ISE-422- Facility Layout and Location
Chapter 5: Layout planning models and
design algorithms
Part 2
Dr. Nabil Nahas
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Mathematical Modeling:
Multiple floors (8)
Two Stage Approach
• Stage 1: Solve Q
• Stage 2: For each floor solve P
How to solve P and Q?
• Exact Methods:
 Branch and bound based methods
 Cutting plane based methods
 Combination of branching, bounding and cutting/pricing...
• Heuristic Methods:
 Greedy heuristic methods
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 Meta-heuristics (simulated annealing, genetic algorithms, etc.)
Heuristic Methods: Notations
Heuristic Methods: Greedy Approach (1)
The objective of our greedy heuristic is to find a
strict local optimal. We assume that we search all
members of the one step neighborhood (all single
swap solutions). However, in some scenarios, the
size of one step neighborhood will become
exponential. In such scenarios, we can restrict
the neighborhood size.
Heuristic Methods: Greedy Approach (2)
Heuristic Methods: Greedy Approach (3)
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Heuristic Methods: Greedy Approach (4)
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Heuristic Methods: Greedy Approach (5)
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Heuristic Methods: Greedy Approach (6)
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Heuristic Methods: Greedy Approach (7)
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Heuristic Methods: Greedy Approach (8)
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Heuristic Methods: Greedy Approach (9)
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Heuristic Methods: Greedy Approach (10)
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Heuristic Methods: Greedy Approach (11)
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Heuristic Methods: Greedy Approach (12)
• In the greedy approach's flow chart, all red
boxes needs problem specific information.
• In the greedy approach's flow chart, all green
boxes do NOT need any problem specific
information.
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Heuristic Methods: Greedy Approach (13)
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Heuristic Methods: Swap mechanism (1)
• Swap: exchange or switch.
• One of the most simple, basic and oldest idea
for a heuristic.
• The idea led to the development of huge
number of permutation based heuristics.
• Simulated annealing is one of the well know
meta-heuristic that has been developed upon
this idea.
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Heuristic Methods: Swap mechanism (2)
The idea:
• Given a solution in a programmable structure,
swap the elements within the structure to
produce an alternate solution.
Difficult or challenging part will be:
• to build a good structure that presents complete
information of the solution
• to develop an efficient swapping mechanism that
explores all possible alternatives
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Heuristic Methods: Swap mechanism (3)
Example: Simple Row Layout
Problem: To find the best row layout for a given ow matrix.
Swap heuristic:
Solution Structure: A VECTOR containing four departments,
representing the row layout.
For example, following structure says department 1 is on the
first in the row sequence. The next department to the left of
dept 1 is dept 2 (similarly 3 next to 2).
Department four is on the end of the row.
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Heuristic Methods: Swap mechanism (4)
• Initial or Current Solution: call it initial solution (if it is the
first iteration) or current solution in general.
• Neighborhood:
Possible swaps from the current solution:
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Heuristic Methods: Swap mechanism (4)
We consider only single swaps for ease of
tractability, i.e., following swaps are only permissible
from the current solution. Example:
We call the SET of ALL such single swap solution
vectors as single exchange neighborhood
The above current solution has a neighborhood
size of 42 =(4x3)/(2x1)=6
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Heuristic Methods: Swap mechanism (4)
Local Best: Any solution (including the current
solution) which has the best objective function value
in the neighborhood is called as local best.
Candidate Solution: A solution from the
neighborhood is called as candidate solution, if we
will select it as the current solution for the next
iteration.
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Heuristic Methods: Greedy strategy
• Start from an initial solution.
• Set current solution as initial solution.
• Begin iterations
• Find the local best from the neighborhood of current
solution.
• Set candidate solution as local best solution.
• If candidate solution is different from current solution
• Then set candidate solution as current solution.
• Else
• Return current solution as local optimal and
terminate.
• Continue iterations
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Greedy heuristic: Example (1)
Question: Given the following flow matrix among
departments 1 to 4, find the best row layout. Assume
distance between the adjacent departments is one
unit.
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Greedy heuristic: Example (2)
Solution:
Iteration 1:
Current solution = Initial Solution = [1; 2; 3; 4],
Current solution's objective function value =
T1;2;3;4 = 10* 1 + 15*2 + 20*3 + 10*1 + 5*2 + 5*1 = 125
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Greedy heuristic: Example (3)
Solution:
Iteration 1:
Current solution's neighborhood:
1
2
1
3
2
1
4
3
2
4
1
3
3
4
3
4
2
2
1
4
1
4
2
4
Cost at [2; 1; 3; 4], will be
T2;1;3;4 = 10x1 + 15x1 + 20x2 + 10x2 + 5x3 + 5x1 = 105
Similarly, we have T2;1;3;4 = 105, T3;2;1;4 = 95, T4;2;3;1 = 120, T1;3;2;4 = 120,
T1;4;3;2 = 105, T1;2;4;3 = 125.
2
3
1
3
Local best
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Greedy heuristic: Example (4)
Solution:
Iteration 2:
Current solution's neighborhood:
3
2
3
1
2
3
4
1
2
4
3
1
1
4
1
Current solution's
objective function value
= 95
4
2
2
3
4
3
4
2
4
Cost at [4; 2; 1; 3], will be
T4;2;1;3 = 10 1 + 15 1 + 20 2 + 10 2 + 5 1 + 5 3 = 105
Similarly, we have T2;3;1;4 = 90, T1;2;3;4 = 125, T4;2;1;3 = 105, T3;1;2;4 = 105,
T3;4;1;2 = 105, T3;2;4;1 = 110.
2
1
3
1
Local best
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Greedy heuristic: Example (5)
Solution:
Iteration 2:
Current solution's neighborhood:
2
1
2
4
1
3
3
2
3
4
3
1
4
2
1
4
3
4
Current solution's
objective function value
= 90
1
2
2
T1;3;2;4 = 120, T4;3;1;2 = 105, T2;1;3;4 = 105, T2;4;1;3 = 100, T3;2;1;4 = 95,
T2;3;4;1 = 105.
Local best = [2; 3; 1; 4], same as the current solution.
Since greedy approach, we stop here.
[2; 3; 1; 4] is a local optimal solution.
2
3
4
1
3
4
1
No
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improvement
Typical Layout
• What is the key element in proposing a heuristic?
 the solution structure
• What was the structure that we picked for row
layout?
 a vector
• What should be the structure for a typical
rectangular layout?
 not a direct answer, a research topic. But lets
discuss some successful approaches
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Typical Layout: Slicing Structure-Idea (1)
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Typical Layout: Slicing Structure-Idea (2)
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Typical Layout: Example (1)
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Typical Layout: Example (2)
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Typical Layout: Example (3)
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Typical Layout: Example (4)
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Typical Layout: Example (5)
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Typical Layout: Example (6)
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Typical Layout: Example (7)
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Typical Layout: Example (8)
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Typical Layout: Example (9)
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Typical Layout: Example (10)
Swap – Type 1
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Typical Layout: Example (11)
Swap – Type 1
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Typical Layout: Example (12)
Swap – Type 2
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Typical Layout: Example (13)
Swap – Type 2
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Typical Layout: Example (14)
Swap – Type 2
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Slicing Structure: Points to Remember
• Total number of columns in the matrix form is
(2m-1), where m is the total number of
departments.
• The number of internal nodes is m - 1.
• The first column should be an internal node.
• The last two columns should be an external
nodes.
• The number of zeroes in the bottom row
before any given column is greater than or
equal to the number of 1's before the given
column.
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Slicing Structure: Example (1)
Is the following slicing structure valid? If yes, then
find the corresponding layout.
Solution:
Verification:
Rule1: The number of columns is odd, equal to 13
(m=7).
Rule2: The number of internal nodes is 6.
Rule3: The first column is an internal node.
Rule4: The last two columns are external nodes.
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Slicing Structure: Example (2)
Is the following slicing structure valid? If yes, then
find the corresponding layout.
Solution:
Rule5: The total number of zeros and ones before
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Slicing Structure: Example (3)
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Slicing Structure: Example (4)
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Slicing Structure: Example (5)
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Slicing Structure: Example (6)
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Slicing Structure: Example (7)
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Slicing Structure: Example (8)
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Slicing Structure: Example (9)
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Slicing Structure: Example (10)
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Slicing Structure: Example (11)
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Slicing Structure: Example (12)
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Slicing Structure: Example (13)
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Typical Layout: MULTIPLE Structure-Idea
• Divide the given area Into small squares.
• The smaller the square area, the good the solution
will be.
• Let us assume that each small square has an area of
1 sq units.
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MULTIPLE: Space Filling Curves (SFC)
• SFCs are used to reconstruct a new layout when any
two departments are exchanged.
• Consider the SFC below which is known as the
Hilbert curve.
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MULTIPLE: Space Filling Curves (SFC) (2)
• The curve connects each grid such that a “dot”
traveling along the curve will always visit a grid that
is adjacent to its current grid
• Each grid is visited exactly once
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MULTIPLE: Example (1)
• Suppose that the layout vector is given by 1-2-3-45-6.
• Start from grid 1 and assign the first 16 grids (along
the SFC) to department 1
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MULTIPLE: Example (2)
1 2 3 4 5 6
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MULTIPLE: Example (3)
1 2 3 4 5 6
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MULTIPLE: Example (4)
1 2 3 4 5 6
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MULTIPLE: Example (5)
1 2 3 4 5 6
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MULTIPLE: Example (6)
1 2 3 4 5 6
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MULTIPLE: Example (7)
1 2 3 4 5 6
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MULTIPLE: Example (8)
1 2 3 4 5 6
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MULTIPLE: Example (9)
1 2 3 4 5 6
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MULTIPLE Structure: Points to Remember
(1)
• (SFCs) can be selected from standard curves, or can
be hand drawn.
• Hilbert Curves are the common SFCs, and can be
easily drawn for any nxn grid, for n a power of 2.
• Typically, the solutions obtained from MULTIPLE
algorithm need to be “massaged” to smooth the
department borders.
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MULTIPLE Structure: Points to Remember
(2)
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MULTIPLE Structure: Points to Remember
(3)
Alternative Layouts
- Type 1: Swap departments for a given SFC.
- Type 2: Change SFC to get new shapes for the
department.
- Typical approach:
• Develop more than one SFC
• For each SFC, find a good heuristic solution.
• Pick the best, from among the different SFCs'
good solutions, as the final best layout.
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MULTIPLE Structure: Points to Remember
(4)
Multiple Floors
Example: Say 1-2-3 are to be placed in floor 1, 4-5-6
are to be placed on floor 2, and 7-8-9 are to be
placed on floor 3.
A practical way to represent the above multi-floor
layout is:
1-2-3-|-4-5-6-|-7-8-9
where | indicates change of floor.
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MULTIPLE Structure: Points to Remember
(5)
Multiple Floors
1-2-3-|-4-5-6-|-7-8-9
Possible swaps:
Reassignment between the floors
1-2-|-3-4-5-6-|-7-8-9
rearrangement within a floor
1-3-2-|-4-5-6-|-7-8-9
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