11, 12 Optimization Based Methods

8/12/2019 11, 12 Optimization Based Methods

1/19

Optimization based methods

.


2/19

1

n

ij k

k

s b

1 1

m m

ij ij

i j

s X

1

1

m

ij

j

X i

1

m

jjj

X p

ij jjX X

Let the similarity between two machines i and j be defined as

where bk=1 if a

ik= a

jkand 0 otherwise.

Let Xjj

= 1 if point (or machine) j is a median.Let Xij= 1 if point (or machine) i is attached to median j

The p-median formulation is as follows:

Maximize

Subject to

Xij =0,1

1

n

ij ik jk

k

d a a

dij= n

sij.

Minimize

1 1

m m

ij iji j

d X


3/19

1 2 3 4 5 6 7 81 1 1 1 12 1 1 1 13 1 1 14 1 1 15 1 1 1 1 16 1 1 1 1 1 1

1 2 3 4 5 61

0

6

1

1

7

6

2 6 0 5 5 3 43 1 5 0 2 8 54 1 5 2 0 6 75 7 3 8 6 0 36 6 4 5 7 3 0

12 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 1 6d

Also d12

= 8s12

= 6.


4/19


5/19


6/19

The optimum solution is

X11

= X31

= X41

= X22

= X52

= X62

= 1; Y11

= Y41

= Y51

= Y61

= Y22

= Y32

= Y72

= Y82

= 1;

Z262

= Z261

= Z551

= Z552

= Z611

= Z612

= Z641

= Z642

= 1.

This means that there are four intercell moves M2-P6, M5-P5, M6-P1 and M6-P4.

This is the same solution obtained using the p median model for p = 2.


7/19

Assignment model


8/19

1 2 3 4 5 6 7 81 1 1 1 12 1 1 1 13 1 1 14 1 1 15 1 1 1 1 16 1 1 1 1 1 1

1 2 3 4 5 61

0

6

1

1

7

6

2 6 0 5 5 3 43 1 5 0 2 8 54 1 5 2 0 6 75 7 3 8 6 0 36 6 4 5 7 3 0

12 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 1 6d

Also d12

= 8s12

= 6.


9/19

The solution to the assignment problem is

X13

= X26

= X34

= X41

= X52

= X65

= 1 with Z = 14.

From this solution we observe that machine 1 is assigned to machine 3,

which is assigned to machine 4. Machine 4 is assigned to machine 1.

We observe a subtour 1-3-4-1 out of the assignments.

Therefore the three machines {1, 3, 4} form a group.

Similarly the machine cell {2, 6, 5} is formed.

Applying the maximum density rule the corresponding part families

are {1, 4, 5, 6} and {2, 3, 7, 8}. There are four intercell moves.


10/19

10

1111111111222222222233333333334444

1234567890123456789012345678901234567890123

1 1 1

2 1 1 1 1 11 1 1

3 1 1 111

4 1 1 1 1 1 1 1

5 1 11 111 1 1 1 1 1 1 1

6 11 111 111 1 1 1 111 1 11 11

7 1 1 1

8 111 11 11 111 11 11 1 11 1 1

9 1 1 1 1 1 1 11 1 1

10 1 11 11 1 1

11 1 1 1 1 1 1

12 1 1 1 1 1

13 1 1

14 1 1 1 1

15 1 1 1 1 1 1 1

16 1 1 1 1 11 1


11/19


12/19

12

An Assignment model Solution

X(1,13) = X(13,1) = 1

X(2,9) = X(9,16) = X(16,2) = 1

X(3,14) = X(14,3) = 1

X(4,5) = X(5,15) = x(15,4) = 1

X(6,8) = X(8,6) = 1

X(7,10) = X(10,7) = 1

X(11,12) = X(12,11) = 1

Seven machine groups

{1,13}, {2,9,16}, {3,14}, {4,5,15}, {6,8}, {7,10} and {11,12}.


13/19

13

Part groups

{1,12}, {2,42,37}, {3,24}, {4,10,18}, {5,14,19,23,8,15,16,29},

{6,17}, {7,34}, {9,20}, {11,22}, {13,25}, {21,41}, {26,31,39},

{27,30}, {28,38}, {32,40}, {33,43}, {35,36}.

Initial Solution

Group Machines Components

123

4567

1,132,9,163,14

4,5,156,87,1011,12

32,42,37,4,10,18,28,38,32,406,17,7,34,35,36

5,14,19,15,16,29,9,21,41,331,8,11,12,20,23,31,39,3,4313,25,2624,22,27,30


14/19

14

Group Components Machines1 2,42,37,4,10,18,28,38,32,40 1,2,9,16

2 6,17,7,34,35,36 3,14

3 5,14,19,15,16,29,9,21,41,33 4,5,15

4 1,8,11,12,20,23,31,39,3,43 6,8,10,75 13,25,26 7

6 24,22,27,30 11,12,13

Intermediate Solution


15/19


16/19

16

11233344 1333 1111222344 112233 122223

2408827802677456589456913931312356193102470

1 1 1

2 1 1 111111

9 111111111

16 1 11 111 1

3 11111

14 1 1 1 1

4 1 11 11111

5 1111111111111

6 1 11 111111 1 11 1 1 1 111 1

8 1 1 11 111 1 111 111 1 111 11

15 1 1 11 111

7 111

10 111111

11 1 1 1 111

12 1 1111

13 1 1

Final block diagonal structure

Intercell move after duplicating 6 and 8


17/19

Measures of performance

Minimize E

Minimize E + B

Grouping efficiency Grouping Efficacy


18/19

1 2 3 4 5 6 7 86 1 1 1 1 1 11 1 1 1 13 1 1 14 1 1 15 1 1 1 1 12 1 1 1 1

2 7 8 3 1 4 5 66 1 1 1 1 1 15 1 1 1 1 12 1 1 1 11 1 1 1 13 1 1 14 1 1 1


19/19

Search Algorithms

Genetic Algorithms

Simulated Annealing

11, 12 Optimization Based Methods

Documents

Transcript of 11, 12 Optimization Based Methods