Detailed layout planning for irregularly-shaped machines with transportation path design

European Journal of Operational Research 177 (2007) 693–718 www.elsevier.com/locate/ejor

Discrete Optimization

Detailed layout planning for irregularly-shaped machines with transportation path design Stefan Bock

a,*

, Kai Hoberg

b

a

b

International Graduate School Dynamic Intelligent Systems, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany Department of Supply Chain Management and Management Science, University of Cologne, Albertus-Magnus-Platz, 50923 Cologne, Germany Received 22 April 2004; accepted 4 November 2005 Available online 10 February 2006

Abstract In order to obtain a competitive level of productivity in a manufacturing system, efficient machine or department arrangements and appropriate transportation path structures are of considerable importance. By defining a production system’s basic structure and material flows, the layout determines its operational performance over the long term. However, most approaches proposed in the literature provide only a block layout, which neglects important operational details. By contrast, in this paper, we introduce approaches to planning layouts at a more detailed level. Hence, this present paper introduces an integrated approach which allows a more detailed layout planning by simultaneously determining machine arrangement and transportation paths. Facilities to be arranged as well as the entire layout may have irregular shapes and sizes. By assigning specific attributes to certain layout subareas, application-dependent barriers within the layout, like existing walls or columns, can be incorporated. We introduce a new mathematical layout model and develop several improvement procedures. An analysis of the computational experiments shows that more elaborate heuristics using variable neighborhoods can generate promising layout configurations.  2006 Elsevier B.V. All rights reserved. Keywords: Facilities planning and design; Combinatorial optimization; Layout planning; Heuristics; Shortest path

*

Corresponding author. Tel.: +49 5251 603363; fax: +49 5251 603511. E-mail address: [email protected] (S. Bock).

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.11.011

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1. Introduction After a dormant period, due to increased global competition and improved information technology, developing decision support systems has again become a vital research area. In particular, layout planning has recently gained renewed interest (e.g., [17,22] or [39]). The renewed awareness of the importance of productivity improvements in companies implies new challenges for layout planning approaches. Because the facility layout determines the basic structure of the production system, it has a considerable impact on the attainable efficiency. However, most approaches for layout planning only determine a block layout. This rough planning neglects many important operational details responsible for the overall productivity of the entire production system. Important details, e.g., aisle structure or transportation flow handling, are strongly dependent on the machine arrangement. Consequently, a separate determination of these details neglects many existing interdependencies and frequently results in inefficient layouts. Hence, this present paper introduces an integrated approach which allows a more detailed layout planning by simultaneously determining machine arrangement and transportation paths. By using a grid-based layout structure, the approach supports a detailed mapping of irregular, but fixed machine shapes. In addition, applicationdependent requirements and attributes can be defined for specific subareas of the underlying layout, which may also have an irregular shape. Furthermore, machines can be allocated within four different orientations. The paper is organized as follows. In Section 2, we provide a brief overview of the relevant literature for layout planning. In Section 3, we introduce the model formulation used throughout the paper. We first describe the main attributes of the model before presenting its full mathematical definition. In Section 4, we introduce several heuristics since a successful application of the proposed approach requires efficient solution procedures. To validate and compare the performance and practical applicability of these heuristics, we present numerical results for randomly generated problems. In Section 5, we conclude with a summary of findings and suggestions for future research.

2. Literature review Layout planning has been a vital research area for many decades [26,22], because the facility layout significantly determines the attainable performance of a manufacturing system [3,16]. Therefore, the literature on facility layout is voluminous [23, p. 100]. Most of the concepts proposed are either algorithmic or procedural [41, p. 128]. While procedural approaches are used to incorporate quantitative as well as qualitative attributes and objectives, algorithmic approaches simplify applied constraints and objective functions in order to define the whole planning process as a combinatorial optimization model. A procedural approach is typically designed as a component approach generating a variety of possible layout alternatives. Its practical application demands substantial experience from the designer, as it incorporates several subjective decisions. Among the procedural approaches, the methods of Apple [3], Reed [32] and Muther [30] in particular, can be highlighted [36, pp. 301–309]. The applicability of Muther’s systematic layout planning procedure to generating a layout in the semiconductor industry was recently shown [42]. An important aspect of procedural approaches addresses the construction of alternative layouts [36, p. 368]. Consequently, evaluation procedures have been formulated to provide an appropriate comparison. For instance, Lin and Sharp propose a generic approach for developing quantitative indices [23]. Besides this, Yang and Kuo [41] propose a hierarchical analysis determining the relative importance of different decision criteria completed by a DEA-based multiple objective decision process. Graph-theory methods are frequently used in procedural approaches to generate alternative layouts. Therefore, the creation process is divided into the adjacency and the block layout problem [37, p. 449]. The adjacency prob-

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lem comprises the definition of a planar graph maximizing the total weighted sum of adjacency benefits deriving from all pairs of machines or departments represented by the respective arc-weights. These arc-weights reflect the benefit from locating pairs of machines close to each other. In the subsequent block layout problem, this planar graph must be transformed into a drawn block plan which additionally incorporates the space requirements of the elements to be arranged. Applicable solution procedures for the adjacency problem can be found, for instance, in [37], while [15] and [38] address the subsequent block layout problem. However, due to their component structure and their neglect of dependencies between the different decision levels, procedural approaches frequently yield only a poor solution for purely quantitatively defined applications. Consequently, most of the proposed approaches in the area of layout planning are designed as algorithmic procedures [16]. Among them, a considerable number are formulated as versions of the wellknown Quadratic Assignment Problem (QAP). Here, layout planning is reduced to the allocation of N uniform elements to M (M P N) positions [6] or [11]. Despite its very compact and simple definition, this model already belongs to the class of NP-complete problems [33]. Thus, the QAP has become a challenging benchmark test for evaluating the efficiency of meta heuristics [35,8,10,34,13]. However, due to its restrictive assumptions, the application of the QAP is limited to only a few specific scenarios. Consequently, several extensions of this model can be found in the literature. These more elaborate approaches, dealing with non-uniform elements to be arranged, can be classified roughly into two different categories, depending on the structure of the considered layout plan [22, p. 880], [21, p. 111]. According to this criterion, a distinction is made between grid-based block plan layout problems and continuous block plan layout problems. In a grid-based problem, the layout is divided into squares, each with a unit area. In contrast, continuous approaches define a layout based on a continuous plane. Considerable contributions have been made in [2,7,25,20,17,1] to the former group of grid-based layouts. To enable an efficient implementation of sophisticated exchange procedures for irregular shaped elements, space-filling curves are used frequently. These curves define a continuous sequence through all neighbored squares in the underlying layout. Defining the elements of each department as a continuous sub-sequence, space-filling curves ensure that a department is never split. However, the itemized approaches do not map specific application-dependent attributes or requirements of subparts of the considered layout. In addition, department shapes are rearranged throughout the exchanges. Thus, fixed machine-shapes cannot be incorporated. Block plans applying a grid-based layout are frequently of irregular shape. Thus, major modifications are necessary before appropriate transportation systems can be integrated. Consequently, Lee and Kim [21] have proposed several procedures for transforming an irregular grid-based block layout into a regular one, while preserving its solution quality. Sophisticated approaches assuming a continuous layout can be found, among others, in [18,28,19,29,40,22,27]. Here, the departments to be arranged are frequently mapped as rectangularly shaped elements. Depending on the particular approach, aspect ratios of the machines are either fixed or variable within predefined intervals. In the approach of Kim and Kim [19], the machines to be arranged can also be rotated in 90 angles. Furthermore, each machine has predefined input and output-points where all flows must end or start. Recent approaches often deal with specific layout features as multiple-floor buildings or dynamic planning issues. In multiple-floor facility layout problems, the approaches model the layout arrangements of several floors simultaneously so as to integrate the respective interdependencies more accurately [22,1,7]. By contrast, dynamic approaches focus on the anticipation of future changes in the production programs to be implemented by the layout. Thus, layout plans are defined as time-dependent so as to map temporary expansions/declines or reallocations of departments [28,5,4,20,29,40]. While most dynamic approaches address arbitrary phases of the system’s life cycle, Montreuil and Venkatadri [28] focus on the expansion phase and its specific requirements.

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Independently of the assumed layout structure (grid-based or continuous), almost all approaches mentioned above generate only a block plan. These plans neglect important operational details like aisle structures and the resulting transportation flows [21, p. 111], [12, p. 145]. Therefore, extended concepts for a more detailed layout planning are proposed by Yang et al. [39] and Chhajed et al. [12]. Here, component approaches are applied, which generate the final layout in multiple iterations. To this end, an initial block layout generated in a first step is subsequently extended by a system of transportation paths. However, by separating the arrangement of machines or departments from the definition of the corresponding transportation paths system, important dependencies are neglected. Hence, this present paper proposes an integrated approach which addresses both decision problems simultaneously. In order to provide more detailed decision support for layout planning, the grid-based layout definition in this paper additionally allows the integration of application-dependent attributes and requirements of irregular layoutsubareas.

3. The new layout planning model In this Section, we introduce the new layout planning model. Before providing the full mathematical definition, we itemize its main attributes. 3.1. Main attributes of the model In order to provide a decision support system for detailed layout planning, the model defines the underlying layout as a grid of uniform squares, denoted in this paper as unit-elements with a predefined side length of slue. Thus, the degree of accuracy can be adapted according to the requirements of the application. Consequently, as illustrated in Fig. 1, every machine and transportation path is mapped as a set of adjacent unit-elements. The shapes of the machines to be arranged are allowed to be irregular, but must be fixed. Hence, independently of the locations, the predefined machine shapes cannot be changed. In addition, specific requirements and attributes can be defined for each unit-element in the layout. By so doing, it is possible to exclude certain subareas of the underlying layout as not usable for transportation flows or for the allocation of machines. Those elements are denoted as ‘‘restricted’’. In addition, machine-dependent requirements for the allocated subarea of the layout can be incorporated into the model. In a similar manner to the model of Kim and Kim [19], every machine can be arranged in a total of four different orientations of 0, 90, 180 and 270. In addition to the machine arrangement, the definition of the layout comprises a detailed determination of the material flow system. Here, for each pair of machines, a path of adjacent unit-elements connecting the respective input- and output-points is computed. It thus becomes

Inputelement

4 E8,1,2 =1

dx 0 1 2

Length 6,11 = 8 Output-element

Machine 4

0 1 dy 2 3 4

1 3 5 6 7 8 9 10 11 2

1 2 3 4 5 6 7

4

Element 8 Machine 6

Output-element

Inputelement

Machine 11

Fig. 1. Definition of irregular machine shapes (left) and a generated transportation path (right).

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possible to integrate already existing transportation systems. Note that such a predefined structure, which frequently exists in industrial applications, can result from already existing systems, practical experience or assumptions about the existing material flows. The following are the characterizing attributes of the new model: • By determining the size of the unit-elements, the degree of accuracy can be adapted according to the requirements of the application. • The machines to be arranged have an irregular, but fixed shape and predefined input- and output-points. • Each machine can be located in one of four different orientations of 0, 90, 180 and 270. • The layout definition entails a detailed determination of the utilized transportation paths. • Specific requirements of the layout locations can be defined according to the requirements of the machines to be allocated. • Relocation costs of already positioned machines can be incorporated into the model. • Specific attributes of layout subareas can be defined. 3.2. Mathematical formulation of the new model Below, a mathematical definition of the new layout model providing a combined machine-arrangement and material flow determination is presented. In order to make it more readily comprehensible, this definition starts with a separate definition of the given parameters, followed by the introduction of the variables to be determined, after which the restrictions and objective function are given. 3.2.1. Parameters The following parameters are used in the mathematical model: • N—Number of machines or machines/departments to be located or reallocated in the layout [–] • X—Number of unit-elements within a horizontal plane [–] X defines the number of existing unit-elements in the horizontal direction, numbered from 1 to X. • Y—Number of unit-elements within a vertical plane [–] Y defines the number of existing unit-elements in the vertical direction, numbered from 1 to Y. To provide a uniform physical distance between the centers of all adjacent unit-elements in the layout, a rectangular neighborhood definition is used. Here, the unit-element (x, y) 2 {1, . . . , X} · {1, . . . , Y} is adjacent to (x 0 , y 0 ) 2 {1, . . . , X} · {1, . . . , Y} if and only if one of the following restrictions is fulfilled: (i) x = x 0  1 and y = y 0 , (ii) x = x 0 + 1 and y = y 0 , (iii) x = x 0 and y = y 0  1, or (iv) x = x 0 and y = y 0 + 1. In addition, a set E of unit-elements is denoted as contiguous, if and only if for each pair e1, e2 2 E there is a sequence of t (P2) unit-elements (e1, e2, . . . , et) 2 Et fulfilling the following restrictions: (i) "i 2 {1, . . . , t  1}: ei and ei+1 are adjacent, (ii) e1 = e1, and (iii) et = e2. • C—Number of conditions [–] C is the number of predefined application-dependent conditions fulfilled by the defined coordinates in the layout in question or required by the machines to be allocated. These conditions are numbered in a predefined sequence from 1 to C. • Fn (1 6 n 6 N)—Number of unit-elements belonging to the nth machine [–] • Restx,y (1 6 x 6 X; 1 6 y 6 Y)—Binary constant defining whether the unit-element with the coordinates (x, y) is a restricted element [–] A restricted element cannot be covered by a machine or transportation-path. Restricted elements can be interpreted, for instance, as walls, stairs or other barriers. Note, that the introduction of restricted

698

•

• •

•

•

•

•

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elements allows an exact mapping of layouts not being rectangular. In this case X and Y are determined by the side lengths of the minimum rectangular cover. This may cause some unrestricted elements to become restricted. Perm_Roadx,y (1 6 x 6 X; 1 6 y 6 Y)—Binary constant defining whether the unit-element with the coordinates (x, y) is a permanent transport element [–] In many applications, there is a predefined, permanent, contiguous transportation system which defines a basic aisle structure. The positions of these permanent transport elements are fixed by the planner using the set of binary constants Perm_Road = {Perm_Roadx,yj1 6 x 6 X; 1 6 y 6 Y}. Thus, these unit-elements cannot be used by machines. Cond cx;y ð1 6 c 6 C; 1 6 x 6 X ; 1 6 y 6 Y Þ—Binary constant indicating whether the unit-element with the coordinates (x, y) provides the cth condition [–] EC nc ð1 6 n 6 N ; 1 6 c 6 CÞ—Binary constant indicating whether the cth layout-condition is necessary for locating machine n [–] The binary parameter EC nc indicates whether the nth machine requires the cth condition to be fulfilled by the chosen layout location. Note that this condition must be provided by all covered unit-elements. For instance, a machine can require an appropriate and sufficiently strong foundation, a minimum ceiling height or a nearby gas supply. Enf ;dx;dy ð1 6 n 6 N ; 1 6 f 6 F n ; X 6 dx 6 X ; Y 6 dy 6 Y Þ—Binary parameters defining the shape of the nth machine [–] As mentioned above, machines have a predefined irregular structure that is mapped as a set of contiguous unit-elements. Accordingly, the shape of the nth machine is defined by a set of binary parameters. As illustrated in Fig. 1, the origin of each machine is given by its input-element which is denoted as the first element. All machine-elements are defined by their relative coordinates (dx, dy) to this origin in the zero-degree orientation. Consequently, the input-element has the relative coordinates dx = dy = 0. Apart from the second element which is kept for the output-point, all other machine-elements are given in an arbitrary sequence. Thus, for the fth element of the nth machine, Enf ;dx;dy indicates whether the this element has the relative coordinates (dx, dy). Hence, for the input element the following holds: 8n . . . ; N g : En1;0;0 ¼ 1. In addition, for all elements: 8n 2 f1; . . . ; N g : 8f 2 f1; . . . ; F n g : PX2 f1;P Y n dx¼X dy¼Y E f ;dx;dy ¼ 1. Note that this shape definition must ensure that for each machine, every relative position is only covered by a single element at most. In addition, it is assumed that each machineshape is a contiguous area. Startnx;y;r ð1 6 n 6 N ; 1 6 x 6 X ; 1 6 y 6 Y ; 0 6 r 6 3Þ—Binary constant indicating the initial position of machine n [–] In the model, a distinction is made between already located and non-located machines. Therefore, if the nth machine is already located, the constants Startnx;y;r define the initial position of its input-point and the current orientation. Specifically, Startnx;y;r indicates whether the input-element of the nth machine is located at the coordinates (x, y) while the machine is arranged in the rth orientation. Thus, PX PY P3 PX n However, if the nth machine is not already located: x¼1 y¼1 r¼0 Start x;y;r ¼ 1. x¼1 PY P3 n Start ¼ 0. x;y;r y¼1 r¼0 Fixedn (1 6 n 6 N)—Binary constant indicating whether the position of machine n is fixed [–] Fixedn indicates whether the already-located machine n can be reallocated in the layout. If Fixedn = 1, this is not allowed. However, if the nth machine does not belong to the already-located machines, Fixedn = 0. MoveCostn (1 6 n 6 N)—Costs for reallocating machine n [currency-units] The parameter MoveCostn defines the costs of moving the nth machine out of its start position. If the nth machine is not already located in the layout, there are no reallocation costs and therefore the parameter is set to zero.

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• FlowCostn,m (1 6 n 6 N; 1 6 m 6 N)—Costs for the transportation of one quantity-unit over a distance of one unit-element from machine n to machine m [currency-units/(quantity-units Æ unit-elements)] The parameter FlowCostsn,m defines the cost rate for transporting one quantity-unit over one unit-element from the nth to the mth machine. Note that these parameters allow machine-pair-dependent cost-rates. • Flown,m (1 6 n 6 N; 1 6 m 6 N)—Estimated total flow to be transported from machine n to machine m throughout the planning horizon [quantity-units] • Pathmax–Maximum length of a transportation path between two machines [distance-units] The parameter Pathmax is introduced to simplify the subsequent definitions. It gives an upper bound for the length of each defined transportation path. Note that this bound is introduced for technical reasons only. Therefore, Pathmax should be given a sufficiently large value so that there are no restrictions. • Min_Dist_Eun,m (1 6 n 6 N; 1 6 m 6 N; n 5 m)—Minimum Euclidian distance between machine n and m [distance-units] Min_Dist_Eun,m defines a minimum Euclidean distance restriction between the machines n and m. This may be necessary, if, for instance, one machine vibrates when operating, whereas the other one needs a vibration-free environment. Note that the minimum distance restrictions must be fulfilled by each arbitrary chosen pair of elements belonging to the respective machines. • Min_Pathn,m (1 6 n 6 N; 1 6 m 6 N; n 5 m)—Minimum length of transportation path from machine n to machine m [distance-units] In analogy to the parameter Min_Dist_Eun,m, Min_Pathn,m defines a minimum length for the defined transportation path connecting the output-point of the nth machine with the input-element of the mth one. Such a restriction may become relevant if, for instance, the transported materials need to cool down during the transportation process. • Max_Pathn,m (1 6 n 6 N; 1 6 m 6 N; n 5 m)—Maximum length of transportation path from machine n to machine m [distance-units] 3.2.2. Variables A solution of the model entails defining the following variables: • Posnx;y;r ð1 6 n 6 N ; 1 6 x 6 X ; 1 6 y 6 Y ; 0 6 r 6 3Þ—Binary variable indicating whether machine n is assigned to the coordinates (x, y) in rotation r [–] By defining these variables, the location and rotation of every machine in the layout is determined. For this, Posnx;y;r is one, if and only if the input-element of the nth machine has the coordinates (x, y) and the nth machine is located in the rth orientation. Note again that apart from the basic orientation of 0, every machine can be rotated in 90 steps. This results in the four possible orientations of 0 (orientation 0), 90 (orientation 1), 180 (orientation 2) and 270 (orientation 3) as shown in Fig. 2. • Pathn;m x;y;w ð1 6 n 6 N ; 1 6 m 6 N ðn 6¼ mÞ; 1 6 x 6 X ; 1 6 y 6 Y ; 1 6 w 6 Pathmax Þ—Binary variable indicating whether the unit-element (x, y) is the wth field of the transportation-path from machine n to machine m [–] In addition to the arrangement of the machines, a solution of the model has to define each transportation-path. This is achieved by the set of variables Pathn;m ¼ fPathn;m x;y;w j1 6 x 6 X ; 1 6 y 6 Y ; 1 6 w 6 Pathmax g for each pair (n, m) 2 {1, . . . , N} · {1, . . . , N} (with n 5 m) of different machines. Therefore, Pathn;m x;y;w is 1, if the unit-element (x, y) is the wth element of the chosen transportation-path starting at the output-element of the nth machine and ending at the input-element of the mth machine. By reducing path definitions to a single plane, the model does not allow simultaneous usage, as, for instance, would be possible by combining floor with overhead transport. Note that each transportation path must allow for the specific requirements of the conveyance executing the respective transports. Consequently, by defining them on the basis of the unit-elements, an application-

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Fig. 2. Deriving the value of the variable Locnf ;x;y while rotating the nth machine.

dependent basic transportation factor may be introduced. This factor determines the minimum width of a path measured in unit-elements. For convenience, in the following this parameter is assumed to be 1. 3.3. Auxiliary operations and derived parameters Before formulating various restrictions, several parameters are introduced in order to allow a simplified definition of the model. • Greater(x, y) ðx; y 2 NÞ—Binary parameter indicating whether x is greater than y [–] This can be computed by 8x; y 2 N : Greaterðx; yÞ ¼

maxfx; yg  y . maxf1; x  yg

• Ident(x, y) ðx; y 2 NÞ—Binary parameter indicating whether x is identical with y [–] This can be computed by 8x; y 2 N : Identðx; yÞ ¼ ð1  Greaterðx; yÞÞ  ð1  Greaterðy; xÞÞ. • Movedn (1 6 n 6 N)—Derived binary variable indicating whether machine n is relocated [–] The derived parameter movedn equals 1, if the nth machine was already located before, but is allocated a modified position or orientation in the generated layout. Thus, Movedn can be computed by 8n 2 f1; . . . ; N g : Moved n ¼

X X Y X 3 X ðStartnx;y;r  ð1  Posnx;y;r ÞÞ. x¼1 y¼1

r¼0

• Lengthn,m (1 6 n 6 N; 1 6 m 6 N (n 5 m))—Length of the transportation path from machine n to machine m [unit-elements] The identifier Lengthn,m gives the number of unit-elements belonging to the transportation path leading from the output-element of the nth machine to the input-element of machine m. This can be computed as follows:

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8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : Lengthn;m ¼ 1 þ

X X Y Path max X X x¼1 y¼1

701

Pathn;m x;y;k .

k¼1

As can be seen from the above calculation, the length of each path also contains the input point of the destination machine. Note further that for each k 2 {1, . . . , Pathmax} and each pair of machines (n, m) there is at most one unit-element (x, y) for which Pathn;m x;y;k equals 1. • Temp_Roadx,y (1 6 x 6 X; 1 6 y 6 Y)—Derived binary variable indicating whether the unit-element with the coordinates (x, y) is a temporary transport field [–] If a non-permanent transport field is used in any path between two machines, it is defined as a temporary transport field. Thus, this field cannot be used for the assignment of machines. The respective parameter Temp_Roadx,y can be derived by 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : Temp Road x;y ! N X N Path max X X n;m ¼ Greater Pathx;y;k ; 0  ð1  Perm Road x;y Þ. n¼1 m¼1

k¼1

• Roadx,y (1 6 x 6 X; 1 6 y 6 Y)—Derived binary variable indicating whether the unit-element with the coordinates (x, y) is a transport field [–] A unit-element (x, y) in the layout is defined as a transport field, if it is either a permanent or a temporary transport element. Therefore 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : Road x;y ¼ Perm Road x;y þ Temp Road x;y . • Locnf ;x;y ð1 6 n 6 N ; 1 6 f 6 F n ; 1 6 x 6 X ; 1 6 y 6 Y Þ—Derived binary variable indicating whether the fth element of machine n is assigned to the unit-element with the coordinates (x, y) [–] Through this set of derived binary parameters, the position of each machine-element in the new layout is derived. This information can be determined by the following computation: Locnf ;x;y ¼

x1 X

y1 X 3 X

Identðr; 0Þ  Posnxdx;ydy;r  Enf ;dx;dy þ

dx¼xX dy¼yY r¼0

 Enf ;dx;dy þ

X x X

y1 X

X x X 3 X

Identðr; 1Þ  Posnxþdy;ydx;r

dx¼yY dy¼1x r¼0 Y y X

3 X

Identðr; 2Þ  Posnxþdx;yþdy;r  Enf ;dx;dy þ

dx¼1x dy¼1y r¼0

Y y X

x1 3 X X

Identðr; 3Þ

dx¼1y dy¼xX r¼0

 Posnxdy;yþdx;r  Enf ;dx;dy . To illustrate the computation itemized above, Fig. 2 shows the resulting position of a unit-element with the relative coordinates (dx = 2, dy = 5) for the four possible orientations of the respective machine. • Innx;y and Outnx;y ð1 6 n 6 N ; 1 6 x 6 X ; 1 6 y 6 Y Þ—Derived binary variables indicating whether the unitelement (x, y) is the input or the output point of the nth machine [–] These additional parameters can be calculated by Input: 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : Innx;y ¼ Locn1;x;y and Output: 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : Outnx;y ¼ Locn2;x;y . Note that it is possible to define combined input- and output-elements. In this specific situation: 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : Innx;y ¼ Outnx;y . • ELnx;y ð1 6 n 6 N ; 1 6 x 6 X ; 1 6 y 6 Y Þ—Derived binary variable indicating whether the unit-element (x, y) belongs to the nth machine [–] The binary variable ELnx;y equals one, if and only if the unit-element with the coordinates (x, y) is covered by an element of machine n. This parameter can be calculated by

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8n 2 f1; . . . ; N g : 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : ELnx;y ¼ Greater

Fn X

! Locnf ;x;y ; 0 .

f ¼1

• Dist Eux;y;x0 ;y 0 ð1 6 x 6 X ; 1 6 y 6 Y ; 1 6 x0 6 X ; 1 6 y 0 6 Y Þ—Euclidian distance between two unit-elements [distance-units] The operation Dist Eux;y;x0 ;y 0 calculates the Euclidian distance between the coordinates (x, y) and (x 0 , y 0 ). Therefore Dist Eux;y;x0 ;y 0 can be substituted by the following computation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8x; x0 2 f1; . . . ; X g : 8y; y 0 2 f1; . . . ; Y g : Dist Eux;y;x0;y 0 ¼ slue  ðx  x0 Þ2 þ ðy  y 0 Þ2 . 3.4. Restrictions A feasible solution of the considered layout problem fulfills the following restrictions: 1. Each machine has a clearly defined position and orientation in the layout 8n 2 f1; . . . ; N g :

X X Y X 3 X x¼1 y¼1

Posnx;y;r ¼ 1.

r¼0

2. Every element of each machine has a clearly defined location in the layout 8n 2 f1; . . . ; N g : 8f 2 f1; . . . ; F n g :

X X Y X

Locnf ;x;y ¼ 1.

x¼1 y¼1

3. Each unit-element of the layout can be covered only by a single machine 8n; m 2 f1; . . . ; N g with n 6¼ m :

X X Y X

ELnx;y  ELmx;y ¼ 0.

x¼1 y¼1

4. No element of a machine is placed on a restricted or transport element X X Y X N X

ELnx;y  ðRoad x;y þ Restx;y Þ ¼ 0.

x¼1 y¼1 n¼1

5. All elements of the layout covered by a machine fulfill its respective predefined conditions 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : 8n 2 f1; . . . ; N g : 8c 2 f1; . . . ; Cg : EC nc  ELnx;y ¼ EC nc  ELnx;y  Cond cx;y . 6. Unambiguity of each defined transportation path 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8l 2 f1; . . . ; Pathmax g :

X X Y X

Pathn;m x;y;l 6 1;

x¼1 y¼1

8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8l 2 f2; . . . ; Pathmax g :

X X Y X x¼1 y¼1

7. No restricted element is covered by a transport element X X Y X x¼1 y¼1

Road x;y  Restx;y ¼ 0.

Pathn;m x;y;l1 P

X X Y X x¼1 y¼1

Pathn;m x;y;l .

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8. Transportation paths are defined only for temporary or permanent transport elements 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ :

Path max X

n;m Pathx;y;l 6 Road x;y .

l¼1

9. Start, end and connectivity of each transportation path Note that in the following computations, non-existent parameters must be deleted. A non-existent parameter occurs if the index-value x or y does not belong to the interval [1, X] or [1, Y], respectively. The path commences with the output-element of machine n: 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : n;m n;m n;m m m m m Outnx;y  ðPathn;m x1;y;1 þ Pathx;y1;1 þ Pathxþ1;y;1 þ Pathx;yþ1;1 þ Inx1;y þ Inx;y1 þ Inxþ1;y þ Inx;yþ1 Þ n ¼ Outx;y . Coherence of intermediate path steps: 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : Identð1; Lengthn;m Þ  Inmx;y  ðOutnx1;y þ Outnx;y þ Outnxþ1;y þ Outnx;yþ1 Þ ¼ Identð1; Lengthn;m Þ  Inmx;y ; and 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8l 2 f2; . . . ; Pathmax g : 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : n;m n;m n;m n;m ð1  Greaterðl; Lengthn;m  1ÞÞ  Pathn;m x;y;l1  ðPathx1;y;l þ Pathx;y1;l þ Pathxþ1;y;l þ Pathx;yþ1;l Þ n;m ¼ ð1  Greaterðl; Lengthn;m  1ÞÞ  Pathx;y;l1 . The path concludes with the input-element of machine m: 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8l 2 f1; . . . ; Pathmax g : 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : n;m n;m n;m Identðl; Lengthn;m  1Þ  Inmx;y  ðPathn;m x1;y;l þ Pathx;y1;l þ Pathxþ1;y;l þ Pathx;yþ1;l Þ ¼ Identðl; Lengthn;m  1Þ  Inmx;y . 10. Euclidean distance and path length restrictions 8n; m 2 f1; . . . ; N g ðn 6¼ mÞ : 8x 2 f1; . . . ; X g : 8y 2 f1; . . . ; Y g : 8x0 2 f1; . . . ; X g : 8y 0 2 f1; . . . ; Y g : ELnx;y  ELmx0 ;y 0  GreaterðMin Dist Eun;m ; Dist Eux;y;x0 ;y 0 Þ ¼ 0; 8n; m 2 f1; . . . ; N gðn 6¼ mÞ : GreaterðMin Pathn;m ; Lengthn;m  slue Þ ¼ 0 and 8n; m 2 f1; . . . ; N gðn 6¼ mÞ : GreaterðLengthn;m  slue ; Max Pathn;m Þ ¼ 0. 11. No machine with a fixed position can be moved from its start position 8n 2 f1; . . . ; N g : Fixed n  Moved n ¼ 0. 3.5. Objective function of the model Each feasible layout L is evaluated by the following objective function which computes the sum of the resulting transportation and relocation costs: CostsðLÞ ¼

N N X X n¼1 m¼1;m6¼n

FlowCostn;m  Flown;m  Lengthn;m þ

N X n¼1

Moved n  MoveCostn .

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4. Solution approaches In this Section, we introduce several approaches for layout construction and improvement. Owing to the complexity of the model, we introduce heuristics. Therefore, the following Section 4.1 introduces some basic procedures used as subroutines in all applied heuristics. These include a method for efficient path computation, a fast iterative feasibility check and an accelerated computation of the objective value. In Section 4.2, we introduce a construction approach for generating an initial solution, after which various improvement approaches are described. Section 4.3 therefore introduces heuristics which rearrange an existing layout by the application of a single operation. Taking those operations as basis, Section 4.4 illustrates two extended procedures combining different methods in a multi-state procedure. By combining possible advantages of the different operations in a variable neighborhood, further improvements should be possible. In the final Section 4.5, numerical results from several computational tests are analyzed. 4.1. Procedures Because the layout under consideration has to be modified frequently throughout the improvement process, efficient procedures for layout modification and objective value determination are important. Consequently, specific procedures for path computation, feasibility testing and objective function value determination are introduced below. 4.1.1. Path computation The definition of the used transportation paths constitutes an important feature of the approach introduced here. This allows specific application-dependent circumstances and restrictions in the considered layout to be incorporated. This is not possible in known approaches using predefined metrics like the l1-, l2- or the lp-distance measures [24, pp. 5–7 and pp. 255–271]. Therefore, apart from the determination of the machine arrangement, a layout definition additionally comprises the transportation paths. This has to be done for each pair (n, m) 2 {1, . . . , N} · {1, . . . , N} (n 5 m) of machines. In many applications, it is useful or prescribed to allocate a certain set of unit-elements in the layout as permanent transport elements, inde-

Temporary transport element M2

Permanent transport element

Relevant transport element M3 M1

3 2 3 1 2 3 3

Fig. 3. Generating a connection to the permanent transportation path system.

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pendent of the position of the machines to be arranged. Consequently, we distinguish between two different sets of transport elements in the layout: permanent transport elements that are predefined and temporary transport elements which are generated by the paths determination (see Fig. 3). By using the set of permanent transport elements as the basic transportation system, the path generation procedure computes the shortest possible connection between the input- and output-element of the considered machines and this predefined path-system. Therefore, an appropriate predefined system simplifies the integration of additional transportation paths into the aisle system which is developed. The path computation is conducted in the form of a breadth-first search, commencing, for each material flow, with the respective output- and input-element. The search is always continued until a connection with the permanent transportation path system or the respective destination or source itself is established or no longer possible. The search process is executed in two separate phases: the search-phase and the generation-phase. In the search-phase, the search is conducted initially by an iterative numbering of the unit-elements in the layout. Therefore, in the tth iteration, all unit-elements in the layout that can be reached within t movements are assigned the label t, while the entire process has been started at the respective input- or output-element. As mentioned before, the four-element or rectangular neighborhood is used. Hence, in the tth iteration of this search process, for each t  1-numbered unit-element, the (at most) four neighboring unit-elements are labeled with t. To incorporate existing restrictions, numbering can be executed only if the potential neighbor is neither a restricted element nor occupied by another machine. The search-phase is stopped if either a permanent transport element or the destination itself is numbered or if there is no further element to be numbered. In the latter case, no transportation path between the input- and the output-element of the respective machines can be generated. It is evident that this layout is not feasible and is therefore omitted. Fig. 3 shows how the search-phase is carried out for the output-element of machine 1. By contrast, if the search-phase ends successfully, the generating-phase for establishing the transportation path can be started. Here, the search-process is executed in the opposite direction back to the origin. It therefore follows the stored numbering in descending order. Finally, the generated path is stored as a dynamic list in addition to its calculated length. Due to the use of the predefined transportation system, this computation does not guarantee the shortest path determination. To simplify the following definitions, "n, m 2 {1, . . . , M} Path ElemOut n!m is introduced as the set of unit-elements belonging to the computed path starting at the output-element of the nth machine and terminating at a permanent transport element or at the input-element of the mth machine. Additionally, 8n; m 2 f1; . . . ; Mg : Path ElemIn n!m is introduced as the set of unit-elements belonging to the computed path starting at the input-element of the mth machine and ending at a permanent transport element or at the output-element of the nth machine. Note that neither the In source nor the destination belong to the paths defined by PathOut n!m and Pathn!m . Therefore, the length of the Out Out path Pathn!m is equal to jPathn!m j þ 1 while the length of the path PathIn n!m can be calculated by jPathIn j þ 1. Note that the procedure described so far assumes the basic transportation factor to be 1. n!m However, by combining unit-elements throughout the computation process this procedure can be extended with minor modifications for application scenarios requiring larger factors. Since, in most cases, the computed transportation paths include major parts of the predefined path-system (i.e. the permanent transport elements), it is useful to develop an efficient procedure for calculating the lengths of these sub-paths. In the case of component transportation paths, the length of these operational sub-paths are unknown. To prepare an efficient length generation, relevant permanent path-fields are introduced (see Fig. 3): Definition 4.1. A permanent path element (x, y) 2 {1, . . . , X} · {1, . . . , Y} is called a relevant permanent path-field if and only if one of the following conditions holds: • Perm_Roadx1,y + Perm_Roadx+1,y + Perm_Roadx,y1 + Perm_Roadx,y+1 P 3 (i.e. the field (x, y) is a crossing-point of the permanent transportation paths system).

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• Perm_Roadx1,y + Perm_Roadx+1,y + Perm_Roadx,y1 + Perm_Roadx,y+1 = 2 ^ Perm_Roadx,y1 + Perm_Roadx,y+1 = 1 ^ Perm_Roadx1,y + Perm_Roadx+1,y = 1 (i.e. the field (x, y) is a rectangular turning point of a permanent transportation path). • Perm_Roadx1,y + Perm_Roadx+1,y + Perm_Roadx,y1 + Perm_Roadx,y+1 = 1 (i.e. the field (x, y) is an ending point of a permanent transportation path). SRPPE defines the set of all relevant permanent path-fields. For a permanent path element (x, y) 2 {1, . . . , X} · {1, . . . , Y} the following identifiers are introduced: • RNS(x, y) = {(x + k, y)jk 2 {0, . . . , X  x} ^ (x + k, y) 2 SRPPE} and RN(x, y) 2 RNS(x, y) as the element with the minimum distance to (x, y). If RNS(x, y) = ; then RN(x, y) = 1. • LNS(x, y) = {(x  k, y)jk 2 {0, . . . , x  1} ^ (x  k, y) 2 SRPPE} and LN(x, y) 2 LNS(x, y) as the element with the minimum distance to (x, y). If LNS(x, y) = ; then LN(x, y) = 1. • ANS(x, y) = {(x, y + k)jk 2 {0, . . . , Y  y} ^ (x, y + k) 2 SRPPE} and AN(x, y) 2 ANS(x, y) as the element with the minimum distance to (x, y). If ANS(x, y) = ; then AN(x, y) = 1. • BNS(x, y) = {(x, y  k)jk 2 {0, . . . , y  1} ^ (x, y  k) 2 SRPPE} and BN(x, y) 2 BNS(x, y) as the element with the minimum distance to (x, y). If BNS(x, y) = ; then BN(x, y) = 1. • N(x, y) = {RN(x, y), LN(x, y), AN(x, y), BN(x, y)}. Additionally, a permanent path element (x, y) is called the neighboring permanent path element NPPE(x 0 , y 0 ) for a given unit-element (x 0 , y 0 ) if and only if (x, y) is the permanent path element which is found first by the breadth-first-search procedure described above, starting from the unit-element (x 0 , y 0 ). If this search ends without success or at the input- or output-element of the respective destination-machine itself, NPPE(x 0 , y 0 ) is set to ‘‘1’’. By using these relevant path-fields and the distances of the shortest paths between them, it becomes possible to compute the distance between two arbitrary permanent transportation fields in O(1). For this purpose, the following rules can be applied: Lemma 4.2. Let n and m be arbitrary machines located in a feasible layout. Furthermore, let (xn, yn) 2 {1, . . . , X} · {1, . . . , Y} be the layout location of the output-element of the nth machine and (xm, ym) 2 {1, . . . , X} · {1, . . . , Y} be the layout location of the input-element of the mth machine. Additionally, it is assumed that transport between both machines is possible. In addition, for i 2 {1, 2} and a unit-element of the layout (x1, x2), the projection pi2 ðx1 ; x2 Þ ¼ xi gives the value of its ith coordinate. Then, the path length Distn!m of the transportation path starting at (xn, yn) and ending at (xm, ym) can be computed as follows: 8 In Out In jPathOut > n!m j þ jPathn!m j þ 2  2  jPathn!m \ Pathn!m j; > > n n m m > > if NPPEðx ; y Þ ¼ NPPEðx ; y Þ 6¼ 1; > > > > jPathIn > n!m j þ 1; > > > if NPPEðxn ; y n Þ ¼ 1; > > > Out > > > jPathn!m j þ 1; > > > < if NPPEðxm ;y m Þ ¼ 1; In 1 n n 1 m m 8n; m 2 f1; .. . ;N g : Distn!m ¼ jPathOut n!m j þ jPathn!m j þ 2 þ jp 2 ðNPPEðx ; y ÞÞ  p 2 ðNPPEðx ; y ÞÞj > > 2 n n 2 m m > þ jp2 ðNPPEðx ; y ÞÞ  p2 ðNPPEðx ;y ÞÞj; > > > > if NPPEðxn ; y n Þ 6¼ NPPEðxm ; y m Þ ^ N ðNPPEðxn ; y n ÞÞ ¼ NðNPPEðxm ;y m ÞÞ; > > > In > 1 n n 2 n n > jPathOut > n!m j þ jPathn!m j þ 2 þ minfjp 2 ðNPPEðx ;y ÞÞ  x1 j þ jp 2 ðNPPEðx ;y ÞÞ  y 1 j > > 1 m m > > þDistððx1 ; y 1 Þ;ðx2 ; y 2 ÞÞ þ jp2 ðNPPEðx ;y ÞÞ  x2 j > > > > þjp22 ðNPPEðxn ; y n ÞÞ  y 2 j j ðx1 ;y 1 Þ 2 N ðNPPEðxn ;y n ÞÞ ^ ðx2 ; y 2 Þ 2 NðNPPEðxm ; y m ÞÞg; > : otherwise

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In this calculation, Dist((x1, y1), (x2, y2)) gives the distance in the predefined path system between the two relevant permanent path-fields (x1, y1) and (x2, y2) measured in unit-elements. Proof. We commence by analyzing the first configuration NPPE(xn, yn) = NPPE(xm, ym) 5 1. In this situation, the transportation path contains a sub-path proceeds on permanent fields, while both temporary In sub-paths PathOut n!m and Pathn!m end at the same permanent element. Therefore, to compute the total path length, the sum of the lengths of both sub-paths running on temporary transport elements is generated. Since both paths can have several steps in common, these elements must be subtracted twice. However, in order to establish a connection of both fields, a single element must be added. Finally, the path length emerges from this resulting number of intermediate transport elements, with one more added. Therefore, in the specific situation defined by the first case we get the formula defined above. The second and third case is defined by NPPE(xn, yn) = 1 _ NPPE(xm, ym) = 1. Because it is assumed that a path connection between (xn, yn) and (xm, ym) is possible, these cases represent the situation where the found transportation path contains no permanent transport elements. Therefore, jPathIn n!m j þ 1 and jPathOut j þ 1 yield the required path-length. n!m The fourth case is characterized by NPPE(xn, yn) 5 NPPE(xm, ym) ^ N(NPPE(xn, yn)) = N(NPPE(xm, ym)). Here, the temporary paths end with different permanent transport elements whose neighboring relevant permanent path-fields are identical. Therefore, the entire path contains a sub-path running on permanent transport elements connecting NPPE(xn, yn) and NPPE(xm, ym). The length of this sub-path is equal to the respective rectangular distance of both elements since the coordinates of NPPE(xn, yn) and NPPE(xm, ym) differ only in one dimension. As a result, the total length of the entire transportation path can be calculated by the sum of the lengths of both temporary paths and the rectangular distance between NPPE(xn, yn) and NPPE(xm, ym), corrected by the addition of two further steps resulting from the insertion of NPPE(xn, yn) and NPPE(xm, ym). The fifth and last case differs from the fourth case only by virtue of a complex computation of the length of the sub-path running on permanent transportation elements, since NPPE(xn, yn) and NPPE(xm, ym) no longer have identical neighboring relevant permanent path fields. Hence, the path length computation must find the shortest connection between NPPE(xn, yn) and NPPE(xm, ym) in the contiguous system of permanent transportation paths. Therefore, all existing different combinations of relevant permanent transportation elements neighboring with NPPE(xn, yn) and NPPE(xm, ym) must be compared because they are all possible transition points. For each generated configuration the resulting length of the arising permanent transportation path between NPPE(xn, yn) and NPPE(xm, ym) can be computed by the rectangular distance of both elements to the respective neighboring relevant permanent transport elements, added to the minimum length of a permanent transportation path connecting these chosen neighboring relevant permanent transport fields. The length of the shortest path from a permanent transport element (x, y) to every element e 2 N(x, y) is equal to the respective rectangular distance of both elements, since the coordinates of (x, y) and e differ in only one dimension. Note further that for every permanent transportation sub-path, there are at most four different configurations to be generated. Therefore, the formula given above can also be applied to the fifth case. h Since the distances between the elements belonging to SRPPE are known in advance by an offline computation and the needed rectangular distances can be computed in constant time, the defined generation of the path-lengths can be achieved very efficiently. 4.1.2. Feasibility test During the application of the improvement procedure, each implemented modification of the layout currently stored must be completed by means of a feasibility check. Since the consideration of some of the restrictions defined in the model (especially those concerning the path generation) can become very costly, the following gradual procedure is applied for this purpose. In order to reduce its average computational

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complexity, this process starts verifying the simple restrictions before the more complex ones are analyzed. Note that the following process is stopped immediately if one of the restrictions is not fulfilled. 1–8 below contain the steps of the testing procedure: 1. Check restriction 4 reduced to the restricted elements for all reallocated machines. Note that if the current solution is an initial one, the verifications in this and the following steps are extended to all machines allocated in the layout. 2. Check restriction 4 reduced to the existing permanent transport elements for all reallocated machines. 3. Check restriction 3 for all reallocated machines. 4. Check restriction 5 for all reallocated machines. 5. In this step restriction 4 reduced to the currently defined transportation paths is verified. If a unit-element is found that is used simultaneously by a machine and a temporary path, the next steps depend on the predefined binary parameter move_temp_paths. If this parameter equals zero, the check process is stopped, returning with the result of an infeasible solution. Otherwise, the respective temporary transportation paths have to be recomputed. 6. The breadth-first-search path generating procedure is started to compute a temporary path from the output- and the input-element of the moved machines to the permanent path system. Note that if during this process a direct connection between the input and output-point of two machine is established this path is stored additionally. If one of these paths cannot be generated the solution is infeasible. 7. Check the existing path-lengths- and distance-restrictions to the already placed machines. 8. If temporary paths have been deleted these paths have to be recomputed next. If this is not possible anymore the current solution is dismissed. After generating the new paths the verification of the respective path-lengths- and distance-restrictions is executed. Note that the compliance with the remaining restrictions 1, 2, 6, 7, 8, 9 and 11 in each generated solution is guaranteed by the definition of the applied operations. 4.1.3. Computation of the objective value During the improvement process a considerable amount of time is spent on the computation of the objective values. Therefore, efficient calculation methods are essential. In order to avoid unnecessary computations, the objective value of the current solution is always derived from the value of the previous solution. Thus, only changes caused by the executed move-operation are examined. For this purpose, a specific list of machines causing a potential change of the current objective value is maintained. While executing a change operation on the current layout, every moved machine or machine with modified transportation paths is added to this list. Afterwards, the length of the transportation paths starting or ending at one of the machines stored in the list is compared with its previous value. The resulting difference is weighted with the respective predefined flow quantity. For this simple calculation, the procedure maintains an additional matrix of the current path-lengths L = (li,j)(16i,j6N) which is updated in each move. Assume that Lnew is the path-length-matrix after the move execution and Lold be the one before. Then, the difference matrix DL = (Dli,j)(16i,j6N) = Lnew  Lold P can be used to calculate the transportation cost changes. This is calculated by the formula DTCðS new Þ ¼ ði;jÞ2fði;jÞjDli;j 6¼0g Dli;j  Flowi;j which respects only the modified matrix-values. In contrast to this simplified computation applied during the improvement process, the transportation costs of an initial solution are computed by multiplying the matrix of the initial path-lengths L with the quantities to be transported. PN Additionally, the reallocation costs are computed for an initial solution by i¼1 Moved n  MoveCostn as defined in the objective function of the model. However, during the improvement process, the current reallocation costs are corrected only for machines whose positions have been modified during the last move. Therefore, the following revised amount DMC(Snew) is added to the stored objective value:

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8 MoveCostn ; > > > > X P Y P 3 P > > > if Startnx;y;r ¼ 1 ^ Startn 6¼ Posold ^ Startn ¼ Posnew ; > > > x¼1 y¼1 r¼0 N < X MoveCostn ; DMCðS new Þ ¼ > n¼1 > X P Y P 3 > P > > if Startnx;y;r ¼ 1 ^ Startn ¼ Posold ^ Startn 6¼ Posnew ; > > > x¼1 y¼1 r¼0 > > : 0; otherwise. In this computation, Startn defines the starting position of the nth machine while Posold and Posnew defines its current location in the emerging layout before and after the considered move. 4.2. Computing an initial solution Before commencing an improvement process, an initial layout must be computed. Therefore, at the beginning of the solution process, a fast generation-procedure is used. This generation-procedure computes a first layout iteratively, by locating the different machines one-by-one in a random sequence. However, this process starts with the machine whose location is assumed to be the most important one according to its flow-relationships. To measure this importance of a specific machine n 2 {1, . . . , N}, we compute its standardized cost-weighted flow-intensity, which is defined below: Definition 4.3. For every machine n 2 {1, . . . , N} the standardized cost-weighted flow-intensity SCWFIn is defined by PN PN i¼1;i6¼n FlowCostn;i  Flown;i þ i¼1;i6¼n FlowCost i;n  Flowi;n . SCWFI n ¼ 2 N Note that it is reasonable to locate machines with a large SCWFI-value in a more centralized position. To generate an initial layout, the following simple procedure is applied: 1. Locate the machine with the maximum SCWFIn-value in the free location nearest the center of the layout. 2. Select the machine currently unplaced with the maximum SCWFIn-value. In order to locate the machine in question, the potential locations of the respective considered machine are searched in concentric circles around the current center. This center is updated from iteration to iteration by using the average coordinates of the already located machines. Using this rule, the location of the machine is always selected as close as possible to the center of the layout currently under consideration. 3. Use the feasible alternative that leads to the best objective value according to the machines already placed. 4. If no feasible location for the currently considered machine can be found, another unplaced machine is taken, while with a predefined probability, one of the machines already placed is removed. 5. If a complete solution cannot be generated within a predefined number of iterations, the computation is terminated. 6. If all machines are located in the layout, the generated solution is regarded as the result of the procedure. If the execution of the procedure defined above does not result in a feasible initial layout, the different machines are now processed in a random sequence. In addition, the different machines are located again one after another in the layout that develops, while a position for the current machine is selected randomly.

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4.3. Single-method improvement approaches The improvement approaches described in this subsection modify an existing layout by the repeated application of a single operation, for which the efficiency of these procedures depends mainly on the efficiency of the respective method. Consequently, these approaches are named according to the nature of the applied modification-operation. Fig. 4 shows some of these operations. Note that an efficient number of tested layouts in the subsequent approaches depends on various issues, including the number of unit-elements, occupation of the layout and intensity of restrictions. The numbers given performed well in the problems considered. 4.3.1. Local relocation search (LRS) This routine is intended to improve a layout currently stored, by moving a randomly chosen machine from its present position to a new location in its immediate neighborhood. Therefore, the neighboring locations of the machine in question are evaluated in a systematic examination process that is reduced to the evaluation of no more than 200 tested and 50 feasible destinations. For a considered input-element (x, y) 2 {1, . . . , X} · {1, . . . , Y} the possible destinations of the local relocation operation are chosen randomly one after the other from the set LN 6ðx;yÞ ¼ fðx þ k; y þ lÞ 2 f1; . . . ; X g  f1; . . . ; Y gjk; l 2 N with 6 6 k, l 6 6}. Additionally, the orientation of the machine at its new location is determined randomly. After examining the different configurations, the best found position is achieved if this move leads to an improvement. Consequently, as a pure improvement routine, the LRS-operation does not allow any deterioration. The example in Fig. 4 shows how a solution can be improved by a local relocation of Machine 3. The machine is moved slightly and rotated.

Machine 2

Machine 3 M3 M2

M1

Machine 1 (a) Original Layout

(b) LRS of machine 3 M1

M2

M2

M3

M1

(c) GRS of machine 3

M3 (d) TEES of machines 1 and 3

Fig. 4. Layouts resulting from the application of the LRS, GRS and the TEES-operation.

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4.3.2. Global relocation search (GRS) As an analog to the LRS-operation, the GRS-procedure attempts to improve a current solution by moving a machine that had been selected randomly. However, in contrast to the LRS-operation, the Global Relocation Search does not reduce the search for the new position to the direct neighborhood of the selected machine. While the moves executed by the LRS-procedure only consider a minor local area of the solution space around the current position in the GRS-operation, a new location of the input-element of a machine being considered can be allocated to every feasible position in the layout. To do so, a potential new position and orientation of a considered machine is chosen randomly from the set {1, . . . , X} · {1, . . . , Y} · {0, 1, 2, 3}, while the optimal layout of altogether 800 generated and 100 feasible layouts is achieved if it leads to a cost reduction. The example in Fig. 4 shows how the solution can be improved by a global relocation of Machine 3. The machine is moved from its original position to the bottom left corner. 4.3.3. Two elements exchange search (TEES) This operation is aimed at improving the current layout by exchanging the location of two randomly chosen machines. However, due to irregular machine-shapes, a simple exchange of the respective inputpoint-positions can result in a non-feasible layout. Therefore, the TEES-procedure attempts to find a feasible location for the exchanged machines in the immediate neighborhood of the respective positions. To do this, the search-procedure executed for the computation of the destination-position during the local relocation search procedure (LRS) is used. Starting from a given location (x, y) of the input-element of the respective machine, this procedure considers the neighborhood LN 6ðx;yÞ ¼ fðx þ k; y þ lÞjk; l 2 N with 6 6 k, l 6 6}, while possible locations and orientations are chosen randomly until a total of 1000 layouts or 100 feasible locations of both machines are considered. If some feasible layout can be generated during this search process, the best alternative of those that were found is achieved if it results in an improvement. Otherwise, the previous solution is restored. The example in Fig. 4 shows how the solution can be improved by a two-element exchange of Machines 1 and 3. 4.3.4. Multiple elements cyclic exchange search (MECES) In each step of this procedure, between two and five machines are chosen randomly so as to generate a cyclical exchange of their current locations. In this cyclical exchange, a permutation of the current locations of the c (3 6 c 6 5) randomly determined machines (numbered from m1 to mc) is executed. In order to generate such a permutation, the c machines are removed from the layout currently considered, while their positions are stored as potential destinations for the subsequent reinsertion process. The procedure then attempts to relocate the removed machines one-by-one in the layout. In order to achieve this, in every executed reinsertion step conducted for a machine mt (1 6 t 6 c), a possible destination is generated by a random selection of a still vacant position defined by the previous location of the input points of the deleted machines. Starting from this drawn position (x, y), the operation tries to find a new location and orientation for the machine mt by the random choice of an element ((x, y), r) from the set LN 6ðx;yÞ  f0; 1; 2; 3g. To do so, the procedure generates 300 randomly chosen layouts from which the best is implemented if this modification leads to a layout with reduced costs. Otherwise, if this search does not result in at least one feasible configuration with reduced total costs, the following measures are undertaken in the given sequence: 1. Repeat the search process with an appropriate location for mt, commencing from a currently vacant input point of another machine belonging to the set of machines whose locations are currently exchanged. If this insertion process fails again, the search is restarted at different locations for a total of six times. 2. If all six attempts fail, a random number of relocated machines are deleted again from the layout in question to restart afterwards the relocation process described above. This additional measure is repeated twice before the execution of the MECES-procedure is abandoned completely.

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4.3.5. Global diversification through controlled deterioration (DET) While all the procedures described above are designed as pure improvement approaches, in the following Section we additionally consider procedures temporarily changing the solution currently under consideration to worse. In the following, the temporary application of these steps is denoted as a (generated) deterioration (process). Such a deterioration can be useful since pure improvement processes can get ‘‘stuck’’ in a local optimum. By contrast, temporary deteriorations can lead the solution process to promising parts of the solution space which would otherwise remain unexamined. In this context, we distinguish between deterministic procedures and operations with a random use of deterioration. While the former generate a deterioration process only in predefined time intervals, the latter execute such a deterioration randomly. Independently of this distinction, for each executed deterioration process, a specific procedure is used in order to implement a predefined deterioration rate iteratively. By using several iterations instead of only a single one, it becomes more likely that the application of this procedure reaches new parts of the solution space. This is because a solution currently considered is changed more extensively during the deterioration process. To describe the operation in detail, we introduce additional parameters. Therefore, lcurr and lcurr define the interval for the targeted deterioration rate per iteration. In addition, lobj gives the lower bound for the deterioration rate to be attained by the entire deterioration process. In each iteration of the deterioration procedure, one of the operations LRS, GRS, TEES or MECES is applied to modify the current solution. Consequently, if Ut denotes the objective value of the solution considered in advance of the tth iteration, the new objective function value Ut+1 must lie within the interval ½Ut  ð1 þ lcurr Þ; Ut  ð1 þ lcurr Þ. If this deterioration rate cannot be achieved within a predefined number of trials, the deterioration procedure is terminated. In such a case, the solution process tracks back to the layout considered before the deterioration process. Otherwise, the iterations are executed until the targeted overall deterioration rate lobj has been achieved. By combining the deterioration procedure with the single-method improvement approaches, we obtain the following extended approaches using deterministic deterioration: • LRS–D (Local Relocation Search with Deterioration): This approach is an extension of the LRS-procedure. Always after 3 Æ N unsuccessful (no improvement) executions of the LRS-operation, the deterioration procedure commences with the parameters lcurr , lcurr , and lobj . While lcurr and lcurr are always predefined, lobj is drawn randomly for each deterioration process. If after several executions of the deterioration procedure no improvement of the best found solution is attained, the subsequent deterioration routine is started with an increased targeted deterioration rate to as to achieve more substantial modifications. • GRS–D (Global Relocation Search with Deterioration): This approach uses the GRS-operation to improve the layout currently considered, while the deterioration-process is generated if the best known solution is not improved within 8 Æ N iterations. Note that the same parameter set for (lcurr , lcurr and lobj ) is used as described for the deterioration procedure applied in the LRS–D-approach. • TEES–D (Two-Element Exchange Search with Deterioration): This approach uses the TEES-operation to modify the current layout while a deterioration process commences if the layout currently considered cannot be improved within N iterations. Again, the same parameter set for (lcurr , lcurr and lobj ) is used as described for the deterioration procedure applied in the LRS–D-procedure. In addition to the approaches with a deterministic use of the deterioration procedure, there are two further approaches applying such kind of operations randomly: • TEES–DC (Two-Element Exchange Search with Dominance Criteria): In this approach, the operation TEES is applied to modify the considered layout. During the enumeration process, every improvement is

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achieved while a deterioration is accepted only with the probability Dp. The parameter Dp is always increased after i consecutive steps which yield no improvement. However, either an improvement or an executed deterioration process provokes that this probability is reset to its initial value. To modify the current layout more comprehensively, each realized deterioration is completed by a specific exchange operation. Thus, after a deterioration, the two exchanged machines are exchanged once again with N/8 randomly chosen machines in the layout. Afterwards, deteriorations are always inactive for at least the next 60 iterations. • GRS–GD (Global Relocation Search with Great Deluge Characteristic): The GRS–GD-approach uses the global relocation operation to modify an existing layout. While an improvement in the current solution is always accepted, a deterioration in the resulting objective value must not exceed the lower bound Ubound = (1 + r) Æ Ubest while Ubest is the objective value of the best currently known solution. 4.4. Multi-state improvement procedures In contrast to the procedures described in Section 4.3, the multi-state approaches no longer execute only a single operation for optimization, but apply several depending on the results achieved by previous computations. By doing so, this kind of improvement procedure results in a specific variable neighborhood search whose efficiency for different applications was shown by Hansen and Mladenovic´ [14]. The different neighborhoods are defined by the application of the modification operations described in Sections 4.3.1, 4.3.2, 4.3.3, 4.3.4 and 4.3.5. To decide about the current neighborhood, the multi-state improvement procedures work in different states which control the application of the different operations. According to the applied transition function controlling the change-over of the different possible states during the computation process, a random multi-state search is distinguished from a guided multi-state search. 4.4.1. Random multi-state search (RMSS) In order to achieve a state transition during the improvement process, the Random Multi-State Search procedure (RMSS) applies an arbitrator. After being started, the arbitrator always generates two random integers r1 2 {1, 2, 3, 4, 5} and r2 between 10 and 40, whereas every possible number has the same probability. While the first integer defines the subsequent state of the algorithm, the latter gives the number of moves to be executed by the algorithm in this new state. As illustrated in Table 1, the current state of the RMSSprocedures determines the operations to be applied for modifying the considered layout. While in states 1, 2 and 3, the respective operation is applied r2 times, whereas, due to its higher complexity, the MECES-operation provided for the fourth state is applied only r2/3 times. In the fifth state, a predefined combination of altogether 2 Æ N steps of GRS, followed by N steps of MECES and completed by N steps of LRS, is executed only once. After the application of the respective predefined number of moves, the arbitrator again generates a change-over of the current state. In addition, if the computation cannot attain an improvement of the best found solution within 30 Æ N steps, the deterioration process (DET) is applied to lead the search into a new part of the solution space. Therefore, the DET-process is executed with the parameters lcurr and Table 1 States and the respective applied operations of the random multi-state search procedure State

Used procedures

I II III IV V VI

Local relocation search (LRS) Global relocation search (GRS) Two elements exchange search (TEES) Multiple elements cyclic exchange search (MECES) 2 Æ N steps of GRS followed by N steps of MECES and N steps of LRS Deterioration DET

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lcurr , while lobj is drawn randomly. In each step of the DET-procedure either a TEES- or a GRS-operation is applied while this decision is taken at random. Here, both operations have an equal probability of being selected. Note that after the application of the DET-procedure, the computation changes immediately to state five, so as to generate substantial additional modifications of the stored layout by the application of the predefined sequence of operations (2 Æ N steps of GRS followed by N steps of MECES completed by N steps of LRS). 4.4.2. Guided multi-state search (GMSS) Analogously to the RMSS-procedure, the GMSS-approach works in different states which control the operations for modifying the considered layout. However, in contrast to the RMSS-approach, the change-over of the states is not achieved randomly, but by applying predefined rules. Thus, the use of an arbitrator is replaced by a direct transition function defining the subsequent state, depending on predefined scenarios occurring during the computation. The GMSS-procedure functions in a total of seven states, while in each state, each possible operation (LRS, GRS, TEES, MECES or DET) is applied with a predefined probability given in Table 2. The algorithm changes from state i 2 {1, 2, 3, 4, 5} to the subsequent state i + 1 2 {2, 3, 4, 5, 6} if the current solution was not improved within a predefined number of iterations. Specifically, the procedure generates the transition from state 1 to state 2 after 2 Æ N unsuccessful iterations, from 2 to 3 after 2 Æ N further steps, from 3 to 4 after 8 Æ N further iterations, from 4 to 5 again after 8 Æ N further steps, and finally from 5 to 6 (the pure deterioration state) after 10 Æ N further unsuccessful iterations. After applying the deterioration-routine (DET) with the parameter set as used in the RMSSprocedure in the sixth state, the algorithm changes for the execution of a total of 2 Æ N iterations to state seven, followed by the subsequent return to the first state. In addition, in the first five states, each improvement causes an immediate return to the first state. 4.5. Numerical results In order to validate the performance of the various proposed solution procedures, 13 different randomly generated instances of the model were solved by applying the introduced improvement approaches. These include two instances with 15, six with 30, four with 60 and one with 100 machines to be arranged in the layout. All numerical results are based on problems that were randomly generated with some predefined settings. The probability of an element in the Flow matrix to be non-zero was set to 20%. The values of these non-zero elements of the Flow matrix were drawn from a uniform distribution between 0 and 100. The elements of the FlowCost matrix were drawn from a set of four values with equal probability reflecting different transport vehicles. The number of machines with restrictions is defined as about 20%. The number of restrictions for a machine is drawn randomly from a uniform distribution between 0 and 3. The types of restriction are also drawn randomly. The machine shapes are predefined and selected randomly. The buildTable 2 The predefined probabilities for the application of the possible operations (LRS-, GRS-, TEES-, MECES- or DET-operation) in the different states in the guided multi-state procedure State

PLRS

PGRS

PTEES

PMECES

PDET

I II III IV V VI VII

0.60 0.20 0.09 0.05 0.02 0.00 0.00

0.15 0.55 0.15 0.10 0.04 0.00 0.40

0.25 0.25 0.75 0.83 0.91 0.00 0.50

0.00 0.00 0.01 0.02 0.03 0.00 0.10

0.00 0.00 0.00 0.00 0.00 1.00 0.00

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ings used for the problems are predefined, i.e. the permanent paths, locations of restricted elements or the conditions of the elements are given. The area occupancy on the sample instances varies between 50% and 95% based on the selected building and the number of machines. Note that in order to guarantee uniform distributions, a random number generator, based on the principles proposed by Park and Miller [31], has been used for the creation of all problem instances. The computational tests were executed on a Sun SPARC Ultra-1, running with the operating system Solaris 5.6, while the different solution procedures defined above were implemented in the programming language C. After computing an initial solution sini by applying the procedure described in Section 4.2, the particular improvement process of the different approaches commences to modify this first layout. For every tested instance, the best solution found for each procedure after 5000 seconds elapsed, is taken as the result of the respective approach. Since a layout planning process is not generally influenced by short term time restrictions, this time horizon for the application of the solution procedures seems to be realistic. To analyze the improvement process arising during the computation, the best solution found for each approach is additionally stored after 50 and 500 seconds. In order to minimize the impact of the generated random numbers on the yielded solution quality, each computation is repeated several times while the average is taken as the result. A first look at the measured results summarized in Table 3 shows that all implemented procedures were able to achieve substantial improvements in the initial layout that were generated by the construction procedure described in Section 4.2. Despite the fact that this construction procedure already considers various layouts while inserting each machine in the emerging layout, the improvement approaches can achieve cost reductions between 16% and in excess of 31% by a targeted application of appropriate modification-operations. By comparing the average solution quality of the different approaches in Table 3, it becomes obvious that the pure-improvement approaches yield a significantly poorer result in comparison with the algorithms using the deterioration process as well. This seems to be explained by the fact that the pureimprovement procedures will not be able to proceed beyond a local minimum. In particular, this becomes particularly obvious by comparing the results of the LRS- and the LRS–D-approach where the highest relative difference of the average solution quality between a pure-improvement procedure and the respective deterministic ‘‘D-approach’’ is achieved. Due to only small modifications generated by the local relocation operation, the LRS-algorithm seems to be the approach most likely to ‘‘get stuck’’ in a local optimum. The next important cognition is obtained from analyzing the measurements given in Table 3. The random application of the deterioration process as conducted in the procedures TEES–DC or in GRS–GD can result in further substantial improvements of the solution in comparison to the pure deterministic use in the

Table 3 Yielded average solution quality of the various approaches measured by the relative improvement in the initial layout Procedure

LRS GRS TEES MECES LRS–D GRS–D TEES–D TEES–DC GRS–GD RMSS GMSS

All experiments

Specific groups

davg;50s ð%Þ Start

davg;500s ð%Þ Start

davg;5000s ð%Þ Start

davg;N¼15 ð%Þ Start

¼30 davg;N ð%Þ Start

davg;N¼60 ð%Þ Start

¼100 davg;N ð%Þ Start

14.08 11.93 9.05 3.51 14.68 13.62 10.73 11.66 10.47 12.36 14.55

16.30 18.18 19.01 13.11 20.68 20.33 19.91 20.09 18.04 20.60 22.99

16.56 20.02 24.70 20.07 23.94 24.17 25.94 26.09 23.06 25.13 27.45

8.33 15.98 24.17 24.27 30.60 29.01 30.07 30.11 20.62 29.00 31.16

16.37 21.26 25.37 22.15 25.17 25.78 26.19 27.29 23.34 25.17 27.73

16.51 17.58 20.88 15.82 17.70 19.31 20.91 21.07 20.28 20.58 22.32

23.91 24.05 24.92 12.96 24.44 24.48 26.85 26.41 25.66 28.29 30.54

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approaches LRS–D, GRS–D and TEES–D. This can be attributed to the fact that a deterministic ‘‘Dapproach’’ reacts only if a local optimum has already been reached, while the random application can prevent its computation so as to use the saved time to find new areas of the solution space. In comparison to all other approaches, the elaborated multi-state procedures yield a substantially higher solution quality for the considered instances. In particular, the GMSS-approach achieves considerable improvements in comparison to the other solution procedures. By analyzing the results of the procedures using only one single operation to modify the considered layout, some possible reasons for the superiority of the guided multi-state search can be identified. While the LRS-operation is able to achieve quick improvements in a specific local area of the layout (illustrated by the average results for a computational time limit of 50 seconds), its application is not efficient in the long run. Here, operations such as GRS, TEES, or MECES which all operate more globally, become preferable. This observation is transformed most efficiently in the GMSS-procedure, where every local improvement immediately leads to a more local search by the transition to the first state where the LRS-operation is applied for 60% of the achieved modifications. Also, only after local improvements are no longer probable, does the algorithm change to higher numbered states where the application of the other operations obtains a substantially higher proportion of the computation time. Therefore, on the (unsuccessful) path from state two to the fifth state, the application probability of the more complex operations is increased iteratively before, in the sixth and seventh state, it is assumed that substantial modifications have become necessary which even justify deteriorations of the objective value of the current solution. Due to these definitions and the measured results, it can be concluded that the GMSS-procedure uses the advantages of the different applied operations LRS, GRS, TEES and MECES most consistently. This observation has been underlined additionally by further measurements where a modified GMSS-algorithm was applied with a change-over to the first state only if a new bestfound solution has been generated. Since this approach cannot yield the solution quality of the original GMSS-procedure, the importance of a systematic consideration of the neighboring solutions of each local improvement becomes obvious. In addition, due to the fact that this systematic consideration of the neighboring solutions starts with only local modifications and enlarges the scanned neighborhood iteratively by applying more and more global operations, this subsequent examination yields substantial improvements.

5. Conclusions The current research addresses layout planning problems. It introduces a new problem definition and the generation of appropriate solution approaches. With respect to the requirements of realistic layout planning, the proposed model allows an application-dependent level of detail to map existing layouts. Accordingly, fixed irregular machine shapes, their rotation and a detailed transportation path planning are integrated into the model. By combining these attributes within a single model, the proposed approach provides decision support for complex layout planning problems. No such procedures have so far been published in the literature. To be able to solve this very complex new model efficiently, we introduce several solution approaches as improvement procedures modifying an initial layout. These approaches use various different operations to move and exchange machines in order to find new layout configurations. By combining the advantages of the different operations, a more elaborate approach has been designed as a multi-state procedure. Thus, an efficient examination of the solution space becomes possible, in contrast to the simple single-operationapproaches. The superiority of the more sophisticated approach was evaluated by computational experiments executed for randomly generated instances of the proposed model. While the results of the research on hand are promising, future research can be extended in two main directions. On the one hand, an important area for further research would deal with a possible extension of the model to include various further aspects required by industrial applications, as, for instance, multiple floor

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layouts or overhead transports. Apart from them, in particular, the consideration of dynamic issues seems to be reasonable. On the other hand, a second challenging research area can address further improvements of the attained solution quality through automated parameter tuning and a parallelization of the procedures. Because demands and flows in facilities are constantly changing, the movement of machines that are already placed becomes inevitable. Therefore, the model could be extended in future to cover these dynamic aspects. Thus, according to several possible scenarios, overall movement costs, transportation costs and downtimes can be optimized simultaneously. With regard to solution quality, the numerical results indicate that construction methods are limited and it is therefore more useful to optimize the improvement approaches. In particular, the guided multi-state heuristic which has a high number of parameters that need to be coordinated offers considerable potential. An obvious extension of this approach is to make it selfadapting, that is, to allow its parameters to tune themselves during the optimization. In this context, one can think of an autonomous correction of the predefined probabilities applied in the respective states. These corrections should be based on the current efficiency of each operation. By doing so, more efficient operations acquire an increased execution probability. While quality could be improved by using this selfadapting approach, the computational burden remains high. Therefore, an efficient parallelization should also be considered. Currently, almost all companies use Local Area Networks connecting modern personal computers. Within these networks, very often, only ordinary office communications or text processing programs are executed. However, these applications use only a small fraction of the available computational capacity. Thus, it seems reasonable to use the available parts of the performance of these systems to achieve a more efficient layout planning. For this purpose, off-peak times of the processors can be used to generate more accurate results by exploring different areas of the solution space simultaneously. Despite the fact that the parallel use of the distributed system by ordinary office applications results in an unpredictable background load, it can be shown that an efficient distributed program with a specific dynamic load-balancer achieves substantial improvements of the yielded solution quality [9]. This was possible even for extremely dynamically changing background load scenarios.

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