Facility Layout
Emma Ross
September 2, 2010
Emma Ross () Facility Layout September 2, 2010 1 / 21
The Problem
Q: Where do we put machines on the production floor?
What makes a good layout?
I Companies want to run their factories ascheaply as possible to maximise profits
I But how do we calculate this cost?
Emma Ross () Facility Layout September 2, 2010 2 / 21
The Problem
Q: Where do we put machines on the production floor?
What makes a good layout?
I Companies want to run their factories ascheaply as possible to maximise profits
I But how do we calculate this cost?
Emma Ross () Facility Layout September 2, 2010 2 / 21
How do we find the cost? (1)
Cost (per unit) for transporting materials between places in thefactory. cij=cost per unit flow from place i to place j .
Do not want machines far apart, transportation will be complicatedand hence costly.
I c12 = c21=10
I c13 = c31=30
I c23 = c32=20
Emma Ross () Facility Layout September 2, 2010 3 / 21
How do we find the cost? (1)
Cost (per unit) for transporting materials between places in thefactory. cij=cost per unit flow from place i to place j .
Do not want machines far apart, transportation will be complicatedand hence costly.
I c12 = c21=10
I c13 = c31=30
I c23 = c32=20
Emma Ross () Facility Layout September 2, 2010 3 / 21
How do we find the cost? (1)
Cost (per unit) for transporting materials between places in thefactory. cij=cost per unit flow from place i to place j .
Do not want machines far apart, transportation will be complicatedand hence costly.
I c12 = c21=10
I c13 = c31=30
I c23 = c32=20
Emma Ross () Facility Layout September 2, 2010 3 / 21
How do we find the cost? (2)
Total cost of the layout:
n∑i=1
n∑j=1
cij fij
where fij= flow between machine i and j, and n=number of machines.
Example
Total Cost= (f21 ∗ c21) + (f23 ∗ c23)= (6∗10)+(2∗20)= 100
Emma Ross () Facility Layout September 2, 2010 4 / 21
How do we find the cost? (2)
Total cost of the layout:
n∑i=1
n∑j=1
cij fij
where fij= flow between machine i and j, and n=number of machines.
Example
Total Cost= (f21 ∗ c21) + (f23 ∗ c23)= (6∗10)+(2∗20)= 100
Emma Ross () Facility Layout September 2, 2010 4 / 21
The Model
Objective Function:
MinN∑i=1
N∑j=1
Ni∑ni=1
Mj∑mj=1
K∑k=1
K∑l=1
fnimj ∗ ckl ∗ xnik ∗ xmj l (1)
Where
xnik =
{1 if nth machine of type i is assigned to location k0 otherwise
Emma Ross () Facility Layout September 2, 2010 5 / 21
Constraints:
K∑k=1
xnik = 1, (2)
N∑i=1
Ni∑ni=1
xnik = 1, (3)
N∑i=0
Ni∑ni=1
fnimj tmjp ≤ cmj , (4)
Ni∑ni=1
Nj∑mj=1
fnimj = fij , (5)
N∑i=0
Ni∑ni=1
fnimj =N∑
q=0
Nq∑rq=1
fmj rq . (6)
Each machine is only at onelocation. . .Each loaction has only 1machine. . .Use of machine doesn’t exceedits capacity. . .Total flow for machine type =sum over copies. . .Input flow = Output flow,nothing lost inside.
Emma Ross () Facility Layout September 2, 2010 6 / 21
But there’s so much more to consider!
This problem in the real world is VERY complicated.Our model is extremely simple - there are many other variables and factorswhich can be (and are) encorporated into other models;
Different sized machines
Different layout of positions
Range of products (not just the one)
Preparing for changing demand -Robustness
Machine copies worked equally. . . etc.
Emma Ross () Facility Layout September 2, 2010 7 / 21
But there’s so much more to consider!
This problem in the real world is VERY complicated.Our model is extremely simple - there are many other variables and factorswhich can be (and are) encorporated into other models;
Different sized machines
Different layout of positions
Range of products (not just the one)
Preparing for changing demand -Robustness
Machine copies worked equally. . . etc.
Emma Ross () Facility Layout September 2, 2010 7 / 21
But there’s so much more to consider!
This problem in the real world is VERY complicated.Our model is extremely simple - there are many other variables and factorswhich can be (and are) encorporated into other models;
Different sized machines
Different layout of positions
Range of products (not just the one)
Preparing for changing demand -Robustness
Machine copies worked equally. . . etc.
Emma Ross () Facility Layout September 2, 2010 7 / 21
But there’s so much more to consider!
This problem in the real world is VERY complicated.Our model is extremely simple - there are many other variables and factorswhich can be (and are) encorporated into other models;
Different sized machines
Different layout of positions
Range of products (not just the one)
Preparing for changing demand -Robustness
Machine copies worked equally. . . etc.
Emma Ross () Facility Layout September 2, 2010 7 / 21
But there’s so much more to consider!
This problem in the real world is VERY complicated.Our model is extremely simple - there are many other variables and factorswhich can be (and are) encorporated into other models;
Different sized machines
Different layout of positions
Range of products (not just the one)
Preparing for changing demand -Robustness
Machine copies worked equally. . . etc.
Emma Ross () Facility Layout September 2, 2010 7 / 21
But there’s so much more to consider!
This problem in the real world is VERY complicated.Our model is extremely simple - there are many other variables and factorswhich can be (and are) encorporated into other models;
Different sized machines
Different layout of positions
Range of products (not just the one)
Preparing for changing demand -Robustness
Machine copies worked equally. . . etc.
Emma Ross () Facility Layout September 2, 2010 7 / 21
Approaches
Traditional: simple e.g. Functional Layout
More modern: algorithmic. Increasingly complicated, including manyinfluencing factors
Traditional is too simple but the more complex methods are socomplicated that they can only solve small problems.
Emma Ross () Facility Layout September 2, 2010 8 / 21
Approaches
Traditional: simple e.g. Functional Layout
More modern: algorithmic. Increasingly complicated, including manyinfluencing factors
Traditional is too simple but the more complex methods are socomplicated that they can only solve small problems.
Emma Ross () Facility Layout September 2, 2010 8 / 21
Where our model comes in
Intent:
Apply this basic optimisation model to numerically small problemswhich can be easily solved.
Experiment by changing variables such asI Demand Volume,I Job type,I Machine capacities.
Try to spot emerging patterns in the optimal layouts.
Find a way to characterise an optimal layout by words.
Apply results to larger more realistic problems.
Emma Ross () Facility Layout September 2, 2010 9 / 21
Where our model comes in
Intent:
Apply this basic optimisation model to numerically small problemswhich can be easily solved.
Experiment by changing variables such asI Demand Volume,I Job type,I Machine capacities.
Try to spot emerging patterns in the optimal layouts.
Find a way to characterise an optimal layout by words.
Apply results to larger more realistic problems.
Emma Ross () Facility Layout September 2, 2010 9 / 21
Where our model comes in
Intent:
Apply this basic optimisation model to numerically small problemswhich can be easily solved.
Experiment by changing variables such asI Demand Volume,I Job type,I Machine capacities.
Try to spot emerging patterns in the optimal layouts.
Find a way to characterise an optimal layout by words.
Apply results to larger more realistic problems.
Emma Ross () Facility Layout September 2, 2010 9 / 21
Experimentation
MPL Optimisation Software
Input
Model (objective, constraints and a demand volume)
Datafile for costs
Datafile for the machines’ capacities
Output
A table indicating where machines should go (the layout design)
A table describing the flow
Emma Ross () Facility Layout September 2, 2010 10 / 21
Experimentation
MPL Optimisation Software
Input
Model (objective, constraints and a demand volume)
Datafile for costs
Datafile for the machines’ capacities
Output
A table indicating where machines should go (the layout design)
A table describing the flow
Emma Ross () Facility Layout September 2, 2010 10 / 21
Experimentation
MPL Optimisation Software
Input
Model (objective, constraints and a demand volume)
Datafile for costs
Datafile for the machines’ capacities
Output
A table indicating where machines should go (the layout design)
A table describing the flow
Emma Ross () Facility Layout September 2, 2010 10 / 21
Experimentation
MPL Optimisation Software
Input
Model (objective, constraints and a demand volume)
Datafile for costs
Datafile for the machines’ capacities
Output
A table indicating where machines should go (the layout design)
A table describing the flow
Emma Ross () Facility Layout September 2, 2010 10 / 21
Unhelpful output format
Machine Place Activity Reduced Cost
0 pO 1.0000 0.00000 p1 0.0000 0.00000 p2 0.0000 0.00000 p3 0.0000 0.00000 pT 0.0000 0.0000
m1 pO 0.0000 0.0000m1 p1 0.0000 -1000000.0000m1 p2 1.0000 0.0000m1 p3 0.0000 -0.0000m1 pT 0.0000 0.0000
. . .
(n + 2)2 rows in the design table - so 3 machines: 21 rows of numbers, 10machines: 144 rows.
Emma Ross () Facility Layout September 2, 2010 11 / 21
I get by with a little help from R
R Program: Layout
Input
Design table from MPL
Flow Table from MPL
Character string for how many copies of each machine there are: e.g.2-3 = 2 type 1 machines and 3 type 3 machines
Output
Flow diagram with machine layout and flow
Emma Ross () Facility Layout September 2, 2010 12 / 21
I get by with a little help from R
R Program: Layout
Input
Design table from MPL
Flow Table from MPL
Character string for how many copies of each machine there are: e.g.2-3 = 2 type 1 machines and 3 type 3 machines
Output
Flow diagram with machine layout and flow
Emma Ross () Facility Layout September 2, 2010 12 / 21
I get by with a little help from R
R Program: Layout
Input
Design table from MPL
Flow Table from MPL
Character string for how many copies of each machine there are: e.g.2-3 = 2 type 1 machines and 3 type 3 machines
Output
Flow diagram with machine layout and flow
Emma Ross () Facility Layout September 2, 2010 12 / 21
Layout’s Output
Example
Emma Ross () Facility Layout September 2, 2010 13 / 21
Results (1)
Parameters which were varied;
Machine capacities
Demand volume
Tasks required - e.g. type 1 → type 2
Collection of machines - how many of each type
Recall. . .
Aim was to characterise patterns in the layout by words.
Observations;
Varying demand voume has little effect on layout
Copies of a machine type most often distanced from their dupliacte(s)
Varying the capacity does have an effect;I Largest capacity copies of a machine placed in ”prime” positions.I Smaller copies sit more on outside positions - only used for overspill.
Emma Ross () Facility Layout September 2, 2010 14 / 21
Results (1)
Parameters which were varied;
Machine capacities
Demand volume
Tasks required - e.g. type 1 → type 2
Collection of machines - how many of each type
Recall. . .
Aim was to characterise patterns in the layout by words.
Observations;
Varying demand voume has little effect on layout
Copies of a machine type most often distanced from their dupliacte(s)
Varying the capacity does have an effect;I Largest capacity copies of a machine placed in ”prime” positions.I Smaller copies sit more on outside positions - only used for overspill.
Emma Ross () Facility Layout September 2, 2010 14 / 21
Results (1)
Parameters which were varied;
Machine capacities
Demand volume
Tasks required - e.g. type 1 → type 2
Collection of machines - how many of each type
Recall. . .
Aim was to characterise patterns in the layout by words.
Observations;
Varying demand voume has little effect on layout
Copies of a machine type most often distanced from their dupliacte(s)
Varying the capacity does have an effect;I Largest capacity copies of a machine placed in ”prime” positions.I Smaller copies sit more on outside positions - only used for overspill.
Emma Ross () Facility Layout September 2, 2010 14 / 21
Results (1)
Parameters which were varied;
Machine capacities
Demand volume
Tasks required - e.g. type 1 → type 2
Collection of machines - how many of each type
Recall. . .
Aim was to characterise patterns in the layout by words.
Observations;
Varying demand voume has little effect on layout
Copies of a machine type most often distanced from their dupliacte(s)
Varying the capacity does have an effect;I Largest capacity copies of a machine placed in ”prime” positions.I Smaller copies sit more on outside positions - only used for overspill.
Emma Ross () Facility Layout September 2, 2010 14 / 21
Results (1)
Parameters which were varied;
Machine capacities
Demand volume
Tasks required - e.g. type 1 → type 2
Collection of machines - how many of each type
Recall. . .
Aim was to characterise patterns in the layout by words.
Observations;
Varying demand voume has little effect on layout
Copies of a machine type most often distanced from their dupliacte(s)
Varying the capacity does have an effect;I Largest capacity copies of a machine placed in ”prime” positions.I Smaller copies sit more on outside positions - only used for overspill.
Emma Ross () Facility Layout September 2, 2010 14 / 21
Results (1)
Parameters which were varied;
Machine capacities
Demand volume
Tasks required - e.g. type 1 → type 2
Collection of machines - how many of each type
Recall. . .
Aim was to characterise patterns in the layout by words.
Observations;
Varying demand voume has little effect on layout
Copies of a machine type most often distanced from their dupliacte(s)
Varying the capacity does have an effect;I Largest capacity copies of a machine placed in ”prime” positions.I Smaller copies sit more on outside positions - only used for overspill.
Emma Ross () Facility Layout September 2, 2010 14 / 21
A more useful model?
Useful starting point but a better model will represent
The ability to make multiple products in one factory and,
The stochasticity of demand.
Our simple model can be adapted to include more than one scenario andfind the best layout given that any of them might occur. This gives amodel which
Gives a more flexible, robust design which will cope with more tasksand possible changes in demand, and
Will make a compromise between optimal layouts for individualscenarios.
Emma Ross () Facility Layout September 2, 2010 15 / 21
A more useful model?
Useful starting point but a better model will represent
The ability to make multiple products in one factory and,
The stochasticity of demand.
Our simple model can be adapted to include more than one scenario andfind the best layout given that any of them might occur. This gives amodel which
Gives a more flexible, robust design which will cope with more tasksand possible changes in demand, and
Will make a compromise between optimal layouts for individualscenarios.
Emma Ross () Facility Layout September 2, 2010 15 / 21
Adapting the Model
Input
Unchanged: 2 datafiles for capacity and cost
Different: Model now has multiple scenarios each with a different taskand/or demand volume.
New Objective:
MinS∑
s=1
N∑i=1
N∑j=1
Ni∑ni=1
Mj∑mj=1
K∑k=1
K∑l=1
πs fnimj scklxnikxmj l (7)
Where S is the total number of scenarios and πs is the probability ofscenario s occuring.
Output
Unchanged: One table representing the optimal layout design
Different: Now have multiple tables representing the flow for eachscenario
Emma Ross () Facility Layout September 2, 2010 16 / 21
Adapting the Model
Input
Unchanged: 2 datafiles for capacity and cost
Different: Model now has multiple scenarios each with a different taskand/or demand volume. New Objective:
MinS∑
s=1
N∑i=1
N∑j=1
Ni∑ni=1
Mj∑mj=1
K∑k=1
K∑l=1
πs fnimj scklxnikxmj l (7)
Where S is the total number of scenarios and πs is the probability ofscenario s occuring.
Output
Unchanged: One table representing the optimal layout design
Different: Now have multiple tables representing the flow for eachscenario
Emma Ross () Facility Layout September 2, 2010 16 / 21
Adapting the Model
Input
Unchanged: 2 datafiles for capacity and cost
Different: Model now has multiple scenarios each with a different taskand/or demand volume. New Objective:
MinS∑
s=1
N∑i=1
N∑j=1
Ni∑ni=1
Mj∑mj=1
K∑k=1
K∑l=1
πs fnimj scklxnikxmj l (7)
Where S is the total number of scenarios and πs is the probability ofscenario s occuring.
Output
Unchanged: One table representing the optimal layout design
Different: Now have multiple tables representing the flow for eachscenario
Emma Ross () Facility Layout September 2, 2010 16 / 21
Analysis of the Results
Now have even more tabular data to make sense of, so a diagramdrawing progam will save much time and potential for mistakes.
Layout is adapted to give outputs such as;
Emma Ross () Facility Layout September 2, 2010 17 / 21
Analysis of the Results
Now have even more tabular data to make sense of, so a diagramdrawing progam will save much time and potential for mistakes.
Layout is adapted to give outputs such as;
Emma Ross () Facility Layout September 2, 2010 17 / 21
Results (2)Parameters which were varied;
Number of scenarios
Probability of each scenario
Basis of Results. . .
Results are based on comparison of
Optimal layout from each scenario individually (first model), and
The stochastic model’s result,
for very small problems only.
Observations;
Demand volume again has very little effect on the layout.
If scnerios differ only by demand volume then no compromise made.
Seems that compromise evident only when the scenrio’s tasks aredifferent,e.g. Scenario 1(type 1 → type 2), Scenario 2(type 2 → type 1).
Emma Ross () Facility Layout September 2, 2010 18 / 21
Results (2)Parameters which were varied;
Number of scenarios
Probability of each scenario
Basis of Results. . .
Results are based on comparison of
Optimal layout from each scenario individually (first model), and
The stochastic model’s result,
for very small problems only.
Observations;
Demand volume again has very little effect on the layout.
If scnerios differ only by demand volume then no compromise made.
Seems that compromise evident only when the scenrio’s tasks aredifferent,e.g. Scenario 1(type 1 → type 2), Scenario 2(type 2 → type 1).
Emma Ross () Facility Layout September 2, 2010 18 / 21
Results (2)Parameters which were varied;
Number of scenarios
Probability of each scenario
Basis of Results. . .
Results are based on comparison of
Optimal layout from each scenario individually (first model), and
The stochastic model’s result,
for very small problems only.
Observations;
Demand volume again has very little effect on the layout.
If scnerios differ only by demand volume then no compromise made.
Seems that compromise evident only when the scenrio’s tasks aredifferent,e.g. Scenario 1(type 1 → type 2), Scenario 2(type 2 → type 1).
Emma Ross () Facility Layout September 2, 2010 18 / 21
Results (2)Parameters which were varied;
Number of scenarios
Probability of each scenario
Basis of Results. . .
Results are based on comparison of
Optimal layout from each scenario individually (first model), and
The stochastic model’s result,
for very small problems only.
Observations;
Demand volume again has very little effect on the layout.
If scnerios differ only by demand volume then no compromise made.
Seems that compromise evident only when the scenrio’s tasks aredifferent,e.g. Scenario 1(type 1 → type 2), Scenario 2(type 2 → type 1).
Emma Ross () Facility Layout September 2, 2010 18 / 21
Results (2)Parameters which were varied;
Number of scenarios
Probability of each scenario
Basis of Results. . .
Results are based on comparison of
Optimal layout from each scenario individually (first model), and
The stochastic model’s result,
for very small problems only.
Observations;
Demand volume again has very little effect on the layout.
If scnerios differ only by demand volume then no compromise made.
Seems that compromise evident only when the scenrio’s tasks aredifferent,e.g. Scenario 1(type 1 → type 2), Scenario 2(type 2 → type 1).
Emma Ross () Facility Layout September 2, 2010 18 / 21
Limitations and Thoughts
Experimenting with just some combinations of variables is time costly.Cannot think of a quicker way to analyse without losing the detailsneeded to understand patterns.
Too simple?I Precisely the point! Awkward problem which merits this approach.I Goes some way to predicting good layouts for larger problems - can
always be built up in complexity over time.
Wordy results?I Better than figurative results in this caseI More useful to manufacturers?I Can only go so far in predicting demand and problem is very awkward
Emma Ross () Facility Layout September 2, 2010 19 / 21
Limitations and Thoughts
Experimenting with just some combinations of variables is time costly.Cannot think of a quicker way to analyse without losing the detailsneeded to understand patterns.
Too simple?I Precisely the point! Awkward problem which merits this approach.I Goes some way to predicting good layouts for larger problems - can
always be built up in complexity over time.
Wordy results?I Better than figurative results in this caseI More useful to manufacturers?I Can only go so far in predicting demand and problem is very awkward
Emma Ross () Facility Layout September 2, 2010 19 / 21
Limitations and Thoughts
Experimenting with just some combinations of variables is time costly.Cannot think of a quicker way to analyse without losing the detailsneeded to understand patterns.
Too simple?I Precisely the point! Awkward problem which merits this approach.I Goes some way to predicting good layouts for larger problems - can
always be built up in complexity over time.
Wordy results?I Better than figurative results in this caseI More useful to manufacturers?I Can only go so far in predicting demand and problem is very awkward
Emma Ross () Facility Layout September 2, 2010 19 / 21
Last words
Have been reminded of how useful going back to basics and understandingthe fundamentals of a problem are - can go a surprisingly long way witheven very complex problems.
Thank you for asking easy questions.
Emma Ross () Facility Layout September 2, 2010 20 / 21