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Facility Design-Week 8 BASIC ALGORITHMS FOR THE LAYOUT PROBLEM

Facility Design-Week 8 BASIC ALGORITHMS FOR THE LAYOUT PROBLEM. Anastasia Lidya Maukar. Introduction. Heuristic Algorithm Construction Algorithm: MST, CORELAP, ALDEP Improvement Algorithm: CRAFT, 2-Opt & 3 Opt Algorithm Hybrid Algorithm: BLOCPLAN

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Facility Design-Week 8 BASIC ALGORITHMS FOR THE LAYOUT PROBLEM

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  1. Facility Design-Week 8BASIC ALGORITHMS FOR THE LAYOUT PROBLEM Anastasia LidyaMaukar

  2. 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

  3. 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

  4. 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.

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

  6. Example 1 - Solution Figure 1. Adjancency Weight Matrix – Example1

  7. Example 1- Solution Figure 2. Single Row Layout of Example 1

  8. Graph Theoretic Method • A Heuristic Algorithm for Identifying Maximal PAG • Terminology • Graph • Complete graph • Planar Graph • Maximal Planar Graph

  9. Planar Nonplanar Planar Graph • A Planar Graph is a graph that can be drawn in two dimensions with no arc crossing. • A graph is nonplanar if it contains either one of the two Kuratowski graphs:

  10. IF1 IF2 IF3 Maximally Planar Graph (MPG) • A planar graph with exactly 3M-6 arcs is called Maximally Planar Graph (MPG). MPG since has 6 arcs Not MPG since has only 5 arcs (5 < 6 = 3M-6) • 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. The tetrahedron has three interior faces (IF1, IF2 and IF3) and an exterior face (EF) EF

  11. Layout…. And its dual… Graph Theoretic Method

  12. 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.

  13. 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

  14. GTA – Example 2

  15. 8 11 5 7 4 14 7 7 4 4 14 14 8 8 11 5 9 5 0 GTA – Example 2 Iteration 0 Iteration 2 Iteration1

  16. GTA – Example 2

  17. 8 9 2 3 11 1 5 10 6 12 13 4 7 GTA – Example 2 Figure 3. Maximal PAG of Example 2

  18. GTA – Example 2 Figure 4. Dual Solution for Example 2

  19. Graph Theoretic Method* Figure 5. Layout Alternative for Example 2

  20. GraphBasedMethod Therearetwostrategiesfordeveloping a maximallyweightedplanaradjacencygraph. Therearetwomethods. • Start withthegraphfromrelationshipdiagramandselectivelypruneconnectingarcs. • Constructiteratively an adjacenygraphvia a nodeinsertationalgortihmwhileretainingplanarity at alltimes. Prepared by: Asst.Prof.Dr. Nevra AKBILEK

  21. Example 3 1.Directors conferenceroom 2.President 3.Sales 4.Personnel 5. Plantmanager 1 2 3 4 5 1 2 3 4 1.Given therelationshipchart 2. Relationshipdiagram 5

  22. AdjacencyGraph-VersionA and B Version A VersionB • Adjacencygraphsforalternativeblocklayouts • Scoreeachbloc plan layoutbysummimngthenumericalweightsassignedtoeacharc. B is betterthan A withscores of 71 and 63, respectively.

  23. Example 3 - Solution Strategy: iteration is based on inserting a new node Step -1: Largest weight-pair departments Step -2: Largest weight-pair departments with respect to 3-4 20 3 4

  24. Example 3 – Solution Step2 2 (Best) 3 4

  25. Example 3 – SolutionStep3 (Best) 1

  26. GraphedBasedProcedure-Step4 2 5 1 Best 3 4 Best

  27. BlockLayoutFromThe Final AdjacencyGraphs

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