Facility Design-Week 8
BASIC ALGORITHMS FOR THE LAYOUT
PROBLEM
Anastasia Lidya Maukar
1
Introduction
•
Heuristic Algorithm
•
Construction Algorithm: MST, CORELAP, ALDEP
•
Improvement Algorithm: CRAFT, 2-Opt & 3 Opt Algorithm
•
Hybrid Algorithm: BLOCPLAN
•
Others: Simulated Annealing, Tabu Search, Genetic Algorithm,
Graph Theoretic Approach.
•
Optimal Algorithm: Branch and Bound algorithms,
Decomposition algorithms, and Cutting plane algorithms
2
Modified Spanning Tree (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
3
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.
4
MST Algorithm-Example 1
M
Machine
1
2
3
4
5
6
Lengths (in feet)
a
1
-
12
3
6
0
20
20
c
2
12
-
5
5
5
0
10
h
3
3
5
-
10
4
2
16
i
4
6
5
10
-
2
12
20
n
5
0
5
4
2
-
6
10
5
e
6
20
0
2
12
6
-
10
Example 1 - Solution
M
1
2
3
4
5
6
a
1
-
204
60
132
0
340
c
2
204
-
75
85
60
0
h
3
60
75
-
200
60
30
i
4
132
85
200
-
34
204
n
5
0
60
60
34
-
72
e
6
340
0
30
204
72
-
Figure 1. Adjancency Weight Matrix – Example1
6
Example 1- Solution
5
2
1
6
4
3
Figure 2. Single Row Layout of Example 1
7
Graph Theoretic Method
•
A Heuristic Algorithm for
Identifying Maximal PAG
Terminology
•
•
•
•
•
Graph
Complete graph
Planar Graph
Maximal Planar Graph
9
1
2
4
5
3
6
7
8
8
Planar Graph
• A Planar Graph is a graph that can be drawn in two
dimensions with no arc crossing.
Planar
Nonplanar
● A graph is nonplanar if it contains either one of the
two Kuratowski graphs:
9
Maximally Planar Graph (MPG)
• A planar graph with exactly 3M-6 arcs is called Maximally Planar Graph (MPG).
Not MPG since
has only 5 arcs
(5 < 6 = 3M-6)
MPG since
has 6 arcs
● The interior faces of a graph are the bounded regions formed by its
arcs, and its exterior face is the unbounded region formed by its
outside arcs.
EF
IF1
IF2
IF3
The tetrahedron has three interior faces (IF1, IF2
and IF3) and an exterior face (EF)
10
Graph Theoretic Method
• Layout….
• And its dual…
1
2
9
1
2
4
5
4
3
3
5
7
6
8
6
7
8
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.
12
Graph Theoretic Method*
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
13
GTA – Example 2
Machine
1
2
3
4
5
6
7
8
9
10
11
12
1
-
1
0
8
0
2
3
0
0
0
0
0
2
1
-
0
1
1
1
0
0
0
0
0
0
3
0
0
-
0
2
0
0
0
0
0
0
0
M 4
8
1
0
-
0
4
14
11
0
0
0
0
a
5
0
1
2
0
-
1
0
0
0
0
0
0
c
6
2
1
0
4
1
-
3
0
0
3
0
0
h
7
3
0
0
14
0
3
-
5
5
9
8
2
i
8
0
0
0
11
0
0
5
-
8
0
0
0
n
9
0
0
0
0
0
0
5
8
-
0
0
0
e
10
0
0
0
0
0
3
9
0
0
-
6
0
11
0
0
0
0
0
0
8
0
0
6
-
4
12
0
0
0
0
0
0
2
0
0
0
4
-
14
GTA – Example 2
Iteration 0
7
4
14
Iteration 2
Iteration1
8
8
8
11
0
11
9
5
7
4
4
5
5
7
14
14
15
GTA – Example 2
Iteration Arcs of
selected face
Nodes Available
Node
Sum of
selected Flows
3
7-8, 7-9, 8-9
2, 3, 5, 6, 10, 11, 12
10
9
4
4-7, 4-9, 7-9
2, 3, 5, 6, 11, 12
11
8
5
1-4, 1-9, 4-9
2, 3, 5, 6, 12
6
6
6
1-8, 1-9, 8-9
2, 3, 5, 12
2
1
7
7-9, 7-10, 910
3, 5, 12
12
2
8
8-9, 8-10, 910
3, 5
3
0
9
7-8, 7-10,8-10 5
5
0
16
GTA – Example 2
8
2
1
3
3
1
6
5
1
0
9
1
2
1
1
4
7
17
Figure 3. Maximal PAG of Example 2
GTA – Example 2
8
13
2
3
1
6
10
9
5
12
11
4
7
Figure 4. Dual Solution for Example 2
18
Graph Theoretic Method*
1
2
3
8
6
9
11 12
10
4
7
5
Figure 5. Layout Alternative for Example 2
19
1. Start with the graph from relationship diagram
and selectively prune connecting arcs.
2. Construct iteratively an adjaceny graph via a
node insertation algortihm while retaining
planarity at all times.
ENM 324 Facilities Planning
There are two strategies for developing a
maximally weighted planar adjacency graph. There
are two methods.
Prepared by: Asst.Prof.Dr.
Nevra AKBILEK
Graph Based Method
20
Example 3
1
1.Directors conference room
2
3
2.President
4
5
3.Sales
1
4.Personnel
2
3
4
5. Plant manager
5
1.Given the relationship chart
2. Relationship diagram
21
Adjacency Graph-Version A and B
Version A
VersionB
Adjacency graphs for alternative block layouts
• Score each bloc plan layout by summimng the numerical weights assigned to each
arc. B is better than A with scores of 71 and 63, respectively.
22
Example 3 - Solution
Strategy: iteration is based on inserting a new node
Step -1: Largest weight-pair departments
3
2
0
4
Step -2: Largest weight-pair departments with respect
to 3-4
23
Example 3 – Solution
Step2
2
(Best
)
3
4
24
Example 3 – Solution
Step3
(Best)
1
25
Graphed Based ProcedureStep4
2
5
1
Best
Best
3
4
26
Block Layout From The Final
Adjacency Graphs
27