CHAPTER SIX

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CHAPTER SIX

• Computational Facility Layout

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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

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Heuristic and Optimality

• Consider the knapsack problem
• Z max 5 x1 7 x2 11 x3 12 x4 17 x5
• Subject to 2 x1 3 x2 4 x3 5 x4 7
  x5 lt 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

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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
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A Quadratic Assignment Model

• Decision Variables
• X(i,j) -- 1 Department I will be located at
  position j
• 0 Otherwise
• Constraints
• Each one position can hold exactly one department
• SUM( i in 110) x(i,j) 1
• Each department has to be assigned exactly one
  position
• SUM( j in 110) x(i,j) 1
• Objective
• SUM(I, j, m, n, all in 1..10 ) x(i,j)
  x(m,n)f(i,m)d(j,n)
• This is an integer quadratic assignment problem.

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Pair-Wise Exchange Heuristic
From\To 1 2 3 4
1 -- 10 15 20
2 -- 10 5
3 -- 5
4 --

• 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

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Pair Wise Exchange Heuristic
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Pair Wise Exchange Heuristic
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Pair Wise Exchange Heuristic
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Pair Wise Exchange Heuristic
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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

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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.

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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

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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

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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

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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.

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Graph-based Method
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Graph-based Method
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Graph-based Method
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Graph-based Method
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Graph-based Method
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Graph-based Method
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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

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A Sample Problem
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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
3 3
3 3
3
3 3
3 3 3
3 3
3 3 3
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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

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CRAFT
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CRAFT
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CRAFT
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CRAFT
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CRAFT
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CRAFT

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

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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

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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
• http//www.me.utexas.edu/jensen/ORMM/frontpage/je
  nsen.lib/index_omie.htmlormm
• Sequence
• Create a Plant
• Define the Facility
• Optimum Sequence
• Craft Method
• Fixed Point
• Optimize

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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

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MIP(Mixed Integer Program)

• Parameters

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MIP(Mixed Integer Program )

• Decision variables

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MIP model setup
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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.

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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 ) lt R (yi-yi) or
• (yi yi ) lt 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
• 810 are the typical size solvable in a
  reasonable amount of time

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Commercial Facility Layout Packages

• In the Instructors 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

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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,
  155, 351-366,1996
• Kusiak, A. and S. S. Heragu, The Facility Layout
  Problem, European Journal of Operational
  Research, v29, 229-251, 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,

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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, 296, 1263-1278,
  1991
• Irvine, S. A. and I. R. Melchert, A New Approach
  to the Block Layout Problem, International
  Journal of Production Research, 358, 2359-2376,
  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,
  407, 451-467 1994
• Tate, D.M. and A. E. Smith, Unequal Area
  Facility Layout Using Genetic Search, IIE
  Transactions, 274, 465-472, 1995
• Your Contribution In The Future !!

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Assignments

• Using Excel Add-ins as well as graph based method
  to solve the following problems
• 6.8, 6.9, 6.10, 6.11 6.14, 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.

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Thanks
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BLOCPLAN

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

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BLOCPLAN
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BLOCPLAN
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BLOCPLAN
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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

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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

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LOGIC
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LOGIC
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LOGIC
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LOGIC
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LOGIC
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LOGIC
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LOGIC