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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 !