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Dr. Daniel Xing Email: [email protected]

EBUS-504

Operations Modelling and Simulation

Lecture 4

Bottleneck analysis-2

University of Liverpool

Management School,

Key learning outcomes

1. Bottleneck analysis recap;

2. Understand steady-state;

3. Analysis under uncertainty;

4. Setup uncertain parameters in Witness

5. Use variables and extract data for analysis

Why bottleneck is so important?

1. It defines the maximal throughput rate of your system.

2. It helps modellers quickly locate queues

3. It helps you identify the total outputs at a certain point.

4. It defines the maximal utilisation rate for each entity of your system.

5. Most importantly, it provides further improvement directions.

Any more?

Let’s do with some exercise

1. The production of product A requires a sequential processes and their

operation time is 3mins, 5mins and 7mins respectively;

2. Product A is assembled by 4 components (2Bs, 1C and 1D) with 2mins.

Each type of component requires a pre-processing operation with

machine time 3mins, 5mins, and 4mins respectively.

3. A supermarket has three tills to serve its customers. Each till needs

2mins on average to finish the service and customers arrive the store

every 1min.

4. A line production is comprised by 3 machines with operation time

6mins, 8mins and 4mins respectively. Every machine needs a 2mins

setup by L1 (there is only one labour available) and part arrives every

2mins.

Analytical-based methods Input rate vs. output rate

Bottleneck of your system is always identified when input rate is faster than

output rate

The busiest resource (capacity analysis)

Filter out the entity in the system which takes the longest time to complete a job

The most congested place (throughput analysis)

The place where a part takes the longest time to enter and leave it.

Product production lifecycle analysis

From raw material(s) until the completion of an end-product, analysing how

much time in percentage that each component needs to be operated with.

Analytical-based methods The ultimate rule to determine a bottleneck under a deterministic setting:

“It is the only resource which makes all other resource waiting”:

✓ Any reduction in its utilisation can reduce the overall throughput

✓ Double its capacity can double the utilisation of any other resources if their

current utilisations are below 50%.

Analytical-based methods

Pros:

✓ Easy to build an overall understanding of your system;

✓ Helpful for model validation purposes;

✓ Light up the initial system improvement plans;

✓ Very effective for deterministic models;

Cons:

❖ Hard to identify the bottleneck when system structure is complex;

❖ Ineffective for stochastic models;

❖ Hard to capture all model details (good for long-term planning but not

short-term)

❖ Can be time consuming and potential human errors

Simulation-based methods a) Methods based on machine utilisations

Simulating your model and determine the bottleneck based on the largest

average machine cycle time or the largest machine utilisation.

b) Methods based on waiting times or queue lengths

Simulating your model and the bottleneck should be at the process with the

largest DROP in waiting time or LONGEST queue length.

Simulation-based methods c) The Arrow Method Based on Starving and Blocking

If the frequency of manufacturing blockage of machine m_i is larger than the

frequency of manufacturing starvation of machine m_i+1, the bottleneck is

downstream of machine of machine m_i. If the frequency of the manufacturing

starvation of machine m_i is larger than the frequency of manufacturing

blockage of m_i-1, the bottleneck is upstream of machine m_i.

Simulation-based methods d) The “turning point” method

Key concepts:

1. a bottleneck machine will often make the upstream machines blocked and

downstream machines starved.

2. a bottleneck machine will also have a lower overall sum of blockage and

starvation time.

Simulation-based methods d) The “turning point” method

Simulation-based methods e) machine production rate sensitivity analysis

the most critical bottleneck machine has the highest sensitivity value of the

system production rate to a machine’s production rate.

i.e. system system

i j

PR PR j i

PR PR

    

 

Increase the production rate of

the potential bottleneck with the same percentage

Compare the increase of system

production rate from each round

Determine the bottleneck which leads the highest system production

rate increase

Simulation-based methods e) machine production rate sensitivity analysis

the most critical bottleneck machine has the highest sensitivity value of the

system production rate to a machine’s production rate.

Simulation-based methods

Pros:

✓ Easy to detect the bottleneck;

✓ Effective for large scale problems;

✓ It can handle stochastic systems;

Cons:

❖ Hard to see the rationale behind the results, it tells you “where” and

“what” but not “why”;

❖ Errors of simulation model can lead to wrong detection;

❖ It requires analytical support to draw the final conclusion in some

cases.

Resume our discussion about queues

1. How do we know the waiting time for the 10th part? 50th part? Or the nth

part?

2. How do we know the queue size at minute 50? Or minute 100? Or the

nth minute?

2mins 1min 8mins 4mins

Resume our discussion about queues

1. Mathematical induction

➢ The 2nd part: 6 mins

➢ The 3rd part: 3mins + 8mins

➢ The 4th part: 8mins + 8mins

➢ The 5th part: 5mins + 8mins + 8mins

➢ The 6th part: 2mins + 8mins + 8mins + 8mins

➢ The 7th part: 7mins + 8mins + 8mins + 8mins

➢ The 8th part: 4mins + 8mins + 8mins + 8mins + 8mins

What do you see from above? Can you conclude a generic formulation using

nth part, bottleneck cycle time T_b and part inter-arrival time T_a?

Resume our discussion about queues

When n=1

W_n=0;

When n>=2

W_n = W_(n-1) – T_a + T_b

Why?

Resume our discussion about queues

When n=1

W_n=0;

When n>=2

W_n = W_(n-1) – T_a + T_b

Why?

Resume our discussion about queues

Let’s change our angle

The nth part

Arriving at what

time makes it no

waiting?

What is the actual

arriving time?

Why the first one is different? - steady state analysis

Steady state:

In systems theory, a system or a process is in a steady state if the

variables which define the behaviour of the system or the process are

unchanging in time

1. Why do we need to exclude warm-up period?

2. How do we determine warm-up period?

Steady-state analysis

1. In a deterministic setting:

The system is deemed to show a repetitive pattern as long as the

simulation run is long enough.

2. In a stochastic setting:

The stochastic processes associated with the output variables of interest

become stationary.

o For example, if the inter arrival time of a part is following a Poisson

distribution or a normal distribution.

Steady-state analysis

Source from: Birta and Arbez (2013)

Steady-state analysis

✓ Selecting the size of the time cells (e.g. Every 5 minutes)

✓ Determine the total number of time cells

✓ Determine the number of replications

✓ Obtain the value as the average over the n replications of the i th cell

averages ( i.e. assuming i is the index for the i th cell and j is the index

for the j replication), then we have:

,i jy

Steady-state analysis

Welch’s moving average method:

If plotted against index i, the resulting graph is ‘choppy’ and difficult to interpret

We introduce a smoothing operation to smooth out the rapid variations to

obtain a smoother curve that captures the long-run trend.

is a moving average value and parameter w represents a window

size that controls the smoothing operation.

( )ia w

Steady-state analysis

When i <= w, there

are not enough

values preceding

time cell I to fill the

window so we use (i-

1) replace w.

Steady-state analysis

ia (3)ia

(5)ia The larger the window size, the smooth

plot you’ll get, but less details can be

captured.