1. Computational Facility Layout
CHAPTER SIX
Computational Facility Layout

2. The Facility Layout Problem
Given the activity relationship as well as the space of the department, how to construct plan the layout of the facility
The basis of the layout planning is the closeness ratings or material flow intensities
minimize the flow times distance
Maximize the closeness (adjacency)
For most practical real world instances, the computational complexity has results in various heuristics
What is heuristic?
Construction Heuristic
Improvement Heuristic

3. Heuristic and Optimality
Consider the knapsack problem
Z = max 5 x1 + 7 x x x x5
Subject to 2 x1 + 3 x x x x5 <= 10
Heuristic: An intuitive problem solving method/procedure
Constructive
Heuristic 1, Pick sequentially the ones with the best benefit
Heuristic 2, Pick sequentially the ones with the best benefit per unit
Greedy Improvement:
Exchange two items in a solution
Meta-Heuristic: Simulated Annealing, Genetic Algorithm
Optimal Solution
Mathematical Programming and Optimization
Linear Programming, Integer Programming, Nonlinear Programming

4. A Simple Facility Layout Problem
Suppose we have 10 identical sized departments and the flow intensity between these 10 department is fij
Find the best arrangement of the 10 department along an aisle so that the total travel (flow intensity times distance) is minimized
A Quadratic Assignment Model is necessary to Optimally solve the problem 1 2 3 4 5 6 7 8 9 10
Position
Department

5. A Quadratic Assignment Model
Decision Variables
X(i,j) : Department I will be located at position j
0: Otherwise
Constraints
Each one position can hold exactly one department
SUM( i in 1…10) x(i,j) = 1
Each department has to be assigned exactly one position
SUM( j in 1…10) x(i,j) = 1
Objective
SUM(I, j, m, n, all in ) x(i,j)* x(m,n)*f(i,m)*d(j,n)
This is an integer quadratic assignment problem.

6. Pair-Wise Exchange Heuristic
From\To 1 2 3 4 -- 10 15 20 5
Phase I: Construct Phase Initial Solution (1,2,3,4)
Phase II: Improvement – Pair Wise Exchange
a) Exchange two departments
b) If results in better solution, accept; go to a)
otherwise stop

7. Pair Wise Exchange Heuristic

8. Pair Wise Exchange Heuristic

9. Pair Wise Exchange Heuristic

10. Pair Wise Exchange Heuristic

11. Pair-Wise Exchange Heuristic
Limitations
No guarantee of optimality,
The final solution depends on the initial layout
Leads to suboptimal solution
Does not consider size and shape of departments
Additional work has to be re-arrange the department if shaper are not equal

12. Graph Based Method
Graph based method dates back to the later 1960s and early 1970s.
The method starts with an adjacency relationship chart
Then, we assign weight to the adjacency relationships between departments
A graph, called adjacency graph is constructed
Node: to represent department
Arc : to represent adjacency, weight on arc represents the adjacency score
Goal: To find a graph with maximum sum of arc weights
However, not all the adjacency relations can be implemented in such a graph, that is the graph may not be planar.

13. A Planar Graph
Planar graph: A graph is a planar if it can be drawn so that each edge intersects no other edges and passes through no other vertices
Intuitively, a planar graph is a graph where there is no intersection of arcs (flow of material)
To find a maximum weight planar graph

14. Procedure to Find Maximum Weight Adjacent Planar Graph
Step 1: Select a department pair with largest weight
Step 2: Select a third department based on the sum of the weights with the two departments selected.
Step 3: Select next unselected department to enter by evaluating the sum of weights and place the department on the face of the graph.
Here, a face of a graph is a bounded region of a graph
Step 4: Continuing the Step 3 until all departments are selected
Step 5: Construct a block layout from the planar graph

15. From Graph to Block Design
Let us “blow” air into each node in the planar graph
Nodes explode
Interior faces becomes a dot
The edge in primal graph becomes the boundary between departments
Dual Graph
Nodes in dual  faces in primal
Edge in dual : if two faces connects in the primal graph
The faces in dual represents the department
Draw Block Design

16. Limitation of Graph Based Method
Limitations
The adjacency score does not account for distance, nor does it account for distance other than adjacent department
Although size is considered in this method, the specific dimension is not, the length between adjacent departments are also not considered.
We are attempting to construct graphs, called planar graphs, whose arcs do not intersect.
The final layout is very sensitive to the assignment of weights in the relationship chart.

17. Graph-based Method

18. Graph-based Method

19. Graph-based Method

20. Graph-based Method

21. Graph-based Method

22. Graph-based Method

23. Computer Relative Allocation of Facility Techniques (CRAFT)
Discrete or Continuous Representation
Discrete Representation
A two-dimension array with numbers
Each cell represents a unit area & numbers represent the department occupied the cell

24. A Sample Problem

25. Valid Discrete Representation
Valid Representation
Contiguous: If an activity is represented by more than one unit, every unit of the must share at least one edge with at least one other unit
Connectedness: The perimeter of an activity must be a single closed loop
No Enclosed Void: No activity shape shall contain an enclosed void 3 3 3

26. Computer Relative Allocation of Facility Techniques -- CRAFT (1963)
Algorithm
1) Any Incumbent Layout
Describe a tentative layout in blocks
Determine centroids of each department
Cost= S distance (in the from-to matrix) X unit cost
Distance can be Euclidian or Rectilinear
2) Improvement: make pair wise or three way exchanges
equal area only
adjacent (generally)
3) If better solution exists; Choose the best, go to 1)
Otherwise Stop

27. CRAFT

28. CRAFT

29. CRAFT

30. CRAFT

31. CRAFT

32. CRAFT
In the original design, exchange has to be departments of equal area or adjacent departments.

33. Shape Consideration Shape Consideration
Shaper Ratio Rule: The ratio of a feasible shape should be with specified limits
Corner Counter: The number of corners for a feasible shaper may not exceed specified maximum

34. Excel Add-ins for facility Planning
The Excel Add-In
Written by Prof. Paul Jensen (UT-Austin)
Contains an implementation of CRAFT and can be downloaded at
Sequence
Create a Plant
Define the Facility
Optimum Sequence
Craft Method
Fixed Point
Optimize

35. Mixed Integer Program
The work begins latterly in the 1990s by Montreuil
The departments are assumed to be rectangular within a rectangular plant.
Plant
Length Bx, Width By
Shape consideration:
Area,
The (minimum, maximum) width of a department
The (minimum, maximum) length of a department
Decisions: Where to put the Departments (Centroid) and the shape (length,width) of the department
Objective = flow_intensity* cost *distance

36. MIP(Mixed Integer Program)
Parameters

37. MIP(Mixed Integer Program )
Decision variables

38. MIP model setup

39. MIP model setup II
Constraint (6.13) ensures the upper corner of j is less than the lower corner of i if z_ij(x) =1 . i.e., to the east of i. Note if z_ij(x) = 0, (6.13) is redundant.
Constraint (6.14) ensures to the north-south relationship
Constraint (6.15) ensures that no two departments overlap by forcing a separation at least in the east-west or north-south direction.

40. MIP Models Benefit of MIP Model
Department shapes as well as their area can be modeled through individually specified lower and upper limits !!!
It might be able to control length-width ratio as well
(xi’’ – xi’ ) <= R (yi’’-yi’) or
(yi’’ – yi’ ) <= R (xi’’-xi’)
Heuristically, we can combine CRAFT with MIP.
Get a initial layout using CRAFT, use MIP to find the best rectangular layout design
Solving the problem exactly (optimal solution) is hard
8~10 are the typical size solvable in a reasonable amount of time

41. Commercial Facility Layout Packages
In the Instructor’s Opinion, there is no commercial package that will suit all the needs, partly due to the difficult of the problem, but more due to the fact that Facility Layout is a combination of Science and Art.
There has been a trend to combine optimization techniques with interactive graphic procedures, especially people have an unique pattern reorganization capability than computers.
We encourage the reader to use the web to keep abreast of new developments, resort to professional publications, which periodically publish survey of software packages for facilities planning, and new techniques

42. References Literature – Presentation topics General Survey
Meller, R.D. and K. Gau, “The Facility Layout Problem: Recent and Emerging Trends and Perspective,” Journal of Manufacturing Systems, 15:5, ,1996
Kusiak, A. and S. S. Heragu, “The Facility Layout Problem,” European Journal of Operational Research, v29, , 1987
Mixed Integer Programming
Montreuil, B., “A Modeling Framework for Integrating Layout Design and Flow Network Design,” Proceedings of the Material Handling Research Colloquium, Hebron, KY, 1990
Assignment Problem and the Location of Economic Activities, Econometrica,

43. Reference Reference (Continue) Graph Based Approach
Hassan, M. M. D and G. L. Hogg, “On Constructing a Block Layout by Graph Theory,” International Journal of production Research, 29:6, , 1991
Irvine, S. A. and I. R. Melchert, “A New Approach to the Block Layout Problem,” International Journal of Production Research, 35:8, , 1997
Computerized Layout Design
Bozer, Y.A., R.D. Meller and S.J. Erlebacher, “An Improvement Type Layout Algorithm for Single and Multiple Floor Facilities,” Management Science, 40:7,
Tate, D.M. and A. E. Smith, “Unequal Area Facility Layout Using Genetic Search,” IIE Transactions, 27:4, , 1995
Your Contribution In The Future !!

44. Assignments
Using Excel Add-ins as well as graph based method to solve the following problems
6.8, 6.9, 6.10, , 6.15, 6.19, 6.20
Compare the results and see if they make sense or not.
Work in group, select one of the papers and present it in class at the end of the quarter.

45. Thanks

46. BLOCPLAN Set up all departments in bands (2or3)
Continuous areas not blocks
Use From to or a relationship chart
Uses two way exchanges

47. BLOCPLAN

48. BLOCPLAN

49. BLOCPLAN

50. MIP(Mixed Integer Program)
Generally a construction type model
Requires some knowledge of linear and integer programming
Solutions to these types of problems are difficult
We will examine the general formulation

51. LOGIC Layout Optimization with Guillotine Induced Cuts
Slice the area to partition the plant between departments
Supersedes BLOCPLAN, because all BLOCPLANS are LOGIC plans
Improved by pair wise exchange or simulated annealing

52. LOGIC

53. LOGIC

54. LOGIC

55. LOGIC

56. LOGIC

57. LOGIC

58. LOGIC