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COMPARACIÓN DE LOS MODELOS DEL PROBLEMA DE APROVISIONAMIENTO CONJUNTO CON DEMANDA DINÁMICA

JULIO 2016

José Mezquita Zapico

DIRECTOR DEL TRABAJO FIN DE GRADO:

Miguel Ortega Mier

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TRABAJO FIN DE GRADO PARA

LA OBTENCIÓN DEL TÍTULO DE

GRADUADO EN INGENIERÍA EN

TECNOLOGÍAS INDUSTRIALES

Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica

José Mezquita Zapico 1

RESUMEN

Este Trabajo de Fin de Grado es un estudio sobre los diferentes modelos de optimización

utilizados para resolver el problema de aprovisionamiento conjunto, especialmente cuando el

tamaño de las instancias de datos es grande. Cuatro modelos extraídos de la literatura

académica han sido comparados. Primero se ha realizado un experimento con 2592 problemas

generados de pequeño tamaño. A partir de estos resultados se ha analizado la influencia de

diversos factores en el tiempo computacional. Posteriormente se ha realizado un experimento

con 152 problemas de gran tamaño. Analizado el tiempo computacional, la diferencia de

optimalidad y la memoria requerida se concluye que en la mayoría de los casos el modelo de

Robinson y Gao es el más eficiente. Sin embargo, en el caso de problemas de gran tamaño

(específicamente con un gran número de periodos de tiempo) con elevados costes fijos y

elevada probabilidad de demanda el modelo de Joneja presenta mejores resultados que el

modelo de Robinson y Gao.

Palabras clave: problema de aprovisionamiento conjunto, MILP, optimización


2 Escuela Técnica Superior de Ingenieros Industriales (UPM)

ÍNDICE

RESUMEN .............................................................................................................. 1

ÍNDICE ................................................................................................................... 2

1 INTRODUCCIÓN ......................................................................................... 3

1.1 Motivación e impacto ........................................................................... 3

1.2 Definición del problema de aprovisionamiento conjunto .................... 3

1.3 Objetivos .............................................................................................. 4

1.4 Impacto medioambiental ...................................................................... 4

2 DESARROLLO DEL TRABAJO ................................................................. 6

2.1 Modelos utilizados ............................................................................... 6

2.1.1 Modelo Básico (BM) ......................................................................... 6

2.1.2 Modelo de Joneja (JON) ................................................................... 7

2.1.3 Modelo de Robinson y Gao (R&G) .................................................. 8

2.1.4 Modelo de requerimientos exactos (ERF) ......................................... 8

2.2 Metodología .......................................................................................... 9

2.2.1 Factores estudiados ........................................................................... 9

2.2.2 Adición de una restricción ............................................................... 10

2.2.3 Diseño del experimento ................................................................... 10

2.3 Resultados .......................................................................................... 12

2.3.1 Experimento preliminar ................................................................... 12

2.3.2 Experimento con problemas grandes .............................................. 14

2.3.3 Análisis de requerimientos de memoria .......................................... 16

3 CONCLUSIONES ....................................................................................... 18

4 INFORMACIÓN DEL TRABAJO ............................................................. 19

4.1 Universidad de destino ....................................................................... 19

4.2 Presupuesto ......................................................................................... 19

4.3 Planificación temporal ........................................................................ 20

TRABAJO EN UNIVERSIDAD DE DESTINO .................................................. 22



1 INTRODUCCIÓN

1.1 Motivación e impacto

En las últimas décadas las empresas han ido adoptando métodos de modelización y

simulación para la toma de decisiones. La programación lineal y la programación entera

permiten encontrar la solución óptima en muchos de los problemas de gestión industrial.

Debido a la competitividad presente en el actual mercado global, la adopción de estas técnicas

de optimización es imprescindible para las empresas que quieran tener éxito.

Sin embargo, la resolución de estos problemas en un ordenador puede tardar mucho tiempo o

incluso no llegar a concluirse nunca si el tamaño del problema, es decir el número de

variables y restricciones, es muy grande.

Este tiempo computacional, el tiempo necesario para llegar a la solución óptima, depende de

muchos factores, y entre ellos, del modelo que se haya utilizado para resolver el problema.

Para resolver un problema de optimización se pueden desarrollar diferentes modelos. Todos

ellos deben de llegar a la misma solución óptima, pero el tiempo empleado difiere

enormemente.

Es de gran importancia para las empresas que utilizan estos métodos elegir el modelo

correcto. Este estudio se centra en comparar los diferentes modelos utilizados para resolver el

problema de aprovisionamiento conjunto.

1.2 Definición del problema de aprovisionamiento conjunto

El problema de aprovisionamiento conjunto, también conocido por sus siglas en inglés JRP

(joint replenishment problem) consiste en determinar el plan óptimo de repuesto de diferentes

productos para satisfacer la demanda de éstos a lo largo de un horizonte temporal. Este

problema asume que hay costes fijos a la hora de hacer un repuesto. Por un lado se incurre en

un coste fijo mayor cada vez que se hace un repuesto, sin importar la cantidad. Por otro lado,

se incurre en un coste fijo menor por cada tipo de producto que se repone. El objetivo es

minimizar el coste total considerando, además de estos costes fijos, los costes de compra y de

almacenamiento del inventario. En el JRP con demanda dinámica (JRPDD) la demanda varía

en los intervalos de tiempo, aunque se asume que es conocida de antemano.



Este problema es adecuado, por ejemplo, para compañías que repongan más de un producto

del mismo proveedor, como las compañías de retail. También puede servir para optimizar las

operaciones de un centro de distribución.

Este problema también puede ser aplicado para las industrias manufactureras, siendo

interpretado como un problema de planificación de la producción sin restricciones de

capacidad. En este caso el objetivo es determinar cuándo producir los diferentes productos y

la cantidad de unidades del lote. El coste fijo mayor puede constituir cambiar la disposición

de las máquinas o las tareas de limpieza y mantenimiento de éstas. Mientras que el coste fijo

menor puede representar el tiempo necesario para programar las máquinas y utillaje para cada

producto o el desperdicio de material. El coste de compra puede interpretarse como el coste

unitario de producción.

Este problema es de la clase de complejidad computacional NP-completo (Arkin, Joneja, &

Roundy, 1989), lo que significa que es improbable que se pueda hallar un algoritmo que lo

resuelva en una duración de tiempo polinómica dependiendo del número de variables.

1.3 Objetivos

El objetivo de este trabajo es comparar los cuatro modelos más importantes que se han creado

para resolver este problema. Por un lado, se analiza la influencia de cada factor en el tiempo

computacional empleado por cada modelo. Por otro lado, se analiza cuáles son los modelos

más eficientes, en términos de tiempo computacional y de requerimientos de memoria, a la

hora de resolver problemas de gran tamaño. Además, se añade una restricción a tres de los

modelos con el objeto de mejorarlos y se analiza su efecto.

1.4 Impacto medioambiental

Las técnicas de optimización han conseguido diseñar planes de producción industrial más

eficientes. Esto no sólo se traduce en una ventaja económica por el ahorro de costes, sino que

también constituye un ahorro de energía y materiales.

Dado que el objetivo de este trabajo es determinar cuál es el modelo más conveniente para

resolver un problema que se puede aplicar a muchas situaciones industriales, las empresas a

las que les resulte de utilidad podrían contribuir a la sostenibilidad. Supongamos el caso de un

centro de distribución de mediano tamaño que no está familiarizado con la programación

entera. De adoptar el modelo conveniente para la optimización de sus operaciones de



transporte de acorde al problema de aprovisionamiento conjunto ahorraría costes entre los

cuales se encuentra la gasolina y por lo tanto reduciría sus emisiones de gases. Sin embargo,

de adoptar el modelo incorrecto podría no obtenerse solución alguna lo que posiblemente

llevaría a la empresa a descartar el uso de la programación entera.



2 DESARROLLO DEL TRABAJO

2.1 Modelos utilizados

2.1.1 Modelo Básico (BM)

Este modelo está basado en la idea de que la demanda de un producto en un período debe ser

cubierta con el inventario más la cantidad que se repone en ese período. A partir de aquí se

utilizarán las siglas “BM” para referirse a “Modelo Básico”.

Las variables de decisión de este modelo son:

𝑥𝑘𝑡 : cantidad repuesta del producto k en el período t.

𝑌𝑘𝑡 : indicador de coste fijo menor. Es una variable binaria que toma el valor de 1 si

está programado un repuesto del producto k en el período t, de otra forma toma el

valor de 0.

𝑍𝑡 : indicador de coste fijo mayor. Es una variable binaria que toma el valor de 1 si

está programado un repuesto para el período t, de otra forma toma el valor de 0.

𝐼𝑘𝑡 : nivel de inventario del producto k al final del período t.

La formulación del modelo es:

min ∑ ∑(𝑠𝑘𝑡 · 𝑌𝑘𝑡 + 𝑐𝑘𝑡 · 𝑥𝑘𝑡 + ℎ𝑘𝑡 · 𝐼𝑘𝑡) + ∑ 𝑆𝑡 · 𝑍𝑡

𝑇

𝑡=1

𝑇

𝑡=1

𝐾

𝐾=1

(1)

Sujeto a:

𝐼𝑘𝑡 = 𝐼𝑘,𝑡−1 + 𝑥𝑘𝑡 − 𝑑𝑘𝑡 (𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇) (2)

𝐼𝑘0 = 0 (𝑘 = 1,2, … , 𝐾) (3)

𝑥𝑘𝑡 ≤ 𝑀 · 𝑌𝑘𝑡 (𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇) (4)

𝑌𝑘𝑡 ≤ 𝑍𝑡 (𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇) (5)

𝑥𝑘𝑡 , 𝐼𝑘𝑡 ≥ 0; 𝑌𝑘𝑡, 𝑍𝑡 ∈ {0, 1} (𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇) (6)

El término 𝑠𝑘𝑡 representa el coste fijo menor del producto k en el período t. 𝑆𝑡 es el coste fijo

mayor en el período t. Los términos 𝑐𝑘𝑡 y ℎ𝑘𝑡 representan el coste de compra y el coste de

mantenimiento de inventario, respectivamente, de una unidad del producto k en el período t.

M representa un número muy grande, es suficiente con que sea la suma de la demanda de

todos los productos durante todos los períodos. Estos términos son iguales para el resto de

modelos.



Por simplicidad en este trabajo se asume que el nivel de inventario al comienzo del primer

período es nulo.

2.1.2 Modelo de Joneja (JON)

Joneja (1990) propone un modelo basado en la formulación del problema del camino más

corto. Se utilizan las siglas “JON” para referirse a este modelo. Las variables utilizadas en

este modelo son las siguientes:

𝑥𝑘𝑖𝑗 : definida en 1 ≤ 𝑖 < 𝑗 ≤ 𝑇 + 1

Es una variable binaria que es igual a 1 si y sólo si el producto k es repuesto en el

período i y en el período j y en ninguno más entre ellos.

𝑍𝑖 : indicador de coste fijo mayor que toma el valor de 1 si y sólo si se realiza un

repuesto en el período i.

El coste total por reponer el producto k en el período i y satisfacer la demanda hasta el período

j-1 es:

𝐶𝑘𝑖𝑗 = 𝑠𝑘𝑖 + ∑ 𝑐𝑘𝑖 𝑑𝑘𝑟

𝑗−1

𝑟=𝑖

+ ∑ ( ∑ ℎ𝑘𝑛) 𝑑𝑘𝜏

𝜏

𝑛=𝑖+1

𝑗−1

𝜏=𝑖+1

(7)


min ∑ ∑ ∑ 𝐶𝑘𝑖𝑗 𝑥𝑘𝑖𝑗

𝑇+1

𝑗=𝑖+1

𝑇

𝑖=1

𝐾

𝑘=1

+ ∑ 𝑍𝑖 𝑆𝑖

𝑇

𝑖=1

(8)

Sujeto a:

∑ 𝑥𝑘𝑖𝑗

𝑇+1

𝑗=𝑖+1

− ∑ 𝑥𝑘𝑠𝑖

𝑖−1

𝑠=1

= 0 (𝑘 = 1, … , 𝐾 ; 𝑖 = 𝛿𝑘 + 1, … , 𝑇) (9)

∑ 𝑥𝑘,𝑖,𝑇+1

𝑇

𝑖=1

= 1 (𝑘 = 1, … , 𝐾) (10)

𝑍𝑖 − ∑ 𝑥𝑘𝑖𝑗

𝑇+1

𝑗=𝑖+1

≥ 0 (𝑘 = 1, … , 𝐾; 𝑖 = 1, … , 𝑇) (11)

𝑥𝑘𝑖𝑗 , 𝑍𝑖 ∈ {0,1} (𝑘 = 1, … , 𝐾; 𝑖 = 1, … , 𝑇; 𝑗 = 𝑖 + 1, … , 𝑇 + 1) (12)

El término 𝛿𝑘 que aparece en la restricción (9) representa el primer período en el que el

producto k experimenta demanda.



2.1.3 Modelo de Robinson y Gao (R&G)

Se utilizan las siglas “R&G” para referirse al modelo de Robinson y Gao. Las variables de

este modelo son:

𝑥𝑘𝑖𝑡 : representa la fracción de la demanda del producto k en el período t que es

repuesta en el período i.

𝑍𝑖 : indicador de coste mayor (igual que en el modelo anterior).

𝑌𝑘𝑖 : indicador de coste menor (igual que en el BM).

El coste de satisfacer toda la demanda del producto k en el período t mediante un repuesto en

el período i es:

𝐶𝑘𝑖𝑡 = (𝑐𝑘𝑖 + ∑ ℎ𝑘𝑗)

𝑡−1

𝑗=𝑖

· 𝑑𝑘𝑡 ( 𝑖 ≤ 𝑡 ) (13)


min ∑ 𝑆𝑡 · 𝑍𝑡 + ∑ ∑ 𝑠𝑘𝑡 · 𝑌𝑘𝑡

𝐾

𝑘=1

𝑇

𝑡=1

+ ∑ ∑ ∑ 𝐶𝑘𝑖𝑡 · 𝑥𝑘𝑖𝑡

𝑇

𝑡=𝑖

(14)

𝑇

𝑖=1

𝐾

𝑘=1

𝑇

𝑡=1

Sujeto a:

∑ 𝑥𝑘𝑖𝑡 = 1 (𝑘 = 1, …

𝑡

𝑖=1

, 𝐾; 𝑡 = 𝛿𝑘, 𝛿𝑘 + 1, … , 𝑇) (15)

𝑥𝑘𝑖𝑡 ≤ 𝑌𝑘𝑖 (𝑖 = 1, … , 𝑇; 𝑘 = 1,2, … , 𝐾; 𝑡 = 𝑖, … , 𝑇) (16)

𝑌𝑘𝑖 ≤ 𝑍𝑖 (𝑘 = 1,2, … , 𝐾; 𝑖 = 1, … , 𝑇) (17)

𝑍𝑖 , 𝑌𝑘𝑖 ∈ {0,1} (𝑖 = 1, … , 𝑇; 𝑘 = 1, … , 𝐾) (18)

Este modelo a diferencia de los anteriores permite incorporar en su estructura el backordering,

esto es satisfacer la demanda después de la fecha de entrega.

2.1.4 Modelo de requerimientos exactos (ERF)

Las siglas utilizadas para referirse a este modelo son “ERF”. Las variables de este modelo

son:

𝑥𝑘𝑖𝑡: variable binaria que toma el valor de 1 si y sólo si la demanda del producto k

desde el período i hasta el período t es cubierta por un repuesto en el período i.

𝑍𝑡: es el indicador de coste fijo mayor (igual que en los casos anteriores).

El coste de satisfacer la demanda del producto k desde el período i hasta el período t es:

𝑐𝑘𝑖𝑡 = 𝑠𝑘𝑖 + 𝑐𝑘𝑡 · ∑ 𝑑𝑘𝑟 + ∑ (∑ ℎ𝑘𝜏

𝑟−1

𝜏=𝑖

) 𝑑𝑘𝑟

𝑡

𝑟=𝑖+1

𝑡

𝑟=𝑖

(19)




min ∑ 𝑆𝑡 · 𝑍𝑡 + ∑ ∑ ∑ 𝑐𝑘𝑖𝑡 · 𝑥𝑘𝑖𝑡

𝑇

𝑡=𝑖

𝐾

𝑘=1

𝑇

𝑖=1

𝑇

𝑖=1

(20)

Sujeto a:

∑ ∑ 𝑥𝑘𝑖𝑡 = 1

𝑇

𝑡=𝜏

𝜏

𝑖=1

(𝑘 = 1, … , 𝐾; 𝜏 = 𝛿𝑘, … , 𝑇) (21)

∑ 𝑥𝑘𝑖𝑡 ≤ 𝑍𝑖

𝑇

𝑡=𝑖

(𝑘 = 1, … , 𝐾; 𝑖 = 1, … , 𝑇) (22)

𝑥𝑘𝑖𝑡 , 𝑍𝑡 ∈ {0,1} (𝑘 = 1, … , 𝐾; 𝑖 = 1, … , 𝑇; 𝑡 = 1, … , 𝑇) (23)

2.2 Metodología

2.2.1 Factores estudiados

Los factores estudiados, a parte del número de productos y períodos que son los que definen

el tamaño del problema, son los siguientes:

Tiempo entre órdenes (TBO): definido en la teoría del EOQ (Economic Order

Quantity), en este problema sirve como una medida indirecta del total de los costes

fijos relativo a los costes de mantenimiento de inventario:

𝑇𝐵𝑂 = √𝐴 · �̅� · ℎ̅

2

(24)

o A total de los costes fijos

o �̅� demanda media por período

o ℎ̅ coste unitario de mantenimiento medio por período

Ratio de coste fijo mayor (MSR): es el ratio del coste fijo mayor al total de los costes

fijos. Si toma el valor de uno, significa que los costes fijos menores son despreciables

en comparación con el coste fijo mayor. Por otro lado, si no hay coste fijo mayor, este

ratio tomará el valor de 0.

Probabilidad de demanda: es la probabilidad de que un producto experimente

demanda en un período.

Índice de dispersión de la demanda: el índice de dispersión (I), también conocido

como varianza-media ratio, es el cociente entre la varianza y la media de una

distribución. Introduciendo este factor medimos como afecta la variabilidad en la

demanda sobre el tiempo computacional. En este experimento la demanda toma



valores enteros. Tres distribuciones estadísticas han sido testeadas: distribución de

Poisson (𝐼 = 1), distribución binomial (𝐼 < 1) y distribución negativa binomial (𝐼 >

1).

2.2.2 Adición de una restricción

Una de las propiedades que una solución óptima del JRP debe satisfacer es que si el coste de

hacer un repuesto en el período t para satisfacer la demanda del producto k en el período q es

mayor que reponerlo directamente en el período q incluyendo los costes fijos (𝑑𝑘𝑞 ∑ ℎ𝑘𝑟 +𝑞−1𝑟=𝑡

𝑐𝑘𝑡 > 𝑆𝑞 + 𝑠𝑘𝑞 + 𝑐𝑘𝑞) entonces no es óptimo reponer la demanda del producto k para el

período q en el período t.

Esta propiedad ha sido añadida a tres de los modelos analizados (BM, R&G y ERF), aquellos

que permitían añadirla en su estructura, con el fin de intentar hacerlos más eficientes. Esta

propiedad que ha sido añadida en forma de restricción reduce el número de variables y

conlleva una solución de la relajación lineal más ajustada en el BM. A los modelos que

incluyen esta restricción se les ha denominado BM-modified, R&G-modified y ERF-modified

2.2.3 Diseño del experimento

Hacer un experimento factorial completo con problemas de gran tamaño llevaría una enorme

cantidad de tiempo. Por ello, se ha realizado primero un experimento preliminar con

problemas de menor tamaño de diseño factorial completo. Posteriormente, en base a los

resultados de este experimento preliminar, se ha diseñado y realizado el experimento con

problemas de gran tamaño.

Figura 1: Proceso experimental



En el experimento preliminar los factores toman los siguientes valores:

Productos: 5, 10, 20, 40

Periodos: 12, 24, 36, 48

TBO: 1.5, 2.5, 4.5

MSR: 0.3, 0.6, 0.9

Índice de dispersión de la demanda (distribución): 0.5 (binomial), 1 (Poisson), 1.5

(negativa binomial)

Probabilidad de demanda: 0.35, 0.75, 1

Los costes de mantenimiento de inventario unitarios y los costes de compra se han fijado a 1

para todos los productos y períodos. Los productos se han dividido en tres categorías de

acorde a su demanda: baja, media y alta demanda con demanda media de 5, 100 y 200

respectivamente.

Para cada combinación se han realizado dos réplicas. En total 2592 problemas se han

generado y resuelto por siete modelos: BM, JON, R&G, ERF, BM-modified, R&G-modified

y ERF-modified. Este experimento fue llevado a cabo en un ordenador ACER Intel® Core™

i3-2310M CPU 2.10GHz con memoria instalada (RAM) 4.0GB.

El experimento de problemas de gran tamaño sigue un diseño factorial solo en determinados

factores y el resto se fijan a un valor medio.

Productos: 100, 500, 800, 1000, 1500

Períodos: 50, 100, 150

TBO: 1.5, 4.5

Probabilidad de demanda: 0.35, 0.75

Para cada combinación factorial dos 2 replicaciones se han realizado. Este conjunto de

problemas se ha dividido en dos subconjuntos. El primero incluye todos los problemas menos

las combinaciones que contienen el número de productos 800 y 1500. El primer subconjunto

de problemas (72 problemas) ha sido resuelto por los modelos BM-modified, R&G y JON,

mientras que el segundo subconjunto contiene el resto de problemas (48 problemas) que han

sido resueltos por los modelos R&G y JON.

Los resultados del R&G son generalmente los mejores. Sin embargo, hay determinados

problemas en los que este modelo no fue capaz de llegar a la solución óptima ni tampoco a

una solución factible. Estos problemas, en los cuales el JON presenta mejores resultados,



tienen en común elevado número de productos y períodos, elevado TBO y elevada

probabilidad de demanda. Para investigar este hecho se realizaron pruebas adicionales. Esta

vez el límite de tiempo se extendió a dos horas. Las combinaciones de productos-períodos

usadas esta vez son: 800-150, 1000-150 y 1500-100, todas ellas con TBO 4.5, MSR 0.6 e

índice de dispersión de la demanda 1. La probabilidad de la demanda tiene dos niveles: 0.75 y

1. Para cada combinación factorial se realizaron 4 replicaciones (en total 24 problemas) y se

resolvieron mediante el R&G y el JON.

El experimento con los problemas de gran tamaño se llevó a cabo utilizando un ordenador

Intel® Xeon® CPU E7-4830 v3 2.10GHz con memoria instalada (RAM) 32.0 GB.

Tanto en el experimento preliminar como en el de problemas de gran tamaño el solver

utilizado fue Xpress 7.9. Los resultados fueron analizados utilizando Microsoft Excel 2013 y

MatLab 2015R.

Los recursos computacionales se han medido fundamentalmente por el tiempo computacional.

Si en un problema no se alcanza la solución óptima durante el tiempo permitido, entonces se

evalúa la calidad de la mejor solución factible obtenida, si es que la hay, a través de la

diferencia porcentual de optimalidad. También se han comparado los requerimientos de

memoria de los distintos modelos.

2.3 Resultados

2.3.1 Experimento preliminar

De los 2592 problemas generados en el experimento preliminar, 36 no pudieron ser resueltos

por el BM ni el BM-modified porque el ordenador se quedó sin memoria. Todos estos

problemas tienen en común el número de productos (40), períodos (48), TBO (4.5) y MSR

(0.3). El resto de los modelos llegaron a la solución óptima en todos los problemas.

Como se puede ver en la figura 2, el modelo R&G y su versión modificado obtuvieron la

solución óptima en el menor tiempo en el 80.05% de los problemas, el BM lo hizo en el

14.47% de los problemas, mientras que el ERF y su versión modificada sólo lo hicieron el

0.31% y el 0.27% de las veces respectivamente.



La adición de la propiedad antes mencionada (versiones modificadas) tiene un efecto positivo,

aunque no lo suficiente como para hacer a un modelo más eficiente que otro. La media y las

varianzas de los tiempos computacionales en este experimento preliminar del BM, el R&G y

el ERF y sus versiones modificadas pueden observarse en la tabla 1. El impacto de añadir esta

propiedad es más notorio en el BM, mientras que el efecto en el R&G y en el ERF no es muy

relevante.

Figura 2: Porcentaje de problemas que un modelo resolvió más rápido que el resto

Tabla 1: Comparación de los modelos con sus versiones modificadas

Tiempo computacional BM R&G ERF

Media 4.91

(4.53)

0.89

(0.88)

3.05

(2.87)

Varianza 1286.26

(439.42)

10.35

(10.66)

146.79

(147.40)

Observaciones 2556 2592 2592

*Valores en segundos

*Los valores en los paréntesis pertenecen a las versiones modificadas

Para realizar el análisis factorial se han realizado análisis de regresión para cada modelo. Las

conclusiones más relevantes de este análisis son:

En todos los modelos el tiempo computacional aumenta exponencialmente con el

aumento del número de productos y períodos. Estos son los factores que más

influencia tienen.



El resto de factores son significativos en casi todos los modelos. Siendo el TBO el

siguiente de mayor influencia especialmente en el BM donde su influencia es casi

similar a la del número de productos y períodos. Un aumento del TBO produce un

aumento del tiempo computacional en todos los modelos.

El MSR está negativamente correlacionado en todos los modelos con el tiempo

computacional. Un aumento del MSR produce una reducción del tiempo

computacional.

El índice de dispersión presenta una correlación muy baja en todos los modelos. En

todo caso, un aumento de éste produce un ligero aumento en el tiempo computacional

de casi todos los modelos.

La probabilidad de demanda tiene efectos adversos dependiendo del modelo. Un

incremento de ésta produce un aumento del tiempo computacional en los modelos BM

y R&G y una reducción en los modelos JON y ERF.

2.3.2 Experimento con problemas grandes

Los resultados del primer subconjunto de problemas muestran que el R&G obtuvo la solución

óptima en menor tiempo que los otros dos modelos el 91.7% (66/72) de los casos, mientras

que el JON lo hizo el 4.2% (3/72) de los casos y el BM solo lo hizo en un problema. En los

otros dos casos ninguno de los modelos obtuvo la solución óptima.

El BM pudo obtener la solución óptima en un número limitado de casos que tienen en común

las más bajas cantidades de productos y períodos y TBO bajo. La media de la diferencia

porcentual de optimalidad en los casos en los que no obtuvo la solución óptima es 43.2%,

incrementando linealmente con el tamaño del problema y el TBO. Estos resultados

comparados con los otros dos modelos lo convierten en ineficiente a la hora de resolver

problemas de gran tamaño.

Los resultados del segundo subconjunto de problemas que solo fueron resueltas por el R&G y

el JON muestran que el R&G obtuvo la solución óptima en el menor tiempo el 85.4% (41/48)

de las veces. En este subconjunto hay cuatro problemas en los cuales ningún modelo llegó a la

solución óptima.

La tabla 2 presenta el número de problemas que cada modelo resolvió óptimamente (o con

una diferencia de optimalidad menor que 0.1%), la media de diferencia porcentual de

optimalidad en los problemas en los que no se obtuvo la solución óptima y el número de

problemas que fueron resueltos óptimamente en menor tiempo que los otros dos modelos.



Tabla 2: Comparación general del BM, el JON y el R&G

BM R&G JON

Problemas resueltos óptimamente* 19/72 114/120 103/120

Diferencia de optimalidad 43.2% 83.5% 55.8%

Modelo más rápido* 1/72 107/120 6/120

*Número de problemas/Total de problemas testeados con un modelo

*Diferencia de optimalidad = |𝑉𝑎𝑙𝑜𝑟 𝑜𝑏𝑗𝑒𝑡𝑖𝑣𝑜 𝑚𝑒𝑗𝑜𝑟 𝑠𝑜𝑙𝑢𝑐𝑖ó𝑛−𝐿í𝑚𝑖𝑡𝑒 𝑖𝑛𝑓𝑒𝑟𝑖𝑜𝑟

𝑉𝑎𝑙𝑜𝑟 𝑜𝑏𝑗𝑒𝑡𝑖𝑣𝑜 𝑚𝑒𝑗𝑜𝑟 𝑠𝑜𝑙𝑢𝑐𝑖ó𝑛|

En los 102 casos en los que el JON y el R&G obtuvieron la solución óptima, el JON requirió

de media 2.9 veces más tiempo que el R&G (𝐽𝑂𝑁 𝑡𝑖𝑒𝑚𝑝𝑜−𝑅&𝐺 𝑡𝑖𝑒𝑚𝑝𝑜

𝑅&𝐺 𝑡𝑖𝑒𝑚𝑝𝑜).

A pesar de obtener la solución óptima en la mayoría de los problemas y hacerlo generalmente

en menos tiempo que el resto de los modelos, hay 5 problemas en los que el R&G no pudo

obtener ni siquiera una solución factible, en estos casos la diferencia de optimalidad equivale

al 100%, lo que explica la media tan alta que aparece en la tabla 2. Como se ha explicado

antes, se han hecho pruebas adicionales para investigar este hecho.

Esta vez el límite de tiempo se fija en 2 horas, por lo que todos los problemas fueron resueltos

óptimamente por los modelos R&G y JON.

La media de los tiempos computacionales para cada combinación estudiada en estas pruebas

se presenta en la tabla 3.

Tabla 3: Tiempos computacionales del R&G y el JON

𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑜𝑠 × 𝑃𝑒𝑟í𝑜𝑑𝑜𝑠 R&G* JON*

800 × 150 46.9

(57.7)

39.3

(35.1)

1000 × 150 95.8

(77.1)

62.8

(52.0)

1500 × 100 54.4

(43.2)

57.8

(47.4)

*Tiempo CPU expresado en minutos

Número sin paréntesis es la media del tiempo CPU con probabilidad de demanda

p=0.75; número con paréntesis es la media del tiempo CPU con p=1

El JON presenta mejores resultados cuando el número de períodos es 150 y el número de

productos es suficientemente elevado, con elevado TBO y probabilidad de demanda. El R&G

requiere un 45.6% más tiempo computacional en estos casos que el JON (𝑅&𝐺 𝑡𝑖𝑚𝑒−𝐽𝑂𝑁 𝑡𝑖𝑚𝑒

𝐽𝑂𝑁 𝑡𝑖𝑚𝑒).



En estos casos incrementando el número de productos se produce un incremento en la

diferencia computacional de ambos modelos.

2.3.3 Análisis de requerimientos de memoria

Aparte de tardar mucho tiempo, otro inconveniente al resolver problemas de gran tamaño es

que la memoria requerida puede exceder la capacidad del ordenador. La memoria requerida

depende también del modelo utilizado.

Para estudiar este aspecto se han realizado unas pruebas con el objetivo de averiguar cuál es el

modelo que requiere menor memoria. Estas pruebas se han llevado a cabo con un ordenador

ACER Intel® Core™ i3-2310M CPU 2.10GHz. Dos combinaciones de períodos han sido

seleccionadas (50 y 84) y el número de productos se ha ido aumentando progresivamente

hasta que todo los modelos fallaran por insuficiencia de memoria. Cuando un modelo se

quedaba sin memoria al resolver un problema era descartado para el siguiente problema con

mayor número de productos. La tabla 5 muestra los resultados de estas pruebas.

Tabla 4: Comparación de los requerimientos de memoria

Productos-

Períodos*

BM BM-

Modified

JON R&G R&G-

Modified

ERF ERF-

Modified

500-50 (-) (-) ✓ ✓ ✓ ✓ ✓

800-50 ✓ ✓ ✓ ✗ ✗

1500-50 ✓ ✓ ✓

1750-50 ✓ ✗ ✗

2000-50 ✗ ✗ ✗

400-84 (-) (-) ✓ ✓ ✗

500-84 ✗ ✗ ✗

(✓) Problema resuelto óptimamente

(✗) Problema no puede ser resuelto porque excede la memoria del ordenador

Problem could not be solved because the computer run out of memory

(-) Procedimiento de ejecución detenido tras una hora

* Todos los problemas tienen MSR=0.6; TBO=2.5; Probabilidad de demanda=0.75;

Índice de dispersión de la demanda=0.5

El BM no agotó la memoria en los problemas que fue testeado, sin embargo no pudo resolver

ninguno de los problemas a los que se aplicó en menos de una hora, presentando una

diferencia porcentual de optimalidad en el mejor de los casos de 34%. El resto de modelos

llegaron a la solución óptima o agotaron la memoria en una duración menor de 25 min en

todos los casos. Los modelos JON y R&G son los que permiten resolver problemas de mayor



tamaño, siendo el primero ligeramente superior en este aspecto. Las versiones modificadas de

los modelos no constituyen una mejora en este apartado. Un mayor número de restricciones

aumenta el número de cálculos en cada nodo del árbol de Branch and Bound haciendo que sea

más pesado, es decir requiera más memoria.



3 CONCLUSIONES

La mejora de los solvers y de los ordenadores permite resolver problemas de programación

entera de gran tamaño. La importancia de elegir el modelo correcto es clara. Elegir un modelo

incorrecto puede llevar a la imposibilidad de alcanzar una solución óptima. En concreto en el

JRP el tiempo computacional aumenta exponencialmente con el número de productos y

períodos. El resto de los factores en este trabajo analizados también tienen influencia en dicho

tiempo.

Los resultados experimentales muestran que el R&G es el más eficiente a la hora de resolver

problemas de gran escala. El siguiente modelo más eficiente (JON) requirió en este

experimento de media 2.9 veces más de tiempo. Solo en determinadas circunstancias (muy

elevado número de períodos, elevados costes fijos y probabilidad de demanda) el R&G no es

el más eficiente. En estos casos el JON requiere significativamente menos tiempo.

De todas formas, en este experimento el R&G pudo resolver los problemas de gran tamaño

con estas condiciones adversas en un tiempo máximo de dos horas. Otra ventaja del R&G es

que permite el backordering en su estructura.

En el caso de no disponer de un ordenador para resolver un problema mediante el R&G, el

JON debería ser probado ya que requiere de menos memoria. Sin embargo, esta diferencia de

requerimientos de memoria es pequeña.

Futuras investigaciones deberían centrarse en el análisis de los heurísticos para resolver el

JRP, analizando la diferencia de optimalidad cuando el tamaño de los problemas es grande.

Por otro lado en este modelo hay muchas asunciones. A la hora de aplicarlo en una empresa

generalmente hay que hacer algunas modificaciones. Algunos estudios interesantes incluyen

la adaptación del JRP bajo demanda estocástica (Khouja y Goyal, 2008), restricciones de

capacidad y de inversión de capital (Hoque, 2006).



4 INFORMACIÓN DEL TRABAJO

4.1 Universidad de destino

La Universidad Técnica de Munich (TUM) fue fundada en 1868 y ha sido decisiva en el

desarrollo industrial de la región de Baviera. Actualmente cuenta con 39.000 estudiantes, una

de las más grandes de Alemania. 13 premios Nobel han estudiado o dado clase en esta

universidad, además de personalidades importantes del ámbito científico como Rudolf Diesel

o Carl von Linde.

La Escuela de Management está situada en el centro de la ciudad en el campus principal. A

esta facultad pertenece el Departamento de Logística y Gestión de la Cadena de Suministro,

en el que fue realizado este trabajo.

El tutor de este trabajo fue Florian Taube, profesor adjunto.

4.2 Presupuesto

El presupuesto de este trabajo viene desglosado en la tabla 5. El sueldo del tutor en la TUM se

establece como el equivalente a 30 horas de trabajo con un sueldo de 22,8€/h1 el establecido

en el estado de Bavaria para los PhD. El sueldo del autor es nulo. La licencia completa de

Xpress2 tiene un precio de 8.123,07€ ($8.995,00). La licencia de Microsoft Office® 365

University3 tiene un precio de 79,00€. La licencia académica de MatLab4 tiene un precio de

500,00€.

Por último el ordenador en el que se ha realizado el experimento pertenece a la TUM y tiene

un precio de 1.960,91€ ($2.170,00). Este experimento ha dispuesto exclusivamente de él 3

semanas. Se supone un ciclo de vida de 2 años. Todos los precios citados contienen IVA.

1 Freistaat Bayern: Landesamt für Finanzen. http://www.lff.bayern.de/

download/bezuege/arbeitnehmer/entgelttabelle_tvl.pdf, a fecha del 12.07.2015. No se ha encontrado el

documento del año 2016 por lo que se supone el mismo salario. 2 FrontlineSolvers®: http://www.solver.com/catalog/solver-engines, consultado el 18.07.2016. 3 Microsoft Store: https://www.microsoftstore.com/store/msde/de_DE/pdp/Office-365-

University/productID.283492900, consultado el 18.07.2016. 4 MathWorks: http://de.mathworks.com/pricing-

licensing/index.html?intendeduse=edu&prodcode=ML, consultado el 18.07.2016.

http://www.solver.com/catalog/solver-engines

https://www.microsoftstore.com/store/msde/de_DE/pdp/Office-365-University/productID.283492900

https://www.microsoftstore.com/store/msde/de_DE/pdp/Office-365-University/productID.283492900

http://de.mathworks.com/pricing-licensing/index.html?intendeduse=edu&prodcode=ML

http://de.mathworks.com/pricing-licensing/index.html?intendeduse=edu&prodcode=ML



Tabla 5: Costes del trabajo

Concepto Gasto (€)

Gastos salariales

Tutor 684,00

Licencias de programas

Xpress 8.123,07

Microsoft Office 79,00

MatLab 500,00

Gastos de equipamiento

Ordenador 56,41

Total 9.442,48

4.3 Planificación temporal

En la siguiente página se muestra el diagrama de Gantt de este trabajo que muestra la

duración de las diferentes tareas. Comenzó en marzo y se finalizó en julio de 2016. Las tareas

principales fueron la búsqueda bibliográfica, la implementación de los modelos, los

experimentos y los análisis de los resultados de éstos.



TRABAJO EN UNIVERSIDAD DE DESTINO

A continuación se presenta el trabajo completo presentado en la universidad de destino. El

idioma del trabajo es inglés. Este trabajo pertenece a la categoría de Bachelor Thesis

reconocida con 12 ECTS.

Bachelor of Science

an der Technischen Universität München

Comparison of the joint

replenishment problem approaches

with dynamic demand

Referent: Logistics and Supply Chain Management

Prof. Dr. Stefan Minner

Technische Universität München

Betreuer: Dipl.-Kfm. Florian Taube

Studiengang: Techn.-u.Managem.BWL

Eingereicht von: José Mezquita Zapico

Connollystr. 3

80809 München

Matrikelnummer 03672276

Eingereicht am: 20.07.2016

Comparison of the joint replenishment problem approaches with dynamic

demand ii

Abstract

This Bachelor thesis is a study of the optimization models of the joint replenishment

problem focusing on the comparison of the different computational requirements of

each one when the scale of the problem is large. Four models extracted from the

literature have been compared. Firstly, an experimental study of 2592 generated

problems has been conducted. From the results of this experiment the influence of

the different factors on the computational time has been analyzed. Secondly, an

experimental study of 152 large scale problems has been conducted. By analyzing

the computational time, optimality gap and memory requirements it is concluded

that in most of the cases the model of Robinson and Gao is the most efficient.

Nevertheless in large scale problems (specifically large number of periods) with

high TBO and demand probability the model of Joneja outperforms systematically

the model of Robinson and Gao.

Keywords: joint replenishment problem, MILP, optimization


demand iii

Table of Contents

List of Figures .......................................................................................................... v

List of Tables .......................................................................................................... vi

List of Abbreviations ............................................................................................. vii

List of Symbols ..................................................................................................... viii

1 Introduction .................................................................................................... 1

2 Review of Literature and Research ................................................................ 3

2.1 JRPDD terms definition ........................................................................ 3

2.2 Basic model (BM) ................................................................................. 3

2.3 The model of Joneja (JON) ................................................................... 4

2.4 The model of Robinson and Gao (R&G) .............................................. 6

2.4.1 Without backordering ........................................................................ 7

2.4.2 With backordering ............................................................................. 8

2.5 The Exact Requirements formulation (ERF) ........................................ 8

2.6 Properties of the optimal solution ......................................................... 9

2.7 Comparison of the models .................................................................. 10

2.8 Research gap covered ......................................................................... 12

3 Methodology ................................................................................................ 13

3.1 Factors ................................................................................................. 13

3.1.1 Time-between-order (TBO) ............................................................. 13

3.1.2 Major to minor setup cost ratio (MSR)............................................ 13

3.1.3 Demand distribution ........................................................................ 13

3.1.4 Demand probability ......................................................................... 15

3.2 Addition of Property 4 ........................................................................ 15

3.3 Experimental process .......................................................................... 16

4 Preliminary experiment ................................................................................ 18

4.1 Preliminary experiment design ........................................................... 18

4.2 Preliminary experiment results ........................................................... 19

4.3. Regression analysis .......................................................................... 22

4.3.1. JON computational time regression analysis ................................... 23

4.3.2. BM computational time regression analysis .................................... 25

4.3.3. R&G computational time regression analysis ................................. 27

4.3.4. ERF computational time regression analysis ................................... 28


demand iv

5. Large scale problems experiment ................................................................. 30

5.1. Experimental design ........................................................................... 30

5.2. Results ................................................................................................. 31

5.3 Memory Requirements Analysis ............................................................ 35

6. Conclusion .................................................................................................... 38

Reference List ........................................................................................................ 39

Appendices ............................................................................................................ 41

Appendix A .................................................................................................. 41

Appendix B .................................................................................................. 42

Appendix C .................................................................................................. 45

Appendix D .................................................................................................. 47

Appendix E ................................................................................................... 48

Appendix F ................................................................................................... 50

Ehrenwörtliche Erklärung ...................................................................................... 51


demand v

List of Figures

Figure 1 "The item-network for item i" Joneja (1990) ............................................ 5

Figure 2: Experimental process ............................................................................. 16

Figure 3: Percentage of problems that a model was faster than the others............ 19

Figure 4: Interaction effect between the TBO and MSR ....................................... 21

Figure 5: JON computational time depending on the number of nodes ................ 34

Figure 6: R&G computational time depending on the number of nodes ............... 34

Figure 7: Computational time depending on the number of products ................... 42

Figure 8: Computational time depending on the number of periods ..................... 42

Figure 9: Computational time depending on the TBO .......................................... 43

Figure 10: Computational time depending on the MSR ........................................ 43

Figure 11: Computational time depending on the index of dispersion of the

demand................................................................................................... 44

Figure 12: Computational time depending on the demand probability ................. 44


demand vi

List of Tables

Table 1: Number of decision variables and constraints of each model ................. 10

Table 2: Average computational times .................................................................. 20

Table 3: Maximun computational times ................................................................ 20

Table 4: Comparison of the models with their modified version .......................... 22

Table 5: Correlation table of the JON computational time with the factors .......... 24

Table 6: Exponential regression analysis of the JON computational time ............ 24

Table 7: Correlation table of the BM computational time with the factors ........... 25

Table 8: Exponential regression analysis for the BM computational time ............ 26

Table 9: Correlation table of the R&G computational time with the factors ........ 27

Table 10: Exponential regression analysis of the R&G computational time......... 27

Table 11: Correlation table with the ERF computational time with the factors .... 28

Table 12: Exponential regression analysis of the ERF computational time .......... 29

Table 13: General comparison of the BM, the JON and the R&G ........................ 32

Table 14: Computational time and branch and bound nodes of the R&G and the

JON depending on the scale of the problems ........................................ 33

Table 15: Computational times of the R&G and JON for very large scale

problems ................................................................................................ 35

Table 16: Comparison on computational memory requirements .......................... 36

Table 17: Exponential regression model of the BM-modified computational time

............................................................................................................... 45

Table 18: Exponential regression analysis of the R&G-modified computational

time ........................................................................................................ 45

Table 19: Exponential regression analysis of the ERF-modified computational

time ........................................................................................................ 46

Table 20: Exponential regression analysis of the JON computational time of large

scale problems ....................................................................................... 48

Table 21: Exponential regression analysis of the R&G computational time of large

scale problems ....................................................................................... 48


demand vii

List of Abbreviations

B&B

BM

BM-modified

ERF

ERF-modified

JON

MSR

R&G

R&G-modified

TBO

Branch and Bound

Basic model

Basic model modified version

Exact Requirements formulation

Exact Requirements formulation modified version

Model of Joneja

Major setup cost ratio

Model of Robinson and Gao

Model of Robinson and Gao modified version

Time-between-order


demand viii

List of Symbols

(𝑛

𝑘)

𝛿𝑘

𝜇

σ

A

𝑐𝑘𝑡

𝑐𝑘𝑖𝑡

𝑑𝑘𝑡

ℎ𝑘𝑡

𝐼𝑘𝑡

I

K

ln(𝑥)

M

N(µ, σ)

𝑃(𝑥)

𝑝𝑘𝑡

𝑆𝑡

𝑆𝑘𝑡

𝑠𝑞𝑟𝑡(𝑥)

T

𝑉𝑎𝑟(𝑥)

⌊𝑥⌋

⌈𝑥⌉

�̅�

𝑥∗

𝑥𝑘𝑡

𝑌𝑘𝑡

𝑍𝑡

Binomial coefficient indexed by 𝑛 and 𝑘

First period when product 𝑘 experiences demand

Mean

Standard deviation

Total setup costs

Purchasing cost of the product 𝑘 in the period 𝑡

Cost associated with maintaining one unit of product 𝑘 in

inventory from the period 𝑖 until the period 𝑡

Demand of product 𝑘 in period 𝑡

Inventory holding cost of product 𝑘 in period 𝑡

Inventory level of product 𝑘 at the end of period 𝑡

Index of dispersion

Number of product types

Natural logarithm of 𝑥

Large number

Normal distribution with mean µ and standard deviation σ

Probability mass function of 𝑥

Backordering penalty cost of product 𝑘 in period 𝑡

Major setup cost

Minor setup cost

Square root of 𝑥

Number of periods

Variance of a distribution

Nearest lower integral value from 𝑥

Nearest higher integral value from 𝑥

Mean of the value 𝑥

Optimal value of a variable 𝑥

Quantity of product 𝑘 replenished in period 𝑡

Minor setup indicator

Major setup indicator

Introduction 4

1 Introduction

The objective of this Bachelor thesis is to extend the research in the joint

replenishment problem with dynamic demand. In concrete, four different mixed-

integer linear programming (MILP) approaches to solve this problem are compared

to determine which is the most efficient when the data instances are large. Although

there is extensive literature research in this problem, to our concern there is no study

analyzing the case of large data instances. As nowadays there is a global tendency

of adopting Big Data procedures, it is important to fill the research gap of solving

this problem with large data instances.

The joint replenishment problem (JRP) consists of determining the optimal

replenishment policy of various products for satisfying the demand over a time

horizon. This problem assumes that there are fixed costs when a replenishment is

done. Whenever a replenishment is done a major setup cost is incurred. Another

setup cost is incurred whenever any product of a type is replenished. This is the

minor setup cost. The objective is to minimize the total cost taking into account also

the purchasing cost and the inventory holding cost. In the joint replenishment

problem with dynamic demand (JRPDD) the demand varies in the different time

periods, though it is considered to be known beforehand.

This problem is suitable for companies that replenish more than one product from

one supplier such as retailing companies. It is also useful to optimize the

transportation operations of a distribution center.

This problem can be also applied to manufacturer companies being interpreted as a

production scheduling problem without capacity constraints, being sometimes

referred in the literature as the lot-sizing problem. In this case the objective is to

determine when to produce the different products and the size of the lots in order to

minimize costs. The major setup cost may be interpreted as the fixed costs for

cleaning and maintaining the machines before producing or designing the layout of

the machines, while the minor setup cost could represent for example the cost of

the setup times of the specific machines needed to produce a product or the waste

of material. The purchasing cost represents the production cost of a product in a

period.

The class of this problem is NP-complete (Arkin, Joneja, & Roundy, 1989) which

means that it is unlikely to be solved in polynomial time depending on the number

of variables. The different structures of the models compared lead to different

computational resources required.


demand 2

In the section 2 of this paper four modelling approaches to solve the JRPDD

extracted from the literature are reviewed, as well as the most relevant studies on

this topic found. Next, the section 3 includes a description of the factors that are

analyzed in this study and the experimental process which includes a preliminary

experiment and the large scale problems experiment. The preliminary experiment

consist of solving a large set of problems of small scale using a full factorial design,

while the large scale experiment is conducted with problems of big size. The

number of problems tested in the later experiment is considerably smaller because

these large scale problems take very long time to be solved. Each of these

experiments and their major findings are described in separate sections (section 4

for the preliminary experiment and section 5 for the large scale problems

experiment). At the end, in section 6 the conclusions are presented and further

investigations suggested.


demand 3

2 Review of Literature and Research

2.1 JRPDD terms definition

The different types of products are denoted by the index k, being K the total

number of product types (k=1,…, K).

The time period is denoted by the index t, being T the total number of time

periods (t=1,…, T). A replenishment can only be made at the beginning of

a period.

The demand 𝑑𝑘𝑡is independent for each product and each period. In this

study the demand has been modelled as discrete (i.e. 𝑑𝑘𝑡 takes only integral

values). Though being variant over the periods, it is assumed to be known

beforehand.

The purchasing cost 𝑐𝑘𝑡 is independent for each product and each period. It

is assumed that there are no quantity discounts neither capital investment

limitations.

The inventory holding costs are assumed to be linear in the number of

products and the time. The term ℎ𝑘𝑡 represents the cost of storaging one unit

of product k in period t. In this study it has been assumed that there are no

storage capacity constraints.

When making an order of product k at time period t to satisfy the demand

of period t+n, it is assumed that the product is received at the beginning of

the period t and delivered at the beginning of the period t+n. Therefore, the

total inventory holding cost of this product is the sum of its single holding

costs from period t until period t+n-1.

The major setup cost in period t is denoted by𝑆𝑡.

The minor setup cost of product k in period t is denoted by 𝑠𝑘𝑡.

Only one of the models studied allows including the backordering (i.e.

satisfying demand after its due date) in its structure. In order to measure the

models’ efficiency in equal terms, the backordering has not been

considered.

2.2 Basic model (BM)

This is the most intuitive model. This model is based in the idea that the demand of

a product in a period must be fulfilled with the inventory plus the quantity

replenished in that period. The decision variables of this formulation are:


demand 4

𝑥𝑘𝑡: order quantity for product k in period t.

𝑌𝑘𝑡 : minor setup indicator. It is a binary variable which takes the value of 1

if a replenishment order for product k is scheduled for period t, otherwise it

takes the value of 0.

𝑍𝑡 : major setup indicator. It is a binary variable which takes the value of 1

if a replenishment order is scheduled for period t, otherwise it takes the value

of 0.

𝐼𝑘𝑡 : inventory level of product k at the end of period t.

The model formulation is:

min∑∑(𝑠𝑘𝑡 · 𝑌𝑘𝑡 + 𝑐𝑘𝑡 · 𝑥𝑘𝑡 + ℎ𝑘𝑡 · 𝐼𝑘𝑡) +∑𝑆𝑡 · 𝑍𝑡

𝑇

𝑡=1

𝑇

𝑡=1

𝐾

𝐾=1

(1)

Subject to

𝐼𝑘𝑡 = 𝐼𝑘,𝑡−1 + 𝑥𝑘𝑡 − 𝑑𝑘𝑡(𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇)(2)

𝐼𝑘0 = 0(𝑘 = 1,2, … , 𝐾)(3)

𝑥𝑘𝑡 ≤ 𝑀 · 𝑌𝑘𝑡(𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇)(4)

𝑌𝑘𝑡 ≤ 𝑍𝑡(𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇)(5)

𝑥𝑘𝑡, 𝐼𝑘𝑡 ≥ 0;𝑌𝑘𝑡, 𝑍𝑡 ∈ {0, 1}(𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇)(6)

The constraint (2) determines the inventory level at the end of a period which is

equal to the inventory level at the end of the previous period plus the quantity order

at the beginning of the period minus the demand in that period. Constraint (3) gives

an initial inventory level. For sake of simplicity in this study it has been assumed

that the inventory level at the beginning of the first period is null for all the products.

The parameter M that appears on constraint (4) is a large number so it ensures that,

whenever any quantity of a product is ordered, the corresponding minor setup cost

is accounted. Setting M as the overall sum of the demand of all products for all

periods is enough to ensure that this constraint functions properly. Constraint (5)

ensures that the major setup cost is incurred whenever any product is ordered. The

non-negative constraint of the inventory level ensures that there is no backordering.

2.3 The model of Joneja (JON)

Joneja (1990) proposes a model based on the shortest path formulation problem.

This problem consist of finding a minimum distance path connecting a set of nodes

in a graph. Applied to the JRP, there is a set of 𝑡 = {1,… , 𝑇 + 1} nodes for each

product, where T is the number of periods. Each node represent a time period and


demand 5

there is a cost for flowing from one period to another. The cost of flowing from one

period t to another period t’ represents the cost of making a replenishment at the

beginning of period t and serving the demand until the period t’-1. The objective is

to find a path which connects the node 1 (or the first node when there is demand)

with the last node and minimizes the cost. The succession of nodes connected

represent the periods when a product is replenished. Intuitively, if the setup costs

are very large relatively to the inventory holding costs, the first node would be

connected directly with the last node. On the other hand, if the setup costs were

insignificant compared with the inventory holding costs, each node would be

connected with the following node.

Figure 1 "The item-network for item i" Joneja (1990)

In this model, the decision variables are:

𝑥𝑘𝑖𝑗 : is defined for 1 ≤ 𝑖 < 𝑗 ≤ 𝑇 + 1

It is equal to 1 if product k is ordered in time period i and time period j and

nowhere in between, otherwise it is equal to 0

𝑍𝑖 : major setup indicator. It is equal to 1 if any product is ordered at

period i, otherwise it is 0.

The total cost for replenishing the product k at time period i and serving the demand

of this product through the time periods until j-1 is:

𝐶𝑘𝑖𝑗 = 𝑠𝑘𝑖 +∑𝑐𝑘𝑖𝑑𝑘𝑟

𝑗−1

𝑟=𝑖

+ ∑ ( ∑ ℎ𝑘𝑛)𝑑𝑘𝜏

𝜏

𝑛=𝑖+1

𝑗−1

𝜏=𝑖+1

(7)

The major setup cost is included in the objective function.

The problem formulation is:

min∑∑ ∑ 𝐶𝑘𝑖𝑗𝑥𝑘𝑖𝑗

𝑇+1

𝑗=𝑖+1

𝑇

𝑖=1

𝐾

𝑘=1

+∑𝑍𝑖 𝑆𝑖

𝑇

𝑖=1

(8)

Subject to:


demand 6


𝑇+1

𝑗=𝑖+1

−∑𝑥𝑘𝑠𝑖

𝑖−1

𝑠=1

= 0(𝑘 = 1,… , 𝐾; 𝑖 = 2,… , 𝑇)(9)

∑𝑥𝑘,𝑖,𝑇+1

𝑇

𝑖=1

= 1(𝑘 = 1,… , 𝐾)(10)

𝑍𝑖 − ∑ 𝑥𝑘𝑖𝑗

𝑇+1

𝑗=𝑖+1

≥ 0(𝑘 = 1,… , 𝐾; 𝑖 = 1, … , 𝑇)(11)

𝑥𝑘𝑖𝑗 , 𝑍𝑖 ∈ {0,1}(𝑘 = 1,… , 𝐾; 𝑖 = 1,… , 𝑇; 𝑗 = 𝑖 + 1,… , 𝑇 + 1)(12)

The constraints (9) and (10) represent the flow constraints. The constraint (11)

ensures that the corresponding major setup cost is incurred whenever any product

is ordered.

This model fails to yield the optimal result if any product is not ordered in the first

time period. Minner (2003) introduces a small readjustment to solve this matter.

First, a new variable 𝛿𝑘 is introduced. It determines the first period when the product

k has positive demand:

𝛿𝑘 = min{𝑡|𝑑𝑘𝑡 > 0}

Secondly, the domain of the constraint (9) has to be defined slightly different:


𝑇+1

𝑗=𝑖+1

−∑𝑥𝑘𝑠𝑖

𝑖−1

𝑠=1

= 0(𝑘 = 1,… , 𝐾; 𝑖 = 𝛿𝑘 + 1,… , 𝑇)(9′)

In this study it has been considered the possibility that a product does not experience

demand in the first period and therefore this formulation has been used.

2.4 The model of Robinson and Gao (R&G)

Robinson and Gao (1996) propose a model based on the uncapacitated facility

location problem formulation. The facility location problem aims to determine

where to open the facilities (given some potential locations) and which quantity of

the markets demand is fulfilled by each facility. To apply this formulation to the

JRP, the facilities correspond to the periods in which an order occurs and the

objective is to determine which quantity of the successive demand for each product

is fulfilled from this order.

According to words of Robinson and Gao (1996) “the single sourcing of this

formulation and the hierarchical structure of the fixed-charge and continuous

variables yield an extremely tight linear programming relaxation for the problem”.

This statement is confirmed in the results later shown in this study.


demand 7

One of the advantages of this model is that its structure allows the backordering

possibility (note that the formulation of the model changes slightly if it is

considered).

2.4.1 Without backordering

The decision variables of this model are:

𝑥𝑘𝑖𝑡: it represents the fraction of demand of product k for time period t that

is replenished in time period i. It is defined for 𝑖 = 1,… , 𝑇; 𝑘 = 1,… , 𝐾; 𝑡 =

𝑖, … , 𝑇.

𝑍𝑖 : major setup indicator. It is a binary variable which takes the value of 1

if a replenishment order is scheduled for period t, otherwise it takes the value

of 0.

𝑌𝑘𝑖: minor setup indicator. It is a binary variable which takes the value of 1

if a replenishment order for product k is scheduled for period t, otherwise it

takes the value of 0.

The cost for supplying all the demand for product k in period t from a replenishment

in period i (given a major setup cost is already fixed) is:

𝐶𝑘𝑖𝑡 = (𝑐𝑘𝑖 +∑ℎ𝑘𝑗)

𝑡−1

𝑗=𝑖

· 𝑑𝑘𝑡(𝑖 ≤ 𝑡)(13)

The problem formulation is:

min∑𝑆𝑡 · 𝑍𝑡 +∑∑𝑠𝑘𝑡 · 𝑌𝑘𝑡

𝐾

𝑘=1

𝑇

𝑡=1

+∑∑∑𝐶𝑘𝑖𝑡 · 𝑥𝑘𝑖𝑡

𝑇

𝑡=𝑖

(14)

𝑇

𝑖=1

𝐾

𝑘=1

𝑇

𝑡=1

Subject to:

∑𝑥𝑘𝑖𝑡 = 1(𝑘 = 1, …

𝑡

𝑖=1

, 𝐾; 𝑡 = 𝛿𝑘, 𝛿𝑘 + 1,… , 𝑇)(15)

𝑥𝑘𝑖𝑡 ≤ 𝑌𝑘𝑖(𝑖 = 1,… , 𝑇; 𝑘 = 1,2, … , 𝐾; 𝑡 = 𝑖, … , 𝑇)(16)

𝑌𝑘𝑖 ≤ 𝑍𝑖 (𝑘 = 1,2, … , 𝐾; 𝑖 = 1,… , 𝑇)(17)

𝑍𝑖 , 𝑌𝑘𝑖 ∈ {0,1}(𝑖 = 1,… , 𝑇; 𝑘 = 1,… , 𝐾)(18)

The constraint (15) ensures that all the demand is fulfilled. Note that the domain of

t in this constraint starts in 𝛿𝑘 which is equal to the first period with demand for

product k (same as explained before in the JON). The constraint (16) ensures that

the minor setup cost is incurred whenever the correspondent product is replenished.

The constraint (17) ensures that the major setup cost is incurred whenever any

product is replenished.


demand 8

2.4.2 With backordering

The backordering consists of fulfilling part or the whole demand after the due date.

This means that the demand for certain period can be replenished in successive

periods. Usually when this happens there is a penalty cost.

The problem formulation with backordering includes the following changes:

The cost for supplying all the demand for product k in period t from a

replenishment in period i depends whether there is backordering or not:

𝐶𝑘𝑖𝑡 =

{

(𝑐𝑘𝑖 +∑ℎ𝑘𝑗) · 𝑑𝑘𝑡(𝑖 ≤ 𝑡)

𝑡−1

𝑗=𝑖

(𝑐𝑘𝑡 +∑𝑝𝑘𝑗) · 𝑑𝑘𝑡 (𝑖 > 𝑡)

𝑖−1

𝑗=𝑡

(19)

The term 𝑝𝑘𝑗 is the penalty cost for product k in period j.

The sum domain of the constraint (15) is changed:

∑𝑥𝑘𝑖𝑡 = 1𝑘 = 1,2, … , 𝐾; 𝑡 = 1,2, … , 𝑇(20)

𝑇

𝑖=1

The domain of the constraint (16) is changed to consider 𝑡 = 1,2, … , 𝑇

2.5 The Exact Requirements formulation (ERF)

This model is presented by Boctor et al. (2004). It is based on the JON. The decision

variables are:

𝑥𝑘𝑖𝑡: is a binary variable which takes the value of 1 if and only if the demand

for product k from period i until period t is covered by an order in period i.

𝑍𝑡: is the major setup indicator. It takes the value of 1 if any product is

replenished in period t, otherwise it is equal to 0.

The cost for fulfilling the demand of product k until period t by a replenishment

made in period i is:

𝑐𝑘𝑖𝑡 = 𝑠𝑘𝑖 +𝑐𝑘𝑡 ·∑𝑑𝑘𝑟 + ∑ (∑ℎ𝑘𝜏

𝑟−1

𝜏=𝑖

)𝑑𝑘𝑟

𝑡

𝑟=𝑖+1

𝑡

𝑟=𝑖

(21)

The formulation of the model is:

min∑𝑆𝑡 · 𝑍𝑡 +∑∑∑𝑐𝑘𝑖𝑡 · 𝑥𝑘𝑖𝑡

𝑇

𝑡=𝑖

𝐾

𝑘=1

𝑇

𝑖=1

𝑇

𝑖=1

(22)

Subject to:

∑∑𝑥𝑘𝑖𝑡 = 1

𝑇

𝑡=𝜏

𝜏

𝑖=1

(𝑘 = 1,… , 𝐾; 𝜏 = 𝛿𝑘, … , 𝑇)(23)


demand 9

∑𝑥𝑘𝑖𝑡 ≤ 𝑍𝑖

𝑇

𝑡=𝑖

(𝑘 = 1,… , 𝐾; 𝑖 = 1,… , 𝑇)(24)

𝑥𝑘𝑖𝑡 , 𝑍𝑡 ∈ {0,1}(𝑘 = 1,… , 𝐾; 𝑖 = 1,… , 𝑇; 𝑡 = 1,… , 𝑇)(25)

The constraint (23) is the flow constraint. It ensures that each node (in this case a

period and product combination) is connected with another. The constraint (24)

ensures that the major setup cost is incurred whenever any product is replenished.

Note that constraint (24) could be replaced by:

∑∑𝑥𝑘𝑖𝑡

𝑇

𝑡=𝑖

𝐾

𝑘=1

≤ 𝐾 · 𝑍𝑖 (𝑖 = 1,… , 𝑇)(24′)

However, the disaggregate constraint (24) is better because it yields a tighter lower

bound provided by the solution in the LP relaxation what decreases significantly

the computational time (Narayanan & Robinson, 2006).

2.6 Properties of the optimal solution

Boctor et al. (2004) review the properties that any optimal solution of the JRPDD

must fulfil:

Property 1: Any optimal solution must fulfil that 𝑥𝑘𝑡∗ · 𝐼𝑘𝑡−1

∗ = 0 being 𝑥𝑘𝑡∗ the

quantity of products of type k replenished in period t and being 𝐼𝑘𝑡−1∗ the

inventory level of product k at the end of the previous period of t. This means

that if a product is replenished in a period it is not optimal to hold this product

in the inventory during the previous period.

Property 2: The order quantity at the beginning of a period is equal to the sum

of the demand for that item in a number of periods. In other words, the demand

of one item for a single period is fully replenished in a single period and not

split in different periods.

𝑥𝑘𝑡∗ ∈ {𝑑𝑘𝑡, 𝑑𝑘𝑡 + 𝑑𝑘,𝑡+1, … ,∑ 𝑑𝑘𝑞

𝑇

𝑞=𝑡}

(25)

Because of this property the variable 𝑥𝑘𝑖𝑡∗ of the R&G takes only values of 1 or

0, so it can be defined as binary. In this property are based the models JON and

ERF.

Property 3: the inventory level for a product takes one of the following values:

𝐼𝑘,𝑡−1∗ = {0, 𝑑𝑘𝑡, 𝑑𝑘𝑡 + 𝑑𝑘,𝑡+1, … ,∑ 𝑑𝑘𝑞

𝑇

𝑞=𝑡}

(26)

Property 4: If the cost of making an order at time t to supply the demand of

product k for period q (purchasing cost at time t and inventory holding costs


demand 10

until period q-1) is higher than the cost of purchasing it at time q including the

set-up costs (major and minor):

𝑑𝑘𝑞∑ℎ𝑘𝑟 + 𝑐𝑘𝑡 > 𝑆𝑞 + 𝑠𝑘𝑞 + 𝑐𝑘𝑞

𝑞−1

𝑟=𝑡

(27)

then is not optimal to replenish the demand of product k for period q in period

t.

The property 4 can be added as constraint to the BM, R&G and ERF to reduce the

number of variables.

2.7 Comparison of the models

As the structure of each model is different the number of variables and constraints

in each of them is different, though they have the same number of parameters. The

table 1 shows the number of decision variables and constraints that are generated

by each model:

Table 1: Number of decision variables and constraints of each model

Decision variables Constraints

BM T(3K+1) 3KT

JON 1

2KT(T+1)+T K(2T+1)-∑ 𝛿𝑘

𝐾𝑘=1

R&G 1

2KT(T+3)+T KT(2+

𝑇+1

2) +K-∑ 𝛿𝑘

𝐾𝑘=1

ERF 1

2KT(T+1)+T K(2T+1)-∑ 𝛿𝑘

𝐾𝑘=1

The numbers of constraints of the JON, the R&G and the ERF depend on 𝛿𝑘 while

their numbers of decision variables do not. This term is influenced by the demand

probability (a factor which is later explained). Given the demand probability an

estimation of 𝛿𝑘 can be calculated using the negative binomial distribution.

The number of constraints and decision variables of the JON and the ERF are

identical. This is because of the big similarity of their formulation. The numbers of

constraints and decision variables are bigger in the R&G than in the JON and ERF.

The computational time needed to solve a MILP problem is very difficult to

estimate. Generally solvers use the branch and bound (B&B) algorithm. This

algorithm starts solving the linear relaxation of the problem (i.e. deleting the

integrality restrictions). If this optimal solution does not violate any integrality

constraint, it is the optimal solution of the MILP problem. If not, an integer variable

𝑥𝑖 taking a fractional value in the linear relaxation solution 𝑥𝑖∗ is picked. Then the

initial problem is divided in two problems (B&B nodes). One includes the


demand 11

constraint 𝑥𝑖 ≤ ⌊𝑥𝑖∗⌋ and the other one includes the constraint 𝑥𝑖 ≥ ⌈𝑥𝑖

∗⌉. Then the

process is repeated with each node. Considering a minimization problem the upper

bound is determined by the feasible integer solutions and the lower bound by the

lowest linear relaxation solution among all current nodes. These bounds allow to

cut some nodes, avoiding the calculation of all of them.

Usually higher numbers of integral variables increase the computational time. The

more constraints there are, the longer time the calculations at each B&B node last.

However, the addition of constraints or variables sometimes has a beneficial effect.

It is the quality of the constraints what influences significantly. Constraints which

provide tight bounds are more useful reducing the computational times because they

allow to cut the branches of the branch and bound tree at an early stage and therefore

less nodes are needed.

Gao et al. (2008) made a comparison on the computational time between the JON

and the R&G. Their results show that the R&G requires less computational time to

obtain the optimal solution. Their study also shows the correlation between the

computational time and some factors of the problem: the computational time

increases with the number of product types, the number of periods and high setup

costs. The ratio of the major to minor setup costs produces different effects

depending on the model (high ratios increase the computational time of the JON

and decrease it in the R&G). They point out that the impact of increasing the number

of periods and products is bigger on the JON. In this model the computational time

increases exponentially with the number of products and periods while in the R&G

it increases linearly.

Boctor et al. (2004) compared the BM, the ERF and another model very similar to

the R&G. In this variant of the R&G the minor setup indicator is eliminated and the

constraints (16) and (17) are substituted by:

∑𝑥𝑘𝑡𝑡 ≤

𝐾

𝑘=1

𝐾 · 𝑍𝑡 (𝑡 = 1,… , 𝑇)

(16′)

𝑥𝑘𝑖𝑡 ≤ 𝑥𝑘𝑖𝑖 (𝑘 = 1,… , 𝐾; 𝑖 = 1,… , 𝑇 − 1; 𝑡 = 𝑖 + 1,… , 𝑇) (17′)

Their results show that the ERF is on average the fastest model. The dependency

on the factors of the problem (number of products, number of periods, setup costs

to total cost ratio and the major to the minor setup cost) is evidenced. Their results

show that the factor with the biggest effect is the problem scale (i.e. number of

products and periods).


demand 12

The computational times are significantly higher in the study of Boctor et al. (2004)

than in the study of Gao et al. (2008). For example, using the R&G the average

computational time for solving a 26-periods problem was 12.21 seconds on the

former study whereas the average time for a 48-periods problem was 1.87 seconds

on the latter study (even that the average of products is also higher in the later

study). This may be explained by the fact that Gao et al. (2008) used a more

powerful computer.

Narayanan and Robinson (2006) compared the R&G and the ERF. Their results

showed that the ERF is faster. These results also evidenced the influence of the

factor demand probability which is the probability that a product type experiences

demand in a period.

2.8 Research gap covered

All these studies have used relative small instances. The largest products-periods

problems are 20-26 in the Boctor et al. study (2004), and 40-48 in the Gao et al.

(2004) as well as in the Narayanan and Robinson (2006) studies. The aim of this

study is to cover the gap of large scale problems.

The evolution of solvers such as XPRESS, CPLEX and Gurobi by including

algorithms that speed-up the B&B procedure has enable to solve large scale MILP

problems (Lima & Grossmann, 2011; Bixby, et al. 2000). The improvements of the

computers have also contributed to this fact.

Concurrent with previous studies a factor analysis is performed. We introduce

another factor which has not been previously studied: the variance of the demand,

in this case measured by the index of dispersion. In total six factors are analyzed:

number of products, number of periods, TBO, MSR, demand probability and index

of dispersion of the demand.


demand 13

3 Methodology

3.1 Factors

The idea is to choose some variables used in the inventory management theories to

cover all the parameters contained in the problem.

Apart from the number of products and periods, the factors included in this study

are: time-between-order (TBO), major to minor setup ratio (MSR), demand

probability and the distribution of the demand. The first two cover the setup costs

while the other two are used to model the demand.

3.1.1 Time-between-order (TBO)

The TBO is a term of the EOQ-principles used in the JRP by some authors (Kirca,

1995; Gao et al., 2008).

The TBO is an indirect measure of the setup costs altogether. A high TBO value

means that the setup costs are large compared to the cost of the inventory holding

costs. It is defined as:

𝑇𝐵𝑂 = √𝐴 · �̅� · ℎ̅

2

(28)

A is the total of setup costs

�̅� is the mean demand per product and period

ℎ̅ is the mean inventory holding cost per unit and period

3.1.2 Major to minor setup cost ratio (MSR)

The MSR is a measure of the relation of the major and the minor setup costs. It

takes values between 0 and 1. The higher it is the larger the major setup cost is

compared to the minor. If it takes the value of 0, it means that there is no major

setup cost. On the other side, if it is equal to 1 it means that there are no minor setup

costs.

3.1.3 Demand distribution

To our concern no study has investigated how the demand variability through the

planning horizon impacts the computational time of this problem. As we wanted to

explore whether the variance of the demand influences the result, we tested three

statistical distribution forms to model the demand. These distribution forms are

Poisson, binomial and negative binomial. Using this distributions we can model the

demand to take integral values.


demand 14

These three distributions have been selected because they cover the whole range of

the index of dispersion. The index of dispersion, also called variance-to-mean ratio,

is a measure of the variability of a distribution form. The binomial distribution has

𝐼 < 1, the Poisson distribution has 𝐼 = 1 and the negative binomial distribution

has 𝐼 > 1.

From the index of dispersion the parameters of the binomial and negative binomial

distributions can be determined.

Poisson distribution: the probability that a specific demand value following the

Poisson distribution is 𝑑𝑘𝑡 = 𝑥 is given by the probability mass function:

𝑃(𝑥) =𝜆𝑥

𝑥!𝑒−𝜆 (𝑥 = 0, 1, 2, … ) (29)

The parameter λ is equal to the mean. This distribution form has the characteristic

that the variance is equal to the mean.

Binomial distribution: the probability that a specific demand value following the

binomial distribution is 𝑑𝑘𝑖 = 𝑥 is given by the probability mass function:

𝑃(𝑥) = (𝑛

𝑥)𝑝𝑥(1 − 𝑝)𝑛−𝑥 (𝑥 = 0,1,2, … , 𝑛) (30)

The parameters n and p must be determined from the mean and the variance

according to the following characteristics of the binomial distribution:

𝜇 = 𝑛𝑝 (31)

𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1 − 𝑝) (32)

Negative binomial: the probability that a specific demand value following the

negative binomial distribution is 𝑑𝑘𝑡 = 𝑥 is given by the probability mass function:

𝑃(𝑥) = (𝑛 + 𝑥 − 1

𝑥)𝑝𝑛(1 − 𝑝)𝑥 (𝑥 = 0,1,2, … , 𝑛) (33)

The parameters n and p must be determined from the mean and the variance

according to the following characteristics of the binomial distribution:

𝜇 =𝑛(1 − 𝑝)

𝑝

(34)

𝑉𝑎𝑟(𝑋) =𝑛(1 − 𝑝)

𝑝2

(35)


demand 15

The procedure used to generate random variables of these distributions for the

experimental study is explained in the Appendix A.

3.1.4 Demand probability

The demand probability, also known as demand density, is the fraction of periods

in which a product experiences demand. This factor is used in the inventory

management to model the demand (Bagchi et al., 1984).

As mentioned before, its influence in the computational time of the R&G and the

ERF has been proven (Narayanan & Robinson, 2006). It has a direct influence in

the number of constraints in the models of R&G, JON and ERF according to table

1. The lower it is, the fewer constraints there are.

3.2 Addition of Property 4

The property 4 of any optimal solution (before explained) have been added to the

models of BM, R&G and ERF to investigate whether it does impact the

computational times. It decreases the number of decision variables of the R&G and

the ERF and it provides a tight constraint for the BM.

To implement this constraint in the BM a binary variable is needed:

𝑧𝑘𝑡𝑞: is equal to 1 for the first period q greater than the period t in which the

inequality 𝑑𝑘𝑞∑ ℎ𝑘𝑟 + 𝑐𝑘𝑡 > 𝑆𝑞 + 𝑠𝑘𝑞 + 𝑐𝑘𝑞𝑞−1𝑟=𝑡 is fulfilled. Otherwise it is

equal to zero. Therefore 𝑧𝑘𝑡𝑞 takes the value of 1 at maximum once for a

product k and a period t.

Then the constraint added is:

𝑥𝑘𝑡 ≤ ∑ (∑ 𝑑𝑘𝜏

𝑞−1

𝜏=𝑡

)𝑧𝑘𝑡𝑞 + (1 −∑ 𝑧𝑘𝑡𝑞)𝑀

𝑇

𝑞=𝑡

𝑇

𝑞=𝑡+1

(𝑘 = 1,… , 𝐾; 𝑖 = 1,… , 𝑇)

(36)

The term ∑ (∑ 𝑑𝑘𝜏𝑞−1𝜏=𝑡 )𝑧𝑘𝑡𝑞

𝑇𝑞=𝑡+1 enforces 𝑥𝑘𝑡 to take a value lower or equal to the

demand for product k from period t until period q-1, being q the first period greater

than t in which the inequality 𝑑𝑖𝑞∑ ℎ𝑖𝑟 + 𝑐𝑖𝑡 < 𝑆𝑞 + 𝑠𝑖𝑞 + 𝑐𝑖𝑞𝑞−1𝑟=𝑡 is fulfilled. The

term (1 − ∑ 𝑧𝑖𝑡𝑞)𝑀𝑇𝑞=𝑖 ensures that if there is no period q greater than t in which

the inequality mentioned is fulfilled the constraint does not apply. M is a large

number (e.g. overall sum of the demand for all products in all periods).

To apply this property to the R&G the following binary variable 𝑧𝑘𝑖𝑡 is needed:

𝑧𝑘𝑖𝑡 is 0 if 𝑑𝑘𝑡∑ ℎ𝑘𝑟 + 𝑐𝑘𝑖 > 𝑆𝑡 + 𝑠𝑘𝑡 + 𝑐𝑘𝑡𝑡−1𝑟=𝑖 , otherwise is 1.

Then the following constraint is added:


demand 16

𝑥𝑘𝑖𝑡 ≤ 𝑧𝑘𝑖𝑡 (𝑘 = 1,… ,𝑁; 𝑖 = 1, … , 𝑇; 𝑡 = 𝑖, . . . , 𝑇) (37)

The same formulation has been used for the ERF. The models including this

property are referred as Basic Model modified (BM-modified), Robinson and Gao

Model modified (R&G-modified) and Exact Requirements Formulation modified

(ERF-modified).

3.3 Experimental process

Doing a full factorial experimental design with large scale problems would take an

enormous quantity of time. Therefore, a preliminary experiment with small scale

problems has been conducted. Based on the results of this preliminary experiment

the experiment with the large scale problems was designed. First, the design and

the results of the preliminary experiment are explained and then the design and the

results of the experiment with large scale problems are explained.

Either in the preliminary and in the large scale problems experiment the tools used

were the same. The optimization software package used is Xpress 7.9 and the

models were implemented using the Mosel language. This optimization software

uses the B&B algorithm to solve MILP problems. It also uses a presolve procedure

to reduce the size of the problem, heuristics and concurrent solve with dual, primal

and barrier algorithms in order to tight the upper and the lower bounds. To extract

conclusions from the output data the software packages Matlab R2015b and

Microsoft Excel 2013 were used.

The computational resources are measured primarily by the computational time that

takes a model to obtain the optimal solution. If the optimal solution is not reached

Figure 2: Experimental process


demand 17

within the permitted time, then the quality of the best feasible solution provided is

evaluated with the optimality gap. Also the computational memory requirements of

the different models have been compared. This is valuable in case of having a

computer with insufficient memory capacity to solve a certain problem.


demand 18

4 Preliminary experiment

4.1 Preliminary experiment design

This experiment is a replication of the one conducted by Gao et al. (2008). The

same scale problems are tested, as well as the TBO and MSR values. However, our

experiment also includes other factors: the demand distribution and the demand

probability.

The experiment design is a full factorial in which the parameters take the following

values:

Products: 5, 10, 20, 40

Periods: 12, 24, 36, 48

TBO: 1.5, 2.5, 4.5

MSR: 0.3, 0.6, 0.9

Index of dispersion of the demand (distribution): 0.5 (binomial), 1

(Poisson), 1.5 (negative binomial)

Demand probability: 0.35, 0.75, 1

The inventory holding cost of one unit per period and the purchasing cost of one

unit are both set to 1 for all the products and periods.

The products are divided in three categories: low demand-, medium demand-, and

high demand-products, with mean demand 5, 100 and 200 respectively. Despite

having different mean demands, all the products have the same index of dispersion

of the demand (consequently the same distribution) in a problem.

The mean setup costs are generated from the TBO and the MSR. First the mean

values are calculated using the following equations:

𝐴𝑘 =𝑇𝐵𝑂2 · 𝑑𝑘̅̅ ̅ · ℎ𝑘̅̅ ̅

2 (38)

�̅�𝑘 = (1 −𝑀𝑆𝑅)𝐴𝑘

(39)

𝑆̅ = 𝑀𝑆𝑅 · ∑𝐴𝑘

𝐾

𝑘=1

(40)

Then the major and minor setup costs for each period and product are randomly

generated following a normal distribution where the standard deviation is set to

10% of the mean:

𝑆𝑡~𝑁(𝑆̅, 0.1𝑆̅) (𝑡 = 1,… , 𝑇) (41)


demand 19

𝑠𝑘𝑡~𝑁(�̅�𝑘, 0.1�̅�𝑘) (𝑘 = 1, … , 𝐾; 𝑡 = 1,… , 𝑇) (42)

According to these equations, major setup costs vary across the planning horizon

and the minor setup costs vary across the products and the planning horizon.

For each factor combination two problems were generated and solve to optimality

by the BM, JON, R&G, ERF, BM-modified, R&G-modified and ERF-modified. In

total 2592 problems. The experiment was conducted using a computer ACER

Intel® Core™ i3-2310M CPU 2.10GHz with installed memory (RAM) 4.0GB.

4.2 Preliminary experiment results

Among the 2592 problems, 36 could not be solved by the BM and the BM-modified

because the computer run out of memory. All these problems have in common high

number of products (40), periods (48) and TBO (4.5) and low value of MSR (0.3).

The rest of the models could yield the optimal solution in all the problems.

The figure 3 presents the percentages of problems that each model solved faster

than all the others.

Figure 3: Percentage of problems that a model was faster than the others

The R&G and its modified variant together yielded the optimal solution in the

shortest time in the 80.05% of the problems. The BM yielded the optimal solution

in the shortest time in the 14.47% of the problems, all of them with low or medium

values of TBO (1.5 or 2.5). The ERF and its modified variant are the fastest models

in only the 0.31% and 0.27% respectively of the problems, being the model that is

the least times the fastest.


demand 20

Table 2: Average computational times

Factor Level BM JON R&G ERF BM-

Modified

R&G-

Modified

ERF-

Modified

Index of

dispersion

0.5 3.82 1.76 0.87 2.72 4.59 0.83 2.56

1 4.56 1.89 0.95 3.07 4.41 0.96 2.83

1.5 6.34 2.19 0.86 3.38 4.92 0.86 3.21

Demand

probability

0.35 4.38 2.67 0.97 4.36 4.51 1.00 4.20

0.75 6.53 1.93 1.14 2.75 5.42 1.07 2.62

1 3.83 1.23 0.57 2.06 4.00 0.57 1.79

TBO

1.5 0.47 1.00 0.46 1.97 0.57 0.42 1.58

2.5 2.33 2.22 1.09 3.15 2.50 1.04 2.96

4.5 12.24 2.61 1.14 4.04 11.12 1.18 4.06

MSR

0.3 4.49 3.17 1.70 5.06 4.78 1.64 4.74

0.6 7.42 1.29 0.50 2.07 6.13 0.51 1.93

0.9 2.81 1.37 0.49 2.04 3.03 0.50 1.94

Products

5 0.25 0.21 0.13 0.47 0.28 0.11 0.35

10 0.88 0.51 0.28 1.07 0.93 0.27 0.89

20 4.55 1.41 0.75 2.43 4.36 0.73 2.17

40 14.50 5.64 2.41 8.26 13.48 2.41 8.07

Periods

12 0.37 0.17 0.09 0.19 0.38 0.09 0.19

24 2.17 0.95 0.60 1.18 2.24 0.58 1.17

36 3.67 2.48 1.06 3.47 4.27 1.04 3.34

48 13.65 4.18 1.82 7.38 11.89 1.81 6.78

Total Average 4.91 1.94 0.89 3.05 4.64 0.88 2.87

*The CPU-time is measured in seconds

Table 3: Maximun computational times

Factor Level BM JON R&G ERF BM-

Modified

R&G-

Modified

ERF-

Modified

Index of

dispersion

0.5 253 66 40 177 482 40 168

1 283 175 66 240 271 80 250

1.5 1560 203 45 328 354 41 331

Demand

probability

0.35 283 203 66 328 482 80 331

0.75 1560 133 45 123 354 44 129

1 243 10 4 15 303 5 14

TBO

1.5 4 9 3 15 4 3 14

2.5 63 133 45 141 118 41 148

4.5 1560 203 66 328 482 80 331

MSR

0.3 147 203 66 328 303 80 331

0.6 1560 15 7 17 482 7 17

0.9 150 11 4 16 145 4 14

Products

5 3 1 1 2 3 1 2

10 11 5 6 6 13 4 6

20 147 31 33 58 207 26 66

40 1560 203 66 328 482 80 331

Periods

12 5 1 1 3 5 2 2

24 99 28 26 56 75 30 50

36 122 97 30 141 303 30 148

48 1560 203 66 328 482 80 331

Maximun 1560 203 66 328 482 80 331


demand 21

In the table 2 the average values of the computational times are presented according

to each factor level. Note that the problems which could not be solved by the BM

and the BM-modified are not included in their averages.

The table 3 shows the maximum computational times for solving a problem

incurred by the different models according to each factor level. The information of

these tables can be seen graphically in the Appendix B. However, to extract valid

conclusions the regression analysis has been done.

The R&G is the fastest model on average and it also presents the smallest maximum

computational time. The BM model is on average the slowest model and it presents

the biggest maximum computational time.

For a given set of products and periods (40 products and 48 periods) different

combinations of TBO and MSR have been analyzed to study their interaction effect.

Figure 4 presents five different TBO-MSR combinations. For each combination the

mean values of the computational time have been calculated with a sample of 18

problems. The combination of TBO=4.5 and MSR=0.3 has not been included for

the BM and BM-modified as these models were not able to solve those problems.

It is this combination which produces the highest increase in the computational time

of all the models. The R&G presents the lowest computational times for all these

combinations.

The addition of the Property 4 has a positive effect although not critical to make a

model more efficient than another. Its impact is more significant in the BM, as the

BM-modified presents a big difference in the maximum computational time

0

10

20

30

40

50

60

70

Com

puta

tional

tim

e (s

)

TBO=4,5; MSR=0,3 TBO=1,5; MSR=0,3 TBO=2,5; MSR=0,6

TBO=4,5;MSR=0,9 TBO=1.5; MSR=0.9

Figure 4: Interaction effect between the TBO and MSR


demand 22

incurred to solve a problem compared to the BM. The differences are not so relevant

in the R&G and the ERF because their original formulations are already very tight.

The mean and the variance values of the computational time of the BM, the R&G

and the ERF and their modified versions are presented in the table 4.

Table 4: Comparison of the models with their modified version

Computational time BM R&G ERF

Mean 4.91

(4.53)

0.89

(0.88)

3.05

(2.87)

Variance 1286.26

(439.42)

10.35

(10.66)

146.79

(147.40)

Observations 2556 2592 2592

*Values in seconds

*The values in the parenthesis belong to the modified versions

The results show contradictory results than in the study of Narayan and Robinson

(2006). They state that the ERF is more efficient than the R&G, whereas our results

show that the R&G is significantly more efficient, even that we have used the

disaggregate constraint in the ERF as they suggest. To further investigate this,

another experiment very similar to theirs was conducted. The results confirm that

R&G is more efficient than the ERF. The explanation of this divergence in the

results is that the software Xpress 7.9 has predetermined a procedure to presolve

the problems, which in the case of the ERF increase substantially the computational

time whereas it does not have such a substantial impact in the other models. This

procedure consist of reducing the scale of the problem and solving it. If this

procedure is deactivated, the results agree with those of Narayanan and Robinson

(2006) showing that the ERF’s computational time is slightly smaller than R&G’s

computational time.

4.3. Regression analysis

The purpose of the regression analysis is to identify which factors are significant

and which have the biggest influence in the computational times. For each model

the influence of each factor have been studied. As the regression model was not

known beforehand, three types have been studied: linear, exponential and quadratic.

It can be that different factors influence the computational times in different modes.

First the correlation between the factors and the computational times is studied. The

natural logarithm and the square root of the computational times have been included

in this analysis. The correlation analysis gives a sense of how each factor influence

the computational time (directly or indirectly and the strength of this influence).


demand 23

The results show that some of the factors have a very small influence on the

computational time related to the others (low correlation values and low regression

coefficients multiplied by the magnitude order of their associated factor). The

factors that explain most of the variability in the three models are the number of

products and the number of periods. It makes sense because these factors determine

the size of the problem (number of variables and constraints). An increase in the

number of products or in the number of periods produces an exponential increase

in the computational time of all the models. The high values of TBO increase the

computational times, with its most notorious effect on the BM. The MSR is

negatively correlated with the computational times. The index of dispersion of the

demand shows a small increasing effect on the computational time. The demand

probability produces different effects depending on the model. High values increase

the computational time in the R&G and BM while decrease it in the JON and ERF.

Note that the regression coefficients presented below depend on the characteristics

of the computer. A more powerful computer will present lower regression

coefficients. However, the proportion between the coefficients would be similar.

The regression analysis of the modified versions of the models can be found in the

Appendix C.

4.3.1. JON computational time regression analysis

The results of the multifactor linear regression analysis for the computational times

obtained with the JON show that the adjusted R square coefficient is 0.135. For this

regression model all the factors are significant, except from the index of dispersion

(p-value=0.213).

Other regression models, like exponential or quadratic, fit better. The correlation

between the factors and computational time has been studied. In the correlation

table 5, “JON” represents the computational time, “ln(JON)” is the natural

logarithm of the computational time and “sqrt(JON)” is the square root of the

computational time. This table gives a broad overview of how each factor

influences the computational time. Values closer to 1 indicate a strong correlation

between high levels of that factor and the largest computational times. On the other

hand, negative correlation values indicate that lowest computational times are

associated with the high levels of that factor. Among the three forms in which the

computational time is expressed, the one with the highest absolute correlation value

for a factor indicates whether that factor influences linearly, exponentially or in a

quadratic form the computational time.


demand 24

Table 5: Correlation table of the JON computational time with the factors

Products Periods TBO MSR

Index of

dispersion

Demand

Probability

JON 0.2773 0.1969 0.0790 -0.0959 0.0228 -0.0756

ln(JON) 0.6530 0.6615 0.0579 -0.0197 0.0283 -0.0109

sqrt(JON) 0.5681 0.4633 0.0947 -0.0743 0.0242 -0.0592

According to these results, the computational time varies exponentially with the

number of products, the number of periods and the index of dispersion, in a

quadratic form with the TBO and linearly with the MSR and the demand

probability. The measure of the effect is clearly higher for the number of products

and periods than for the rest of the factors, being the index of dispersion of the

demand the factor with the least effect.

Selecting a multifactor exponential regression model, the following results are

obtained:

Table 6: Exponential regression analysis of the JON computational time

Regression statistics

Multiple R 0.932

R Square 0.869

Adjusted R Square 0.868

Standard error 0.587

Observations 2592

ANOVA

Degrees of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F

Regression

statistics 6 5886.05 981.01 2849.39 0

Residual 2585 889.98 0.34

Total 2591 6776.04

Coefficients

Standard

error t Stat p-value Lower 95%

Upper

95%

Intercept -4.734 0.065 -72.666 0 -4.862 -4.606

Products 0.079 0.001 91.606 0 0.077 0.080

Periods 0.080 0.001 92.806 0 0.078 0.081

TBO 0.075 0.009 8.116 0.000 0.057 0.093

MSR -0.130 0.047 -2.762 0.006 -0.222 -0.038

Index of

dispersion 0.112 0.028 3.975 0.000 0.057 0.168

Demand

Probability -0.066 0.043 -1.525 0.127 -0.150 0.019


demand 25

This model fits to the experiment results much better than the linear one (adjusted

R square=0,868). For this model, the index of dispersion of the demand is

significant while the demand probability is not (p-value=0,127). Large p-values

indicate that the probability of that coefficient being null is high. Although being

significant, the influence of the index of dispersion of the demand and the MSR is

small. For example, according to this regression model if there were two problems

with the only varying factor being the index of dispersion, one of them with index

of dispersion of 1.5 and the other one with 0.5; the computational time of the

former would be a 12% larger than the latter’s computational time. If the only

varying factor was the MSR and the problems took the extreme values of 0.3 and

0.9 respectively, the difference in the computational times would be 8% (these

calculations are explained in the Appendix D). Almost all the adjusted R square

coefficient in this model is determined by the number of products and periods

(99,4%).

The number of products and the number of periods have nearly the same influence

in the computational times, as their regression coefficients are very similar (0.079

and 0.080 respectively).

4.3.2. BM computational time regression analysis

The correlation values between the factors and the computational time for this

model are shown in the table 7:

Table 7: Correlation table of the BM computational time with the factors


Index of

dispersion

Demand

Probability

BM 0.155 0.129 0.141 -0.020 0.029 -0.002

ln(BM) 0.611 0.475 0.465 -0.009 0.035 0.044

sqrt(BM) 0.420 0.323 0.354 -0.038 0.033 0.003

From this table it is determined that the computational time varies exponentially

with the number of products, the number of periods, the TBO, the index of

dispersion and the demand probability; and in a quadratic form with the MSR.

However, the correlation values for the MSR, the index of dispersion and the

demand probability are very low.

Again the exponential regression model (adjusted R square = 0.859) fits better than

the linear regression model (adjusted R square = 0.063) and the quadratic regression

model (adjusted R square = 0.430). Using the exponential regression model, the

following results are obtained:


demand 26

Table 8: Exponential regression analysis for the BM computational time


Multiple R 0.927

R Square 0.860



Observations 2556

ANOVA

Degrees of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F

Regression

statistics 6 7501.368 1250.228 2599.39 0

Residual 2549 1225.994 0.481

Total 2555 8727.362

Coefficients

Standard

error t Stat p-value

Lower

95%

Upper

95%

Intercept -6.421 0.077 -82.916 0.000 -6.572 -6.269

Products 0.089 0.001 85.747 0.000 0.087 0.091

Periods 0.069 0.001 67.015 0.000 0.067 0.071

TBO 0.737 0.011 66.569 0.000 0.715 0.759

MSR -0.349 0.056 -6.197 0.000 -0.459 -0.238

Index of

dispersion 0.159 0.034 4.744 0.000 0.094 0.225

Demand

Probability 0.301 0.051 5.882 0.000 0.201 0.402

All the factors are significant in this regression analysis. The influence on the

computational time of the number of products is bigger than the influence of the

number of periods. The computational time increase with the demand probability,

unlike in the JON.

The factors products and periods together account for the 71.1% of the R square

coefficient of the exponential regression model, while products, periods and TBO

together account for the 99.4%. This reflects the important effect of the TBO on the

computational time in this model.

The results for the BM-modified are very similar. The exponential regression

coefficient of the number of products is slightly lower for the BM-modified

(0.086) than for the BM (0.089), while the exponential regression coefficient of

the number of periods is slightly higher for the BM-modified (0.070) than for the

BM (0.069). These small differences make big effects when the number of

products or periods is high enough, as they are exponential coefficients. The

influence of the TBO is smaller for the BM-modified (coefficient of 0.673) than

for the BM (coefficient of 0.737). According to this comparison, it is better to use


demand 27

the BM-modified when the number of products is larger than the number of

periods and when the TBO is high.

4.3.3. R&G computational time regression analysis

The table 9 shows the correlation values between the factors and the computational

time expressed in different forms:

Table 9: Correlation table of the R&G computational time with the factors


Index of

dispersion

Demand

Probability

R&G 0.276 0.194 0.077 -0.155 -0.001 -0.046

ln(R&G) 0.572 0.666 0.051 -0.078 0.026 0.111

sqrt(R&G) 0.486 0.436 0.081 -0.151 0.013 0.016

According to these correlation values, the computational time varies exponentially

with the number of products, the number of periods, the index of dispersion and the

demand probability; with a quadratic form with the TBO and linearly with MSR.

The correlation value of the index of dispersion is very low. Similar to the BM, the

demand probability increases the computational time, unlike in the JON.

Again the exponential regression model (adjusted R square = 0.809) fits better than

the linear regression model (adjusted R square = 0.146) and the quadratic regression

model (adjusted R square = 0.470). The results of the exponential regression

analysis are shown below:

Table 10: Exponential regression analysis of the R&G computational time


Multiple R 0.899

R Square 0.809



Observations 2559

ANOVA

Degrees of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F

Regression

statistics 6 4730.464 788.411 1801.709 0

Residual 2552 1116.731 0.438

Total 2558 5847.195

Coefficients

Standard


Upper

95%

Intercept -5.404 0.075 -72.362 0.000 -5.550 -5.257

Products 0.066 0.001 67.603 0.000 0.064 0.068

Periods 0.077 0.001 78.357 0.000 0.075 0.079

TBO 0.069 0.010 6.556 0.000 0.048 0.089


demand 28

MSR -0.478 0.053 -8.967 0.000 -0.583 -0.374

Index of

dispersion 0.105 0.032 3.275 0.001 0.042 0.167

Demand

Probability 0.680 0.049 13.898 0.000 0.584 0.776

All the factors are significant. In this case, the influence on the computational time

of the number of periods is bigger than the influence of the number of products.

The number of periods and the number of products together account for the 96.4%

of the adjusted R square coefficient of the exponential regression model.

The results of the R&G-modified are very similar. The exponential regression

coefficient of the number of products is higher in the R&G-modified (0.071) than

for the original version (0.066), while the exponential regression coefficient of the

number of periods is nearly the same for the R&G-modified (0.076) than for the

original version (0.077). The influence of the demand probability is smaller for

the modified version (exponential regression coefficient of 0.607) than for the

original one (0.680).

4.3.4. ERF computational time regression analysis

In the ERF the biggest correlation between the computational time and a factor is

shown for the number of periods. The number of products presents also a strong

correlation with the logarithm of the computational time. It is interesting to note

that in this model the correlation between the TBO and the computational time is

very low. The demand probability shows a negative correlation with it, like in the

JON.

Table 11: Correlation table with the ERF computational time with the factors


Index of

dispersion

Demand

Probability

ERF 0.251 0.220 0.067 -0.102 0.022 -0.079

ln(ERF) 0.551 0.781 0.027 -0.040 0.030 -0.013

sqrt(ERF) 0.495 0.569 0.057 -0.095 0.030 -0.061

The regression type which best fits the ERF computational time is the exponential,

with an adjusted R square of 0.917, while the linear and the quadratic types yield

lower adjusted R square coefficients (0.131 and 0.584, respectively). The table 12

presents the results of the regression analysis using an exponential type. All the

factors are significant using α=0.05. The coefficient of the periods (0.098) is larger

than in any other model. The coefficient of the products is lower than in the JON

and the BM, but larger than in the R&G. The rest of the factors have little impact


demand 29

in the computational time. The number of products and periods together account

for the 99.7% of the adjusted R square.

Table 12: Exponential regression analysis of the ERF computational time

The ERF-modified presents larger coefficients for the number of products (0.074),

the TBO (0.111), the MSR (-0.151) and the index of dispersion (0.131), whereas

the coefficients of the number of periods (0.092) and the demand probability

(-0.287) are lower.


Multiple R 0.958

R Square 0.917



Observations 2592

ANOVA

Degrees of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F

Regression

statistics 6 6785.75 1130.96 4785.61 0

Residual 2585 610.90 0.24

Total 2591 7396.65

Coefficients

Standard


Lower

95%

Upper

95%

Intercept -4.495 0.054 -83.280 0.000 -4.601 -4.389

Products 0.069 0.001 97.483 0.000 0.068 0.071

Periods 0.098 0.001 138.220 0.000 0.097 0.100

TBO 0.036 0.008 4.702 0.000 0.021 0.051

MSR -0.276 0.039 -7.091 0.000 -0.353 -0.200

Index of

dispersion 0.124 0.023 5.310 0.000 0.078 0.170

Demand

Probability -0.083 0.036 -2.340 0.019 -0.153 -0.014


demand 30

5. Large scale problems experiment

5.1. Experimental design

The experimental design is a full factorial with 4 factors varying: products, periods,

TBO and demand probability. The TBO has an important effect on the BM. In the

preliminary experiment low values of TBO made the BM the fastest model many

times. The demand probability has been included in the factorial design because it

causes different effects depending on the model becoming a critical factor in certain

situations. The index of dispersion of the demand and the MSR have not been

included in the factorial design. Including them with only 2 factor levels would

have increased the time of the experiment by four times longer. Although they have

significant effects on the computational times, these effects are very similar in all

the models and small compared to the other factors’ effects. The factorial factors

take the following values:

Products: 100, 500, 800, 1000, 1500

Periods: 50, 100, 150

TBO: 1.5, 4.5

Demand probability: 0.35, 0.75

The MSR is fixed to a medium level of 0.6 and the index of dispersion of the

demand to 1 (Poisson distribution). The inventory holding cost of one unit per

period and the replenishment cost of one unit are both set to 1 for all the products

and periods.

For each factor combination two replications were done. This sums a total of 120

problems.

The time limit is set to one hour of computational time.

The models tested in this experiment are the BM-modified, R&G and JON. The

ERF was not included because in the preliminary experiment presented an average

computational time considerably higher than the R&G and the JON and it yielded

the optimal solution in the shortest time only the 0.57% of the times. Although

disabling the presolve procedure of the Xpress reduced significantly the

computational time of this model in the preliminary experiment, the impact of the

presolve time when the scale of the problems is large is not so relevant. To test this

fact, 26 problems of the above mentioned were chosen randomly and solved by the

ERF (with the presolve procedure disabled) and the R&G. The ERF only

outperformed the R&G in two problems. The average computational time was 3.2

times larger in the ERF than in the R&G. Moreover, because of its very similar


demand 31

structure to the JON, it makes sense to include only one of them. The BM was

included because in the preliminary experiment it yielded the optimal solution in

the shortest time the 15.82% of the times. The aim of including it was to test whether

there is any factor combination which makes it outperform the others. It was used

its modified version as its computational time presented smaller regression

coefficients for the number of products and TBO.

First, a subset of 72 problems was tested. These problems exclude the instances

with number of products 800 and 1500. These problems were solved using the BM-

modified, the R&G and the JON. The results of this first subset showed poor

performance of the BM-modified. Therefore, it was not tested in the following

subsets of problems.

Secondly, the subset of 48 problems remaining (the factorial combinations with the

number of products taking the values of 800 and 1500) was tested using the R&G

and the JON.

The results of the R&G were generally the best. However, there were some cases

where it did not reach neither the optimal solution nor a feasible solution within the

allowed time while the JON presented better results. These problems coincide in

having large number of periods, products, high TBO and high demand probability.

The demand probability is a critical factor these times because for similar factor

combinations with low demand probability (0.35) the R&G largely outperforms the

JON. To further investigate this case additional tests were conducted. This time

expanding the time limit to two hours. The products-periods combinations used

were: 800-150, 1000-150 and 1500-100. All of them with TBO equal to 4.5, MSR

equal to 0.6 and index of dispersion equal to 1. The demand probability took the

values of 0.75 and 1. For each factor combination four problems were generated (in

total 24) and solved by the R&G and JON.

This large scale problems experiment was performed using a computer Intel®

Xeon® CPU E7-4830 v3 2.10GHz with installed memory (RAM) 32.0 GB.

5.2. Results

The results of the first subset of problems show that the R&G yielded the optimal

solution faster than the other two models 91.7% (66/72) of the cases. The JON

yielded the optimal solution faster than the other two models 4.2% (3/72) of the

cases. The BM only yielded the optimal solution faster than the other two models

once. In the remaining two cases none of the models yielded the optimal solution

within the time allowed.


demand 32

The BM could yield the optimal solution within the allowed time only for

combinations of low quantity of products and periods and low TBO. The optimality

gap of the instances not solved to optimality by this model is on average 43.2%,

increasing linearly with the number of products, periods and TBO. The poor results

of the BM compared to the other two models make it totally inefficient for problems

with large quantities of products and periods.

The results of the second subset of problems, which were only solved by the R&G

and JON, show that the R&G yielded the optimal solution faster than the JON the

85.4% (41/48) of the times. In this subset of problems there were 4 which were not

solved to optimality within the allowed time by neither the R&G nor the JON.

The table 13 presents the number of problems that each model yielded the optimal

solution (or a feasible solution with optimality gap lower than 0.1%), the average

optimality gap of the problems when it could not yield that optimal solution (or a

feasible solution with optimality gap lower than 0.1%) and the number of problems

which were solved faster than the other models.

Table 13: General comparison of the BM, the JON and the R&G

BM R&G JON

Problems solved to optimality* 19/72 114/120 103/120

Average optimality gap 43.2% 83.5% 55.8%

Fastest model* 1/72 107/120 6/120

*Number of problems/Total number of problems tested with a model

*𝑂𝑝𝑡𝑖𝑚𝑎𝑙𝑖𝑡𝑦𝑔𝑎𝑝 = |𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒𝑣𝑎𝑙𝑢𝑒𝑜𝑓𝑏𝑒𝑠𝑡𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛−𝐿𝑜𝑤𝑒𝑟𝑏𝑜𝑢𝑛𝑑

𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒𝑣𝑎𝑙𝑢𝑒𝑜𝑓𝑏𝑒𝑠𝑡𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛|

In the 102 cases where both the JON and the R&G yielded the optimal solution, the

JON required on average 2.9 times more computational resources than the R&G

(𝐽𝑂𝑁𝑡𝑖𝑚𝑒−𝑅&𝐺𝑡𝑖𝑚𝑒

𝑅&𝐺𝑡𝑖𝑚𝑒).

The table 14 presents the detailed results for each product-period combination

studied, including the mean computational times and the number of branch and

bound nodes of the problems solved to optimality by the models R&G and JON.

In 87.7% (100/114) of the times that the R&G yielded the optimal solution it was

reached in the first B&B node. The maximum number of branch and bound nodes

that an instance needed using this model was 35. In 92.2% (95/103) of the times

that the JON yielded the optimal solution it was reached in the first B&B node. The

maximum number of B&B nodes that an instance needed using this model was 37.

These numbers of nodes are very low when compared to those of the BM, which

needed in average 6986. These results support the fact that the JON and the R&G


demand 33

provide an extremely tight linear relaxation (lower bound). Indeed, these models

provide the same lower bound. This fact was previously pointed out by Gao et al.

(2003).

Table 14: Computational time and branch and bound nodes of the R&G and the JON

depending on the scale of the problems

R&G JON

𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠

× 𝑃𝑒𝑟𝑖𝑜𝑑𝑠

Time* B&B nodes Time* B&B nodes

100 × 50 0.1 1 0.1 1

100 × 100 0.3 2 0.7 2

100 × 150 0.7 1 2.1 1

500 × 50 0.6 1 1.7 1

500 × 100 6.3 3 14.1 7

500 × 150 11.4 2 22.8 1

800 × 50 1.2 1 3.5 1

800 × 100 12.8 7 22.3 2

800 × 150 25.1 1 45.7 1

1000 × 50 2.0 1 5.2 1

1000 × 100 19.2 2 31.5 1

1000 × 150 27.4 1 58.0 1

1500 × 50 3.8 1 10.8 1

1500 × 100 22.2 1 49.6 1

1500 × 150 31.0 1 - -

*CPU-time expressed in minutes.

This values are the average of the problems solved to optimality or optimality gap

lower than 0.1%.

(-)Means that none of the problems was solved to optimality.

For a given factor combination, those solutions which were found in the first B&B

node required considerably less computational time than the others which needed

more than one node. The figures 4 and 5 show the average computational time for

a given set of products-periods comparing those problems which were solved in the

first B&B node with those that needed more than one node to be explored. As the

number of constraints in this problems is very large the calculations at each node

are heavy. The tight lower bound provided by the linear relaxation in the models of

R&G and JON avoid exploring a large number of nodes. This fact makes the BM

inefficient compared to the R&G and JON, as its linear relaxation does not provide

a tight lower bound needing a larger amount of nodes to be explored.


demand 34

Figure 5: JON computational time depending on the number of nodes

Figure 6: R&G computational time depending on the number of nodes

Despite solving to optimality most of the problems within the time limit and doing

it generally faster than the other models, there are five cases in which the R&G

could not yield the optimal solution neither a feasible solution. In this cases the

optimality gap is equivalent to 100%, what explains the high average shown in

table13. In some of these cases the JON yielded a the optimal solution or at least a

feasible solution. As mentioned before, this cases were further analyzed by

conducting tests on the factor combination of 800-150, 1000-150 and 1500-100

products periods; 0.75-1 demand probability and high TBO (4.5). The index of

dispersion of the demand and the MSR are the same as in the previous tests (1 and

0.6, respectively). For each factor combination four problems were generated (in

total 24) and solved by the R&G and JON. The time limit was expanded to 2 hours.

00

10

20

30

40

50

100x100 500x50 500x100 1000x100

CP

U-t

ime

(min

)

Products x Periods

R&G

1 node More than one node

00

05

10

15

20

25

30

100x100 500x50 500x100C

PU

-tim

e (m

in)

Products x Periods

JON

1 node More than one node


demand 35

The computational times are presented in the table 15. Due to the expansion of the

time limit all the problems were solved to optimality.

Table 15: Computational times of the R&G and JON for very large scale problems

𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠 × 𝑃𝑒𝑟𝑖𝑜𝑑𝑠 R&G* JON*

800 × 150 46.9

(57.7)

39.3

(35.1)

1000 × 150 95.8

(77.1)

62.8

(52.0)

1500 × 100 54.4

(43.2)

57.8

(47.4)

*CPU-times are expressed in minutes

Number without brackets is the mean CPU-time with demand probability p=0.75;

number with brackets is the mean CPU-time with p=1

The JON outperforms the R&G when the number of periods is 150 and the number

of products is also large, given high TBO and demand probability. The R&G

consumes 45.6% more computational resources in these cases than the JON

(𝑅&𝐺𝑡𝑖𝑚𝑒−𝐽𝑂𝑁𝑡𝑖𝑚𝑒

𝐽𝑂𝑁𝑡𝑖𝑚𝑒). In these circumstances, increasing the number of products

produces an increase in the computational times difference. However, when the

number of periods is not so large (e.g. equal or lower to 100) the R&G outperforms

the JON for any factor combination studied.

The Appendix E contains the regression analysis of the JON and the R&G with the

data collected from this experiment. As in the preliminary experiment, the

regression model which fits better the computational time in both models is the

exponential model. For the two models the adjusted R square coefficient is very

high. When the scale of the problem is large, the TBO is not significant (using

α=0.05) in the JON. All the factors considered are significant in the R&G. As the

previous regression analysis pointed out, the demand probability has different

effects in the two models. An increase of it causes an increase in the computational

time in the R&G, whereas a decrease in the JON.

5.3 Memory Requirements Analysis

Apart from lasting long time, another inconvenient when solving large scale

problems is that the memory requirements can exceed the computer capacity. The

memory requirements for a problem depend on the model used because of the

different number of constraints and variables.

Different scale problems were solved by all the models previously mentioned

including their modified versions. This time the tests were performed with the


demand 36

computer ACER Intel® Core™ i3-2310M CPU 2.10GHz. This computer has less

installed memory than the other used for the experiment above explained (4.0 GB

vs. 32.0 GB) so the problems it can solve are smaller. When the computer runs out

of memory, it stops the running procedure. The table 16 shows a comparison

between the models memory requirements depending on the scale of the problem.

Two numbers of periods were selected (50 and 84) and the number of products were

increased consecutively until none of the models could solve the problem. It has

been assumed that if the computer runs out of memory solving a problem with a

model, it runs out of memory solving a larger problem with the same model.

Therefore, when a model made the computer run out of memory it was discarded

for the larger size problem. The BM did not run out of memory in the studied

problems, however it was stopped after one computing hour having a large

optimality gap in all of the cases (more than 34%). The B&B nodes of this model

do not require as much capacity as in the other models, however the linear

relaxation is less tight what makes the calculation of a very large number of nodes

needed and thus increasing slowly the memory used.

Table 16: Comparison on computational memory requirements

Products-

Periods*

BM BM-

Modified

JON R&G R&G-

Modified

ERF ERF-

Modified

500-50 (-) (-) ✓ ✓ ✓ ✓ ✓

800-50 ✓ ✓ ✓ ✗ ✗

1500-50 ✓ ✓ ✓

1750-50 ✓ ✗ ✗

2000-50 ✗ ✗ ✗

400-84 (-) (-) ✓ ✓ ✗

500-84 ✗ ✗ ✗

(✓) Problem could be solved to optimality by the specific model

(✗) Problem could not be solved because the computer run out of memory

(-) Running procedure was stopped after one hour * All the problems tested have: MSR=0.6; TBO=2.5; Demand Probability=0.75;

Demand Index of Dispersion=0.5

The rest of the models either solved the problem or run out of memory in less than

25 min. The JON and the R&G are the models which allow to solve the largest

problems, being the former slightly better in this aspect. The addition of Property 4

does not have a positive effect in this aspect. It makes sense because the more


demand 37

constraints, the more calculations are in each node of the B&B tree what increases

the memory required.


demand 38

6. Conclusion

Current solvers enable to solve very large problems. In concrete, in the JRP the

computational time increases exponentially with the number of products and

periods. The rest of the factors have less influence, though significant.

The experimental results show that the R&G is the most efficient model when the

scale of the problem is large. The following most efficient model (JON) consumed

on average 2.9 times more time in the problems tested. The ERF despite being the

most efficient when the scale of the problem is smaller and the presolve procedure

of the solver is deactivated, it is not when the problems are large. Only under very

specific circumstances (high number of products, periods, setup costs and demand

probability) the R&G is not the most efficient. In these cases the JON yields the

solution in less time (average of 45.6% less time in the problems tested).

Nevertheless, the R&G could solve very large problems with these adverse

conditions in a maximum time of two hours. Another advantage of the R&G is that

it is the only model which allows the backordering in its structure.

In the event of having a computer without enough memory to solve a problem by

the R&G, the JON should be tried. However, the memory requirements difference

is small.

The importance of choosing the correct model is clear when the problems are large.

Choosing the wrong can lead to the impossibility of obtaining the optimal solution

or obtaining a solution with an unacceptable optimality gap.

Further investigations should focus on the analysis of the JRPDD heuristics

algorithms with large scale problems. Many studies have been done on the topic of

the JRPDD heuristics (Boctor, et al. 2004; Khouja & Goyal, 2008; Joneja, 1990;

Kirca, 1995; Robinson, et al. 2009), however all of these studies use small scale

problems. Another promising area of investigation is the joint replenishment

problem under stochastic demand reviewed by Khouja and Goyal (2008) as well as

other extensions of the problem including capacity constraints and limitation of

capital investment (Hoque, 2006).


demand 39

Reference List

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uncapacitated multi-echelon production planning problems. Operations

Research Letters, 8, 61-66.

Bagchi, U., Hayya, J. C., & Ord, J. K. (1984). Concepts, Theory and Techniques:

Modelling demand during lead time. Decision Sciences, 15, 157-176.

Bixby, R., Fenelon, M., Gu, Z., Rothberg, E., & Wunderling, R. (2000). MIP:

Theory and practice closing the gap. In M. Powell, & S. Scholtes, System

Modeling and Optimization: Methods, Theory and Applications (pp. 19-50).

Norwell, MA.

Boctor, F. F., Laporte, G., & Renaud, J. (2004). Models and algorithms for the

dynamic-demand joint replenishment problem. International Journal of

Production Research, 42, 2667-2678.

Gao, L.-L., Altay, N., & Robinson, E. P. (2008). A comparative study of modeling

and solution approaches for the coordinated lot-size problem with dynamic

demand. Mathematical and Computer Modelling, 47, 1254-1263.

Hoque, M. A. (2006). An optimal solution technique for the joint replenishment

problem with storage and transport capacities and budget constraints.

European Journal of Operational Research, 175, 1033-1042.

Joneja, D. (1990). The joint replenishment problem: new heuristics and worst case

performance bounds. Operations Research, 38, 711-723.

Khouja, M., & Goyal, S. (2008). A review of the joint replenishment problem

literature: 1989-2005. European Journal of Operational Research, 186, 1-

16.

Kirca, O. (1995). A primal-dual algorithm for the dynamic lot-sizing problem with

joint set-up costs. Naval Research Logistics, 42, 791-806.

Lima, R. M., & Grossmann, I. E. (2011). Computational advances in solvind Mixed

Integer Linnear Programming problems. Chemical Engineering Commons.

Minner, S. (2003). A note on computational aspects of the dynamic joint

replenishment problem. Working paper.

Narayanan, A., & Robinson, E. P. (2006). More on ‘Models and algorithms for the

dynamic-demand joint replenishment problem. International Journal of

Production Research, 44, 383-397.

Robinson, E. P., & Gao, L.-L. (1996). A dual ascent procedure for multiproduct

dynamic demand coordinated replenishment with backlogging.

Management Science, 42(11), 1-9.


demand 40

Robinson, P., Narayanan, A., & Sahin, F. (2009). Coordinated deterministic

dynamic demand lot-sizing problem: A review of models and algorithms.

The International Journal of Management Science, 37, 3-15.


demand 41

Appendices

Appendix A

This appendix includes the procedure used to generate the random variables

according to each of the distributions used.

To generate a random variable 𝑥 following the Poisson distribution, the following

algorithm has been used:

1. Generate a random variable uniformly-distributed 𝑈[0,1)

2. Set

𝑥 = 𝑗𝑖𝑓𝐹(𝑗 − 1) < 𝑈 ≤ 𝐹(𝑗)

Where F is the cumulative distribution function

To generate a random variable 𝑥 following the binomial distribution, the following

algorithm has been used:

1. Generate n Bernoulli(p) random variables: 𝑌1, … , 𝑌𝑛

To generate a Bernoulli(p) random variable:

1.1. Generate a random variable uniformly-distributed U[0,1)

1.2.If 𝑈 ≤ 𝑝, then 𝑌 = 1; else 𝑌 = 0

2. Set 𝑥 = 𝑌1 + 𝑌2 +…+𝑌𝑛

To generate a random variable 𝑥 following this negative binomial distribution, the

following algorithm has been used:

1. Generate 𝑧 Bernoulli(p) random variables, with the procedure

explained before, until they sum n successes

2. Set 𝑥 = 𝑧 − 𝑛


demand 42

Appendix B

The following figures show the average computational times of the different models

from the preliminary experiment depending on the factors levels. Note that the 32

problems which were not solved by the BM and BM-modified have not been

included in the average.

Figure 7: Computational time depending on the number of products

Figure 8: Computational time depending on the number of periods

0,00

2,00

4,00

6,00

8,00

10,00

12,00

14,00

16,00

5 10 20 40

Com

puta

tional

tim

e (s

)

Products

Products

BM Joneja R&G

ERF BM-Modified R&G-Modified

ERF-Modified

0

2

4

6

8

10

12

14

16

12 24 36 48

Com

puta

tional

tim

e(s)

Periods

Periods

BM Joneja R&G


ERF-Modified


demand 43

Figure 9: Computational time depending on the TBO

Figure 10: Computational time depending on the MSR

0

2

4

6

8

10

12

14

1,5 2,5 4,5Com

puta

tional

tim

e (s

)

TBO

TBO

BM Joneja R&G


ERF-Modified

0

2

4

6

8

0,3 0,6 0,9

Com

puta

tional

tim

e (s

)

MSR

MSR

BM Joneja R&G


ERF-Modified


demand 44

Figure 11: Computational time depending on the index of dispersion of the demand

Figure 12: Computational time depending on the demand probability

0

1

2

3

4

5

6

7

0,5 1 1,5

Com

puta

tional

tim

e (s

)

Index of dispersion

Index of dispersion of the demand

BM Joneja R&G ERF

BM-Modified R&G-Modified ERF-Modified

0

1

2

3

4

5

6

7

0,35 0,75 1

Com

puta

tional

tim

e (s

)

Demand probability

Demand probability

BM Joneja R&G


ERF-Modified


demand 45

Appendix C

This appendix includes the regression analysis of the modified versions of the

models. In all of them the regression model which presents a higher adjusted R

square is the exponential.

Table 17: Exponential regression model of the BM-modified computational time


Multiple R 0.929

R Square 0.863

Adjusted R

Square 0.863


Observations 2557

ANOVA

Degrees of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F

Regression

statistics 6 7081.316 1180.219 2684.506 0

Residual 2550 1121.085 0.440

Total 2556 8202.401

Coefficients

Standard


Lower

95% Upper 95%

Intercept

-

6.207 0.074 -83.849 0.000 -6.353 -6.062

Products 0.086 0.001 87.121 0.000 0.084 0.088

Periods 0.070 0.001 71.425 0.000 0.068 0.072

TBO 0.673 0.011 63.604 0.000 0.652 0.694

MSR

-

0.341 0.054 -6.340 0.000 -0.446 -0.235

Index of

dispersion 0.164 0.032 5.097 0.000 0.101 0.227

Demand

Probability 0.423 0.049 8.637 0.000 0.327 0.519

Table 18: Exponential regression analysis of the R&G-modified computational time


Multiple R 0.905

R Square 0.819



Observations 2561

ANOVA

Degrees

of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F


demand 46

Regression

statistics 6 5004.54 834.09 1922.35 0

Residual 2554 1108.15 0.43

Total 2560 6112.70

Coefficients

Standard


Lower

95%

Upper

95%

Intercept -5.750 0.074 -77.440 0.000 -5.896 -5.605

Products 0.071 0.001 72.776 0.000 0.069 0.073

Periods 0.076 0.001 78.192 0.000 0.074 0.078

TBO 0.129 0.010 12.321 0.000 0.108 0.149

MSR -0.315 0.053 -5.921 0.000 -0.419 -0.210

Index of

dispersion 0.107 0.032 3.367 0.001 0.045 0.169

Demand

Probability 0.607 0.049 12.460 0.000 0.511 0.702

Table 19: Exponential regression analysis of the ERF-modified computational time


Multiple R 0.950

R Square 0.902

Adjusted R

Square 0.902


Observations 2592

ANOVA

Degrees of

freedom

Sum of

squares

Mean of

the

squares F

Significance

F

Regression

statistics 6 6607.95 1101.32 3972.89 0.00

Residual 2585 716.58 0.27

Total 2591 7324.53

Coefficients

Standard


Lower

95%

Upper

95%

Intercept -4.657 0.058 -79.664 0.000 -4.772 -4.542

Products 0.074 0.001 96.247 0.000 0.073 0.076

Periods 0.092 0.001 119.573 0.000 0.091 0.094

TBO 0.112 0.008 13.479 0.000 0.096 0.128

MSR -0.151 0.042 -3.573 0.000 -0.234 -0.068

Index of

dispersion 0.131 0.025 5.155 0.000 0.081 0.180

Demand

Probability -0.287 0.039 -7.422 0.000 -0.362 -0.211


demand 47

Appendix D

Given the regression model for the JON shown in the section 4.3.1, the difference

in the computational time can be estimated. The expression of the computational

time is:

𝑡 = 𝑒(−4.73+0.08𝑝𝑟+0.08𝑝𝑒𝑟+0.08𝑇𝐵𝑂−0.13𝑀𝑆𝑅+0.11𝐼−0.07𝐷𝑃) (App.1)

In the case of two problems with same factor values except from the index of

dispersion of the demand (𝐼1 = 1.5; 𝐼2 = 0.5), we calculate the estimated

difference.

Applying the natural logarithm:

ln(𝑡1) = − 4.73 + ⋯+ 0.11𝐼1 (App.2)

ln(𝑡2) = − 4.73 + ⋯+ 0.11𝐼2 (App.3)

ln(𝑡1) − ln(𝑡2) = 0.11(𝐼1 − 𝐼2) (App.4)

Due to the properties of the logarithm:

ln𝑡1𝑡2= 0.11(𝐼1 − 𝐼2) (App.5)

𝑡1𝑡2= 𝑒0.11(𝐼1−𝐼2)

(App.6)

Substituting the values of 𝐼1and 𝐼2:

𝑡1 = 0.12𝑡2 (App.7)

The other example (varying factor is MSR) can be calculated using the same

procedure.


demand 48

Appendix E

The following tables include the regression analysis of the R&G and JON

computational times of the large scale problems experiment. Only the problems

which were solved to optimality have been included. As mentioned before the

coefficients differ from those of the preliminary experiment because the computer

used is different. Even if both experiments would have been performed with the

same computer the coefficients may differ because the regression model is just an

approximation. The computational time can take different values in problems with

exactly the same parameters as the demand and setup cost generation have a random

component.

Table 20: Exponential regression analysis of the JON computational time of large scale

problems


Multiple R 0.932

R Square 0.868

Adjusted R

Square 0.863


Observations 103

ANOVA

Degrees of

freedom

Sum of

squares

Mean

of the

squares F

Significance

F

Regression

statistics 4 274.63 68.66 161.71 0.00

Residual 98 41.61 0.42

Total 102 316.23

Coefficients

Standard


Upper

95%

Intercept 0.979 0.300 3.269 0.001 0.385 1.574

Products 0.003 0.000 22.104 0.000 0.003 0.003

Periods 0.027 0.002 16.648 0.000 0.024 0.031

TBO 0.074 0.043 1.709 0.091 -0.012 0.160

Demand

Probability -0.269 0.324 -0.832 0.408 -0.911 0.373

Table 21: Exponential regression analysis of the R&G computational time of large scale

problems


Multiple R 0.920

R Square 0.847




demand 49

Observations 104

ANOVA

Degrees of

freedom

Sum of

squares

Mean

of the

squares F Significance F

Regression

statistics 4 285.58 71.39 136.56 0.00

Residual 99 51.76 0.52

Total 103 337.34

Coefficients

Standard


Lower

95%

Upper

95%

Intercept -1.564 0.344 -4.543 0.000 -2.247 -0.881

Products 0.003 0.000 19.594 0.000 0.003 0.003

Periods 0.025 0.002 13.744 0.000 0.022 0.029

TBO 0.417 0.049 8.515 0.000 0.320 0.514

Demand

Probability 1.119 0.355 3.154 0.002 0.415 1.823


demand 50

Appendix F

On the CD there are the following files:

basicmodel.mos: BM implemented in Mosel language

joneja.mos: JON implemented in Mosel language

Rg.mos: R&G implemented in Mosel language

Exact-requirements-formulation: ERF implemented in Mosel language

Basicmodel-modified: BM-Modified implemented in Mosel language

RG-modified: R&G-modified implemented in Mosel language

ERF-Modified: ERF-Modified implemented in Mosel language

SetupCostGeneration.mos: code used to generate the setup costs

binomial.mos: code used to generate the demand following a binomial

distribution

poisson.mos: code used to generate the demand following a Poisson distribution

negativebinomial.mos: code used to generate the demand following a negative

binomial distribution

Large-problems.xlsx: Excel file containing the output values of the large scale

problems experiment

Preliminary-experiment.xlsx: Excel file containing the output values of the

preliminary experiment

Ehrenwörtliche Erklärung

Ich erkläre hiermit ehrenwörtlich, dass ich die vorliegende Arbeit selbständig angefertigt

habe. Die aus fremden Quellen direkt und indirekt übernommenen Gedanken sind als

solche kenntlich gemacht.

Ich weiß, dass die Arbeit in digitalisierter Form daraufhin überprüft werden kann, ob

unerlaubte Hilfsmittel verwendet wurden und ob es sich – insgesamt oder in Teilen – um

ein Plagiat handelt. Zum Vergleich meiner Arbeit mit existierenden Quellen darf sie in eine

Datenbank eingestellt werden und nach der Überprüfung zum Vergleich mit künftig

eingehenden Arbeiten dort verbleiben. Weitere Vervielfältigungs- und Verwertungsrechte

werden dadurch nicht eingeräumt.

Die Arbeit wurde weder einer anderen Prüfungsbehörde vorgelegt noch veröffentlicht.

München, den 20.07.2016

José Mezquita Zapico

comparaciÓn de los modelos del problema de ...oa.upm.es/43917/1/tfg_jose_mezquita_zapico.pdf ·...

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