V5 - Smapling Plans

• Control during production

  • Real-time control of the production output, to check whether process is in control (use of control charts).

  • More desirable since it gives the possibility to notice and adjust the process within a short time

• Acceptance control

  • Inspection of the products received by the supplier and decision whether to accept or refuse them

  • Supplier can be external or internal

• Outgoing inspection

  • inspection is performed immediately following production.

• Incoming inspection

  • lots of products are inspected as they are received from the supplier.

• Intermediate inspection

  • semi-finished products can be inspected before moving from one shop to another, within the same factory

• Checking the products’ (dimensional and functional) conformity to specifications

• Estimate the percentage of defective elements (p).

• Can be performed by

  • customer

  • supplier

• but inspection cost is generally charged to supplier.

• 100% inspection

• Acceptance sampling

• Accept with no inspection

• inspect every item in the lot, removing all defective units found

• The advantage is that we are completely sure of the quality of the production output

• Do not require statistics

• a sample, which is a subset of the production output, is selected from the lot

• then sample units are inspected with respect to a specific quality characteristic

• On the basis of the sample information, the lot is then accepted or rejected

• Do require statistics

• It is useful in situations where the supplier’s process is very good, with almost no defective units

• Do not require statistics

• Testing is destructive

• Cost of 100% inspection is extremly high

• When a 100% inspection not feasible or require a lot of time

• When the supplier has an excelent quality history

• When the inspection error rate is high, so that a 100% inspection would cause a higher percentage of defective untis to be passed

• Risk of accepting bad lots and rejecting good lots

• Less information is generated

• Requires planning and documentation (100% inspection does not)

• the decision whether to accept or refuse the lot is made considering just a portion of the whole lot, using statistical techniques.

• It can be interpreted as a hypothesis test

  • null hypothesis (Hp0) => “the supplier’s lot/material is good”.

• Null hypothesis (Hp0) => “the supplier’s lot/material is good”.

• Type 1 error (alpha)

  • Player => Supplier

  • Risk => Seeing good lots being rejacted

  • alpha => risk of rejection of Hp0 when it is actually true

• Type 2 error ( beta)

  • Player => Customer

  • Risk => Accepting bad quality lots

  • beta => risk of failing to reject Hp0 when it is actually false.

• For attributes:

  • the quality characteristic is expressed through a binary parameter (good/bad, conforming/non-conforming)

  • Can be classified into sampling plans:

    • for defectives => product that fail to meet specifications since they have one or more defects

    • for defects => nonconformities that are serious enough to significantly affect the safe/effective use of the product unit

• For variables: the quality characteristic is measured on a continuous numerical scale (e.g., the length of a mechanical component)

• Measurements of attributes are

  • quicker and more practical than those of variables

  • but with lower information content

• Example: In order to control the diameter of a metallic bar, we could use a hard gauge to check its conformity to specifications, or a calliper to determine its real diameter.

• Group of units of the same product, which have been produced under homogeneous conditions

  • i.e., same machines, same operators, same materials, approximately in the same time

The Operating Characteristic (OC) curve depicts the discriminatory power or severity or protection of the sampling plan, representing the probability of acceptance (Pa) as a function of the lot fraction nonconforming (p).

There are two types of OC curves:

Type-A when Pa is calculated referring to the Hypergeometric distribution;

Type-B when Pa is calculated referring to the Binomial (approx.) distribution.

The OC curve is a monotonically decreasing curve of Pa with respect to p, which asymptotically tends to 0.

The shape of the curve depend on the parameters that characterize the sampling plan (N, n, c).

• If n<<N (or n/N ≤ 0.10), the two OC curves are virtually undistinguishable because the hypergeometric distribution is well approximated by the binomial one.

• Supplier wants to avoid that lots with relatively high quality are rejected

  • relatively lower p

  • Concept of Serverity => Sampling plan should not be too severe

• Customer wants to avoid accepting lots with relatively low quality

  • relatively higher p

  • Concept of Protection => Sampling plan will need to adequately protect him/her from this risk

• The sample size (n) is a parameter affecting the OC curve significantly

• While n increases, the curve approaches the ideal “step shape”

  • Step shape OC curve is acieved when: n → N (100% inspection)

  • Discriminates “good” and “bad” lots perfectly

  • for relatively low p values => 𝑃_𝑎 = 1

  • for larger p values => 𝑃_𝑎 = 0.

• Increasing n will also make the sampling plan more expensive and time-consuming

• Sampling plans with small c values are more discriminatory for lower p values => OC curve in these cases are said to be more severe

• The severity of a sampling plan is the propensity to obtain a relatively low 𝑃_𝑎 value for a given p value (defectiveness)

• Sampling plans with small c values are more discriminatory for lower p values => OC curve in these cases are said to be more severe

• The severity of a sampling plan is the propensity to obtain a relatively low 𝑃_𝑎 value for a given p value (defectiveness)

  • In the example, three different OC curves (with different c values) are intersected with a vertical line corresponding to p = 2%.

  • The 𝑃_𝑎 values of the three OC curve are radically different => 𝑃_𝑎(𝑐=2) >𝑃_y(𝑐=1) >𝑃_a(𝑐=0)

  • While the curve with c = 0 is relatively severe (fully convex), that one with c = 2 is relatively indulgent.

• Nearly no impact

• The use of a sample size that are a fixed percentage of the lot size (e.g., n/N = constant) is a conceptually wrong practice.

• The reason is that n and N differently influence the OC-curve behaviour

  • Effect of n predominates with respect to that of N.

• Consequence => severity of the OC curve vary in an uncontrolled manner

• alpha => risk of rejection of Hp0 when it is actually true

• beta => risk of failing to reject Hp0 when it is actually false

• For each rejected lot (N), the inspection is extended from the sample (n) to the rest of the lot (N – n);

• Any defective unit found during the inspection, both of the sample (irrespectie of acceptance or rejection) and the rest of the lot (N – n), must be removed from the lot and then replaced with a good/conforming unit.

• For double sampling plans two points of the OC-curve (AQL, 1 – a and LTPD, b) are not enough

  • there are only two equations for four unknown factors to be determined (i.e., n1, c1, n2 and c2).

  • Two additional equations can be added by imposing some (additional) constraints. The most common ones are

    • n2 = 2∙n1

    • c2 = 2∙c1 or c1 + c2 = const.

• For skip-lot plans only some fractions of the submitted lots are inspected.

• Generally, skip-lot sampling plans are used where it is necessary to reduce the average amount of inspection required

• When production is continuous (the product is not formed into lots) a segment of production can be conventionally marked off as a “lot”.

• In this case – in order to avoid a 100% inspection – only a fraction of the units can be inspected.

• These sample units are selected one at a time at random from the flow of production