An efficient multiple-stage mathematical programming method for advanced single and multi-floor facility layout problems

Accepted Manuscript

An Efficient Multiple-stage Mathematical Programming Method for Advanced Single and Multi-Floor Facility Layout Problems Abbas Ahmadi, Mohammad Reza Akbari Jokar PII: DOI: Reference:

S0307-904X(16)30002-6 10.1016/j.apm.2016.01.014 APM 10974

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

11 March 2014 23 December 2015 13 January 2016

Please cite this article as: Abbas Ahmadi, Mohammad Reza Akbari Jokar, An Efficient Multiple-stage Mathematical Programming Method for Advanced Single and Multi-Floor Facility Layout Problems, Applied Mathematical Modelling (2016), doi: 10.1016/j.apm.2016.01.014

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Highlights • A multiple stage mathematical programming method is presented for

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FLPs.

• Both single and multi-floor problems can be handled.

• A wide variety of high quality layouts at competitive cost can be obtained.

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• A short amount of time is consumed to approach the real-life problems. • Flexibility of the method to cover various real-life limitations has en-

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hanced it.

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An Efficient Multiple-stage Mathematical Programming Method for Advanced Single and Multi-Floor Facility Layout Problems Abbas Ahmadia,∗, Mohammad Reza Akbari Jokara a

Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran

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Abstract

Single floor facility layout problem (FLP) is related to finding the arrangement of a given number of departments within a facility; while in multi-floor FLP, the departments should be imbedded in some floors inside the facility. The significant influence of layout design on the effectiveness of any orga-

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nization has turned FLP into an important issue. This paper presents a three- (two-) stage mathematical programming method to find competitive

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solutions for multi- (single-) floor problems. At the first stage, the departments are assigned to the floors through a mixed integer programming model

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(the single floor version does not require this stage). At the second stage, a nonlinear programming model is used to specify the relative position of the

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departments on each floor; and at the third stage, the final layouts within the floors are determined, through another nonlinear programming model. The

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multi-floor version is studied in the states in which the locations of the elevators are either formerly specified or not. Computational results show that ∗

Corresponding author. Tel: +98 912 448-6907 Email addresses: [email protected] (Abbas Ahmadi), [email protected] (Mohammad Reza Akbari Jokar)

Preprint submitted to Applied Mathematical Modelling

January 22, 2016

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this framework can find a wide variety of high quality layouts at competitive cost (up to 43% reduction) within a short amount of time for small and espe-

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cially large size problems, compared to the existing methods in the literature. Also, the proposed method is flexible enough to accommodate the complicated and real-world problems, because of using mathematical programming model and solving it directly. Keywords:

Facility layout, Single floor, Multi-floor, Optimization, Nonlin-

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ear programming, Mixed-integer programming 1. Introduction

A facility refers to anything that facilitate the work; in a manufacturing environment, it may be a workstation, a warehouse, a department, a ma-

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chine tool, etc. Facility layout is the arrangement of all facilities needed for producing a product or delivering a service [1]. Facility layout problem has

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many applications in the real world, including layout design for manufacturing systems, hospitals, schools and airports, printed circuit board, backboard

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wiring problems, typewriters, warehouses, hydraulic turbine design, and so on [2, 3, 4]. A suitable structure for facility layout can be beneficial for any

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organization. For example, Francis et al. [5] stated that, in industrial environments, an appropriate layout could reduce total operating expenses by up

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to 15%.

Unequal-areas facility layout problem is concerned with determining the

arrangement of a given number of departments within a facility (hereafter, a facility is the land space, in which departments must be embedded) so that, under some constraints, a given qualitative or quantitative objective function 3

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would be optimized [6]. This objective function might be minimization of material handling costs [7, 8], maximization of desirable relations between

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departments [9], or a combination of different objectives [10, 11]. Most of the effort conducted in the field of FLP thus far are concerned

with single floor problems. But, nowadays, the rapid growth of industries and population, and consequently land shortage have led to an increase in land prices [12, 13]. This challenge has necessitated the use of lands in multi-floor

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structures.

Quadratic assignment problem (QAP) is a restricted version of single floor FLP, in which the shape of all departments are identical and fixed (see [14, 15, 16, 17] for example). Considering computational effort, this problem is an NP-complete problem [18, 6]. Therefore, other complicated

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problems with unequal-area departments and further constraints, such as the existence of a number of floors and elevators (for the interaction between

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floors), are placed in the NP-complete class. This matter has engaged many researchers to present heuristic methods such as CRAFT, ALDEP, CORE-

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LAP [19], and planar graph technique[16], as well as to employ metaheuristic algorithms such as genetic algorithms, simulated-annealing algorithm, and

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tabu-search algorithm [20, 21, 22, 23] for solving this problem. Nevertheless, these methods do not always provide good solutions. Different models of multi-floor FLP are found in the literature, most

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of which have been examined in a deterministic state, and a limited numbers [24, 25] have been studied in a dynamic state, as well. In terms of the objective function, most of the presented models have one objective,

whereas the rest [26, 12, 27, 28] are multi-objective. Also, some models

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have been restricted, such that they can only consider one elevator [29, 30] and require fixed location elevators [28, 30, 31], equal area departments [29],

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as well as regular and fixed shapes for departments or/and floors [32, 13]. Several models can accommodate the structure of aisles [28, 33], two types of elevators [13], irregularly shaped departments and/or floors [12, 34], location

and number of elevators as decision variables [12, 27], and also number of floors [26].

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The methods offered for solving these problems operate in one or more stages. The single stage methods use the techniques, including mixed integer

programming [13], heuristic exchange procedures [31], genetic algorithms [35, 28], and simulated-annealing algorithm [36] to find a good layout. On the other hand, the multiple-stage methods often assign the departments to the

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floors at their first stage, and then determine the layout of the floors at the next stage(s). Indeed, multiple-stage methods may include one (or more) fur-

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ther stage(s) to obtain the final layout. The number and type of these stages may be different from one method to another. For example, one method first

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determines the block layout within each floor, and then tries to find the location of elevators [12, 39]; while the other method only obtains the lay-

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out of departments, in one stage, by considering fixed located elevators [32]. Most of the methods that initially assign the departments to the floors use a mixed integer programming model [32, 37, 38] called FAF (floor assignment

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formulation) [37] for this purpose. However, other techniques like K-mean

algorithm [29] have been also deployed. At the next stage(s), in order to find the detailed layout design, the techniques such as nonlinear programming [32], simulated-annealing algorithm [37, 36, 39], tabu-search algorithm [38],

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and heuristic exchange procedures [29] are employed. Most of the frameworks presented in the literature often provide one

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solution (layout) for any problem, which may not be applicable. Furthermore, they mainly have many difficulties in computational time (and so, they are appropriate only for small size problems) and also the quality of the obtained layout.

In this paper, an exhaustive multiple-stage framework is presented for

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single- and multi-floor cases, in which the mathematical programming techniques are used to find a wide variety of high quality solutions with the

lowest possible cost and within a short amount of time compared to the earlier frameworks in the literature. At the first stage (but only for multifloor problems), a mixed integer linear programming is employed to assign

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departments to floors by minimizing vertical interaction cost between the departments. The second stage is a nonlinear programming model which

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establishes the relative position of departments on each floor using the same approach employed in [6], except that with some improvements (the model

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presented by Jankovits et al. [6] is entitled JLAV model, originated from the name of its authors, which proposed for single-floor problems). At the third

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stage, using the solution obtained from the last stage, another nonlinear programming model is developed, in order to find the detailed layout of all floors simultaneously. The procedure for solving single-floor problems is similar to

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that of multi-floor problems, by this difference that it does not require stage

one.

Other features of the proposed framework are its capability to accommo-

date the problems with fixed shapes and locations for departments, unequal-

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area and non-rectangular floors, as well as finding the location of elevators. In addition, the presented method is capable to incorporate the elevators

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with pre-determined and also not-pre-determined locations. The rest of paper is organized as follows. In Section 2, the background

required for stage two of the presented method is provided. In Section 3, the method is proposed for single-floor cases. The extended version of single-

floor to multi-floor cases is given in Section 4. Computational results that

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validate the strength and effectiveness of this method constitute Section 5.

Finally, conclusions, managerial insights and potential directions for future work are discussed in Section 6.

2. Background to JLAV model

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In this section, first, the single-floor FLP problem is explained (for solving of which many methods have been proposed in the literature). Afterward,

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JLAV model is described.

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2.1. Original problem

Throughout this paper, it is assumed that there are N departments la-

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beled i = 1, ..., N . Department i has a rectangular shape with area ai and its position is expressed by the coordinates of its center, i.e., (xi , yi ). Cost per unit distance between departments i and j is indicated by cij (it is assumed

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that cij =cji ). Distances between departments are considered as the rectilinear distance from their centers. Also, the facility (and equivalently the floors) must be rectangular (however, in the presented method, it is possible to consider non-rectangular floors, given that each floor could be converted

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into a rectangle by adding some smaller rectangular shapes). It is notable that the flows between the departments are applied through cij .

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The aim of Jankovits et al. [6] and many other researchers, in single-floor FLPs, is to find the optimal solution of the following model: min

(xi ,yi ),hi ,wi ,hF ,wF

X

1≤i
cij (|xi − xj | + |yi − yj |),

∀i,

wi hi = ai βi ≤ β ∗

∀i,

and hmin ≤ hF ≤ hmax F F .

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wFmin ≤ wF ≤ wFmax

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1 1 s.t. |xi − xj | ≥ (wi + wj ) or |yi − yj | ≥ (hi + hj ) 2 2 1 1 1 1 w i − xi ≤ w F ∀i, xi + wi ≤ wF and 2 2 2 2 1 1 1 1 yi + hi ≤ hF and hi − yi ≤ hF ∀i, 2 2 2 2

(1)

∀i < j,

(2) (3) (4) (5)

(6) (7)

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The aim of this model is to obtain the optimal block layout of departments. The objective function (1) tires to minimize the interaction cost between departments. Constraints (2) ensure that there is no overlap between

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any two departments. Here, wi and hi are indicating the width and height of facility i, respectively. Constraints (3) and (4) limit all departments to

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be entirely embedded inside the facility. Equation (5) impose the constraint related to areas of departments. In constraint (6), βi = max{wi /hi , hi /wi }

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is called aspect ratio and represents the shape of department i (in terms of the extent which is close to square) and β ∗ shows the maximum acceptable value for all βi . The range of the dimensions of the facility are specified in constraint (7), where wF , wFmin , and wFmax are the width of the facility and its minimum and maximum acceptable values, respectively. These notations 8

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max are also used for the height of the facility; i.e. hF , hmin F , hF . It should be

noticed that center of the facility has been considered the origin of the x − y

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plane. This model is equivalent to vCCV model (named by Anjos and Vannelli[40], according to the initials of its authors’ name), which was presented

for the first time in [41]. The difference between model (1)-(7) and vCCV model is in constraint (6). Indeed, by writing this constraint in the afore-

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mentioned form, there is a better control over the shape of all departments simultaneously.

The difficulty which is encountered in solving this model arises from constraint (2), such that this constraint has made the solution process complicated for large size problems. Therefore, in order to solve the model,

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Jankovits et al. [6] proposed a framework composed of two stages. The first

difficulty.

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stage was a model named JLAV model, which coped with the aforementioned

2.2. JLAV model

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The framework presented for single-floor problems by Jankovits et al. [6] is a two-step method. The purpose of the first step (i.e., JLAV model) is

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to find the relative position of the departments within the facility, so that by knowing these relative positions, constraint (2) is converted into a linear

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constraint. The next stage is a semi-definite programming (SDP) model which determines the exact position and shape of the departments, but is not dealt with here. In this subsection, the concept and formulation of the first stage of this framework is described. Here, each department is approximated by a circle 9

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with radius ri . The distance between any two circles is considered to be the squared-Euclidian distance between their centers, i.e., Dij = (xi −xj )2 +(yi −

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yj )2 . The aim is to arrange these circles so that their arrangement can be considered as a good approximation of the departmental layout.

To achieve the layout of the circles, Anjos and Vannelli [40] presented a

model called AR (attractive-repeller). The advantage of AR model is having

only linear constraints , but nonlinear objective function. This model is given

min

(xi ,yi ),hF ,wF

X

cij Dij + f (

1≤i
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as follows: Dij ), tij

wFmin ≤ wF ≤ wFmax

∀i,

∀i,

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1 1 s.t. xi + ri ≤ wF and ri − xi ≤ wF 2 2 1 1 yi + ri ≤ hF and ri − yi ≤ hF 2 2

and hmin ≤ hF ≤ hmax F F ,

(8) (9) (10) (11)

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where f (z) = 1/z − 1 for z > 0, and tij = α(ri + rj )2 for a given α and 1 ≤ i < j ≤ N (α is overlapping parameter of the circles).

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The first term of the objective function tries to get the circles closer to each other through reducing Dij to Dij = 0; so, it acts as an attractor

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element. But, the second term pushes the circles away and prevents their overlap. In other words, this term appears as a repeller component. As a

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result of these two components, Dij recieves a balanced value and the circles are dispersed reasonably within the facility. The best separation of the circles in AR model occurs at optimality, when Dij /tij = 1. Therefore, if α = 1,

the circles should intersect at exactly one point in the optimal state. For α < 1, some overlap is allowed, while α > 1 enforces the circles to be farther. 10

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Hence,

√

tij is target distance between a pair of circles i and j, such that in

the optimal case, the distance between these two circles is equal to that.

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The presence of the second term in the objective function of AR model has turned it into a non-convex objective function. Hence, Anjos and Vannelli

[40], by dealing with those elements of the objective function that caused it

to be non-convex, convexified the model and entitled it CoAR (Convexified p AR) model. If we notice, when Dij = tij /cij there is no force between

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circles i and j. The analysis provided in [40] motivates the idea of definp ing Tij = tij /(cij + ) as generalized target distance, where adding a small number  > 0 provides the opportunity for using cij = 0. By employing Tij , CoAR model is given below:

(xi ,yi ),hF ,wF

X

Fij (xi , xj , yi , yj ),

1≤i
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(9) − (11),

s.t. where

  c D + tij − 1, ij ij Dij Fij (xi , xj , yi , yj ) =  2√c t − 1,

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(12)

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min

ij ij

Dij ≥ Tij 0 ≤ Dij < Tij .

(13)

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Indeed, this model, by using the concept of generalized target distance,

considers a constant value for the non-convex parts of the objective function.

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The readers are referred to [40] in order to find the theoretical aspect of this model. CoAR model is not a practical model in terms of computation, because a

fairly specialized algorithm is required to stop at solutions that are on or near the flat portion of the objective function, but are farthest from the origin, 11

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i.e., where Dij ≈ Tij . Hence, Anjos and Vannelli [42] proposed a new model by adding the term −Kln(Dij /Tij ) to the objective function and entitled it

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ModCoAR (Modified CoAR) model. The reason for choosing this function is inspired by the log-barrier functions in interior-point methods for convex op-

timization. This new practical model is not a convex model; however, it can

be efficiently solved and still aims to achieve the generalized target distance (i.e., Dij ≈ Tij at optimality). ModCoAR model is as follows: (xi ,yi ),hF ,wF

X

1≤i
[Fij (xi , xj , yi , yj ) − Kln(

(9) − (11),

s.t.

Dij )], Tij

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min

where

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  c D + tij − 1, ij ij Dij Fij (xi , xj , yi , yj ) =  2√c t − 1, ij ij

Dij ≥ Tij

0 ≤ Dij < Tij ,

(14)

(15)

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where constant K is a penalty factor influencing the aggregation of the circles.

(How to choose an appropriate value for constant K and other related pa-

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rameters, which will be introduced later, is discussed in Section 5). Jankovits et al. [6] adopted a new strategy to improve ModCoAR model.

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In this approach, the objective function does not improve as the circles start to overlap. This action is done based on ri + rj , such that α is removed from

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tij (but, it is taken into account in the objective function in another way), i.e., tij = (ri + rj )2 . Also, the new target distance τij and parameter νij are defined. This strategy finally leads to the subsequent model (which is called

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JLAV model):

(xi ,yi ),hF ,wF

1≤i
[Fij (xi , xj , yi , yj ) − Kln(

Dij )], tij

(9) − (11),

where   νij =    τij = 

cij tij , √ 2 cij tij +  − 1, tij , q

tij , cij +

tij ≥

tij ≥

tij cij +

(17)

o.w. , q

tij cij +

o.w. ,

  c D + ij ij Fij (xi , xj , yi , yj ) =  ν ,

αtij Dij

− 1,

Dij ≥ τij

0 ≤ Dij < τij .

(18)

(19)

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ij

2.2.1. Radii of circles

q

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s.t.

(16)

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X

min

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Since the circles are an approximation for the departments, their areas must depend on the areas of the departments. Earlier models calculated the p radius of circle i via ri = ai /π. But, Jankovits et al. [6] assert that using

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this expression will result in a layout with large aspect ratio. On the other

hand, by obtaining the relation βi ≤ ai /µ2i , where µi is the smallest edge

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of department i, they concluded that, through changing the smallest edge of each department, it is possible to control the upper bound of its aspect ratio.

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As is known, the only controllable factor of a circle is its radius. Con-

sidering the two recent conclusions leads to the idea of making the radius of the circles dependent on the smallest desired edge of departments. The

intuition that large departments require larger circles than the small ones (to p form themselves into near-square shapes) leads to ri = ai /π log2 (1 + φa2i ), 13

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where parameter φ is the smallest desired edge for all departments. Since all circles have been enlarged, facility dimensions are also adjusted through ai )} φ2

(i.e. to the extent that the largest

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multiplying by χ = maxi {log2 (1 +

circle have been enlarged), in order for the circles to be nicely spread within the facility. In fact, this parameter will contribute to reduce overlap. As will

be seen in the next Section, it leads to obtain the departments with small aspect ratios.

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3. Proposed method for single-floor problems

In this section, the new framework is presented for single-floor problems, which is a two-stage method. At the first stage, the same approach as JLAV model is employed, but by some improvements. The improved model is called

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IJLAV (improved JLAV). At the second stage, a nonlinear programming

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model is employed to determine the final layout. 3.1. First stage: IJLAV model

As observed in the last section, increasing the radius of each circle must

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help the corresponding department to reach a near-square shape. In other words, if the aim is to reduce the aspect ratio and, equivalently, to enlarge the

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smallest edge of departments, the radii of the circles should be enlarged. But, p in the term ri = ai /π log2 (1 + φa2i ), any increase in φ (the smallest desired

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edge for departments) will result in the decreased radii of the circles; i.e. the

following illogical relationships are established between the parameters: φ ↓⇒ radius ↑⇒ area ↑⇒ aspect ratio ↓

φ ↑⇒ radius ↓⇒ area ↓⇒ aspect ratio ↑ . 14

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(20)

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Consequently, the radius of the circles is modified as: r ai ai log2 (1 + ri = ). π mini {ai } − φ2

This modification leads to the following relations among the parameters: φ ↓ (↑) ⇒ radius ↓ (↑) ⇒ area ↓ (↑) ⇒ aspect ratio ↑ (↓)

Moreover, parameter χ is modified as χ = averagei (log2 (1 + mini {aaii }−φ2 )). more than the required level.

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Because, in JLAV model, the dimensions of the facility have been enlarged

Also, Jankovits et al. [6] fixed φ = 2 for all problems (except for the problems in which the smallest edge of the departments is already known). This issue debilitates the model in obtaining a low aspect ratio. Hence, a

edge of the departments.

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suitable procedure is presented for providing a better control over the smallest

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As previously mentioned, φ has a tight relationship with βi . On the

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other hand, if the small edge of department i is hi (or wi ), we have hi = p p ai /βi (or wi = ai /βi ). Meaning that the small edge of department i p is always equal to ai /βi . Replacing βi and ai with β = maxi {βi } and

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mini {ai }, respectively, gives a lower bound for the edge size of all depart-

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ments; i.e., the following can be given: s mini {ai } wi , hi ≥ ∀i. β As a result, φ can be defined as φ =

p mini {ai }/β and. Now, it is pos-

sible to directly control the aspect ratio by setting a value for β, instead of controlling it via φ. It should be noticed that β = 4, for example, does not 15

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'$

∆x



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i

@ @  &% ∆y @ @j

Figure 1: The relative position of departments i and j

mean that the final aspect ratio (i.e., β ∗ ) will be equal to 4; instead, it is a

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parameter to control radii of the circles and subsequently β ∗ .

3.2. Second stage: Nonlinear programming model to obtain final layout By obtaining the solution of IJLAV model, the relative position of the departments to each other is determined using the same approach presented in [6]. In this methodology, if circles i and j have the position illustrated

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in Figure (1) and ∆x ≤ ∆y (∆x ≥ ∆y), it is supposed that these two departments are vertically (horizontally) separated and department i is above

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(to the left of) department j. Knowing this information, one of the two inequalities of Constraint (2) can be selected and its absolute values can be

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eliminated.

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After resolving the difficulty of Constraint (2), the final layout can be

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found using the following nonlinear programming model1 : X

min

cij (uij + vij ),

(21)

1≤i
s.t. uij ≥ xi − xj

∀i < j,

uij ≥ xj − xi

∀i < j,

vij ≥ yi − yj

∀i < j,

vij ≥ yj − yi

∀i < j,

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(xi ,yi ),hi ,wi ,hF ,wF

(22)

(23)

(24)

(25)

1 1 1 1 w i − xi ≤ w F xi + wi ≤ wF and 2 2 2 2 1 1 1 1 yi + hi ≤ hF and hi − yi ≤ hF 2 2 2 2

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wi hi = ai

∀i,

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wi − β ∗ hi ≤ 0 and hi − β ∗ wi ≤ 0 wFmin ≤ wF ≤ wFmax

∀i,

∀i,

∀i,

and hmin ≤ hF ≤ hmax F F ,

(26) (27) (28) (29) (30)

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Of course, this model does not contain non-overlapping constraints. Because, as explained above, according to the relative position of the circles

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associated with each two departments, we should select one of the two in-

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equalities in Constraint (2) and appropriately set aside its absolute values. 1

jankovits et al.[6] proposed an SDP model to find the final solution. But, it is note-

worthy that the constraints related to locating the departments within the facility were

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wrongly inserted in the context, because those constraints belonged to the circles, not to the departments!

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4. Proposed method for multi-floor problems In this section, the method presented in the last section (which was for

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single-floor problems) is extended to multi-floor cases. Here, it is assumed that there are K floors labeled l = 1, 2, ..., K. Floor l has area Al with

dimensions wl and hl , which can vary in a given range. Also, the vertical distance between any two adjacent floors is h. The floors interact with each

other through elevators; so, there are E elevators denoted by e = 1, 2, ..., E.

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Costs per unit distance between a pair of departments i and j in vertical and horizontal directions are shown by cVij and cH ij , respectively. As well as, it is assumed that all elevators are identical and each of them can provide service for all floors. Other notations and assumptions are similar to those of the single-floor version.

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Two versions of this problem are investigated. In the first case, it is supposed that the elevators are fixed, i.e., their locations are predetermined.

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In the second state, the location of elevators is considered as decision variable, which must be determined. The aim is to determine the floor of all depart-

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ments, their locations and dimensions within the floors ; and also the location of elevators in the second version.

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The first difference between single- and multi-floor problems is in estab-

lishing the floor related to each department. Therefore, the departments are

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first assigned to the floors, then the interior layout of the floors is specified. Indeed, the present framework is composed of three steps: (1) Assigning de-

partments to floors, (2) Finding relative position of departments within the floors, and (3) Determining the final layout for each floor. The first stage is identical for both versions. So, this stage is first de18

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scribed; afterwards, other stages related to each of the versions are separately explained.

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4.1. Stage one: Assigning departments to floors

At this stage, FAF model presented in [37] is employed under the notions

of [32], in order to allocate the departments to the floors through minimizing the flow cost in vertical direction. This model is as follows: min

X

Vij ,

(31)

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1≤i
s.t. Vij ≥ (yi − yj )hcVij and Vij ≥ (yj − yi )hcVij

l=1 N X i=1

∀i,

lxil = yi

∀i,

xil = 1 ai xil ≤ Al

∀l,

(32) (33) (34) (35)

if department i is assigned to floor l

(36)

o.w.

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  1, xil =  0,

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l=1 K X

ED

K X

∀i, j,

where yi is a variable which represents the floor of department i, and recieves

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one of the numbers in set {1, 2, ..., K}. The objective function (31) is originally

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1≤i
|yi − yj |hcVij , which has

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been converted to a linear form using Constraints (32). Constraint (33) establishes the relationship between variables yi and xil . Each department should only be assigned to one floor in Constraint (34). Constraint (35) does not allow many departments to be assigned to a floor more than its available area. 19

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4.2. Multi-floor problems with fixed elevators 4.2.1. Stage two: Finding relative position of departments

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The purpose of this stage is to determine the relative position of the departments within the floors by approximating them as circle (for the same reasons explained for the single-floor version).

Due to the effects of parameters K and α on dispersion of the circles, these

parameters are set individually for each floor, which provide better control

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over the arrangement of the circles within the floors. Therefore, in order to obtain the flow cost between the same-floor departments (the departments

located on the same floor), an objective function as the objective function of IJLAV model is considered for each floor. Parameter β is considered to be identical for all floors, because of the desire to control their aspect ratios

M

simultaneously. Another advantage of this policy is the reduced number of parameters.

ED

The distance between any two same-floor departments is calculated as the horizontal distance from their centers. But, for different-floor departments

PT

(the departments situated on different floors), it consists of both horizontal and vertical distances. Indeed, in order to move from department i to de-

CE

partment j (which are located on different floors), movement should be from department i to one of the elevators, and after arriving at the desired floor

AC

(using the elevator), department j should be headed for. Since attempts are made to reduce distances, for the purpose of minimizing costs, it is necessary to select an elevator such that the aforementioned distance gets the lowest possible value. So, the distance between any two different-floor departments V i and j can be formulated as dH ij + dij = min{die + dej } + h|yi − yj |, where e

20

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V dH ij , dij and die are horizontal and vertical distances between departments i

and j, and distance from department i to elevator e, respectively.

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Since the departments are not allowed to be exchanged among the floors, the vertical flow cost has a constant value (which is equal to the objective

function value of FAF model). Therefore, it is not required to be included directly in the model hereinafter. This cost is added to other layout costs at the end.

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¯ ie = (xi − Distance between circle i and elevator e can be expressed as D

¯ ij = mine {D ¯ ie + D ¯ ej } is defined as the horizontal xe )2 + (yi − ye )2 . Now, D

distance between the circles i and j situated on different-floors. This equation equivalent to: ∀e.

(37)

M

¯ ej D¯ij ≤ D¯ie + D

¯ ij = 0, a negative value is set as the coefficient of In order to prevent D

ED

¯ ij is to get its this variable in the objective function, because the aim for D upper bound, according to Constraint (37), to express the smallest horizontal

AC

CE

PT

distance between circles i and j. Hence, the model proposed for finding the

21

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layout of the circles is as follows: X

¯ ij ≤ D ¯ ie + D ¯ ej s.t. D

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Dij )] + ... [Fij1 (xi , xj , yi , yj ) − K 1 ( (xi ,yi ),hF ,wF tij i
∀e and i < j on different floors,

(9) − (11), where

ij

αl tij Dij

− 1,

M

  cH D + ij ij Fijl (xi , xj , yi , yj ) =  v ,

(39)

τij ≤ Dij

τij > Dij .

ED

Here, αl and K l are the parameters associated with floor l. The objective function (38) consists of an objective function of single-floor

PT

version problem (i.e. Equation (16)) for each of the floors, in addition to a ¯ ij = 0. Constraint (39) is the extension of penalty term in order to prevent D

CE

inequality (37) for all i and j. The main problem in [32] was how to model the constraints related to

AC

horizontal distance between different-floor departments (or circles) and they finally failed to cope with this difficulty. Also, they did not directly accommodate aspect ratio in their model.

22

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4.2.2. Stage three: Determining final layout In this step, the nonlinear programming model presented in Section (3.2)

the final layout of the floors.

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is extended to multi-floor problems with fixed elevators, in order to determine

Distances for the same-floor departments are the same as those of vCCV model; but, for different-floor departments, we have:

¯ = min{|x − x | + |y − y | + |x − x | + |y − y |} D ij i e i e e j e j e

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¯ + Dy ¯ + Dx ¯ + Dy ¯ }. = min{Dx ie ie ej ej e

(40)

This equation can be rewritten using less than or equal constraints and negative coefficients for the corresponding parts in the objective function (i.e.,

ij

M

similar to inequality (37)). On the other hand, in order to prevent the ar¯ via increasing the right hand side of its constraint, bitrary increasing of D some penalty is assigned, in the objective function, to the variables located

AC

CE

PT

ED

on the right hand side of that constraint (see the third term of the objective

23

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function (41)). Finally, the intended model can be formulated as follows:

(xi ,yi ),hi ,wi ,hF ,wF i
+M

on same floors

X

cH ij (uij + vij ) −

¯ + Dy ¯ ), (Dx ie ie

i
X

¯ cH ij Dij

on different floors

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X

min

(41)

i,e

¯ ≤ Dx ¯ ¯ ie + Dy ¯ ie + Dx ¯ ej + Dy s.t. D ij ej

∀ e and i < j on different floors, (42)

¯ ≥x −x Dx ie i e ¯ ≥x −x Dx ie e i

∀ i and e,

¯ ≥y −y Dy ie i e

∀ i and e,

¯ ≥y −y Dy ie e i

∀ i and e,

(43) (44)

(45) (46)

∀ i < j on same floor,

M

(22) − (25)

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∀ i and e,

ED

(26) − (30),

where M is a penalty constant value.

PT

This model is an extension of the second stage model in the single-floor problem. The aim of adding the second term to the objective function is similar to that of the objective function (38), i.e. in order to satisfy Equation (40)

CE

in the form of Constraint (42) ,which is its expanded form. Also, Constraints (43)-(46) are added to eliminate the absolute values of Equation (40).

AC

Finally, by subtracting the penalties and adding the interaction costs in

vertical direction, the cost of the obtained layout (Zf inal ) can be calculated

24

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as follows:

i
X

on different floors

¯ cH ij Dij − M

X

¯ + Dy ¯ )+Z (Dx ie ie F AF ,

(47)

i,e

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Zf inal = Z + 2

where Z and ZF AF are the objective function values for the last model and FAF model, respectively. Coefficient 2 in the second term is because of the

negative coefficient of that term in objective function (41); indeed, this cost

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has been subtracted once from Z, so it should be added twice now. 4.3. Multi-floor problems with variable elevators

In this subsection, an advanced version of the previous problem is examined, in which the location of elevators should also be established. Therefore,

M

the models of the second and the third stages differ from the former. It should be noted that the variable elevator problem refers to a problem in which the

ED

location of the elevators are decision variables. 4.3.1. Stage two: Finding relative positions of departments

PT

In the new problem, due to unknown location of elevators and further ¯ ij in the objective function, we take the logarithm (a concave freeness of D

CE

function) of the flow cost of different-floor departments (see the forth term of the objective function (48)). This action decreases the intensity of the ¯ ij . Also, in order to prevent the elevators objective function to increase D

AC

¯ ij , a penalty from being located on each other, which leads to increasing of D

is assigned to the objective function, when this happens. In other words, the aim is to disperse the elevators within the facility (see the fifth term of the objective function (48)). The intended model is given as follows: 25

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X

i
on different floors X

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Dij [Fij1 (xi , xj , yi , yj ) − K 1 ( )] + ... tij i
(xi ,yi ),hF ,wF

¯ ie + D ¯ ej ) (D e and i
(49) (50) (51)

ED

M

+

(48)

e
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min

(9) − (11),

PT

¯ ee0 = (xe − xe0 )2 + (ye − ye0 )2 is the distance between elevators e where D and e0 . That term of objective function (48) which has log10 depends on

CE

¯ ij , and its own D ¯ ij also depends on the distance of departments from the D ¯ ij , through locatnearest elevator. As a result, the model tries to increase D

AC

ing all elevators on each other. But, adding the next term to the objective function does not allow it to happen. Using a concave function (i.e. ln) in this term will result in a good dispersion of the elevators throughout the facility; because if a non-concave function is used, the elevators are divided

into two groups, such that the elevators of each group are exactly situated 26

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on each other. Also, these two groups of elevators are located at the farthest two points, i.e., two opposite corners of the facility. This fact is easier to

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comprehend when studying the features of concave functions in a minimization knapsack problem subjected to the following conditions: (1) All variables are nonzero, and (2) Objective function value is positive for all values of the

variables. In optimal solution of this problem, all variables have positive and moderate values [43].

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The last term of the objective function is added to prevent an arbitrary

increase of the right hand side of Constraint (51). Without this penalty ¯ ij in the objective function will lead to term, the negative coefficient of D its increase, which is not favorable. Constraints (49) and (50) enforce the

M

elevators to be located inside the facility

4.4. Stage three: Determining final layout

ED

In order to determine the final layout, the model developed for the third stage of fixed elevators problem can be extended using the concepts provided in the last model (which determined the layout of circles). This model will

AC

CE

PT

be as follows:

27

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min

on same floors X X X ¯ ¯ + Dy ¯ ) − ln ¯ ee0 , cH (Dx D ie ie ij Dij + M i,e e
i
− log10

i
cH ij (uij + vij )

s.t. (22) − (25)

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(xi ,yi ),hi ,wi ,hF ,wF

X

∀ i < j on same floor,

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(22) − (30), (42) − (46), (49) − (50),

With regard to the previous description and similarity of this model to model

M

of the fixed elevators problem, the presented model will be intelligible. The only difference is that the new model contains an additional term in the

ED

objective function, in order to disperse the elevators throughout the facility (with the same meaning described for model of the circles). Also, here, there

PT

are the constraints to limit the elevators to be located inside the facility. By subtracting the penalties and adding other related costs, similar to Equation

CE

(47), cost of the final layout will be calculated as follows:

AC

Zf inal = Z + log10

X

¯ cH ij Dij +

X

on different i
i
e
28

¯ cH ij Dij

(53)

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5. Computational results In order to solve the presented models, CPLEX 12.5.0.0 and KNITRO

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8.0.0 solvers were employed for the first stage and for the second and third stages, respectively, through GAMS modeling language. All tests were exe-

cuted on a system equipped with an Intel Core2Duo 2.40 GHz CPU and 3 GB of RAM while utilizing Microsoft Windows Vista.

In order to solve the second stage model in multi-floor (and equivalently

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the first stage model in single-floor) cases, an initial solution is required for the circles. Hence, for single-floor problems, the manner proposed in [42] was adopted, in which the centers of the circles are placed with equal intervals around a large circle whose radius is (wF +hF ). Therefore, the initial solution

xi = (wF + hF ) cos θi

M

for circle i can be expressed as follows:

and yi = (wF + hF ) sin θi ,

(54)

ED

where θi = 2π(i − 1)/N . But, for multi-floor problems, using initial random solutions could provide better layouts.

So, the coordinates of the

PT

circles’ centers were set via xi = χ unif orm(−wF /2 + ri , wF /2 − ri ) and yi = χ unif orm(−hF /2 + ri , hF /2 − ri ). Moreover, this new procedure could

CE

easily handle bad scale problems, while the previous way got into trouble in these situations, such that KNITRO solver could not continue the solution process. It also should be noticed that, at the second stage of fixed elevator

AC

cases, the coordinates of the elevators were scaled by χ. In order to obtain better results, an initial solution was defined for elevators in variable elevator

cases as follows: xe = R cos θe and ye = R sin θe and R = 0.8 min{χwF /2, χhF /w}. 29

(55)

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Indeed, due to the sameness of elevators, this way is similar to the method used for single-floor cases. Except that the radius of the large circle is smaller,

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such that all elevators will be imbedded inside the facility. 5.1. Tuning the parameters

Value of parameters K, α, and β had a tremendous effect on quality and cost of the final solution. Parameters K and α were related to dispersion and

overlapping of the circles, respectively, such that a small value for K put the

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circles on each other and a large value pushed them away toward the bounds of the facility. Also, the higher the value of α, the less the overlap would be.

The second modification in IJLAV model provided a basis for establishing a new systematic procedure for tuning the aforementioned parameters.

M

Dimensions of facility, in IJLAV model, were enlarged enough, such that if the circles were ideally dispersed (i.e. with small overlapping and good dis-

ED

persion), almost all the facility regions would be covered with the circles. This fact was used to set the parameters. First, initial values were arbitrarily assigned to K, α and β. Then, K was

PT

adjusted so that the circles were arranged with a proper dispersion. Next, α was amended such that the objective (i.e. covering most of the facility

CE

regions) was fulfilled. At this time, in order to achieve a better layout for the circles, α and K were modified by small changes. After reaching the

AC

desired layout, β could be altered to the extent of decimal values. Figure (2)

illustrates this way. In order to find other layouts, another value can be set to β and the procedure can be repeated. In fact, a good layout cost can be found for another arrangement of the circles. In other words, the proposed strategy is a sufficient, but not the necessary, condition for obtaining a good 30

(a) K = 660, α = 664

(b) K = 3960, α = 664 (c) K = 3960, α = 1860 Cost=21889.425

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Cost=23554.314

Cost=24089.612

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Figure 2: An example for setting the parameters (β

=

2). The cost shown is cost of the final layout.

solution.

The experiments showed that the closer the value of β to 1, the easied

M

the finding of layouts with low cost would be. Also, for small values of β, parameters α and K required larger values. Moreover, these two parameters

ED

depended on density of the flows between the departments, while β did not. In some problems, a number of circles may not accommodate within the

PT

facility because of the large size of their radii; so, the model would not be infeasible. To obviate this issue, radii of all the circles should be multiplied

CE

by a positive and less than one number.

AC

5.2. Single-floor instances Since the proposed framework for single-floor problems was obtained by

modifying the method presented in [6], most of the current attention in this subsection is on spotlighting the superiority of the improved method over the previous one. Hence, the problems solved in [6] were employed. However,

31

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Table 1: Comparison of presented framework for small size problems

Best cost by

Best cost in

Optimal or

Gap (%) (to

our frame-

[6] (less than

best known

best known

work (less

1.36 s)

cost (re-

solution)

than 0.48 s)

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Instance

ported in [6])

264.50

10-department [41] 23892.93 280.04

FO10 [45]

31.94

FO11 [45]

33.11

235.95

20396.19

17.14

320.07

238.27

17.53

35.71

29.41

8.60

35.33

33.93

-2.42 (im-

7416.00

5004.00

provement) 21.05

ED

14-department [46] 6057.49

12.10

29193

M

O10 [45]

298.30

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9-department [44]

due to the large number of problems, their details are not described here.

PT

Table (1) illustrates the results for some small size problems. The time consumed to obtain the presented solutions (measured in sec) is shown in the

CE

header of the table by character ”s”. As can be seen in the table, the proposed framework significantly has outperformed the framework of Jankovits

AC

et al. [6]. Even for problem FO11, a solution has been obtained which is better than the best known solution. Tables (2)-(4) are related to three large size problems, where Armour

and Buffa [47] and Nagent [48] with 20 and 30 departments, respectively, are the best known large benchmarks in FLP field. The displayed aspect

32

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Table 2: Comparison of presented framework for Armour and Buffa problem

Best cost

Best cost in

Best cost in

Best cost in

Best cost in

by our

[49] (2640

[50]

[42] (18 s)

[6] (17.37 s)

framework

s)

2708.0

(0.61 s)

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β∗

2853.3

-

-

-

5

2817.6

5524.7

5397.6

4591.3

4

2875.1

5743.1

3

2986.6

5832.6

2

2960.6

6171.1

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6

3009

5370.6

4786.4

2960.5

5594.3

5140.1

*

6023.2

5224.7

*

M

* This framework has not been able to achieve the given β ∗

ratios in the tables show the rounded values to the nearest integer amount.

ED

Also, the provided computational time shows the average time consumed for all the presented solutions. Figure (3) displays a layout of the circles and

PT

departments for Armour and Buffa problem. Tables (2)-(4) indicate that the proposed method provides better solu-

CE

tions within very little time than the other methods, even for large size problems. Indeed, the nonlinear programming model presented in the sec-

AC

ond stage and also the systematic method for tuning the parameters have enhanced the proposed framework in terms of cost and computational time. On the other hand, the first modification in IJLAV model not only has helped to attain low aspect ratios, but also rarely a layout with high aspect ratio is obtained. Another evidence which proves the priority of the presented 33

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Table 3: Comparison of presented framework for Nugent problem

Best cost by

Best cost in

Best cost in

Best cost in [6]

our framework

[51] (876.6 s)

[52]

(258.1 s)

(1.8 s) 20565.4

-

-

5

20337.9

-

-

4

20627.0

-

-

3

20684.8

-

2

20825.0

23416.5

AN US

6

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β∗

23770 24916 25000

-

*

21560.6

*

M

* This framework has not been able to achieve the given β ∗

Best cost by our framework

Best cost in [6] (180-300 s)

PT

β∗

ED

Table 4: Comparison of presented framework for problem B with 30 department [6]

(1.02 s) 9188.4

10604

CE

6

9243.2

10424

4

9186.9

10199

AC

5

3

9296.9

*

2

9421.0

*

* This framework has not been able to acheive the given β ∗

34

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(b) Final layout

cles (first stage)

(second stage)

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(a) Layout of cir-

Figure 3: Solutions of Armour and Buffa problem (β ∗ = 2 and cost=2960.6)

method is that this framework provided good layouts by most values of the

M

parameter and rarely yielded a poor layout.

5.3. Multi-floor with fixed elevators instances

ED

5.3.1. Instance 1: 15-department problem The first instance is a 15-department and 3-floor problem with 6 fixed

PT

elevators, in which department 15 is fixed on floor one at the bottom right hand corner with size 5*5 [34]. Area of the departments can vary in a given

CE

range, and their shapes are not necessarily rectangle. Here, the same as Bernardi and Anjos [32], value of the department’s area is fixed as the final layout obtained in [34].

AC

The best solution of MULTIPLE (name of the method presented by Bozer

et al. [34]), for this problem, was obtained in 37.9 sec with cost 125,822.50 and without any shape constraints. The results provided in Table (5) indicate that the proposed framework has performed 12.05% better than MULTIPLE,

35

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Table 5: Comparison of presented framework for15-department problem with fixed elevators

Framework of [32]

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Our framework β ∗ (f1, f2,

Best cost

β ∗ (f1, f2,

Best cost

Cost im-

f3)

(0.62 s)

f3)

(0.88 s)

provement (%)

110 658.625

7.06, 4.42,

123 501.53

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3.12, 5.00, 4.17

10.40

5.76

3.12, 4.17,

110 772.002

8.00, 7.05,

4.17

124 763.39

11.21

126 936.07

4.08

4.32

3.12, 2.06,

3.14, 4.26,

121 752.824

4.32

M

3.23

ED

even within less computational time.

Three results of [32] are illustrated in Table (5), such that the first two

PT

solutions are the lowest obtained costs, and the third one is the lowest aspect ratio achieved for all floors. As can be seen, the proposed method not only

CE

could generate layouts with low cost in less computational time, but also has obtained smaller aspect ratios. For example, it has generated a layout with maximum aspect ratio of 3.227, which is a very low value, while its cost is

AC

also less than the lowest cost reported in [32]. 5.3.2. Instance 2: 40-department problem The next instance is a 40-department and 4-floor problem with 3 fixed elevators, in which department 40 having a non-rectangular shape is fixed 36

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Table 6: Comparison of presented framework for 40-department problem with fixed elevators

Framework of [32]

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Our framework β ∗ (f1, f2,

Best cost

β ∗ (f1, f2,

Best cost

Cost im-

f3, f4)

(1.9 s)

f3, f4)

(2.48 s)

provement (%)

14115.248

4.26, 7.86,

4.50, 4.50 2.76, 3.70,

20446.32

AN US

2.76, 3.70,

31.0

8.00, 7.99 14135.511

3.25, 4.53,

3.70, 3.70

21704.10

34.9

5.00, 5.00

M

on the first floor [53]. Since the proposed framework requires rectangular departments, the assumptions taken by Bernardi and Anjos [32] are adopted.

ED

MULTIPLE and SABLE (name of the method presented by Meller and Bozer [36]) achieved the costs with the average and standard deviation of

PT

(23348.29,2355.15) and (21622.7,423.2) for 10 initial layouts, respectively. Their best cost was 20441.46.

CE

Table (6) shows the strong performance of the proposed framework in terms of cost, aspect ratio and computational time, in which two of the best solutions for cost and aspect ratio are presented. Comparison of cost improve-

AC

ment in this problem with the previous one confirms the fact that capability of the proposed framework is more prominent for large problems. This feature is another advantage of the proposed method in terms of practicality,

due to the large size of real-world problems.

37

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Also, all runs of the proposed framework yielded feasible solutions unless aiming to achieve a very low aspect ratio, e.g. 3 or 4, while feasible solutions

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of [32] were few, even for large aspect ratios. For example, in 30 iterations for β ∗ = 5 and β ∗ = 8, only 3 and 12 iterations produced feasible layouts, respectively. 5.4. Multi-floor with variable elevators instances 5.4.1. Instance 1: 11-department problem

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The first instance is a 11-department problem, named Irohara11F3 [13],

with 3 floors and 2 elevators. All departments are rectangular with fixed dimensions and maximum aspect ratio of 1.3, because the method presented by Goetschalckx and Irohara [13] could not determine dimensions of depart-

M

ments. Also, they assumed that elevators should be located at the boundaries of departments. Here, these two assumptions are not taken into considera-

ED

tion.

The best cost reported for this problem in [13] is 123,319.55, which has bee attained in 10998 sec. In fact, their method provided only one layout by

PT

consuming a large amount of time, even for a small size problem. Table (7) shows the potency of the presented framework to obtain a variety of layouts

CE

with lower costs within the smallest amount of time. Although, the aspect

AC

ratios are larger than those of Irohara’s layout. 5.4.2. Instance 2 and 3: 15- and 40-department problems Here, the 15- and 40-department problems which were previously solved

are deployed again, by this difference that the locations of their elevators are

considered as decision variable. The last columns of Tables (8) and (9) show 38

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Table 7: Results of presented framework for 11department problem with variable elevators

Cost of our solution (0.58 s)

6.27, 5.56, 5.61 116875.015 4.10, 4.10, 4.10 117709.404 3.10, 3.10, 3.10 118987.475

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β ∗ (f1, f2, f3)

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Table 8: Results of presented framework for 15department problem with variable elevators

β ∗ (f1, f2, f3)

Cost of our solution (1.58 s)

108294.959

3.12, 4.30, 4.17

108372.726

12.3

ED

M

3.12, 5.00, 4.17

Cost improvement (%)

the percentage of cost improvement for our best solution compared with the

PT

best solution reported in [32] (in which the elevators were fixed), even though

CE

our aspect ratios are considerably less than theirs. 5.5. Instance 4 and 5: New instances

AC

Finally, to demonstrate the capability of the presented framework in more

complex cases, two new hard problems are generated. The first one is made by adding 10 additional departments to 40-department problem in the last subsection. The area of these 10 new departments (numbered from 41 to 50) and their flows can be seen in Tables (10) and (11), respectively. The flows 39

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Table 9: Results of presented framework for 40department problem with variable elevators

β ∗ (f1, f2, f3, f4)

Cost improvement (%)

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Cost of our solution (1.58 s)

4.51, 3.85, 2.83, 4.51

11637.550

43.1

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Table 10: The area of new departments in 50-department problem

Department

41

42

43

44

45

46

47

48

49

50

Area

8

8

4

12

12

12

16

4

4

20

M

are randomly generated, so that the flow density remains constant. Also, each floor is a square with the area of 139. The results for three layouts of

ED

this problem are illustrated in Table (12). The second new instance is the same as 40-department problem with the difference that all floors are square with different areas of 95, 115, 106, and

PT

104, respectively, except that the floor one has a rectangular empty space with width 3.8 and height 1.05 on the right hand up corner. In order to en-

CE

able the elevators to serve all floors, they must be located inside the smallest floor, i.e. the ground floor. This instance is called 40C-department problem

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.

Table (13) and Figure (4) display the result and shape of the best layout

obtained for this problem, respectively. These two problems demonstrates the strength and flexibility of the proposed methodology for large and complicated problems in addition to its fast computation and consistency with 40

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Table 11: The flows generated for new departments in 50-department problem

Flow i to j

Flow i to j

Flow i to j

41 to 1

7

43 to 49

13

46 to 26

8

41 to 3

15

44 to 23 5

46 to 27

70

41 to 17

25

44 to 26

8

46 to 32

22

41 to 44

17

44 to 48

8

46 to 50

13

41 to 45

12

45 to 6

16

47 to 1

4

42 to 6

81

45 to 33 12

47 to 2

42 to 8

17

45 to 47 8

42 to 24

19

46 to 8

43 to 40

48

46 to 20

Flow

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i to j

13

49 to 30

15

49 to 33

33

50 to 6

15

50 to 14

12

19

50 to 16

18

48 to 23

12

50 to 19

11

17

48 to 50

10

50 to 35

16

14

49 to 18

95

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49 to 27

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Table 12: Results of presented framework for 50-department problem

Cost of our solution (5.8 s)

3.20, 4.14, 4.96, 3.80

18462.844

3.20, 3.97, 3.97, 3.79

18564.850

2.73, 2.73, 2.73, 2.73

19043.895

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β ∗ (f1, f2, f3, f4)

Table 13: Results of presented framework for 40C-department problem

β ∗ (f1, f2, f3, f4)

Cost of our solution (3.9 s)

3.80, 5.00, 5.00, 2.84

11426.05

41

(a) Floor 1

(c) Floor3

(b) Floor2

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(d) Floor4

Figure 4: The best layout obtained for 40C-department problem

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quality of the layout. 6. Conclusions and future research

This paper proposed a three-stage mathematical programming method for multi-floor facility layout problems. At the first stage, the departments

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were assigned to the floors; at the second the relative position of the departments on each floor was determined; and finally, at the third stage the

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layout of the departments and also the location of the elevators (if they were decision variable) were established. Moreover, this method could easily be

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implemented for single-floor problems, as a special case of multi-floor problems. Computational results corroborated the capability of this framework

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in finding a wide variety of layouts with low and competitive costs and also aspect ratios within the least possible time, even for large size problems,

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in comparison with other existing methods in the literature. Furthermore, using mathematical programming approach for finding the solution, made this framework flexible enough to easily accommodate different real-life con-

straints, including fixed location departments, non-rectangular floors, floors with different areas, fixed shape departments, limitations to locations of ele42

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vators and departments (due to technical constraints), and so on. The wide variety of layouts provided by this framework proposes a large number of al-

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ternatives for designers and decision makers to select a suitable layout which encompasses other aspects and limitations of their design. Therefore, it can

be claimed that this method is easily applicable to real-life problems, which is a great advantage from the managerial perspective.

In summary, the managers and practitioners, using this method, can

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effortlessly provide an efficient and effective plant configuration for their

firm/organization. Utilizing this technique would enable the firms/organizations to reach a smooth flow of work, material, information, and personnel throughout the system. This aim is attained through reducing unnecessary material handling, minimizing production/service delays, avoiding bottlenecks, using

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the available area effectively, preventing unnecessary and costly changes, decreasing process inventory, etc.

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Determining the relative position of elevators at the second stage can be considered in future work. This matter may improve the final layout

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cost. Also, combining the propsed framework with other methods (such as MULTIPLE), to investigate the exchange of departments between different

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floors, may result in cost saving, especially when vertical direction cost is considerable. Finally, taking into account the area of elevators and their capacities, determining their numbers (while considering establishment cost),

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and using several types of elevators can be investigated in future work.

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