When a process variable has only random variation each output is independent of the previous ones. This is what happens in a lottery.  In some processes this independence does not happen. If we control our daily weight, for instance, our weight today is correlated to the weight of the previous days: it has autocorrelation

A similar effect happens when you control a heavy aircraft or a ship: the weight prevents you from making a sharp turning to change the course. This opposing force to change is what is called Inertia

The inertia definition applies to moving objects and it is proportional to the object mass. But inertia also applies to fluids: a tank accumulating a fluid will also have this inertia effect.

 If we try to control a process with only random variation by reacting to every output we can see in Process Control that the process will get worse: variation will increase. 

We will now experience how to control a process with inertia with a simulator of the tilt control in a plane:

Download Inertia simulator.xlsm

Close other Excels and allow macros to run it.

Press Start to start simulating and place the cursor on top of the vertical arrows to adjust the tilt by shifting up/ down (do not click).

You should try to keep tilt as close to zero as possible.

The graphs below will show you the adjustments you made and the actual tilt evolution along the 50 runs

The Average and StdDev on top will tell you the extent of your success.

Response delay

The first thing you will notice is that there is a delay between your actions and the tilt response 

This is the result of inertia: the response is slow. 

Stabilising effect of inertia

 In this graph we can see the random source of variation and the resulting tilt without adjustments. We notice that inertia has produced a stabilising effect reducing drastically variability. This can be confirmed by the histograms:

We notice that inertia has caused a reduction of standard deviation from 18.4 to 2.6. We also notice that Tilt (with autocorrelation) passes the normality test (p = 0.21).

Now let's look at the stability of tilt: 

Variation has been drastically reduced but the source of variation was in control and now we have situations of out of control as shown by the Individuals Control Chart.

These tilt out-of-control situations have very limited range compared with the source range of variation.


Another effect of inertia is autocorrelation:

We notice that the random source of variation had no autocorrelation but inertia has caused significant tilt autocorrelation.

Automatic Control of Process with Inertia

If we automatically balance by making Adjustment = - Tilt: 

the result will be reasonable as compared to the case of no adjustment: 

We obtain very similar standard deviations: 3.35 Vs 4.00.

 Now we can experience the difficulties of achieving an effective balancing:

It is difficult to predict when adjustment is required and how much to adjust. 

Real Life: Unstable Source of Variation

So far we have assumed that the source of variation was random with an average of zero.

Now let us try something closer to reality:

Download Inertia variation.xlsm

Try to achieve balance now. 

With automatic Adjust = - Tilt we can achieve:

You can try other automatic adjustment formulas as a function of tilt by modifying the formula in A6.

Continuous Process Balancing by Accumulation in a Tank

By accumulating a fluid in a tank we achieve the autocorrelation effect which is useful to reduce the standard deviation of a critical metric.

Factories often need to drain water to a river or to the sea in which case they have to comply with regulations about its pH. 

Pure water has a pH of 7. Some local regulations require that water pH should be between 5.5 and 9.5 before it can be drained into a river.

In the following example accumulation in a tank has been used in order to reduce the pH standard deviation and meet the required specs. We have done a capability analysis of both the input to the tank and the output drained to the river:

Ppk has increased from 0.26 (totally unacceptable) to 1.15. Total ppms are reduced from 342,594 to 285.

Looking at autocorrelation:

We can see significant autocorrelation produced by the tank accumulation.

Looking at pH stability:

We notice clustering and trends but this happens well within the spec limits.

Body Weight Autocorrelation

 Looking at the data used in


Let's check it for autocorrelation:

We confirm that body weight is indeed autocorrelated: our weight today is correlated to the weight we had in the previous days: it has inertia. 


  • Process inertia shows with autocorrelation: metric values are dependent of previous values
  • Inertia causes a delay between cause and effect
  • The standard deviation of the sources of variability is reduced in the effects
  • Mechanical inertia is proportional to the mass of the object
  • Fluid inertia is proportional to the volume of its storage in a tank or pond
  • Mechanical inertia is used by flywheels to smooth rotation speed
  • Fluid accumulation in a tank is used to reduce the standard deviation of its parameters


When we need to process different products in the same line we want to work out the best job sequence to minimize lead time and maximize capacity utilization.

For non repetitive processes you can use project management scheduling:


The ideal of "Lean Production" is lot size of 1 but this is not always possible when there is a change over time. This change over involves loading different components and changing jigs or molds. When change over or setup time is significantly higher than process time we are forced to work with larger lot sizes. 

 Scheduling Constraints

 In a sequential line products are processed through a number of workstations following a certain sequence but process parameters are typically different for each product. Each workstation, in order to do the work needs:

  • equipment available
  • operator ready
  • WIP (Work In Process) from the previous workstation available

And we want to do the job with:

  • minimum equipment time
  • minimum operator time
  • minimum WIP waiting 

The problem is that unless everything is perfect someone has to wait:

In the case of a space ship in order to minimize the astronaut time: a huge team of ground personnel is available just in case.

In an airport, in order to maximize the runway utilization (airport bottleneck): planes, pilots, personnel, customers are waiting for take off.

In Formula I races 7 second service time is achieved with a team of 10 people doing nothing most of the time during the race.

In our sequential line in a workstation either the equipment or the operator will be idle or there will be excess WIP.

 If process time for all the sequential operations are the same:

The process is perfect: nobody waits. While WS A is processing Order 2 WS B is processing Order 1 and the total lead time for each order is 4h.

If process time for operation A is shorter than B:

Order 2 will have to wait for operation B to finish with order 1 and therefore Order 2 will take 4h instead of 3 to complete and this will affect the total lead time seen by the customer waiting for this order. Also it will increase the amount of WIP in the line.

If time B is shorter:

Workstation B will be idle until order 2 arrives and this will decrease capacity utilization.

Excel Scheduler

You can download an Excel scheduler for 4 sequential operations:


 It is used to schedule an electronic circuit assembly line with 4 operations in this sequence:

  1. SMT: Surface Mount Technology automatic component placement and reflow sodering
  2. INS: Manual pin-thru-hole component insertion and wave soldering
  3. ICT: In-Circuit-Test
  4. FUN: Functional Test

Process Parameters

 In sheet Times we must input Equipment times, Operator times and Number of operators for each of the 4 operations and for each product. 

In the existing data we want to maximize equipment utilization by adding as many operators as required to do this. Therefore:

Number of operators = Operator time / Equipment time   (rounding up)

For instance number of operators for Prod A INS = 18:30 / 01:00  = 19 (rounded up)

Cycle time for each product and operation = maximum of Equipment time and (Operator time/ Number of operators)

The scheduler will use these calculated cycle times: Job operation time = Setup time + Cycle time x Quantity

Setup times for each operation are input in the Main sheet and are considered the same for all products. 

With the current number of operators you can notice that Cycle times are identical to equipment times: by adding enough operators we have insured that the line capacity is limited by equipment times and not operator times. 

Obviously the number of operators calculated may not be feasible, in which case we enter the available operators and processing will take longer as we will experience later.  

 Main Sheet

Column A has the order in which we want to run the jobs. It should have numbers 1 to 10 in the order we decide.

Column B has the products to be produced selected for the ones we defined in sheet Times.

Column C holds the number of operations already completed (0 to 4)

Column D has the quantities we need to produce of each product

Finally the starting time stamp is the day and time we will start producing

Gantt Chart

The Gantt chart shows the scheduling result:

By placing the cursor on the chart we can see the starting or ending time stamp for any operation and product.

WIP waiting times are in light colors and these are what we want to minimize to reduce lead times.

We can also notice when workstations are idle.

In this case, if we start on 6/2/2019 at 6:00 and all 10 jobs will be finished before 12/2/2019.

Additional Inputs and Results

The number of work cells in each operation is 1 by default but we may have up to 5. 

Setup times need to be defined for each operation. In this example it is the time it takes to unload all component reels and load the new ones for the next product. In the case of test it is the time it takes to change the test jigs and load the new program. We assume all products have the same setup times.

The next 2 lines are outputs:

  • Total process times of each workstation (no waiting times included) 
  • Capacity utilization of each workstation. You can notice that SMT with 100% utilization is, in this case, the bottleneck for the whole line. 

Total lead time and average lead time are outputs that will affect the customers which we want to minimize.

 Schedule for Each Workstation

A detailed schedule is produced for each workstation with starting and ending timestamps and number of operators required.

The actual times can be added on the right columns to be used on the next reschedule.

Bottleneck Scheduling

Looking at the capacity utilization of the 4 workstations we notice that SMT has 100% utilization but all the rest are well below.

This indicates that the bottleneck of this line with these required quantities is SMT. In this case the bottleneck happens to be the first workstation so the following stations are starved of products so they can't process as much as they could. 

The order in which we have scheduled the 10 jobs has been from lowest use of SMT (bottleneck) time to highest use. 

Let's see what would happen if we reverse the order: Highest use of bottleneck first:

We notice that the total lead time has increased from 5 days 17h to 7 days: an increase of 1 day and 7 hours.

Average job lead time has also increased from 1d 10h 24m to 1d 13h 16m

It is obvious that it makes no sense to schedule products A and B at the end since they do no use the bottleneck so if we just put them first keeping the rest of the sequence as it is we would get a total lead time of 6 days 11h: still worse than the first option. 

The first conclusion is that we should schedule focusing on the bottleneck workstation and process from least to most bottleneck use

Bottleneck in the last operation

 In this case, with the same products but different quantities to be processed we notice that this sequence has a lot of waiting time before the INS and FUN operations. Capacity utilization is very good but average lead time is very high: all jobs except the first have a lot of waiting and these long lead times will be unacceptable to the customer. 

The bottleneck in this case is FUN: total time for all jobs is 114h 51m 40s much higher than the other workstations. 

We notice that with this sequence we are scheduling from highest bottleneck (FUN) use to lowest, so let's try the reverse order:

Total lead time has increased 1d 4h but average lead time has decreased from 3 days 16h to 1 day 10h (less than half).

This, therefore, reinforces our previous conclusion: schedule from less bottleneck use to more.

 Schedule with operators constraint

 In the examples so far we have assumed that there are enough operators in each workstation so that the equipment never stops but this assumption may not be feasible.

Looking at the schedule for the INSERT operation we are using 19 operators for product A and the number decreases down to 1 for product J. 

Let's assume that we only have 3 operators available to perform this operation for all products:

In the Times sheet we enter 3 in the number of operators for all products and the result is that now cycle times have increased above equipment times. This means that now the operators will be the constraint rather than the equipment in the INS workstation.

The resulting schedule is:

Now INS has become the bottleneck and there is an accumulation of products waiting to be processed in this workstation causing waiting time in all jobs. Average lead time has increased from 1 day 10h to 2 days 21h (it has doubled). Total lead time has only increased 10 hours. 

 Cell duplication

In a line we will always have a bottleneck which limits the capacity of the whole line. In some cases we might be able to duplicate the bottleneck workstation to increase the overall line capacity. 

Let's duplicate the SMT workstation which was the bottleneck in our first example to see the effect:

 Total lead time has been reduced from 5 days 17h to 3 days 15h (more than 2 day reduction)

Average lead time has increased 4 hours: we have waiting in INS and FUN.

We notice that SMT capacity utilization has decreased from 100% to 86% while utilization in the other 3 workstations has increased. 

Let's see if duplicate INS and FUN as well:

We notice that total lead time has not improved but average lead time has decreased 9 hours: practically no waiting time.

Since we have increased all capacities capacity utilizations have decreased.


Going back to our first schedule starting on 6/2/2019 we normally have to update this schedule at least daily or when a major change has taken place. We need to remove what has already been done and add any new jobs. Let's see how do we reschedule on the next day 7/2/2019:

We change our starting timestamp to 07/02/19 06:00 and then input in the OK column: Products A and B completed (4 operations finished) C and D only FUN is missing (3 completed), etc.

If there are any new jobs we would add them at the end shifting all jobs up.

 Job Splitting

Due to the setup times every time we change product we may think that the larger the lot size the better. Let's see this effect with a new example:

We need to produce these 5 products with quantities of 500 each. What would be the effect of splitting the jobs in two?

In spite of duplicating setup times we see average lead times have been halved and total lead time is also reduced.


  • In order to commit a delivery time to the customer we need to schedule taking into account existing jobs already in process
  • When a job is given priority we need to know the impact to all other current jobs
  • Job scheduling enables lead time reduction while increasing resource utilization
  • To minimize total lead time we should focus on the bottleneck
  • Schedule jobs from lowest to highest bottleneck utilization
  • After total lead time is minimized small changes in sequence will allow the reduction of waiting times and therefore the reduction of average lead time. 
  • By meeting the resulting schedule on each workstation we will insure on-time delivery to the customer.
  • Regular rescheduling is required to adapt to the real current situation

Production based on forecast uses resources to produce items which will not eventually be sold while there is a shortage of those items the market actually demands.

Demand lead times are getting shorter in most business while supply lead times are not able to keep pace. This often leaves just one alternative: "make to plan", also called "Push" logistics. The result is that in spite of our excess inventories we are unable to satisfy the demand: we have plenty of what nobody wants and no enough of what they want.

"Make to order" or "Pull" logistics produces only what has been ordered by the customer so we avoid dedicating resources to produce unwanted items. The problem usually is lead time: can the customer wait until our supply chain is able to deliver?

One way to implement "Pull" logistics is to use Kanban. 

A Kanban (Japanese for a card) is a token generated by the consumption of an item (or fixed lot of items) which authorises the production of a similar one to replace it.

Download:  Kanban Logistics Simulator

In the model above the blue workstation can produce one item A or one B every 4 minutes. There is no setup time when changing product. "a" is a kanban which enables the production of "A" and "b" a kanban to produce "B". 

When an item "A" is consumed its corresponding kanban "a" is released and it joins the queue at the production station. Kanbans "a" and "b" will be produced in the order of arrival to the queue.

Market demand is defined by the Takt times of A and B: one item A is required every 10 minutes and one B also every 10 minutes. 

The number of kanbans in circulation determines the maximum inventory we could accumulate.


Supply Capacity Greater than Demand

In this case Takt of both A and B is 20 minutes which is equivalent to 3 items per hour. Since we need 3 A's and 3 B's every hour this gives a total of 6 items per hour to be produced. Production process time is 5 minutes therefore production capacity is 12 items/ hour. Kanbans are limiting production to the consumption rate of 6/ hour therefore we are producing at 50% capacity.

In this case we have decided to put into circulation 10 "a" kanbans and 10 "b" therefore we have accumulated an inventory of 10 A's and 10 B's.


Balanced Supply and Demand

 In this case supply capacity 12/h is the sum of demand thruputs (6 + 6)/h therefore supply capacity utilization is 100% and demand fulfilment of both A and B is also 100%.


Short Supply

 In this case Supply is the bottleneck: we are unable to supply the market demand which is 12 A's plus 12 B's' per hour. Since total capacity is 12/h fulfilment of both demand A and demand B is 50%. 

We see no inventory of A's or B's because as soon as they are produced they are consumed.


Priority Allocation

In this case of limited supply we might decide to give priority to one item (or customer) Vs the other. We can do this with the number of kanbans in circulation. In this case by releasing 10 a's and 5 b's we are able to increase fulfilment of A to 70% at the expense of fulfilment of B which will drop to 30%.


Item B Discontinued

With this Just-In-Time approach a drop in the demand of one item (customer) could be balanced by an increase in another one.

In this example B in no longer required so the full capacity is available to produce A. 


Balanced Fulfilment

In this case of short supply the demand of A and B is different. We may decide to manage this situation by balancing the fulfilment of A and B. Since demand of A is double of B we release double number of "a" kanbans Vs "b".

In this way we achieve 80% fulfilment both for A and B.


Number of Kanbans

 The more kanbans we release the more inventory we could accumulate so we want to keep the number of kanbans as low as possible.

On the other hand we may have variation both in process time and in the demands of both A and B. To compensate for this variation we will need to increase the number of kanbans and therefore the level of inventory.


Kanban and Variation

Variation causes an accumulation of WIP in the Value Stream:

As shown in:  https://polyhedrika.com/2-uncategorised/12-process-simulation-2

Variation in process capacity §2 causes an accumulation of WIP both before and after it. This increases the lead time for the whole value stream.

If we eliminate this excess WIP by applying kanban:

We have, indeed, eliminated the excess WIP but at the expense of reducing the effective capacity of the total value stream (Average thruput) and exposing the customer to the variation of step §2: drastic drop in on-time-delivery.

So this is a case where kanban is NOT recommended. 



  • Kanban is a practical way to implement pull logistics in our supply chain
  • Managing the number of kanbans we put a limit in the inventory level of each item
  • High variation in the supply or the demand side will require a higher number of kanbans and might make pull logistics impossible



In this new situation of the Coronavirus impact on the whole world one of the difficulties to recover production in manufacturing lines could be having to share line terminals by different employees while avoiding virus transmission.  

One alternative to solve this problem is to have each employee use his/ her own smartphone to do all the reporting.  

One major advantage of using your own phone is that the skill is already there: a new IT application would require training. 

Some Possible Business Uses for an Employee's Own Smartphone:

  1. Workplace data collection in real time
  2. Incident reporting anywhere any time
  3. Maintenance repair action reporting from anywhere when it is complete
  4. Value Stream Map building and validation in Gemba (where the action is)
  5. Real-time Statistical Process Control
  6. Real time manufacturing status display for all those concerned

 Workplace data collection

Pencil and paper data collection is still very common even in high-tech manufacturing industries:


This approach, apart from being time consuming, makes it impossible to control the process in real time.

Pencil and paper collection doesn't meet the current health requirements under Coronavirus: several people need to handle the paper.

An alternative is to collect the data in a spreadsheet in the cloud (Google Drive, Onedrive, Intranet, etc.) using our own smartphone connected via WIFI.

Defects can be detected anywhere along the process but it may be inconvenient to go to the nearest terminal to report. With our own smartphone we can report the defect in a spreadsheet in the cloud but also make a photo and attach it to the file so you may avoid any additional explanations.

This approach also enables data entry validation avoiding data entry errors.

Data entry could be done with pull-down menus to avoid typing.

Timestamps can automatically be added to register the reporting time. 


In case of having to enter a written clarification, text can be entered with voice from the smartphone.

 Incident and Repair Action Reporting

Incidents affecting the process may take place anywhere any time. If reporting is done at a later stage time information is lost: when exactly did it happen. This information is critical when it comes to failure analysis: correlation with process parameters or other incidents, etc.

By using our smartphone we are able to report it when it happened and the timestamp can be automatically added. 

The maintenance spreadsheet in the cloud can keep track both of incidents reported and repair actions by the maintenance department. 

Pull down menus can be used to report the equipment, component, failure, etc. and this will enable the maintenance department to track failure frequencies and get to the root cause of problems. 

Incidents can be classified by urgency to enable maintenance assign priorities. Pending actions already late are marked in red and due actions in orange. 

Using a smartphone simplifies reporting of both incidents and repair actions.

Association of defect type and repair action is also possible with this integrated worksheet helping Maintenance in their diagnostic.

Value Stream Map Building and Validation

A Value Stream Map of the process we want to improve is built after visiting the line where the process takes place. 

This VSM can be built in a spreadsheet in the cloud by using a smartphone from the line itself:


  • The VSM is built directly in the cloud (Google Drive) and all participants visiting the line have write access with their smartphone.
  • As they move along the line they all agree on who is doing what and when.
  • Typing of the process steps in the Excel sheet can be done by voice. 
  • The VSM, as it is being built, can be shown to the line operators in order to validate it.
  • After the VSM is complete the same method can be used to collect process parameters (process time, WIP, etc.) in the VSM Excel visiting the line.
  • An Excel file can be used to measure and collect process times with the smartphone:

Real Time Statistical Process Control

 Statistical Process Control (SPC) requires real-time data collection in the different control points. It may be difficult to provide terminals in all these points to report so our own smartphone may be a valid alternative by building our SPC chart in a spreadsheet in the cloud (Google Drive, etc.)


This Individuals-Moving Range SPC chart applies the Western Electric rules to detect out-of-control situations in real time as the data is being entered. 

This chart is visible to the operator entering the data with his smartphone and also to the operators responsible for that process step with a tablet or PC connected via WIFI to the Google Drive spreadsheet.

In this example there is a significant drop in yield between 10:00 and 16:00 (4 out of 5 consecutive values more than 1 sigma below the average)

 Real Time Manufacturing Status

In order to keep everyone involved in the process informed on how are we doing we need this information to be available and up to date at all times.

 A Gantt chart can show the plan and it could be updated by the production system or by the person responsible with a smartphone. 


 An Andon can be displayed in the line to show the status but also accessed from the smartphones to see how are we doing. 

 In this example:

  • Component manual insertion DPMO (Defects per million insertions) is 4124: above the target of 1500 (RED)
  • Pin-Thru-Hole wave solder DPMO (Defective solder joints per million joints) is 32: below the target of 190 (GREEN)
  • Current throughput: 13 cards/ hour at 13:55 on 15/11/2005 at the PTHHW station

With this real-time information the operators involved can react before it is too late and correct the situation. 


The smartphones owned by employees in the line often use a much more advanced technology than the reporting workstations which are often not very user friendly.

The employees' own smartphones may be used for real time data collection to avoid sharing terminals and comply with health requirements in the current situation of a Coronavirus Pandemic.

Data collection in real time is one of the most useful smartphone applications even with no phone line connection other than WIFI.

Since everyone already has the skills you can easily implement these proposed solutions to control your process in real time and keep everyone informed at all times.