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8/10/2019 Initial Layout (1)
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
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
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Relationship (REL) Chart
A Relationship (REL) Chartrepresents
M(M-1)/2 symmetric qualitative
relationships, i.e.,
where
rij{A, E, I, O, U}: Closeness Value
(CV) between activities i and j; rijis an
ordinal value.
A number of factors other than material
handling flow (cost) might be of primary
concern in layout design.
rijvalues 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.
r12r
13
r23
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Adjacency
Two activities are (fully) adjacentin a layout if they share a common border of positive
lenght, i.e., not just a point.
Two activities are partially adjacentin 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.
Example: (Fully) adjacent: a12= a13= a24= a34= a45= 1,
Partially adjacent: a14= a23= a25= , and
Non-adjacent: a15= a25= 0.
.adjacentnotaretheyif
and,adjacentpartiallyaretheyif)10(
,adjacentarejandiactivitiesif
0
1
aij
3
1 2
4 5
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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:
where dij= distance between activities i and j.
Layout score based on adjacency:
where aij[0, 1]: adjacency coefficient between activities i and j.
1M
1i
M
1ijijij
d d)r(VLS
1M
1i
M
1ijijij
a a)r(VLS
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Traditional Layout Configuration
An Activity Relationship Diagramis 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.
A Block Planrepresents 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.
A Rating
E Rating
I Rating
O Rating
U Rating
X Rating
Legend
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
Code Reason
1 Flow of material
2 Ease of supervision
3 Common personnel
4 Contact Necessary5 Conveniences
Rating Definition
A Absolutely Necessary
E Especially Important
I Important
O Ordinary Closeness OK
U Unimportant
X Undesirable
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
U
U
E
3
U
U
E
3
E
5
O
4
U
O
4
U
U
E
3
A
1
O
3
I
2
U
U
U
I
4
U
U
I
2
U
U
U
U
U
I
2
U
U
A
1
U
O
2
U
I
1
U
I
2
U
U
I
2
U
REL chart:
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Example (Cont.)
10
5 8 7
9 6
4 2 3
1Activity Relationship
Diagram
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Example (Cont.)
2
(125)
Space Relationship
Diagram
3
(125)
1(1000)
4
(350)
6
(75)
9(500)
10(1750)
5(500)
8(200)
7(575)
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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 CVs:
V(A) = 10,000, V(O) = 10,
V(E) = 1,000, V(U) = 1,
V(I) = 100, 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.
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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 As.
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.
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
1. Receiving
2. Shipping
3. Raw Materials Storage
4. Finished Goods Storage
5. Manufacturing
6. Work-In-Process Storage
7. Assembly
8. Offices
9. Maintenance
AA
E
OU
UA
O
E
E
E
A
A
X
X
AU
U
A
O
O
A
O
A
O
U
E
A
U
E
U
E
AU
O
A
1. Receiving
2. Shipping
3. Raw Materials Storage
4. Finished Goods Storage
5. Manufacturing
6. Work-In-Process Storage
7. Assembly
8. Offices
9. Maintenance
CV values:
V(A) = 125
V(E) = 25
V(I) = 5
V(O) = 1
V(U) = 0
V(X) = -125
Partial adjacency:= 0.5
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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 A E I O U X
TCR Order
1
2
3
4
5
6
7
8
9
- A A E O U U A O
A - E A U O U E A
A E - E A U U E A
E A E - E O A E U
U O 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 U A U A O A X -
3 1 0 2 2 0
2 2 0 1 3 0
3 3 0 0 2 0
2 4 0 1 1 0
4 1 0 2 1 0
2 0 0 4 2 0
4 0 0 0 3 1
1 3 0 2 0 2
3 0 0 2 2 1
402
301
450
351
527
254
625
452
502
(5)
(7)
(4)
(6)
(2)
(8)
(1)
(9)
(3)
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (cont.)
7
125
125
125 125
62.5 62.5
62.562.5
7 125
62.5 62.5
62.5187.5
5125
62.5 187.5
187.5 187.5
7 0
62.5 0
5
187.5
187.5
9187.5
62.5 125 62.5
0
62.5125
7 0
125.5 0
5
1.59126.5
0.5 1 0.5
0
163.5
3125
62.5
62.5
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Example (cont.)
7 1255
137.5925 0
100
337.5
37.5
12.5
112.5 12.5
62.5
62.5137.537.5
7
125
5
9125
12.5
387.5
137.5
12.5
162.5 125
62.5
0025
4125 62.5
75
9
1
125
31
0
1
1 1.5
125
188
4
1.5 0.5
21
0.5
0.5
63.5
62.562.5
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (cont.)
75
9
75
-60.5
3112.5
1
87.5 -62.5
-112
4
-37.5 12.5
225
12.5
12.5
-37.5
-61.525.5 612.5
0.5 10.5 0.5
75
9
3
1 42
6
8
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Planar Graph
Assumption:
A Planar Graphis a graph that can be drawn in two dimensions with no arc crossing.
.otherwise
,adjacentfullyarejandiactivitiesif
0
1aij
NonplanarPlanar
A graph is nonplanar if it contains either one of the two Kuratowski graphs:
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Relationship (REL) Graph
Given a (block) layout with M activities, a corresponding planar undirected graph,
called the Relationship (REL) Graph, can always be constructed.
REL Graph
1 2
543
6(Exterior)
1 2
543
Block Layout
A REL graph has M+1 nodes(one node for each activity and a node for the exterior ofthe layout. The exterior can be considered as an additional activity. Thearcscorrespond
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.
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Relationship (REL) Diagram
A Relationship (REL) Diagramis 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.
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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 facesof a graph are the bounded regions formed by its arcs, and its
exterior faceis the unbounded region formed by its outside arcs.
IF1 IF2
IF3
EF The tetrahedron has three interior faces (IF1, IF2and IF3) and an exterior face (EF)
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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.
Layout REL Graph
Not an MPG
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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)sas 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 Graphis the dual of the REL graph.
Given a graph G, its dual graph GDhas 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.
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Layout Graph
Example.
The number of nodes in G (primal graph) is the same than the number of faces in GD
(dual graph), and vice versa. In addition,(GD)D= G.
Primal Graph is Planar Dual Graph is planar.
G GD
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Layout Graph (Cont.)
Given a layout, the corresponding layout graphcan 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.,
Layout Graph
1 2
54
3
(Exterior)
Given a REL graph (RG), its corresponding layout graph (LG)is LG = RGD. E.g.,
Layout
c g
a b
d f
e
h
1 2
54
3
6
RG LG
RGD
LGD
Only activity 3 and
exterior meet here
Activities 3, 5, and
exterior meet here
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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 willsatisfy all the adjacency requirements. In practice, this is not straightforward because the
space requirements of the activities are difficult to handle.
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
Space Requirements:
Dept. Area
A 300
B 200
C 100
D 200
E 100
F (exterior)
REL graph (Primal graph):
A B
C D
F G
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
Layout graph (Dual graph):
A B
C D
F G
1
2
3
4
5
7
6
8
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example (Cont.)
A corner pointis 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 layoutgraph. 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.
Block Layout:Square Block Layout:
(areas are not considered)
A
D
BC
E
8 1 6
7 2 3 4
5
A
D
B
C
E
7
8
8 4
1 5
2 3
F
F
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Heuristic Procedure to Construct a Relationship Graph
1. Rank activities in non-increasing order of TCRk, k = 1, ,M, where
TCRk=
(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 TCRks).
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)thnode into the exterior face of the REL graph.
Max{0, V(r )} Max{0, V(r )}.iki 1
k-1
kjj=k+1
M
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Copyright 1999. Created by Jose Ventura for the College-Industry Council for Material Handling Education
Example
OI
OA
X
U
U
O
UU
E
E
A
B
C
D
E
F
I
E
E
CV values:
V(A) = 81V(E) = 27
V(I) = 9
V(O) = 3
V(U) = 1
V(X) = -243
REL chart:
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Table of TCR Values
Department Summary
Dept.
A B C D E F A E I O U X
TCR Order
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
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Example (Cont.)
Step 2:
A
C
F D
A
O
E U
E
I = rADV(rAD) = 9
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Example (Cont.)
Step 3: Insert B.
A
C
F D
EF
IF1 IF2
IF3
I
I I
E
E
E
U
U
U
X X
X
Face 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
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Example (Cont.)
Step 3 (Cont.): Insert B.
A
C
F D
B
IF1
IF2 IF3
IF4
IF5
EF
Face LSa
EF 5
IF1 7
IF2 33 *
IF3 31
IF4 31
IF5 5
Insert E in IF2
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Example (Cont.)
Step 4: Call exterior activity EX.
A
C
F D
B EX
E
Since arcs (AB), (BD),
and (DA) are the outside
arcs, EX connects to
nodes A, B, and D.
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Example (Cont.)
LSaUBis 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.
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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
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Example
Step 1: (from before)
A
C
F D
B EX
E
REL Graph
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Example (Cont.)
Step 2: take the dual of RG
C
F
D
EX
E
A
B
Layout Graph
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Example (Cont.)
Step 3:
Initial layout isdrawn as a square,
but could be any
other shape.
Only A and B are
nonrectangular.
B D
A
F
E C
Initial Layout
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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 layoutshould be usually be rectangularif it
corresponds to an entire building because rectangular buildings are usually cheaperto
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
spaceto enclose and provide more layout possibilities in interiors as compared to other
shapes.
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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 LSaof 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 tetrahedronformed in step 2 of the Heuristic procedure to construct a new REL graph.