An Application of the Quality Transfer Function – part 1

by Andy Urquhart on 16th December 2009

In a previous article, y=f(x) was disproved as a transfer function, and a new formula for quality as a transformation proposed. This white paper describes its application in innovative systems such as kinetic recovery system (KERS). Part 1 looks at Useful work versus Worthwhile work on a single-piece flow line.

Introduction

In a previous article¹, we proposed a Quality transfer function as a general form of the transfer functions used by engineers to describe the Quality of various characteristics, such as optical images, acoustic waveforms, feedback control responsiveness, and electronic filter response, as follows:

Y (ω) = Г (ω) X (ω) Eqn. 1

Where, in using Equation (1) Y (ω) represents an output, X (ω) an input, and Г (ω) as the set of Quality transfer functions, all functions having an argument denoting frequency.

Although some may consider Equation (1) somewhat daunting, it only represents the product of two variables, which some people already use intuitively when inquiring about the frequency response of a stereo amplifier.

With one particular frequency in mind, instead of Equation (1) we could simply write:

Y = Г X Eqn. 2

Where, in using Equation (2) Y represents an output, Г a scalar transfer function, and X an independent variable.

In the following sections, we will describe how we can use one particular member of the set of Quality transfer functions – a work transfer function (WTF) – to study the principles of operation of systems, such as film handling systems; one of the many sub-systems comprising Image Setters: but before we do, to clarify our usage, we will describe how we use the WTF to gain insights into simpler systems, such as a single-flow manufacturing line and a Kinetic Energy Recovery System (KERS.)

Once the reader has become accustomed to our terminology, we will then use the WTF to investigate the film handling system in rather more detail, with the goal of understanding the principles of operation and the root cause of a family of failures modes . We will also mention how we incorporate other methods and approaches into our studies, including: emulation, the scientific method, and the systems approach; which we use to reduce complexity.

Although we completed the film handler study several years ago² and the company has already sold more than 3,000 units of their award winning Drum Image Setter world-wide, our use of the WTF is contemporary and germane given the resurgent interest in rotating systems, including kinetic energy recovery systems (KERS) 2 and high-efficiency renewable energy systems, such as windmills, hybrid magnetic motors, and permanent magnetic generators. Of course, in working with these systems, we cannot always assume 100% efficiency when transforming energy from one form to another, which is why we introduce the concepts of useful work and worthwhile work.

The Work Quality Transfer Function

As we mentioned in our earlier article¹, we base our general form of the Quality transfer function, Equation (1) on the optical transfer function (OTF) and our justification for using the OTF as a template is the knowledge that lenses transform light, and since light comprises electromagnetic energy, we have inadvertently stumbled upon a transfer function suitable for describing the Quality of work. (The reader may recall energy and work share the same physical dimensions: Joules.)

As with the response of a stereo amplifier, the WTF (ω) is a function of frequency, or period of repetition. Therefore, in using the optical transfer function as a template, we invoke a WTF having the following form:

Y (ω) = WTF (ω) x X (ω) Eqn. (3)

Where, in using Equation (3) Y (ω) represents an output, X (ω) an input, and WTF (ω) the work transfer function – a particular member of the set of Quality transfer functions, Г(ω).

Since we can associate the work transfer function WTF with efficiency by virtue of the relationship:

η = Work_out ÷ Work_in Eqn. (4)

We can rewrite Equation (4) as follows:

Y (ω) = η (ω) x X (ω) Eqn. (5)

We further clarify the relevance of the frequency ω in Equation (5) in the context of a single-flow manufacturing line, in which we process items one by one, sometimes through numerous stages, in sympathy with a takt cycle. If the takt period equals τ then the frequency related to the period is:

ω = 1 ÷ τ Eqn. (6)

Although American Lean production lines use takt times, our usage is limited to production lines using the method of one-by-one confirmation of the Toyota Production System (TPS) as described by Kitano-san ². Under the TPS, each stage uses autonomous inspection (process self-assurance) to trigger an alarm that stops production as soon as the stage encounters a problem, or defect. Therefore, under the TPS, the Quality of a product or service cannot be defined in terms of defects, process capability, or any other inspection metric, because in principle it implies a zero-defect system.

If the reader is happy to accept efficiency as a member of the set of Quality transfer functions, we can better appreciate how another transformation – the 5S transformation – can improve Quality.

Useful Work versus Worthwhile Work

As optical engineers often factorise the Optical Transfer function (OTF) into two parts: a Modulation transfer function (MTF) part and a Phase transfer function (PTF) part, as indicated by Equation (7) as follows:

OTF (ω) = MTF (ω) x PTF (ω) Eqn. (7)

Where, in using Equation (7) the MTF (ω) represents the bandwidth or spectrum or useful energies available to an optical imaging system due to the physical limitation of a finite lens aperture, or light collecting ability. The PTF (ω) on the other hand represents the bandwidth of worthwhile energies remaining after degradation in passing through a malformed lens, the malformation due to a lens aberration, such as a spherical aberration.

In sympathy with the OTF, we can also factorizing the WTF into two parts:

Y (ω) = η_u(ω) x η_w(ω) x X (ω) Eqn. (8)

Where, in using Equation (8) the term η_u(ω) denotes the useful efficiency, or the spectrum of useful work, while the other term η_w(ω) denotes the worthwhile efficiency, or spectrum of worthwhile work. (We often use the terms ‘frequency response’ and ‘spectrum’ interchangeably.)

Our distinction between useful work and worthwhile work can be better understood through the following: How often does useful work bring forth little worthwhile.³

Useful Work

Using the example of a single-flow manufacturing line again and bearing in mind the following relationship:

Work = Power x time Eqn. (9)

Or, rewriting Equation (9)⁴ as:

Work = Effort x time Eqn. (10)

Accordingly, we are able to estimate the amount of useful work W_U performed by a stage in a single-flow manufacturing line from the standard time, provided we assume an average effort. (In the author’s experience, Japanese companies calculate standard times theoretically using software⁵.)

Worthwhile Work

In manufacturing, the only worthwhile work is that which delivers a fully functional unit or part. Accordingly, in a single flow line, this only occurs after completing a successful takt because as soon as someone on the production line encounters a problem, an alarm sounds and the production line stops.

If the line is balanced and there are m stages, T successful takts produce:

W_U= m x T Eqn. (11)

Where in using Equation (11) we determine the useful work by multiplying the number of stages, m, by the total number of takt periods, T.

Useful Efficiency versus Worthwhile Efficiency

Since it is unreasonable to expect someone performing manual labour to work flat out all the time with maximum effort, we have to recognise that periods of hard labour require periods of rest – otherwise mistakes happen. This ‘no unreasonableness’ principle is called “Muri” in Japanese companies because it has a major impact on waste, which explains their preference for calculating standard times – not to make people work harder – but to ensure a balance between effort and sufficient time to complete the task correctly. How different from most Western companies who always try to ‘crack the whip’ at the end of the month.

Useful Efficiency

For reasons already mentioned, the takt time should comprise the standard time and a period of rest, the rest time also sufficient to allow an operator to move from one stage to another: if necessary, which is often the case on a new line. Taking this in account, we can calculate the useful efficiency using the ratio:

η_u(ω) = Power out x Std Time / Power in x Takt Time = Std Time / Takt Time Eqn. (12)

Thus, if the effort only goes towards producing a part, the Power Out equals the Power In and the useful efficiency is just the ratio of the standard time to the takt time.

For the sake of simplicity, we will ignore the possibility of the actual working time being greater than the standard time because under TPS this would also cause considerable concern. Indeed, we now see how we can improve the efficiency of a production line using power tools, hoists, or other means to improve the efficiency of useful work. Furthermore, we conclude the use of these can actually improve Quality and not just productivity.

The reader should note we cannot always increase the frequency of production by reducing the period of the takt while maintaining the same standard time, since this would diminish the rest period. If we want to reduce the takt time, we must first reduce the standard time by increasing the division of labour, through the creation of a greater number of stages. We would then have to work out how much rest operators need to maintain stamina while working under the arrangements.

Worthwhile Efficiency

As mentioned previously, the amount of worthwhile work equates to the number of good items produced through each stage and since the potential for good items is the theoretical number of takts per day, we can calculate the worthwhile efficiency using the following ratio:

η_u(ω) = (m x T_G ) ÷ n τ Eqn. (13)

Where, in using Equation (13) m represents the number of stages, T_G the number of good takts, τ = Takt time and n = number of planned daily takts. More conventionally, we refer to the useful efficiency as the production efficiency and the worthwhile efficiency as yield. Thus, in our use of the WTF we combine both these efficiencies.

For processes not involved in replication, such as original works of art, literary works, or even custom designed software, the Quality of these processes, means of production, and services must be characterised by different means.

Part 2 of this article will be published tomorrow, when the notion of the QTF will be integrated with other methods, such as the systems approach, the scientific method, statistical modelling, and emulation to better understand the system’s principles of operation, with a view to discovering new and better systems without limiting the scope of investigation to optimization alone. If you have any questions, queries or opinions on this paper, then please use the form below to contact the author.

Acknowledgements

Although the author takes full responsibility for any conceptual flaws or mistakes in this article, he would like to thank Giuseppe (Peppe) Calcara and Matt Moore for proof reading the article.