Optimal Multi-Floor Layout for Adjacency Maximization
Hossein Neghabi*1, Farhad Ghassemi Tari
*
Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran
Abstract: Multi-floor facility layout problem concerns the arrangement of departments on the different floors. In this paper, a new mathematical model is proposed for multi-floor layout with unequal department. Maximizing the number of useful adjacencies among departments is considered as the objective function. The adjacencies are divided into two chief categories: horizontal and vertical adjacencies. The former may be occurred between the departments assigned to same floors. The latter can be happened between departments assigned to any consecutive floors. A minimum common boundary length (surface area) between any two horizontal (vertical) adjacent departments is specified. The efficiency of the model is demonstrated by two illustrative examples where all of the satisfactory adjacencies are created. The proposed model is practical in multi-floor factories where the existence of adjacencies between departments is useful or essential due to possible establishment of conveyor, transferring pipe, lift truck route etc.
Keywords: Adjacency, Mathematical model, Multi-floor facility layout, unequal department
1
- Email addresses:
[email protected] (H.Neghabi),
[email protected] (F.Ghasemi Tari)
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1. Introduction Facility Layout Problem (FLP) concerns finding the appropriate arrangement of departments. Generally, a good layout decreases material handling cost and increases availability among departments. Since the cost of rearrangement of facilities is relatively high; the decision about this issue is classified as a strategic decision and primary layout for facilities is very important. In some situations, there is not sufficient space for establishing the manufacturing plant or the land cost for creating the factories is high, and it seems that the total cost of making a multi-floor factory can be less than a single-floor one. In addition, sometimes due to the nature of material or the process specification, it is more practical to arrange departments into multiple floors rather than single floor. Multi-floor facility layout problem (MFFLP) is one category of layout problems which facilities can be located in the limited numbers of floors so that the factory space can be utilized efficiently. In MFFLP, there are two general types of flow between facilities: (I) horizontal flow that occurs between facilities located on the same floor, and (II) vertical flow that can be seen between facilities on different floors. Usually transferring the materials between two departments, located on different floors are done using elevators, conveyors, pipes etc. Generally, it is impossible to develop a layout model which can capture all the relevant aspects of the problem; this partly explains why a huge numbers of papers consider only a particular aspect of this problem. One of the interesting and applicable approaches for the assessment of different layouts is the number (or value) of created useful adjacencies among departments or facilities. It therefore stands to reason that maximizing the number of useful adjacencies between facilities is a desirable objective function considered in layout problems. Typically, most papers in the related literature utilize graph theory for maximizing the number of adjacencies. Graph theory is used to maximize the number of adjacencies between departments with total disregard for the physical shape of departments; therefore, in a layout drawing based on the graph theory, departments can be 2
or
shaped, and it is difficult to
actualize such a layout in the real world. This constraint seems to be reason enough for us to present a new mathematical model for maximizing the number of useful adjacencies among rectangular departments or facilities. Therefore, this study proposes a new model for maximizing the number of horizontal and vertical adjacencies among different rectangular departments in multi-floor layouts. 2. Literature Review Since the facility layout problem has proven to be a NP-complete problem [1], various heuristic and meta-heuristic algorithms have been developed in last decades. MFFLP is categorized as a particular case of FLP and all of solution approaches for MFFLP can be classified into two main groups: one-stage algorithms and two-stage algorithms. In the following paragraphs brief reviews have been presented. In the one stage approaches, all departments, except the fixed ones, are assigned to the floors during running the algorithm. Johnson was the first researcher who introduced the MFFLP in 1982 [2]. In 1994, Meller and Bozer studied MFFLP and attempt to minimize the total vertical and horizontal material flow costs. They also developed a Simulated Annealing (SA) algorithm to solve their model [3]. A considerable number of well-known researchers have endeavored to solve MFFLP by applying different genetic algorithms (GAs) with diversity in fitness functions and constraints [4-6]. In many algorithms, minimizing the total material handling costs is considered as the objective function. In some models, a multi objective function approach is followed considering other interests such as maximizing the total adjacencies value [7], improving the safety factor [8-9], and minimizing the total cost of installing the elevators [10], minimizing facility building cost [11], in addition to minimizing the material handling cost. Moreover, in the related literature, there are several exact approaches for solving the MFFLP [12-14]. With regard to the complexity nature of MFFLP, the exact algorithms and models are used for small size problems. In two-stage approaches, first each department is assigned to a specific floor, and then, the algorithm arranges the layout of a floor according to the assigned departments. Meller and Bozer are the 3
first researchers who used a mathematical model to assign the departments to the floors and then, utilized a heuristic algorithm to arrange the assigned departments in each floor [15]. Abdinnour and Hadley compared two different models, such that in the first one, a heuristic algorithm was used to assign the departments to specific floor while in the second one, a deterministic mathematical model was utilized for this purpose. Moreover, in both models they planned the layout of the departments within each floors, by using a Tabu Search (TS) algorithm. They concluded the solutions obtained by mathematical model are better than the heuristic one [16]. Bernadi and Anjos have developed a deterministic mathematical model which assigns departments to the floors. They compared two solution approaches of MFFLP. They concluded that simultaneous arrangement of departments in all floors leads to better solution than independent layout of each floor [17]. 3. Statement of the Problem This study takes the shape of all departments as rectangular norm and adjacencies can occur along with
,
or
axis. This paper attempts to determine the center point coordination and assigned
floors of each facility/department such that horizontal and vertical adjacencies are maximized. In a single floor layout, each department can be adjacent with other departments in four directions (please see Fig.1.a); but in multi-floor layout, the maximum possible direction is 6 (consider Fig.1.b where a top and a bottom adjacencies may be added).
Fig.1.a.
Fig.1.b.
Fig.1: Difference between all possible directions for creating adjacencies in single floor (Fig.1.a) and Multi-floor (Fig.1.b)
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Our purpose from horizontal adjacencies is all possible adjacencies that can be created among the departments located on a specified floor. For creating the horizontal adjacencies, it is sufficient that two departments having a common boundary length along
or
axis with each other (Fig.3.a).
The vertical adjacency occurs when any two departments , are located on two consecutive floors, and have a common surface with each other. In other words, when department department
lies on floor
or
, and two departments
and
lies on floor , and
have a common surface along
axis, two departments are vertically adjacent (Fig.2).
𝑍 axis 𝑋 axis
𝑗
𝑌 axis
Floor 𝑘
𝑆2 𝑤2
𝑖
Floor 𝑘
Fig.2: Two departments i and j are vertically adjacent.
4. Formulation In this study, all departments can be located anywhere on each floor in view of the non-overlapping constraints. One further assumption is that all parameters are predetermined and certain. The objective function and constraints are as follows: 4.1. Objective Function
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As mentioned earlier, maximizing the number of useful adjacencies is taken as the objective function of the proposed model; hence considering the
(a positive number) as the adjacent value, the
objective function can be written as:
∑∑
Where
(
)
indicates the number of facilities that should be located in the given floors.
and
are two binary variables that imply horizontal and vertical adjacencies, respectively, as follows:
{
{
Noticeably,
is a correction factor that prevents to double enumeration of the horizontally
adjacencies. In general, the contribution (importance) of the horizontal and the vertical adjacencies in the objective function may be weighted up or down; but, in this study they are assumed to be in same order.
4.2. Floor Constraints Generally, the number of floors and departments is specified and each department should be assigned to exactly one floor. To this end, a new binary variable is introduced as follows:
{
In view of the mentioned variable, the floor constraint can be written as:
∑
6
indicate the number of available floors. For formulating some constraints such as non-overlapping, minimum common boundary length, identifying horizontal adjacencies, it is important to determine whether two departments assigned to the same floor or not. Hence a new binary variable
and
are
is introduced, which is equal to 1
when two departments are located on the specific floor. The value of
is obtainable through Eqs. (3)-
(5) [13].
Eq. (3) represent when two departments the
and
are located on the same floor (i.e
takes one, otherwise if at least one of two variables
or
take zero, then
),
will be zero too.
4.3. Non-Overlapping Constraints The constraints Eqs. (6)-(9) are added to the model in order to ensure all departments are assigned to the same floor; do not overlap with each other. To this end, two new binary variables introduced and 4 disjunctive constraints are included in the model. satisfied when at least one of them is active.
(
) (
(
( )
) (
) (
( )
) )
(
)
7
and
are
Non-overlapping constraint is
is the center point coordination of department department , furthermore
and
/
represent the length/width of
is a huge positive number. Note that, if
, it means two departments
and are located on different floors and all constraints (6)-(9) are inactive. 4.4. Distance between two departments Distance between any pair of departments is used in other constraints; distance along with
and
axis can be obtained by defining 4 new parameters. These parameters are:
{
{
{
{
Value of these variables can be obtained by:
Regarding the definition of the new parameter, the distance along with calculated as (
and (
and
axis is
) respectively.
4.5. Horizontal adjacencies constraints: The Eqs. (12) and (13) are used for determining horizontal adjacencies. Clearly, when the distance along
axes between center points of two departments
and , allocated to the same floor, is
less than half of the sum of its lengths/width, the possibility of the existence of an adjacency along the
8
axes arises. Hence, the Eqs. (12) and (13) are added to the model to identify the horizontal adjacencies; using two new types of binary variables
(
)
(
( )
and
.
) (
)
For satisfying the adjacent condition, it is necessary for both binary variables equal to 1 simultaneously. Note that in both Fig.3.a and Fig.3.b, the value of in Fig.3.a, the value of
and
is equal to 1, but only
is 1 too; therefore two departments and are horizontally adjacent. In other
words, the two binary variables
and
should be equaled 1 simultaneously, to claim that two
departments and are horizontally adjacent with each other.
𝑆 𝑤
𝑗
to be
𝑗
𝑗 𝑖
𝑖 𝑍 𝑌
𝑁𝑋𝑖𝑗
𝑁𝑌𝑖𝑗 Fig.3.a
𝑋
𝑁𝑋𝑖𝑗
𝑁𝑌𝑖𝑗 Fig.3.b
Fig.3: One of necessary condition for horizontally adjacencies of two departments i and j.
In view of the above explanation, the following constraints are included in the model:
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is a binary variable indicating whether two departments and are horizontally adjacent or not. Therefore, its value is 1 when both binary variables zero when at least one of the two variables
and
and
are equal to 1; also
is equal to
are zero.
4.6. Minimum Common boundary Length Constraints In this study, it is assumed that two departments
and
are adjacent when they have a minimum
common boundary length with each other. This common boundary can be used for handling equipment and pipes that transfer material between two departments. To this end, two new parameters introduced; representing the minimum common boundary length along
and
(Fig.3.a). For modeling the relevant constraints, two new types of binary variables
and
are
axis respectively and
are
introduced, too. Therefore, the following Eqs. (16)-(19) should be added to the model.
(
)
(
( (
)
) )
(
(
( )
) )
(
)
Considering the non-overlapping constraints (6)-(9) versus minimum common boundary length constraints (16)-(19), it is concluded that the two variables
and
can not be equal to 1,
simultaneously. Therefore, if one of the two variables becomes equal to 1, it means that two departments have a common boundary with each other. Therefore the following constraint is added to model for determining that the constraint of the minimum common boundary length has been met or not.
Where boundary along
is a binary variable which indicates whether two departments have a common or
axis, or not. 10
4.7. Floor Space Constraints In this study, a rectangular shape of land area is assumed to be used whose length and width are and
respectively. To this end, the following constraints are included.
4.8. Vertical adjacencies constraints When two departments and are located on two sequential floors, it is possible to create vertical adjacencies with each other. To this end, it is sufficient that two departments surface with each other along the types of binary variables two departments
and
having a common
axis (Fig.2). These constraints can be modeled by defining two new and
.
(
) is a binary variable equal to 1 when any
and , belong to any two sequential floors
and (
),
overlap along the
axis. In addition, in order to define common surface specification, two new parameters 2 are
introduced (Fig. 2) so that for any two vertically adjacent departments and ,
minimum length/width of the common surface along the
(
)
(
)
2
(
)
(
)
2
)
2
)
(
11
and
define the
axis. The model includes the following
constraints (25)-(28) as
(
2
2
(
)
(
)
In constraints (25)-(28) the two parameters Obviously,
and
and
2
, play the same role of big-M or a huge number.
, can be equal to 1 at only one floor. Therefore, the following constraints
(29) and (30) are added as
∑
∑ If the value of
(
)is equal to 1, it means that two departments and probably overlap
each other between two assuming sequential floors such as Consequently, when the two variables
and
and
; along the
axis.
are simultaneously equal to 1; two departments
and are vertically adjacent. Therefore the following Eqs. (31)-(32) are added to model.
Indicates vertically adjacent variables and is equal to 1 when two departments
and
have
vertically adjacent conditions. 4.9. Logical constraints
Regarding the definition of variables and the structure of mathematical model, the following constraints (33)-(35) are inevitable.
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5. Numerical Result The performance of the proposed mathematical modeling was evaluated using CPLEX, version 9.1.0, on a portable computer (2 GB RAM and 2GHz CPU Intel Core 2 Duo). For this purpose the proposed model was put to test using the two following illustrative examples available in the literature. 5.1. Example 1: This example was first proposed in [18] which include 11 departments that must be located on different floors. In our model, the adjacent matrix is substituted by mentioned material flow. Therefore, the adjacent value is represented in Table 1 and departments dimension is shown in Table 2. In this example the minimum common boundary length along the
and
axis is the same and equal to
and the minimum common surface dimension is taken as
2
2
. Also, it is
assumed that three quadrangular floors are available for the assignment of departments with the dimension
.
Table 1: Adjacent value for example 1.
1 2 3 4 5 6 7 8 9 10
2 342 -
3 0 838 -
4 0 0 829 -
5 322 320 0 461 -
6 0 0 0 0 439 -
7 0 0 0 0 0 474 -
8 0 0 0 0 259 0 0 -
9 0 0 0 0 532 0 0 27 -
10 0 0 0 0 680 0 0 0 231 -
11 0 461 0 0 438 0 0 0 0 658
The optimal center point coordination and the optimal assignment of floor for each department are summarized in Table 2.
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Table 2: departments dimension and optimal solution for example 1
Dimension
Departments 1 2 3 4 5 6 7 8 9 10 11
1.8 1.9 2.0 1.7 1.9 1.9 1.9 2.0 1.8 2.0 1.8
Optimal Location 1.7 1.8 1.6 1.5 1.8 1.5 1.8 1.9 1.6 1.5 1.7
1.1 2.95 1.2 0.95 2.75 1.05 1.05 1.0 2.9 3.0 3.1
0.85 1.4 3.1 1.75 1.6 3.25 3.1 1.25 2.8 3.25 3.15
Assigned Floor 1 1 1 2 2 2 3 3 3 2 1
Fig.4 shows optimal layout for example 1. Since, The optimal value is 6482, implying that in the final solution all useful adjacencies are created, in other words the obtained solution is an upper bound for creating the useful adjacencies.
Fig.4: Optimal layout for example 1.
Fig.5 depicts a schematic view of optimal solution obtained for example 1.
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7
9 8 Floor 3
6 10 5
4 3
Floor 2
11
2 1 Floor 1
Fig.5: schematic view of optimal layout for example 1.
5.2. Example 2: This problem is the 11-unite instance of a production system introduced by Georgiadis et al [19]. It is assumed that three potentials floors are available and its dimension is taken as
In order to
solve this problem, the connection and pumping cost matrix is taken as that adjacent value matrix. This assumption sounds logical, because any increase/decrease of the connection and the pumping cost correspond to increase/decrease in the adjacent value between them. The adjacent value and the associated dimension of each department are shown respectively in Table 3 and Table 4. Also, the value of the new parameters are set
2
.
2
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Table 3: Adjacent value for example 2. 2
2
0 -
950 0
0 380
0 0
0 570
0 0
0 0
0 0
0 0
0 0
-
-
0 -
570 285
0 0
190 0
0 0
0 0
0 0
0 0
-
-
-
-
0 -
0 456
456 0
304 0
0 0
0 0
-
-
-
-
-
-
0 -
0 0 -
285 0 0 -
285 0 0 0
Table 4 contains the optimal solution including center point coordination and the proper assignment of floor for each department. The total adjacent value is 4731, meaning all valuable adjacencies between departments occur. Table 4: departments dimension and optimal solution for example 2.
Dimension
Departments
2
5.0 6.0 4.0 6.5 6.0 4.0 4.0 5.0 4.0 2.0 3.0
Optimal Location 3.0 5.0 6.0 5.0 3.0 5.5 5.0 3.0 6.0 1.0 2.0
3.5 3.0 6.0 3.25 7.0 2.0 2.5 7.0 8.0 3.0 2.5
The optimal layout for example 2 is shown in Fig.6.
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8.5 7.5 7.0 2.5 2.5 7.25 2.5 1.5 6.0 4.0 1.0
Assigned Floor 1 3 2 3 2 2 1 1 1 2 2
Fig. 6. Optimal layout for example 2.
6. Conclusion In this paper, a new mathematical model for multi-floor facility layout problem has been proposed. For further compatibility with the real world, the departments are considered quadrangle which may be vertically or horizontally adjacent with each other. The constraints of minimum common boundary length (surface area) between any two horizontal (vertical) adjacent departments are taken into account. The objective function of the optimization problem is set as maximizing the number of useful adjacencies among departments. The efficiency of the model is demonstrated by two illustrative examples. The authors are hopeful that the presented model may be helpful for optimal arrangement of departments in multi-floor factories where the existence of adjacencies between departments is useful or essential due to possible establishment of conveyor, transferring pipe, lift truck route etc.
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[17] Bernadi, S., Anjos, M.F., (2012). A two-stage mathematical-programming method for the multi-floor facility layout problem, Journal of the Operational Research Society, 1, 1-13. [18] Goetschalckx, M., Irohara, T., (2007). Efficient formulations for the multi-floor facility layout problem with elevators, Optimization Online. [19] Georgiadis, M. C., Rotstein, G. E., & Macchietto, S. (1997). Optimal layout design in multipurpose batch plants. Industrial and Engineering Chemistry Research, 36, 4852–4863.
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