Contenido

Recuérdese que el objetivo primordial del CEP (Control Estadístico de Procesos) consiste en monitorizar la evolución de un proceso de producción para detectar cambios de manera rápida y efectiva.

Los datos necesarios para definir un diagrama de Shewhart son los siguientes: en puntos temporales equidistantes, se toma una muestra pequeña (generalmente de un tamaño que ronda las 4 o 5 unidades) y se hace una medición de la variable de control que interesa monitorizar. Dichas muestras son conocidas como subgrupos racionales (traducción directa del inglés rationale subgroups). En general, la elección de estos subgrupos se hace de tal manera que las observaciones sean independientes además de incorporar la información suficiente para detectar variaciones en el corto plazo.

Es común utilizar la media aritmética como variable de control. El diagrama de Shewhart se construye representando gráficamente las medias de los grupos de control frente al tiempo. Una suposición adicional que es importante realizar es que . Como la colección de datos es una secuencia de variables aleatorias independientes e idénticamente distribuidas se cumple que donde es el tamaño de cada grupo de control.

Una vez se han tenido en cuenta las consideraciones anteriores es importante crear el diagrama utilizando tres conceptos capitales en esta área de conocimiento: (línea central), (límite de control superior) y (límite de control inferior). Las líneas y se encuentran a una distancia de la línea central. El sistema de control está diseñado de tal manera que si alguna de las observaciones queda fuera de los límites establecidos se activa una alarma para llevar a cabo la vigilancia del proceso.

Se puede formular el problema en términos estadísticos con las siguientes hipótesis nula y alternativa para cada uno de los subgrupos de control.

Bajo estas consideraciones, la hipótesis nula se corresponde con una situación bajo control mientras que la hipótesis alternativa describe un proceso fuera de control por el cual la media cambia de manera persistente. Aunque este escenario es algo simplista (puede haber cambios epidémicos en la media), suele ser el más utilizado en la literatura.

Es importante comentar que los diagramas de Shewhart están pensados para tener una duración finita, es decir, por la propia formulación del problema ha de existir un momento en el cual se produzca la parada (que es equivalente a que el proceso esté fuera de control).

Se define la longitud de este diagrama de control como sigue:.

La duración esperada de un diagrama de control puede calcularse teniendo en cuenta que sigue una distribución geométrica de parámetro , que se corresponde con la probabilidad que tener un primer fracaso o, equivalentemente, situarse fuera de los límites de control.

En consecuencia la duración esperada del proceso es inversamente proporcional a la probabilidad de situarse fuera de los límites de control.

Statistical process control and Markov chains

Once we use control rules in Shewhart graphs, it acquires memory and the statistics (measurements on the control groups) are no longer independent. Consequently, the expected duration of a control diagram no longer has the geometric distribution that was previously seen. The idea behind the procedure we are going to use today is to represent the control diagram as a Markov chain through an appropriate selection of the state space. We will work on the following specific example that is worth mentioning.

Let be a diagram with the rule "two consecutive points outside the control limits." The states of the chain are as follows:

Note that state 2 is absorbing, when you enter the region out of control it is impossible to change the region, remember that we assume that there are no epidemic changes. The state transition matrix of the Markov chain is as follows:

Remember that the Chapman-Kolmogorov equations relate the change from one distribution of states to another in the Markov chain. Translated into equations, this means that:.

where is a row vector.

The following description refers more to the industrial sector than to the service sector, although the principles of SPC can be applied to both sectors. For a description and example of how to apply SPC to the service sector, refer to Roberts (2005). SPC has also been successfully applied to detect changes in organizational behavior with Social Network Change Detection introduced by McCulloh.

Traditionally, in mass production processes, the quality of the finished part was controlled through product inspections at the end of the process; accepting or rejecting each part (or production samples) based on specification criteria. The difference between Statistical Process Control is that it uses statistical tools to observe the performance of the production process to predict important deviations that may result in rejected product.

There are two types of variations in all industrial processes and both variations cause subsequent variations in the final product. The first are variations of natural or common cause and can be variations in temperature, specifications in raw materials or electricity, etc. These variations are small and are usually close to the mean value. The variation pattern would be similar to the patterns found in nature and the distribution forms the normal distribution curve (bell shape). The latter are known as special causes and occur less frequently than the former.

For example, a cereal box production line may be designed to fill each cereal box with 500 grams of product, but some boxes may have a little more than 500 grams, and others may have a little less, depending on the net weight distribution. If the production process, its inputs, or its environment changes (for example, production machines show signs of wear), this layout may change. For example, if the pulleys wear out, the machine that fills boxes with cereal may begin to put more cereal into each box than specified. If this change is allowed to continue unchecked, more and more products will be produced that do not fall within the manufacturer's or consumer's tolerances, resulting in waste. While in this case, the waste is present in the form of a “free” product for the consumer, normally the waste consists of rework or scrap.

By observing at the right moment what has happened in the process that has caused a change, the quality engineer or any member of the team responsible for the production line can solve the main cause of the variation that has entered the process and the problem is corrected.

The SPC also indicates when an action should be taken within a process, but it also indicates when actions should NOT be taken. An example is a person who would like to maintain a balanced weight and takes weight measurements every week. A person who does not understand SPC concepts may start a diet every time their weight increases, or eat more every time their weight decreases. This type of action can be detrimental and can lead to more variation in weight. SPC is justified by a variation from normal weight and an improved indication of when the person is gaining or losing weight.

Contenido

Recuérdese que el objetivo primordial del CEP (Control Estadístico de Procesos) consiste en monitorizar la evolución de un proceso de producción para detectar cambios de manera rápida y efectiva.

Los datos necesarios para definir un diagrama de Shewhart son los siguientes: en puntos temporales equidistantes, se toma una muestra pequeña (generalmente de un tamaño que ronda las 4 o 5 unidades) y se hace una medición de la variable de control que interesa monitorizar. Dichas muestras son conocidas como subgrupos racionales (traducción directa del inglés rationale subgroups). En general, la elección de estos subgrupos se hace de tal manera que las observaciones sean independientes además de incorporar la información suficiente para detectar variaciones en el corto plazo.

Es común utilizar la media aritmética como variable de control. El diagrama de Shewhart se construye representando gráficamente las medias de los grupos de control frente al tiempo. Una suposición adicional que es importante realizar es que . Como la colección de datos es una secuencia de variables aleatorias independientes e idénticamente distribuidas se cumple que donde es el tamaño de cada grupo de control.

Una vez se han tenido en cuenta las consideraciones anteriores es importante crear el diagrama utilizando tres conceptos capitales en esta área de conocimiento: (línea central), (límite de control superior) y (límite de control inferior). Las líneas y se encuentran a una distancia de la línea central. El sistema de control está diseñado de tal manera que si alguna de las observaciones queda fuera de los límites establecidos se activa una alarma para llevar a cabo la vigilancia del proceso.

Se puede formular el problema en términos estadísticos con las siguientes hipótesis nula y alternativa para cada uno de los subgrupos de control.

Bajo estas consideraciones, la hipótesis nula se corresponde con una situación bajo control mientras que la hipótesis alternativa describe un proceso fuera de control por el cual la media cambia de manera persistente. Aunque este escenario es algo simplista (puede haber cambios epidémicos en la media), suele ser el más utilizado en la literatura.

Es importante comentar que los diagramas de Shewhart están pensados para tener una duración finita, es decir, por la propia formulación del problema ha de existir un momento en el cual se produzca la parada (que es equivalente a que el proceso esté fuera de control).

Se define la longitud de este diagrama de control como sigue:.

La duración esperada de un diagrama de control puede calcularse teniendo en cuenta que sigue una distribución geométrica de parámetro , que se corresponde con la probabilidad que tener un primer fracaso o, equivalentemente, situarse fuera de los límites de control.

En consecuencia la duración esperada del proceso es inversamente proporcional a la probabilidad de situarse fuera de los límites de control.

Statistical process control and Markov chains

Once we use control rules in Shewhart graphs, it acquires memory and the statistics (measurements on the control groups) are no longer independent. Consequently, the expected duration of a control diagram no longer has the geometric distribution that was previously seen. The idea behind the procedure we are going to use today is to represent the control diagram as a Markov chain through an appropriate selection of the state space. We will work on the following specific example that is worth mentioning.

Let be a diagram with the rule "two consecutive points outside the control limits." The states of the chain are as follows:

Note that state 2 is absorbing, when you enter the region out of control it is impossible to change the region, remember that we assume that there are no epidemic changes. The state transition matrix of the Markov chain is as follows:

Remember that the Chapman-Kolmogorov equations relate the change from one distribution of states to another in the Markov chain. Translated into equations, this means that:.

where is a row vector.

The following description refers more to the industrial sector than to the service sector, although the principles of SPC can be applied to both sectors. For a description and example of how to apply SPC to the service sector, refer to Roberts (2005). SPC has also been successfully applied to detect changes in organizational behavior with Social Network Change Detection introduced by McCulloh.

Traditionally, in mass production processes, the quality of the finished part was controlled through product inspections at the end of the process; accepting or rejecting each part (or production samples) based on specification criteria. The difference between Statistical Process Control is that it uses statistical tools to observe the performance of the production process to predict important deviations that may result in rejected product.

There are two types of variations in all industrial processes and both variations cause subsequent variations in the final product. The first are variations of natural or common cause and can be variations in temperature, specifications in raw materials or electricity, etc. These variations are small and are usually close to the mean value. The variation pattern would be similar to the patterns found in nature and the distribution forms the normal distribution curve (bell shape). The latter are known as special causes and occur less frequently than the former.

For example, a cereal box production line may be designed to fill each cereal box with 500 grams of product, but some boxes may have a little more than 500 grams, and others may have a little less, depending on the net weight distribution. If the production process, its inputs, or its environment changes (for example, production machines show signs of wear), this layout may change. For example, if the pulleys wear out, the machine that fills boxes with cereal may begin to put more cereal into each box than specified. If this change is allowed to continue unchecked, more and more products will be produced that do not fall within the manufacturer's or consumer's tolerances, resulting in waste. While in this case, the waste is present in the form of a “free” product for the consumer, normally the waste consists of rework or scrap.

By observing at the right moment what has happened in the process that has caused a change, the quality engineer or any member of the team responsible for the production line can solve the main cause of the variation that has entered the process and the problem is corrected.

The SPC also indicates when an action should be taken within a process, but it also indicates when actions should NOT be taken. An example is a person who would like to maintain a balanced weight and takes weight measurements every week. A person who does not understand SPC concepts may start a diet every time their weight increases, or eat more every time their weight decreases. This type of action can be detrimental and can lead to more variation in weight. SPC is justified by a variation from normal weight and an improved indication of when the person is gaining or losing weight.