1. Basic Algorithms and Software for the Layout Problem
Chapter 5

2. Algorithms
Optimal
Heuristic

3. Algorithms Construction Improvement Hybrid MST Graph Theoretic Method
2-opt
Greedy
Steepest Descent
3-opt
Hybrid
Modified Penalty Algorithm

4. MST Algorithm
Step 1: Given the flow matrix [fij], clearance matrix [dij] and machine lengths li, compute an adjacency weight matrix where:
f’ij = (fij)(dij+0.5(li+lj)).
Step 2: Find the largest element in [f’ij] and the corresponding i, j. Denote this pair of i, j as i*, j*. Connect machines i*, j*. Set f’i*j* =f’i *i* =-infinity

5. MST Algorithm
Step 3: Find the largest element f’i*k,f’j*l in row i*, j* of matrix If f’i*k*>f’j*l* connect k to i*, remove row i*, column i* from matrix and set i* = k. Otherwise, connect l to j*, remove row j*, column j* from matrix and set j* = l. Set f’i*j* =f’i *i* =-infinity
Step 4: Repeat step 3 until all machines are connected. The sequence of machines obtained determines the arrangement of machines.

6. Example 1 M 1 2 3 4 5 6 a - 12 20 c 10 h 16 i n e
M a c h i n e Lengths (in feet)

7. Example 1 Solution M 1 2 3 4 5 6 a - 204 60 132 340 c 75 85 h 200 30 i
340 c 75 85 h
200 30 i 34 n 72 e

8. Example 1 Solution 5 2 1 6 4 3

9. Graph Theoretic Method
Terminology
Graph
Complete graph
Planar Graph
Maximal Planar Graph

10. Graph Theoretic Method
Layout….
And its dual…

11. Graph Theoretic Method*
Step 1: Identify the department-pair in the flow matrix with the maximum flow. Place the corresponding nodes in a new PAG and connect them.
Step 2: From the rows corresponding to the connected nodes in the flow matrix, select the node which is not yet in the PAG and has the largest flows with the connected nodes.
Step 3: Update PAG by connecting the selected node to those in Step 2. This forms a triangular face in the PAG.
Step 4: For each column of the flow matrix corresponding to a node not present in the PAG, examine the sum of flow entries in the rows corresponding to the nodes of the triangular face selected in step 3. Select the column for which this sum is the largest. Update PAG by placing the corresponding node within the selected face and connect it to nodes of the face. This forms three new triangular faces.
Step 5: Arbitrarily select one of the faces formed and go to Step 4. Repeat Step 5 until all the nodes have been included in the PAG.
* Based on the result that the maximum number of arcs in a planar graph with n nodes 3n-6

12. Graph Theoretic Method1 2 3 4 5 6 7 8 9 10
1 1 12 - M a 14 11 c h i n e
Do Example 2

13. Graph Theoretic Method
Example 2

14. Graph Theoretic Method
Example 2

15. Graph Theoretic Method
Example 2

16. 2-opt algorithm
Step 1: Let S be the initial solution provided by the user and z its OFV. Set i=1; j=i+1=2.
Step 2: Consider the exchange between the positions of departments i and j in the solution S. If the exchange results in a solution S’ that has an OFV z’< z, set z*=z’ and S*=S’. If j < mn, set j=j+1; otherwise, set i=i+1, j=I+1. If i < mn, repeat step 2; otherwise, go to step 3.
Step 3: If S not =S*, set S=S*, z=z*, i=1, j=i+1=2 and go to step 2. Otherwise, return S* as the best solution to the user. Stop.
Do Example 3 using SINROW or MULROW

17. 2-opt algorithm . mn
m+1
m+2 1 2 m

18. Example 3 O 1 2 3 4 f - 17 12 11 S
[fij]=
[dij]= i t c e

19. 3-opt algorithm
Step 1: Let S be the initial solution and z its OFV; Set S*=S, z*=z, i=1; j=i+1; k=j+1.
Step 2: Consider changing the position of department i to that of j, j to that of k, and k to that of i, simultaneously. If the resulting solution SN has OFV z’ < z, set z*=z’ and S*=S’.
Step 3: If k < mn, set k = k +1, and repeat step 2. Otherwise, set j=j+1 and check if j < mn-1.
If j < mn-1, set k=j +1, and repeat step 2. Otherwise, set i = i +1, j=i+1, k = j +1, and check if i < mn-2.
If i < mn -2, repeat step 2. Otherwise, go to step 4.
Step 4: If S not = S*, set S=S*, z=z*, i=1, j=i+1, k=j+1 and go to step 2. Otherwise, return S* as the best solution to the user. Stop.

20. Layout Software CRAFT BLOCPLAN PFAST FactoryFLOW Layout-iQ VIP-PLANOPT
Flowpath Calculator

21. CRAFT in Excel
Do Example 4

22. BLOCPLAN Adjacency score Rel-dist score
R-score = 1- (rel-dis score- lower bound)/(upperbound-lower bound)
Rij numeric value assigned to the relationship code between departments i and j,
n total number of departments, and
dij rectilinear distance between the centers of departments i and j

23. BLOCPLAN
Do Example 5

24. PFAST

25. Layout-iQ

26. VIP-PLANOPT

27. Flowpath Calculator

28. BLOCPLAN
Re-layout
CRAFT-M
Multi-Floor Layout
BLOCPLAN