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 Others: Simulated Annealing, Tabu Search, Genetic Algorithm, Graph Theoretic Approach. Optimal Algorithm: Branch and Bound algorithms, Decomposition algorithms, and Cutting plane algorithms

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

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.

MST Algorithm-Example 1 M a c h i n e Lengths (in feet) M 1 2 3 4 5 6 a - 12 20 c 10 h 16 i n e

Example 1 - Solution M 1 2 3 4 5 6 a - 204 60 132 340 c 75 85 h 200 30 i 34 n 72 e Figure 1. Adjancency Weight Matrix – Example1

Example 1- Solution 5 2 1 6 4 3 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

Planar Graph A Planar Graph is a graph that can be drawn in two dimensions with no arc crossing. Nonplanar Planar 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). 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. IF1 IF2 IF3 EF The tetrahedron has three interior faces (IF1, IF2 and IF3) and an exterior face (EF)

Graph Theoretic Method 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.

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

GTA – Example 2 Machine 1 2 3 4 5 6 7 8 9 10 11 12 - M 14 a c h i n e

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

GTA – Example 2 Iteration Arcs of selected face Nodes Available Node selected Sum 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 1-8, 1-9, 8-9 2, 3, 5, 12 2 1 7 7-9, 7-10, 9-10 3, 5, 12 12 8-9, 8-10, 9-10 3, 5 7-8, 7-10,8-10

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

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

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

Graph Based Method There are two strategies for developing a maximally weighted planar adjacency graph. There are two methods. Start with the graph from relationship diagram and selectively prune connecting arcs. Construct iteratively an adjaceny graph via a node insertation algortihm while retaining planarity at all times. ENM 324 Facilities Planning Prepared by: Asst.Prof.Dr. Nevra AKBILEK

Example 3 2 3 4 5 1 1.Directors conference room 2.President 3.Sales 4.Personnel 5. Plant manager 1.Given the relationship chart 2. Relationship diagram

Adjacency Graph-Version A and B 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 node Step -1: Largest weight-pair departments Step -2: Largest weight-pair departments with respect to 3-4 20 3 4

Example 3 – Solution Step2 (Best) 3 4

Example 3 – Solution Step3 1 (Best)

Graphed Based Procedure-Step4 3 4 2 5 1 Best

Block Layout From The Final Adjacency Graphs