facility layout - lancaster university · facility layout emma ross september 2, 2010 emma ross...

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?

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?

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

I c12 = c21=10

I c13 = c31=30

I c23 = c32=20

I c12 = c21=10

I c13 = c31=30

I c23 = c32=20

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

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

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)

xnik =

{1 if nth machine of type i is assigned to location k0 otherwise

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∑

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.

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.

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.

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.

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.

Intent:

Experimentation

MPL Optimisation Software

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

Experimentation

Datafile for costs

Output

Experimentation

Datafile for costs

Output

Experimentation

Datafile for costs

Output

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.

I get by with a little help from R

R Program: Layout

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

R Program: Layout

Flow Table from MPL

Output

R Program: Layout

Flow Table from MPL

Output

Layout’s Output

Example

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.

Results (1)

Machine capacities

Demand volume

Recall. . .

Observations;

Results (1)

Machine capacities

Demand volume

Recall. . .

Observations;

Results (1)

Machine capacities

Demand volume

Recall. . .

Observations;

Results (1)

Machine capacities

Demand volume

Recall. . .

Observations;

Results (1)

Machine capacities

Demand volume

Recall. . .

Observations;

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.

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.

Adapting the Model

Unchanged: 2 datafiles for capacity and cost

Different: Model now has multiple scenarios each with a different taskand/or demand volume.

New Objective:

MinS∑

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

Adapting the Model

Different: Model now has multiple scenarios each with a different taskand/or demand volume. New Objective:

MinS∑

N∑i=1

N∑j=1

Ni∑ni=1

Mj∑mj=1

K∑k=1

K∑l=1

Output

Adapting the Model

Different: Model now has multiple scenarios each with a different taskand/or demand volume. New Objective:

MinS∑

N∑i=1

N∑j=1

Ni∑ni=1

Mj∑mj=1

K∑k=1

K∑l=1

Output

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;

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;

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).

Number of scenarios

Observations;

Number of scenarios

Observations;

Number of scenarios

Observations;

Number of scenarios

Observations;

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

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.

facility layout - lancaster university · facility layout emma ross september 2, 2010 emma ross...

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