Models for the Layout Problem

1
Models for the Layout Problem

• Chapter 7

2
Models

• Physical
• Analog
• Mathematical

3
Analog Model
4
Algorithms
Computation time requirement comparison of
polynomial and nonpolynomial algorithms1

1 Based on data in Garey and Johnson
(1979).
5
Generic Modeling Tools

• Mathematical Programming
• Queuing and Queuing Network
• Simulation

6
Single-row layout
7
Multi-row layout
8
Airport terminal gates
9
Department shape approximation
10
Single-row layout modeling
11
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)

12
ABSMODEL 1
Subject to


13
Do Example 1 in LINGO
14
LMIP 1?

Minimize

Subject to
15
LMIP 1
Minimize
Subject to


16
LINGO
Machine Dimensions Horizontal Clearance
Matrix Flow Matrix

• 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
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


i1,2,...,n
Subject to
j1,2,...,n
i, j1,2,...,n
19
Do Example 3 in LINGO
Office Site
20
ABSMODEL 2
Minimize
xi xj yi yj gt 1 i1,2,...,n1
ji1,...,n xi, yi integer i1,...,n
Subject to


21
Do Example 4 in LINGO
Office Site
22
ABSMODEL 3
Minimize
xi xj Mzijgt 0.5(lilj)dhij
i1,2,...,n1 ji1,...,n yi yj M(1-zij)gt
0.5(bibj)dvij i1,2,...,n1
ji1,...,n zij(1-zij) 0
i1,2,...,n1 ji1,...,n xi, yi gt
0
i1,...,n
Subject to


23
Do Example 5 in LINGO
Office
Trips Matrix
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