangle of Zone 4 (or Zone 1) aisle width angle of the diagonal aisle

In this standard geometry, the diagonal aisles always end at the upper corners of the warehouse. Cardona et al. (2012) presented an optimal value for the slope of the diagonal aisles depending on the dimensions of the warehouse. They found that, for practical designs, it is preferable to have the diagonal aisles ending at the upper corners of the warehouse, even if the theoretical model recommends something different. They also proved that when diagonal aisles end at the upper corners, instead of having the optimal slope, the deviation of the expected travel distance is at most 0.6% of the optimal value for practical designs. Fig. 4 shows the four zones of a fishbone layout. Note that Zone 1 mirrors Zone 4 and Zone 2 mirrors Zone 3, therefore both half picking spaces are symmetrical on the vertical axis (Fig. 4). But, Zone 1 not necessarily mirrors Zone 2, nor Zone 4 necessarily mirrors Zone 3. For example, Fig. 4 shows a detailed design where Zone 3 and Zone 4 are not symmetric.

Fig. 3. Standard geometry of fishbone layouts.

Fig. 4. Definitions of zones of a fishbone layout.

Please cite this article as: Cardona, L.F., et al., Detailed design of fishbone warehouse layouts with vertical travel. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.03.006i

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Now, we define a detailed three-dimensional design of a warehouse layout DW as a set of openings that belongs to the warehouse W, where an opening O with dimensions ew, el and eh (Fig. 5) is defined by its location (x; y; z). We call ni the number of lines of zone i, ωij the number of positions in line j of zone i and τ the number of tiers of the layout; then the detailed design of a fishbone warehouse is defined in the following equation: DW ¼ fOijkt j 1 r i r 4; 1 rj r ni ; 1 rk r ωij ; 1 r t r τg:

ð1Þ

Following this definition, the detailed design of a fishbone layout is completely defined by its features ωij, ni, τ, and the location ðx; y; zÞ of each opening Oijkt . Fig. 6 shows an example of the notation. In Fig. 6(c), lines are numbered top to bottom and positions left to right. In the same way, Fig. 6(a), (b) and (d) shows the notation for lines and positions of its zones. Fig. 1 shows a fishbone layout with six tiers,

that is t ¼ 1…6 where tiers are numbered from bottom to top, that is t ¼1 is the tier on the floor. This way, the highlighted item of Fig. 6(c) will be denoted by O4;5;6;1 . Now, we define how openings are arranged within each zone depending on the four primary characteristics of the layout. This way, all the features of the layout will depend on these four characteristics. Fig. 1 shows certain regularities or patterns in the arrangement of the openings of the fishbone layout. We represent these patterns with mathematical finite sequences and use them to define ωij, ni, τ, and the location of each opening Oijkt .

3.1.1. Definition of ωij Here, we introduce three of the four primary characteristics of the layout:

η s I2

Fig. 5. Dimensions of the openings.

the number of lines of Zone 1 (or Zone 4) the number of openings of the first line of Zone 1 (or Zone 4) increment in the number of openings between two consecutive odd lines in Zone 1 (or Zone 4)

To represent the arrangement of openings on Zone 1 (or Zone 4), we have stated that lines of Zone 1 (or Zone 4) have a constant increment between odd lines that we call I2. That is, I2 is the increment in openings between the first line and the third line, that in turn is the same difference in openings between the third line and the fifth line, and so on.

Fig. 6. Notation of zones of a fishbone layout. (a) Zone 3. (b) Zone 2. (c) Zone 4. (d) Zone 1.

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Now we introduce other feature of the fishbone layout that depends on I2: the increment in openings between the line j and the line j  1, when j is even. We called this feature I1. Fig. 7 shows an example of the Zone 1 of a fishbone layout. In this example, η ¼ 4 because it has four lines; s ¼1, given that the first line has one opening; I 2 ¼ 6, given that the first line has one opening and the third line has five openings; and I 1 ¼ 1, given that the increment in openings of the second line with respect to the first line is one. With the primary characteristics η, s and I2, we will determine θ, I1, β and α. Fig. 7 shows that the slope of the diagonal aisle (m) is defined as m¼

2el þap : I 2 ew

ð2Þ

It follows that

θ ¼ tan  1 ðmÞ:

ð3Þ

Eq. (4) shows how to find I1 with respect to the previously defined m
 el I1 ¼ : ð4Þ mew In the example of Fig. 7, el ¼ 1, ew ¼0.7 and ap ¼2, hence m ¼0.95, θ ¼ 43:61 and I 1 ¼ ⌊1:5c ¼ 1. Next, we calculate β. Let us call u the length of the line segment that joins the upper vertex of the triangle of Zone 1 and the point where the first line intersects the vertical boundary of Zone 4. (Fig. 8). From top to bottom, β begins with u, next, there are two lines; next, there is an aisle, next, there are two lines; next, there is an aisle, and the process is repeated until Zone 1 has completed η lines. Following that structure, we can calculate β in terms of u, η and el as follows:

β ¼ u þ ηel þ

ap ð2η þð  1Þη  1Þ: 4

ð5Þ

In this equation, the term ð  1Þη is needed because when η is even, we need an additional aisle after the last line, whereas such an aisle is not needed when η is odd. We calculate u as the maximum between the minimum space that the forklift needs to pick an item in the first openings of the first line ðap  ap cos ðθÞÞ, and the corresponding cathetus to the length of the first line ðsew mÞ, and it is calculated as u ¼ maxðsew m; ap  ap cos ðθÞÞ: Finally, we calculate

ð6Þ

α in terms of β and m as

β α¼ :

5

adjacent to the central aisle, next, there is an aisle, next, there are two lines, next, there is an aisle, and the process is repeated until no more lines fit in α. Finally, ω2j is calculated as follows:
  a m ω2j ¼ α  jel  p ð2j  ð  1Þj  3Þ : el 4 Finally, let us call W to the set of all

ωij and define it as

W ¼ fωij j 1 r i r4; 1 rj r ni g:

ð8Þ

3.1.2. Definition of ni We know that n1 ¼ n4 ¼ η (remember that η is one of the primary characteristics of the fishbone layout). Additionally, from the symmetry between the triangles of Zone 2 and Zone 3 of the standard geometry, we know that n2 ¼ n3 . Hence, we only need to calculate n2, to completely define ni. As we stated in the definition of ωij, once we define η, s and I2, we also define the size of the triangles that define the four Zones (αand β). And once α is defined, we can calculate the maximum number of lines that fit in α. We call that number c, then, considering the aisle arrangement of Zone 2 explained in the definition of ω2j, c is calculated as follows:   ap ð9Þ c ¼ max qel þ ð2q þ ð 1Þq  1Þ r α q 4 As in Eq. (5), the term 14ð2q þ ð 1Þq  1Þ is used to add an aisle when q is even. In most cases, the number of lines that fit in α will be the number of lines that fit in Zone 2 (or Zone 3), but it is not always the case. Note that when m is too small, there can be some ω2j (or ω3j) equal to zero, hence there will be some lines with zero openings. Fig. 1 shows a fishbone layout where the number of lines that fit in α is 10 ðc ¼ 10Þ and it actually has 10 lines in Zone 2 (or Zone 3). On the other hand, Fig. 10 shows a fishbone layout where the number of lines that fit in α is 7 ðc ¼ 7Þ, but there cannot be any openings in the last line, therefore, in Fig. 10, we can only see six lines in Zone 2 (or Zone 3), even if c¼7. Finally, ( η; i ¼ 1 3 i ¼ 4 ni ¼ ð10Þ c; i ¼ 2 3 i ¼ 3:

3.1.3. Definition of τ The number of tiers is the fourth primary characteristic of the fishbone layout.

ð7Þ

m

In consequence, once we define η, s and I2, the values of β and α are settled, and with them, the dimensions of the triangles that define the four Zones. Now, we calculate ωij as follows: ( ω1j ; i ¼ 1 3 i ¼ 4 ωij ¼ ω ; i ¼ 2 3 i ¼ 3: 2j Let us begin with its definition for Zone 1 (ω1j). The first line ðj ¼ 1Þ has s openings, next, j¼ 2 has s þ I 1 openings, next, j¼3 has s þ I 1 þ ðI 2  I 1 Þ openings, next, j¼4 has s þ I 1 þ ðI 2  I 1 Þ þ I 1 openings, and so on (Fig. 8). This way, ω1j is defined as follows: 1 4

ω1j ¼ s þ ð2j þ ð  1Þj 1ÞI1 þ 14 ð2j  ð  1Þj  3ÞðI2  I1 Þ: In this case, 14ð2jþ ð 1Þj  1Þ increment I1 openings when j is even and 14ð2j  ð  1Þj  3Þ increment I 2  I 1 openings when j is odd. To define ω2j, we put as many openings as will fit in each line of Zone 2, according to the following structure: we locate two lines

3.1.4. Definition of the location of each opening Oijkt We define the location of an opening as Oijkt ¼ ðX ijkt ; Y ijkt ; Z ijkt Þ; where ðX ijkt ; Y ijkt ; Z ijkt Þ are the coordinates of the opening Oijkt . We take the left-bottom corner of the warehouse as the reference point ð0; 0; 0Þ, and the coordinates of the opening Oijkt as the point located in the left bottom corner of the opening. We take into account the geometries and notation as shown in Figs. 3 and 6, to define the coordinates ðX ijkt ; Y ijkt ; Z ijkt Þ of an opening Oijkt as 8 i¼1 A  kew ; > > > > > α þ ap sin ðθÞ þ ap þ ðj  1Þel > > > > ap < þ ð2j ð 1Þj  3Þ; i¼2 ð11Þ X ijkt ¼ 4 > > ap > j > > α þ ap sin ðθÞ  jel  ð2j  ð  1Þ  3Þ; i ¼ 3 > > 4 > > : ðk  1Þe ; i ¼ 4; w

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Y ijkt

8 < β  u  je  ap ð2j  ð  1Þj  3Þ; l 4 ¼ : B  ke ; w

i ¼ 13i ¼ 4 i ¼ 2 3 i ¼ 3;

Z ijkt ¼ eh ðt  1Þ;

3.2. Optimization model ð12Þ

ð13Þ

where A is the width of the warehouse and B is the length of the warehouse (Fig. 9), and they are given by A ¼ 2α þ ap þ 2ap sin ðθÞ;

ð14Þ

B ¼ β þ ap cos ðθÞ:

ð15Þ

3.1.5. Algorithm Algorithm 1 generates a detailed fishbone layout given the four primary characteristics of the layout (η, s, I2, τ) and the parameters (ew ; el ; eh ; ap ).

In this section, we present an optimization model for the detailed design of the fishbone layout that minimizes the total operational cost and complies with a desired storage capacity. The model takes the following parameters: the dimensions of the openings ðew ; el ; eh Þ, the width of the aisles ðap Þ, the speeds of the forklift ðV l ; V z Þ, a desired storage capacity ðCÞ, a maximum number of tiers ðlh Þ and two constants ðC 1 ; C 2 Þ that weight the material handling cost and the cost of leasing and maintaining the warehouse area. The variables of the model are the four primary characteristics ðη; s; τ; I 2 Þ of the fishbone layout that we defined in Section 3.1. The objective of the model is to minimize the total operational cost of the fishbone layout (Eq. (16)), which we define as the expected monthly expenses generated by the material hand-

Algorithm 1. Detailed design fishbone. 1:procedure DETAILEDDESIGN ðη; s; I 2 ; τ; ew ; el ; eh ; ap Þ 2: m’CalculateSlopeðew ; el ; ap ; I 2 Þ 3: θ’CalculateAngleðmÞ 4: I 1 ’CalculateIncrementðew ; el ; mÞ 5: u’CalculateUðew ; s; m; ap ; θÞ 6: β ’CalculateBetaðel ; ap ; u; ηÞ 7: α’CalculateAlphaðβ; mÞ 8: A’CalculateTotalWidthðap ; α; θÞ 9: B’CalculateTotalLengthðβ; ap ; θÞ 10: c’CalculateCðel ; ap ; αÞ 11: ni ’CalculateNiðc; ηÞ 12: ωij ’CalculateOpeningsðew ; el ; ap ; s; I 1 ; I 2 ; m; α; ni Þ 13: DW ’∅ 14:

▹ Eq. (2) ▹ Eq. (3) ▹ Eq. (4) ▹ Eq. (6) ▹ Eq. (5) ▹ Eq. (7) ▹ Eq. (14) ▹ Eq. (15) ▹ Eq. (9) ▹ Eq. (10) ▹ Eq. (8) ▹ Initialize DW

15: for t¼1 to τ do 16: 17: for i¼ 1 to 4 do 18: 19: for j ¼1 to ni do 20:

▹ Tiers ▹ Zones ▹ Lines

21: for k¼ 1 to ωij do 22: x’CalculateXcoordinateðew ; el ; i; j; k; α; ap ; θ; AÞ 23: y’CalculateYcoordinateðew ; el ; ap ; β ; u; i; j; k; BÞ 24: z’CalculateZcoordinateðeh ; tÞ 25: Oijkt ’CreateOpeningðx; y; zÞ 26: DW ’DW [ Oijkt 27: 28: end for 29: 30: end for 31: 32: end for 33: 34: end for 35: return DW 36end procedure

To illustrate the algorithm, we use it with the following input data: η ¼ 6, s ¼1, I 2 ¼ 3, τ ¼ 1, ew ¼ 1, el ¼1, eh ¼1, ap ¼1 Fig. 10 shows the resulting fishbone layout generated with the algorithm and Table 1 shows the features of the layout.

▹ Positions ▹ Eq. (11) ▹ Eq. (12) ▹ Eq. (13)

ling system and the cost of leasing and maintaining the warehouse area. The cost of the warehouse area will be the marginal cost per area unit ðC 1 Þ multiplied by the total area of the warehouse (Eq.

Please cite this article as: Cardona, L.F., et al., Detailed design of fishbone warehouse layouts with vertical travel. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.03.006i

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Fig. 9. Notation of the areas of the fishbone layout.

Fig. 7. Definition of I2 and I1.

Fig. 10. Example of a fishbone layout generated with Algorithm 1.

To satisfy the desired number of openings ðCÞ, we make the total number of openings of the layout greater than or equal to the desired storage capacity (Eq. (17)). To limit the number of the tiers, we bind τ to its maximum value lh (Eq. (18)), where lh represents the maximum height for the warehouse because of the limits of the civil works or the maximum height that the forklifts can reach. To sum up, the complete optimization model is as follows: Parameters

Fig. 8. Definition of β.

Table 1 Features of Fig. 10. Feature

Value

m θ I1 u β α A B c ½ni 

1 451 1 1 10 10 22.41 10.70 7 [6,7,7,6]

ew ; el ; eh ap V l; V z C lh C1 C2 Variables

η

the number of lines of Zone 1 the number of openings of the first line of Zone 1 the number of tiers of the layout. regular increment of openings between odd lines of Zone 1 (or Zone 4)

s (19)). If the company owns the land, C1 can be considered as the opportunity cost of the money. This way, the area cost is ðC 1 ÞnArea. The material handling costs are two times4 the product of the expected travel time of picking an arbitrary item in the warehouse (Eq. (20)), the expected number of picks in a monthly operation, and the marginal cost of the material handling system per minute. Here, C2 is the expected number of picks in a monthly operation multiplied by the marginal cost of the material handling system per minute. This way, material handling cost is 2nðC 2 ÞnE½T. The constraints of this model are two: to satisfy the desired storage capacity ðCÞ, and to limit the number of tiers of the layout.

4

The expected value of the travel is multiplied by 2 considering a round-trip.

dimensions the openings aisle width forklift speeds desired storage capacity in openings maximum number of tiers area cost material handling cost

τ

I2

Objective function Z ¼ ðC 1 ÞnArea þ 2nðC 2 ÞnE½T

ð16Þ

Constraints S  C Z0

ð17Þ

lh  τ Z 0

ð18Þ

where S is the total number of openings in the layout (Section 3.2.3). Finally, in the following Sections 3.2.1, 3.2.2 and 3.2.3, we show how to calculate the area of the warehouse ðAreaÞ, the expected

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8

Finally, we defined E½T as the arithmetic average of T ðOP ijkt Þ for all i, j, k and t. That is E½T ¼

ωij ni X τ X 4 X 1X T ðOP ijkt Þ; St ¼1i¼1j¼1k¼1

ð20Þ

where S is the total number of openings of the fishbone layout (Section 3.2.3).

Fig. 11. Picking points of a fishbone layout.

travel time ðE½TÞ and the total number of openings of the layout ðSÞ.

3.2.3. Number of openings in a fishbone layout ðSÞ The number of openings in a fishbone layout is the number of openings of each tier of the layout multiplied by the number of tiers, where the number of openings of each tier of the layout is the sum of all ωij. Then, S is calculated as follows: S¼τ

ni 4 X X

ωij :

i¼1j¼1

3.2.1. Area of the warehouse (Area) The area of the warehouse is defined by A, the width of the warehouse (Eq. (14)), and B, the length of the warehouse (Eq. (15)) Area ¼ AB:

ð19Þ

3.2.2. Expected travel distance ðE½TÞ First, for each opening, we calculate the coordinates ðx; y; zÞ of the point where the forklift picks up or deposits an item on the opening Oijkt (Fig. 11). Analogous to what we did in Section 3.1, here we provide a mathematical finite sequence to calculate the picking point of an opening Oijkt , denoted as OP ijkt ¼ ðXP ijkt ; YP ijkt ; ZP ijkt Þ where 8 i¼1 A ðk  0:5Þew ; > > > > > A ap þ 2el > j > ð2j þ ð  1Þ  1Þ; i ¼ 2 < þ 2 4 XP ijkt ¼ A ap þ 2el > > >  ð2j þ ð  1Þj  1Þ; i ¼ 3 > > 2 4 > > : ðk  0:5Þe ; i ¼ 4; w

YP ijkt

8 < β  u þ ap  ap þ 2el ð2jþ ð 1Þj  1Þ; 2 4 ¼ : B  ðk  0:5Þe ; w

i ¼ 13i ¼ 4 i ¼ 2 3 i ¼ 3;

ZP ijkt ¼ eh ðt  1Þ: We then use OPijkt to calculate the time needed to pick up or deposit an item at opening Oijkt, denoted as T ðOP ijkt Þ. Cardona et al. (2012) created an analytical expression for the expected travel distance of a fishbone layout in a continuous domain, here we extend it for a three-dimensional layout and a discrete domain. In Appendix A, we show how to calculate T ðOP ijkt Þ, here we present the resulting function ( T 1 ðOP ijkt Þ; i ¼ 2 3 i ¼ 3 T ðOP ijkt Þ ¼ T 2 ðOP ijkt Þ; i ¼ 1 3 i ¼ 4

3.3. Heuristic approach The optimization model presented in Section 3.2 is an integer nonlinear/non-smooth optimization model with non-linear constraints. Given the combination of integer constraints and a non-linear objective function, we propose a genetic algorithm (GA) to solve the problem and we use MATLAB as the mathematical programming environment. In the following sections, we present the specifics of our customized genetic algorithm. For further material on genetic algorithms, refer to Goldberg (1989).

3.3.1. Initialization The initialization procedure uses a random procedure to create m individuals from the solution domain following a genetic representation. The genetic representation for the variables of the problem will be the array ½s; τ; I 2  where each entry is an integer number. The optimization model (Section 3.2) has four variables which are the primary characteristics of the fishbone layout ½η; s; τ; I 2 , but here we use only ½s; τ; I 2  in the genetic representation and find the minimum η such that S 4 C. To find η, Algorithm 2 uses a binary search given that the greater the value of η, the greater the storage capacity S – ceteris paribus. The algorithm uses the functions UpperBoundðs; η; I 2 ; CÞ and CalculateCAPðs; τ; I 2 ; ηÞ, where UpperBoundðs; η; I 2 ; CÞ calculates the minimum number such that the layout has more openings than the required storage capacity when s¼1, I 2 ¼ 1, and τ ¼ 1, and CalculateCAPðs; τ; I2 ; ηÞ calculates the storage capacity ðSÞ for given values of s, τ, I2 and η (Section 3.2.3).

where 1 ðð1 þ m2 Þ0:5 mÞj XP ijkt  0:5Aj Vl YP ijkt ZP ijkt þ ; þ Vl Vz

T 1 ðOP ijkt Þ ¼

and 1 j XP ijkt  0:5Aj Vl ZP ijkt 1 ðð1 þ m2 Þ0:5  1ÞYP ijkt þ : þ mV l Vz

T 2 ðOP ijkt Þ ¼

Fig. 12. Definition of ubs.

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Algorithm 2. Find η. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:

procedure BINARYSEARCH ðs; η; I 2 ; CÞ lb’1; ub’UpperBoundðs; η; I 2 ; CÞ while ðub  lbÞ 4 1 η’ðub þ lbÞ=2; S’CalculateCAPðs; τ; I 2 ; ηÞ; if ðS ¼ CÞ break elseif ðS 4 CÞ ub’η else lb’η end if end while η’up; end procedure

The solution domain of the problem is the cartesian product ½1; ubs   ½1; ubτ   ½1; ubI2  where ubs, ubτ and ubI2 are the upper boundaries of s, τ and I2 respectively, and ½1; ub is the set of integer numbers between 1 and ub including both extremes. The upper boundary of s is the minimum number such that the first line is large enough so that there can fit another two lines before the “first line” (Fig. 12). In that case, as can be seen in Fig. 12, there are two lines before the first line, which makes no sense  2el q þ ap q ubs ¼ min Z 2el þ 2ap q ubI2  ðap þ 2el ÞðubI2 ew Þ 4 sew  Z ew ð21Þ 2el þ ap The upper boundary of τ is given directly by ubτ ¼ lh . Finally, the upper boundary of I2 is the minimum number such that the layout has more openings than the desired when s¼ 1, η ¼ 3, and τ ¼ 1. Once the genetic representation and the solution domain has been defined, the initialization is a random procedure that creates m individuals that belong to ½1; ubs   ½1; ubτ   ½1; ubI2 .

3.3.2. Recombination The GA uses the fitness function (Eq. (16)) to evaluate the goodness of each individual created in the initialization. Next, a roulette-wheel operator (Kazarlis et al., 1996) selects x percentage of individuals of the population for recombination (recombination fraction) where each individual has a probability of being selected proportional to the goodness of its fitness value (note that one individual can be selected more than once). Then, the recombination population is ordered through a random function, then the parent i is paired up with the parent i þ 1. Afterwards, for each duple of paired parents, the GA performs a single point crossover operator (Srinivas and Patnaik, 1994) to create a child for the next generation. Finally, the GA replaces the bottom x percentage of the worst individuals of the population for the children created on the recombination process.

3.3.3. Mutation The GA randomly selects y percentage of the individuals of the population and mutates them (mutation fraction). For each entry of each individual, the algorithm adds the integer part of a random number from a log-normal distribution – with parameters based on the characteristics of the current population and the domain of the variables. This operator is based on the Power mutation operator introduced by Deep et al. (2009).

9

3.3.4. Stopping criteria Finally, the new population is used in the next iteration of the algorithm. Generally, a GA terminates when a maximum of generations has been performed, the best solution of the population has not changed in n generations (Stall generations), or a maximum time (T) is reached.

4. Numerical example and discussion In this section, we illustrate the use of the method – through a hypothetical numerical example – and the insight we gained. The numerical example is a greenfield warehouse project where all the cost and operation parameters were estimated based on a typical distribution center that operates in Kentucky, USA. The distribution center is a unit load warehouse that handles pallets of 50-kg bags of bulk material. The material is received – in trucks of 33 ton filled with 50-kg bags – at a rate of 77 trucks per day. Then, the material is unitized in pallets of 42 bags and moved by forklifts from the docks to the reserve area. In the reserve area, each pallet is stored in an opening rack with dimensions: el ¼ 1.524 m, ew ¼ 1.524 m and eh ¼1.524 m. Next, the unit loads are picked up by forklifts to be shipped on 26 ton intermodal containers at a rate of 90 containers per day. The DC operates 20 days per month, in 8 h shifts and 1 shift per day with a 1 hour break at noon, two breaks of 15 min – one in the morning and one in the afternoon – and a daily meeting in the morning of 15 min with the plant supervisor. All forklifts in the warehouse are equal. They have to be able to lift unit loads of 2.1 ton up to seven levels ðlh Þ. The required aisle width ðap Þ is 4.572 m, the average travel speed ðV l Þ is to be considered 9.656 km/h and the lifting speed ðV z Þ is 5.486 m/min. The cost of one forklift is estimated to be 47,130 USD amortized to five years, the salary of the forklift operator is 39,900 USD/year and the maintenance of the forklift cost 2000 USD/year. The efficiency of the operator is assumed to be 85% and the reliability of the equipment 99.4%. The discount rate was estimated as 4%. With this information, we calculate the cost of a minute of travel (0.60706 USD) and the number of travels per month (46,772.1 USD). Finally, we multiply the cost of a minute of travel by the number of travels per month, and obtained C2 equals to 28,394 USDntravel=ðyearnminÞ. The marginal area cost was estimated based on a leasing of 8241 USD/month for a 4273.53 m2 warehouse. In this way, C1 is 23.14 USD=ðm2 nyearÞ. Now that we have values for ew, el, eh, ap, Vl, Vz, lh, C1 and C2, we fix these parameters and vary the storage capacity ðCÞ in a similar experiment to the one performed by Gue et al. (2012) for small ðC o 9400Þ, medium ðC o 21; 600Þ and large ðC o 38; 800Þ scenarios. The computational experiments were performed in a grid of five computers with processors Intel(R) Core(TM) i7-4500 CPU @ 1.8 GHz 2.4 GHz and 8 GB of RAM memory. For the small scenarios, we simulated 121 warehouses with C from 3450 to 9400 in steps of 50. For each instance, we executed the GA and also performed an exhaustive search to find a threedimensional fishbone design. In addition we found an equivalent three-dimensional traditional aisle design for comparison purposes. The GA was performed5 with a maximum running time (T) of 90 s, a population (m) of 20, a recombination fraction (x) of 0.8, a mutation fraction (y) of 0.02 and stall generations (n) of 10. The exhaustive search evaluates all the elements that belong to the solution domain ½1; ubs   ½1; ubτ   ½1; ubI2 , so it guarantees optimality of the solution. The equivalents three-dimensional traditional aisle designs were found with the method proposed by Onut et al. (2008). Fig. 13(a) presents the percentage of 5

The running parameters of the GA were selected based on preliminary runs.

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10

0.5

Relative Frequency

Relative Frequency

0.45

0.3

0.15

0.61

1.24

1.87 2.50 % Deviation

0.4 0.3 0.2 0.1 0

3.13

1.33

1.73

2.14 % Deviation

2.54

2.95

Relative Frequency

0.4

Relative Frequency

0.3 0.2 0.1 0

1.04

1.37

1.71

2.04

2.37

0.75

0.5

0.25

0.23

0.61

1

1.37

1.77

2.35

% Deviation over the exhaustive search value

% Deviation

Fig. 13. Numerical example insights. (a) Deviation of the total cost of the traditional design over the fishbone design – small scenario. (b) Deviation of the total cost of the traditional design over the fishbone design – medium scenario. (c) Deviation of the total cost of the traditional design over the fishbone design – large scenario. (d) Deviation of the GA over the exhaustive search value.

Time (sec)

2500

1800

Exhaustive Search GA

900

0.34

0.94

1.54

Storage Capacity

2 x 10

Fig. 14. Running time comparison.

deviation of the total cost of the traditional aisle design over the total cost of the fishbone design. In this hypothetical distribution center, the fishbone design was, on average, 2.06% better than the traditional design for small scenarios. Additionally, we found that for this numerical example the optimal solution always belongs to the cartesian product ½1; C1=3   ½1; C1=3   ½1; C1=3 . For the medium scenarios, we simulated 61 warehouses with C from 9600 to 21,600 in steps of 200. For each instance, we performed the GA, an exhaustive search and found an equivalent threedimensional parallel aisle design (Onut et al., 2008). The GA was performed with a maximum running time (T) of 180 s, a population (m) of 20, a recombination fraction (x) of 0.8, a mutation fraction (y) of 0.02 and stall generations (n) of 10. It is not practical to perform a complete exhaustive search in this case, therefore we limited the search to the domain ½1; C1=3   ½1; C1=3   ½1; C1=3 . Even if optimality is not guaranteed in the exhaustive search, a near optimal solution is expected given that for small scenarios the optimal solution was always on that region. Fig. 13(b) presents the percentage of deviation of the total cost of the traditional aisle design over the total cost of fishbone design. In this hypothetical distribution center, the fishbone

design was, on average, 1.98% better than the traditional design for medium scenarios. For the large scenarios, we simulated 50 warehouses with C from 21,950 to 38,750 in steps of 350. For each instance, we performed the GA and found an equivalent three-dimensional parallel aisle design (Onut et al., 2008). The GA was performed with a maximum running time (T) of 270 s, a population (m) of 20, a recombination fraction (x) of 0.8, a mutation fraction (y) of 0.02 and stall generations (n) of 10. For large scenarios, we could not perform an exhaustive search because its impractical computational times. Fig. 13(c) presents the percentage of deviation of the total cost of the traditional aisle design over the total cost of fishbone design. In this hypothetical distribution center, the fishbone design was, on average, 1.8% better than the traditional design for large scenarios. The computational experiments present significant opportunities for further analysis. Using the data of the 232 fishbone designs created for all the scenarios (small, medium, and large)

 we evaluate the goodness of the solutions found by GA,  we compare the running time of the GA with the one of the exhaustive search,

 we analyzed the slope of the diagonal aisle of the fishbone designs compared to the study of Cardona et al. (2012), and

 the impact that changes in the cost parameters on the layout structure for a specific situation.

4.1. Goodness of solutions Fig. 13(d) shows the deviation of the GA over the optimum value for small scenarios or over the value found by the exhaustive search for the medium scenarios. Here, we can see that 87.8% of the solutions found by the GA were deviated only by 0.61%, which suggests a good performance of the GA. Considering only the small

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scenarios, the GA was deviated only by 0.587% over the optimum value in 91.74% of the cases. 4.2. Running time One of the important points of using a genetic algorithm is that it saves significant computational time. Here, we compared the running time of the exhaustive search with that of the genetic algorithm for small and medium scenarios. Fig. 14 presents an increasing behavior of the running time for both algorithms given that the bigger the desired storage capacity, the bigger the search space. Moreover, we see that the running time for the exhaustive search is significantly higher then for the genetic algorithm. For a desired storage capacity of 3450, the running time of the exhaustive search is 2.8 times the one of the genetic algorithm. And for a desired storage capacity of 3450, the running time of the exhaustive search is 10.5 times the one the genetic algorithm. To sum up, the GA reaches optimal or near optimal solutions – in most of the cases deviated at most by 0.587% – in a reasonable computational time – 270 s for large scenarios. 4.3. Slope analysis Cardona et al. (2012) presented a continuous optimization model to determine the best slope of diagonal cross aisle. For greenfield projects, they found that the slope of the diagonal aisle should be 1.00 (π =4 rad). As we stated in the literature review, it is not always possible to obtain the optimal layout recommend by Cardona et al. (2012) when the arrangement of the openings is considered. Here, we calculated the slope of the diagonal cross aisle of the 232 fishbone designs of the computational experiments. We found that 198 designs have a slope of 1.00 (π =4 rad), 31 designs have a slope of 1.25 (0.896 rad), 2 designs have a slope of 0.833 (0.694 rad), and 1 design has a slope of 1.667 (1.0303 rad). In conclusion, these computational experiments evidence the applicability the results of Cardona et al. (2012) and show a very good performance of the algorithm presented in this paper to achieve the optimal slope recommend by Cardona et al. (2012). 4.4. Impact of changes in cost parameters Here, we evaluate how sensitive the layout is to changes in cost parameters. We simulated 100 scenarios based on the same hypothetical distribution center presented in this section. We fixed parameters for the simulation were: el ¼1.524 m, ew ¼ 1.524 m, eh ¼ 1.524 m, ap ¼4.572 m, Vl ¼9.656 km/h, ðV z Þ ¼ 5:486 m=min, lh, C 2 ¼ 28; 394, and C ¼ 10; 000. And, we varied the area cost (C1) from 0.1075 (a 10% of the original value) to 21.5 (10 times its original value) in equally space steps. For each scenario, we found the best fishbone design with the GA, and for each of these 100 designs we calculated the total cost for each of the 100 scenarios. In this way, we ended up with a 100  100 matrix with the total cost of each design in each scenario. From this data, we see that the bigger is the area cost, the higher is the warehouse, given that the algorithm is trying to minimize the cost of the land. For area cost from 10% to 110% of the original value (between 0.1057 and 2.365), we see that changes of 10% in the area cause increments of 1 or 2 levels until the design achieves its maximum number of levels (lh), which in this case is 15. We called threshold value to the value of the area cost above which τ ¼ lh . In this case the threshold value is 2.365. For changes in the area cost under the threshold value, the average increment in the total cost is 56.01% and the maximum increment is 89.9%. For changes in the area cost above the threshold value, the average increment in the total cost is 3.70%

11

and the maximum increment is 4.04%. When the changes in the area cost are large to move between the lower region and the upper region of the threshold value, the average increment in the total cost is 165.92% and the maximum is 210.39%. From this preliminary simulations, we conducted further simulation with smaller steps in the changes of the area cost and concluded that if the change in the area cost is small enough to remain the optimal value of τ then the expected increment in the total cost is less than 4.78%, but if the change of the area cost imply a change in the optimal value of τ then the increment in the total cost could be up 210.39%. Finally, we compare the difference between the fishbone designs and traditional designs in response to changes in the cost parameters. For each of the 100 fishbone designs used in the computational experiments of this sections, we found an equivalent traditional design. For this hypothetical case, the results show that on average fishbone designs have an expected travel time 4.43% lower than traditional design; on the other hand, fishbone designs present an area 14.8% bigger than traditional designs. The convenience of the fishbone design should be evaluated case-by-case depending on the cost structure of the warehouse, but this results suggest that for regions with high land cost, fishbone designs can have total cost until 14.9% higher than traditional designs; and on regions with high labor cost, fishbone designs can have total cost until 4.43% lower than traditional designs.

5. Conclusions and future research In this paper, we provide a method that generates a detailed threedimensional design of a fishbone layout given a desired storage capacity. Our method is useful for practitioners when all openings of the warehouse are equal, a random storage policy is adopted and the material handling system is constituted only by forklifts (all forklifts are equal). Contrary to several approaches where usually only one aspect is taken into account for defining the warehouse layout (generally the expected travel distance), our model is more complete as it delivers the detailed design of the layout, optimizing the total operational cost of the warehouse, considering both area cost and material handling cost. Additionally, our method provides its output in a practical way for a warehouse practitioner as it delivers the coordinate ðx; y; zÞ for each opening of the layout. We proposed an algorithm based on mathematical finite sequences to generate the detailed design of a fishbone layout given four primary characteristics of the layout (Section 3). In addition, we provide an optimization model (integer non-linear/non-smooth optimization problem with non-linear constraints) that finds a layout that complies with desired storage capacity and optimizes the total operational cost (Section 3.2). Finally, we proposed a genetic algorithm to perform the optimization procedure (Section 3.3). We demonstrated the use of our method through a hypothetical numerical example (Section 4) based on a typical operation of a distribution center in Kentucky, USA. We tested the performance of the optimization procedure concluding that the GA was able to find a near optimum value – deviated only in 0.587% – over the optimum value in 91.74% of the cases. Additionally, the GA presented a significant improvement in computational time, it was between 2.8 and 10.5 times faster than an exhaustive search. Furthermore, the computational experiments showed that our method is very good in achieving the optimal slope of the diagonal cross aisle recommended by Cardona et al. (2012), in 85% of the cases, we obtained fishbone designs with a slope of π =4. Since the introduction of the fishbone design (Gue and Meller, 2009), the expected improvement on the expected travel distance is 20% leading to lower material handling cost compared with parallel

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aisles design. But it is also expected that the area of a fishbone design will be greater than that of a parallel aisles design for the same desired storage capacity, in consequence, higher area cost is expected. Therefore when comparing the fishbone layout with the parallel aisles design, there is a trade off between area cost and material handling cost. For the numerical example of this paper, we found that the fishbone design presents an improvement on the total cost of only 2.06% for small projects and 1.8% for large warehouses. This opens a research line on the suitability of non-traditional designs for industrial contexts. Additionally, the results of the computational experiments showed that on average fishbone designs have an expected travel time 4.43% lower than traditional design; on the other hand, fishbone designs present an area 14.8% bigger than traditional designs. This opens a research line to evaluate the convenience of fishbone design that weight the improvements in the expected travel time and the losses in area cost. Our method is useful for warehouse practitioners as it reduces the dependence upon experts and human decision making in the implementation of fishbone layouts. It considers the most critical factors impacting the operational cost of a warehouse to generate the detailed three-dimensional design and it delivers an optimum or a near optimum solution in a reasonable time (maximum 4.5 min). For future research, we recommend to analyze the robustness of the total operational cost. We want to determine how critical it is to choose an optimum value, and in the same way, to analyze the sensitivity of the total operational cost to variations in each of the primary characteristics of the layout. We propose to enhance the optimization procedure. We used a genetic algorithm that does not guarantee the optimality of the solution. The problem can be explored from the combinatorial optimization point of view or can be considered as a non-smooth non-linear optimization problem with non-linear constraints. Another research avenue that opens this paper is to test the findings of Cardona et al. (2012) about the optimal characteristics of fishbone layouts. They used a continuous analysis and stated that a fishbone design should have a central P&D point and the size width should be the double of its length (the width is the side of the warehouse where the P&D point is located). It is worth it to verify if what Cardona et al. (2012) found with a continuous analysis is still valid in a discrete model as the one proposed in this paper. Finally, we propose to extend our method for different contexts. As we explained in Section 3 (detailed design algorithm), our method has some limitations; in future research, these limitations can be surpassed aiming to make our method more useful for warehouse practitioners. For example, we propose to consider the storage assignment policy to analyze how sensible the optimum layout is to product allocation. Also, to consider different ways of operation of the material handling system: multiple kinds of forklifts and different metric distance as Tchebycheff (Clark and Meller, 2013). Lastly, we recommend to consider a detailed design with different types of openings in the warehouse.

ðx; y; zÞ the location of an item in the warehouse Vl the linear speed of the forklift Vz the lifting speed of the forklift A the width of the warehouse m the slope of the diagonal aisle then, T is defined as follows: T V l ;V z ;A;m ðx; y; zÞ ¼

where DA;m ðx; yÞ is the distance from the P&D point of the warehouse to the point ðx; y; 0Þ. Cardona et al. (2012) created a function for DA;m ðx; yÞ, here we adapt it to our notation. Let be

 D1A;m ðx; yÞ – the distance from the P&D point of the warehouse to the point ðx; y; 0Þ, if ðx; yÞ A Zone 2 3 Zone 3.

 D2A;m ðx; yÞ – the distance from the P&D point of the warehouse to the point ðx; y; 0Þ, if ðx; yÞ A Zone 1 3 Zone 4. Then ( DA;m ðx; yÞ ¼

Cardona et al. (2012) created an analytical expression for the expected travel distance of fishbone layouts, here, we extend it for a three-dimensional layout and a discrete domain. Here, we try to preserve the notation of Cardona et al. (2012), as much as possible. Fig. A1 shows an example of the path that the forklift has to travel and the vertical distance that the forklift has to lift the forks in order to pick an item on the right-hand half-picking space. The travel time of a pick operation is defined by the length of the path, the height of the location of item, the linear speed and lifting speed of the forklift. Let be

D1A;m ðx; yÞ

ðx; yÞ A Zone2 3 Zone3

D2A;m ðx; yÞ

ðx; yÞ A Zone1 3 Zone4;

where 2 D1A;m ðx; yÞ ¼ ððð0:5AÞ2 þ ð0:5mAÞ2 Þ0:5  0:5mAÞj x  0:5Aj þ y; A D2A;m ðx; yÞ ¼ j x  0:5Aj þ

2 ððð0:5AÞ2 þ ð0:5mAÞ2 Þ0:5  0:5AÞ: mA

Simplifying D1A;m ðx; yÞ ¼ ðð1 þ m2 Þ0:5  mÞj x  0:5Aj þ y; 1 D2A;m ðx; yÞ ¼ j x  0:5Aj þ ð1 þ m2 Þ0:5  1Þy: m Finally, the travel time is defined in terms of OPijkt by ( T 1 ðOP ijkt Þ; i ¼ 2 3 i ¼ 3 T ðOP ijkt Þ ¼ T 2 ðOP ijkt Þ; i ¼ 1 3 i ¼ 4 where T 1 ðOP ijkt Þ ¼

1 ðð1 þ m2 Þ0:5  mÞj XP ijkt  0:5Aj Vl

þ

YP ijkt ZP ijkt þ ; Vl Vz

and T 2 ðOP ijkt Þ ¼

Appendix A. Travel time T

DA;m ðx; yÞ z þ ; Vl Vz

1 j XP ijkt  0:5Aj Vl

þ

ZP ijkt 1 ðð1 þm2 Þ0:5  1ÞYP ijkt þ : mV l Vz

Fig. A1. Travel path for picking an item in a fishbone warehouse.

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References Bartholdi, J.J., Hackman, S.T., 2011. Warehouse & Distribution Science: Release 0.95. The Supply Chain and Logistics Institute, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA. Bassan, Y., Roll, Y., Rosenblatt, M.J., 1980. Internal layout design of a warehouse. AIIE Trans. 12, 317–322. http://dx.doi.org/10.1080/05695558008974523. Berry, J.R., 1968. Elements of warehouse layout. Int. J. Prod. Res. 7, 105–121. http: //dx.doi.org/10.1080/00207546808929801. Cardona, L.F., Rivera, L., Martnez, H.J., 2012. Analytical study of the fishbone warehouse layout. Int. J. Logist. Res. Appl. 15, 365–388. http://dx.doi.org/ 10.1080/13675567.2012.743981. Caron, F., Marchet, G., Perego, A., 2000. Layout design in manual picking systems: a simulation approach. Integr. Manuf. Syst. 11, 94–104. Çelik, M., Süral, H., 2014. Order picking under random and turnover-based storage policies in fishbone aisle warehouses. IIE Trans. 46, 283–300. http://dx.doi.org/ 10.1080/0740817X.2013.768871. Clark, K.A., Meller, R.D., 2013. Incorporating vertical travel into non-traditional cross aisles for unit-load warehouse designs. IIE Trans. 45, 1322–1331. Deep, K., Singh, K.P., Kansal, M., Mohan, C., 2009. A real coded genetic algorithm for solving integer and mixed integer optimization problems. Appl. Math. Comput. 212, 505–518. Dukic, G., Opetuk, T., 2012. Warehouse layouts. In: Manzini, R. (Ed.), Warehousing in the Global Supply Chain. Springer, London, pp. 55–69. http://dx.doi.org/ 10.1007/978-1-4471-2274-6_3. Francis, R.L., 1967. On some problems of rectangular warehouse design and layout. J. Ind. Eng. 18, 595–604. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Machine Learning, 1st ed.. Addison-Wesley Longman Publishing Co., Inc, Boston, MA, USA. Gue, K.R., Ivanovi, G., Meller, R.D., 2012. A unit-load warehouse with multiple pickup and deposit points and non-traditional aisles. Transp. Res. Part E: Logist. Transp. Rev. 48, 795–806 〈http://dx.doi.org/10.1016/j.tre.2012.01.002http://dx. doi.org/10.1016/j.tre.2012.01.002〉. Gue, K.R., Meller, R.D., 2009. Aisle configurations for unit-load warehouses. IIE Trans. 41, 171–182. Hsieh, L., Tsai, L., 2006. The optimum design of a warehouse system on order picking efficiency. Int. J. Adv. Manuf. Technol. 28, 626–637.

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Kazarlis, S., Bakirtzis, A., Petridis, V., 1996. A genetic algorithm solution to the unit commitment problem. IEEE Trans. Power Syst. 11, 83–92. Meller, R., Gue, K., 2009. The application of new aisle designs for unit-load warehouses. In: Proceedings of the 2009 NSF CMMI Engineering Research and Innovation Conference, vol. 1, Honolulu, Hawaii, pp. 1–8. Meller, R., Gue, K., 2010. Fishbones, herringbones and chevrons: new ways to design a warehouse. In: 2010 Informs Conference on O.R. Practice, vol. 1, Orlando, FL. Onut, S., Tuzkaya, U.R., Doa, B., 2008. A particle swarm optimization algorithm for the multiple-level warehouse layout design problem. Comput. Ind. Eng. 54, 783–799 〈http://dx.doi.org/10.1016/j.cie.2007.10.012〉. Öztürkoğlu, O., Gue, K.R., Meller, R.D., 2012. Optimal unit-load warehouse designs for single-command operations. IIE Trans. 44, 459–475. http://dx.doi.org/ 10.1080/0740817X.2011.636793. Parikh, P.J., Meller, R.D., 2010. A travel-time model for a person-onboard order picking system. Eur. J. Oper. Res. 200, 385–394 〈http://dx.doi.org/10.1016/j.ejor. 2008.12.031〉. Park, Y.H., Webster, D.B., 1989. Modelling of three-dimensional warehouse systems. Int. J. Prod. Res. 27, 985–1003. http://dx.doi.org/10.1080/00207548908942603. Pohl, L.M., Meller, R.D., Gue, K.R., 2009. Optimizing fishbone aisles for dualcommand operations in a warehouse. Naval Res. Logist. (NRL) 56, 389–403. http://dx.doi.org/10.1002/nav.20355. Pohl, L.M., Meller, R.D., Gue, K.R., 2011. Turnover-based storage in non-traditional unit-load warehouse designs. IIE Trans. 43, 703–720. Roodbergen, K.J., Sharp, G.P., Vis, I.F.A., 2008. Designing the layout structure of manual order picking areas in warehouses. IIE Trans. 40, 1032–1045. Roodbergen, K.J., Vis, I.F., 2006. A model for warehouse layout. IIE Trans. 38, 799–811. Rouwenhorst, B., Reuter, B., Stockrahm, V., Van Houtum, G., Mantel, R., Zijm, W., 2000. Warehouse design and control: framework and literature review. Eur. J. Oper. Res. 122, 515–533. Srinivas, M., Patnaik, L., 1994. Genetic algorithms: a survey. Computer 27, 17–26. White, J.A., 1972. Optimum design of warehouses having radial aisles1. AIIE Trans. 4, 333–336. http://dx.doi.org/10.1080/05695557208974871. Zhang, G.Q., Xue, J., Lai, K.K., 2002. A class of genetic algorithms for multiple-level warehouse layout problems. Int. J. Prod. Res. 40, 731–744.

Please cite this article as: Cardona, L.F., et al., Detailed design of fishbone warehouse layouts with vertical travel. International Journal of Production Economics (2015), http://dx.doi.org/10.1016/j.ijpe.2015.03.006i