{"id":12366,"date":"2012-06-15T14:52:26","date_gmt":"2012-06-15T13:52:26","guid":{"rendered":"http:\/\/www.decidesoluciones.es\/descuentos-y-penalizaciones-pwl-no-continuas-2\/"},"modified":"2020-03-27T13:17:53","modified_gmt":"2020-03-27T13:17:53","slug":"discounts-and-penalties-discontinuous-pwl","status":"publish","type":"post","link":"https:\/\/decidesoluciones.es\/en\/discounts-and-penalties-discontinuous-pwl\/","title":{"rendered":"Discounts and Penalties. Discontinuous PWL"},"content":{"rendered":"<p>Companies that need to procure large amounts of a certain product usually get what is known as sales volume discounts from the providers. Volume discounts establish an amount above which a discount is applied for any additional units bought, or for the total purchase.<\/p>\n<p>The total cost function, depending on whether this discount is applied to additional units or to the total purchase looks like this:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-COSTE-21.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-610\" src=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-COSTE-21.png\" alt=\"\" width=\"477\" height=\"339\" \/><\/a><\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-COSTE-1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-611\" src=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-COSTE-1.png\" alt=\"\" width=\"480\" height=\"320\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>Una vez adquirido el producto, su almacenamiento tambi\u00e9n tiene un coste asociado que en muchos casos se ver\u00e1 incrementado por una penalizaci\u00f3n si se supera cierta capacidad disponible. De este modo, una funci\u00f3n de coste de almacenamiento podr\u00eda tener la siguiente forma:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-COSTE-3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-613\" src=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-COSTE-3.png\" alt=\"\" width=\"467\" height=\"331\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>Decidir las cantidades de las que abastecerse en cada periodo teniendo en cuenta los costes asociados a esa adquisici\u00f3n (compra, almacenamiento, transporte, &#8230;) suele suponer un gran reto para las empresas y de ah\u00ed que hagan uso de herramientas de optimizaci\u00f3n para ayudarse en la toma de decisiones.<\/p>\n<p>&nbsp;<\/p>\n<h2>PWL no continuas<\/h2>\n<p>En su <a href=\"http:\/\/www.decidesoluciones.es\/como-formular-una-funcion-lineal-a-trozos-piecewise-linear-function\/\">\u00faltima entrada<\/a>, Javier nos hablaba de las funciones lineales a trozos (PWL). En particular, nos hablaba de c\u00f3mo formular PWL continuas. En ocasiones, nos encontraremos con PWL no continuas como las que aparecen arriba. Al igual que ocurr\u00eda con las continuas, muchos de los productos que resuelven problemas de Programaci\u00f3n Lineal Entera tienen esta formulaci\u00f3n para las no continuas. Sin embargo, es interesante conocer c\u00f3mo se modelizan.<\/p>\n<p>Una PWL no continua tiene la siguiente forma:<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-PWL1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-617\" src=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/FUNCION-PWL1.png\" alt=\"\" width=\"467\" height=\"326\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>Y su expresi\u00f3n ser\u00eda:<\/p>\n<p><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/EXPRESION-PWL.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-619\" src=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/EXPRESION-PWL-300x156.png\" alt=\"\" width=\"300\" height=\"156\" \/><\/a><\/p>\n<p>De modo que en cada intervalo, la funci\u00f3n es lineal.<\/p>\n<h2><\/h2>\n<h2>C\u00f3mo <em>linealizar <\/em>una PWL no continua<\/h2>\n<p>La idea consiste en escribir x como suma de n-1 variables (una por cada intervalo) del mismo tipo que x, de las cuales s\u00f3lo una tomar\u00e1 un valor distinto de cero y, en tal caso, su valor estar\u00e1 en el intervalo correspondiente. Con esta definici\u00f3n de la variable x, podremos escribir la PWL no continua como una <a href=\"http:\/\/es.wikipedia.org\/wiki\/Combinaci%C3%B3n_lineal\">combinaci\u00f3n lineal<\/a> de las n-1 variables, de modo que, para cada una de estas variables, si su valor es cero, \u00a0su aportaci\u00f3n al valor de la funci\u00f3n sea nula<span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">, y en caso contrario:<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">[math]f(x)=frac{(x_i-a_i)(b^i_{i+1}-b^i_i)}{a_{i+1}-a_i}+b^i_ i ; a_ileq x_i leq a_{i+1}[\/math]<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\"><br \/>\n<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">As\u00ed, suponiendo que x es una variable continua positiva, comenzaremos escribiendo la variable como sigue:<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">[math]x=sum_{i=1}^{n-1}x_i; x_iin mathbb{R}^{+};forall{i=1..n-1}[\/math]<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\"><br \/>\n<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">donde, como dec\u00eda, s\u00f3lo una de las variables del sumatorio ser\u00e1 distinta de cero y en tal caso, con un valor dentro de su correspondiente intervalo. Esta caracter\u00edstica se puede modelizar f\u00e1cilmente haciendo uso de lo que se conoce como <a href=\"http:\/\/en.wikipedia.org\/wiki\/Special_ordered_set\">Special Ordered Sets<\/a> de tipo 1 (<a href=\"http:\/\/en.wikipedia.org\/wiki\/Special_ordered_set\">SOS1<\/a>) y como comentaba Javier con los de tipo 2, todos los motores de Programaci\u00f3n Lineal Entera tienen una representaci\u00f3n interna de este tipo de conjuntos. <\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">Una modelizaci\u00f3n de esta \u00faltima propiedad sin hacer uso de los <a href=\"http:\/\/en.wikipedia.org\/wiki\/Special_ordered_set\">SOS1 <\/a>ser\u00eda la siguiente:<\/span><\/span><\/p>\n<p style=\"text-align: center;\">[math]a_iy_i leq x_i leq a_{i+1}y_i; ; forall{i=1..n-1}[\/math]<\/p>\n<p style=\"text-align: left;\"><span style=\"font-family: Calibri, sans-serif; font-size: 15px; line-height: 17px;\">donde<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif; font-size: 15px; line-height: 17px;\">[math]y_iin {0,1}; ; forall{i=1..n-1}; ; sum_{i=1}^{n-1}y_i=1[\/math]<\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\"><br \/>\n<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">Con esto, ya estar\u00edamos en condiciones de poder escribir nuestra PWL como una expresi\u00f3n lineal del siguiente modo:<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">[math]f(x)= sum_{i=1}^{n-1}(m_ix_i-c_iy_i)[\/math] <\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">donde<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">[math]m_i=frac{b^i_{i+1}-b^i_i}{a_{i+1}-a_i} ;; y ;; c_i=m_ia_i-b^i_i; forall{i=1..n-1}[\/math]<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\"><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/EXPRESION-LINEAL-FUNCION3.png\"><br \/>\n<\/a><br \/>\n<\/span><\/span><\/p>\n<p>Se comprueba f\u00e1cilmente que si una de las n-1 variables que hemos utilizado para definir x toma valor 0, su aportaci\u00f3n a la funci\u00f3n ser\u00eda nula. Si por el contrario es distinta de cero, su aportaci\u00f3n a la funci\u00f3n ser\u00eda:<\/p>\n<p style=\"text-align: center;\">[math]m_ix_i-c_i=m_i(x_i-a_i)+b^i_i=frac{(x_i-a_i)(b^i_{i+1}-b^i_i)}{a_{i+1}-a_i}+b^i_i[\/math]<\/p>\n<p style=\"text-align: left;\">como quer\u00edamos.<\/p>\n<p>&nbsp;<\/p>\n<h2>Consideraciones Adicionales<\/h2>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">Al igual que ocurre con las PWL continuas, incluir una no continua en un problema lo hace m\u00e1s complejo ya que como hemos visto, para una PWL no continua con n-1 &#8220;trozos&#8221; se necesitan n-1 variables del mismo tipo que x, n-1 variables binarias y 2*n restricciones. En particular, a\u00f1ade mayor complejidad al problema que las PWL continuas.<\/span><\/span><\/p>\n<p><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">Con la formulaci\u00f3n que presentaba Javier para las PWL continuas, se vio que si la funci\u00f3n est\u00e1 en el objetivo de un problema de maximizaci\u00f3n y era c\u00f3ncava, la complejidad del problema no se ve\u00eda afectada al incluir la PWL ya que se pod\u00eda prescindir de la modelizaci\u00f3n del <a href=\"http:\/\/en.wikipedia.org\/wiki\/Special_ordered_set\">SOS2<\/a>. Aunque esto no es as\u00ed para esta modelizaci\u00f3n, s\u00ed hay casos concretos en los que podremos prescindir de alguna de las restricciones. Por ejemplo, si nuestra PWL no continua est\u00e1 en la funci\u00f3n objetivo de un problema de maximizaci\u00f3n, es creciente en cada &#8220;trozo&#8221; y se verifica que<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">[math]frac{b^i_{i+1}-b^i_i}{a_{i+1}-a_i}leq frac{b^{i+1}_{i+2}-b^{i+1}_{i+1}}{a_{i+2}-a_{i+1}}; forall{i=1..n-2}[\/math]<\/span><\/span><\/p>\n<p style=\"text-align: left;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">y<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: Calibri, sans-serif;\"><span style=\"font-size: 15px; line-height: 17px;\">[math]b^{i+1}_{i+1}leq b^i_{i+1}; forall{i=1..n-1}[\/math]<\/span><\/span><\/p>\n<p>entonces podemos prescindir de las restricciones<\/p>\n<p style=\"text-align: center;\">[math]a_iy_ileq x_i; forall{i=1..n-1}[\/math]<\/p>\n<p style=\"text-align: left;\">Una PWL con estas caracter\u00edsticas tendr\u00eda la siguiente forma:<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/PWL-ESPECIAL.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-646\" src=\"http:\/\/www.decidesoluciones.es\/wp-content\/uploads\/2012\/06\/PWL-ESPECIAL.png\" alt=\"\" width=\"485\" height=\"317\" \/><\/a><\/p>\n<p style=\"text-align: center;\">\n","protected":false},"excerpt":{"rendered":"<p>Companies that need to procure large amounts of a certain product usually get what is known as sales volume discounts from the providers. Volume discounts establish an amount above which a discount is applied for any additional units bought, or for the total purchase.<\/p>\n","protected":false},"author":30,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[262],"tags":[],"class_list":["post-12366","post","type-post","status-publish","format-standard","hentry","category-uncategorized","category-262","description-off"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Discounts and Penalties. 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