Facility Design-Week 8BASIC ALGORITHMS FOR THE LAYOUT PROBLEM

Anastasia Lidya Maukar

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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

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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

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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.

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MST Algorithm-Example 1M a c h i n e Lengths (in feet)

M 1 2 3 4 5 6

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

e 6 20 0 2 12 6 - 10

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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

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Example 1- Solution

5 2 1 6 4 3

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Figure 2. Single Row Layout of Example 1

Graph Theoretic Method

• A Heuristic Algorithm for Identifying Maximal PAG

• Terminology• Graph• Complete graph• Planar Graph• Maximal Planar Graph

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Planar Graph• A Planar Graph is a graph that can be drawn in two

dimensions with no arc crossing.

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NonplanarPlanar

● A graph is nonplanar if it contains either one of the two Kuratowski graphs:

Maximally Planar Graph (MPG)

• A planar graph with exactly 3M-6 arcs is called Maximally Planar Graph (MPG).

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Not MPG sincehas only 5 arcs(5 < 6 = 3M-6)

MPG sincehas 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.

IF1 IF2

IF3

EF The tetrahedron has three interior faces (IF1, IF2 and IF3) and an exterior face (EF)

Graph Theoretic Method

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• Layout…. • And its dual…

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.

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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

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GTA – Example 2Machine

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 -

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GTA – Example 2

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4 7 14

Iteration 0

Iteration 2

4 714

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Iteration1

GTA – Example 2Iteration Arcs of

selected faceNodes Available Node

selectedSum of 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, 9-10

3, 5, 12 12 2

8 8-9, 8-10, 9-10

3, 5 3 0

9 7-8, 7-10,8-10 5 5 016

GTA – Example 2

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Figure 3. Maximal PAG of Example 2

GTA – Example 2

Figure 4. Dual Solution for Example 2

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Graph Theoretic Method*

Figure 5. Layout Alternative for Example 2

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Graph Based Method

There are two strategies for developing a maximally weighted planar adjacency graph. There are two methods.

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.

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Example 3

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1.Directors conference room

2.President

3.Sales

4.Personnel

5. Plant manager

2. Relationship diagram1.Given the relationship chart

Adjacency Graph-Version A and B

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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.

Example 3 - Solution

Strategy: iteration is based on inserting a new nodeStep -1: Largest weight-pair departments

Step -2: Largest weight-pair departments with respect to 3-4

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3 4

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Example 3 – Solution Step2

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(Best)

Example 3 – SolutionStep3

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(Best)1

Graphed Based Procedure-Step4

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3 4

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Best

Best

Block Layout From The Final Adjacency Graphs

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