where aij [0, 1]: adjacency coefficient between activities i and j.
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.
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.
Example
Example (Cont.)
Example (Cont.)
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(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.
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.
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”.
Example
Table of TCR Values
Example (cont.)
Example (cont.)
Example (cont.)
Planar Graph
Assumption:
A Planar Graph is a graph that can be drawn in two dimensions with no arc crossing.
Relationship (REL) Graph
Given a (block) layout with M activities, a corresponding planar undirected graph, called the Relationship (REL) Graph, can always be constructed.
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.
Maximally Planar Graph (MPG)
A planar graph with exactly 3M-6 arcs is called Maximally Planar Graph (MPG).
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.
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.
Layout Graph
Example.
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.,
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.
Example
Example (Cont.)
Example (Cont.)
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 TCRk‘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.
Example
Table of TCR Values
Example (Cont.)
Step 2:
Example (Cont.)
Step 3: Insert B.
Example (Cont.)
Step 3 (Cont.): Insert B.
Example (Cont.)
Step 4: Call exterior activity EX.
Example (Cont.)
LSaUB is the sum of the 3M - 6 ( 3 6 - 6 = 12), largest V(rij)’s.
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.
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.
Example
Step 1: (from before)
Example (Cont.)
Step 2: take the dual of RG
Example (Cont.)
Step 3:
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.
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.
