Unit 6 (2017)Apunte Inglés
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Topic 6: Quality control
Evaluates the results of the process by comparing to the ideal results. If there is any
difference between them, then the objective is to minimise it. Apart from separating correct
products (the ones that comply specifications) from defective products that must be redone,
includes the prevention concept (actions to guarantee the expected results).
All the efforts dedicated to obtain products or services that comply design specifications at minimum cost.
Seven Ishikawa’s basic tools - - Pareto chart. Based on the idea that, in general, most defects in an article can be attributed to a reduced number of causes (Pareto Law 20-80). It classifies few vital causes from the rest of trivial causes. Pareto diagrams identify the causes of a quality problem rapidly and easily. Example: A company produces an article which presents several manufacturing defects. The objective is to remove them. Management wishes to know which are the causes of most defective items.
If the two main causes of the problem are removed (grated surface and arm rupture), 84% of defective articles are avoided.
Cause-and-effect diagrams. They are also known as Ishikawa diagram or fishbone, is used to classify and clear the causes that originate an effect. It is necessary to identify and face the causes (and NOT the effects) to solve a problem. The basic structure of these diagrams is a central arrow and the studied effect is placed on the right.
Consequently, firstly the quality problem must be defined and the effect that measures it. Then the causes are classified. The causes are placed tidily in the main branches: Inside these main branches, causes are placed in little branches. A brainstorming session can be performed previously to identify the causes.
- Final comment: the identified and classified causes are potential causes. This diagram is the starting point to verify and confirm the real causes.
Check sheets. Check sheets are printed sheets that allow data collection in a simple and precise way so that collection tasks are easier for the operators. The fundamental objectives are: To ease data collection and organize data for further analysis. There are different type of templates according to its application.
Check sheets for defective articles They’re used to detect the type of defects and their frequency percentages in defective products in order to reduce them.
Check sheets for defects location Sketches of the manufactured piece where defects are located. They allow to detect if defects are always placed in the same place.
Check sheets to control the distribution of the production process.
They’re used to collect data of continuous variables such as weight, diameter, volume,… Then, it is possible to draw a histogram to study the distribution of the production process, and calculate the average and dispersion.
- Histograms. A histogram is a graphical representation of the distribution of data. Data is organised to study frequency of occurrence. Example: Consider 100 measurements of the diameter of a cylindrical piece. The number of measurements n should be (at least) between 50 and 100 to study a certain characteristic. Firstly, data is divided in 10 groups of 10 measurements. For each set of data, we determine the maximum and minimum values.
Secondly, the amplitude of all data is determined: Maximum value – Minimum value = 7,58 – 7,19 = 0,39 This value is divided by k = 10 to obtain the number of classes (number of groups or bars) of the graphic: 𝑀𝑎𝑥 − 𝑀𝑖𝑛 0,39 ℎ= = = 0,039 → 0,04 → 0,05 𝑘 10 The interval h is the unit to adjust the horizontal axis (bar width). In this case, we consider 0,05. The number of classes k depends on data collection: The value that limits the first bar is fixed taking into account the extreme of the amplitude + half of the accuracy of collected data. In this case, the minimum value is 7,19 accuracy of real data is 0,01. Consequently, the limits of first bar is : 7,19 0,01/2 = 7,185 Rest of limits of bars will be: 7,185 - 7,235; 7,235 – 7,285; 7,285 – 7,335; ...
Then, the frequencies table must be calculated accounting data that belongs to each interval: These data allows to draw the histogram. In Cartesian axis, horizontal axis represents a quality characteristic and the vertical axis represents the frequency (number of data inside one bar).
Each bar limit is a class. Bar width is a class interval. The central value is the average value. Consider that the diameter tolerances are between 7,15 cm and 7,55 cm. In this case the process is off-centre. A certain number of pieces are produced outside specifications limits.
These two cases show the same frequency distribution BUT case 1 is centered inside the tolerance limits. Nevertheless, case 2 is an off-centre distribution and shows that some articles are produced outside specifications (grey bars).
Case 3 is a bell-shaped distribution that represents a variability due to random causes.
Ideal distribution. Case 4 shows a right-skewed histogram. The right tail is longer and mass of distribution is concentrated on the left. This indicates that data does NOT follow normal law.
- Case 5 is a bimodal histogram because it presents two peaks. In some cases indicates that data can be divided in two subsets of data that differ from each other in some way. Case 6 is distribution that shows a small peak on the right. This indicates defects or errors because these data does not follow the general behaviour. Probably an assignable cause can be determined.
Functions of the histograms: 1) Verify if production is inside specifications.
2) Determine the behaviour of the distribution of data by observing the histogram shape.
3) Analyse if stratification is necessary due to interference of different factors that can affect variability. In this case, data are separated is subsets to differentiate causes of dispersion and to identify the origin of the problem easily.
Scatter diagram. These diagrams are useful to analyse whether a quality characteristic and a factor are related. Also, they’re called correlation diagrams.
Steps to make a scatter plot: 1. Identify the factors that seem to be correlated 2. Take 50 pairs of data approximately 3. Draw Cartesian axis to place the pairs of data 4. The quality characteristic is located in Y-axis Variable X is a possible cause and Y is the quality characteristic that seems correlated.
Both variables present certain positive correlation. In other words, the quality characteristic is related to the cause as suspected.
It is possible to calculate the correlation coefficient quantitively: 𝑟= 𝑆𝑥𝑦 √𝑆𝑥𝑦 , 𝑆𝑦𝑦 𝑛 : 𝑆𝑥𝑥 𝑛 = ∑(𝑋𝑖 − 𝑋̅)2 = ∑ 𝑋𝑖2 − 𝑖=1 𝑖=1 𝑛 𝑛 𝑆𝑦𝑦 = ∑(𝑌𝑖 − 𝑌̅) = ∑ 𝑌𝑖2 − 2 𝑖=1 𝑛 𝑆𝑥𝑦 𝑖=1 𝑛 = ∑(𝑋𝑖 − 𝑋̅) ∗ (𝑌𝑖 − 𝑌̅) = ∑ 𝑋𝑖 𝑌𝑖 − 𝑖=1 𝑖=1 (∑𝑛𝑖=1 𝑋𝑖 )2 𝑛 (∑𝑛𝑖=1 𝑌𝑖 )2 𝑛 (∑𝑛𝑖=1 𝑋𝑖 ) ∗ (∑𝑛𝑖=1 𝑌𝑖 ) 𝑛 In the example: SXX = 71,56; SYY = 122,53; SXY = 74,46; r = 0,7952 (positive correlation) The correlation coefficient takes values between -1 and 1. If the resulting value is: - A. close to 1, this indicates there is a strong positive correlation.
B. close to -1, then the correlation is negative C. close to 0, then the correlation is weak.
Stratification. It’s a method to identify the origins of variability of collected data. For example, when an article is manufactured by different machines, by different operators or using different materials, then it’s advisable to classify the data by machinery, operators or materials. This way, it’s possible to identify the origin of the problem. Maybe this could not be detected if all data is mixed.
Stratification is one of the seven basic Ishikawa tools. Normally complements the rest of methodologies.
Statistical Process Control (SPC).
Is the application of statistical techniques to measure and analyse the variations of a production process. Causes of these variations can be: random and assignable.
a. Random causes: They cannot be controlled and they appear at random. They affect all production processes and always they can be considered.
b. Assignable causes: They can be studied and are the ones that contribute most to the variability of the process. Usually, they’re due to tiredness of workers, different grade of experience/training, different behaviour of materials,… so that it’s NOT possible to obtain identical products. The same happens in service industries: a cook CANNOT obtain two identical dishes or a lecturer CANNOT repeat two identical classes,… Process capacity Process capacity is defined as the variability amplitude of a process when this is under control; in other words, when variability is not due to assignable causes. When a product is designed, the nominal value and a tolerance margin is defined. The tolerances interval defines the limits inside which the product is considered correct. This interval of tolerances is limited by an upper tolerance limit (LTS) and lower tolerance limit (LTI). For example: Control graphics The objective of a control graphic is to differentiate variations due to random causes or due to assignable causes. The vertical axis represents the range of the studied quality characteristic.
Also, the lower tolerance limit, the upper tolerance limit and the average are placed on the graphic. The horizontal axis represents time. This way, the evolution of the quality characteristic can be observed vs. time. Also, it can be compared with respect to established control limits.
Types: - Control graphic by variables. This means that measurements belong to a continuous quality characteristic such as weigth, length, speed, density, volume,… In this case, the control graphics for the average and range will be plotted. To start with, at least 100150 elements must be measured (and grouped in samples of 4-5 units) to guarantee that the sample is significant.
25 samples of 5 measurements.
Examples: ∑𝑛 𝑖=1 𝑋𝑖 o Sample average: o Sample range: 𝑅 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒. Example: 15-13=2 o ∑ ∑ 𝑋 𝑅 Global average for X and R: 𝑋̅ = 𝑘=1 𝑖 𝑅̅ = 𝑘=1 𝑖 . Example: 𝑘 𝑘 14 + 14 + 13,2 + ⋯ + 14,8 + 13 + 15,2 𝑋̅ = = 14,56 25 2+ 4+ 4+ ⋯+5+ 5+ 4 𝑅̅ = = 3,72 25 𝑛 . Example: 14 𝑘 ̅ 𝑘 Graphic X: shows variation of the average of the process. Graphic R: shows variations of the process dispersion. Consequently, control graphics are very interesting because they show variations of the average and dispersion at the same time and indicate process anomalies.
In this example, both average and range graphics show that the process is under control because: 1. All the points are inside the control limits.
2. Points do not follow any particular tendency.
- Control graphic by attributes. An attribute is a quality characteristic that cannot be measured. Control graphics by attributes classify the product as Acceptable or Defective. For example, if a piece presents spots, we wish to know if the product shows this characteristic to determine if it is acceptable or defective. Since a defective products can have several defects, these graphics monitor the number of detected defects or the number of defects. According to this criteria and the sample, there are several types of graphics.
Acceptance sampling The acceptance sampling determines the percentage of products that verify specifications. It is used to inspect elements that the company purchase to suppliers (raw material, components,…) or also pieces that have been processed in one step of the process and are evaluated before going into the next step. This technique involves taking random lots of products, measuring a certain characteristic and then comparing it with a established standard. This is much cheaper than 100% inspection. The quality of the sample is used to judge the quality of the complete production lot. Sampling can be defined by variables or by attributes.
The acceptance sampling is performed through a sampling plan. A simple sampling plan is defined by the sample size, n, and by the acceptance number, c, ( maximum number of defects that can be found in the sample before rejecting the lot),...
- If the inspected sample has a number of defects < c , the lot is accepted.
If the number of detected defects is higher, then the lot is rejected or then it’s performed a 100% inspection.
Example: Consider that we wish to accept all production lots whose number of defective products < 2,5% and reject the rest. Imagine a lot of 1.000 pieces that has a 4% of defective products. The inspection has taken a sample of 20 pieces and none is defective Lot is accepted. In this case, the sampling plan gives a wrong result. The 20 pieces sample could have showed at random, one, two, three,…defective pieces. This fact is fundamental in sampling plans. It’s possible to reject good production lots or accept defective lots.
Simple sampling plan by attributes. Characteristic curve Each sampling plan has associated a characteristic curve that describes the ability of the plan to distinguish between correct lots and defective lots. Indicates the probability that the plan accepts lots of different qualities.
AOQ, Average Output Quality: AOQ = w . Pa (w) AQL, Acceptable Quality Level: - Maximum number of defective articles inside a lot so that it is considered acceptable.
Lowest level of quality that the company accepts.
Lots with this level of quality will be accepted.
LTPD, Lot Tolerance Percent Defective: - Level of quality of a defective lot.
Lots with this quality level will be rejected.
From the producer perspective, a good sampling plan is one that has a low probability to reject correct lots.
Producer risk (): Probability that the lot is rejected even though the number of defective products is lower than AQL. From customers or consumers perspective, the sampling plan, should have a low probability to accept defective lots: Consumer risk (): Probability that the lot is accepted although the number of defective products ≥ LTPD Sampling plan. Considerations Normally, the sampling plans are designed considering a producer risk of 5% ( = 0,05) and a consumer risk of 10% ( = 0,10).
The selection of specific values AQL, LTPD, and is an economic decision based on costs, company policies or contract requirements.