Applied Mathematical Modelling 40 (2016) 5605–5620

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier.com/locate/apm

An efficient multiple-stage mathematical programming

method for advanced single and multi-floor facility layout

problems

Abbas Ahmadi ∗, Mohammad Reza Akbari Jokar

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

a r t i c l e i n f o

Article history:

Received 11 March 2014

Revised 23 December 2015

Accepted 13 January 2016

Available online 22 January 2016

Keywords:

Facility layout

Single floor

Multi-floor

Optimization

Nonlinear programming

Mixed-integer programming

a b s t r a c t

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 organization has turned FLP into an important issue. This paper

presents a three- (two-) stage mathematical programming method to find competitive so-

lutions for multi- (single-) floor problems. At the first stage, the departments are assigned

to the floors through a mixed integer programming model (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 departments on each floor; and at the third stage, the

final layouts within the floors are determined, through another nonlinear programming

model. The multi-floor version is studied in the states in which the locations of the eleva-

tors are either formerly specified or not. Computational results show that 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 especially 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.

© 2016 Elsevier Inc. All rights reserved.

1. Introduction

A facility refers to anything that facilitates the work; in a manufacturing environment, it may be a workstation, a ware-

house, a department, a machine tool, etc. Facility layout is the arrangement of all facilities needed for producing a product or

delivering a service [1] . Facility layout problem has many applications in the real world, including layout design for manufac-

turing systems, hospitals, schools and airports, printed circuit board, backboard wiring problems, typewriters, warehouses,

hydraulic turbine design, and so on [2–4] . A suitable structure for facility layout can be beneficial for any organization.

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

expenses by up 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

∗ Corresponding author. Tel.: +98 912448-6907.

E-mail addresses: abbas_ahmadi@alum.sharif.edu , a.ahmadi@engineer.com (A. Ahmadi), reza.akbari@sharif.edu (M.R. Akbari Jokar).

http://dx.doi.org/10.1016/j.apm.2016.01.014

S0307-904X(16)30 0 02-6/© 2016 Elsevier Inc. All rights reserved.

5606 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

constraints, a given qualitative or quantitative objective function would be optimized [6] . This objective function might be

minimization of material handling costs [7,8] , maximization of desirable relations between departments [9] , or a combina-

tion 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 structures.

Quadratic assignment problem (QAP) is a restricted version of single floor FLP, in which the shapes of all departments

are identical and fixed (see [14–17] for example). Considering computational effort, this problem is an NP-complete prob-

lem [6,18] . Therefore, other complicated problems with unequal-area departments and further constraints, such as the ex-

istence of a number of floors and elevators (for interaction between floors), are placed in the NP-complete class. This mat-

ter has engaged many researchers to present heuristic methods such as CRAFT, ALDEP, CORELAP [19] , and planar graph

technique [16] , as well as to employ metaheuristic algorithms such as genetic algorithms, simulated-annealing algorithm,

and tabu-search algorithm [20–23] 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 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 [12,26–28] are multi-objective. Also, some models 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] , as well as regular and fixed shapes for departments or/and floors [13,32] . In this respect, a limited number

of 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] .

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 [28,35] , and

simulated-annealing algorithm [36] to find a good layout. On the other hand, the multiple-stage methods often assign the

departments to the 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) further 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 determines the block layout within

each floor, and then tries to find the location of elevators [12,37] ; while the other method only obtains the layout of depart-

ments, 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 [32,38,39] called FAF (floor assignment formulation) [38] for this

purpose. However, other techniques like K-mean algorithm [29] have also been deployed. At the next stage(s), in order to

find the detailed layout design, the techniques such as nonlinear programming [32] , simulated-annealing algorithm [36–38] ,

tabu-search algorithm [39] , and heuristic exchange procedures [29] are employed.

Most of the frameworks presented in the literature often provide one 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 single- and multi-floor cases, in which the math-

ematical 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

multi-floor problems), a mixed integer linear programming is employed to assign departments to floors by minimizing ver-

tical interaction cost between the departments. The second stage is a nonlinear programming model which establishes the

relative position of departments on each floor using the same approach employed in [6] , except that with some improve-

ments (the model presented by Jankovits et al. [6] is entitled JLAV model, originated from the name of its authors, which

was proposed for single-floor problems). At the third stage, using the solution obtained from the last stage, another non-

linear 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 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 accommodate the problems with fixed shapes and loca-

tions for departments, unequal-area and non-rectangular floors, as well as finding the location of elevators. In addition, the

presented method is capable to incorporate the elevators with pre-determined and also not-pre-determined locations.

The rest of the 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 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

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

in the literature). Afterward, JLAV model is described.

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5607

2.1. Original problem

Throughout this paper, it is assumed that there are N departments labeled i = 1 , . . . , N. Department i has a rectangular

shape with area a i and its position is expressed by the coordinates of its center, i.e., ( x i , y i ). Cost per unit distance between

departments i and j is indicated by c ij (it is assumed that c ij = c ji ). 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 into a rectangle

by adding some smaller rectangular shapes). It is notable that the flows between the departments are applied through c ij .

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

(x i ,y i ) ,h i ,w i ,h F ,w F

∑

1 ≤i< j≤N

c i j (| x i − x j | + | y i − y j | ) , (1)

s.t. | x i − x j | ≥ 1

2

(w i + w j ) or | y i − y j | ≥ 1

2

(h i + h j ) ∀ i < j, (2)

x i +

1

2

w i ≤1

2

w F and

1

2

w i − x i ≤1

2

w F ∀ i, (3)

y i +

1

2

h i ≤1

2

h F and

1

2

h i − y i ≤1

2

h F ∀ i, (4)

w i h i = a i ∀ i, (5)

βi ≤ β∗ ∀ i, (6)

w

min F ≤ w F ≤ w

max F and h

min F ≤ h F ≤ h

max F . (7)

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 any two departments.

Here, w i and h i are indicating the width and height of facility i , respectively. Constraints (3) and (4) limit all departments

to be entirely embedded inside the facility. Eq. (5) impose the constraint related to areas of departments. In constraint (6) ,

βi = max { w i /h i , h i /w i } 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 β i . The range of the dimensions of the facility are specified

in constraint (7) , where w F , w

min F

and w

max F

are the width of the facility and its minimum and maximum acceptable values,

respectively. These notations are also used for the height of the facility; i.e. h F , h min F

and h max F

. It should be noticed that

center of the facility has been considered the origin of the x − y 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 aforementioned form, there is a better control over the shapes 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, Jankovits et al. [6] proposed

a framework composed of two stages. The first stage was a model named JLAV model, which coped with the aforementioned

difficulty.

2.2. JLAV model

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 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 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 with radius r i . The distance between any two circles is considered to be the squared-Euclidean

distance between their centers, i.e., D i j = (x i − x j ) 2 + (y i − y j )

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 ad-

vantage of AR model is having only linear constraints , but nonlinear objective function. This model is given as follows:

min

(x i ,y i ) ,h F ,w F

∑

1 ≤i< j≤N

c i j D i j + f

(D i j

t i j

), (8)

s.t. x i + r i ≤1

w F and r i − x i ≤1

w F ∀ i, (9)

2 2

5608 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

y i + r i ≤1

2

h F and r i − y i ≤1

2

h F ∀ i, (10)

w

min F ≤ w F ≤ w

max F and h

min F ≤ h F ≤ h

max F , (11)

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

circles).

The first term of the objective function tries to get the circles closer to each other through reducing D ij to D i j = 0 ; so, it

acts as an attractor 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 result of these two components, D ij receives 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 D i j /t i j = 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. Hence, √

t i j 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.

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 AR) model. If we notice, when D i j =

√

t i j /c i j there is no force

between circles i and j . The analysis provided in [40] motivates the idea of defining T i j =

√

t i j / ( c i j + ε) as generalized target

distance , where adding a small number ε > 0 provides the opportunity for using c i j = 0 . By employing T ij , CoAR model is

given below:

min

(x i ,y i ) ,h F ,w F

∑

1 ≤i< j≤N

F i j (x i , x j , y i , y j ) ,

s.t. (9) − (11) , (12)

where

F i j (x i , x j , y i , y j ) =

{c i j D i j +

t i j

D i j − 1 , D i j ≥ T i j

2

√

c i j t i j − 1 , 0 ≤ D i j < T i j . (13)

Indeed, this model, by using the concept of generalized target distance, considers a constant value for the non-convex

parts of the objective function. 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, i.e. where D ij ≈T ij . Hence, Anjos and Vannelli [42] proposed a new model by adding the term −Kln (D i j /T i j ) to the objective function and

entitled it ModCoAR (Modified CoAR) model. The reason for choosing this function is inspired by the log-barrier functions

in interior-point methods for convex optimization. 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. D ij ≈ T ij at optimality). ModCoAR model is as

follows:

min

(x i ,y i ) ,h F ,w F

∑

1 ≤i< j≤N

[F i j (x i , x j , y i , y j ) − Kln

(D i j

T i j

)],

s.t. (9) − (11) , (14)

where

F i j (x i , x j , y i , y j ) =

{c i j D i j +

t i j

D i j − 1 , D i j ≥ T i j

2

√

c i j t i j − 1 , 0 ≤ D i j < T i j , (15)

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 parameters, which will be introduced later, is discussed in Section 5 ).

Jankovits et al. [6] adopted a new strategy to improve ModCoAR model. In this approach, the objective function does not

improve as the circles start to overlap. This action is done based on r i + r j , such that α is removed from t ij (but, it is taken

into account in the objective function in another way), i.e., t i j = (r i + r j ) 2 . Also, the new target distance τ ij and parameter ν ij

are defined. This strategy finally leads to the subsequent model (which is called JLAV model):

min

(x i ,y i ) ,h F ,w F

∑

1 ≤i< j≤N

[F i j (x i , x j , y i , y j ) − Kln

(D i j

t i j

)],

s.t. (9) − (11) , (16)

where

νi j =

{

c i j t i j , t i j ≥√

t i j

c i j + ε2

√

c i j t i j + ε − 1 , o.w., (17)

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5609

τi j =

⎧ ⎨

⎩

t i j , t i j ≥√

t i j

c i j + ε√

t i j

c i j + ε , o.w.,

(18)

F i j (x i , x j , y i , y j ) =

{c i j D i j +

αt i j

D i j − 1 , D i j ≥ τi j

νi j , 0 ≤ D i j < τi j . (19)

2.2.1. Radii of circles

Since the circles are an approximation for the departments, their areas must depend on the areas of the departments.

Earlier models calculated the radius of circle i via r i =

√

a i /π . But, Jankovits et al. [6] assert that using this expression will

result in a layout with large aspect ratio. On the other hand, by obtaining the relation βi ≤ a i /μ2 i , where μi is the smallest

edge 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.

As we know , the only controllable factor of a circle is its radius. Considering 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 form themselves into near-square shapes) leads to

r i =

√

a i /π log 2 (1 +

a i φ2 ) , where parameter φ is the smallest desired edge for all departments. Since all circles have been

enlarged, facility dimensions are also adjusted through multiplying by χ = max i { log 2 (1 +

a i φ2 ) } (i.e. to the extent that the

largest 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.

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 IJLAV (improved

JLAV). At the second stage, a nonlinear programming 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 help the corresponding department to reach a

near-square shape. In other words, if the aim is to reduce the aspect ratio and, equivalently, enlarge the smallest edge of

departments, the radii of the circles should be enlarged. But, in the term r i =

√

a i /π log 2 (1 +

a i φ2 ) , any increase in φ (the

smallest desired edge for departments) will result in the decreased radii of the circles; i.e. the following illogical relation-

ships are established among the parameters:

φ ↓⇒ radius ↑⇒ area ↑⇒ aspect ratio ↓

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

Consequently, the radius of the circles is modified as:

r i =

√

a i π

log 2

(1 +

a i min i { a i } − φ2

). (20)

This modification leads to the following relationships among the parameters:

φ ↓ (↑ ) ⇒ radius ↓ (↑ ) ⇒ area ↓ (↑ ) ⇒ aspect ratio ↑ (↓ )

Moreover, parameter χ is modified as χ = a v erage i ( log 2 (1 +

a i min i { a i }−φ2 )) . Because, in the JLAV model, the dimensions of

the facility have been enlarged more than the required level.

Also, Jankovits et al. [6] fixed φ = 2 for all problems (except for the problems in which the smallest edge of departments

is already known). This issue debilitates the model in obtaining a low aspect ratio. Hence, a suitable procedure is presented

for providing a better control over the smallest edge of departments.

As previously mentioned, φ has a tight relationship with β i . On the other hand, if the small edge of department i is

h i (or w i ), we have h i =

√

a i /βi ( or w i =

√

a i /βi ) . Meaning that the small edge of department i is always equal to √

a i /βi .

Replacing β i and a i with β = max i { βi } and min i { a i }, respectively, gives a lower bound for the edge size of all departments;

i.e., the following can be given:

w i , h i ≥√

min i { a i } β

∀ i.

As a result, φ can be defined as φ =

√

min i { a i } /β . Now, it is possible 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 mean that the final aspect

ratio (i.e., β∗) will be equal to 4; instead, it is a parameter to control the radii of the circles and subsequently β∗.

5610 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

Fig. 1. The relative position of departments i and j .

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 in Fig. 1 and x ≤ y

( x ≥ y ), it is supposed that these two departments are vertically (horizontally) separated and department i is above (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 eliminated.

After resolving the difficulty of Constraint (2) , the final layout can be found using the following nonlinear programming

model 1 :

min

(x i ,y i ) ,h i ,w i ,h F ,w F

∑

1 ≤i< j≤N

c i j (u i j + v i j ) , (21)

s.t. u i j ≥ x i − x j ∀ i < j, (22)

u i j ≥ x j − x i ∀ i < j, (23)

v i j ≥ y i − y j ∀ i < j, (24)

v i j ≥ y j − y i ∀ i < j, (25)

x i +

1

2

w i ≤1

2

w F and

1

2

w i − x i ≤1

2

w F ∀ i, (26)

y i +

1

2

h i ≤1

2

h F and

1

2

h i − y i ≤1

2

h F ∀ i, (27)

w i h i = a i ∀ i, (28)

w i − β∗h i ≤ 0 and h i − β∗w i ≤ 0 ∀ i, (29)

w

min F ≤ w F ≤ w

max F and h

min F ≤ h F ≤ h

max F . (30)

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

position of the circles associated with each two departments, we should select one of the two inequalities in Constraint (2)

and appropriately set aside its absolute values.

4. Proposed method for multi-floor problems

In this section, the method presented in the last section (which was for 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 A l with dimensions w l and h l ,

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. Costs per unit distance between a pair of

departments i and j in vertical and horizontal directions are shown by c V i j

and c H i j , 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.

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. In the second state, the location of elevators is considered as decision variable, which must be

determined. The aim is to determine the floors of all departments, their locations and dimensions within the floors ; and

also the location of elevators in the second version.

1 Jankovits et al. [6] proposed an SDP model to find the final solution. But, it is noteworthy that the constraints related to locating the departments

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

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5611

The first difference between single- and multi-floor problems is in establishing the floor related to each department.

Therefore, the departments are 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 departments to floors, (2) Finding relative position of depart-

ments 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 described; afterwards, other stages related to each of

the versions are separately explained.

4.1. Stage one: Assigning departments to floors

At this stage, FAF model presented in [38] 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

∑

1 ≤i< j≤N

V i j , (31)

s.t. V i j ≥ (y i − y j ) hc V i j and V i j ≥ (y j − y i ) hc V i j ∀ i, j, (32)

K ∑

l=1

lx il = y i ∀ i, (33)

K ∑

l=1

x il = 1 ∀ i, (34)

N ∑

i =1

a i x il ≤ A l ∀ l, (35)

x il =

{1 , if department i is assigned to floor l 0 , o.w.

(36)

where y i is a variable which represents the floor of department i , and recieves one of the numbers in set {1, 2,…, K }.

The objective function (31) is originally ∑

1 ≤i< j≤N | y i − y j | hc V i j , which has been converted to a linear form using Con-

straints (32) . Constraint (33) establishes the relationship between variables y i and x il . Each department should only be as-

signed to one floor in Constraint (34) . Constraint (35) does not allow many departments to be assigned to a floor more than

its available area.

4.2. Multi-floor problems with fixed elevators

4.2.1. Stage two: Finding relative position of departments

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 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 similar

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

of parameters.

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

for different-floor departments (the departments situated on different floors), it consists of both horizontal and vertical

distances. Indeed, in order to move from department i to department 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 (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 i and j can be formulated as d H i j

+ d V i j

= min

e { d ie + d e j } + h | y i − y j | , where d H

i j , d V

i j and d ie are

horizontal and vertical distances between departments i and j , and distance from department i to elevator e , respectively.

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.

Distance between circle i and elevator e can be expressed as D̄ ie = (x i − x e ) 2 + (y i − y e )

2 . Now, D̄ i j = min e { ̄D ie + D̄ e j } is

defined as the horizontal distance between the circles i and j situated on different floors. This equation is equivalent to:

D̄ i j ≤ D̄ ie + D̄ e j ∀ e. (37)

5612 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

In order to prevent D̄ i j = 0 , a negative value is set as the coefficient of this variable in the objective function, because

the aim for D̄ i j is to get its upper bound, according to Constraint (37) , to express the smallest horizontal distance between

circles i and j . Hence, the model proposed for finding the layout of the circles is as follows:

min

(x i ,y i ) ,h F ,w F

∑

i< j on floor 1

[F 1 i j (x i , x j , y i , y j ) − K

1

(D i j

t i j

)]+ ... +

∑

i< j on floor l

[F l i j (x i , x j , y i , y j ) − K

l

(D i j

t i j

)]

+ ... +

∑

i< j on floor k

[F k i j (x i , x j , y i , y j ) − K

k

(D i j

t i j

)]−

∑

i< j on different floors

c H i j D̄ i j , (38)

s.t. D̄ ij ≤ D̄ ie + D̄ ej ∀ e and i < j on different floors ,

( 9 ) − ( 11 ) ,

where

F l ij

(x i , x j , y i , y j

)=

{

c H ij D ij +

αl t ij

D ij

− 1 , τij ≤ D ij

v ij , τij > D ij .

(39)

Here, αl and K

l are the parameters associated with floor l .

The objective function (38) consists of an objective function of single-floor version problem (i.e. Eq. (16) ) for each of the

floors, in addition to a penalty term in order to prevent D̄ i j = 0 . Constraint (39) is the extension of inequality (37) for all i

and j .

The main problem in [32] was how to model the constraints related to horizontal distance between different-floor de-

partments (or circles) and they finally failed to cope with this difficulty. Also, they did not directly accommodate aspect

ratio in their model.

4.2.2. Stage three: Determining final layout

In this step, the nonlinear programming model presented in Section (3.2) is extended to multi-floor problems with fixed

elevators, in order to determine the final layout of the floors.

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

have:

¯̄D i j = min

e {| x i − x e | + | y i − y e | + | x e − x j | + | y e − y j |}

= min

e { ¯̄Dx ie +

¯̄Dy ie +

¯̄Dx e j +

¯̄Dy e j } . (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., similar to inequality (37) ). On the other hand, in order to prevent the arbitrary increasing of¯̄D i j via increasing the right hand side of its constraint, some penalty is assigned, in the objective function, to the variables

located on the right hand side of that constraint (see the third term of the objective function (41) ). Finally, the intended

model can be formulated as follows:

min

(x i ,y i ) ,h i ,w i ,h F ,w F

∑

i< j on same floors

c H i j (u i j + v i j ) −∑

i< j on different floors

c H i j ¯̄D i j + M

∑

i,e

( ¯̄Dx ie +

¯̄Dy ie ) , (41)

s.t. ¯̄D ij ≤ ¯̄Dx ie +

¯̄Dy ie +

¯̄Dx ej +

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

¯̄Dx ie ≥ x i − x e ∀ i and e, (43)

¯̄Dx ie ≥ x e − x i ∀ i and e, (44)

¯̄Dy ie ≥ y i − y e ∀ i and e, (45)

¯̄Dy ie ≥ y e − y i ∀ i and e,

(22) − (25) ∀ i < j on same floor ,

(26) − (30) , (46)

where M is a penalty constant value.

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 Eq. (40) in the form of

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5613

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

Eq. (40) .

Finally, by subtracting the penalties and adding the interaction costs in vertical direction, the cost of the obtained layout

( Z final ) can be calculated as follows:

Z f inal = Z + 2

∑

i< j on different floors

c H i j ¯̄D i j − M

∑

i,e

( ¯̄Dx ie +

¯̄Dy ie ) + Z F AF , (47)

where Z and Z FAF 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 the objective function (41) ; indeed, this cost 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, the models of the second and third stages differ from the former. It should be noted that the

variable elevator problem refers to a problem in which the location of the elevators are decision variables.

4.3.1. Stage two: Finding relative positions of departments

In the new problem, due to unknown location of elevators and further freeness of D̄ i j in the objective function, we

take the logarithm (a concave 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 objective function to increase D̄ i j . Also, in order to prevent the

elevators from being located on each other, which leads to increasing of D̄ i j , a penalty 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:

min

(x i ,y i ) ,h F ,w F

∑

i< j on floor 1

[F 1 i j (x i , x j , y i , y j ) − K

1

(D i j

t i j

)]+ ... +

∑

i< j on floor l

[F l i j (x i , x j , y i , y j ) − K

l

(D i j

t i j

)]

+ ... +

∑

i< j on floor k

[F k i j (x i , x j , y i , y j ) − K

k

(D i j

t i j

)]− log 10

∑

i< j on different floors

c H i j D̄ i j − ln

∑

e<e ′ D̄ ee ′

+

∑

e and i< j on different floors

( ̄D ie + D̄ e j ) (48)

s.t. x e ≤ 1

2

w F and − x e ≤ 1

2

w F ∀ e, (49)

y e ≤ 1

2

h F and − y e ≤ 1

2

h F ∀ e, (50)

D̄ ij ≤ D̄ ie + D̄ ej ∀ e and i < j on different floors ,

( 9 ) −( 11 ) , (51)

where D̄ ee ′ = (x e − x e ′ ) 2 + (y e − y e ′ ) 2 is the distance between elevators e and e ′ . That term of objective function (48) which

has log 10 is dependent on D̄ i j , and its own D̄ i j is also dependent on the distance of departments from the nearest elevator.

As a result, the model tries to increase D̄ i j , through locating 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 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 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] .

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 term, the negative coefficient of D̄ i j in the objective function will lead to its increase, which is not

favorable. Constraints (49) and (50) enforce the elevators to be located inside the facility

4.4. Stage three: Determining final layout

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 be as follows:

5614 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

min

( x i ,y i ) ,h i ,w i ,h F ,w F

∑

i< j on same floors

c H ij

(u ij + v ij

)− log 10

∑

i< j on different floors

c H ij ¯̄D ij + M

∑

i,e

(¯̄Dx ie +

¯̄Dy ie

)− ln

∑

e<e ’

D̄ ee ’ ,

s.t. ( 22 ) − ( 25 ) ∀ i < j on same floor ,

( 22 ) −( 30 ) ,

( 42 ) −( 46 ) ,

( 49 ) −( 50 ) ,

(52)

With regard to the previous description and similarity of this model to model 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 objective function,

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

here, there are the constraints to limit the elevators to be located inside the facility. By subtracting the penalties and adding

other related costs, similar to Eq. (47) , cost of the final layout will be calculated as follows:

Z f inal = Z + log 10

∑

i< j on different floors

c H i j ¯̄D i j +

∑

i< j on different floors

c H i j ¯̄D i j − M

∑

i,e

( ¯̄Dx ie +

¯̄Dy ie ) + ln

∑

e<e ′ D̄ ee ′ + Z F AF . (53)

5. Computational results

In order to solve the presented models, CPLEX 12.5.0.0 and KNITRO 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 executed on a system equipped

with an Intel Core2Duo 2.40 GHz CPU and 3GB of RAM while utilizing Microsoft Windows Vista.

In order to solve the second stage model in multi-floor (and equivalently 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 (w F + h F ) . Therefore,

the initial solution for circle i can be expressed as follows:

x i = (w F + h F ) cos θi and y i = (w F + h F ) sin θi , (54)

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

coordinates of the circles’ centers were set via x i = χ uni f orm (−w F / 2 + r i , w F / 2 − r i ) and y i = χ uni f orm (−h F / 2 + r i , h F / 2 −r i ) . Moreover, this new procedure could 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 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:

x e = R cos θe and y e = R sin θe and R = 0 . 8 min { χw F / 2 , χh F /w } . (55)

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, 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 αwere related to dispersion and overlapping of the circles, respectively, such that a small value for K put the 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. Dimensions of facility, in IJLAV model, were enlarged enough, such that if the circles were

ideally dispersed (i.e. with small overlapping and good dispersion), 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 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 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 desired layout, β could be altered to the extent of decimal values. Fig. 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 solution.

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5615

Fig. 2. An example for setting the parameters ( β = 2 ). The cost shown is cost of the final layout.

Table 1

Comparison of presented framework for small size problems.

Instance Best cost by our framework

(less than 0.48 s)

Best cost in [6] (less

than 1.36 s)

Optimal or best known

cost (reported in [6] )

Gap (%) (to best known

solution)

9-department [44] 264.50 298.30 235.95 12 .10

10-department [41] 23892.93 29193.00 20396.19 17 .14

O10 [45] 280.04 320.07 238.27 17 .53

FO10 [45] 31.94 35.71 29.41 8 .60

FO11 [45] 33.11 35.33 33.93 −2 .42 (improvement)

14-department [46] 6057.49 7416.00 50 04.0 0 21 .05

Table 2

Comparison of presented framework for Armour and Buffa problem.

β∗ Best cost by our framework (0.61 s) Best cost in [49] (2640 s) Best cost in [50] Best cost in [42] (18 s) Best cost in [6] (17.37 s)

6 2853.3 – – – 2708.0

5 2817.6 5524.7 5397.6 4591.3 3009.0

4 2875.1 5743.1 5370.6 4786.4 2960.5

3 2986.6 5832.6 5594.3 5140.1 ∗

2 2960.6 6171.1 6023.2 5224.7 ∗

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

The experiments showed that the closer the value of β to 1, the eased 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 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 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 by a positive and

less than one number.

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

described here.

Table 1 illustrates the results for some small size problems. The time consumed to obtain the presented solutions (mea-

sured in sec) is shown in the header of the table by character “s”. As can be seen in the table, the proposed framework

significantly has outperformed the framework of Jankovits 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 ratios in the tables show

the rounded values to the nearest integer amount. Also, the provided computational time shows the average time consumed

for all the presented solutions. Fig. 3 displays a layout of the circles and departments for Armour and Buffa problem.

Tables 2 –4 indicate that the proposed method provides better solutions within very little time in comparison to the other

methods, even for large size problems. Indeed, the nonlinear programming model presented in the second stage and also the

systematic method for tuning the parameters have enhanced the proposed framework in terms of cost and computational

5616 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

Table 3

Comparison of presented framework for Nugent problem.

β∗ Best cost by our framework (1.8 s) Best cost in [51] (876.6 s) Best cost in [52] Best cost in [6] (258.1 s)

6 20565.4 – – 23770.0

5 20337.9 – – 24916.0

4 20627.0 – – 250 0 0.0

3 20684.8 – – ∗

2 20825.0 23416.5 21560.6 ∗

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

Table 4

Comparison of presented framework for problem B with 30 department [6] .

β∗ Best cost by our framework (1.02 s) Best cost in [6] (180–300 s)

6 9188.4 10604.0

5 9243.2 10424.0

4 9186.9 10199.0

3 9296.9 ∗

2 9421.0 ∗

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

(a) Layout of cir-

cles (first stage)

(b) Final layout

(second stage)

Fig. 3. Solutions of Armour and Buffa problem ( β∗ = 2 and cost = 2960.6).

Table 5

Comparison of presented framework for15-department problem with fixed elevators.

Our framework Framework of [32]

β∗ (f1, f2, f3) Best cost (0.62 s) β∗ (f1, f2, f3) Best cost (0.88 s) Cost improvement (%)

3.12, 5.00, 4.17 110 658.625 7.06, 4.42, 5.76 123 501.53 10.40

3.12, 4.17, 4.17 110 772.002 8.00, 7.05, 4.32 124 763.39 11.21

3.12, 2.06, 3.23 121 752.824 3.14, 4.26, 4.32 126 936.07 4.08

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 method is that this

framework provided good layouts by most values of the parameter and rarely yielded a poor layout.

5.3. Multi-floor with fixed elevators instances

5.3.1. Instance 1: 15-department problem

The first instance is a 15-department and 3-floor problem with 6 fixed 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 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] .

The best solution of MULTIPLE (name of the method presented by Bozer et al. [34] ), for this problem, was obtained in

37.9 s 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, even within less computational time.

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

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5617

Table 6

Comparison of presented framework for 40-department problem with fixed elevators.

Our framework Framework of [32]

β∗ (f1, f2, f3, f4) Best cost (1.9 s) β∗ (f1, f2, f3, f4) Best cost (2.48 s) Cost improvement (%)

2.76, 3.70, 4.50, 4.50 14115.24 4.26, 7.86, 8.00, 7.99 20446.32 31.0

2.76, 3.70, 3.70, 3.70 14135.51 3.25, 4.53, 5.00, 5.00 21704.10 34.9

Table 7

Results of presented framework for 11-department problem with variable elevators.

β∗ (f1, f2, f3) 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

Table 8

Results of presented framework for 15-department problem with variable elevators.

β∗ (f1, f2, f3) Cost of our solution (1.58 s) Cost improvement (%)

3.12, 5.00, 4.17 108294.95 12.3

3.12, 4.30, 4.17 108372.72

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 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 on the first floor [53] . Since the proposed framework requires rectangular departments, the

assumptions taken by Bernardi and Anjos [32] are adopted.

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

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

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 improvement in this problem

with the previous one confirms the fact that the 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 the real-world

problems.

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 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

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 departments. Also, they assumed that elevators should be located at

the boundaries of departments. Here, these two assumptions are not taken into consideration.

The best cost reported for this problem in [13] is 123,319.55, which has been attained in 10998 s. In fact, their method

provided only one layout by 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 with lower costs within the smallest amount of time. Although,

the aspect 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 the percentage

of cost improvement for our best solution compared with the best solution reported in [32] (in which the elevators were

fixed), even though our aspect ratios are considerably less than theirs.

5618 A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620

Table 9

Results of presented framework for 40-department problem with variable elevators.

β∗ (f1, f2, f3, f4) Cost of our solution (1.58 s) Cost improvement (%)

4.51, 3.85, 2.83, 4.51 11637.55 43.1

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

Table 11

The flows generated for new departments in 50-department problem.

i to j Flow i to j Flow i to j Flow i to j Flow

41 to 1 7 43 to 49 13 46 to 26 8 49 to 27 13

41 to 3 15 44 to 23 5 46 to 27 70 49 to 30 15

41 to 17 25 44 to 26 8 46 to 32 22 49 to 33 33

41 to 44 17 44 to 48 8 46 to 50 13 50 to 6 15

41 to 45 12 45 to 6 16 47 to 1 4 50 to 14 12

42 to 6 81 45 to 33 12 47 to 2 19 50 to 16 18

42 to 8 17 45 to 47 8 48 to 23 12 50 to 19 11

42 to 24 19 46 to 8 17 48 to 50 10 50 to 35 16

43 to 40 48 46 to 20 14 49 to 18 95

Table 12

Results of presented framework for 50-department problem.

β∗ (f1, f2, f3, f4) Cost of our solution (5.8 s)

3.20, 4.14, 4.96, 3.80 18462.84

3.20, 3.97, 3.97, 3.79 18564.85

2.73, 2.73, 2.73, 2.73 19043.89

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

(a) Floor 1 (b) Floor2 (c) Floor3 (d) Floor4

Fig. 4. The best layout obtained for 40C-department problem.

5.5. Instance 4 and 5: New instances

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

The results for three layouts of this problem are illustrated in Table 12 .

The second new instance is the same as 40-department problem by the difference that all floors are square with different

areas of 95, 115, 106, and 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 enable 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 .

A. Ahmadi, M.R. Akbari Jokar / Applied Mathematical Modelling 40 (2016) 5605–5620 5619

Table 13 and Fig. 4 display the result and shape of the best layout obtained for this problem, respectively. These two

problems demonstrate the strength and flexibility of the proposed methodology for large and complicated problems in

addition to its fast computation and consistency with 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 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 layout of the departments and also the location of the elevators (if

they were decision variable) were established. Moreover, this method could easily be implemented for single-floor problems,

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

including fixed location departments, non-rectangular floors, floors with different areas, fixed shape departments, limitations

to locations of elevators and departments (due to technical constraints), and so on. The wide variety of layouts provided by

this framework proposes a large number of alternatives 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 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 unnec-

essary material handling, minimizing production/service delays, avoiding bottlenecks, using the available area effectively,

preventing unnecessary and costly changes, decreasing process inventory, etc.

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

improve the final layout cost. Also, combining the proposed framework with other methods (such as MULTIPLE), to inves-

tigate the exchange of departments between different 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), and using several types of elevators can be investigated in future work.

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