Team #2

Mingang FuLin Ben

Kuowei Chen

Traditional Manufacturing ProcessesProduct LayoutFunctional Layout

G ML D

Lathedepartment

Millingdepartment

Drillingdepartment

Grindingdepartment

(a) Product layout

(b)Functional layout

Introduction

(c) Group layout

Group Technology

Introduction

A part family may consist of groups of parts requiring similar and sometimes identical operation processes, materials, and tools.

A manufacturing cell is formed by the machines which are required to produce a part family.

Goal: form a manufacturing system that consists of cells to maximize the moves of parts processed within

the cells, at the same time, to minimize the parts flow between cells

Problem Description

Example:

parts

1 2 3 4 5 6 7

1 1 1

2 1 1

3 1 1 1

machines 4 1 1 1

5 1 1

6 1 1 1

7 1 1

parts

1 7 3 4 6 2 5

2 1 1

5 1 1

3 1 1 1

machines 4 1 1 1

6 1 1 1

1 1 1

7 1 1

Problem Description

• Notations:i and j are machine indexes (i, j = 1, 2,…, Nm).

k is a part index (k = 1, 2, 3,…, Np)

c is a cell index (c = 1, 2, 3,…,Nc) is the production volume of part k is available transfer units per trip for part k using a transfer device is the upper limit of cell size is the number of trips made by part k between machines i and j: Where indicates the smallest integer value greater than or equal to

w.

• Variables:

kpv

kcd

cU

ijknt kijk

k

pvnt

cd

w

Formulation 1

1, if machine is assigned to cell 0, otherwise

i cicx

Objective function:

Constraints:

1 1 1 1

1max

2

pc m mNN N N

ijk ic jcc i j k

z nt x x

1

1

. . 1,

1 ,

x 0,1 , ,

c

m

N

icc

N

ic ci

ic

s t x i

x U c

i c

Formulation 1

parts

1 2 3 4 5 6 7

1 1 1

2 1 1

3 1 1 1

machines 4 1 1 1

5 1 1

6 1 1 1

7 1 1

X11=1, X21=1X32=1, X42=1, X52=1X63=1, X73=1

Formulation 1 - Example

Nm = 7, Np = 7, Nc = 3

Assume 1 and 1, k kpv cd k

Objective function value = 2

parts

1 7 3 4 6 2 5

2 1 1

5 1 1

3 1 1 1

machines 4 1 1 1

6 1 1 1

1 1 1

7 1 1

Formulation 1 - Example

X21=1, X51=1X32=1, X42=1, X62=1X13=1, X73=1

Objective function value = 10

m machines and n parts with k cells and there are a total of k(m+n) variables and (m+n) constrains.

parts ofnumber then

machines ofnumber them

specified (families) cells ofnumber the

...1,1

...1,1

,0

,1

,0

,1

1

1

k

njy

mix

otherwise

lfamilyparttoassignedisjpartify

otherwise

lcelltoassignedisimachineifx

k

ljl

k

lil

jl

il

Formulation 2

With the size of problem increases, the model becomes too large to handle. To overcome this problem, we can change the integer programming model with following declaration:

Then we define “group efficiency” as following and maximize it.

ljly

lilx

j

i

family part toassigned is part ,

cell toassigned is machine ,

)(

elements lexceptioan ofnumber the

blocks diagonal in the viodsofnumber the

matrix data in the operations ofnumber the

1

1

1

1

0

0

0

e

e

e

ee

ee

e

ee

e

v

vv

parts

1 2 3 4 5 6 7

1 1 1

2 1 1

3 1 1 1

machines 4 1 1 1

5 1 1

6 1 1 1

7 1 1

Cell 1, Family 1

X11=1, X21=1, X31=0, …X71=0X12=0, X22=0, X32=1, X42=1, X52=1, X62=0, X72=0X13=0, …X53=0, X63=1, X73=1

Y11=1, Y21=1, Y31=0, …Y71=0Y12=0, Y22=0, Y32=1, Y42=1, Y52=1, Y62=0, Y72=0Y13=0, …Y53=0, Y63=1, Y73=1

Cell 2, Family 2

Cell 3, Family 3

X1=1, X2=1, X3=2, X4=2, X5=2, X6=3, X7=3;Y1=1, Y2=1, Y3=2, Y4=2, Y5=2, Y6=3, Y7=3

e=17, ev=10, eo=10, Gamma=7/27

Formulation 2 - Example

parts

1 7 3 4 6 2 5

2 1 1

5 1 1

3 1 1 1

machines 4 1 1 1

6 1 1 1

1 1 1

7 1 1

X1=3, X2=1, X3=2, X4=2, X5=1, X6=2, X7=3;Y1=1, Y2=3, Y3=2, Y4=2, Y5=3, Y6=2, Y7=1e=17, ev=0, eo=0, Gamma=17/17=1, best

Formulation 2 - Example

Thank you !