Initial Layout Construction
• Preliminaries
– From-To Chart / Flow-Between chart
– REL Chart
– Layout Scores
• Traditional Layout Construction
• Manual CORELAP Algorithm
• Graph-Based Layout Construction
–
–
–
–
REL Graph, REL Diagram, Planar Graph
Layout Graph, Block Layout
Heuristic Algorithm to Construct a REL Graph
General Procedure
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
From-To and Flow-Between Charts
Given M activities, a From-To Chart
represents M(M-1) asymmetric quantitative
relationships.
Example:
A Flow-Between Chart represents
M(M-1)/2 symmetric quantitative
relationships, i.e.,
gij = fij + fji, for all i > j,
f12 f13
f21
f31
f23
where
gij = material flow between activities i and j.
f32
where
fij = material flow from activity i to activity j.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Relationship (REL) Chart
A Relationship (REL) Chart represents
M(M-1)/2 symmetric qualitative
relationships, i.e.,
r12
r13
r23
where
rij {A, E, I, O, U}: Closeness Value
(CV) between activities i and j; rij is an
ordinal value.
A number of factors other than material
handling flow (cost) might be of primary
concern in layout design.
rij values when comparing pairs of activities:
A = absolutely necessary
 5%
E = especially important
 10 %
I = important
 15 %
O = ordinary closeness
 20 %
U = unimportant
 50 %
X = undesirable
 5%
V(rij) = arbitrary cardinal value assigned to rij,
e.g., V(U) = 1, etc.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Adjacency
•
Two activities are (fully) adjacent in a layout if they share a common border of positive
lenght, i.e., not just a point.
•
Two activities are partially adjacent in a layout if they only share one or a finite
number of points, i.e., zero length.
•
Let aij  [0, 1]: adjacency coefficient between activities i and j.
 1 if activities i and j are adjacent ,

a ij   (0    1) if they are partially adjacent , and
 0 if they are not adjacent .

•
Example:
1
2
3
4
5
(Fully) adjacent: a12 = a13 = a24 = a34 = a45 = 1,
Partially adjacent: a14 = a23 = a25 = , and
Non-adjacent: a15 = a25 = 0.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Layout Scores
Two ways of computing layout scores:
•
Layout score based on distance:
M 1 M
LSd    V(rij )  d ij
i 1 ji 1
where dij = distance between activities i and j.
•
Layout score based on adjacency:
M 1 M
LSa    V(rij )  a ij
i 1 ji 1
where aij  [0, 1]: adjacency coefficient between activities i and j.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Traditional Layout Configuration
• An Activity Relationship Diagram is developed from information
in the activity relation chart. Essentially the relationship diagram is a
block diagram of the various areas to be placed into the layout.
• The departments are shown linked together by a number of lines. The
total number of lines joining departments reflects the strength of the
relationship between the departments. E.g., four joining lines indicate
a need to have two departments located close together, whereas one
line indicates a low priority on placing the departments adjacent to
each other.
• The next step is to combine the relationship diagram with
departmental space requirements to form a Space Relationship
Diagram. Here, the blocks are scaled to reflect space needs while
still maintaining the same relative placement in the layout.
Legend
A Rating
E Rating
I Rating
O Rating
U Rating
X Rating
• A Block Plan represents the final layout based on activity
relationship information. If the layout is for an existing facility, the
block plan may have to be modified to fit the building. In the case of
a new facility, the shape of the building will confirm to layout
requirements.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
REL chart:
1. Offices
2. Foreman
3. Conference Room
4. Parcel Post
5. Parts Shipment
6. Repair and Service Parts
7. Service Areas
8. Receiving
9. Testing
10. General Storage
O
4
I
5
U
E
5
O
4
U
U
U
O
3
I
2
U
O
4
U
E
3
U
I
2
I
U
2
U
U
U
U
E
3
U
U
U
U
U
I
4
O
2
U
U
U
U
A
1
Reason
1
Flow of material
2
Ease of supervision
3
Common personnel
4
Contact Necessary
5
Conveniences
U
I
2
U
I
1
Code
U
I
2
Rating
Definition
A
Absolutely Necessary
E
Especially Important
A
I
Important
1
O
Ordinary Closeness OK
U
Unimportant
E
3
U
U
X
Undesirable
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material
Handling Education
Example (Cont.)
5
8
7
10
9
6
4
2
3
Activity Relationship
Diagram
1
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
5
(500)
7
(575)
8
(200)
9
(500)
6
(75)
10
(1750)
4
(350)
Space Relationship
Diagram
3
(125)
2
(125)
1
(1000)
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Manual CORELAP Algorithm
•
CORELAP is a construction algorithm to create an activity relationship (REL) diagram
or block layout from a REL chart.
•
Each department (activity) is represented by a unit square.
•
Numerical values are assigned to CV’s:
V(A) = 10,000,
V(E) = 1,000,
V(I) =
100,
•
V(O) =
10,
V(U) =
1,
V(X) = -10,000.
For each department, the Total Closeness Rating (TCR) is the sum of the absolute
values of the relationships with other departments.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Procedure to Select Departments
1. The first department placed in the layout is the one with the greatest TCR value. I|f a tie
exists, choose the one with more A’s.
2. If a department has an X relationship with he first one, it is placed last in the layout. If a
tie exists, choose the one with the smallest TCR value.
3. The second department is the one with an A relationship with the first one. If a tie exists,
choose the one with the greatest TCR value.
4. If a department has an X relationship with he second one, it is placed next-to-the-last or
last in the layout. If a tie exists, choose the one with the smallest TCR value.
5. The third department is the one with an A relationship with one of the placed departments.
If a tie exists, choose the one with the greatest TCR value.
6. The procedure continues until all departments have been placed.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Procedure to Place Departments
•
Consider the figure on the right. Assume that a department is placed
in the middle (position 0). Then, if another department is placed in
position 1, 3, 5 or 7, it is “fully adjacent” with the first one. It is
placed in position 2, 4, 6 or 8, it is “partially adjacent”.
8
7
6
1
0
5
2
3
4
•
For each position, Weighted Placement (WP) is the sum of the numerical values for all
pairs of adjacent departments.
•
The placement of departments is based on the following steps:
1. The first department selected is placed in the middle.
2. The placement of a department is determined by evaluating all possible locations
around the current layout in counterclockwise order beginning at the “western edge”.
3. The new department is located based on the greatest WP value.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
1. Receiving
A
2. Shipping
A
E
3. Raw Materials Storage
E
A
E
4. Finished Goods Storage
A
5. Manufacturing
U
A
8. Offices
9. Maintenance
X
A
E
U
A
O
U
E
O
O
E
U
A
A
U
A
A
U
O
O
7. Assembly
U
U
E
6. Work-In-Process Storage
O
CV values:
V(A) = 125
V(E) = 25
V(I) = 5
V(O) = 1
V(U) = 0
V(X) = -125
Partial adjacency:
 = 0.5
O
A
X
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Table of TCR Values
Department
Summary
Dept.
1
2
3
4
5
6
7
8
9
TCR Order
1 2 3
4
5
6
7
8
9
A E I O U X
A
A
E
U
U
U
A
O
E
A
E
E
O
A
E
U
O
U
A
E
A
A
O
A
U
O
U
O
A
A
O
O
U
U
U
A
A
A
X
A
A
E
E
E
O
O
X
X
O
A
A
U
A
O
A
X
-
3
2
3
2
4
2
4
1
3
A
E
A
O
O
U
E
U
A
E
E
A
U
U
E
A
1
2
3
4
1
0
0
3
0
0
0
0
0
0
0
0
0
0
2
1
0
1
2
4
0
2
2
2
3
2
1
1
2
3
0
2
0
0
0
0
0
0
1
2
1
402
301
450
351
527
254
625
452
502
(5)
(7)
(4)
(6)
(2)
(8)
(1)
(9)
(3)
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (cont.)
62.5 125
7
125
62.5 125
62.5 125
62.5
62.5 187.5 187.5 62.5
125
125
62.5
62.5 187.5 187.5 62.5
5
125
62.5
0
62.5 125.5 63.5
1
0
3
5
7
0
9
1.5
0
1
0.5
187.5
5
7
0
125
187.5
9
187.5
0
62.5 126.5
62.5 125
7
62.5
0.5
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (cont.)
12.5 37.5 100 137.5 62.5
37.5
3
5
37.5
1
9
12.5
25
12.5
12.5
25
12.5
0
0
125
87.5
3
5
7
62.5
137.5 62.5
137.5
1
9
4
125
62.5 125
125
125
62.5
7
0
0
62.5 125
188 62.5
0.5
1
3
5
7
125
1
2
1
9
4
63.5
0.5
1
1
1.5
1.5
0.5
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (cont.)
0.5
1
0.5
12.5 25.5 -60.5
6
-61.5
3
5
7
-112
8
3
5
7
1
9
4
-37.5
2
1
9
4
12.5 112.5
25
2
12.5 87.5
75
6
-62.5 -37.5 12.5
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Planar Graph
•
Assumption:
•
A Planar Graph is a graph that can be drawn in two dimensions with no arc crossing.
1 if activities i and j are fully adjacent ,
a ij  
0 otherwise .
Planar
•
Nonplanar
A graph is nonplanar if it contains either one of the two Kuratowski graphs:
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Relationship (REL) Graph
•
Given a (block) layout with M activities, a corresponding planar undirected graph,
called the Relationship (REL) Graph, can always be constructed.
3
1
2
1
4
(Exterior)
Block Layout
5
3
2
4
5
6
REL Graph
•
A REL graph has M+1 nodes (one node for each activity and a node for the exterior of
the layout. The exterior can be considered as an additional activity. The arcs correspond
to the pairs of activities that are adjacent.
•
A REL graph corresponding to a layout is planar because the arcs connecting two
adjacent activities can always be drawn passing through their common border of
positive length.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Relationship (REL) Diagram
•
A Relationship (REL) Diagram is also an undirected graph, generated from the REL
diagram, but it is in general nonplanar.
•
A REL diagram, including the U closeness values, has M(M-1)/2 arcs. Since a planar
graph can have at most 3M-6 arcs, a REL diagram will be nonplanar if M(M-1)/2 >
3M-6.
M(M-1)/2 > 3M-6

M  5.
•
A REL graph is a subgraph of the REL diagram.
•
For M  5, at most 3M-6 out of M(M-1)/2 relationships can be satisfied through
adjacency in a REL graph.

An upper bound on LSa, LSaUB, is the sum of the 3M-6 longest V(rij)’s.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
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
The tetrahedron has three interior faces (IF1, IF2
and IF3) and an exterior face (EF)
IF3
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Maximally Planar Graph (MPG)
•
The interior faces and the exterior face of an MPG are triangular, i.e., the faces are
formed by three arcs.
Not triangular
Not an MPG
•
The REL graph of a given a (block) layout may not be an MPG.
Not an MPG
REL Graph
Layout
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Maximally Planar Weighted Graph (MPWG)
•
An MPG whose sum of arc weights is as large as any other possible MPG is called a
Maximally Planar Weighted Graph (MPWG).
•
Using the V(rij)’s as arc weights, a REL graph that is a MPWG has the maximum
possible LSa, close to LSaUB.
•
Since it is difficult to find an MPWG, a Heuristic (non-optimal) procedure will be used
to construct a REL graph that is an MPG, but may not be an MPWG (although its LSa
will be close to LSaUB).
•
The Layout Graph is the dual of the REL graph.
•
Given a graph G, its dual graph GD has a node for each face of G and two nodes in GD
are connected with an arc if the two corresponding faces in G share an arc.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Layout Graph
•
Example.
G
•
GD
The number of nodes in G (primal graph) is the same than the number of faces in G D
(dual graph), and vice versa. In addition,
(GD)D = G.
•
Primal Graph is Planar  Dual Graph is planar.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Layout Graph (Cont.)
•
Given a layout, the corresponding layout graph can always be constructed by placing
the nodes at the corners of the layout where three or more activities meet (including the
exterior of the layout as an activity). The arcs in the graph are the remaining portions of
the layout walls. E.g.,
a
1
2
3
d
5
g
f
Activities 3, 5, and
exterior meet here
h
(Exterior)
Layout
•
Only activity 3 and
exterior meet here
e
c
4
b
Layout Graph
Given a REL graph (RG), its corresponding layout graph (LG) is LG = RGD. E.g.,
6
2
1
3
RGD
5
4
LGD
RG
LG
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Layout Graph (Cont.)
•
If LG is given, then RG = LGD, but for layout construction, the layout is not known
initially, so LG cannot be constructed without RG.
•
If a planar REL graph (primal graph) exist, the corresponding layout graph (dual graph)
is also planar. Therefore, it is possible theorectically to construct a block layout that will
satisfy all the adjacency requirements. In practice, this is not straightforward because the
space requirements of the activities are difficult to handle.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
REL graph (Primal graph):
A
Space Requirements:
B
F
C
D
G
Dept.
Area
A
300
B
200
C
100
D
200
E
100
F
(exterior)
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Layout graph (Dual graph):
6
A
B
4
1
8
3
2
C
F
G
5
D
7
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Square Block Layout:
(areas are not considered)
Block Layout:
8
8
A
1
C
7
2
D
3
F
5
E
4
B
A
6
B
4
1
8
E
5
2
C
3
D
F
7
•
A corner point is a point where at least three departments meet, including the exterior
department.
•
Note that each corner point in the block layout corresponds to a node in the layout
graph. In the first block layout, each corner point is defined by “exactly” three
departments. In this case, there is a one-to-one correspondence between corner points
and nodes in the layout graph. In the square block layout, there are two corner points
defined by four departments, i.e., (A, B, C, D) and (B, D, E, F). Each of these two corner
points corresponds to two nodes in the layout graph.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Heuristic Procedure to Construct a Relationship Graph
1. Rank activities in non-increasing order of TCRk, k = 1, …,M, where
k-1
TCRk =
 Max{0, V(r
i 1
ik
M
)} 
 Max{0, V(r
kj
)}.
j=k+1
(Note that the negative values of V(rik) and V(rkj) are not included in TCRk).
2. Form a tetrahedron using activities 1 to 4 (i.e., the activities with the four largest TCR k‘s).
3. For k = 5, …, M, insert activity k into the face with the maximum sum of weights (V(rij))
of k with the three nodes defining the face (where “insert” refers to connecting the inserted
node to the three nodes forming the face with arcs).
4. Insert (M+1)th node into the exterior face of the REL graph.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
REL chart:
CV values:
A
B
I
O
X
C
D
E
F
I
O
U
U
A
U
E
E
U
E
E
V(A) = 81
V(E) = 27
V(I) = 9
V(O) = 3
V(U) = 1
V(X) = -243
O
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Table of TCR Values
Department
Summary
Dept.
TCR Order
A
B
C
D
E
F
A
E
I
O
U
X
A
-
I
O
I
O
A
1
0
2
2
0
0
105
2
B
I
-
X
U
U
E
0
1
1
0
2
1
38
5
C
O
X
-
U
E
E
0
2
0
1
1
1
58
3
D
I
U
U
-
U
E
0
1
1
0
3
0
39
4
E
O
U
E
U
-
O
0
1
0
2
2
0
35
6
F
A
E
E
E
O
-
1
3
0
1
0
0
165
1
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Step 2:
A
O
I = rAD  V(rAD) = 9
A
C
E
F
U
E
D
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Step 3: Insert B.
A
I
EF
I
I
IF1
IF2
X C
E
E
X
X
E
F
Face
IF3
U
U
D
LSa
EF
9 + 1 + 27 = 37
IF1
9 + 27 - 243 = -207
IF2
9 - 243 + 1 = -233
IF3
27 - 243 + 1 = -215
*
 Insert B in EF
U
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Step 3 (Cont.): Insert B.
A
B
EF
IF1
IF3
IF2
C
IF4
F
D
IF5
Face
LSa
EF
5
IF1
7
IF2
33
IF3
31
IF4
31
IF5
5
*
 Insert E in IF2
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Step 4: Call exterior activity EX.
A
EX
B
E
C
F
Since arcs (AB), (BD),
and (DA) are the outside
arcs, EX connects to
nodes A, B, and D.
D
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
•
LSaUB is the sum of the 3M - 6 ( 3  6 - 6 = 12), largest V(rij)’s.
In the last example,
LSaUB = V(rAF) + V(rBF) + V(rCE) + V(rCF) + V(rDF) + V(rAB) + V(rAD) + V(rAC)
+ V(rAE) + V(rEF) + V(rBD) + V(rBE) = 81 + 27 + 27 + 27 + 27 + 9 + 9 + 3
+ 3 + 3 + 1 + 1 = 218.
•
For the final REL graph, LSa = 218.
•
•
LSaUB = LSa  The final REL graph is an MPWG  It is optimal.
LSaUB > LSa  The final REL graph may not be an MPWG  It may not be optimal.
•
Using the Heuristic procedure, the generated REL graph will always be an MPG since
each face is triangular.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
General Procedure for
Graph Based Layout Construction
1. Given the REL chart, use the Heuristic procedure to construct the REL graph.
2. Construct the layout graph by taking the dual of the REL graph, letting the facility
exterior node of the REL graph be in the exterior face of the layout graph.
3. Convert (by hand) the layout graph into an initial layout taking into consideration the
space requirement of each activity.
REL Chart
REL Graph
Layout Graph
Initial Layout
Space
Requirements
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
Step 1: (from before)
A
EX
B
E
C
F
D
REL Graph
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Step 2: take the dual of RG
A
B
EX
E
C
F
D
Layout Graph
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Step 3:
A
E
C
B
D
F
• Initial layout is
drawn as a square,
but could be any
other shape.
• Only A and B are
nonrectangular.
Initial Layout
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Comments
1. If an activity is desired to be adjacent to the exterior of a facility (e.g., a shipping/receiving
department), then the exterior could be included in the REL chart and treated as a normal
activity, making sure that, in step 1 of the general procedure, its node is one of the nodes
forming the exterior face of the REL graph.
2. The area of each interior face of the layout graph constructed in step 2 does not correspond to
the space requirements of its activity.
3. In step 3, the overall shape of the initial layout should be usually be rectangular if it
corresponds to an entire building because rectangular buildings are usually cheaper to
build; even if the initial layout corresponds to just a department, a rectangular shape would
still be preferred, if possible.
4. In step 3, the shape of each activity in the initial layout should be rectangular if possible, or at
most L- or T-shaped (e.g., activities A and B), because rectangular shapes require less wall
space to enclose and provide more layout possibilities in interiors as compared to other
shapes.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Comments (Cont.)
5. All shapes should be orthogonal, i.e., all corners are either 90 or 270 (e.g., a triangle is not
an orthogonal shape since its corners could all be 60 ).
6. In step 1, if the LSa of the REL graph is less than LSaUB, then the REL graph may not be
optimal. The following three steps may improve the REC graph for the purpose of increasing
LSa:
a) Edge Replacement: replace an arc in the REL graph with a new arc not previously
in the graph, without losing planarity, if it increases LSa.
b) Vertex Relocation: move a node in the REL graph connected to three arcs to
another triangular face if it increases LSa.
c) Use a different activity to replace one of the four activities of the tetrahedron
formed in step 2 of the Heuristic procedure to construct a new REL graph.
Copyright © 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education