1. Models for the Layout Problem
Chapter 10

2. Models
Physical
Analog
Mathematical

3. Analog Model

4. 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
0.008 seconds
0.064 seconds
0.216 seconds 2n
1.0 seconds
12.7 days
366 centuries
NP-complete
[1] Based on data in Garey and Johnson (1979).

5. Generic Modeling Tools
Mathematical Programming
Queuing and Queuing Network
Simulation
Queue
Server(s)
Arrival Process
Departure Process

6. Single-row layout

7. Multi-row layout

8. Airport terminal gates

9. Department shape approximation

10. Single-row layout modelingxj li lj xi
Dept i
Dept j

11. Parameters and variables for the single-row layout model
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)

12. ABSMODEL 1
Subject to
. . . li lj xi
Dept i
Dept j xj

13. Do Example 1 in LINGO General repair area Customer service
Parts display R o m 1 2 3 4 5
Room
Number
Room Name
Dimensions
(in feet) - 12 8 20
TV/VCR
20 x 10 6
Audio
10 x 10
[fij]= 10
Microwave
Computer
Parts
15 x 10

14. LMIP 1?
Minimize
Subject to

15. LMIP 1
Minimize
Subject to

16. LINGO Do Example 2 in LINGO without integer variables
Machine Dimensions Horizontal Clearance Matrix Flow Matrix 1 2 3 4 5
25x20 -
3.5
5.0 25 35 50
35x20
3.0 10 15 20
30x30
40x20
35x35
Do Example 2 in LINGO without integer variables
Do Example 2 in LINGO with integer variables

17. QAP Parameters: n total number of departments and locations1 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:

18. QAP
i=1,2,...,n
Subject to
j=1,2,...,n
i, j=1,2,...,n

19. Do Example 3 in LINGO Office Site O 1 2 3 4 f - 17 12 11 S [fNij]=
[dij]= i t c e

20. ABSMODEL 2
Minimize
Subject to
|xi – xj| + |yi – yj| > 1 i=1,2,...,n–1; j=i+1,...,n
xi, yi = integer i=1,...,n

21. Do Example 4 in LINGO Office Site O 1 2 3 4 f - 17 12 11 S [fNij]=
[dij]= i t c e

22. ABSMODEL 3
Minimize
|xi – xj| +Mzij> 0.5(li+lj)+dhij i=1,2,...,n–1; j=i+1,...,n
|yi – yj| +M(1-zij)> 0.5(bi+bj)+dvij i=1,2,...,n–1; j=i+1,...,n
zij(1-zij) = i=1,2,...,n–1; j=i+1,...,n
xi, yi > i=1,...,n
Subject to

23. Do Example 5 in LINGO Office Trips Matrix O 1 2 3 4 5 Office
Dimensions (in feet) f - 10 15 20
25 x 2 30 35
25 x 20
[fij] = i
35 x 30 c
30 x 20 e
35 x 20

24. LMIP 2
Subject to

25. LP for generating blockplan
Parameters
Upper and lower bounds on the length of department i
Upper and lower bounds on the width of department i
Upper and lower bounds on the perimeter of department i
Set of department pairs adjacent in the horizontal and vertical dimensions, respectively
Decision Variables
x, y coordinates of upper right corner of department i
x, y coordinates of lower left corner of department i

26. LP for generating blockplan (cont.)
Subject to