Product and Equipment Analysis
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Transcript Product and Equipment Analysis
Models for the Layout
Problem
Chapter 7
Models
• Physical
• Analog
• Mathematical
Analog Model
Algorithms
Computation time requirement comparison of polynomial and nonpolynomial algorithms[1]
TCF
Problem Size
P or NP
Complete
10
20
40
60
N
0.001 seconds
0.002 seconds
0.004 seconds
0.006 seconds
P-complete
N3
0.001 seconds
0.008 seconds
0.064 seconds
0.216 seconds
P-complete
2n
0.001 seconds
1.0 seconds
12.7 days
[1]
366 centuries
Based on data in Garey and Johnson (1979).
NP-complete
Generic Modeling Tools
• Mathematical Programming
• Queuing and Queuing Network
• Simulation
Server(s)
Arrival
Process
Queu
e
Departure
Process
Single-row layout
Multi-row layout
Airport terminal gates
Terminal
Gates
Department shape approximation
Single-row layout modeling
li
lj
Dept i
Dept j
xi
xj
Parameters and variables for the
single-row layout model
Parameters:
• n
number of departments in the problem
• cij cost of moving a unit load by a unit distance between
departments i and j
• fij
number of unit loads between departments i and j
• li
length of the horizontal side of department i
• dij minimum distance by which departments i and j are to be
separated horizontally
• H
horizontal dimension of the floor plan
Decision Variable:
• xi
distance between center of department i and vertical reference
line (VRL)
ABSMODEL 1
n 1
n
Minimize cij f ij xi x j
i 1 j i 1
Subject to
xi x j 0.5 li l j dij
li
i 1, 2,..., n 1
lj
...
Dept i
Dept j
xi
xj
H 0.5li xi 0.5 li
i 1,2,..., n
Do Example 1 in LINGO
General
repair
area
Customer
service
Parts
display
area
[fij]
=
R
o
o
m
1
2
3
4
5
Room
Number
Room
Name
Dimensions
(in feet)
R
1
-
12
8
20
0
1
TV/VCR
20 x 10
o
2
12
-
4
6
2
2
Audio
10 x 10
o
3
8
4
-
10
0
3
Microwave
10 x 10
m
4
20
6
10
-
3
4
Computer
20 x 10
5
0
2
0
3
-
5
Parts
15 x 10
LMIP 1?
xi x j if xi x j 0
x
otherwise
0
ij
x j xi if xi x j 0
x
otherwise
0
ij
1 if xi x j
zij
0 otherwise
n 1
Minimize
c
f
x
x
ij ij ij ij
n
i 1 j i 1
Subject to
x
ij
xij 0.5 li l j dij i 1, 2,..., n 1; j i 1,..., n
xi x j xij xij
i 1,2,..., n 1; j i 1,..., n
LMIP 1
c f x x
n 1
Minimize
n
i 1 j i 1
Subject to
ij ij
ij
ij
xi x j Mzij 0.5 li l j dij , i 1, 2,...n 1; j i 1,..., n
x j xi M (1 zij ) 0.5 li l j dij i 1, 2,..., n 1; j i 1,..., n
xi x j xij xij i 1,2,..., n -1; j i 1,...n
xij , xij i 1,2,..., n -1; j i 1,...n
zij 0 or 1
xi 0
i=1, 2,..., n -1; j i 1,...n
i=1, 2,..., n
LINGO
Machine Dimensions Horizontal Clearance Matrix
1
2
3
4
5
Flow Matrix
1
2
3
4
5
25 35 50
0
1
25x20
1
-
3.5
5.0
5.0
5.0
1
-
2
35x20
2
3.5
-
5.0
3.0
5.0
2
25
-
3
30x30
3
5.0
5.0
-
5.0
5.0
3
35
10
4
40x20
4
5.0
3.0
5.0
-
5.0
4
50
15 50
5
35x35
5
5.0
5.0
5.0
5.0
5.0
5
0
20 10 15
10 15 20
-
50 10
-
15
-
• Do Example 2 in LINGO without integer variables
• Do Example 2 in LINGO with integer variables
QAP
1
2
a
b
b,1
d,2
3
4
c
d
c,3
a,4
Parameters:
n
total number of departments and locations
aij net revenue from operating department i at location j
fik flow of material from department i to k
cjl cost of transporting unit load of material from location j to l
Decision Variable:
1 if department i is assigned to location j
xij
0 otherwise
QAP
n
Maximize
n
n
n
n
a x f
ij ij
i 1 j 1
n
Minimize
n
n
a x
ij ij
i 1 j 1
n
n
n
i 1 j 1 k 1 l 1
i k j l
n
n
n
n
ik
f
i 1 j 1 k 1 l 1
i k j l
c jl xij xkl
ik
c jl xij xkl
n
Minimize fik c jl xij xkl
i 1 j 1 k 1 l 1
n i k j l
Subject to
x
j 1
n
ij
x
i 1
ij
1
i=1,2,...,n
1
j=1,2,...,n
xij 0 or1
i, j=1,2,...,n
Do Example 3 in LINGO
1
Office
2
3
4
1
Site
2
3
4
O
[fNij]=
f
1
-
17
12
11
f
2
17
-
12
4
i
3
12
12
-
c
4
11
4
4
e
S
1
-
1
1
2
i
2
1
-
2
1
4
t
3
1
2
-
1
-
e
4
2
1
[dij]
=
-
ABSMODEL 2
c f x
n1
Minimize
n
i 1 j i 1
ij
ij
i
x j yi y j
|xi – xj| + |yi – yj| > 1 i=1,2,...,n–1; j=i+1,...,n
xi, yi = integer
i=1,...,n
Subject to
H
xi
Facility i
V
yi
Facility j
yj
xj
HRL
VRL
Do Example 4 in LINGO
1
Office
2
3
4
1
Site
2
3
4
O
[fNij]=
f
1
-
17
12
11
f
2
17
-
12
4
i
3
12
12
-
c
4
11
4
4
e
S
1
-
1
1
2
i
2
1
-
2
1
4
t
3
1
2
-
1
-
e
4
2
1
[dij]
=
-
ABSMODEL 3
c f x
n 1
Minimize
n
i 1 j i 1
Subject to
ij
ij
i
x j yi y j
|xi – xj| +Mzij> 0.5(li+lj)+dhij
|yi – yj| +M(1-zij)> 0.5(bi+bj)+dvij
zij(1-zij) = 0
xi, yi > 0
i=1,2,...,n–1; j=i+1,...,n
i=1,2,...,n–1; j=i+1,...,n
i=1,2,...,n–1; j=i+1,...,n
i=1,...,n
xj
bj
Facility j
dhij
lj
dvij
li
bi
xi
VRL
yj
Facility i
yi
HRL
Do Example 5 in LINGO
Office
O
[fij] =
Trips Matrix
1
2
3
4
5
Office
Dimensions
(in feet)
f
1
-
10
15
20
0
1
25 x 2
f
2
10
-
30
35
10
2
25 x 20
i
3
15
30
-
10
20
3
35 x 30
c
4
20
35
10
-
15
4
30 x 20
e
5
0
10
20
15
-
5
35 x 20
LMIP 2
n
Minimize
n
c
i 1 j 1
Subject to
f h
ij ij ij
hij Myij L xi x j i, j n; i j
hij M 1 yij x j xi i, j n; i j
xi x j Myij dij i, j n; i j
xi x j Myij dij M i, j n; i j
xi x j Myij dij M i, j n; i j
xi x j Myij d ij i, j n; i j
xi 0
yij 0 or 1; hij 0 i, j n; i j
in
MCj
MCk
MCi
MCi
LP for generating blockplan
Parameters
Lui , Lli
Upper and lower bounds on the length of department i
Wi u ,Wi l
Upper and lower bounds on the width of department i
Pi u , Pi l
Upper and lower bounds on the perimeter of department i
HA, VA
Set of department pairs adjacent in the horizontal and vertical dimensions, respectively
Decision Variables
xiu , yiu
x, y coordinates of upper right corner of department i
xil , yil
x, y coordinates of lower left corner of department i
LP for generating blockplan (cont.)
n 1
Minimize cij fij xij xij yij yij
n
i 1 j i 1
Subject to xi x j xij xij i 1,2,..., n 1; j i 1,..., n
Lli xiu xil Lui i 1, 2,..., n
yi y j yij yij i 1,2,..., n 1; j i 1,..., n
Wi l yiu yil Wi u i 1, 2,..., n
Pi l 2 xiu xil yiu yil Pi u i 1, 2,..., n
xil xi xiu i 1,2,..., n
yil yi yiu i 1,2,..., n
xil xuj i, j VA
yil yuj i, j HA