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Layout design problems with heterogeneous area constraints
Accepted Manuscript Layout Design Problems with Heterogeneous Area Constraints Junjae Chae, Amelia Regan PII: DOI: Reference:
S0360-8352(16)30385-0 http://dx.doi.org/10.1016/j.cie.2016.10.016 CAIE 4501
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
28 April 2016 19 September 2016 20 October 2016
Please cite this article as: Chae, J., Regan, A., Layout Design Problems with Heterogeneous Area Constraints, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.10.016
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Layout Design Problems with Heterogeneous Area Constraints Junjae Chae1 and Amelia Regan2 * 1
School of Air Transport, Transportation and Logistics, Korea Aerospace University, South Korea
2
Department of Computer Sciences, University of California at Irvine, USA
Abstract The facility layout problem (FLP) is one of the design problems involving the assignment of facilities (e.g., machines, departments) to planar region (e.g., a plant), so as to achieve the objectives such as to minimize the cost of projected interaction between facilities or to maximize the closeness rating, etc. In this study, we deal with one of the FLP models that minimize the material handling cost between rectangular departments. Each department has an area restriction that specifies the total area that it must occupy while the specific lengths and widths are determined by the model. However, some department do not have flexibility and the dimension are predetermined. Thus, we proposed the to solve the layout design problem with two types of department. We call these constraints heterogeneous area constraints. The type A department, flexible length and width, and type B department, fixed length and width, are assigned to given floor space. Using a well-known data set which has been previously used by many other researchers, and modified version of the data aimed at testing our model, we show that the proposed model properly generates layout design for these two types of departments.
Keywords: FLP (facility layout problem); MIP (mixed integer programming); Heterogeneous area constraint; Block layout design with unequal areas.
*
Corresponding author; [email protected]
1. Introduction The facility layout problem (FLP) is to find a non-overlapping arrangement of work areas (e.g., departments, workstations or service areas) within a given space with respect to a variety of objectives (e.g., minimizing total distance of material flow or maximizing the closeness effect). The departments or workstations are considered to be of rectangular shapes by many researchers, though others have assumed that the departments or workstations have irregular-shaped areas (Bock & Hoberg, 2007; Bukchin & Tzur, 2014). Irregularly shaped areas arise in application such as the design of shopping malls, zoos or ships, while factories, warehouses and cross-docking facilities typically have rectangular areas. For the objectives, minimizing the cost due to material flow between departments or workstations is common, but some research has been conducted for maximization problems or multi-objective problems. As mentioned, the FLP has been studied by many researchers and it has various problem types. For example, Kusiak and Heragu (1987), Meller and Gau (1996) provide detailed surveys of the different variants and solution approaches for the problem. Drira et al. (2007) provides a tree representation for problem types, models, and approaches for facility layout problems. Other review papers focus on solution approaches tailored to specific aspects of layout problems (Levary & Kalchik, 1985; Liggett, 2000; Singh & Sharma, 2006; Tao, Wang, Qiao, & Tang, 2012). The review of facility layout problems by Singh and Sharma (2006) and Drira et al. (2007) categorize two mathematical formulations for facility layout design: one is QAP (Quadratic Assignment Problem) for discrete formulations, and the other is MIP (Mixed Integer Programming) for continuous formulations. In this study, we deal with the continuous space of a facility assignment, MIP model, and have the objective of minimizing the total weighted distance of material flow in the space. The weighted distance between a paired departments is measured by the rectilinear distance multiplied by the number of trips between the centroids of the departments. Further, each department has an area restriction that specifies the total area that it must occupy, while the specific lengths and widths are determined by the model. The facility layout problem is referred to as the block layout design problem when it deals with only rectangular blocks for departments. The problem has two main restrictions: the area requirement for the departments and floors, and the department location restrictions. The location restrictions are based on the continuous department positions in the given space and the area requirement for the department needs to handle the varying dimensions. The models and solution approaches for this block layout design problem has been studied by a number of researchers and some of studies are focused on the direct solution of the optimization problem with unequal (non-uniform) area requirements. On the other hand, a model could be required to specify the orientation of the department in the case in which departments have fixed dimensions. The models for this block layout design problem usually handle unequal area departments either with flexible dimension or with fixed dimension. In our research, we consider departments with unequal areas with changeable dimensions in the case of type “A” departments and fixed dimensions in the case of type “B” departments. To handle these two types of departments in a model, we need to analyze which part of variables and parameters can be used to represent both types of departments and which variables and parameters are necessarily used for specific types of departments. We follow the path of the modeling technique for direct 1
solution of optimization of unequal area block layout design problems. In other word, the FLP model introduced by Sherali et al.(2003) and Castillo and Westerlund (2005) is used to assign the type A departments. Then we provide a method to embed the different types of department restrictions, type B, into the model. Several methods to represent the type B department are evaluated and one of the methods is selected to use for embedding into the FLP model for type A department. The FLP model for type A department presented by Sherali et al.(2003) and Castillo and Westerlund (2005) uses the linearization technique for non-linear department area constraints which is represented by the horizontal length multiplied by the vertical length. Thus this model has an area feasibility issue if there is no tolerance of restrictions for maximum aspect ratio of department. That means the layout configuration generated by the model could become infeasible when the underestimated department is expanded to its intended size. The chance that the model is feasible is increased when the model includes the type B department because of inflexibility of the department area. We test the feasibility of the layout configuration of the given problem set and show the possible number of linear segments of non-linear area constraint to have a feasible layout. Additional tests were done for relaxation on floor space to see the impact of area compactness on feasibility. Specifically, a larger problem set was tested to check the applicability of this relaxation. More details of the problem are explained in Section 2 and the background and the motivation for our research is also discussed in that Section. The modeling procedure and the methodology including heterogeneous department types in a single model are presented in Section 3, and the results generated by the proposed model are discussed in Section 4. In the last Section, the conclusion and directions for future research are provided.
2. Background Most facility layout research resorts to developing heuristic solution methods since these problems are very difficult to solve optimally. Various solution approaches have been developed to solve these layout problems and several review papers present summaries of these (Drira et al., 2007; Kusiak & Heragu, 1987; Levary & Kalchik, 1985; Meller & Gau, 1996; Singh & Sharma, 2006). Adding the unequal area requirements with varying dimensions makes the problem even more difficult to solve. However, there have been attempts to solve the problem through traditional modeling and optimization techniques. Several studies present improved techniques to model problems in this category (Castillo & Westerlund, 2005; Lacksonen, 1994; Meller et al., 1999; Montreuil, 1990; Sherali et al., 2003). These research efforts provide methods to linearize the unequal area constraint so that a mixed integer linear programming (MILP) model can be solved. The models in those papers assume that all the departments have variable width and length dimensions. However, as discussed in Sherali et al.(2003), some departments could have fixed dimensions because of some specific physical characteristics of the activities of that department. Thus, the modified model considering both fixed and variable departments is necessary. As mentioned, we set type A departments as those with flexible dimensions and type B departments as those with fixed width and length. Initially, this categorization was introduced in Chae (2014). The characteristics of 2
these department types are shown in Figure 1. Different approaches are needed to model those different types of department layout problems. For the layout problem for type A departments, the lengths and widths are decision variables as are the centroids of each department. In the case of layout problem with type B departments, the position of the centroid of each department is the main concern. However, the latter model is not easier since the model needs to determine the precise orientation of each department. Thus handling these two types of departments in a model increases the difficulty because of the addition of new binary variables.
lix Equal area
liylong
liy l jy l
x short i
x j
l jxlong
l
Type A
Type B
Figure 1. Characteristics of type A and B departments There are two basic approaches to include these two different types of departments in a model. The first is that the constraints for type B departments are inserted to the layout model for type A departments. The second is the opposite: type A department constraints are inserted into the model for type B departments. The former method is more reasonable since starting from a model which can handle more flexible layouts and constraining some of these to be less flexible, is more straightforward than adding flexibility to a model with more constrained layouts. Therefore, that is the approach we use in our work. The base model for type A department layout used in this study is based on the model introduced by Sherali et al. (2003) and Castillo & Westerlund (2005). The parameters and variables of the model and the model itself for type A, FLP-A, are as follows. Parameters:
f ij : material flow volume from department i to j ai : area requirement for department i
lix , liy : horizontal and vertical length of department i Lx , Ly : horizontal and vertical length of floor space (facility)
lbix , ubix , lbiy , ubiy : lower and upper bound of lix , liy Variables: d ij : distance from department i to j (cix , ciy ) : coordinates for centroid of department i
zijx 1 : if department i is to the left of department j ( zijx 0 , otherwise) zijy 1 : if department i is lower than department j ( zijy 0 , otherwise)
FLP-A:
3
Min
f d
s.t.
(1)
ij ij
i
j
ai lix liy i
(2)
s
l ls c i c sj j Ls (1 zijs ) i j, s x, y 2 2 s i
y
( z
s ij
z sji ) 1 i j
(3)
(4)
s x
lis ls cis Ls i 2 2 lbis lis ubis
i, s
(5)
i, s
dij cix c xj ciy c jy
(6) i j
zijs 0,1 i j
(7) (8)
In the objective function shown in (1), the total weighted distance describes the sum of the product of the flows ( f ij ) and the distances ( d ij ) separating departments i and j . A rectilinear distance metric is used for measuring travel distance between departments’ centroids ( cis , c sj ) as shown in (7). The area of department i is expressed in (2) where its width and length are ( l ix l iy ) and the department areas should not overlap in a restricted space, ( Lx Ly ), as shown in (3). The binary variable, zijs , indicates the relative position of departments. Constraints (4) control the relative position of departments. Constraints (5) make sure that all the departments are in the given space. Each department is restricted by an aspect ratio ( i 1 ), which, practically, defines the maximum permissible ratio between its longest and shortest sides. The lower and upper bounds of each department’s dimension is based on its aspect ratio as in (6). Montreuil (1990) introduced a MIP model for facility layout design and used bounded perimeter constraint for area control. However, the constraint generates the department area much smaller than the actual. Lacksonen (1994) proposed linearization of area constraint which generates area between 0% to 3% larger than required area when i 2 . However, this linearization requires two additional binary variables for each department. Meller et al.(1999) proposed the model with linearization of non-linear unequal area constraints using perimeter and area relationship. This improved the previous perimeter constraint introduced by Montreuil (1990) and generated maximum 2.5% error when i 2 . As demonstrated in figure 1 (type A) the non-linear line indicates that departments of equal area and this non-linear line can be linearized by using several linear expression. The process of linearization generates the measureable gap with the actual area. Sherali et al. (2003) and Castillo & Westerlund (2005) suggested MIP models for block layout problem, which is a modification of the basic FLP model shown above. These two models provide the method to linearize the unequal area curve using tangential support as shown in figures 2 and 3. Castillo & Westerlund (2005) use the number of tangential supports as the factor controlling the maximum error generated by linearizing the unequal area curve. Sherali et al. (2003) tested the sensitivity of the errors generated by varying the number of tangential supports in the model. 4
The variable xik ( lbix xik ubix ) in figure 3 indicates the point in the x axis of tangential supports and the model includes up to k linearized constraints. The number k is the factor for controlling the approximation error in Castillo & Westerlund (2005). The approaches of these two studies are almost identical in terms of area representation with linearized equations. As mentioned above, the lower and upper bound of a department’s dimension is based on its aspect ratio. We can tighten these bounds by setting
ubi min ai i , Ls , i, s x, y s
lbi s
(9)
ai , i, s x, y ubis
(10)
These new constraints are derived from constraints (2) and (6), and ls Ls , s x, y presented using somewhat different notation.
l iy
ubiy
a
b
lbiy
lbix
ubix
l iy
Figure 2. Outer approximation to the area constraints using tangential supports (re-drawn based on Sherali et al. (2003)) l iy
ubiy
a
ai lix (lbix )2 liy 2ai lbix ai lix xik2 liy 2ai xik ailix (ubix )2 liy 2aiubix
b
lbiy
lbix x k
ubix
lix
Figure 3. Linearized constraints for the area approximation (re-drawn based on Castillo & Westerlund (2005))
Constraints (2) are replaced with linearized area constraints as follows ailix xik2 liy 2ai xik , lbix xik ubix , i, k
5
(11)
and the values of xik equal to
k ubix lbix , k 0,1,...,K 1, for any selected integer K 2 . K 1
xik lbix
(12)
This approach is purely linear and no binary variables are involved. Because the problem is to minimize the distance, the tendency of the problem is to underestimate the area, thus the equality constraints in (2) can be substituted by (11) as inequality constraints. The larger integer K makes the formulation generate results closer to that of non-linear area constraint. The error generated by underestimating the area is controlled in (Castillo & Westerlund, 2005) by setting K to a large number. Meller et al. (2007) proposed another FLP formulation using location relationship for each department, sequence-pair, which is inspired by Murata et al. (1996). The basic frame of this model is same as the one in Sherali et al. (2003) and the redundancy in solution set is reduced through the proposed model. The model by Sherali et al. (2003) also was used as a base work for the model in Saraswat et al. (2015) and Ridwan et al. (2016) for applying various objectives in FLP. Konak et al. (2006) presented the formulation for FLP in different structure – flexible bay – with keeping the basic form as MIP. Kim (2006) used the linearization technique similar to Sherali et al. (2003) and modification method to configure less turn material handling path.
A recent research, Huchette et al. (2016), presented a case study on
formulations for the floor layout problem. They showed the comparison of results generated by several different approaches for FLP using MIP formulation and observed impact of varying inputs to the model. The model for type B department layout is different from that of type A. The orientation of the departments should be considered when it is assigned to a floor space and this increases the solution time since the binary variables are used to express its orientation in the model. The basic model for FLP with fixed departments, FLPB, is as following. FLP-B:
Min
(13)
f d
ij ij
i
s.t.
j
l short l long l short l long j j cix i (1 wi ) i wi c xj (1 w j ) w j Lx (1 zijx ), i j 2 2 2 2
(14)
l short l long lishort l long j j wi i (1 wi ) c yj wj (1 w j ) Ly (1 zijy ), i j 2 2 2 2
(15)
ciy
lishort l long l short l long (1 wi ) i wi cix Lx i (1 wi ) i wi , i 2 2 2 2
(16)
lishort l long l short l long wi i (1 wi ) ciy Ly i wi i (1 wi ), i 2 2 2 2
(17)
y
( z
s ij
z sji ) 1 i j
(18)
s x
dij cix c xj ciy c jy i j
(19)
zijs 0,1 i j
(20) 6
wi 0,1 i
(21)
Some additional variable and parameters are introduced to represent the characteristics of this specific problem:
wi controls department orientation, lishort , lilong indicate the length of short and long sides of departments. The vertical and horizontal length of each department are the parameters in this model. However, the orientation is a decision variable, and the binary variable, wi , indicates this orientation. If wi 1 , then the department is horizontally long rectangular shape.
Constraints (14) and (15) prevent overlapping departments and determine
orientation. Two constraints selectively work for best relative location and the orientation assignment to floor space. The floor space has boundaries indicated by Lx and Ly . Constraints (16) and (17) insure that all the departments and their segments are in the boundary of the given floor space. However, without the flexibility of the department dimension, it is not easy to find a feasible layout if the floor space is not enough to hold the assigned fixed type department. This model is similar to MLP (Machine Layout Problem) or cell block layout in FMS (Flexible Manufacturing System) in that the machine or cell block have fixed dimension. Heragu & Kusiak (1990) presented a model in MIP form for machine allocation in given row in available area. Das (1993) presented the FMS layout design model extended from the MLP, and the dimension of each cell block and pickup/ drop-off (p/d) point on each cell block is predefined and the orientation of each cell block is a decision variable. This model is used as a basic model for other research. Yang & Peters (1998) presented MLP model with different environment: dynamic and uncertain production. However, the handling the orientation and overlap prevention constraints are as in Das (1993). Kim & Kim (2000) used the term ‘fixed shape facility’ instead of ‘machine’ and presented a model in MIP form and similar model presented in Deb et al. (2005), Leno et al. (2013) and Leno et al. (2015), and all those researches used the coordinates of block corner as an indication of block position in the floor space instead of using centroid of block. The model controls the p/d point located on the boundary of block while the model in Das (1993) defined the p/d position on center line in block.
3.
Embedding constraints for type B department to the model for type A department
layout As it is mentioned, the basic model for layout holding two different types of department is the model for type A department layout. There are several ways to represent the orientation of departments for type B departments in a given type A department layout model. Type B departments do not have the upper and lower limit for the area since these already have their own dimensions. However, the dimension of type A departments are expressed and determined by the area constraint (9) to (11) in previous section so we need lbix , ubix , lbiy , ubiy to express the dimension of type B departments.
7
l iy
a (lbix , ubiy )
ubiy
lbiy
b (ubix , lbiy )
lix lbix
ubix
Figure 4. Possible points of type B department
The dimension of a type B department can be either a or b in Figure 4. If the department is vertically long then the dimension of the department is lbix ubiy , and the point a , and if the department horizontally long, then it has lbiy ubix as the dimensions, the point b .
l iy
ubiy
a
(ubiy lbiy )l ix
lbiy
(ubix lbix )l iy
b
ubix ubiy lbix lbiy
lbix
ubix
lix
Figure 5. Two possible points specifying dimensions for a type B Department
The straight line passing through two point a and b can be shown as follows
(ubiy lbiy )lix (ubix lbix )liy ubixubiy lbixlbiy
i t y pB e
(22)
The shaded area on triangle formed by constraints (6) and (22) in figure 5 indicates the feasible region of possible determination of type B department area. Since the triangle satisfies the convex condition and the LP (Linear Programming) solution finds the solution at extreme point, and the model tends to find the minimum value, it is expected that the solution could be generated at point a or b .
If this fact is true or all the case of
type B departments, the additional binary variables selecting either the point a or b would not need to be introduced to the model.
However, our model is a MIP and the solution is not necessarily at the extreme points. 8
Thus, it is necessary to adopt the binary variables for determining the orientation of the department which is either point a or b . (liy ubiy ) m1(lix lbix )
l iy
ubiy
l iy
a
a
ubiy
lbix lix
lbiy
b
lbiy
b
lbiy liy
(liy lbiy ) m2 (lix ubix )
lbix
ubix
lbix
lix
(a)
ubix
lix
(b) ai lix (lbix )2 liy 2ai lbix
l iy
a
ubiy
lbiy
b a l (ub ) l 2ai ub x i i
x 2 y i i
x i
lbix
ubix
lix
(c) Figure 6. type B department orientation determination Our research considers three different cases of modeling for determination of type B department’s orientation. These three models choose the orientation of the type B department by selectively activating the constraints which are chosen by binary variables. The first case shown in Figure 6 a) needs two additional constraints as following
(liy ub y ) m1(lix lbix )
i typeB
(23)
(liy lb y ) m2 (lix ubix )
i typeB
(24)
where 0 m1, m2 . Only one constraint is active when the model determines the layout. The second and third case shown in Figures 6 b) and c) need three additional constraints to form the triangular feasible region and make the decision of department orientation, which is determined at the two vertices of isosceles triangle.
lbix lix
i typeB
9
(25)
lbiy liy
i typeB
(26)
ai lix (lbix )2 liy 2ai lbix
i typeB
(27)
ai lix (ubix )2 liy 2ai ubix
i typeB
(28)
Either constraint (25) or (26) works with (22) to determine the orientation as shown in Figure 6 b), or constraints (27) and (28) are alternatively active as (25) and (26). We tested the three methods mentioned above to compare the time to optimal found. The problem set is solved using IBM ILOG CPLEX Optimization Studio 12.6 in intel core i7-4500U CPU at 1.80 GHz computer with 8Gb memory. Table 1. Comparison of time to solve the problems CPU time(sec) Problem
OFV
FO7 FO7B1_1 FO7B1_2 FO7B2_1 FO7B2_2 FO8 FO8B1_1 FO8B1_2 FO8B2_1 FO8B2_2 FO9 FO9B1_1 FO9B1_2 FO9B2_1 FO9B2_2
20.45 22.04 22.81 25.49 26.80 22.25 28.83 25.75 31.5 29.15 23.46 27.87 27.69 32.12 28.75
a
b
C
OFV (Sherali et al)
17.48 6.81 20.33 68.95 65.63 40.52 136.94 246.55 268.98 225.81 6479.28 837.51
17.63 6.84 20.80 71.33 61.64 40.80 135.97 247.64 271.42 241.66 6781.22 968.69
17.86 6.66 20.95 69.41 60.73 40.55 136.73 250.59 273.47 229.64 6630.58 921.59
20.94 22.27 23.46 -
OFV (Castillo and Westerlund) 20.71 22.29 23.46 -
The basic test was performed using the well-known data for the layout design problem introduced by Meller et al. (1999) and used by several other researchers (for example Castillo & Westerlund, (2005) and Sherali et al. (2003)). Since there are no comparable results for the model in this research, a new modifed problem set from exising problems are generated. FO7 to FO9 are 7 to 9 type A department problem sets with one or two type A departments in the problem changed to type B departments. FO7B1_1 indicates a 7 department problem including 1 type B department, and the last 1 means the first problem in two constructed problems. The starting model without consideration of type B departments generates the optimal layout idential to Castillo & Westerlund (2005) and Sherali et al. (2003). The objective function values (OFVs) are a little different because of the different number of segments for linearizing the non-linear area constraint. There is not much difference in search time for the three models. However, the method a) is slightly better for FO9 problem set, and the method only adopts two more constraints. Thus, the best model is as follows. (1), (3) – (8) 10
ailix xik2 liy 2ai xik M (1 i ) , lbix xik ubix , i, k
(29)
(liy ub y ) (lix lbix ) M (1 i i )
(30)
(liy lb y ) (lix ubix ) M (i i ) i {0,1} ,
i
i
i
i {0,1}
(31) (32)
i is the parameter indicating the type of department, i 0 if i typeB and i 1 if i type A , and i is the decision variables for the type B department orientation. If i 0 , then the department is horizontally
long, and if i 1 , then the department is vertically long rectangular.
4. Feasibility Test The department area is underestimated because of linearization of non-linear area constraints. Thus, there is no guarantee that the final layout will properly represent the allocation of original departments. Once the model generates the optimal layout, then it needs to test the layout configuration to see whether it is feasible when the underestimated area is expanded to the original.
7 1
3
2
6
4
7
4 5 6 3
2 5
1
7
2
6 3
4
1 5
OFV = 22.04
(a)
OFV = 25.57
OFV = 27.4
(b) Figure 7. FO7B1_1 final layout configuration
(c)
Figure 7 shows that the final layout of 7-department problem with 1 type B department; department 5. The number of segments, the tangential support for the area constraint, to linearize area constraint is initially set to 10 and this model generates the layout show in Figure 7 (a). However, the layout turned out to be infeasible after expanding the shrunken area. To reduce the linearization error and keep the accuracy to a certain level, the number of segments can be increased and makes the layout closer to what it is intended (Castillo & Westerlund, 2005). Sherali et al. (2003) test this number to 50 and it is found that the problems used in that paper are not 11
sensitive in the number of linearization segments for their objective function value. The OFV of FO7 with 10 tangential supports of the area constraint is 20.94 and it increases to 20.95 with 50 supports in Sherali et al. (2003). In the case of increasing the number of support for the heterogeneous layout design model, the OFV is increased 14% and 24% after the number of segment set to 30 and 50 for FO7B1, respectively. This is because one type B department affects the flexibility of the floor space. Thus, the model to solve this type of problem needs to set the number of supports for the area constraints large enough to reduce the possibility of generating an infeasible layout. Figure 7 (b) shows the result when the number of supports are set to 30 and (c) shows the case of 50 supports. Table 2 shows the area of each department and the assigned area after the model solve the problem. Notice that the layout configuration of Figure 7 (b) and (c) are totally different from Figure 7 (a) and only (c) is feasible if the underestimated department area is expanded to their original area. The last column in Table 2, % error, indicates the difference between the original area and the model generated area which is underestimated by the linearization. Table 2. Area comparison of final layout for FO7B1_1 Dept.# Original (a) (b) (c)
1 16 15.75 16 16
2 16 15.75 15.99 16
3 16 16 16 16
4 36 36 36 36
5(fixed) 9 9 9 9
6 9 9 9 9
7 9 9 9 9
% error 0.45 0.02 0.00
Most of the results from the problem with 10 supports turned to infeasible after department area expansion as shown in Table 3; 2 out of 12 are feasible layout indicated OFV in bold. The results from the 50 supports for the area constraint show that the number of feasible configurations is increased but there still is no guarantee that the model can give the feasible layout even with 50-support. The model with 30 supports generate the solution closed to optimal and the % error also very closed to that of the model with 50 supports. However, it is found that the two third of problem set could not get the feasible layout in terms of original area constraints; 4 out of 12 are feasible. The layout configuration of the problem FO7B1_2 is identical to three different number of support in Table 3. That means 10 support model can find solutions as good as the 30 or 50 support model. For the problem FO7B2_1 and FO7B2_2, there is no feasible layout for both of 30 and 50 support models though it found the optimal layout for 10 support model. The area compactness, the ratio of the sum of all the department areas to the given floor space of the problems introduced in this research is closed to 1. This leads to the layout configuration with no flexibility. Furthermore, the problems are holding two fixed dimensioned departments among seven-department. Thus, it is easily lost its chance to have feasible solution. On the other hand, the problem FO8B2_1 and FO8B2_2 have found the optimal solution for 30 and 50 support problem and so problem FO9B2_1 and FO9B2_2 do. This is because the total area is relatively large than the fixed area so it has higher chance to have feasible layout. The time taken for the final solution was greatly increased as increasing the number of support for most of the problem set. In the case of FO7B1_2 and FO8B1_1, the time to solve was not increased as other problem sets were. It happened that the final layout of every case of the model is feasible including the ones of 100support model as shown Table 4.
12
Table 3 Effect of area constraints on feasibility 10-support Problem OFV Time(sec) % error OFV FO7B1_1 22.04† 17.48 0.45 25.57† FO7B1_2 6.81 0.02 22.81 22.81 FO7B2_1 25.49† 20.33 0.18 No sol. FO7B2_2 26.80† 68.95 0.48 No sol. FO8B1_1 65.63 0.15 28.83 28.89 FO8B1_2 25.75† 40.52 0.23 26.93† FO8B2_1 31.50† 136.94 0.16 38.91† † FO8B2_2 29.15 246.55 0.35 33.25 FO9B1_1 27.87† 268.98 0.31 29.48 FO9B1_2 27.69† 225.81 0.59 28.76 FO9B2_1 32.02† 6479.28 1.25 36.45 FO9B2_2 28.75 837.51 0.31 28.99† † the solution turned to infeasible after area expansion ‡ additional test for feasibility check is performed
30-support Time(sec) 121.06 16.05 343.80 782.86 58.23 113.33 2368.08 1236.13 1202.69 3737.22 45124.20 3260.11
% error 0.02 0.02 0.02 0.03 0.02 0.02 0.03 0.02 0.01 0.01
OFV 27.40 22.81‡ No sol. No sol. 28.89‡ 28.72 39.42‡ 33.26‡ 29.51‡ 28.82‡ 36.45 33.50
50-support Time(sec) 255.14 12.03 467.94 904.75 196.66 521.41 2515.47 2836.48 2559.00 3144.31 53866.44 6681.34
% error 0.00 0.01 0.02 0.00 0.02 0.02 0.02 0.01 0.00 0.00
It is noticeable that the % error generated by 30-support model is very close to that of 50-support model but it does not make the layout feasible as often as 50-support model. The % error of 30-support model for FO8B2_1 is 0.02 which is almost same as 50-support model. However, the % error of these two are not equal. The values were rounded off to two decimal points and it turned out to be different after the point of round off was changed. The % error of 30-support for this problem is 0.0223 while it is 0.0154 for 50-support problem. It might be assumed that these two layouts are identical, because the OFV is almost same and the % error shown in table 3 are equal, but the layouts are quite different from each other and that of 10-support model as shown in Figure 8. Additionally, the 100-support model is tested and found that there is no feasible solution if it uses the original dimensions for the problem.
(a) OFV=31.50 % error = 0.16
(b) OFV=38.91, % error = 0.02
(c) OFV=39.42, % error = 0.02
Figure 8. Layout comparison for FO8B2_1 (Department 4 and 8 have fixed dimension)
To make sure that the model generates a feasible solution, a 100-support model is run for the problem set, which the % error of underestimated area is greater than 0.00 in 50-support model. Table 4 shows the result of the test of the 100-support model for the selected problem as mentioned above and compare to that of 50-support model. The model with 100-support generates the layout without area underestimation if there exist the feasible solution.
13
Table 4 additional tests for feasibility with 100-support Problem FO7B1_2 FO8B1_1 FO8B2_1 FO8B2_2 FO9B1_1 FO9B1_2
50-support OFV 22.81 28.89 39.42† 33.26 29.51† 28.82†
Time(sec) 12.03 196.66 2515.47 2836.48 2559.00 3144.31
% error 0.01 0.02 0.02 0.02 0.02 0.01
OFV 22.82 28.90 No Sol. 33.26 32.36 32.20
100-support Time(sec) 28.19 148.06 45337.17 4935.97 15388.61 24514.06
% error 0.00 0.00 0.00 0.00 0.00
†
turned to infeasible after department expansion to original area
The greater number of type B departments affects the feasibility of the layout. The test for 3-type B departments imbedded in the existing problem set has been examined and the results are shown in Table 5. Adding just one more type B department to the problem set greatly reduces the chance to have feasible configuration. This is because the area of floor space is restricted with high area compactness, which is close to 1, and the possible cases of arrangement for all the department in the floor space are reduced as increasing the portion of type B department area in restricted space. Table 5 additional tests for 3 type B departments 10-support OFV Time(sec) % error FO7B3_1 0.74 41.57 314 FO7B3_2 No sol. 48.92 FO8B3_1 46.34 2498.59 0.51 FO8B3_2 No sol. 1427.0 FO9B3_1 0.24 38.58 22871.38 FO9B3_2 0.46 38.91 4014.63 † No feasible solution found in time limit (24hrs) Problem
OFV No sol. No sol. No sol. No sol. No sol. † No sol. †
30-support Time(sec) 141.58 65.11 4835.40 2729.56 86400 86400
% error -
It is expected that the problem set with more than 3 type B department included in FO7 and FO8 would generate infeasible solutions. Thus, only the 9 department problem, FO9, that includes 4 type B departments are tested as the extended problem. For FO9B4_1, the four smallest departments in FO9 are selected and set to fixed type, and four randomly selected departments are set to type B for FO9B4_2 to test its feasibility related to the fixed area portion in given space. These two problems generate the solution for 10-support but there is no solution for 30-support for both of problem in given condition as shown in Table 6. In this result, FO9B4_2 generates the optimal solution for 10-support problem much faster than that of FO9B4_1 which has less portion of fixed area. Thus,
the feasibility does not appear to be directly related to the portion of the fixed department area.
Problem
% of type B area at given space 23.1 49.4
Table 6 additional tests for FO9B4 30-support 10-support OFV Time(sec) % error OFV Time(sec) † † 0.57 No sol. 86400 45.29 86400 0.53 No sol. 26934.31 59.61 21046.22
FO9B4_1 FO9B4_2 † Best solution found in time limit(24hrs): 17.76% of optimality gap ‡ No feasible solution found in time limit(24hrs)
% error -
More than 4 type B department cannot be included in FO9 to get the feasible solution. A couple more experiments for FO9B5 were conducted and no feasible solution was found. It is mentioned in Liu & Meller 14
(2007) that it is difficult to have a feasible layout when a relatively small departmental aspect ratio is used along with a 100% area compactness, and the area relaxation allows many of the layout representations to be feasible even if it is limited in 5% of the total facility size, and this can greatly reduce the solution time. Bozer & Wang (2012) adopts the explanation of Liu & Meller (2007) and applied to their problem. Thus, the additional tests for the 5% area extension has been done for FO9B4 to evaluate its effects as shown in Table 7. Table 7 test for extended floor space No. of support
OFV 10† 45.29‡ 10 28.15 30 28.15 50 28.15 100 28.15 † No area extension applied ‡ Best solution found in 24hrs
FO9B4_1 Time(sec) 86400 63.23 85.80 45.17 77.48
% error 0.57 0.00 0.00 0.00 0.00
OFV 59.61 33.16 33.19 33.19 33.20
FO9B4_2 Time(sec) 21046.22 108.39 199.50 202.64 304.36
% error 0.53 0.13 0.01 0.01 0.00
The first row of the results in Table 7 is brought from Table 6 to compare the results. Only 5% of facility area relaxation can drop the computational time dramatically and provide chance to have low-cost configuration. The layout configuration generated by this extension cannot be applicable in the case that the physically inflexible boundary is used as a floor space but this gives great alternative if this extension is acceptable. Bozer & Wang (2012) stated “If no empty space is provided and department shape constraints are used, potentially many lowcost layouts that would have been adopted by the decision-maker would be needlessly rejected by the algorithm”, and the heterogeneous department layout problem is face to same issue and even in tighter condition in that the type B department have no flexibility. Since we expect that adding a small amount of empty space to candidate facilities in this type of problem would give the chance to solve bigger problem, we modified FO9 to make 10 to 13 heterogeneous department problem and set the facility area is 5% greater than the total department area. Additional test has been done with 30-supports since the departmental area discrepancy with 30-supports is greatly reduced comparing to that of 10-support. Table 8 Additional tests for larger problems Problem OFV Time(sec) % error CR10B3 28.53 36.59 0.03 CR11B4 37.51 1307.97 0.03 CR12B5 41.20 15865.17 0.03 CR13B6 44.65† 86400 0.03 † Best solution found in time limit(24hrs): 4.84% of optimality gap
The results in additional test for larger problem are shown in Table 8. The area relaxation allows the larger problem have the solution much faster than the one without floor space extension. The model provides the optimal solution for 12-department problem with 5 type B departments. For 13-department problem with 6 type B departments, the model generates the solution with 4.84% of optimality gap in 24 hours. The optimal layout for 12-department is provided in Figure 9.
15
Figure 9 Optimal layout for 12-department problem (OFV = 41.2, Type B = {1, 2, 5, 10, 12})
5. Conclusion In this research, a facility layout problem with unequal area department is presented. We propose a model for layout design with heterogeneous area constraints which includes the two different types of department area constraints; one for flexible width and length for the specific area defined as type A department, the other for fixed dimension defined as type B department. Three models to choose the orientation of type B departments have been tested, and one of the methods has been chosen to efficiently represent the problem. The model generates the final layout configuration considering these two types of departments. However, the feasibility issues arise when the area compactness is close to 1. The model was derived from the basic facility layout problem, which is for all type A departments, by embedding decision making constraints for type B departments. The linearization for non-linear area constraints is very critical for type A departments and the number of linear segments affects solution accuracy. The layout solution with a small number of linear support constraints, (up to 10) could not insure that the layout configuration is feasible if the underestimated area is expanded to the original area. This issue reported by previous researches which deal with only type A departments. Adding type B department to the model makes the layout lose its flexibility much more than the model of type A and increases the chance of an infeasible solution. Thus, it is important to set the level of flexibility when the layout design with heterogeneous area constraints is applied to the problem with high area compactness. The issue will be released if the floor space is much bigger than the total area of the departments. In this research, it is tested that how the model works for solution after small amount of empty space is added to original floor space. Precisely, the solution generated by this floor space expansion is not feasible for the original facility, but it gives great alternatives to have cost-effective layout in favorable time. According to the results generated by the model, the time taken to solve the problem increases as the area accuracy is increased. In some cases, there is no feasible solution for original area. And there are important trade-offs between solution time, the gap between the sum of the areas of the individual departments and the total floor space, the allowable error introduced by the area linearization. These optimization models are restricted to solving problems with fewer than 10 departments with non-flexible floor space. Thus, an important future research direction for these problems is to develop effective heuristic solutions for larger departments.
16
References Bock, S., & Hoberg, K. (2007). Detailed layout planning for irregularly-shaped machines with transportation path design. European Journal of Operational Research, 177, 693–718. http://doi.org/10.1016/j.ejor.2005.11.011 Bozer, Y. A., & Wang, C. T. (2012). A graph-pair representation and MIP-model-based heuristic for the unequal-area facility layout problem. European Journal of Operational Research, 218(2), 382–391. http://doi.org/10.1016/j.ejor.2011.10.052 Bukchin, Y., & Tzur, M. (2014). A new MILP approach for the facility process-layout design problem with rectangular and L/T shape departments. International Journal of Production Research, 7543(January 2015), 37–41. http://doi.org/10.1080/00207543.2014.930534 Castillo, I., & Westerlund, T. (2005). An ε-accurate model for optimal unequal-area block layout design. Computers and Operations Research, 32(3), 429–447. http://doi.org/10.1016/S0305-0548(03)00246-6 Chae, J. (2014). Mixed type layout design model for minimizing material handling movement. Korean Journal of Logistics, 22(3), 51–60. Das, S. K. (1993). A Facility Layout Method for Flexible Manufacturing System. International Journal of Production Research, 31(2), 279–297. Deb, S. K., Bhattacharyya, B., & Bhattacharyyaz, B. (2005). Solution of facility layout problems with pickup/drop-off locations using random search techniques. International Journal of Production ResearchOnline) Journal International Journal of Production Research International Journal of Production Research, 4322(22), 20–7543. http://doi.org/10.1080/00207540500057506 Drira, A., Pierreval, H., & Hajri-Gabouj, S. (2007). Facility layout problems: A survey. Annual Reviews in Control, 31(2), 255–267. http://doi.org/10.1016/j.arcontrol.2007.04.001 Heragu, S. S., & Kusiak, A. (1990). Machine layout: an optimization and knowledge-based approach. International Journal of Production Research, 28(4), 615–635. http://doi.org/10.1080/00207549008942744 Huchette, J., Dey, S. S., & Vielma, J. P. (2016). Strong mixed-integer formulations for the floor layout problem, 1–43. Retrieved from http://arxiv.org/abs/1602.07760 Kim, J.-G. (2006). A Mixed Integer Programming Model to Modify a Block Layout for Obtaining Smooth Material Flow Paths. Asia Pacific Management Review, 11(1), 21–27. Kim, J.-G., & Kim, Y.-D. (2000). Layout planning for facilities with fixed shapes and input and output points. International Journal of Production Research, 38(18), 4635–4653. http://doi.org/10.1080/00207540050205550 Konak, A., Kulturel-Konak, S., Norman, B. A., & Smith, A. E. (2006). A new mixed integer programming formulation for facility layout design using flexible bays. Operations Research Letters, 34(6), 660–672. http://doi.org/10.1016/j.orl.2005.09.009 Kusiak, A., & Heragu, S. S. (1987). The facility layout problem. European Journal of Operational Research, 29, 229–251. Lacksonen, T. A. (1994). Static and Dynamic Layout Problems with Varying Areas. The Journal of the Operational Research Society, 45(1), 59–69. http://doi.org/10.1057/jors.1994.7 Leno, I. J., Sankar, S. S., & Ponnambalam, S. G. (2013). An elitist strategy genetic algorithm using simulated annealing algorithm as local search for facility layout design. International Journal of Advanced Manufacturing Technology, 1–13. http://doi.org/10.1007/s00170-013-5519-3 Leno, I. J., Sankar, S. S., & Ponnambalam, S. G. (2015). MIP model and elitist strategy hybrid GA–SA algorithm for layout design. Journal of Intelligent Manufacturing, (August), 1–19. http://doi.org/10.1007/s10845-015-1113-x 17
Levary, R. R., & Kalchik, S. (1985). Facilities layout-a survey of solution procedures. Computers & Industrial Engineering, 9(2), 141–148. Liggett, R. S. (2000). Automated facilities layout: Past, present and future. Automation in Construction, 9(2), 197–215. http://doi.org/10.1016/S0926-5805(99)00005-9 Liu, Q., & Meller, R. D. (2007). A sequence-pair representation and MIP-model- based heuristic for the facility layout problem with rectangular departments. IIE Transactions, 8830(March), 377–394. http://doi.org/10.1080/07408170600844108 Meller, R. D., Chen, W., & Sherali, H. D. (2007). Applying the sequence-pair representation to optimal facility layout designs. Operations Research Letters, 35(5), 651–659. http://doi.org/10.1016/j.orl.2006.10.007 Meller, R. D., & Gau, K. Y. (1996). The facility layout problem: Recent and emerging trends and perspectives. Journal of Manufacturing Systems, 15(5), 351–366. http://doi.org/10.1016/0278-6125(96)84198-7 Meller, R. D., Narayanan, V., & Vance, P. H. (1999). Optimal facility layout design. Operations Research Letters, 23, 117–127. Montreuil, B. (1990). A Modeling Framework for Integrating Layout Design and Flow Network Design. In Progress in Material Handling Research Colloquium (pp. 43–58). Hebron, KY. Murata, H., Fujiyoshi, K., Nakatake, S., & Kajitani, Y. (1996). VLSI module placement based on rectanglepacking by the sequence-pair. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 15(12), 1518–1524. http://doi.org/10.1109/43.552084 Ridwan, M., Purnomo, A., & Wiwoho, Y. S. (2016). Multi-objective Mixed Integer Programming approach for facility layout design by considering closeness ratings , material handling , and re-layout cost. Materials Science and Engineering, 105, 0–10. http://doi.org/10.1088/1757-899X/105/1/012045 Saraswat, A., Venkatadri, U., & Castillo, I. (2015). A framework for multi-objective facility layout design. Computers and Industrial Engineering, 90, 167–176. http://doi.org/10.1016/j.cie.2015.09.006 Sherali, H. D., Fraticelli, B. M. P., & Meller, R. D. (2003). Enhanced Model Formulations for Optimal Facility Layout. Operations Research, 51(4), 629–644. http://doi.org/10.1287/opre.51.4.629.16096 Singh, S. P., & Sharma, R. R. K. (2006). A review of different approaches to the facility layout problems. International Journal of Advanced Manufacturing Technology, 30(5-6), 425–433. http://doi.org/10.1007/s00170-005-0087-9 Tao, J., Wang, P., Qiao, H., & Tang, Z. (2012). Facility layouts based on intelligent optimization approaches. 2012 IEEE 5th International Conference on Advanced Computational Intelligence, ICACI 2012, 502–508. http://doi.org/10.1109/ICACI.2012.6463215 Yang, T., & Peters, B. A. (1998). Theory and Methodology Flexible machine layout design for dynamic and uncertain production environments. European Journal of Operational Research, 108, 49–64.
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Acknowledgement This research was partially supported by the MSIP(Ministry of Science, ICT and Future Planning), Korea, under the ITRC(Information Technology Research Center) support program ((IITP-2016-H8601-16-1010)) supervised by the IITP(Institute for Information & communications Technology Promotion). Amelia Regan was partially supported by a Rueben Smeed Visiting Professorship at the Centre for Transport Studies at University College London during the period of this research.
l1x
liy ubiy
lilong
l1y
l l2x
lishort long i
l
y 2
lishort
a′
(ubiy − lbiy )lix y i
lb
+ (ubix − lbix )liy
b′
≥ ubix ubiy − lbix lbiy
lbix
ubix
lix
We propose a new model to solve the layout design problem with flexible and fixed departments.
These heterogeneous departments significantly increase problem complexity.
Fine grained tangential support for area constraints is sometimes required.
S0360-8352(16)30385-0 http://dx.doi.org/10.1016/j.cie.2016.10.016 CAIE 4501
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
28 April 2016 19 September 2016 20 October 2016
Please cite this article as: Chae, J., Regan, A., Layout Design Problems with Heterogeneous Area Constraints, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.10.016
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Layout Design Problems with Heterogeneous Area Constraints Junjae Chae1 and Amelia Regan2 * 1
School of Air Transport, Transportation and Logistics, Korea Aerospace University, South Korea
2
Department of Computer Sciences, University of California at Irvine, USA
Abstract The facility layout problem (FLP) is one of the design problems involving the assignment of facilities (e.g., machines, departments) to planar region (e.g., a plant), so as to achieve the objectives such as to minimize the cost of projected interaction between facilities or to maximize the closeness rating, etc. In this study, we deal with one of the FLP models that minimize the material handling cost between rectangular departments. Each department has an area restriction that specifies the total area that it must occupy while the specific lengths and widths are determined by the model. However, some department do not have flexibility and the dimension are predetermined. Thus, we proposed the to solve the layout design problem with two types of department. We call these constraints heterogeneous area constraints. The type A department, flexible length and width, and type B department, fixed length and width, are assigned to given floor space. Using a well-known data set which has been previously used by many other researchers, and modified version of the data aimed at testing our model, we show that the proposed model properly generates layout design for these two types of departments.
Keywords: FLP (facility layout problem); MIP (mixed integer programming); Heterogeneous area constraint; Block layout design with unequal areas.
*
Corresponding author; [email protected]
1. Introduction The facility layout problem (FLP) is to find a non-overlapping arrangement of work areas (e.g., departments, workstations or service areas) within a given space with respect to a variety of objectives (e.g., minimizing total distance of material flow or maximizing the closeness effect). The departments or workstations are considered to be of rectangular shapes by many researchers, though others have assumed that the departments or workstations have irregular-shaped areas (Bock & Hoberg, 2007; Bukchin & Tzur, 2014). Irregularly shaped areas arise in application such as the design of shopping malls, zoos or ships, while factories, warehouses and cross-docking facilities typically have rectangular areas. For the objectives, minimizing the cost due to material flow between departments or workstations is common, but some research has been conducted for maximization problems or multi-objective problems. As mentioned, the FLP has been studied by many researchers and it has various problem types. For example, Kusiak and Heragu (1987), Meller and Gau (1996) provide detailed surveys of the different variants and solution approaches for the problem. Drira et al. (2007) provides a tree representation for problem types, models, and approaches for facility layout problems. Other review papers focus on solution approaches tailored to specific aspects of layout problems (Levary & Kalchik, 1985; Liggett, 2000; Singh & Sharma, 2006; Tao, Wang, Qiao, & Tang, 2012). The review of facility layout problems by Singh and Sharma (2006) and Drira et al. (2007) categorize two mathematical formulations for facility layout design: one is QAP (Quadratic Assignment Problem) for discrete formulations, and the other is MIP (Mixed Integer Programming) for continuous formulations. In this study, we deal with the continuous space of a facility assignment, MIP model, and have the objective of minimizing the total weighted distance of material flow in the space. The weighted distance between a paired departments is measured by the rectilinear distance multiplied by the number of trips between the centroids of the departments. Further, each department has an area restriction that specifies the total area that it must occupy, while the specific lengths and widths are determined by the model. The facility layout problem is referred to as the block layout design problem when it deals with only rectangular blocks for departments. The problem has two main restrictions: the area requirement for the departments and floors, and the department location restrictions. The location restrictions are based on the continuous department positions in the given space and the area requirement for the department needs to handle the varying dimensions. The models and solution approaches for this block layout design problem has been studied by a number of researchers and some of studies are focused on the direct solution of the optimization problem with unequal (non-uniform) area requirements. On the other hand, a model could be required to specify the orientation of the department in the case in which departments have fixed dimensions. The models for this block layout design problem usually handle unequal area departments either with flexible dimension or with fixed dimension. In our research, we consider departments with unequal areas with changeable dimensions in the case of type “A” departments and fixed dimensions in the case of type “B” departments. To handle these two types of departments in a model, we need to analyze which part of variables and parameters can be used to represent both types of departments and which variables and parameters are necessarily used for specific types of departments. We follow the path of the modeling technique for direct 1
solution of optimization of unequal area block layout design problems. In other word, the FLP model introduced by Sherali et al.(2003) and Castillo and Westerlund (2005) is used to assign the type A departments. Then we provide a method to embed the different types of department restrictions, type B, into the model. Several methods to represent the type B department are evaluated and one of the methods is selected to use for embedding into the FLP model for type A department. The FLP model for type A department presented by Sherali et al.(2003) and Castillo and Westerlund (2005) uses the linearization technique for non-linear department area constraints which is represented by the horizontal length multiplied by the vertical length. Thus this model has an area feasibility issue if there is no tolerance of restrictions for maximum aspect ratio of department. That means the layout configuration generated by the model could become infeasible when the underestimated department is expanded to its intended size. The chance that the model is feasible is increased when the model includes the type B department because of inflexibility of the department area. We test the feasibility of the layout configuration of the given problem set and show the possible number of linear segments of non-linear area constraint to have a feasible layout. Additional tests were done for relaxation on floor space to see the impact of area compactness on feasibility. Specifically, a larger problem set was tested to check the applicability of this relaxation. More details of the problem are explained in Section 2 and the background and the motivation for our research is also discussed in that Section. The modeling procedure and the methodology including heterogeneous department types in a single model are presented in Section 3, and the results generated by the proposed model are discussed in Section 4. In the last Section, the conclusion and directions for future research are provided.
2. Background Most facility layout research resorts to developing heuristic solution methods since these problems are very difficult to solve optimally. Various solution approaches have been developed to solve these layout problems and several review papers present summaries of these (Drira et al., 2007; Kusiak & Heragu, 1987; Levary & Kalchik, 1985; Meller & Gau, 1996; Singh & Sharma, 2006). Adding the unequal area requirements with varying dimensions makes the problem even more difficult to solve. However, there have been attempts to solve the problem through traditional modeling and optimization techniques. Several studies present improved techniques to model problems in this category (Castillo & Westerlund, 2005; Lacksonen, 1994; Meller et al., 1999; Montreuil, 1990; Sherali et al., 2003). These research efforts provide methods to linearize the unequal area constraint so that a mixed integer linear programming (MILP) model can be solved. The models in those papers assume that all the departments have variable width and length dimensions. However, as discussed in Sherali et al.(2003), some departments could have fixed dimensions because of some specific physical characteristics of the activities of that department. Thus, the modified model considering both fixed and variable departments is necessary. As mentioned, we set type A departments as those with flexible dimensions and type B departments as those with fixed width and length. Initially, this categorization was introduced in Chae (2014). The characteristics of 2
these department types are shown in Figure 1. Different approaches are needed to model those different types of department layout problems. For the layout problem for type A departments, the lengths and widths are decision variables as are the centroids of each department. In the case of layout problem with type B departments, the position of the centroid of each department is the main concern. However, the latter model is not easier since the model needs to determine the precise orientation of each department. Thus handling these two types of departments in a model increases the difficulty because of the addition of new binary variables.
lix Equal area
liylong
liy l jy l
x short i
x j
l jxlong
l
Type A
Type B
Figure 1. Characteristics of type A and B departments There are two basic approaches to include these two different types of departments in a model. The first is that the constraints for type B departments are inserted to the layout model for type A departments. The second is the opposite: type A department constraints are inserted into the model for type B departments. The former method is more reasonable since starting from a model which can handle more flexible layouts and constraining some of these to be less flexible, is more straightforward than adding flexibility to a model with more constrained layouts. Therefore, that is the approach we use in our work. The base model for type A department layout used in this study is based on the model introduced by Sherali et al. (2003) and Castillo & Westerlund (2005). The parameters and variables of the model and the model itself for type A, FLP-A, are as follows. Parameters:
f ij : material flow volume from department i to j ai : area requirement for department i
lix , liy : horizontal and vertical length of department i Lx , Ly : horizontal and vertical length of floor space (facility)
lbix , ubix , lbiy , ubiy : lower and upper bound of lix , liy Variables: d ij : distance from department i to j (cix , ciy ) : coordinates for centroid of department i
zijx 1 : if department i is to the left of department j ( zijx 0 , otherwise) zijy 1 : if department i is lower than department j ( zijy 0 , otherwise)
FLP-A:
3
Min
f d
s.t.
(1)
ij ij
i
j
ai lix liy i
(2)
s
l ls c i c sj j Ls (1 zijs ) i j, s x, y 2 2 s i
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(3)
(4)
s x
lis ls cis Ls i 2 2 lbis lis ubis
i, s
(5)
i, s
dij cix c xj ciy c jy
(6) i j
zijs 0,1 i j
(7) (8)
In the objective function shown in (1), the total weighted distance describes the sum of the product of the flows ( f ij ) and the distances ( d ij ) separating departments i and j . A rectilinear distance metric is used for measuring travel distance between departments’ centroids ( cis , c sj ) as shown in (7). The area of department i is expressed in (2) where its width and length are ( l ix l iy ) and the department areas should not overlap in a restricted space, ( Lx Ly ), as shown in (3). The binary variable, zijs , indicates the relative position of departments. Constraints (4) control the relative position of departments. Constraints (5) make sure that all the departments are in the given space. Each department is restricted by an aspect ratio ( i 1 ), which, practically, defines the maximum permissible ratio between its longest and shortest sides. The lower and upper bounds of each department’s dimension is based on its aspect ratio as in (6). Montreuil (1990) introduced a MIP model for facility layout design and used bounded perimeter constraint for area control. However, the constraint generates the department area much smaller than the actual. Lacksonen (1994) proposed linearization of area constraint which generates area between 0% to 3% larger than required area when i 2 . However, this linearization requires two additional binary variables for each department. Meller et al.(1999) proposed the model with linearization of non-linear unequal area constraints using perimeter and area relationship. This improved the previous perimeter constraint introduced by Montreuil (1990) and generated maximum 2.5% error when i 2 . As demonstrated in figure 1 (type A) the non-linear line indicates that departments of equal area and this non-linear line can be linearized by using several linear expression. The process of linearization generates the measureable gap with the actual area. Sherali et al. (2003) and Castillo & Westerlund (2005) suggested MIP models for block layout problem, which is a modification of the basic FLP model shown above. These two models provide the method to linearize the unequal area curve using tangential support as shown in figures 2 and 3. Castillo & Westerlund (2005) use the number of tangential supports as the factor controlling the maximum error generated by linearizing the unequal area curve. Sherali et al. (2003) tested the sensitivity of the errors generated by varying the number of tangential supports in the model. 4
The variable xik ( lbix xik ubix ) in figure 3 indicates the point in the x axis of tangential supports and the model includes up to k linearized constraints. The number k is the factor for controlling the approximation error in Castillo & Westerlund (2005). The approaches of these two studies are almost identical in terms of area representation with linearized equations. As mentioned above, the lower and upper bound of a department’s dimension is based on its aspect ratio. We can tighten these bounds by setting
ubi min ai i , Ls , i, s x, y s
lbi s
(9)
ai , i, s x, y ubis
(10)
These new constraints are derived from constraints (2) and (6), and ls Ls , s x, y presented using somewhat different notation.
l iy
ubiy
a
b
lbiy
lbix
ubix
l iy
Figure 2. Outer approximation to the area constraints using tangential supports (re-drawn based on Sherali et al. (2003)) l iy
ubiy
a
ai lix (lbix )2 liy 2ai lbix ai lix xik2 liy 2ai xik ailix (ubix )2 liy 2aiubix
b
lbiy
lbix x k
ubix
lix
Figure 3. Linearized constraints for the area approximation (re-drawn based on Castillo & Westerlund (2005))
Constraints (2) are replaced with linearized area constraints as follows ailix xik2 liy 2ai xik , lbix xik ubix , i, k
5
(11)
and the values of xik equal to
k ubix lbix , k 0,1,...,K 1, for any selected integer K 2 . K 1
xik lbix
(12)
This approach is purely linear and no binary variables are involved. Because the problem is to minimize the distance, the tendency of the problem is to underestimate the area, thus the equality constraints in (2) can be substituted by (11) as inequality constraints. The larger integer K makes the formulation generate results closer to that of non-linear area constraint. The error generated by underestimating the area is controlled in (Castillo & Westerlund, 2005) by setting K to a large number. Meller et al. (2007) proposed another FLP formulation using location relationship for each department, sequence-pair, which is inspired by Murata et al. (1996). The basic frame of this model is same as the one in Sherali et al. (2003) and the redundancy in solution set is reduced through the proposed model. The model by Sherali et al. (2003) also was used as a base work for the model in Saraswat et al. (2015) and Ridwan et al. (2016) for applying various objectives in FLP. Konak et al. (2006) presented the formulation for FLP in different structure – flexible bay – with keeping the basic form as MIP. Kim (2006) used the linearization technique similar to Sherali et al. (2003) and modification method to configure less turn material handling path.
A recent research, Huchette et al. (2016), presented a case study on
formulations for the floor layout problem. They showed the comparison of results generated by several different approaches for FLP using MIP formulation and observed impact of varying inputs to the model. The model for type B department layout is different from that of type A. The orientation of the departments should be considered when it is assigned to a floor space and this increases the solution time since the binary variables are used to express its orientation in the model. The basic model for FLP with fixed departments, FLPB, is as following. FLP-B:
Min
(13)
f d
ij ij
i
s.t.
j
l short l long l short l long j j cix i (1 wi ) i wi c xj (1 w j ) w j Lx (1 zijx ), i j 2 2 2 2
(14)
l short l long lishort l long j j wi i (1 wi ) c yj wj (1 w j ) Ly (1 zijy ), i j 2 2 2 2
(15)
ciy
lishort l long l short l long (1 wi ) i wi cix Lx i (1 wi ) i wi , i 2 2 2 2
(16)
lishort l long l short l long wi i (1 wi ) ciy Ly i wi i (1 wi ), i 2 2 2 2
(17)
y
( z
s ij
z sji ) 1 i j
(18)
s x
dij cix c xj ciy c jy i j
(19)
zijs 0,1 i j
(20) 6
wi 0,1 i
(21)
Some additional variable and parameters are introduced to represent the characteristics of this specific problem:
wi controls department orientation, lishort , lilong indicate the length of short and long sides of departments. The vertical and horizontal length of each department are the parameters in this model. However, the orientation is a decision variable, and the binary variable, wi , indicates this orientation. If wi 1 , then the department is horizontally long rectangular shape.
Constraints (14) and (15) prevent overlapping departments and determine
orientation. Two constraints selectively work for best relative location and the orientation assignment to floor space. The floor space has boundaries indicated by Lx and Ly . Constraints (16) and (17) insure that all the departments and their segments are in the boundary of the given floor space. However, without the flexibility of the department dimension, it is not easy to find a feasible layout if the floor space is not enough to hold the assigned fixed type department. This model is similar to MLP (Machine Layout Problem) or cell block layout in FMS (Flexible Manufacturing System) in that the machine or cell block have fixed dimension. Heragu & Kusiak (1990) presented a model in MIP form for machine allocation in given row in available area. Das (1993) presented the FMS layout design model extended from the MLP, and the dimension of each cell block and pickup/ drop-off (p/d) point on each cell block is predefined and the orientation of each cell block is a decision variable. This model is used as a basic model for other research. Yang & Peters (1998) presented MLP model with different environment: dynamic and uncertain production. However, the handling the orientation and overlap prevention constraints are as in Das (1993). Kim & Kim (2000) used the term ‘fixed shape facility’ instead of ‘machine’ and presented a model in MIP form and similar model presented in Deb et al. (2005), Leno et al. (2013) and Leno et al. (2015), and all those researches used the coordinates of block corner as an indication of block position in the floor space instead of using centroid of block. The model controls the p/d point located on the boundary of block while the model in Das (1993) defined the p/d position on center line in block.
3.
Embedding constraints for type B department to the model for type A department
layout As it is mentioned, the basic model for layout holding two different types of department is the model for type A department layout. There are several ways to represent the orientation of departments for type B departments in a given type A department layout model. Type B departments do not have the upper and lower limit for the area since these already have their own dimensions. However, the dimension of type A departments are expressed and determined by the area constraint (9) to (11) in previous section so we need lbix , ubix , lbiy , ubiy to express the dimension of type B departments.
7
l iy
a (lbix , ubiy )
ubiy
lbiy
b (ubix , lbiy )
lix lbix
ubix
Figure 4. Possible points of type B department
The dimension of a type B department can be either a or b in Figure 4. If the department is vertically long then the dimension of the department is lbix ubiy , and the point a , and if the department horizontally long, then it has lbiy ubix as the dimensions, the point b .
l iy
ubiy
a
(ubiy lbiy )l ix
lbiy
(ubix lbix )l iy
b
ubix ubiy lbix lbiy
lbix
ubix
lix
Figure 5. Two possible points specifying dimensions for a type B Department
The straight line passing through two point a and b can be shown as follows
(ubiy lbiy )lix (ubix lbix )liy ubixubiy lbixlbiy
i t y pB e
(22)
The shaded area on triangle formed by constraints (6) and (22) in figure 5 indicates the feasible region of possible determination of type B department area. Since the triangle satisfies the convex condition and the LP (Linear Programming) solution finds the solution at extreme point, and the model tends to find the minimum value, it is expected that the solution could be generated at point a or b .
If this fact is true or all the case of
type B departments, the additional binary variables selecting either the point a or b would not need to be introduced to the model.
However, our model is a MIP and the solution is not necessarily at the extreme points. 8
Thus, it is necessary to adopt the binary variables for determining the orientation of the department which is either point a or b . (liy ubiy ) m1(lix lbix )
l iy
ubiy
l iy
a
a
ubiy
lbix lix
lbiy
b
lbiy
b
lbiy liy
(liy lbiy ) m2 (lix ubix )
lbix
ubix
lbix
lix
(a)
ubix
lix
(b) ai lix (lbix )2 liy 2ai lbix
l iy
a
ubiy
lbiy
b a l (ub ) l 2ai ub x i i
x 2 y i i
x i
lbix
ubix
lix
(c) Figure 6. type B department orientation determination Our research considers three different cases of modeling for determination of type B department’s orientation. These three models choose the orientation of the type B department by selectively activating the constraints which are chosen by binary variables. The first case shown in Figure 6 a) needs two additional constraints as following
(liy ub y ) m1(lix lbix )
i typeB
(23)
(liy lb y ) m2 (lix ubix )
i typeB
(24)
where 0 m1, m2 . Only one constraint is active when the model determines the layout. The second and third case shown in Figures 6 b) and c) need three additional constraints to form the triangular feasible region and make the decision of department orientation, which is determined at the two vertices of isosceles triangle.
lbix lix
i typeB
9
(25)
lbiy liy
i typeB
(26)
ai lix (lbix )2 liy 2ai lbix
i typeB
(27)
ai lix (ubix )2 liy 2ai ubix
i typeB
(28)
Either constraint (25) or (26) works with (22) to determine the orientation as shown in Figure 6 b), or constraints (27) and (28) are alternatively active as (25) and (26). We tested the three methods mentioned above to compare the time to optimal found. The problem set is solved using IBM ILOG CPLEX Optimization Studio 12.6 in intel core i7-4500U CPU at 1.80 GHz computer with 8Gb memory. Table 1. Comparison of time to solve the problems CPU time(sec) Problem
OFV
FO7 FO7B1_1 FO7B1_2 FO7B2_1 FO7B2_2 FO8 FO8B1_1 FO8B1_2 FO8B2_1 FO8B2_2 FO9 FO9B1_1 FO9B1_2 FO9B2_1 FO9B2_2
20.45 22.04 22.81 25.49 26.80 22.25 28.83 25.75 31.5 29.15 23.46 27.87 27.69 32.12 28.75
a
b
C
OFV (Sherali et al)
17.48 6.81 20.33 68.95 65.63 40.52 136.94 246.55 268.98 225.81 6479.28 837.51
17.63 6.84 20.80 71.33 61.64 40.80 135.97 247.64 271.42 241.66 6781.22 968.69
17.86 6.66 20.95 69.41 60.73 40.55 136.73 250.59 273.47 229.64 6630.58 921.59
20.94 22.27 23.46 -
OFV (Castillo and Westerlund) 20.71 22.29 23.46 -
The basic test was performed using the well-known data for the layout design problem introduced by Meller et al. (1999) and used by several other researchers (for example Castillo & Westerlund, (2005) and Sherali et al. (2003)). Since there are no comparable results for the model in this research, a new modifed problem set from exising problems are generated. FO7 to FO9 are 7 to 9 type A department problem sets with one or two type A departments in the problem changed to type B departments. FO7B1_1 indicates a 7 department problem including 1 type B department, and the last 1 means the first problem in two constructed problems. The starting model without consideration of type B departments generates the optimal layout idential to Castillo & Westerlund (2005) and Sherali et al. (2003). The objective function values (OFVs) are a little different because of the different number of segments for linearizing the non-linear area constraint. There is not much difference in search time for the three models. However, the method a) is slightly better for FO9 problem set, and the method only adopts two more constraints. Thus, the best model is as follows. (1), (3) – (8) 10
ailix xik2 liy 2ai xik M (1 i ) , lbix xik ubix , i, k
(29)
(liy ub y ) (lix lbix ) M (1 i i )
(30)
(liy lb y ) (lix ubix ) M (i i ) i {0,1} ,
i
i
i
i {0,1}
(31) (32)
i is the parameter indicating the type of department, i 0 if i typeB and i 1 if i type A , and i is the decision variables for the type B department orientation. If i 0 , then the department is horizontally
long, and if i 1 , then the department is vertically long rectangular.
4. Feasibility Test The department area is underestimated because of linearization of non-linear area constraints. Thus, there is no guarantee that the final layout will properly represent the allocation of original departments. Once the model generates the optimal layout, then it needs to test the layout configuration to see whether it is feasible when the underestimated area is expanded to the original.
7 1
3
2
6
4
7
4 5 6 3
2 5
1
7
2
6 3
4
1 5
OFV = 22.04
(a)
OFV = 25.57
OFV = 27.4
(b) Figure 7. FO7B1_1 final layout configuration
(c)
Figure 7 shows that the final layout of 7-department problem with 1 type B department; department 5. The number of segments, the tangential support for the area constraint, to linearize area constraint is initially set to 10 and this model generates the layout show in Figure 7 (a). However, the layout turned out to be infeasible after expanding the shrunken area. To reduce the linearization error and keep the accuracy to a certain level, the number of segments can be increased and makes the layout closer to what it is intended (Castillo & Westerlund, 2005). Sherali et al. (2003) test this number to 50 and it is found that the problems used in that paper are not 11
sensitive in the number of linearization segments for their objective function value. The OFV of FO7 with 10 tangential supports of the area constraint is 20.94 and it increases to 20.95 with 50 supports in Sherali et al. (2003). In the case of increasing the number of support for the heterogeneous layout design model, the OFV is increased 14% and 24% after the number of segment set to 30 and 50 for FO7B1, respectively. This is because one type B department affects the flexibility of the floor space. Thus, the model to solve this type of problem needs to set the number of supports for the area constraints large enough to reduce the possibility of generating an infeasible layout. Figure 7 (b) shows the result when the number of supports are set to 30 and (c) shows the case of 50 supports. Table 2 shows the area of each department and the assigned area after the model solve the problem. Notice that the layout configuration of Figure 7 (b) and (c) are totally different from Figure 7 (a) and only (c) is feasible if the underestimated department area is expanded to their original area. The last column in Table 2, % error, indicates the difference between the original area and the model generated area which is underestimated by the linearization. Table 2. Area comparison of final layout for FO7B1_1 Dept.# Original (a) (b) (c)
1 16 15.75 16 16
2 16 15.75 15.99 16
3 16 16 16 16
4 36 36 36 36
5(fixed) 9 9 9 9
6 9 9 9 9
7 9 9 9 9
% error 0.45 0.02 0.00
Most of the results from the problem with 10 supports turned to infeasible after department area expansion as shown in Table 3; 2 out of 12 are feasible layout indicated OFV in bold. The results from the 50 supports for the area constraint show that the number of feasible configurations is increased but there still is no guarantee that the model can give the feasible layout even with 50-support. The model with 30 supports generate the solution closed to optimal and the % error also very closed to that of the model with 50 supports. However, it is found that the two third of problem set could not get the feasible layout in terms of original area constraints; 4 out of 12 are feasible. The layout configuration of the problem FO7B1_2 is identical to three different number of support in Table 3. That means 10 support model can find solutions as good as the 30 or 50 support model. For the problem FO7B2_1 and FO7B2_2, there is no feasible layout for both of 30 and 50 support models though it found the optimal layout for 10 support model. The area compactness, the ratio of the sum of all the department areas to the given floor space of the problems introduced in this research is closed to 1. This leads to the layout configuration with no flexibility. Furthermore, the problems are holding two fixed dimensioned departments among seven-department. Thus, it is easily lost its chance to have feasible solution. On the other hand, the problem FO8B2_1 and FO8B2_2 have found the optimal solution for 30 and 50 support problem and so problem FO9B2_1 and FO9B2_2 do. This is because the total area is relatively large than the fixed area so it has higher chance to have feasible layout. The time taken for the final solution was greatly increased as increasing the number of support for most of the problem set. In the case of FO7B1_2 and FO8B1_1, the time to solve was not increased as other problem sets were. It happened that the final layout of every case of the model is feasible including the ones of 100support model as shown Table 4.
12
Table 3 Effect of area constraints on feasibility 10-support Problem OFV Time(sec) % error OFV FO7B1_1 22.04† 17.48 0.45 25.57† FO7B1_2 6.81 0.02 22.81 22.81 FO7B2_1 25.49† 20.33 0.18 No sol. FO7B2_2 26.80† 68.95 0.48 No sol. FO8B1_1 65.63 0.15 28.83 28.89 FO8B1_2 25.75† 40.52 0.23 26.93† FO8B2_1 31.50† 136.94 0.16 38.91† † FO8B2_2 29.15 246.55 0.35 33.25 FO9B1_1 27.87† 268.98 0.31 29.48 FO9B1_2 27.69† 225.81 0.59 28.76 FO9B2_1 32.02† 6479.28 1.25 36.45 FO9B2_2 28.75 837.51 0.31 28.99† † the solution turned to infeasible after area expansion ‡ additional test for feasibility check is performed
30-support Time(sec) 121.06 16.05 343.80 782.86 58.23 113.33 2368.08 1236.13 1202.69 3737.22 45124.20 3260.11
% error 0.02 0.02 0.02 0.03 0.02 0.02 0.03 0.02 0.01 0.01
OFV 27.40 22.81‡ No sol. No sol. 28.89‡ 28.72 39.42‡ 33.26‡ 29.51‡ 28.82‡ 36.45 33.50
50-support Time(sec) 255.14 12.03 467.94 904.75 196.66 521.41 2515.47 2836.48 2559.00 3144.31 53866.44 6681.34
% error 0.00 0.01 0.02 0.00 0.02 0.02 0.02 0.01 0.00 0.00
It is noticeable that the % error generated by 30-support model is very close to that of 50-support model but it does not make the layout feasible as often as 50-support model. The % error of 30-support model for FO8B2_1 is 0.02 which is almost same as 50-support model. However, the % error of these two are not equal. The values were rounded off to two decimal points and it turned out to be different after the point of round off was changed. The % error of 30-support for this problem is 0.0223 while it is 0.0154 for 50-support problem. It might be assumed that these two layouts are identical, because the OFV is almost same and the % error shown in table 3 are equal, but the layouts are quite different from each other and that of 10-support model as shown in Figure 8. Additionally, the 100-support model is tested and found that there is no feasible solution if it uses the original dimensions for the problem.
(a) OFV=31.50 % error = 0.16
(b) OFV=38.91, % error = 0.02
(c) OFV=39.42, % error = 0.02
Figure 8. Layout comparison for FO8B2_1 (Department 4 and 8 have fixed dimension)
To make sure that the model generates a feasible solution, a 100-support model is run for the problem set, which the % error of underestimated area is greater than 0.00 in 50-support model. Table 4 shows the result of the test of the 100-support model for the selected problem as mentioned above and compare to that of 50-support model. The model with 100-support generates the layout without area underestimation if there exist the feasible solution.
13
Table 4 additional tests for feasibility with 100-support Problem FO7B1_2 FO8B1_1 FO8B2_1 FO8B2_2 FO9B1_1 FO9B1_2
50-support OFV 22.81 28.89 39.42† 33.26 29.51† 28.82†
Time(sec) 12.03 196.66 2515.47 2836.48 2559.00 3144.31
% error 0.01 0.02 0.02 0.02 0.02 0.01
OFV 22.82 28.90 No Sol. 33.26 32.36 32.20
100-support Time(sec) 28.19 148.06 45337.17 4935.97 15388.61 24514.06
% error 0.00 0.00 0.00 0.00 0.00
†
turned to infeasible after department expansion to original area
The greater number of type B departments affects the feasibility of the layout. The test for 3-type B departments imbedded in the existing problem set has been examined and the results are shown in Table 5. Adding just one more type B department to the problem set greatly reduces the chance to have feasible configuration. This is because the area of floor space is restricted with high area compactness, which is close to 1, and the possible cases of arrangement for all the department in the floor space are reduced as increasing the portion of type B department area in restricted space. Table 5 additional tests for 3 type B departments 10-support OFV Time(sec) % error FO7B3_1 0.74 41.57 314 FO7B3_2 No sol. 48.92 FO8B3_1 46.34 2498.59 0.51 FO8B3_2 No sol. 1427.0 FO9B3_1 0.24 38.58 22871.38 FO9B3_2 0.46 38.91 4014.63 † No feasible solution found in time limit (24hrs) Problem
OFV No sol. No sol. No sol. No sol. No sol. † No sol. †
30-support Time(sec) 141.58 65.11 4835.40 2729.56 86400 86400
% error -
It is expected that the problem set with more than 3 type B department included in FO7 and FO8 would generate infeasible solutions. Thus, only the 9 department problem, FO9, that includes 4 type B departments are tested as the extended problem. For FO9B4_1, the four smallest departments in FO9 are selected and set to fixed type, and four randomly selected departments are set to type B for FO9B4_2 to test its feasibility related to the fixed area portion in given space. These two problems generate the solution for 10-support but there is no solution for 30-support for both of problem in given condition as shown in Table 6. In this result, FO9B4_2 generates the optimal solution for 10-support problem much faster than that of FO9B4_1 which has less portion of fixed area. Thus,
the feasibility does not appear to be directly related to the portion of the fixed department area.
Problem
% of type B area at given space 23.1 49.4
Table 6 additional tests for FO9B4 30-support 10-support OFV Time(sec) % error OFV Time(sec) † † 0.57 No sol. 86400 45.29 86400 0.53 No sol. 26934.31 59.61 21046.22
FO9B4_1 FO9B4_2 † Best solution found in time limit(24hrs): 17.76% of optimality gap ‡ No feasible solution found in time limit(24hrs)
% error -
More than 4 type B department cannot be included in FO9 to get the feasible solution. A couple more experiments for FO9B5 were conducted and no feasible solution was found. It is mentioned in Liu & Meller 14
(2007) that it is difficult to have a feasible layout when a relatively small departmental aspect ratio is used along with a 100% area compactness, and the area relaxation allows many of the layout representations to be feasible even if it is limited in 5% of the total facility size, and this can greatly reduce the solution time. Bozer & Wang (2012) adopts the explanation of Liu & Meller (2007) and applied to their problem. Thus, the additional tests for the 5% area extension has been done for FO9B4 to evaluate its effects as shown in Table 7. Table 7 test for extended floor space No. of support
OFV 10† 45.29‡ 10 28.15 30 28.15 50 28.15 100 28.15 † No area extension applied ‡ Best solution found in 24hrs
FO9B4_1 Time(sec) 86400 63.23 85.80 45.17 77.48
% error 0.57 0.00 0.00 0.00 0.00
OFV 59.61 33.16 33.19 33.19 33.20
FO9B4_2 Time(sec) 21046.22 108.39 199.50 202.64 304.36
% error 0.53 0.13 0.01 0.01 0.00
The first row of the results in Table 7 is brought from Table 6 to compare the results. Only 5% of facility area relaxation can drop the computational time dramatically and provide chance to have low-cost configuration. The layout configuration generated by this extension cannot be applicable in the case that the physically inflexible boundary is used as a floor space but this gives great alternative if this extension is acceptable. Bozer & Wang (2012) stated “If no empty space is provided and department shape constraints are used, potentially many lowcost layouts that would have been adopted by the decision-maker would be needlessly rejected by the algorithm”, and the heterogeneous department layout problem is face to same issue and even in tighter condition in that the type B department have no flexibility. Since we expect that adding a small amount of empty space to candidate facilities in this type of problem would give the chance to solve bigger problem, we modified FO9 to make 10 to 13 heterogeneous department problem and set the facility area is 5% greater than the total department area. Additional test has been done with 30-supports since the departmental area discrepancy with 30-supports is greatly reduced comparing to that of 10-support. Table 8 Additional tests for larger problems Problem OFV Time(sec) % error CR10B3 28.53 36.59 0.03 CR11B4 37.51 1307.97 0.03 CR12B5 41.20 15865.17 0.03 CR13B6 44.65† 86400 0.03 † Best solution found in time limit(24hrs): 4.84% of optimality gap
The results in additional test for larger problem are shown in Table 8. The area relaxation allows the larger problem have the solution much faster than the one without floor space extension. The model provides the optimal solution for 12-department problem with 5 type B departments. For 13-department problem with 6 type B departments, the model generates the solution with 4.84% of optimality gap in 24 hours. The optimal layout for 12-department is provided in Figure 9.
15
Figure 9 Optimal layout for 12-department problem (OFV = 41.2, Type B = {1, 2, 5, 10, 12})
5. Conclusion In this research, a facility layout problem with unequal area department is presented. We propose a model for layout design with heterogeneous area constraints which includes the two different types of department area constraints; one for flexible width and length for the specific area defined as type A department, the other for fixed dimension defined as type B department. Three models to choose the orientation of type B departments have been tested, and one of the methods has been chosen to efficiently represent the problem. The model generates the final layout configuration considering these two types of departments. However, the feasibility issues arise when the area compactness is close to 1. The model was derived from the basic facility layout problem, which is for all type A departments, by embedding decision making constraints for type B departments. The linearization for non-linear area constraints is very critical for type A departments and the number of linear segments affects solution accuracy. The layout solution with a small number of linear support constraints, (up to 10) could not insure that the layout configuration is feasible if the underestimated area is expanded to the original area. This issue reported by previous researches which deal with only type A departments. Adding type B department to the model makes the layout lose its flexibility much more than the model of type A and increases the chance of an infeasible solution. Thus, it is important to set the level of flexibility when the layout design with heterogeneous area constraints is applied to the problem with high area compactness. The issue will be released if the floor space is much bigger than the total area of the departments. In this research, it is tested that how the model works for solution after small amount of empty space is added to original floor space. Precisely, the solution generated by this floor space expansion is not feasible for the original facility, but it gives great alternatives to have cost-effective layout in favorable time. According to the results generated by the model, the time taken to solve the problem increases as the area accuracy is increased. In some cases, there is no feasible solution for original area. And there are important trade-offs between solution time, the gap between the sum of the areas of the individual departments and the total floor space, the allowable error introduced by the area linearization. These optimization models are restricted to solving problems with fewer than 10 departments with non-flexible floor space. Thus, an important future research direction for these problems is to develop effective heuristic solutions for larger departments.
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Acknowledgement This research was partially supported by the MSIP(Ministry of Science, ICT and Future Planning), Korea, under the ITRC(Information Technology Research Center) support program ((IITP-2016-H8601-16-1010)) supervised by the IITP(Institute for Information & communications Technology Promotion). Amelia Regan was partially supported by a Rueben Smeed Visiting Professorship at the Centre for Transport Studies at University College London during the period of this research.
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We propose a new model to solve the layout design problem with flexible and fixed departments.
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