I recently had a very unpleasant experience with a notebook that I bought about six months ago. At first, I was having some irritating problems with the LCD (the display). It would be very dim for about five minutes when I turn the computer on; the system had to warm up before the LCD would display correctly. I did bear with that situation until it started to blackout while I was in the middle of my work.
Since it was still under warranty, I sent it back to the manufacturer for repair and decided that I would never buy a product from that manufacturer again.

I      Assessing the cost of Quality
The quality of a product is one of the most important factors that determine a company’s sales and profit. Quality is measured in relation with the characteristics of the products that customers’ expect to find on it, so the quality level of the products is ultimately determined by the customers. The customers’ expectations about a product’s performance, reliability and attributes are translated into Critical-To-Quality (CTQ) characteristics and integrated in the products’ design by the design engineers.
While designing the products, they must also take into account the resources’ capabilities (machines, people, materials…), i.e. their ability to produce products that meet the customers’ expectations. They specify with exactitude the quality targets for every aspect of the products.

But quality comes with a cost. The definition of the Cost Of Quality is contentious. Some authors define it as the cost of non-conformance, i.e. how much producing nonconforming products would cost a company. This is a one sided approach since it does not consider the cost incurred to prevent non conformance and above all in a competitive market, the cost of improving the quality targets.

For instance, in the case of an LCD (Liquid Crystal Display) manufacturer, if the market standard for a 15” LCD with a resolution of 1024x768 is 786,432 pixels and a higher resolution requires more pixels, improving the quality of the 15” LCDs, pushing the company’s specifications beyond the market standards would require the engineering of LCDs with more pixels which would require extra cost.

The cost of quality is traditionally measured in terms of the costs conformance and the cost of nonconformance to which we will add the cost of innovation. The cost of conformance includes the appraisal and preventive costs while the cost of non-conformance includes the costs of internal and external defects.

Cost of conformance

·         Preventive Costs
The costs incurred by the company to prevent non-conformance.
It includes the costs of:

o        Process capability assessment and improvement

o        The planning of new quality initiatives (process changes, quality improvement projects….)

o        Employee training …
 

·         Appraisal Cost.
The cost incurred while assessing, auditing, inspecting products and procedures to conform products and services to specifications. It is intended to detect quality related failures. It includes:

o        Cost of process audits

o        Inspection of products received from suppliers

o        Final inspection audit

o        Design review

o        Pre-release testing

Cost of non conformance
 The cost of non conformance is in fact the cost of having to rework products and the loss of customers that results from selling poor quality products. 

·         Internal Failure

o       Cost of reworking products that failed audit

o       Cost of bad marketing

o       Scrap

·         External Failure

o       Cost of customer support

o       Cost of shipping returned products

o       Cost of reworking products returned from customers

o       Cost of refunds

o       Loss of customer Goodwill

o       Cost of discounts to recapture customers

In the short term, there is a positive correlation between quality improvement and the cost of conformance and a negative correlation between quality improvement and the cost nonconformance. In other words, an improvement in the quality of the products will lead to an increase in the cost of conformance that generated it. This is because an improvement in the quality level of a product might require extra investment in R&D, more spending in appraisal cost, more investment in failure prevention and so on.
But a quality improvement will lead to a decrease in the cost of nonconformance because fewer products will be returned from the customers, therefore less operating cost of customer support and there will be less internal rework.

For instance, one of the CTQs (Critical-To-Quality) for an LCD (Liquid Crystal Display) is the number of pixels it contains. The brightness of each pixel is controlled by individual transistors that switch the backlights on and off. The manufacturing of LCDs is very complex and very expensive and it is very hard to determine the number of dead pixels on an LCD before the end of the manufacturing process. So in order to reduce the number of scrapped units, if the number of dead pixels is infinitesimal or the dead pixels are almost invisible, the manufacturer would consider the LCDs as “good enough” to be sold. Otherwise, the cost of scrap or internal rework would be so prohibitive that it would jeopardize the cost of production. Improving the quality level of the LCDs to zero dead pixels would therefore increase the cost of conformance.
On the other hand, not improving the quality level of the LCDs will lead to an increase in the probability of having returned products from customers and internal rework, therefore increasing the cost of nonconformance.

The following graph plots the relationship between quality improvement and the cost of conformance on one hand and the cost of non-conformance on the other hand.

If the manufacturer determines the quality level at Q2, the cost of conformance would be low (C1), but the cost of nonconformance would be high (C2) because the probability for customer dissatisfaction will be high and more products will be returned for rework therefore increasing the cost of rework, the cost of customers services and shipping and handling.
The Total cost of Quality would be the sum the cost of conformance and the cost of nonconformance, that cost would be C3 for a quality level of Q2.

C3 = C1 + C2.

 

Should the manufacturer decide that the quality level would be at Q1, the cost of conformance (C2) would be higher than the cost of nonconformance (C1) and the Total cost of Quality would be at C3.

The Total Cost Of Quality is minimized only when the cost of conformance and the cost of nonconformance are equal.

It is worth to note that currently, the frequently used graph to represent the throughput yield in manufacturing is the Normal curve. For a given target and specified limits, the normal curve helps estimate the volume of defects that should be expected. So while the Normal curve estimates the volume of defects, the U curve estimates the cost incurred as a result of producing parts that do not match the target.

The following graph represents both the volume of expected conforming and nonconforming parts and the costs associated to them at every level.

 

II     Taguchi's Loss Function
In the now traditional quality management acceptance, the engineers integrate all the CTQs in the design of their new products and they clearly specify the target for their production processes as they define the characteristics of the products to be sent to the customers, but because of unavoidable common causes of variation (variations that are inherent to the production process and that are hard to eliminate) and the high costs of conformance, they are obliged to allow some variation or tolerance around the target. Any product that falls within the specified tolerance is considered as meeting the customers’ expectations, and any product outside the specified limits would be considered as non-conforming.

But according to Taguchi, the products that do not match the target, even if they are within the specified limits do not operate as intended and any deviation from the target, be it within the specified limits or not will generate financial loss to the customers, the company and to society and the loss is proportional to the deviation from the target.

Suppose that a design engineer specifies the length and diameter of a certain bolt that needs to fit a given part of a machine. Even if the customers do not notice it, any deviation from the specified target will cause the machine to wear out faster causing the company financial loss under the form of repair of the products under warranty or a loss of customers if the warranty has expired.

Taguchi Constructed a Loss Function equation to determine how much society loses every time the parts produced do not match the specified target. The Loss Function determines the financial loss that occurs every time a CTQ of a product deviates from its target. The loss function is the square of the deviation multiplied by a constant k, with k being the ratio of the cost of defective product and the square of the tolerance.

The loss Function quantifies the deviation from the target and assigns a financial value to the deviation.

∆ = cost of a defective product

And  m = LSL – T  or m = T - USL

According to Taguchi, the cost of quality in relation with the deviation from the target is not linear because the customers’ frustration increases (at a faster rate) as more defects are found on a product. That’s why the Loss function is quadratic.

The graph that depicts the financial Loss to society that results from a deviation from the target resembles the Total Cost of quality U graph that we built earlier but the premises that helped build them are not the same. While the Total Cost curve was built based on the costs of conformance and nonconformance, Taguchi’s Loss Function is primarily based on the deviation from the target and measures the loss from the customers’ expectation perspective.

Example 1:
Suppose a machine manufacturer specifies the target for the diameter of a given rivet to be 6 inches and the upper and lower limits of 5.98 and 6.02 inches respectively. A bolt measuring 5.99 inches is inserted in its intended hole of a machine. Five months after the machine was sold, it breaks down as a result of loose parts. The cost of repair is estimated at $95, find the loss to society incurred as a result of the part not matching its target.

Solution:
We must first determine the value of the constant k

   T = 6

   USL = 6.02

    m = (USL - T) = 6.02 - 6 = 0.02

   ∆ = 95

   K = (95 / 0.004) = 237500
 

   Therefore

l(y) = 237500 x 0.0001 = 23.75

Not producing a bolt that match the target would have resulted in a financial loss to society that amounted to $23.75.

Introduction to Taguchi Method II