A MATHEMATICAL OPTIMIZATION APPROACH BASED ON LINEARIZED MIP MODELS FOR SOLVING FACILITY LAYOUT PROBLEMS

Braga, Evelyn Michelle Henrique; Salles Neto, Luiz Leduino de

doi:10.1590/0101-7438.2022.042.00261044

Evelyn Michelle Henrique Braga ^**Corresponding author

ICT, Universidade Federal de São Paulo (UNIFESP), São José dos Campos, SP, Brazil - E-mail: evelyn.braga@unifesp.br

Instituto Tecnológico de Aeronáutica (ITA), São José dos Campos/SP, Brazil

https://orcid.org/0000-0003-1318-4593

Luiz Leduino de Salles Neto

ICT, Universidade Federal de São Paulo (UNIFESP), São José dos Campos, SP, Brazil - E-mail: luiz.leduino@unifesp.br

https://orcid.org/0000-0001-8938-5370

*Corresponding author

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Figure 1
Representation of Euclidean

(d = D)

and rectilinear

(d = H + V)

distances.

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Figure 2
Departments with candidate locations for I/O points: a) four possibilities located on middle of the edges of the block; b) four options located on the corners of the block; and c) can assume infinite positions around on the boundaries of the block.

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Figure 3
Block orientations.

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Figure 4
Grid with feasible flow paths for a problem with three facilities and contour distance d1,3 between the output of block 1 and the input of block 3 highlighted.

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Figure 5
Optimal layouts generated for fixed I/O points.

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Figure 6
Optimal layouts generated for variables I/O points.

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Figure 7
Layouts generated for the UAFLP2 Model - Instance: Deb05N12.

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Figure 8
Layouts generated for the UAFLP2 Model - Instance: Dun03N62.

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Table 1
Notation used in the general FLP model.

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Table 2
Notation used in the Models 1, 2, 3 and 4.

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Table 3
Additional notations used in Models 1, 2, 3 and 4 linearized.

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Table 4
Features of the instances available in the literature.

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Table 5
Comparative results for fixed I/O points.

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Table 6
Comparative results for variable I/O points - Part I.

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Table 7
Comparative results for variable I/O points - Part II.

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Indices and input data
N	total number of facilities/locations
i e j	index for facilities = (1,...,N)
r e s	index for locations = (1,...,N)
d _rs	distance between locations r and s
f _ij	material flow between facilities i e j
*Decision variables*
x _ir	1 if facility i is located in r, 0 cc.

Indices, input data and markers
N	total number of facilities
i e j	index for facilities = (1,...,N)
f _ij	material flow between facilities i and j
W	width of the floor space
H	height of the floor space
w	original width of facility i
h	original height of facility i
Ix _i , Iy _i	original coordinates x and y of the input of facility i
Ox _i , Oy _i	original coordinates x and y of the output of facility
A, B,C, D	sides of facilities (bottom, left, top and right)
*Decision variables*
d _ij	rectilinear distance between the output of facility i and the input of facility j
wr _i	real width of facility i (considering the rotation)
hr _i	real height of facility i (considering the rotation)
x _i ;y _i	coordinates x and y of lower left corner of facility i
xs _i ; ys _i	coordinates x and y of upper right corner of facility i
$x_{i}^{I}, y_{i}^{I}$	coordinates x and y of the input of facility i
$x_{i}^{O}, y_{i}^{O}$	coordinates x and y of the output of facility i
$m_{i}^{I}, n_{i}^{I}$	perimeter position in directions x and y of the input of facility i
$m_{i}^{O}, n_{i}^{O}$	perimeter position in directions x and y of the output of facility i
l _{i j}	(relative position) 1 if facility i is placed totally to the left of j, 0 cc.
b _{i j}	(relative position) 1 if facility i is placed totally below j, 0 cc.
u _i ;v _i	(0,0) if facility i is in its original orientation; (1,0) if facility i is rotated 90º clockwise; (0,1) if facility i is rotated 180º clockwise; (1,1) if facility i is rotated 270º clockwise
$p_{i}^{I}, q_{i}^{I}$	defines on which side the input of facility i is located: (0,0) if on side A; (1,0) if on side B; (0,1) if on side C; (1,1) if on side D
$p_{i}^{O}, q_{i}^{O}$	defines on which side the output of facility i is located: (0,0) if on side A; (1,0) if on side B; (0,1) if on side C; (1,1) if on side D

Decision variables
dx _ij	auxiliary variable that replaces the nonlinear binomial $\| x_{i}^{O} - x_{j}^{I} \|$
dy _ij	auxiliary variable that replaces the nonlinear binomial $\| y_{i}^{O} - y_{j}^{I} \|$
lx _ij	auxiliary variable that replaces the nonlinear term l _ij x _i
by _ij	auxiliary variable that replaces the nonlinear term b _i
uv _i	auxiliary variable that replaces the nonlinear term u _i v _i
$p^{I (O)} m_{i}^{I (O)}$	auxiliary variable that replaces the nonlinear term $p_{i}^{I (O)} m_{i}^{I (O)}$
$p^{I (O)} n_{i}^{I (O)}$	auxiliary variable that replaces the nonlinear term $p_{i}^{I (O)} n_{i}^{I (O)}$
p ^I(O) wr _i	auxiliary variable that replaces the nonlinear term $p_{i}^{I (O)} w r_{i}$
q ^I(O) hr _i	auxiliary variable that replaces the nonlinear term $q_{i}^{I (O)} h r_{i}$
p ^I(O) q ^I(O) wr _i	auxiliary variable that replaces the nonlinear term $p_{i}^{I (O)} q_{i}^{I (O)} w r_{i}$
p ^I(O) q ^I(O) hr _i	auxiliary variable that replaces the nonlinear term $p_{i}^{I (O)} q_{i}^{I (O)} h r_{i}$

Reference	Data type	Size	Code
Das (1993DAS S. 1993. A facility layout method for flexible manufacturing systems. The International Journal of Production Research, 31(2): 279-297.)	fixed I/O points	4	Das93N4
		6	Das93N6
		8	Das93N8
		10	Das93N10
		12	Das93N12
Welgama & Gibson (1993WELGAMA P & GIBSON P. 1993. A construction algorithm for the machine layout problem with fixed pick-up and drop-off points. The International Journal of Production Research, 31(11): 2575-2589.)	fixed I/O points	6	Wel93N6
	fixed I/O points	12	Wel93N12
Deb & Bhattacharyya (2005DEB S & BHATTACHARYYA B. 2005. Solution of facility layout problems with pickup/drop-off locations using random search techniques. International journal of production research, 43(22): 4787-4812.)	without previous I/O points	12	Deb05N12
	without previous I/O points	18	Deb05N18
Dunker et al. (2003DUNKER T, RADONS G & WESTKÄMPER E. 2003. A coevolutionary algorithm for a facility layout problem. International Journal of Production Research, 41(15): 3479-3500.)	I/O points in the centroids	62	Dun03N62

Instance	Best solution in literature			Our approach
Instance	Reference	Algorithm method	Solution	Our approach
Das93N4	RAJ1998	Genetic algorithm	1393.6 ¹	1393.6 ¹
Das93N6	RAJ1998	Genetic algorithm	2612.7	2556 ¹
Das93N8	DUN2003	Coevolutionary algorithm	8778.3 ¹	8778.3 ¹
Das93N10			15694.5	15222.9
Das93N12			37396.1	36622.8
Wel93N6	WEL1993	Construction algorithm	421.5	398.5 ¹
Wel93N12	WEL1993	Construction algorithm	5903	5458.5
Dun03N62	DUN2003	Coevolutionary algorithm	3939362	5484080

Model	Instance	Best solution in literature¹		Our approach
Model	Instance	Metric	Solution	Our approach
UAFLP1	Deb05N12	rectilinear distance	21142	11504
UAFLP1	Deb05N18		62902	37382
UAFLP3	Deb05N12		19892	6924
UAFLP3	Deb05N18		61756	27092

Model	Instance	Best solution in literature¹		Our approach
Model	Instance	Metric	Solution	Our approach
UAFLP2	Deb05N12	rectilinear distance	8676	6804
UAFLP2	Deb05N18	rectilinear distance	440124	27780
UAFLP1	Wel93N6	contour distance	325	237 ²
	Wel93N12		5750	5288
	Deb05N12		37894	12404
	Deb05N18		72451	40754
UAFLP2	Wel93N6		266	115 ²
	Wel93N12		5454	2889
	Deb05N12		16404	7028
	Deb05N18		58324	28540
	Dun03N62		4921320	3972940

d_{i j}, w r_{i}, h r_{i}, x_{i}, y_{i}, x s_{i}, y s_{i}, x_{i}^{I (O)}, y_{i}^{I (O)}, m_{i}^{I (O)}, n_{i}^{I (O)}, l x_{i j}, b y_{i j}, d x_{i j}, d y_{i j}, u v_{i}, p^{I (O)} m_{i}^{I (O)}, p^{I (O)} n_{i}^{I (O)}, p^{I (O)} q^{I (O)} w r_{i}, p^{I (O)} q^{I (O)} h r_{i}, p^{I (O)} w r_{i}, q^{I (O)} h r_{i} \geq 0 \forall i

[1] *Corresponding author

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A MATHEMATICAL OPTIMIZATION APPROACH BASED ON LINEARIZED MIP MODELS FOR SOLVING FACILITY LAYOUT PROBLEMS

ABSTRACT