Last Updated on 20 September 2023

The **Taguchi loss function** is a graph that shows how much it costs as your product varies from the target value. It was developed by Genichi Taguchi, and shows that loss is not a ‘yes/no’ function (i.e. you either get a full loss or you get no loss) as is usually assumed in Lean Six Sigma. Instead, there is a target where there is no loss, and the **loss gradually builds** in both directions past that, so a small deviation leads to a small loss. This discovery helped paved the development of Lean and Continuous Improvement or Kaizen.

Even though customers may ‘accept’ products between their two limits, they will get **greater value** from them **closer to the target quality**. This extra value will lead to extra profit for your company from increased sales and being able to charge higher prices. It is primarily used in lean manufacturing, but it has implications in all types of organizations; your manager may accept a sales report (and you’ll still get paid) with some factual or spelling errors in it, but they will get more value and your career will benefit if there are no errors.

## What is it?

The Taguchi loss function is a change from the traditional view that quality is either met as ‘acceptable’, or missed as ‘defective’. It instead teaches that even when your product meets the outer limits of what is acceptable, it’s still worth aiming for the optimal level. In essence:

The further the product deviates from the target, the larger the loss is

There will always be some loss as there is always some variation, but this model says that this loss is reduced every time you reduce the variation. This loss is to the whole of society, so it could affect you or any of your customers or stakeholders. The losses usually make it back to your organization, and so should be removed.

The loss is proportional to the deviation from the target

## The traditional view; goal posts

The traditional view is that quality is like shooting at a goal. The posts that you are aiming between are the lowest acceptable value or Lower Specification Limit (LSL calculated m-d on chart below) and highest acceptable value or Upper Specification Limit (USL calculated m+d).

If you get your metric between the goal posts you have no quality losses and the product is acceptable to sell. If you get outside the posts, there is a full loss, as the product is useless to the customer. This is known in quality as the ‘goalpost view’ as the scoring is much as in scoring a goal.

## The Taguchi view; the loss curve

Taguchi’s view contrasts with the more traditional view in that there is a loss as soon as you are any distance away from the required amount. This model won’t let you describe an almost rejected part as ‘lossless’.

His philosophy teaches that you should continually try to get improvements, as this will provide you with more gains than just being happy with ‘good enough’. This has led to embracing e.g. the techniques of Kaizen in Lean where all improvements are embraced even if they don’t turn a part from ‘rejected’ to ‘acceptable’.

## Taguchi quality loss function formula

The equation is most commonly given as the form:

L = k (y-m)^{2}

where:

- L is the loss function
- y is the value of the characteristic you are measuring (e.g. length of product)
- m is the value you are aiming for (in our example, perfect length for the product)
- k is a proportionality constant (i.e. just a number to quantify the value of the loss)

### Calculate k (proportionality constant) in Taguchi loss function

You will often want to calculate the k value for a given process. For this you will need to know two things:

- d is the allowable tolerance from m that is allowed
- A = the loss if the item needs to be scrapped

You can then calculate the constant k with the following formula:

k = A / d^{2}

It doesn’t even need to be the maximum loss that can be had, as this constant works at all levels. If you know the loss at any level, e.g. at variance ‘d’ you need to repair at cost ‘A’, you can use those figures instead.

## Why are there losses ?

The Taguchi loss function says there are losses where there previously weren’t any in the Six Sigma model, so where are these coming from? Taguchi defines loss broader than many, to include losses to the customer, as if they lose time and money it will usually cost your organization in the long run.

The losses may lead to financial losses in your company in many ways, such as:

- Loss of customer goodwill from receiving poor quality products; this will translate into lower sales and rejection of price increases
- A reputation for lower quality goods that will lose sales and pricing power
- Warranty costs for parts that fail soon into their working life
- Product returns where the product doesn’t work sufficiently well for the customer
- Reduced product effectiveness for the customer. This is loss of value for the customer costing them time and money, which they will reclaim from your organization at some point (e.g. through rejecting price increases).

A lot of these are hard to measure, which is why it hasn’t completely replaced the traditional loss graph as the losses can be hard to quantify. You can of course use the formula above to estimate these losses though.

## How can we use it?

A key use is that it converts in a simple way the variation in the system into a financial loss. This makes it valuable for explaining and justifying process improvement projects to management. Using the formula, if you are aiming for a set % decrease in variation, you can convert that into a $ saving.

This dollar value can also help you quantify predicted project gains for cost-benefit analysis. It can also justify the time and expense going into designing good quality operations.

The philosophy behind it, that all improvement is of value and will reduce losses, can help your organization strive for every-improving your product. This can improve your reputation, and eventually improve sales and market share.

## Worked example

Say your product length needs to be 1,000 mm long, and your customer will accept +/- 10mm. Your loss if the part is rejected is $10. You are currently produce 3,000 a month. The current process makes them approximately 1,008 mm long, which is acceptable to the customer.

You have a project that will cost $30,000, and will reduce the product to 1,003 mm long. As your product is currently being accepted by the customer, management is reluctant to accept the project – should you do it?

### Current loss

We need to first of all work out our constant ‘k’:

k = A / d^{2} = $10 / ( 10mm)^{2} = 10/100 = 0.1 $/mm^{2}

For loss per item we use the loss function:

L = k (y-m)^{2} = 0.1 (1008-1000)^{2} = $6.40 per unit

With 3,000 units per month this is a monthly loss of $19,200.

### New loss after the project

You don’t need to start from scratch for the new loss; you just recalculate L using the new length of 1,003 mm:

L = k (y-m)^{2} = 0.1 x (1003-1000)^{2} = $0.90 per unit

This is a monthly loss of $2,700, a monthly gain of $16,500.

### Project conclusion

Your gains of $16,500 a month will recover your cost of $30,000 in two months, making this a bargain according to the Taguchi loss function. Although your customer is accepting the product you’re making, it will be a lot closer to the parameters they want, giving them much higher usage value.

You may not have instant payback, as your customer is currently accepting the product. Your product will be seen to be of higher quality and utility to your customer, meaning you should get higher sales and be able to justify price increases to your customer.

It should be said though in this circumstances that the value improvement goes to the customer, so it isn’t guaranteed that the value increase will also pass to you. This should be taken into consideration when you are working out whether to do the project. This is because we are looking at value to everyone involved, and the benefits don’t always (immediately) go to the organization doing the improvements.

It is also worth checking with the customer that getting closer to the ‘perfect’ length is of improved use to them, as the Taguchi loss function won’t necessarily work in every case.

### Variable variationâ€¦

In reality, your variation will not be the same for each part you make. You will usually be able to calculate the current losses, as you will have your actual measurements of your product, but future losses can’t be predicted as easily, as you have no specific measurements.

As populations usually follow the normal distribution, you can take your variance to be following the same ‘curve’ as it currently does. Your calculation now becomes a lot easier. Loss is proportional to the square of the variance, so if you are going to reduce variance by 50% (a factor of 2), your losses will reduce by a factor of 4 (2^{2}).

You have the data to calculate your current losses, so you can use this to work out the savings. For a r% reduction in variation, you get:

Savings = old losses – new losses

= old losses x (1- r%^{2})

## Smaller is better

For some quality issues, you want the smallest number possible. This could be e.g. defect size, number of defects etc. You can easily adjust the formula to work for these situations. As you’re aiming for zero, you simply set your target value ‘m’ to zero, making your formula:

L = ky^{2}

## Bigger is better

There can be situations where you want a quality attribute to be as large as possible, such as strength. Zero would now be the worst possible outcome, e.g. zero strength would crumble immediately and have the maximum loss. You can recreate the formula by taking the inverse of y:

L = k (1/y)^{2}

Obviously the loss wouldn’t be infinite at zero; there will be a maximum possible loss that this formula would level off at.

## Video explanation

Sometimes it’s easier to understand from a video, so here is an (unaffiliated) video that explains it well:

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